text stringlengths 4 2.78M |
|---|
---
abstract: 'Upon excitation by a laser pulse, broken-symmetry phases of a wide variety of solids demonstrate similar order parameter dynamics characterized by a dramatic slowing down of relaxation for stronger pump fluences. Motivated by this recurrent phenomenology, we develop a simple non-perturbative effective model of dynamics of collective bosonic excitations in pump-probe experiments. We find that as the system recovers after photoexcitation, it shows universal prethermalized dynamics manifesting a power-law, as opposed to exponential, relaxation, explaining the slowing down of the recovery process. For strong quenches, long-wavelength over-populated transverse modes dominate the long-time dynamics; their distribution function exhibits universal scaling in time and space, whose universal exponents can be computed analytically. Our model offers a unifying description of order parameter fluctuations in a regime far from equilibrium, and our predictions can be tested with available time-resolved techniques.'
author:
- 'Pavel E. Dolgirev'
- 'Marios H. Michael'
- Alfred Zong
- Nuh Gedik
- Eugene Demler
bibliography:
- 'cdw\_lib.bib'
title: '[Universal dynamics of order parameter fluctuations in pump-probe experiments]{}'
---
\#1
In the theory of equilibrium phase transitions, the concept of universality plays a central role because it allows describing a plethora of experimentally studied thermal phase transitions with just a few universality classes [@Goldenfeld1992]. For systems far from equilibrium, the notion of universality is relatively unexplored and has recently emerged as an active field [@schwarz1988three; @micha2003relativistic; @micha2004turbulent; @berges2008nonthermal; @berges2011strong; @nowak2012nonthermal; @orioli2015universal; @bhattacharyya2019universal], partially motivated by recent progress in ultracold-atom [@erne2018universal; @prufer2018observation; @eigen2018universal; @navon2016emergence] and ultrafast pump-probe experiments [@mitrano2019]. In a non-equilibrium context, one of the dramatic manifestations of the universality is the emergence of the self-similar evolution of correlation functions [@barenblatt1996scaling]. In particular, after a strong perturbation, the transient equal-time two-point correlation function $D(|{\bf x} - {\bf y}|, t)$ might depend only on a single evolving length scale $\xi(t)$ and two universal functions: $$D(|{\bf x} - {\bf y}|, t) = g(t) f(|{\bf x} - {\bf y}|/\xi(t)). \label{eqn:first}$$ Functional forms of $f(x)$ and $g(t)$ depend neither on microscopic parameters nor on initial conditions. Typical equations of motion, often a complex system of partial integro-differential equations, represent an interplay between many degrees of freedom such as quasiparticles, order parameter (OP), phonons and/or magnons. If these equations allow for the above self-similar form, the analysis might reduce to just a few differential equations, which is particularly appealing since it eases the interpretation of the involved evolution. From a physical standpoint, the self-similarity suggests that there exists a stabilization-like mechanism responsible for this form.
Several recurrent observations in experiments also hint at the existence of universality in the out-of-equilibrium context. Aided by recent advances in pump-probe techniques, light-induced phase transitions have been investigated in a wide variety of materials, including charge-density-wave (CDW) compounds [@tomeljak2009dynamics; @schmitt2011ultrafast; @zong2019evidence; @kogar2019light; @ruan2019; @MohrVorobeva2011; @laulhe2017ultrafast; @vogelgesang2018phase; @Huber2014; @mitrano2019; @storeck2019hot], excitonic insulators [@hellmann2012time; @okazaki2018photo], magnetically-ordered systems [@Kirilyuk2010; @dean2016ultrafast], and systems that exhibit several intertwined orders [@mariager2014structural; @Beaud2014]. Upon photoexcitation, some general phenomenology is observed: (i) The recovery of a photo-suppressed OP takes longer at stronger pump pulse fluence; (ii) The amplitude of the OP restores faster than the phase, exhibiting a separation of timescales; (iii) Related to (ii), peaks in diffraction experiments remain broadened compared to equilibrium shape long after photoexcitation, showing prolonged suppression of long-range phase coherence. These observations motivate us to search for a unified theoretical description.
![(Color online) (a) Schematics of a non-equilibrium state: electrons (red) and the phononic bath (blue) are thermal with temperatures $T_e(t)$ and $T$, respectively; the OP subsystem (mixed colors) is not assumed to be thermal. (b) Time evolution of the Landau coefficient $r(t)$, cf. Eq. . It mimics a photoexcitation event in (a). (c) Schematics of dynamical stages experienced by the system after a quench. During stages 3 and 4, the system exhibits self-similarity. Green and orange color codes indicate that the scaling exponents $\alpha$ and $\beta$, cf. Eq. , are different in these two stages.[]{data-label="Fig1"}](Figure1v7.pdf){width="1\linewidth"}
A common approach to describing many-body dynamics in pump-probe experiments in states with broken symmetry is based on the so-called three-temperature model (3TM) [@tao2013anisotropic; @mansart2010ultrafast; @perfetti2007ultrafast; @dolgirev2019amplitude], or more generally the $N$-temperature model ($N$TM) [@johnson2017watching]. In this framework, a non-equilibrium state is characterized by assigning different temperatures to different subsystems, such as electrons, phonons, and OP degrees of freedom [@Remark1]. Upon photoexcitation, most incoming light is absorbed by electrons, instantaneously increasing the electronic temperature, $T_e$. The introduction of $T_e(t)$ is justified provided we are only interested in phononic timescales that sufficiently exceed the fast electron-electron scattering time. Subsequent dynamics corresponds to heat exchange between hot electrons and the other two subsystems. In this process, it is often assumed that the lattice heating is negligible because the lattice heat capacity at room temperature is several orders in magnitude larger than that of electrons. Even though the 3TM suggests an intuitive picture about the interplay among different subsystems, it often lacks theoretical justification. In particular, low-energy low-momenta Goldstone modes can be easily excited in the symmetry-broken phase. Hence, one key assumption in the 3TM that the OP subsystem remains thermal is a crude over-simplification.
![(Color online) Long-time self-similarity. (a) Time dependence of the change in transverse momentum distribution $\delta D^{\perp}_{\bf k}$ normalized by the equilibrium value in Eq. . Quench strength is set to be $(r_f-r_i) = 80.$ Dashed vertical lines track the position of the peak, $k_{\perp}^*(t)$; $g(t)$ corresponds to the peak height. (b) Rescaled curves collapse into $f(x)$, cf. Eq. . (c) Evolution of $k_\perp^*(t)\sim t^{-\frac{1}{2}}$ at different quench strengths. Note that $k_\perp^*(t)$ does not depend on quench. (d) The same for the scaling function $g(t)$. From this figure we extract $\alpha \approx 0.7$, cf. Eq. , in the third dynamical stage and $\alpha = -1$ in the final stage. \[Fig2\]](Figure2_2.pdf){width="1\linewidth"}
In this Letter, we go beyond the 3TM and formulate a general theory of out-of-equilibrium OP correlations to account for potentially non-thermal states of the OP subsystem – see Fig. \[Fig1\]a. Our theory focuses on non-linear dynamics of collective bosonic excitations. This should be contrasted to earlier work on the relaxation of quasiparticles in superconductors, in which recombination dynamics can lead to faster relaxation rates for higher quasiparticle densities [@gedik2004single; @kusar2008controlled; @prasankumar2016optical] (see, however, Ref. [@boschini2018collapse]). Within our effective bosonic model, we find that upon photoexcitation, the system passes through four dynamical stages outlined in Fig. \[Fig1\]c. For a strong quench, not only is the OP subsystem far from being thermal but overpopulated slow Goldstone modes fully dominate the intrinsic evolution at long times. Even more strikingly, in the last two dynamical stages in Fig. \[Fig1\]c, the distribution function of these modes exhibits self-similar evolution as in Eq. . With these findings, we can explain all of the mentioned experimental observations.
More specifically, our discovery of self-similarity can be summarized in the following equations. The distribution function of the Goldstone modes follow $$\begin{aligned}
\delta D_{\bf k}^{\perp}(t) \simeq \frac{g(t)}{k^2} f(k/{k^*_\perp(t)}),\label{eqn::scaling_D_perp}\end{aligned}$$ where $\delta D_{\bf k}^{\perp}(t) \equiv (D_{\bf k}^{\perp}(t) - D_{{\bf k}, eq}^{\perp})$ and $D_{{\bf k}, eq}^{\perp}$ is the pre-pulse equilibrium distribution given by Eq. . The form in Eq. is similar to the one in Eq. , though written in momentum space; $\xi_{\perp}(t) \equiv ({k^*_\perp(t)})^{-1}$ represents the emergent time-dependent length scale. We also identify the scaling relations $$\begin{aligned}
g(t)\sim t^{\alpha},\, k^*_\perp \sim t^{-\beta}.\label{eqn::exponents}\end{aligned}$$ Both power-law exponents $\alpha,\, \beta$ and the function $f(x)$ are universal. We find that $\beta = \frac{1}{2}$, while $\alpha \approx 0.7$ at early times and $\alpha = -1$ in the final relaxation stage. The scaling functions $f(x)$, $k^*_\perp(t)$, and $g(t)$ are shown in Fig. \[Fig2\].
We first explore the implications of the self-similarity on the experimental phenomenology. Prior to the arrival of the pump pulse, the system possesses long-range coherence manifested in the macroscopic homogeneous OP $\phi$ and divergent transverse correlation length $\xi_\perp = \infty$. The laser pulse depletes this coherence. The scaling suggests that as the system evolves towards equilibrium, it develops a finite correlation length $\xi_\perp(t)$ that slowly grows in a diffusive manner $\xi_\perp(t) \sim \sqrt{t}$ [@bray2002theory], consistent with recent experiments [@laulhe2017ultrafast; @vogelgesang2018phase]. This physical picture explains the broadening of diffraction peaks observed long after the arrival of the pulse. The slowing-down of the OP recovery can also be deduced from Eq. . The system enters the final dynamical stage with $g(t) \simeq A_Q t^{-1}$, where $A_Q$ is a constant of proportionality that monotonically increases with the quench strength. By contrast, as shown in Fig. \[Fig2\]c, $k^*_\perp (t)$ does not depend on the quench. Therefore, the cumulative effect, expressed in the change of the population of transverse modes $\delta n_{\rm tot}^\perp$, behaves as $$\delta n_{\rm tot}^\perp \equiv \int \frac{d^3 {\bf k}}{(2\pi)^3} \delta D_{\bf k}^{\perp}(t) \sim A_Q t^{-3/2}, \label{eqn::delta_n_tot}$$ i.e. as a [*power-law*]{}. Since the transverse modes dominate the long-time dynamics, from Eq. it follows that characteristic recovery time $\tau_{\rm rec}\sim A_Q^{2/3}$ is a monotonically increasing function of the quench strength – see also Fig. \[Fig3\]b.
We now explicitly formulate our model and derive the above results. We describe spontaneous symmetry breaking (SSB) in the framework of the time-dependent Landau-Ginzburg formalism (model-A [@hohenberg1977theory; @sun2019transient]): $$\begin{aligned}
\frac{d \phi_{\alpha} ({\bf x}, t)}{dt} = -\Gamma\frac{\delta {\cal F}}{\delta \phi_{\alpha}({\bf x}, t)} + \eta_{\alpha}({\bf x}, t). \label{eqn::model_A}\end{aligned}$$ Here $\phi_\alpha$ is an $N$-component vector of real fields representing the OP. The free energy functional reads $$\begin{aligned}
{\cal F}[{\bf \phi}] = \int d^3 {\bf x} \left[ \frac{r}{2} \phi_{\alpha}^2 + \frac{K}{2} (\nabla \phi_{\alpha})^2 + u (\phi_{\alpha}^2)^2 \right],\label{eqn::F}\end{aligned}$$ and the second term in Eq. represents the noise originating from the phononic bath (with temperature $T$): $$\begin{aligned}
\mean{\eta_{\alpha}({\bf x}, t)\eta_{\beta}({\bf x'}, t') } = 2T\Gamma \delta_{\alpha,\beta}\delta({\bf x} - {\bf x'}) \delta(t-t'). \label{eqn::noise}\end{aligned}$$ Here $r,\, K,\, u,$ and $\Gamma$ are the model parameters. For homogeneous quenches, without loss of generality, we assume that SSB occurs along the first direction: $\phi(t) = \mean{\phi_{1}({\bf x}, t)}$. Associated with the OP are longitudinal $D^{\parallel}_{{\bf k}}(t)\equiv \mean{\phi_1({\bf k}; t)\phi_1(-{\bf k}; t) }_c$ and transverse $D^{\perp}_{{\bf k}}( t)\equiv \mean{\phi_{\alpha\neq1}({\bf k}; t)\phi_\alpha(-{\bf k}; t) }_c$ correlation functions. In the language of the CDW theory, these correlators represent momentum distribution functions of amplitudons (Higgs modes) and phasons (Goldstone modes), respectively. The model-A formalism – can be conveniently rewritten in terms of the Fokker-Planck equation: $$\begin{aligned}
\partial_t {\cal P} = T\Gamma \sum_{{\bf k},\alpha} \frac{\delta}{\delta \phi_{\alpha,{\bf k}}}\left[ \frac{{\cal P}}{T} \frac{\delta {\cal F}}{\delta \phi_{\alpha,-{\bf k}}} + \frac{\delta {\cal P}}{\delta \phi_{\alpha,-{\bf k}}} \right],\label{eqn::F_P}\end{aligned}$$ where ${\cal P}([\phi], t)$ is the probability distribution functional of space-dependent field configurations $\phi_\alpha({\bf x})$. To the leading order in $1/N$, ${\cal P}([\phi], t)$ is Gaussian, implying that the OP $\phi(t)$ and the correlators $D^\parallel_{\bf k}(t),\,D^\perp_{\bf k}(t)$ form a closed set of dynamical variables. The self-consistent equations of motion read (see Ref. [@SM] for the details) $$\begin{aligned}
\frac{d \phi(t)}{dt} &=& -\Gamma\, r_{\rm eff}\, \phi, \label{eqn::dt_phi} \\
\frac{d D^{\perp}_{{\bf k}}(t)}{dt} &=& 2T\Gamma - 2\Gamma (K {\bf k}^2 + r_{\rm eff})D^{\perp}_{{\bf k}},\label{eqn::dt_Dperp}\\
\frac{d D^{\parallel}_{{\bf k}}(t)}{dt} &=& 2T\Gamma - 2\Gamma (K {\bf k}^2 + r_{\rm eff} + 8u\phi^2)D^{\parallel}_{{\bf k}}\label{eqn::dt_Dpar}.\end{aligned}$$ Here the self-consistent “mass”-term is defined as $$r_{\rm eff}(t) = r(t) + 4u \left( \phi^2 + n_{\rm tot}^\parallel + (N-1) n_{\rm tot}^\perp \right),\label{eqn::r_eff}$$ where $ n_{\rm tot}^{\perp(\parallel)} \equiv \int^{\Lambda} \frac{d^3 {\bf q}}{(2\pi)^3} D^{\perp(\parallel)}_{{\bf q}}$. Note that quantities such as energy or total number of excitations are not conserved. The presence of the bath, cf. Eq. , will also always result in the thermalization of the system. This should be contrasted to quenches in the isolated $O(N)$ model, where, to the leading in $1/N$ order, the system does not demonstrate equilibration [@chandran2013equilibration; @sciolla2013quantum; @chiocchetta2015short; @maraga2015aging; @chiocchetta2016short].
From the equations of motion, we obtain equilibrium correlators: $$\begin{aligned}
D^{\parallel}_{\bf k} = \frac{T}{Kk^2+8u\phi^2 + r_{\rm eff}},\, D^{\perp}_{\bf k} = \frac{T}{K {\bf k}^2 + r_{\rm eff}}. \label{eqn::D_eq}\end{aligned}$$ This result is a manifestation of the equipartition theorem. In the symmetry broken phase, where $r_{\rm eff} = 0$ and $\phi \neq 0$, we observe that the OP equilibrium value $\phi$ is affected by the thermal fluctuations, cf. Eq. (\[eqn::r\_eff\]). The transverse correlation length $\xi_\perp\propto r_{\rm eff}^{-1/2}$ is indeed divergent. In the disordered phase, $r_{\rm eff} \neq 0$ and $\phi = 0$, the transverse and longitudinal correlations are not distinguishable.
A useful point of view on the above approximations is as follows. The equations of motion – are equivalent to $$\begin{aligned}
\frac{d\delta \phi^{\perp}_{\bf k} (t)}{dt} &=& -\Gamma (K {\bf k}^2 + r_{\rm eff})\phi^{\perp}_{\bf k} + \eta^{\perp}_{\bf k}(t),\label{eqn::perp_k}\\
\frac{d\delta \phi^{\parallel}_{\bf k} (t)}{dt} &=& -\Gamma (K {\bf k}^2 + r_{\rm eff} + 8u\phi^2)\phi^{\parallel}_{\bf k} + \eta^{\parallel}_{\bf k}(t),\label{eqn::par_k}\end{aligned}$$ where $\delta \phi_{\bf k}^{\alpha}$ represents the fluctuating part of the corresponding Fourier mode $\phi^{\alpha}_{\bf k}$. We observe that each of the fluctuating modes lives in an effectively parabolic potential, $\mean{\delta \phi_{\bf k}^{\alpha}} = 0$ and the noise term establishes the equilibrium variances given by Eq. .
We now formulate the quenching protocol. For simplicity, we assume that the electronic temperature $T_e$ cools down to the equilibrium value $T$ with a constant rate $\tau_{\rm QP}$ defined by the electron-phonon coupling. In the usual Landau-Ginzburg theory, the coefficient $r(T_e)$ depends linearly on $T_e$. To mimic a photoexcitation event, we therefore impose the following dynamics on $r(t)$ – see Fig. \[Fig1\]b: $$r(t) = r_i + \theta(t) \exp{(-t/\tau_{\rm QP})} \times (r_f-r_i), \label{eqn::quench}$$ where $\theta(t)$ is the Heaviside theta function, $r_i$ is the pre-pulse value chosen such that $\phi \neq 0$, and $(r_f - r_i)$ characterizes the strength of the pulse. Below we are interested in the dynamics for time delays much beyond $\tau_{\rm QP}$.
![(Color online) Intrinsic dynamics for different quench strengths, $(r_f-r_i)$. (a) Time dependence of the OP $\phi(t)$ normalized by the pre-pulse value $\phi_0$. At long times, $(\phi(t) - \phi_0)\sim t^{-d_\phi}$ with $d_\phi = \frac{3}{2}$. (b) OP recovery time $\tau_{\rm rec}$. (c) Dynamics of $r_{\rm eff}(t)$. Initially large positive value of $r_{\rm eff}$ is quickly suppressed and even becomes negative. Then it slowly restores, as a power law $r_{\rm eff} \sim t^{-d_r}$ with $d_r = \frac{5}{2}$, to the zero value. Inset: Zoomed-in view on the long-time tails. (d) Evolution of $D^{\perp}_{k_0}$, where $k_0 = \frac{2\pi}{L}$ is the lowest wave vector used in our calculations ($L = 1000$). For a strong pulse, initially $D^{\perp}_{k_0}$ is suppressed to almost zero, but then, after $r_{\rm eff}$ changes sign, it exponentially proliferates. Dotted line corresponds to $D^{\parallel}_{k_0}$ for the strongest pulse considered. Note that $D^{\parallel}_{k_0}$ and $D^{\perp}_{k_0}$ very soon merge into a single curve indicating that the OP is melted. Inset: longer time dynamics for the strongest pulse. We observe that $D^{\parallel}_{k_0}$ and $D^{\perp}_{k_0}$ become distinguishable once the OP value $\phi(t)$ becomes appreciable. Throughout the paper, we use the following parameters: $K = u = 1$, $N = 4$, $\Lambda = \pi$, $\Gamma = 0.5$, $\tau_{\rm QP} = 0.3$, $r_i = -15$, $T = 0.1$. All panels share the same color scale in (b) for the quench strengths. \[Fig3\] ](Figure3.pdf){width="1\linewidth"}
We turn to discuss the internal dynamics that happens to the system as a whole after being quenched, cf. Eq. . As mentioned in Fig \[Fig1\]c, we identify four dynamical stages – (i) depletion, (ii) inflation, (iii) mode decoupling and (iv) relaxation to the thermal equilibrium – which we cover below.
In Fig. \[Fig3\]a, we show numerical results for the dynamics of the OP, $\phi(t)$, at different quench strengths. For a weak pump, $\phi(t)$ becomes slightly suppressed and then quickly recovers to the initial value $\phi_0$. This should be contrasted to the case of a strong pulse, for which initially the OP becomes strongly suppressed and then goes through a long recovery process. The recovery takes longer for stronger pulses – see Fig. \[Fig3\]b. This slowing-down is due to the power-law dynamics $\delta \phi(t)\equiv (\phi(t) - \phi_0) \sim t^{-d_\phi},$ $d_\phi = \frac{3}{2}$.
In Fig. \[Fig3\]c, we plot the evolution of $r_{\rm eff}(t)$ for different quenches. Note that upon arrival of a laser pulse, $r_{\rm eff}(+0) = (r_f-r_i)$. This large initial value first decreases due to the time evolution of the “bare value” of $r(t)$, cf. Eq. , and later, $t \gtrsim \tau_{\rm QP}$, due to the dynamics of the OP and collective modes described by Eqs. –. Even though $r(t)$ returns to its equilibrium value $r_i$ during a relatively short time $\tau_{\rm QP}$, dynamics of $r_{\rm eff}$ occurs over much longer time scale where it even changes sign, as shown in Fig. \[Fig3\]c. We find that long-time evolution of $r_{\rm eff}\sim t^{-d_r}$ is power-law-like with $d_r = \frac{5}{2}$. For the fluctuating modes $\delta \phi_{\bf k}^{\alpha}$, a large value of $r_{\rm eff}$ implies that each of the effective parabolic potentials becomes initially steeper, and, as such, the noise term in Eqs. – will tend to depopulate these modes – see also Fig. \[Fig3\]d. Therefore, the first stage – depletion – is characterized by suppression of the OP and correlations $D^{\perp}_{\bf k}$ and $D^{\parallel}_{\bf k}$.
The second stage – inflation – starts when $r_{\rm eff}$ changes its sign. Note that a negative value of $r_{\rm eff}$ implies that each of the effective parabolic potentials becomes more shallow or, as the case for the low-momenta transverse modes, can even become inverted. Therefore, during the inflation, population in each of the modes proliferates, most dramatically for the low-momenta modes – see Fig. \[Fig3\]d. For a given mode $\delta\phi^{\alpha}_{\bf k}$, a useful quantity is the time $t_{\bf k}^{\alpha}$ when the corresponding occupation $D^{\alpha}_{\bf k}$ reaches its maximum: $\frac{d}{dt} { D}^{\alpha}_{\bf k} (t_{\bf k}^{\alpha}) = 0$. One can deduce that (i) $t^{\alpha}_{\bf k}$ is larger for lower ${\bf k}$, (ii) for a given ${\bf k}$, $t^{\alpha}_{\bf k}$ grows with the quench strength, and (iii) $t^{\perp}_{\bf k} > t^{\parallel}_{\bf k}$.
For a strong quench and at the time when the OP becomes completely suppressed, the longitudinal and transverse correlations are no longer distinguishable – see Fig. \[Fig3\]d. This parallels the disordered phase in equilibrium situation. As the OP develops, these modes start to separate. We will associate the end of the inflation stage with the time $t^{\parallel}_{{\bf k} = 0}$, when $D^{\parallel}_{{\bf k} = 0}$ reaches its maximum value.
![(Color online) Separation of timescales. (a) Long-time dynamics of the total population of longitudinal modes, $n^{\parallel}_{\rm tot}(t)$. (b) The same for transverse modes, $n^{\perp}_{\rm tot}(t)$. Note that by the time when $n^{\parallel}_{\rm tot}$ is nearly fully recovered, $n^{\perp}_{\rm tot}$ approaches its maximum.[]{data-label="Fig4"}](Figure4.pdf){width="1\linewidth"}
Because of the additional correction to the quadratic term for the longitudinal correlations in Eq. , the subsequent evolution – the stage called mode decoupling – is very different for the longitudinal and transverse modes; see Fig. \[Fig4\]. The longitudinal correlations start to relax back to the thermal equilibrium value in Eq. , while the transverse modes continue to proliferate, resulting in $\alpha$, cf. Eq. , being positive during the third dynamical stage. Moreover, by the time when $n^{\parallel}_{\rm tot}$ is sufficiently recovered, $n^{\perp}_{\rm tot}$ is about to reach its maximum value. Strong experimental evidence of this separation of timescales was reported in Refs. [@zong2019evidence; @laulhe2017ultrafast; @vogelgesang2018phase].
Just after the mode decoupling, $n^{\perp}_{\rm tot}$ starts to slowly decrease, cf. Eq. , suggesting that the system enters the final relaxation stage. Note that even though lowest-momenta modes $D^{\perp}_{\bf k}$ continue to proliferate at very long times, their relative contribution to $n_{\rm tot}^{\perp}$ is suppressed by the reduced phase space of these modes, which is proportional to $k^2$. The underlying dynamics is reminiscent of an inverse particle cascade in the theory of turbulence [@orioli2015universal; @zakharov2012kolmogorov; @nazarenko2011wave]. The main difference is that in our system the dynamics is overdamped.
All of the long-time power-law exponents: $\beta = \frac{1}{2}$, $\alpha = -1$, $d_\phi = \frac{3}{2}$, and $d_{r} = \frac{5}{2}$ – can be deduced merely from the scaling form , as we outline in Ref. [@SM]. However, it is essential to understand why this self-similarity occurs in the first place. Re-establishing the long-range coherence, which is depleted by the laser pulse, is the slowest process that happens in the system, $k^*_\perp \sim t^{-\frac{1}{2}}$. From Fig. \[Fig2\], we note that the most relevant transverse modes are the ones with wave vectors close to $k^*_\perp$. For these modes, we can safely neglect fast $r_{\rm eff}\sim t^{-\frac{5}{2}}$ in Eq. compared to slow $(k^*_\perp)^2\sim t^{-1}$, resulting in a simple diffusion-like equation with the following solution: $$\begin{aligned}
\delta D^\perp_k = A_k \exp(-2\Gamma k^2 t), \label{eqn:diff_eq_sol}\end{aligned}$$ where $A_k$ is yet unknown function of $k$. As supported by Fig. \[Fig3\]d, $\delta D^\perp_{k}(t)$ does not diverge for $k\rightarrow 0$. One may then Taylor-expand $A_k$ as $A_{k} = A_0 + A_2 k^2 + A_4 k^4 + \dots$ The relevant $k$ vectors, the ones in the vicinity of $k^*_\perp(t)$, are small at long times, and, thus, it is safe to leave only the dominant harmonic $A_0$ in this expansion. For example, one can obtain $$\begin{aligned}
\delta n_{\rm tot}^\perp &\sim& (k_\perp^*)^3 \int dx\, x^2\, {\rm e}^{-x^2}\times \\
&& \times (A_0 + A_2 x^2 (k_\perp^*)^2 + A_4 x^4 (k_\perp^*)^4 + \dots)\notag{}.\end{aligned}$$ Due to $(k^*_\perp(t))^2 \sim t^{-1}$, indeed contributions from higher harmonics soon become irrelevant. At long times, we can therefore approximate $\delta D^\perp_k \sim \exp(-2\Gamma k^2 t),$ consistent with $\beta = \frac{1}{2}$ and $\alpha = -1$. The above analysis has explained all long-time scalings. Note, however, that the self-similarity in Eq. settles much earlier than the final relaxation stage. It is striking that the functional form of $f(x)$, cf. Eq. , is the same for the last two dynamical stages (see Fig. \[Fig2\]), an interesting feature that warrants further investigation.
To test the aforementioned predictions, a variety of experimental setups arranged in a pump-probe scheme could be performed, for example, electron or x-ray diffuse scattering [@chase2016; @wall2018; @stern2018], resonant inelastic x-ray scattering [@mitrano2019], and Brillouin scattering [@demokritov2001]. These experiments give access to momentum- and/or energy-resolved dynamics of bosonic excitations related to OP, so one may specifically search for signatures of: (i) non-thermal population of the transverse modes, (ii) the self-similarity encoded in Eq. , and (iii) different dynamical stages after photoexcitation \[see Fig. \[Fig1\](c)\].
For outlook we suggest three possible research directions. First, one can generalize our analysis to systems that have additional conservation laws. For example, in magnetic systems, one may take into account SU$(2)$ symmetry [@bhattacharyya2019universal] (or approximate symmetry, as is the case for most systems). Second, it is interesting to extend our approach to a fully microscopic model [@babadi2015far; @babadi2017theory; @lemonik2018model; @lemonik2019transport] in which one investigates the dynamics of electrons self-consistently rather than phenomenologically. Such microscopics would allow computing other transient properties of the many-body electron systems, for example, various spectral functions that can be probed in time- and angle-resolved photoemission spectroscopy. It would also provide further insights about the interplay between quasiparticles and OP, cf. Eq . Finally, one can also take into account coherent dynamics [@gagel2014universal; @gagel2015universal], and study, for example, the damping of the Higgs excitations. Exploring the above directions would pave the path towards a more profound understanding of universality in non-equilibrium phase transitions.
The authors would like to thank A. Kogar, B.V. Fine, A.E. Tarkhov, A. Bedroya, V. Kasper, S.L. Johnson, J. Rodriguez-Nieva, J. Marino, A. Schuckert, A. Cavalleri, G. Falkovich, and Z.X. Shen for fruitful discussions. N.G. and A.Z. acknowledge support from the U.S. Department of Energy, BES DMSE and the Skoltech NGP Program (Skoltech-MIT joint project) (analysis and manuscript writing). P.E.D., M.H.M., and E.D. were supported by the Harvard-MIT Center of Ultracold Atoms, AFOSR-MURI Photonic Quantum Matter (award FA95501610323), and DARPA DRINQS program (award D18AC00014).
Supplemental Materials
======================
I. Derivation of the equations of motion
----------------------------------------
Here we provide details of the derivation of the main Eqs. –.
**Dynamics of $\phi$.** Evolution of the field $\phi = \frac{1}{\sqrt{V}}\phi_{1,{\bf q} = 0}$ can be obtained from: $$\partial_t \mean{\phi_{\alpha, {\bf q}}}_t = \int D[\phi] \phi_{\alpha, {\bf q}} \partial_t {\cal P}([\phi],t) = -\Gamma \mean{\frac{\delta {\cal F}}{\delta \phi_{\alpha,-{\bf q}} }},\label{eqn::S2}$$ where in the last equality we used the Fokker-Planck Eq. , and integration by parts. The latter derivative can be calculated from Eq. : $$\frac{\delta {\cal F}}{\delta \phi_{\alpha,-{\bf q}}} = (r + K{\bf q}^2)\phi_{\alpha, {\bf q}} + \frac{4u}{V} \sum_{{\bf k_1},{\bf k_2}} \phi_{\beta,{\bf k_1}}\phi_{\beta,{\bf k_2}}\phi_{\alpha,{\bf q- k_1-k_2}}.$$ Using Wick’s theorem and leaving only terms up to the leading order in $1/N$, we obtain $$\begin{aligned}
\mean{ \sum_{{\bf k_1},{\bf k_2}} \phi_{\beta,{\bf k_1}}\phi_{\beta,{\bf k_2}}\phi_{1,{\bf - k_1-k_2}}} \approx \phi_{1,{\bf q} = 0}^3 + \notag{}\\
+\phi_{1,{\bf q} = 0} \sum_{\bf k} (D^{\parallel}_{\bf k} + (N-1)D^{\perp}_{\bf k}). \label{eqn::S4}\end{aligned}$$ Combining Eq. and Eq. we arrive at Eq. of the main text.
**Dynamics of the correlators.** Applying the same trick as above, we derive: $$\begin{aligned}
&&\partial_t \mean{\phi_{\alpha,{\bf k}}\phi_{\alpha,-{\bf k}}}_c = 2 T \Gamma - 2 \Gamma \times \notag{}\\
&&\times \left[ \mean{\phi_{\alpha,{\bf k}} \frac{\delta F}{\delta \phi_{\alpha,{\bf k}}}} - \mean{\phi_{\alpha,{\bf k}}} \mean{\frac{\delta F}{\delta \phi_{\alpha,{\bf k} }}} \right].\label{eqn::S5}\end{aligned}$$ For the case of the transverse component, in the leading in $1/N$ order we obtain: $$\begin{aligned}
&& \mean{\phi_{\alpha, {\bf k}}\sum_{{\bf k_1},{\bf k_2}} \phi_{\beta,{\bf k_1}}\phi_{\beta,{\bf k_2}}\phi_{\alpha,{\bf -k- k_1-k_2}} } \notag{} \approx\\
&& \approx D^\perp_{\bf k} \left( \phi_{1,{\bf q}=0}^2 + \sum_{\bf q} (D^{\parallel}_{\bf q} + (N-1)D^{\perp}_{\bf q}) \right). \label{eqn::S6}\end{aligned}$$ Combining Eq. and Eq. we arrive at Eq. of the main text. For the case of the longitudinal component, similarly to the above discussion we get $$\begin{aligned}
&&\sum_{{\bf k_1},{\bf k_2}}\Big( \mean{\phi_{1,{\bf k}} \phi_{\beta,{\bf k_1}}\phi_{\beta,{\bf k_2}}\phi_{1,{\bf -k- k_1-k_2}} } - \notag{}\\
&&-\mean{\phi_{1,{\bf k}}}\mean{\phi_{\beta,{\bf k_1}}\phi_{\beta,{\bf k_2}}\phi_{1,{\bf -k- k_1-k_2}} } \Big) \notag{} \approx\\
&& \approx D^\parallel_{\bf k} \left( 3\phi_{1,{\bf q}=0}^2 + \sum_{\bf q} (D^{\parallel}_{\bf q} + (N-1)D^{\perp}_{\bf q}) \right).\end{aligned}$$ This equation leads to Eq. .
II. Long-time self-similarity
=============================
Transverse correlations
-----------------------
In the main text, we presented the derivation of the long-time exponents $\beta = \frac{1}{2}$ and $\alpha = -1$. Here we derive $d_r = \frac{5}{2}$ and $d_\phi = \frac{3}{2}$ starting from the scaling form and using the equations of motion.
To extract the value of $d_r$, we need to consider the interplay between the order parameter and the transverse correlations (longitudinal correlations can be ignored, cf. Fig. \[Fig4\]). Assuming that at long times $r_{\rm eff}\sim t^{-d_r}$, the equation of motion reads $$\begin{aligned}
\frac{d\delta \phi}{dt} = -\Gamma r_{\rm eff}(t) \phi \sim t^{-d_r},\end{aligned}$$ where we implied that the order parameter $\phi(t) = \phi_{\rm eq} + \delta \phi(t)$ is already close to its equilibrium value $\phi_{\rm eq}$. Integrating the above equation, we obtain $\phi^2(t) \approx \phi_{\rm eq}^2 + C t^{-d_\phi}$, where $C$ is some constant and $d_{\phi} = d_r - 1$. Note that since $\phi^2(t)$ enters the definition of $r_{\rm eff}(t)$, cf. Eq. , the more dominant scaling $t^{-d_r +1}$ from the order parameter must be compensated by the transverse correlations. From the scaling we note that $$\begin{aligned}
\delta n_{\rm tot}^\perp \sim t^{\alpha} \int dk f(k t^{\beta}) \sim t^{\alpha - \beta}.\end{aligned}$$ Therefore, we have $$\begin{aligned}
\alpha - \beta = -d_r +1 \Rightarrow d_r = 1 + \beta - \alpha = \frac{5}{2}.\end{aligned}$$ This result also gives $d_\phi = d_r - 1 = \frac{3}{2}$. It is encouraging to see that all of the universal (independent from the microscopic parameters such as $\Gamma$, $\tau_{\rm QP}$, $T$, $u$ and $K$) scaling exponents can be obtained from a single assumption in Eq. .
Longitudinal correlations
-------------------------
During the evolution, the longitudinal correlation function $D^{\parallel}_{{\bf k}}$ remains bell-shaped with a maximum at $k = 0$ suggesting to define $\tilde{g}(t) = D^{\parallel}_{k = 0}(t)$ and $k^*_\parallel(t)$ to be the wave vector corresponding to half width at half maximum in $D^{\parallel}_{{\bf k}}$. Notably, both functions at long times behave as $\tilde{g}(t),\, k^*_\parallel(t) \sim t^{-d_{\phi}}$ – see Fig. \[fig:univ\_longitudinal\]. We also observe that this power-law exponent implies that the longitudinal correlations exhibit the leading scaling (see the previous subsection), i.e. these modes should not be entirely ignored.
![(a) evolution of the scaling function $\delta\tilde{g}(t)\equiv \tilde{g}(t) - \tilde{g}_{\rm eq}$ for different quenches. (b) the same for the longitudinal wave vector $\delta k^*_\parallel(t)\equiv k^*_\parallel(t) - k^*_{\parallel, {\rm eq}}$. The second (inflation) and the third (mode decoupling) stages of the overall dynamics are clearly seen. At long times, both functions scale as $\delta\tilde{g}(t),\, \delta k^*_\parallel(t) \sim t^{-d_{\phi}}$.[]{data-label="fig:univ_longitudinal"}](Scalings_longitudinal.png){width="1\linewidth"}
To explain the above observation, we note that at long times, when the order parameter $\phi(t) = \phi_0 + \delta \phi$ is already close to be recovered, the equation of motion can be approximated to (we fix $K = 1$ for convenience) $$\frac{d \delta D^{\parallel}_{{\bf k}}}{dt} \approx -32 \Gamma u \phi_0 \delta \phi D^{\parallel}_{{\bf k}, {\rm eq}} - 2\Gamma ( k^2 + 8u\phi_0^2) \delta D^{\parallel}_{{\bf k}},$$ where $D^{\parallel}_{{\bf k}}(t) = D^{\parallel}_{{\bf k}, {\rm eq}} + \delta D^{\parallel}_{{\bf k}}(t)$ and we disregarded fast $r_{\rm eff}(t) \sim t^{-d_r}$ compared to slow $\delta \phi(t)\sim t^{-d_{\phi}}$ ($0 <d_{\phi}< d_{r}$, see previous subsection). The above equation can be solved analytically. Indeed, substituting $$\delta D^{\parallel}_{{\bf k}}(t) = {\rm e}^{ - 2\Gamma (k^2 + 8u\phi_0^2) t} h_{{\bf k}}(t)$$ we obtain the following equation on $h_{{\bf k}}(t)$: $$\frac{d h_{{\bf k}}}{dt} = -32\Gamma u\phi_0 \delta \phi D^{\parallel}_{{\bf k}, {\rm eq}} {\rm e}^{ 2\Gamma (k^2 + 8u\phi_0^2) t}.$$ Integration of this equation gives $$h_{{\bf k}}(t) = h_{{\bf k}}(t_0) + \frac{C}{k^2 + 8u\phi_0^2} \int\limits_{t_0}^t dt' \frac{{\rm e}^{ 2\Gamma (k^2 + 8u\phi_0^2) t'}}{(t')^{d_{\phi}}},$$ where $C$ is some constant. We, therefore, conclude that $$\delta D^\parallel_{\bf k} = \delta D^{\parallel, (1)}_{\bf k} + \delta D^{\parallel, (2)}_{\bf k},$$ where $\delta D^{\parallel, (1)}_{\bf k}(t) =h_{{\bf k}}(t_0) {\rm e}^{ - 2\Gamma (k^2 + 8u\phi_0^2) t}$ decays exponentially in time, whereas $$\delta D^{\parallel, (2)}_{\bf k} \sim \frac{{\rm e}^{ - 2\Gamma (k^2 + 8u\phi_0^2) t}}{k^2 + 8u\phi_0^2} \int\limits_{t_0}^t dt' \frac{{\rm e}^{ 2\Gamma (k^2 + 8u\phi_0^2) t'}}{(t')^{d_{\phi}}}\label{eqn::D_par_2}$$ is potentially important.
At long times $t\rightarrow \infty$, we observe that $$F(t) \equiv \int\limits_{t_0}^t dt' \frac{{\rm e}^{ a t'}}{(t')^{b}} \sim \frac{{\rm e}^{ a t}}{t^{b}},\, a,b>0.\label{eqn::general_long_time}$$ Indeed, by differentiating $F(t)$ we note that it satisfies $$\frac{d F}{dt} = \frac{{\rm e}^{ a t}}{t^{b}}.$$ By substituting $F(t) = {\rm e}^{ a t} p(t)$ we separate rapid exponential growth from slow power-law-like dynamics encoded in $p(t)$: $$\frac{d p}{d t} + a p = \frac{1}{t^{b}}.$$ From this equation, we finally see that $p \sim t^{-b}$ (as long as $a \neq 0$). Combining Eqs. and , we conclude that $$\delta D^{\parallel, (2)}_{\bf k} \sim \frac{ t^{-d_\phi}}{k^2 + 8u\phi_0^2},$$ i.e. indeed $\delta D^{\parallel}_{\bf k}$ gets power-law-like contribution with the leading exponent. For completeness, we also note that $$\delta n^{\parallel, (2)}_{\rm tot} = \int\frac{d^3 {\bf k}}{(2\pi)^3} \delta D^{\parallel, (2)}_{\bf k} \sim t^{-d_\phi}$$ also exhibits the same scaling.
|
---
abstract: 'The negatively charged nitrogen-vacancy (NV) center spin in diamond can be used to realize quantum computation and to sense magnetic fields. Its spin triplet consists of three levels labeled with its spin z-components of +1, 0, and -1. Without external field, the +1 and -1 states are degenerate and higher than the 0 state due to the zero-field splitting. By taking the symmetrical and anti-symmetrical superpositions of the +1 and -1 states as our qubit basis, we obtain exact evolution operator of the NV center spin under time-dependent magnetic field by mapping the three-level system on time-dependent quantum two-level systems with exact analytical solutions. With our exact evolution operator of the NV center spin including three levels, we show that arbitrary qubits can be prepared from the starting 0 state and arbitrary rapid quantum logic gates of these qubits can be realized with magnetic fields. In addition, it is made clear that the typical quantum logic gates can be accomplished within a few nanoseconds and the fidelity can be very high because only magnetic field strength needs to be controlled in this approach. These results should be useful to realizing quantum computing with the NV center spin systems in diamond and exploring other effects and applications.'
author:
- 'Wen-Qi Fang and Bang-Gui Liu'
title: 'Exact magnetic field control of nitrogen-vacancy center spin for realizing fast quantum logic gates'
---
Introduction
============
The negatively charged nitrogen-vacancy (NV) center in diamond has been intensively investigated because it can be used for realizing quantum computation and sensing weak magnetic field, electric field, strain etc.[@dobro; @schir; @doher; @mans; @info] As quantum technology evolves, one can manipulate the NV center spin with electromagnetic field [@fuchs; @lond; @mayer], optical field [@tama; @bass; @jacq], and stress field [@macq; @teis]. Its spin ground state is a triplet ($S=1$) with 0 and $\pm 1$ as the spin z exponents, and its zero-field splitting makes the $|0\rangle$ state the lowest, leaving the $|+1\rangle$ and $|-1\rangle$ degenerate. One can apply a longitudinal magnetic field to split the $|+1\rangle$ and $|-1\rangle$ states, adding a Zeeman term to the spin Hamiltonian. Usually, it is preferable to work with a two level system representing a qubit by using the $|0\rangle$ state and one of the $|+1\rangle$ and $|-1\rangle$ states, and then control it with microwave pulse or radio-frequency wave [@fuchs]. In such approaches, however, one usually use only weak driving under the rotating wave approximation, or else the third state cannot be neglected. Additionally, one needs accurate timing to track the relative phase between the $|0\rangle$ and $|+1\rangle$ (or $|+1\rangle$) states due to the zero-field splitting effect. Recently, the NV center spin was used to experimentally realize geometric quantum gates with high fidelity[@arroyo; @czu]. In this approach, the $|+1\rangle$ and $|-1\rangle$ states are used to encode the qubit. The latest report shows that the $|+1\rangle$ and $|-1\rangle$ states can be manipulated with internal strain field[@mech]. Nevertheless, the geometric quantum gates need half microsecond to be accomplished and the mechanical quantum gates require longer time. Therefore, much more rapid quantum logic gates of such qubits are highly desirable.
In this paper, we use $|\pm \rangle=(|+1\rangle \pm|-1\rangle)/\sqrt{2}$ of the NV center spin as the qubit basis and then show how to prepare arbitrary qubits from the starting $|0\rangle$ state and how to construct rapid quantum logic gates for the qubits. On the basis of powerful exact analytical results of a quantum two-level system under a time-dependent magnetic field[@barnes; @barnes1], we construct exact evolution operator for the NV center spin (including three levels) under time-dependent magnetic fields. Then, we use these exact evolution operators to prepare arbitrary qubit states with the basis $|\pm \rangle$ from the starting state $|0\rangle$ and realize arbitrary quantum logic gates. All the typical quantum logic gates can be completed within a few nanoseconds. The fidelity can be made very high because one needs to control magnetic field strength only. More detailed results will be presented in the following.
The rest of the paper is organized as follows. In Set. , we define the Hamiltonian and elucidate the new spin basis. In Sec. , we construct exact evolution operators for the NV center spin under time-dependent magnetic fields. In Sec. , we show how to use the exact evolution operators to prepare arbitrary qubit states with the basis $|\pm \rangle$ from the starting state $|0\rangle$. In Sec. , we construct arbitrary quantum logic gates for the NV center spin qubits by using the exact evolution operators. Finally, we make some necessary discussions and give our conclusion in Sec. .
New spin basis
==============
The Hamiltonian of NV center spin $\vec{S}$, in the presence of time dependent magnetic field $\vec{B}(t)$=$(B_x(t),B_y(t),B_z(t))$, can be written as $$\label{eq1}
H=DS_{z}^{2}+\gamma \vec{S}\cdot\vec{B}(t),$$ where $\hbar=1$ is used, $D=2.87$GHz is the zero-field splitting, and $\gamma=2.8$MHz/G is the electron gyromagnetic ratio. Accordingly, the Schrödinger equation for time-evolution operator $U$ is given by $i\frac{d}{dt}U=HU$.
In the $S_z$ representation, the matrix form of the Hamiltonian (\[eq1\]) can be written as: $$\label{eq2}
H=\left(\begin{array}{ccc}
D+\gamma B_z & \frac{\gamma (B_x-i B_y)}{\sqrt{2}} & 0 \\
\frac{\gamma (B_x+i B_y)}{\sqrt{2}} & 0 & \frac{\gamma (B_x-i B_y)}{\sqrt{2}} \\
0 & \frac{\gamma (B_x+i B_y)}{\sqrt{2}} & D-\gamma B_z \\
\end{array}\right).$$ Introducing the unitary transform $$\label{eq3}
S_{1}=\left(
\begin{array}{ccc}
\frac{1}{\sqrt{2}}& 0 & \frac{1}{\sqrt{2}} \\
0 & 1 & 0 \\
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\end{array}
\right),$$ we can transform the Hamiltonian (\[eq2\]) into $$\label{eq4}
H^{'}=\left(\begin{array}{ccc}
D & \gamma B_x & \gamma B_z \\
\gamma B_x & 0 & i\gamma B_y \\
\gamma B_z & -i\gamma B_y & D \\
\end{array}\right).$$ It means that the basis is changed from ($|+1\rangle$,$|0\rangle$,$|-$1$\rangle$) to ($|+\rangle$,$|0\rangle$,$|-\rangle$), where $|+\rangle$=$\frac{1}{\sqrt 2}$($|+1\rangle
+|-1\rangle$) and $|-\rangle$=$\frac{1}{\sqrt2}$($|+1\rangle-|-1\rangle$). When the magnetic field is turned off, the $D$ term will produce the same overall phase for both $|+\rangle$ and $|-\rangle$. Therefore, there will be no relative phase between $|+\rangle$ and $|-\rangle$, and $|+\rangle$ and $|-\rangle$ can be used to make a stable qubit basis.
exact evolution operators
=========================
With a special magnetic field $\vec{B}(t)$=$(\alpha B_1,\beta B_1,0)$, the Hamiltonian in Eq. (\[eq4\]) becomes $$\label{eq5}
H_{o}=\left(\begin{array}{ccc}
D & \alpha\gamma B_1 & 0 \\
\alpha\gamma B_1 & 0 & i\beta\gamma B_1 \\
0 & -i\beta\gamma B_1 & D \\
\end{array}\right),$$ where $\alpha$ and $\beta$ are adjustable real parameters satisfying $\alpha^{2}+\beta^{2}$=$1$. Because of this condition, $\alpha$ and $\beta$ can be parameterized as $\alpha=\cos \theta$ and $\beta=\sin \theta$, where $-\pi \le \theta \le \pi$. Introducing another unitary transform $$\label{eq6}
S_{2}=\left(\begin{array}{ccc}
\cos \theta & 0 & i \sin\theta \\
0 & 1 & 0 \\
i \sin\theta & 0 & \cos\theta \\
\end{array}\right)$$ we can transform $H_o$ into a block-diagonal matrix $H_t=S_2H_oS_2^{\dag}$, $$\label{eq7}
H_{t}=\left(\begin{array}{ccc}
D & J(t) & 0 \\
J(t) & 0 & 0 \\
0 & 0 & D \\
\end{array}\right)$$ where $J(t)$=$\gamma B_1(t)$. Because $H_t$ consists of a 2$\times$2 block $$\label{eq72}
H_{t2}=\left(\begin{array}{cc}
D & J(t) \\
J(t) & 0 \\
\end{array}\right)$$ and a 1$\times$1 block $H_{t1}=D$, we can focus on $H_{t2}$ in Eq. (\[eq72\]).
Introducing a 2$\times$2 unitary transform $$T_{h}= \frac{1}{{\sqrt 2
}}\left( {\begin{array}{*{20}{c}}
1 & 1 \\
1 & { - 1} \\
\end{array}} \right),$$ we can change $H_{t2}$ into $H_{h2}=H_2+D/2$, with $H_2$ given by $$\label{eqH2}
H_{2}=\left(
{\begin{array}{*{20}{c}}
{ J(t)} & {\frac{D}{2}} \\
{\frac{D}{2}} & { - J(t)} \\
\end{array}} \right).$$ The constant term $\frac{D}{2}$ in $H_{h2}$ contributes only an overall phase to the time evolution operator. For the time-dependent two-level Hamiltonian $H_{2}$, there are many exact solutions such as those[@add1; @add2; @add3; @barnes; @barnes1]. Here, we choose a powerful method to construct its exact time evolution operator[@barnes; @barnes1].
The Schrödinger equation of $H_{2}$ can be exactly solved by the evolution operator[@barnes; @barnes1], $$\label{eq8}
U_2=\left(\begin{array}{ccc}
u_{11} & -u_{21}^{\ast } \\
u_{21} & u_{11}^{\ast }
\end{array}\right), |u_{11}|^2+|u_{21}|^2=1,$$ and the matrix elements $u_{11}$ and $u_{21}$ and the quantity $J(t)$ can be expressed as[@barnes; @barnes1] $$\label{eq9}\left\{
\begin{array}{l}
{u_{11}}(t)=\cos (\chi (t)){e^{i{\xi_-}(t)}} \\
{u_{21}}(t) = i\eta \sin (\chi (t)){e^{i{\xi_+}(t)}} \\
J(t) = \frac{\ddot{\mathop {\chi}}(t)}{{\sqrt {{D^2} -
4\dot{\mathop {\chi }}(t)^2 } }} - \frac{1}{2}\sqrt {{D^2} -
4\dot{\mathop {\chi }}(t)^2 } \cot (2\chi (t))
\end{array}\right.,$$ where $\xi_{\pm}$ is defined as $$\label{eq10}
\xi_{\pm}=\int_{0}^{t}dt^{^{\prime
}}\frac{1}{2}\sqrt{D^{2}-4\dot{\chi}^{2}}\csc (2\chi)\pm
\frac{1}{2}\arcsin (\frac{2\dot{\chi}}{D})\pm \eta \frac{\pi }{4}.$$ Here, $\eta$ can take either $+1$ or $-1$, and $\chi$ must satisfies three conditions: $|{\dot{\mathop{\chi}}(t)}|\le\frac{D}{2}$, $\chi(0)=0$, and $\dot{\mathop{\chi}}(0)=-\eta \frac{D}{2}$. By choosing suitable $\chi(t)$, we can exactly construct $J(t)$ in $H_2$ and the evolution operator $U_2$ in this way.
After adding the phase factor due to the $D/2$ term and making the inverse unitary transformation with $T_h^{\dagger}$, we obtain the evolution operator for $H_{t2}$, $$\label{Ut2}
U_{t2}\left( t\right) =\left(
\begin{array}{cc}
\bar{u}_{11} & -\bar{u}_{21}^{\ast } \\
\bar{u}_{21} & \bar{u}_{11}^{\ast } \\
\end{array}%
\right) e^{-i\frac{D}{2}t}$$ where the matrix elements $\bar{u}_{11}$ and $\bar{u}_{21}$ are expressed as $$\left\{
\begin{array}{c}
{\bar{u}_{11}}(t)=\cos(\chi(t))\cos({\xi_-}(t))+ i\eta\cos({\xi_+}(t))\sin(\chi(t))\\
{\bar{u}_{21}}(t)=\eta\sin(\chi(t))\sin({\xi_+}(t))+i\cos (\chi(t))\sin ({\xi_-}(t))\\
\end{array}\right.$$
Consequently, the whole time evolution operator of the Hamiltonian $H_{t}$ can be written as: $$\label{+eq11}
U_{t}\left( t\right) =\left(
\begin{array}{ccc}
\bar{u}_{11} & -\bar{u}_{21}^{\ast } & 0 \\
\bar{u}_{21} & \bar{u}_{11}^{\ast } & 0 \\
0 & 0 & e^{-i\frac{D}{2}t}%
\end{array}%
\right) e^{-i\frac{D}{2}t}$$ After making the inverse transform $S_2^{\dagger}$, we can get the time evolution operator of the starting Hamiltonian $H_{o}$ in the new basis of $|+\rangle$ and $|-\rangle$: $$\label{eq12}
\begin{split}
&U_{o}(\theta,t)=d(t)\times \\
&\left(
\begin{array}{ccc}
\bar{u}_{11}\alpha^2+d(t)\beta^2 & -\bar{u}_{21}^{*}\alpha &
-i\left(d(t)-\bar{u}_{11}\right)\alpha\beta \\
\bar{u}_{21}\alpha & \bar{u}_{11}^{*} & i\bar{u}_{21}\beta \\
i\left(d(t)-\bar{u}_{11}\right)\alpha\beta & i \bar{u}_{21}^{*}\beta
& d(t)\alpha^2+\bar{u}_{11}\beta^2 \\
\end{array}
\right),
\end{split}$$ where $\alpha=\cos \theta$, $\beta=\sin \theta$, and $d(t)=e^{-i\frac{D}{2}t}$.
From above equations (\[eq9\]) and (\[eq10\]), we can see that if the time-dependent function $\chi(t)$ is specified, the evolution operator (\[eq12\]) will be determined immediately. Using this time evolution operators, we can control a single NV center spin exactly and efficiently, and thus we can initialize and gate arbitrary qubits in the stable basis of $|+\rangle$ and $|-\rangle$.
Exact initializing of arbitrary qubits
======================================
Experimentally, the NV center spin can be easily prepared in state $|0\rangle$. We try to realize state transfer between $|0\rangle$ and $|\pm\rangle$. With the time evolution operator $U_{o}$ applied, the state $|0\rangle$ will become $$\label{eq13}
U_{o}(\theta,t)|0\rangle=d(t)\left(
\begin{array}{ccc}
-\cos\theta(\eta\sin\chi\sin\xi_{+}-i\cos\chi\sin\xi_{-}) \\
\cos\chi\cos\xi_{-}-i\eta\sin\chi\cos\xi_{+} \\
i\sin\theta(\eta\sin\chi\sin\xi_{+}-i\cos\chi\sin\xi_{-}) \\
\end{array}
\right).$$ We require that the function $\chi(t)$ is given by $$\label{eq14}
\chi\left(t\right)=\lambda t-\frac{{2\kappa{\lambda^3}{t^3}}}{3}+\frac{{\kappa{\lambda^3}{t^4}}}{T_f}-\frac{{2\kappa{\lambda^3}{t^5}}}{{5{T_f^2}}},$$ where $\lambda$ is defined as $\frac{D}{2}$, $\kappa$ is an adjustable parameter, and $T_f$ describes the time duration. Using $\eta$=$-1$, we have $\xi_{+}(T_f)$=$\xi_{-}(T_f)$ according to equation (\[eq10\]). Because the target state doesn’t contain state $|0\rangle$, we need to set $\bar{u}_{11}(T_f)=0$, [*i.e.*]{} $\cos(\xi_{+}(T_f))=\cos(\xi_{-}(T_f))=0$. Then the quantity $\chi(T_f)$ contributes an overall phase in the state $U_o(\theta,T_f)|0\rangle$ in Eq. (\[eq13\]). In order to achieve a minimal time value $T_f$ and a finite field pulse in the time interval $t\in(0, T_f)$, we need two conditions: $0<\chi(T_f)\leq\frac{\pi}{2}$ and $$\label{eq15}
\int_{0}^{T_f}dt\frac{1}{2}\sqrt{D^{2}-4\dot{\chi}^{2}}\csc (2\chi)=\frac{\pi}{2}.$$ Once we set $t=T_f$ and choose a value for $\chi(T_f)$, $\kappa$ can be formally solved by using equation (\[eq14\]), reading $\kappa=15(\lambda T_f-\chi(T_f))/(\lambda T_f)^3$. The time duration $T_f$ can be solved from equation (\[eq15\]), and then $\kappa$ can be calculated immediately. In the following, we shall show how to initialize three typical qubits from the spin state $|0\rangle$.
[*Initializing the basis states*]{} $|\pm\rangle$. Choosing $\alpha=0$ (or $\beta=0$) in Eq. (5), we can easily get the target state $U_o(\frac{\pi}{2},T_f)|0\rangle=|-\rangle$ (or $U_o(0,T_f)|0\rangle=|+\rangle$) with an overall phase. In this way, we get $|\pm\rangle$ from $|0\rangle$.
[*Initializing a superposed state*]{} $\cos\theta_1|+\rangle +i\sin\theta_1|-\rangle$. In this cases, we can assume $0\leq\theta_{1} \leq \pi$ without losing any effective information. Choosing $\alpha$ and $\beta$ in Eq. (5) to satisfy the equality $\arctan(\beta/\alpha)=\theta \ge 0$, we can let $\theta = \pi-\theta_1$ in Eq. (17) and thus obtain the final state $$\label{eq16}
U_{o}(\pi-\theta_1,T_f)|0\rangle=-i\left(
\begin{array}{ccc}
\cos\theta_1\\
0 \\
i\sin\theta_1 \\
\end{array}
\right)e^{-i\chi(T_f)}e^{-i\frac{D}{2}T_f}.$$ Neglecting the overall factor, we obtain $\cos\theta_1|+\rangle +i\sin\theta_1|-\rangle$.
[*Initializing arbitrary state*]{} $\cos\theta_{1}|+\rangle +e^{i\varphi}\sin\theta_{1}|-\rangle$. We assume that $0\leq\varphi\leq\pi$. State $\cos\theta_1|+\rangle +i\sin\theta_1|-\rangle$ can be obtained from $|0\rangle$ in the same way as above. The next step is to realize the phase factor $e^{i(\varphi-\frac{\pi}{2})}$ for $|-\rangle$. If $B_1(t)$ in Eq. (\[eq5\]) is a time-independent magnetic field $B_0$, the method presented in Sec. is not applicable, but it is not difficult to derive the evolution operator: $$\label{eq17}
\begin{split}
&P(\tau,\alpha_0,\beta_0)=d(\tau)\times \\
&\left(
\begin{array}{ccc}
\beta_0 ^2 d(\tau)+(-1)^n\alpha_0 ^2& 0 & i \alpha_0\beta_0\left((-1)^n- d(\tau)\right) \\
0 & (-1)^n & 0 \\
-i\alpha_0\beta_0\left((-1)^n-d(\tau)\right) & 0 & \alpha_0^2 d(\tau)+(-1)^n\beta_0^2 \\
\end{array}
\right),
\end{split}$$ where $n$ is a non-negative integer. We assume $\tau$=$\tau_f$ when the evolution ends. Here, $\tau_f$ must satisfy the equality: $\frac{D}{2}\tau_f\sqrt{1+4 (\frac{\gamma B_{0}}{D})^2}$=$n\pi$. If $n=0$, we must have $\tau_f=0$. Consequently, it means $\varphi-\frac{\pi}{2}=0$, and the operator $P(\tau_f,\alpha_0,\beta_0)$ becomes identity operator, or trivial operator. For any nontrivial evolution operator, we have $\tau_f>0$ and $n=1$, Because of $\varphi-\frac{\pi}{2}\in[-\frac{\pi}{2},\frac{\pi}{2}]$, we need nontrivial evolution operators for $|\varphi-\frac{\pi}{2}|>0$. It is easily seen that only if $\alpha_0\beta_0=0$ in (21), can arbitrary relative phases be realized by the operator $P(\tau_f,\alpha_0,\beta_0)$. This means that the magnetic field should be applied in either $x$-axis or $y$-axis. For $\varphi\in[0,\frac{\pi}{2})$, we need to set $\alpha_0=0$ and then obtain $\tau_f=\frac{2}{D}(\varphi+\frac{\pi}{2})$ and $B_0=\frac{D}{2\gamma}\sqrt{(\frac{\pi}{\varphi+\frac{\pi}{2}})^2-1}$; and for $\varphi\in(\frac{\pi}{2},\pi]$, we need to set $\beta_0=0$ and then obtain $\tau_f=\frac{2}{D}(\frac{3\pi}{2}-\varphi)$ and $B_0=\frac{D}{2\gamma}\sqrt{(\frac{\pi}{\frac{3\pi}{2}-\varphi})^2-1}$. We can parameterize $\alpha_0=\cos \theta_0$ and $\beta_0=\sin \theta_0$. In this way, time duration $\tau_f$ will be only few nanoseconds and magnetic field $B_0$ will be reasonable at the same time. Because $\theta_0$ ($\alpha_0$ and $\beta_0$), $\tau_f$, and $B_0$ are determined by $\varphi$, we can use $P_f(\varphi-\frac{\pi}{2})$ to denote the evolution operator.
Therefore, the whole procedure, achieved in two steps, can be represented as the evolution operator $$\label{ui}
U_I(\varphi,\theta_1)=P_f(\varphi-\frac{\pi}{2})U_{o}(\pi-\theta_1,T_f)$$ For practical application, it is useful to adjust the value $\chi(T_f)$ to connect the two magnetic fields $B_1(t)$ and $B_0$ at the time $T_f$ along either $x$-axis or $y$-axis. We show in Fig. 1 that magnetic field $B_1(T_f)$ can vary from $450G$ to $740000G$ when $\chi(T_f)$ changes within $(0, \frac{\pi}{2}]$. Because of the large domain of $B_1(T_f)$, we can likely realize continuous connection of magnetic field in either $x$-axis or $y$-axis.
![\[fig:qtest1\] The final magnetic field $B_1(T_f)$ in unit of $10^3$G depending on parameter $\chi(T_f)$.](fig1.eps){width="0.8\columnwidth"}
realization of quantum logic gates
==================================
In last section, we show how to initiate the basis states $|\pm\rangle$ and their superposed states from the state $|0\rangle$. Our arbitrary qubits are made from the basis states $|\pm\rangle$. In the quantum circuit model of computation, a quantum gate is a basic quantum circuit operating on a small number of qubits. We show how to realize typical quantum gates on the qubit in the following. In some of the cases, the special state $|0\rangle$ can be used as a auxiliary state, or a bridge.
[*$\frac{\pi}{4}$ phase shift gate*]{}: $|+\rangle+|-\rangle \longrightarrow |+\rangle +e^{i\frac{\pi}{2}}|-\rangle$. In this case, we need only a phase factor $e^{i\frac{\pi}{2}}$ for the $|-\rangle$ term. This can be achieved by applying a unitary transformation $P_f(\frac{\pi}{2})$ on the starting state $|+\rangle+|-\rangle$. As a result, we obtain the evolution operator for the $\frac{\pi}{4}$ phase shift gate: $$U_{\frac{\pi}{4}}=P_f(\frac{\pi}{2})e^{-i\pi}.$$ The time duration $\tau_1$ and the time-independent magnetic field $h_1$ can be given by $\tau_1=\frac{\pi}{D} \sim 1.1$ns and $h_1=\frac{\sqrt{3}D}{2\gamma} \sim 888$G.
[*$\frac{\pi}{8}$ phase shift gate*]{}: $|+\rangle+|-\rangle \longrightarrow |+\rangle +e^{i\frac{\pi}{4}}|-\rangle$. It is similar to the $\frac{\pi}{4}$ phase shift gate. The phase factor is $e^{i\frac{\pi}{4}}$ in this case. Applying $P_f(\frac{\pi}{4})$ on $(|+\rangle+|-\rangle)$, we obtain the evolution operator for the $\frac{\pi}{8}$ phase shift gate $$U_{\frac{\pi}{8}}=P_f(\frac{\pi}{4})e^{-i\frac{\pi}{4}}.$$ The time duration $\tau_2$ and the time-independent magnetic field $h_2$ are given by $\tau_2=\frac{3\pi}{2D} \sim1.64$ns and $h_2=\frac{\sqrt{7}D}{6\gamma} \sim 452$G.
[*Pauli-X gate*]{}: $|+\rangle\longrightarrow|-\rangle$. This gate can be realized by applying $U_o^{\dag}(0,t_1)$ on the initial state $|+\rangle$ and then $U_o(\frac{\pi}{2},t_2)$ on the resulting intermediate state $|0\rangle$: $$|-\rangle \propto U_o(\frac{\pi}{2},t_2) U_o^{\dag}(0,t_1) |+\rangle.$$ In this way the gate is realized in two steps. Letting $\chi_1(t_1)=\chi_2(t_2)=\frac{\pi}{2}$, we have $t_1=t_2=T$, and then obtain $\kappa_1=\kappa_2=\frac{{15(\lambda T-\frac{\pi}{2})}}{{{\lambda^3}{T^3}}}$. Using the phase condition in (\[eq15\]), we can obtain $T=\frac{3.93}{D}$. For the first step, the time-dependent field function $J(t)/\lambda$ and the probability $P_{|+\rangle}$ of $|+\rangle$ as functions of time ($\lambda t$) are shown in Fig. 2. For the second step, we have similar time dependence for the field and the probability. After these two steps, the evolution operator for the Pauli-X gate is given by $$V_X=U_o(\frac{\pi}{2},T) U_o^{\dag}(0,T)e^{i\pi/2}.$$ And the total time interval is equivalent to $2T\sim 2.7$ns.
In addition, this state can be realized without applying $U_o(\theta_2,T_1)$ on state $|0\rangle$, where $T_1$ is pulse time duration. At this time, we must guarantee matrix element $\bar{u}_{21}(T_1)$=$0$. After applying operator $U_o(\theta_2,T_1)$ on state $|+\rangle$, we derive $$U_{o}(\theta_2,T_1)|+\rangle=d(T_1)\left(
\begin{array}{ccc}
e^{-i\frac{D}{2}T_1}\sin^2\theta_2-e^{-i\chi_3(T_1)}\cos^2\theta_2\\
0 \\
i(e^{-i\frac{D}{2}T_1}+e^{-i\chi_3(T_1)})\cos\theta_2\sin\theta_2 \\
\end{array}
\right)$$ with $\xi_{\pm}(T_1)$=$\pi+2m\pi$ and $m$ is positive integer. The condition to achieve state $|-\rangle$ is $\sin^2(2\theta_2)[1+\cos(\frac{D}{2}T_1-\chi_3(T_1))]$=$2$. It can be satisfied by setting $\cos(\frac{D}{2}T_1-\chi_3(T_1))$=$1$ and $\theta_2$=$\frac{\pi}{4}$. A reasonable result is $\frac{D}{2}T_1-\chi_3(T_1)$=$2\pi$ and then $\kappa_3$=$\frac{30\pi}{\lambda^3T_1^3}$. Using the phase condition about $\xi_{\pm}(T_1)$ in equation (\[eq15\]) by replacing $\frac{\pi}{2}$ with $\pi+2m\pi$, we can get a self-consistent value, $T_1\sim 5.3$ns from $\frac{D}{2}T_1 \approx7.542$ with $m=4$. Therefore, the evolution operator can be expressed as $$U_X^{\prime}=U_o(\theta_2,T_1)e^{i(DT_1+\pi/2}.$$ The smooth pulse and probability evolution of state $|+\rangle$ is shown in Fig. 3. Comparing it with Fig. 2, we can see that it is not as efficient as that using the intermediate state $|0\rangle$ because the magnetic field as a function of time is irregular.
![\[fig:qtest2\] The time $t$ (in $1/\lambda$) dependence of $J(t)/\lambda$ (upper panel) and $P_{|+\rangle}$ (lower panel), with $\kappa=\frac{{15(\lambda
T_f-\frac{\pi}{2})}}{{{\lambda^3}{T_f^3}}}$ in Eq. (18).](fig2.eps){width="0.8\columnwidth"}
![\[fig:qtest2\] The time $t$ (in $1/\lambda$) dependence of $J(t)/\lambda$ (upper panel) and $P_{|+\rangle}$ (lower panel), with $\kappa=\frac{30\pi}{\lambda^3T_f^3}$ in Eq. (18).](fig3.eps){width="0.8\columnwidth"}
[*Hadamard gate*]{}: $|\pm\rangle\longrightarrow\frac{1}{\sqrt{2}}|+\rangle
\pm\frac{1}{\sqrt{2}}|-\rangle$. In this situation, we can’t use operator (\[ui\]) directly. So in first step, we change it to state $|0\rangle$. It is the same step mentioned in dealing with Pauli-X gate: $U_o^{\dag}(0,T) |+\rangle$. After this step, for case $|0\rangle$ turning to $\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$, we only have to set $\theta_1=\frac{\pi}{4}$ and $\varphi=0$ in operator (\[ui\]). The evolution operator for the Hadamard gate is $$U_H=U_I(0,\frac{\pi}{4}) U_o^{\dag}(0,T).$$ The total time is $2T+\tau_f\approx\frac{2\times3.93+\pi}{D} \approx 3.83$ns.
In addition, this gate can be realized without using $|0\rangle$. In this case, the static field is applied in $z$-axis, and the time evolution operator for the Hamiltonian (\[eq4\]) with magnetic field $(0,0,B_z)$ is given by $$\label{eq30}
Q(t)=\left(
\begin{array}{ccc}
\cos(J_zt) & 0 & -i\sin(J_zt) \\
0 & e^{iDt} & 0 \\
-i\sin(J_zt) & 0 & \cos(J_zt)
\end{array}
\right)e^{-iDt}$$ where $J_z$=$\gamma B_{z}$. If the initial state is $|+\rangle$ and $J_zt_f$=$\frac{\pi}{4}$, where $t_f$ is time duration, $Q(t_f)$ makes the state $|+\rangle$ become $Q(t_f)|+\rangle=\frac{1}{\sqrt{2}}(|+\rangle-i|-\rangle)$. In order to get target state $\frac{1}{\sqrt{2}}(|+\rangle+|-\rangle)$, we need a phase factor $e^{i\pi/2}$ for the $|-\rangle$. It can be shown that the time value and magnetic field is the same as $\tau_1$ and $h_1$. With the phase shift operator $P_f(\frac{\pi}{2})$, the whole process can be represented as $$\label{eq32}
U_H^{\prime}
=P_f(\frac{\pi}{2})Q(t_f)=\left(\begin{array}{ccc}
i\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
0 & ie^{iDt_f} & 0 \\
i\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}%
\end{array}
\right)e^{-iDt_f}.$$ The total time duration is $t_f+\tau_1=
\frac{\pi}{4\gamma B_{z}}+\frac{\pi}{D} \approx 1.375$ns with $B_z \sim 1000$G. Furthermore, this gate can change $|-\rangle$ into $\frac{1}{\sqrt{2}}|+\rangle-\frac{1}{\sqrt{2}}|-\rangle$.
[*Arbitrary gating*]{}. In this case, we need to realize $\cos\theta_{1}|+\rangle +e^{i\varphi _{1}}\sin \theta
_{1}|-\rangle \longrightarrow \cos \theta_{2}|+\rangle+e^{i\varphi
_{2}}\sin \theta_{2}|-\rangle$. Assuming $0\leq\theta_{1},\theta_{2}\leq\pi$ and $0\leq\varphi_{1},\varphi_{2}\leq \pi$. If $\theta_{1}$=$\theta_{2}$, we need only to modify the relative phase. According to the previous subsection, we can realize this control through the intermediate state $|0\rangle$. The first part is actually the inverse process of that in Sec. . As the operator (\[ui\]) is unitary, the first process can be achieved by the operator $U_I^{\dag}(\varphi_1,\theta_1)$, with $T_2$ being the pulse time duration. Then, the target state can be realized by $U_I(\varphi_2,\theta_2)$ with $T_3$ being the pulse time duration. The whole process can be represented by $$\label{eq33}
%\begin{split}
U_A(\varphi_2, \theta_2,\varphi_1,\theta_1)
=U_I(\varphi_2,\theta_2)U_I^{\dag}(\varphi_1,\theta_1)
%\end{split}$$
This gate can be realized without the intermediate state $|0\rangle$. It is clear from operator (\[eq30\]) that $J_zt$ can be seen as one variable because the overall phase factor does not play role in the gate. As a result, the larger the longitudinal field, the smaller the time duration. The initial relative phase $\varphi_1$ should be adjusted before the probability amplitude is changed. At first, we change the relative phase $\varphi_{1}$ into $\frac{\pi}{2}$ by $P_f^{\dag}(\varphi_1-\frac{\pi}{2})$, making a state $|\Phi(0)\rangle=\cos\theta_1|+\rangle
+i\sin\theta_1|-\rangle$. Then, we apply the time evolution operator $Q(t_e)$ on $|\Phi(0)\rangle$, where $t_e$ is the time duration, and obtain the state $|\Phi(t_e)\rangle=(\cos(\theta_1-J_z t_e)|+\rangle+i\sin(\theta_1-J_z t_e)|-\rangle)e^{-iDt_e}$. Here, $J_z t_e$ can be expressed as $$J_zt_e=\left\{\begin{array}{ll}\theta_1-\theta_2 & ~{\bf if}~~ \theta_1\ge \theta_2\\ \theta_1-\theta_2+2\pi & ~{\bf if}~~ \theta_1< \theta_2\end{array}\right.$$ Finally, we apply the phase regulation $P_f(\varphi_2-\frac{\pi}{2})$ to get the phase $\varphi_2$. The whole procedure can be represented as the unitary operator: $$\label{eq34}
U^{\prime}_A(\varphi_2, \theta_2,\varphi_1,\theta_1) =P_f(\varphi_2-\frac{\pi}{2})Q(t_e)P_f^{\dag}(\varphi_1-\frac{\pi}{2})$$
Gate $\frac{\pi}{4}$ phase $\frac{\pi}{8}$ phase Pauli-X Hadamard
------- ----------------------- ----------------------- --------- ----------
$T_G$ 1.1 ns 1.6 ns 2.7 ns 3.8 ns
5.3 ns 1.4 ns
: The time durations ($T_G$) of the $\frac{\pi}{4}$ phase, $\frac{\pi}{8}$ phase, Pauli-X, and Hadamard gates
Discussion and Conclusion
=========================
The time durations $T_G$ of the typical gates are summarized in Table I. It is clear that the gates need at most a few nanoseconds. Furthermore, it can be estimated to take only nanoseconds to complete the initialization and gating of arbitrary qubits. Therefore, the qubits on the basis of $|+\rangle$ and $|-\rangle$ can be fast initialized and gated with magnetic fields, in comparison to those in terms of $|+1\rangle$ and $|-1\rangle$ with strong mechanical driving[@mech]. When the magnetic field is applied along the z axis, the effect of the time evolution operator (\[eq30\]) on our NV center qubit with the basis of $|+\rangle$ and $|-\rangle$ looks like that on spin-$\frac 12$ qubit in “bang-bang” approach[@zhihui], but they are different from each other. Because here one needs to control the magnetic field strength only, the theoretical fidelity can be very high for these gates.
In summary, by choosing $|\pm \rangle$, defined as $(|+1\rangle \pm|-1\rangle)/\sqrt{2}$, of the NV center spin as the qubit basis, we have obtained exact evolution operator of the NV center spin under time-dependent magnetic field by mapping the three-level system of the NV center spin on a two-level spin system under a time-dependent magnetic field and using the existing exact analytical results of the quantum two-level system[@barnes; @barnes1]. With the exact evolution operator of the NV center spin including three levels, we have shown how to pre pare arbitrary qubits with the basis $|\pm \rangle$ from the starting $|0\rangle$ state and to realize arbitrary rapid quantum logic gates for the qubits. It also has been estimated that the typical quantum logic gates can be accomplished within a few nanoseconds. We believe that the fidelity can be very high because only magnetic field strength needs to be controlled. these results are useful to realizing quantum computing with the NV center spin systems in diamond and exploring other quantum effects and applications.
This work is supported by Nature Science Foundation of China (Grant Nos. 11174359 and 11574366), by Chinese Department of Science and Technology (Grant No. 2012CB932302), and by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB07000000).
[\*]{}
V. V. Dobrovitski, G. D. Fuchs, A. L. Falk, C. Santori, and D. D. Awschalom, Annu. Rev. Condens. Matter Phys. **4**, 23 (2013). R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Annu. Rev. Phys. Chem. **65**, 83 (2014). M. W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N. B. Manson, and L. C. L. Hollenberg, Phys. Rev. B **85**, 205203 (2012). N. B. Manson, J. P. Harrison, and M. J. Sellars, Phys. Rev. B **74**,104303 (2006). J. W. Zhou, P. F. Wang, F. Z. Shi, P. Huang, X. Kong, X. K. Xu, Q. Zhang, Z. X. Wang, X. Rong, and J. F. Du, Front. Phys. **9**, 587 (2014). T. P. M. Alegre, C. Santori, G. Medeiros-Ribeiro, and R. G. Beausoleil, Phys. Rev. B **76**, 165205 (2007). P. London, P. Balasubramanian, B. Naydenov, L. P. McGuinness, and F. Jelezko, Phys. Rev. A **90**, 012302 (2014). G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, Science **326**, 1520 (2009). P. Tamarat, N. B. Manson, J. P. Harrison, R. L. McMurtrie, A. Nizovtsev, C. Santori, R. G. Beausoleil, P. Neumann, T. Gaebel, F. Jelezko, P. Hemmer, and J. Wrachtrup, New J. Phys. **10**, 045004 (2008). L. C. Bassett, F. J. Heremans, D. J. Christle, C. G. Yale, G. Burkard, B. B. Buckley, D. D. Awschalom, Science **345**, 1333 (2014). V. Jacques, P. Neumann, J. Beck, M. Markham, D. Twitchen, J. Meijer, F. Kaiser, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, Phys. Rev. Lett. **102**, 057403 (2009). J. Teissier, A. Barfuss, P. Appel, E. Neu, and P. Maletinsky, Phys. Rev. Lett. **113**, 020503 (2014). E. R. MacQuarrie, T. A. Gosavi, N. R. Jungwirth, S. A. Bhave, and G. D. Fuchs, Phys. Rev. Lett. **111**, 227602 (2013). C. Zu, W. B. Wang, L. He, W. G. Zhang, C. Y. Dai, F. Wang, and L. M. Duan, Nature **514**, 72 (2014). A.-C. Silvia A. Lazariev, S. W. Hell, and G. Balasubramanian, Nature Commun. **5**, 4870 (2014). A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, and P. Maletinsky, Nat. Phys. 10.1038/NPHYS3411 (2015).
N. V. Vitanov, New J. Phys. **9**, 58 (2007). A. Gangopadhyay, M. Dzero, and V. Galitski, Phys. Rev. B **82**, 024303 (2010). E. Barnes and S. Das Sarma, Phys. Rev. Lett. **109**, 060401 (2012). G. C. Hegerfeldt, Phys. Rev. Lett. **111**, 260501 (2013). E. Barnes, Phys. Rev. A **88**, 013818 (2013). Z. H. Wang and V. V. Dobrovitski, Phys. Rev. B **84**, 045303 (2011).
|
---
abstract: 'We present an analysis of the deepest pure-UV observations with the highest angular resolution ever performed, a set of 12 exposures with the HST WFPC2 and F160BW filter obtained in parallel observing mode, which cover $\sim$12 square arcminutes in the LMC, North of the bar and in the “general field” regime of the LMC. The 341 independent measurements of 198 objects represent an accumulated exposure of $\geq$2 10$^4$ sec and reveal stars as faint as m$_{UV}\simeq$22 mag. The observations show that $\sim$2/3 of the UV emission from the LMC is emitted by our HST-detected UV stars in the field, [*i.e., not*]{} in clusters or associations. We identified optical counterparts in the ROE/NRL photometric catalog for $\sim$ 1/3 of the objects. The results are used to discuss the nature of these UV sources, to estimate the diffuse UV emission from the LMC as a prototype of dwarf galaxies, and to evaluate the contamination by field stars of UV observations of globular and open clusters in the LMC. We find that the projected density of UV stars in the general field of the LMC is a few times higher than in the Galactic disk close to the Sun. Combining our data with observations by UIT allows us to define the stellar UV luminosity function from m$_{UV}$=8 to 18 mag, and to confirm that the field regions in the LMC have been forming stars at a steady rate during the last 1 Gyr, with an IMF close to the Salpeter law.'
author:
- 'Noah Brosch[[^1]]{}, Michael Shara, John MacKenty, David Zurek and Brian McLean'
title: 'Far-Ultraviolet Imaging of the Field Star Population in the Large Magellanic Cloud with HST[[^2]]{}'
---
\#1\#2\#3\#4\#5\#6\#7
to\#2
------------------------------------------------------------------------
Introduction
============
The Large Magellanic Cloud (LMC) is the nearest galaxy to the Milky Way (with the exception of a merging dwarf galaxy), one in which individual stars can be distinguished and studied fairly easily with instruments of the highest angular resolution such as the Hubble Space Telescope (HST). The LMC is different from the Milky Way (MW) in that it shows more intense star formation (SF), at least when compared with the Solar neighborhood, [*i.e.,*]{} the MW region within $\sim$2 kpc of the Sun where the interstellar extinction is reasonably small and individual stars can be easily studied in the visible region of the spectrum.
The difference in SF, coupled with the much smaller size of the LMC compared with the MW, implies that this is a dwarf irregular or a tidally truncated small spiral, in which the SF processes are probably different from those in the MW. Feitzinger (1987) argued that the LMC is a good example of a galaxy in which the SF proceeds primarily via the stochastic self-propagating SF mechanism (SSPSF: Gerola & Seiden 1978), whereas the conventional view ([*e.g.,*]{} Kaufman 1979) is that the SF in the MW is driven mainly by spiral density waves. The different SF mechanism, together with the possibility of studying individual stars, are the reasons the LMC is a very popular target for studies of SF-related phenomena ([*e.g.,*]{} Geha 1998, Battinelli & Demers 1998).
There is good reason to study the population of field stars in the LMC; these objects represent the unspectacular star formation processes in the LMC and, by inference, in dwarf irregular galaxies in general. If we manage to understand the “quiet” mode of star formation, [*i.e.,*]{} that which is not generated in a starburst, we may better understand the origin of the UV radiation in the Universe. Note that in starburst galaxies only $\sim$20% of the UV light at 2200Å is produced by stars in clusters (Meurer 1995) and the majority of the light originates from a general population of UV stars. This diffuse UV emission might be due to the stars that have been ejected from regions of active SF. Studies of the field star population of the LMC were reported by [*e.g.,*]{} Elson (1997), Geha (1998). These were performed with HST in optical and near-infrared bands, being rather insensitive to massive/hot stars. The stellar populations of the LMC were reviewed recently by Feast (1995), who did not find strong evidence for a starburst triggered by a LMC-SMC-Milky Way interaction. He argued that the SF increase in the LMC was caused by the collapse of the system to a plane about 4 Gyrs ago.
The nature of the fainter ultraviolet (hereafter UV, covering the approximate wavelength range $\lambda\lambda$1000-3000Å) sources has not yet been resolved. This is in part due to the fact that since the TD-1 mission in the late 1960s (Thompson 1978) there has not been an all-sky survey in the UV. There has also not been an instrument capable of providing good imaging in the UV, from which photometric parameters and shape information on extended objects could be derived. Previous deep UV observations have been perfomed by the UIT Shuttle-borne telescope (Stecher 1992) and by the FOCA balloon-borne telescope (Milliard 1992). The latter are in a spectral band centered at 2015Å and $\sim188$Å wide; the width of the band depends essentially on the altitude of the balloon which carried the FOCA telescope to 42+ km altitude. Because of this, the FOCA passband has a slight red leak ($\sim10^{-2}$), which depends on the exact altitude of the instrument during the observation. The results from the diverse FOCA flights have not yet been published in a systematic manner, as the ground-based follow-up is still in its early phases. The few published FOCA results, supplemented by ground-based follow-up observations from large telescopes ([*e.g.,*]{} Treyer 1998), are sufficiently intriguing to warrant concerted attempts to observe faint UV sources with good angular resolution. These results indicate that $\sim$50% of the UV sources in a field centered on the Coma cluster are background galaxies ranging to z$\approx$0.7 with strong emission lines. Similar results were announced by the group headed by R. Ellis (Treyer 1998).
Other UV imaging instruments, such as S201 (Carruthers 1977) or FAUST (Bowyer 1995), had much lower angular resolution than either UIT ($\sim$3 arcsec) or FOCA ($\sim$10 arcsec) and did not reach objects as faint as these. However, these low angular resolution imagers offer unique panoramic views of the LMC in the UV. Previous UV imaging observations of the LMC were reported by Page & Carruthers (1981) using the S201 camera carried to the Moon’s surface during the Apollo 16 flight; by Courtés (1984) with the wide-field camera operated on the Space Shuttle; by Smith (1987) using a rocket-borne UIT prototype; and by Courtés (1995) with the FAUST instrument. In addition, observations in the UV near 30 Dor with photometers were reported by Koornneef (1977) using the ANS satellite with five UV bands and with an angular resolution of 2.5 arcmin; by Nandy (1979) and Morgan (1979) from the TD-1 satellite with four bands and with 1.7 arcmin resolution; and on stellar assocations in the LMC with the Skylab S183 experiment (Vuillemin 1988). Most LMC imaging in the UV was done with very wide field cameras and with low angular resolution, of order 3-5 arcmin. Despite the low resolution, which allows only for the mapping of the general UV emission pattern, these observations generated quantitative measures of the emission from unresolved sources and diffuse nebulosity in the LMC. In particular, Page & Carruthers (1981) measured the total UV emission from the LMC to be S$_{1400}$=3.4 10$^{-7}$ erg sec$^{-1}$ cm$^{-2}$ Å$^{-1}$ ster$^{-1}$ (at 1400Å) and S$_{1300}$=3.8 10$^{-7}$ erg sec$^{-1}$ cm$^{-2}$ Å$^{-1}$ ster$^{-1}$ (at 1300Å).
The 30 Doradus and SN 1987A regions in the LMC were observed with higher angular resolution by the UIT and the results were published by Cheng (1992). The latter observation is relevant, because it relates to a region of the LMC very close to that studied here. Similar UIT observations, of the associations LH 52 and LH 53, and the SNR N49 in the LMC, were reported by Hill (1995). The field population of UV stars was studied by Hill (1994), who found a lower star formation (SF) rate there than in the stellar associations of the LMC, as well as a SF less biased toward high-mass objects. Finally, all the UV sources in the direction of the LMC which have been observed by UIT have been collected in a catalog by Parker (1998). The results from the two UIT flights constitute the deepest catalog of UV objects in the LMC published to date (Parker 1998). The UIT observations yielded photometry from images with reasonable resolution: FWHM=3.4 arcsec. A comparison of the UIT-derived magnitudes and synthetic photometry for 255 stars in common with the IUE observatory show no systematic shift, but a standard deviation of 0.25 mag in the distribution of magnitudes.
It is possible that many UV sources are cosmologically-important objects, as Treyer (1998) have shown. These sources may account for a large part of the extragalactic diffuse UV background (dUVB); this may not vanish completely in directions with very low HI column densities, indicating that not all the dUVB is dust-scattered Galactic UV light (Leinert 1998). This motivated us to study the deepest UV observations obtained in parallel mode by the largest orbiting astronomical telescope, the Hubble Space Telescope. HST is equipped, in principle, with three UV-imaging instruments: WFPC2, FOC, and STIS. The second instrument has an extremely restricted field of view, while the third has a similarly small field but a much higher photometric accuracy, dynamic range, and throughput than the FOC. However, STIS cannot be used (yet) for parallel UV imaging observations because of the possible damage to the MAMA detectors by UV sources, and most of its pointings are in the direction of crowded, well-defined fields such as galactic nuclei and globular clusters where the sources are fainter than the damage limit. It is clear that, at present, only WFPC2 can offer a reasonably large field combined with some UV throughput for the serendipitous study of the UV sky.
A number of papers have been published which analyse data collected with the WFPC2 UV filter set. Most of these filters have significant red leaks and the accuracy of the photometry cannot always be fully estimated. For this reason, many researchers prefer to use the F160BW filter. This “Wood’s filter” separates a pure UV band with an effective wavelength $\lambda_{e}$=1491Å and bandpass 446Å (Holtzman 1995: H95b). The F160BW filter has a very low throughput (approximately one order of magnitude below other UV filters from the WFPC2 complement); however, its bandpass has no red leak, unlike other WFPC2 filters. It also vignettes $\sim$1/8 of the field imaged through each WF CCD (the PC chip is not vignetted). More details about this filter can be found in Watson (1994). Observations using F160BW to study hot stars in globular clusters were reported by [*e.g.,*]{} Mould (1996) and Cole (1997). Hunter (1997) studied the UV emission from another cluster (R136) with the F170W filter, as did Gilmozzi (1994) for the “double cluster” NGC 1850A+B; the F170W filter has a very strong red leak and the interpretation of the results with it is difficult.
In order to realize fully the UV imaging potential of the WFPC2+F160BW combination, a program of parallel imaging was initiated by JM. Some archived observations obtained in this program are used here to study the field population of UV stars in the LMC. We use the results to estimate the diffuse UV emission from the LMC and by inference, from other star-bursting dwarf galaxies, and to evaluate the degree of contamination by field UV stars of the pointed observations of globular and open clusters in the LMC. Our observations and reductions are described in Section 2 and the photometry is discussed in Section 3. Optical counterparts are identified in Section 4, and their nature is discussed in Section 5. The detection of a significant number of UV stars allows us to draw some general conclusions about the SF in the LMC.
Observations
============
The HST observed objects in the N132D region of the LMC with the Faint Object Spectrograph from 23 to 26 August 1995 collecting data for proposal 5607. During these very long pointings the WFPC2 was commanded to observe in parallel with the F160BW filter. These observations are ideal to study the population of hot stars in the crowded environment of the LMC. During the LMC session the WFPC2 acquired 12 images with F160BW with exposure times of 2300 sec to 4900 sec, which we analyze here. A log of the HST data sets used in this paper is given in Table 1.
The region observed by HST is centered at (J2000) 5$^h$ 25$^m$, –69$^{\circ}$ 30’; the foreground extinction in this direction is E(B–V)$\leq$0.1 (Lucke 1974). The area is located near the southeast end of the LMC bar, not on the bar itself but rather on its northern edge, $\sim0^{\circ}.7$ off its major axis, and $\sim1^{\circ}.3$ off the LMC’s kinematic center. The list of UV-bright objects or regions in the LMC detected by S201 (Page & Carruthers 1981) contains no object in this direction, and neither does the catalog of infrared sources in the LMC and SMC (Schwering & Israel 1990). The map of radio continuum produced by Xu (1992) shows that at the location of the HST WFPC2 pointing there are no strong ridges of 6.3 cm continuum emission. The map of diffuse H$\alpha$ emission (Fig. 8 in Xu 1992) shows that the line emission is also minimal in this location, but that 30 Dor, about half a degree to the East, is very bright. The region is one of those studied by Davies (1976) with UK Schmidt plates through a 100Å wide H$\alpha$ filter to search for nebulosities. The N132D region is noted in their Table II along with N132H as being faint and having “knots in envelope”, a size of 7$\times$4 arcmin, and corresponding to the MC39 11cm radio source. MC39 is a radio and Xray-detected supernova remnant (Clark 1982), located $\sim$10 arcmin south of the WFPC2 region imaged in the UV. The wide field imaging by the S183 experiment indicates that the emission at 2600Å (250Å passband) from the location of the HST pointings analyzed here is less than 2.69 10$^{-14}$ erg sec$^{-1}$ cm$^{-2}$ Å$^{-1}$ (square arcmin)$^{-1}$ (Vuillemin 1988, Fig. 1b). This figure also shows that the HST pointing was not towards any of the UV-bright stellar associations in the LMC.
The image of the N132 field in Fig. 3 of Parker (1998) contains the HST field studied here in its lower-left (north of the center) part. No exceptionally high stellar density is apparent, nor is this a very sparse field. A careful comparison between the 11$^{\circ}$.4 diameter image obtained by Smith (1987) at 1590Å and shown as the top figure of their Plate 23 and the location of the N132 field in the R-band image shown by Parker (1998, Fig. 2) emphasizes that the location of the HST region studied here is off the LMC bar. Although the diffuse UV emission in the vicinity of N132 seems to be higher than from the rest of the LMC disk, it is clear that the HST observations pointed to an area of unexceptional (faint) diffuse UV emission, between Shapley Constellations II and V (see Fig. 4 of Smith 1987). As it is not clear how good is the flat fielding of the very wide field image in Smith (1987), we cannot remark on the absolute UV surface brightness from the location of HST observations.
The region observed with the HST is therefore typical of the “field” stellar population of the LMC, [*i.e.,*]{} not containing stellar associations or clusters, HII regions or known supernova remnants. It is, thus, representative of the field stellar population in the nearest dwarf galaxy which shows intense star formation. The LMC images obtained by HST are shown in the mosaic of Figure 1, where the different frames have been combined using the astrometric parameters of the PC chip from set A as produced by the HST reduction pipeline. The other images were linked into the mosaic through stars in common with previously linked images. We indicate the scale of the figure with a 30 arcsec bar and the north and west directions by vectors plotted at the figure’s lower right corner. The mosaic is shown here for display purposes only and it was not used in the photometric reduction. Note also that in its creation we made no attempt to remove distortions, which are small, in general, for the WFPC2.
The images are grouped by coordinates in sets (A through F), which were obtained in the same sky position, with the same guide stars, and at the same HST roll angle. Using the task [**gcombine**]{} of IRAF (in stsdas.toolbox.imgtools), with the CCDCRREJ option which is based on rejection of pixels higher by more than five standard deviations from their immediate surroundings, or with the combination of images based on crrej (in stsdas.hst\_calib.wfpc) where pixels in stacked images are rejected by a similar algorithm, it is possible to eliminate most of the cosmic rays (CRs). These are especially troublesome in very long WFPC2 exposures and it is virtually impossible to see any UV source before combining at least two images and rejecting the CR events.
Even with the CCDCRREJ option enough traces of CR events remained in the combined frame, mainly at locations where one CR track crossed another, to significantly confuse the detection of faint objects. This is because of the low throughput of the WFPC2+F160BW combination, the very low UV sky background in this spectral region, and the susceptibility of the WFPC2 CCDs to CR events. However, these cases could be rejected by blinking the combined images against individual frames used in the same combination. The case for dataset F is particularly difficult to analyze, as it consists of a single long exposure with significant but not identical overlap with set E. For this dataset we identified only objects visible in the original frames which had the proper intensity profile of a star (after some CR-cleaning with the [**cosmicray**]{} task in noao.imred.ccdred).
We used the combined frames to identify genuine UV sources by blinking combined sets of frames with significant overlap. The objects recognized this way are listed with a double or triple entry in Table 2. We accepted only those sources which appeared in a combined image, and which we could recognize on the two (or three) original images which made up the combined image and had the “proper” shape parameters (FWHM$\simeq$0.2 arcsec) expected of genuine stellar images. We calculated the celestial coordinates of all sources and performed aperture photometry with the [**phot**]{} package (in noao.digiphot.apphot) with a 0.5 arcsec round aperture. This is justified, given the PSF FWHM of images obtained with F160BW (up to $\sim$0.26 arcsec; Watson 1994). The [**phot**]{} package performs aperture photometry and subtracts the sky background as estimated from a ring around the aperture. When performing aperture photometry, the undersampling of the WF PSF is not important.
The photometry has not been corrected for charge transfer efficiency (Whitmore & Heyer 1997), because we found this to be insignificant in comparison with the other uncertainties in the data. For the same reason we did not correct for PSF variability. We performed aperture corrections by subtracting 0.1 mag from the [**phot**]{} results. We also corrected for the molecular contamination of the WFPC2 (MacKenty 1995). The photometry is reported in Table 2 as [*monochromatic magnitudes*]{} at $\lambda$=1491Å, with $$m_{UV}=-2.5 \times log f_{1491} -21.1$$ and $f_{1491}=(\frac{DN}{sec}/GR_i) \times F_{\lambda_0}$, where $GR_i$ is the gain ratio and $F_{\lambda_0}$ is the zero point constant from Holtzman (1995).
We did not specifically exclude sources in the areas vignetted by the F160BW filter. These are included in the Table and are marked by a [**V**]{} following the dataset and CCD chip identifiers given in column 6. Among the 341 independent measurements, only 28 are of objects in vignetted locations on the WF chips. In four cases a source with multiple measurements appears vignetted in one image but is not vignetted on others; this allows a comparison of photometry which helps evaluate the effect of vignetting. With one exception (source no. 45) the two photometry measurements agree to within 0.27 mag or better. We conclude that the vignetting by the F160BW filter does not appear to significantly influence our photometry.
Results
=======
We present the raw UV photometry results for all the objects identified in the WFPC2 images in Table 2. Each independent measurement is identified by a source number, which is the running number of UV sources and indicates the order in which the sources were identified. Some objects appear in more than one (combined) image; for each measurement we present the derived value and the photometric error, the image set from which the particular result originated (A, B, C, [*etc.*]{}), and the CCD chip in which the object appeared. For example, the first object in Table 2 was detected in the PC chip of the combined set A, at pixel coordinates (242.117; 260.523) and its raw UV magnitude was 19.190$\pm$0.041.
The on-chip pixel coordinates are listed beside the celestial coordinates, to aid further investigations of the sources. The correlation between different measurements was done through the celestial coordinates (calculated with the IRAF [**metric**]{} task, in stsdas.hst\_calib.wfpc), recognizing two separate observations as belonging to the same object if their image centers were separated by $\leq$0.5 arcsec. The photometric error listed in Table 2 is that given by the [**phot**]{} task in IRAF, [*i.e.,*]{} a formal error depending on the photon count and the local background; this does not reflect the calibration error of the photometry from F160BW images or residual flat field errors. Note that the F160BW flat field correction can be quite noisy (Biretta & Baggett 1998). The calibration error, on the other hand, is $\sim$10% at the center of the CCD chips (Baggett 1998).
We combined the different measurements of the same object in a single entry in Table 3, which has the source coordinates, UV magnitudes and errors. We identify a source by its first listed source number in Table 2. Figure 2 shows the distribution of remeasurements of the same object; approximately half of the objects have been measured more than once. Figure 3 shows the distribution of apparent UV magnitudes from Table 3. Objects with more than one measurement are shown in the lower panel.
A formal comparison of the two magnitude distributions, [*e.g.,*]{} via a Kolmogorov-Smirnov test of the cumulative distributions, shows that they are not drawn from the same parent population. This is probably the combination of differences introduced at the faint end of the distribution by a few spurious sources and at the bright end by the differnt areal coverage; while singly-detected sources can appear anywhere on the mosaic in Fig. 1, the multiply-measured objects [**must**]{} reside in the region of image overlap, which is smaller than the total area observed by HST. Note also that faint sources may register on an individual image but not on another from a different image combination; it would contribute a real source measured only once, although it may reside in an overlap region.
Fig. 3 shows that (a) brighter objects are more frequently seen in multiple exposures, and (b) despite the caveat in the previous paragraph, the faint end of the distribution is not dominated by the singly-measured objects. In fact, the drop in the number-magnitude plots occurs in both histograms at m$_{UV}$=19, indicating a similar degree of completeness for the singly-measured and multiply-measured objects. It is possible that a few faint sources are spurious, but most represent real objects conservatively chosen while adopting stellar images as candidate objects. We have likely missed some fainter LMC objects through this methodology, but we are fairly certain that we excluded virtually all spurious sources. The number-magnitude diagrams indicate that our photometry becomes incomplete at m$_{UV}\simeq$18.5.
Comparing the independent measurements of the same objects, after correcting for WFPC2 contamination, we derive an unbiased estimator of the measurement error, which disregards calibration unknowns but which is more representative of the internal photometry error than the value given by [**phot**]{}. This is the measurement error we adopt and present in Table 3. We show in Figure 4 the distribution of the individual FUV measurements for objects which have been measured independently twice (filled squares) where the error bars are the formal [**phot**]{} errors of these points, and the average value adopted in Table 3 (empty diamonds) with its error bar estimated from the dispersion of the individual measurements. It is clear that the dispersion of the measurements is reasonably small for objects with m$_{UV}\leq$19, but increases for objects at m$_{UV}\geq$19 to one or more magnitudes. This demonstrates once again that the completeness limit of the UV observations reported here is near m$_{UV}\approx$18.5.
We note that none of the $\sim$200 UV objects identified here appear convincingly extended when examined on the HST F160BW images. We conclude tentatively that most of them are probably stars or starlike objects, and that in the direction of the small LMC area sampled by the observations there are no UV-bright background galaxies.
The Ultraviolet Imaging Telescope (UIT) imaged regions of the LMC in FUV and NUV bands during the ASTRO-1 flight. Results of these observations, pertaining to the stellar associations LH52 and LH53, were published by Hill (1995). The results concerning field stars in the LMC were discussed in the context of the IMF in the field and in associations (Hill 1994). Despite the shallowness of the UIT results, by $\sim$3 mag relative to the HST data reported here, it is worthwhile first to consider the relative density of field stars. We find $\sim$200 stars in 12 arcmin$^2$ to m$_{UV}\approx$19, while Hill (1994) find that about 70% of their 1563 stars, to m$_{UV}\approx$15, belong to the field population. This yields a projected density of 0.9 star arcmin$^{-2}$ for stars brighter than m$_{UV}$=16, from which we expect to find 10 stars brighter than m$_{UV}$=16 in our data. Instead, as Fig. 5 demonstrates, we find almost four times this number of stars in our HST images.
An extensive catalog of UV stars in the two MCs, originating from both ASTRO-1 (1990) and ASTRO-2 (1995) observations, was published by Parker (1998). The UIT observations most relevant to the analysis presented here are toward N132 and were obtained during the ASTRO-2 flight with the B1 filter ($\lambda_e$=1521Å, $\Delta\lambda$=354Å). This bandpass is very close to that of the HST WFPC2 F160BW, with a 30Å difference in $\lambda_e$ and the HST bandpass being wider by 92Å than that of the present observations, therefore a direct comparison of measured magnitudes is justified. The LMC area sampled by the HST F160BW mosaic is much smaller than the UIT image and lies close to the Northern edge of the N132 field. The list of Parker has been examined and the objects in common with our data set have been extracted and are presented in Table 4.
A comparison between the UIT B1 magnitudes and those derived here for the same objects indicates that, with the exception of one object, the UIT stars are reported brighter by $\sim$1 magnitude than the HST photometry. This could, in principle, be the effect of a consistent calibration error of the F160BW magnitudes relative to the calibration of UIT, which is derived from objects in common with IUE. We deem it more likely to be the result of confusion contributions in the moderate angular resolution UIT images. In fact, in many cases we can identify the stars which would have been included as single stars in the UIT photometry; these additional contributors are listed on separate lines in Table 4, below the primary HST candidate counterpart. Despite the inclusion of the additional stars, we were not able to reproduce the UIT magnitudes by adding up their measured flux. Obviously, more objects than the stars we detected must be included in order to account for the difference in magnitudes, from which one can assume that there are even more faint UV stars not detected in our WFPC2 images which crowd into the UIT PSF and are measured as a single star.
Note that the UV survey to be conducted by GALEX (Bianchi & Martin 1997) will have similar problems to those of UIT in observing the LMC, as its angular resolution shall be only 3-5 arcsec. These would only be more acute in moderately crowded fields, as GALEX is supposed to reach fainter UV magnitudes than did UIT.
Optical counterparts
====================
One basic requirement for discussing the nature of the fainter UV sources is the correspondence with an optical counterpart. Unfortunately, the F160BW observations in the direction of the LMC were not collected together with HST imaging with other filters, thus no images with comparable resolution in a different spectral band exist in the HST archives. We attempted photometry on the scans of the Second STScI Digitized Sky Survey (DSS-II), but the relatively large pixel size used for these scans, compounded with the plate scale of the Schmidt plates, resulted in extreme crowding of the stellar images. No photometry to a reasonable accuracy could be performed on the stars at the location of the HST region studied here using the DSS scans.
We searched published catalogs with photometric information and could identify 58 UV sources in Table 3 with objects listed in the ROE/NRL catalog (Yentis 1992; Wallin 1994). This catalog is derived from COSMOS 5$^{\circ}.4\times$2’.3 scans of the short red UKST survey plates in the direction of the LMC, which were scanned with a very small laser beam, resulting in a smaller pixel size than the 15$\mu$m of the DSS-II. The astrometry is based on the HST Guide Star Catalog and the positions are accurate to $\sim$2 arcsec. The ROE/NRL catalog also lists magnitudes, which are in a band close to Johnson R, have a typical accuracy of 0.2 mag, and reach objects as faint as $\sim$21.5 mag. Our identification of an optical counterpart for a UV source required a positional coincidence within 2 arcsec; this is justified, given the astrometric accuracy of the ROE/NRL catalog, but may result in some chance coincidences.
The success rate in finding optical counterparts by cross-correlating with the ROE/NRL catalog is 78% for objects with m$_{UV}<$16 and drops to $\sim18$% for m$_{UV}>$18. The dependence of the success rate on the UV brightness, which should in principle be related to the optical brightness of the counterpart, indicates that most optical identifications are probably not chance effects. However, the fact that we could not find optical counterparts for all the bright UV sources is puzzling. One possibility, which we could not test, is that some possible counterparts were rejected by the ROE automatic processing of the COSMOS scans because of extreme crowding, thus they are not listed in the ROE/NRL catalog.
We transformed approximately the ROE/NRL catalog magnitudes to the V band to determine the nature of the LMC UV sources by assuming that the objects are earlier than mid-F and converting their UV-R color indices to UV-V. The LMC objects for which we found optical counterparts in the ROE/NRL catalog are listed in Table 5, along with their UV and optical data. The error in UV–V is obtained as the harmonic mean of the UV photometric error in Table 3 and an assumed 0.2 mag error in V propagated from the “short-red” magnitude listed in the ROE/NRL catalog and the adopted transformation.
We compare the properties of the LMC stars to stars in the Milky Way via an observational color-magnitude diagram based on high galactic latitude objects observed with the FAUST Shuttle-borne telescope. We elected to use observed photometry instead of convolving theoretical model atmosphere spectral energy distributions (SEDs) with the different instrumental responses, or using IUE spectra ([*e.g.,*]{} Fanelli 1992) convolved similarly, because the UV photometric results are unbiased and UV-selected, just as the LMC sample is unbiased and UV-selected. The comparison is made with stars observed by the FAUST experiment in the direction of the North Galactic Pole (NGP), Coma, and Virgo (Brosch 1995, 1997, 1998) which have parallax measurements in the Hipparcos and Tycho catalogs. We consider the LMC stars listed in Table 5 to be at a distance modulus of 18.5 for the purpose of determining their absolute magnitudes, and plot them in the same diagram with the FAUST objects.
The FAUST observations are in a spectral band centered at $\lambda_C$=1650Å and $\Delta\lambda\sim$300Å wide, which is not too different from the WFPC2+F160BW band definition ($\lambda_e$=1491Å, $\Delta\lambda\sim$446Å). The two bands mostly overlap, with the HST one being wider and bluer than that of FAUST. As both bands are not defined extremely accurately, we cannot derive a color term to transform between the two. It is possible that some chromatic effects may be present, mainly in the photometry of the high T$_{eff}$ stars which are very blue. We note also that the FAUST observations are biased [**against**]{} the detection of hot main sequence stars because of the direction of observation (high $\vert$b$\vert$) and the relative rarity of these stars in the immediate Solar neighborhood where Hipparcos parallaxes are available. For this reason, the earliest main sequence object we can locate on the FAUST color-magnitude diagram is late-B.
The UV color-magnitude diagram shown in Fig. 6 indicates that the LMC objects join up smoothly with the Galactic UV stars in Coma, Virgo, and the North Galactic Pole region. These have been identified in the publications mentioned above as main sequence stars. By inference, we deduce that at least the 58 objects with ROE/NRL catalog counterparts are also mostly main sequence objects. Note that the apparent width of the upper main sequence, as defined by the HST data from the LMC field, is similar to that of the lower main sequence defined by the FAUST stars in the MW.
Discussion
==========
Figure 5 shows that the F160BW mini-survey of the LMC field becomes progressively incomplete for m$_{UV}>18.5$. Considering the distance modulus, this indicates that we are observing LMC objects in the general field with M$_{UV}<0$, but we detect objects as faint as M$_{UV}\simeq$+2.5. The faintest objects for which we have UV-optical color information may, therefore, be early-F main sequence stars.
We added the UV emission from all the detected stars and found that these account for 2.53 10$^{-12}$ erg sec$^{-1}$ cm$^{-2}$ Å$^{-1}$ for the entire area surveyed here. This translates into 2.49 10$^{-7}$ erg sec$^{-1}$ cm$^{-2}$ Å$^{-1}$ ster$^{-1}$ at 1500Å from stellar sources not in clusters or associations, and is consistent with the integrated LMC emission at 1400Å from Page & Carruthers (1981); their value is higher by one third, but they include all UV sources, cluster and field, in their estimate. The average UV brightness measured by TD-1 (Morgan 1979) in the immediate vicinity of the HST-imaged region of the LMC is 6.8 10$^{-7}$ erg sec$^{-1}$ cm$^{-2}$ Å$^{-1}$ ster$^{-1}$ at 1550Å. This is higher by a factor of 2.7 than we measure, but the nearest TD-1 measurement (Region III) lies $\sim$20 arcmin South of the region studied here, deeper into the LMC bar.
[*The calculation shows that the large part of the UV radiation in the LMC, at least for $\lambda\geq$1500Å, is produced by field stars and not by objects in clusters or associations.*]{} This is consistent with the estimate by Parker (1998) that $\sim$60% of the UV light in the N132 image originates in stars fainter than 16 mag or from diffuse emission. It is also consistent with the measurement of Meurer (1995) that in starbursts, clusters of young stars provide at most 20% of the UV emission. Our integrated value for the 1500Å emission, combined with the upper limit for the diffuse emission at this location and at 2600Å from Vuillemin (1988), indicates a UV color index \[1500\]–\[2600\]$<$0.26 for the integrated stellar population detected in the UV.
Photometry of field UV sources with the FAUST experiment was done, as mentioned above, in a spectral band very similar to that defined by WFPC2 and F160BW. For comparison purposes, we use here the results for the NGP (Brosch 1995) and the Virgo regions (Brosch 1997), from which we selected only the stellar sources. Data from the Coma field (Brosch 1998) is not used because this FAUST field is not representative of the “general field” of the high-latitude MW, as it contains the Mel 111 open cluster. A comparison with the LMC stars detected here with the HST is valid, because with the sensitivity limit of FAUST it observed mostly Galactic sources. At the NGP and Virgo high (Northern) Galactic latitude of the FAUST exposures referred to here we should have a fair indication of (half) the projected density of UV stars in the Milky Way (MW). In these regions FAUST detected 172 stars to a UV magnitude limit of $\sim$14 and a completeness limit of $\sim$12.5.
The FAUST images considered here cover a solid angle of $\sim$0.05 ster, which projects to $\sim$1.7 10$^4$ pc$^2$ when viewed from outside the MW and adopting a 1 kpc scale height for the UV stars. The coverage of the LMC F160BW mosaic is $\sim$12 square arcmin, which corresponds to $\sim$4400 pc$^2$ at the distance of the LMC, thus about one quarter the projected area in the Milky Way sampled by FAUST. Therefore, the projected number density of UV stars in the HST observation of the LMC field region is n$_*\sim\frac{198}{4400}\simeq$0.045 stars pc$^{-2}$, whereas FAUST counted n$_*\sim\frac{172}{1.7 \, 10^4}\simeq$0.01 stars pc$^{-2}$ in the MW. The latter value should be doubled to account for the full width of the Galactic disk when viewed from outside.
It is possible to compare directly the FAUST results with those of the present HST observations. While FAUST sampled the (thick) Galactic disk and the halo, HST observed the edge of the bar and the disk of the LMC to a similar depth. The reason is that the objects deteced by FAUST reach into the thick disk and the halo of the MW to $\sim$1 kpc, while the HST observed objects at $\sim$60 kpc; the gain in sensitivity with HST makes up for the increased distance to yield approximately similar depths of observation. Thus, if the projected density of UV-bright stars in the LMC and in the disk and halo of the MW would have been the same, similar projected number densities of stars should have been observed, assuming that the LMC thickness is probably not much different than that of the MW. However, note that the value of n$_*$ for the LMC field is higher than that in the MW, as reflected by the FAUST data. The spatial density of hot stars in the LMC field regime is apparently higher by a factor of 2-3 than that in the MW.
[*We emphasize that our observations indicate a significant contamination by young field stars in all cluster UV observations in the LMC.*]{} This is especially important in case one aims to detect blue stragglers, or hot evolved stars which belong to the target cluster ([*e.g.,*]{} Mould 1996, where four blue stragglers were identified); if the detected objects have a projected density of $\sim$17 arcmin$^{-2}$ to m$_{UV}\simeq$19 they may well be only field interlopers, as already remarked by Cole (1997).
Finally, we can use our data, in combination with the similar magnitude estimates by UIT in N132 (Parker 1998) to produce a combined (observational) luminosity function for UV stars from m$_{UV}$=8 to 19. For this purpose we selected the N132 objects from the LMC catalog of Parker and scaled our HST stellar UV magnitude distribution to the total number of stars in the UIT sample with 12$\leq$m$_{UV}\leq$15. This is the magnitude range where the UIT star counts should be essentially complete and where no HST objects were probably missed. There are 2271 UIT stars in the N132 region, whereas there are only 17 HST stars in the same magnitude range, yielding a scale factor for the number count of 133.59. This is close to the ratio of sampled areas between the UIT field and that of the HST mosaic ($\sim$105), providing some confidence for this scaling approach. Note that the possible $\sim$one magnitude discrepancy between the UIT and HST magnitudes would not affect our conclusions significantly, because the slope of the UV luminosity function is defined mainly by its faint end, [*i.e.,*]{} by the HST measurements.
We show in Figure 7 the two luminosity functions plotted on the same scale. It is evident that the scaling procedure continues the trend of higher star counts to m$_{UV}\simeq$18, after which faint stars are progressively lost from the HST images. Figure 8 shows the logarithm of the combined star counts, where for m$_{UV}\leq$13 we adoped the UIT star counts and for the fainter magnitudes those from the (scaled) HST data set. We compare the slope of this observed luminosity function with the models presented in Parker (1998) for the UIT star counts.
The models allow a comparison with an LMC metallicity population formed continuously over a period $\Delta$t with a Salpeter IMF, using Fig. 11 from Parker and comparing the predicted slope of the star count distribution between m$_{UV}$=10 and m$_{UV}$=15 with the same parameter from the actual, combined star count. The closest fit is with a stellar population formed over $\Delta$t$\simeq$1 Gyr, because the younger star formation processes produce a distribution of the fainter UV stars which is much shallower than observed. This is consistent with the finding of Geha (1998) that the SF rate in the LMC did not change appreciably in the last 2 Gyr.
It is similarly possible to compare the slope of predicted luminosity functions for a continuous star formation over 1 Gyr, with different IMF slopes (Fig. 14 of Parker ), with that actually observed. The comparison indicates that the Parker (1998) model with the slope most similar to that measured here is that with an initial mass function slope $\Gamma$ of –1.8, close to the Salpeter value and found also to best fit the UIT data. Although the data do not really warrant it, we note that the measured slope is smaller than that predicted by the models with $\Gamma$=–1.8 but higher that that for $\Gamma$=–1.0. Thus, a Salpeter slope with $\Gamma$=–1.4 would probably be better than the –1.8 value found to best fit the bright end of the UV luminosity function by Parker
Conclusions
===========
We analyzed a set of very deep HST images obtained in a UV spectral band which is not contaminated by visible light leaks. Our findings can be summarized as follows:
1. We identified 198 UV sources with stars in the “general field” of the LMC, at the edge of the bar and not associated with any known star cluster or association.
2. A comparison with Galactic objects for the 30% of the sources for which we found optical catalog counterparts indicates that the objects are main sequence stars.
3. The UV emission from the stellar sources we detected, scaled to the entire solid angle covered by the LMC, accounts for 2/3 of the UV emission from the Cloud.
4. The observed UV luminosity function joins up smoothly with that defined for brighter objects, but on a wider field, from the UIT observation of the N 132 region.
5. The joint luminosity function of UV stars, which covers the UV magnitude range 8 to 19, confirms the claim by Parker (1998) that the LMC “field” domain has been forming stars continuously over (at least) 1 Gyr, with an IMF close to the Salpeter law.
Acknowledgements {#acknowledgements .unnumbered}
================
UV research at Tel Aviv University is supported by grants from the Ministry of Science and Arts through the Israel Space Agency, from the Austrian Friends of Tel Aviv University, and from a Center of Excellence Award from the Israel Science Foundation. NB acknowledges support from a US-Israel Binational Award to study UV sources measured by the FAUST experiment, and partial support from the Space Telescope Science Institute during a sabbatical visit. We thank Stefano Casertano for discussions on WFPC2 calibrations, and an anonymous referee for extensive remarks which improved significantly the presentation.
References {#references .unnumbered}
==========
Baggett, S., Caserano, S. & the WFPC2 group 1998, STScI ISR WFPC2 98-01.
Battineli, P. & Demers, S. 1998, AJ, 115, 1472.
Bianchi, L. & Martin, C. 1997 in “The Ultraviolet Astrophysics beyond the IUE final Archive” (R. Gonzales-Riestra, W. Wamsteker & R.A. Harris, eds.), ESA SP-413.
Biretta, J. & Baggett, S. 1998, STScI STAN/WFPC2-no.33, August.
Bowyer, S., Sasseen, T.P., Wu, X. & Lampton, M. 1995, ApJS, 96, 461.
Brosch, N., Almoznino, E., Leibowitz, E.M., Netzer, H., Sasseen, T., Bowyer, S., Lampton, M. & Wu , X. 1995, ApJ 450, 137.
Brosch, N., Formiggini, L., Almoznino, E., Sasseen, T., Lampton, M. & Bowyer, S. 1997, ApJS, 111, 143.
Brosch, N., Ofek, E.O., Almoznino, E., Sasseen, T., Lampton, M. & Bowyer, S. 1998, MNRAS, 295, 959.
Carruthers, G.R. & Page, T. 1977, ApJ, 211, 728.
Cheng, K.-P., Michalitsianos, A.G., Hintzen, P. Bohlin, R.C., O’Connell, R.W., Cornett, R.H., Morton, R.S., Smith, A.M., Smith, E.P. & Stecher, T.P. 1992, ApJ, 395, L29.
Clark, D.H., Tuohy, I.R., Dopita, M.A., Mathewson, D.S., Long, K.S., Szymkowiak, A.E. & Culhane, J.L. 1982, ApJ, 225, 440.
Cole, A.A., Mould, J.R., Gallagher, J.S. III, Clarke, J.T., Trauger, J.T., Ballester, G.E., Burrows, C.J., Casertano, S., Griffiths, R., Hester, J.J., Hoessel, J.G. Holtzman, J.A., Scowen, P.A., Stapelfeldt, K.R. & Westphal, J.A. 1997, AJ, 114, 1945.
Courtés, G., Viton, M., Sivan, J.-P., Decher, R. & Gary, A. 1984, Science, 225, 179.
Courtés, G., Viton, M., Bowyer, S., Lampton, M., Sasseen, T.P. & Wu, X. 1995, A&A, 297, 338.
Davies, R.D., Elliott, K.H. & Meaburn, J. 1976, Mem. R. astr. Soc., 81, 89.
Elson, R.A.W., Gilmore, G.F. & Santiago, B.X. 1997, MNRAS, 289, 157.
Fanelli, M.N., O’Connell, R.W., Burstein, D. & Wu, C.-C. 1992, ApJS, 82, 197.
Feast, M.W. 1995 in “Stellar Populations” (P.C. van der Kruit & G. Gilmore, eds.) Dordrecht; Kluwer Academic Publishers, pp. 153-163.
Feitzinger, J.V., Haynes, R.F., Klein, U., Wielebinski, R. & Perschke, M. 1987, Vistas in Astr., 30, 243.
Geha, M.C., Holtzman, J.A., Mould, J.R., Gallagher, J.S.III, Watson, A.M., Cole, A.A., Grillmair, C.J., Staplefeld, K.R., Ballester, G.E., Burrows, C.J., Clarke, J.T., Crisp, D., Evans, R.W., Griffiths, R.E., Hester, J.J., Scowen, P.A., Trauger, J.T. & Westphal, J.A. 1998, AJ, 115, 1045.
Gerola, H. & Seiden, P.E. 1978, ApJ, 223, 129.
Gilmozzi, R., Kinney, E.K., Ewald, S.P., Panagia, N. & Romaniello, M. 1994, ApJ, 435, L43.
Hill, R.S., Isensee, J.E., Cornett, R.H., Bohlin, R.C., O’Connell, R.W., Roberts, M.S., Smith, A.M. & Stecher, T.P 1994, ApJ, 425, 122.
Hill, R.S., Cheng, K. -P.; Bohlin, R.C., O’Connell, R.W., Roberts, M.S., Smith, A.M. & Stecher, T.P., 1995, ApJ, 446, 622.
Holtzman, J.A., Burrows, C.J., Casertano, S., Hester, J.J., Trauger, J.T., Watson, A.M. & Worthey, G. 1995, PASP, 107, 156 (H95a).
Holtzman, J.A., Burrows, C.J., Casertano, S., Hester, J.J., Trauger, J.T., Watson, A.M. & Worthey, G. 1995, PASP, 107, 1065 (H95b).
Hunter, D.E., Vacca, W.D., Massey, P., Lynds, R. & O’Neil, E.J. 1997, AJ, 113, 1691.
Kaufman, M. 1979, ApJ, 232, 707.
Koornneeff, J. 1977, A&AS, 29, 117.
Leinert, Ch., Bowyer, S., Haikala, L.K., Hanner, M.S., Hauser, M.G., Levasseur-Regourd, A.-Ch., Mann, I., Mattila, K., Reach, W.T., Schlosser, W., Staude, H.J., Toller, G.N., Weiland, J.L., Weinberg, J.L. & Witt, A.N. 1998, A&AS, 127, 1.
Lucke, P.B. 1974, ApJS, 28, 73.
Lucke, P.B. & Hodge, P.W. 1971, AJ, 75, 171.
MacKenty, J. W., Baggett, S.M., Biretta, J., Hinds, M., Ritchie, C.E., Feinberg, L.D. & Trauger, J.T. 1995, Proc. SPIE, 2478, 160.
Meurer, G.R., Heckman, T.M., Leitherer, C., Kinney, A., Robert, C. & Garnett, D.R. 1995, AJ, 110, 2665.
Milliard, B. Donas, J. & Laget, M. 1991, Adv. Space Res., 11, 135.
Morgan, D.H., Nandi, K. & Carnochan, D.J. 1979, MNRAS, 188, 131.
Mould, J.R., Watson, A.M., Gallagher, J.S. III, Ballester, G.E., Burrows, C.J., Casertano, S., Clarke, J.T., Crisp, D., Griffiths, R., Hester, J.J., Hoessel, J.G. Holtzman, J.A., Scowen, P.A., Stapelfeldt, K.R., Trauger, J.T. & Westphal, J.A. 1996, ApJ, 461, 762.
Nandy, K., Morgan, D.H, & Carnochan, D.J. 1979, MNRAS, 186, 421.
Page, T. & Carruthers, G.R. 1981, “Revised listing - S201 far-ultraviolet atlas of the Large Magellanic Cloud”, NRL Memorandum Report 4660.
Parker, J.Wm., Hill, J.K., Cornett, R.H., Hollis, J., Zamkoff, E., Bohlin, R.C., O’Connell, R.W., Neff, S.G., Roberts, M.S., Smith, A.M. & Stecher, T.P. 1998, AJ, 116, 180.
Schwering, P.B.W. & Israel, F.P. 1990 “Atlas and catalogue of infrared sources in the Magellanic Clouds”, Dordrecht: Kluwer Academic Publishers.
Smith, A.M., Cornett, R.H. & Hill, R.S. 1987, ApJ, 320, 609.
Stecher, T.P., Baker, G.R. 1992, ApJ, 395, L1.
Treyer, M., Ellis, R.S., Milliard, B., Donas, J. & Bridges, T.J. 1998, MNRAS, in press (astro-ph/9806056).
Thompson, G.I., Nandy, K., Jamar, C., Monfils, A., Houziaux, L., Carnochan, D.J. & Wilson, R. 1978 “Catalog of Stellar Ultraviolet Fluxes”, SRC.
Vuillemin, A. 1988, A&AS, 72, 249.
Watson, A.M., Mould, J.R., Gallagher, J.S., Ballester, G.E., Burrows, C.J., Casertano, S., Clarke, J.T., Crisp, D., Griffiths, R.E., Hester, J. J.F, Hoessel, J.G., Holtzman, J.A., Scowen, P.A., Saplefeldt, K.R., Trauger, J.T., Westphal, J.A.199, ApJ, 435, 55.
Whitmore, B. & Heyer, I. 1997, Instrument Science Report WFPC2 97-08.
Xu, C., Klein, U., Meinert, D., Wielebinski, R. & Haynes, R.F. 1992, A&A, 257, 47.
Yentis, D.J., Cruddace, R.G., Gursky, H., Stuart, B.V., Wallin, J.F., MacGillivray, H.T. & Collins, C.A. 1992, in “Digitized optical sky surveys” (H.T. MacGillivray & E.B. Thomson, eds.), Dordrecht: Kluwer, p. 67.
Wallin, J.F., Yentis, D.J., MacGillivray, H.T., Bauer, S.B. & Wong, C.S. 1994, AAS Bull., 184, 2704.
Figure captions {#figure-captions .unnumbered}
===============
[cccccc]{} A & u2ou1001t & 5:25:17.6 & –69:30:26 & 2500 & 230895 A & u2ou1101t & 5:25:17.6 & –69:30:26 & 4400 & 230895 A & u2ou1201t & 5:25:17.6 & –69:30:26 & 4400 & 230895 F & u2ou1301t & 5:25:31.4 & –69:30:52 & 4400 & 230895 B & u2ou1501t & 5:25:22.7 & –69:30:09 & 2300 & 260895 B & u2ou1502t & 5:25:22.7 & –69:30:09 & 2300 & 260895 C & u2ou1601t & 5:25:36.8 & –69:30:30 & 4900 & 260895 C & u2ou1701t & 5:25:36.8 & –69:30:30 & 4400 & 260895 D & u2ou1e01t & 5:25:17.8 & –69:30:30 & 4600 & 240895 D & u2ou1f01t & 5:25:17.8 & –69:30:30 & 4600 & 240895 E & u2ou1g01t & 5:25:31.8 & –69:30:51 & 4600 & 240895 E & u2ou1h01t & 5:25:31.8 & –69:30:51 & 4600 & 240895
[ccccccc]{} 1 & 242.117 & 260.523 & 5:25:12.886 & –69:30:37.13 & A-pc & 19.190 (0.041) 1 & 140.889 & 329.237 & 5:25:12.881 & –69:30:37.22 & D-pc & 18.847 (0.035) 2 & 308.600 & 288.237 & 5:25:12.468 & –69:30:39.53 & A-pc & 18.112 (0.020) 2 & 205.689 & 361.042 & 5:25:12.461 & –69:30:39.65 & D-pc & 18.051 (0.021) 3 & 393.000 & 795.000 & 5:25:08.099 & –69:30:35.52 & A-pc & 19.072 (0.034) 4 & 442.279 & 556.773 & 5:25:09.893 & –69:30:41.22 & A-pc & 13.752 (0.002) 4 & 322.418 & 637.817 & 5:25:09.888 & –69:30:41.32 & D-pc & 13.444 (0.002) 5 & 531.523 & 665.313 & 5:25:08.753 & –69:30:43.39 & A-pc & 18.934 (0.033) 5 & 405.079 & 751.607 & 5:25:08.749 & –69:30:43.54 & D-pc & 18.779 (0.032) 6 & 597.824 & 177.715 & 5:25:12.537 & –69:30:53.58 & A-pc & 17.590 (0.010) 6 & 500.647 & 268.901 & 5:25:12.533 & –69:30:53.72 & D-pc & 17.279 (0.013) 7 & 373.000 & 066.000 & 5:25:14.509 & –69:30:50.67 & D-pc & 16.583 (0.009) 8 & 758.794 & 236.243 & 5:25:12.185 & –69:31:05.32 & D-pc & 19.942 (0.081)
10 & 095.365 & 263.299 & 5:25:15.811 & –69:30:07.98 & A-wf2 & 18.220 (0.020) 10 & 106.265 & 306.270 & 5:25:15.812 & –69:30:08.10 & D-wf2 & 18.086 (0.019) 11 & 152.362 & 529.463 & 5:25:16.430 & –69:29:41.14 & A-wf2 & 16.467 (0.008) 11 & 146.375 & 575.363 & 5:25:16.436 & –69:29:41.27 & D-wf2 & 16.221 (0.007) 12 & 182.786 & 067.505 & 5:25:15.895 & –69:29:40.17 & A-wf2 & 19.190 (0.036) 12 & 205.649 & 116.813 & 5:25:13.054 & –69:30:23.56 & D-wf2 & 18.972 (0.034) 13 & 201.310 & 428.790 & 5:25:14.938 & –69:29:49.07 & A-wf2 & 15.717 (0.006) 13 & 201.505 & 477.811 & 5:25:14.953 & –69:29:49.10 & D-wf2 & 15.484 (0.005) 14 & 261.667 & 457.343 & 5:25:14.043 & –69:29:44.40 & A-wf2 & 15.525 (0.005) 14 & 260.222 & 509.980 & 5:25:14.040 & –69:29:44.47 & D-wf2 & 15.318 (0.005) 15 & 292.361 & 174.331 & 5:25:11.744 & –69:30:10.03 & A-wf2 & 19.755 (0.052) 15 & 309.946 & 229.913 & 5:25:11.726 & –69:30:10.14 & D-wf2 & 18.927 (0.032) 16 & 317.258 & 626.340 & 5:25:14.077 & –69:29:26.70 & A-wf2 & 19.116 (0.041) 16 & 305.174 & 682.373 & 5:25:14.078 & –69:29:26.78 & D-wf2 & 18.845 (0.042) 17 & 330.646 & 558.050 & 5:25:13.428 & –69:29:32.66 & A-wf2 & 17.274 (0.013) 17 & 322.711 & 614.787 & 5:25:13.431 & –69:29:32.86 & D-wf2 & 17.088 (0.012) 18 & 336.000 & 789.620 & 5:25:14.726 & –69:29:10.86 & A-wf2 & 17.309 (0.016) 19 & 346.644 & 172.694 & 5:25:10.765 & –69:30:08.47 & A-wf2 & 19.033 (0.032) 19 & 362.435 & 231.432 & 5:25:10.769 & –69:30:08.54 & D-wf2 & 18.731 (0.029) 20 & 531.974 & 152.208 & 5:25:07.318 & –69:30:04.32 & A-wf2 & 18.099 (0.019) 20 & 548.410 & 222.324 & 5:25:07.333 & –69:30:04.46 & D-wf2 & 17.855 (0.017) 21 & 714.051 & 479.000 & 5:25:06.085 & –69:29:27.74 & A-wf2 & 17.795 (0.021) 21 & 710.531 & 559.679 & 5:25:06.095 & –69:29:27.93 & D-wf2 V & 17.718 (0.030) 22 & 719.335 & 417.689 & 5:25:05.614 & –69:29:33.39 & A-wf2 & 21.144 (0.282) 22 & 729.462 & 514.000 & 5:25:05.523 & –69:29:31.72 & D-wf2 & 17.370 (0.021) 23 & 750.094 & 434.000 & 5:25:05.170 & –69:29:30.81 & A-wf2 & 19.497 (0.084) 23 & 716.000 & 499.988 & 5:25:05.687 & –69:29:33.40 & D-wf2 & 19.820 (0.134) 24 & 089.643 & 459.087 & 5:25:17.123 & –69:29:49.75 & A-wf2 & 19.532 (0.045) 24 & 088.011 & 501.180 & 5:25:17.119 & –69:29:49.89 & D-wf2 & 19.260 (0.039) 25 & 645.560 & 028.878 & 5:25:04.619 & –69:30:20.24 & D-wf2 & 15.771 (0.006) 26 & 771.340 & 076.698 & 5:25:02.585 & –69:30:12.39 & D-wf2 & 17.360 (0.015)
31 & 054.017 & 397.253 & 5:25:21.550 & –69:30:41.15 & A-wf3 & 17.643 (0.015) 31 & 072.656 & 372.330 & 5:25:21.545 & –69:30:41.39 & D-wf3 & 17.381 (0.013) 31 & 178.305 & 274.530 & 5:25:21.556 & –69:30:40.94 & B-wf4 & 17.330 (0.015)
32 & 067.471 & 563.700 & 5:25:24.590 & –69:30:45.32 & A-wf3 & 15.895 (0.006) 32 & 075.996 & 539.289 & 5:25:24.602 & –69:30:45.46 & D-wf3 & 15.672 (0.006) 32 & 344.651 & 269.488 & 5:25:24.597 & –69:30:45.08 & B-wf4 & 15.584 (0.006)
33 & 091.434 & 478.313 & 5:25:23.224 & –69:30:40.35 & A-wf3 & 18.351 (0.022) 33 & 104.822 & 455.424 & 5:25:23.223 & –69:30:40.48 & D-wf3 & 18.103 (0.020) 33 & 260.864 & 241.980 & 5:25:23.220 & –69:30:40.07 & B-wf4 & 17.907 (0.020)
34 & 275.671 & 633.669 & 5:25:27.136 & –69:30:28.18 & A-wf3 & 18.748 (0.033) 34 & 279.313 & 621.811 & 5:25:27.154 & –69:30:28.35 & D-wf3 & 18.488 (0.030) 34 & 425.554 & 065.323 & 5:25:27.152 & –69:30:27.96 & B-wf4 & 18.497 (0.032)
35 & 303.764 & 447.109 & 5:25:23.991 & –69:30:19.50 & A-wf3 & 18.142 (0.020) 35 & 318.739 & 437.390 & 5:25:23.987 & –69:30:19.67 & D-wf3 & 17.937 (0.019) 120 & 043.898 & 242.704 & 5:25:23.966 & –69:30:19.70 & B-wf3 & 17.988 (0.024)
36 & 361.684 & 320.639 & 5:25:22.082 & –69:30:09.89 & A-wf3 & 18.525 (0.025) 36 & 384.462 & 314.714 & 5:25:22.079 & –69:30:10.12 & D-wf3 & 18.351 (0.024) 121 & 109.647 & 119.213 & 5:25:22.071 & –69:30:10.18 & B-wf3 & 18.411 (0.029)
37a & 453.913 & 587.461 & 5:25:27.433 & –69:30:09.89 & A-wf3 & 19.250 (0.054) 37a & 459.992 & 586.638 & 5:25:27.429 & –69:30:10.07 & D-wf3 & 18.830 (0.044) 122 & 187.228 & 392.523 & 5:25:27.417 & –69:30:10.17 & B-wf3 & 19.176 (0.054)
37b & 457.481 & 572.207 & 5:25:27.185 & –69:30:09.12 & A-wf3 & 21.788 (0.498) 37b & 463.361 & 566.679 & 5:25:27.081 & –69:30:09.24 & D-wf3 & 20.348 (0.153) 38a & 466.876 & 556.314 & 5:25:26.955 & –69:30:07.75 & A-wf3 & 19.474 (0.064) 38a & 474.728 & 557.458 & 5:25:26.974 & –69:30:07.95 & D-wf3 & 19.555 (0.077) 38b & 471.438 & 549.874 & 5:25:26.862 & –69:30:07.05 & A-wf3 & 20.002 (0.098) 38b & 483.000 & 551.000 & 5:25:26.911 & –69:30:06.92 & D-wf3 & 21.066 (0.297) 39 & 535.855 & 227.626 & 5:25:21.510 & –69:29:50.51 & A-wf3 & 19.375 (0.044) 39 & 563.767 & 232.487 & 5:25:21.514 & –69:29:50.69 & D-wf3 & 19.160 (0.044)
40 & 551.226 & 293.726 & 5:25:22.788 & –69:29:51.16 & A-wf3 & 19.865 (0.066) 40 & 574.895 & 299.707 & 5:25:22.813 & –69:29:51.36 & D-wf3 & 19.649 (0.066) 126 & 301.585 & 104.793 & 5:25:22.770 & –69:29:51.48 & B-wf3 & 19.584 (0.069)
41 & 591.000 & 588.000 & 5:25:28.300 & –69:29:57.07 & A-wf3 & 18.666 (0.042) 41 & 594.612 & 595.292 & 5:25:28.275 & –69:29:57.50 & D-wf3 & 20.071 (0.159) 42 & 612.000 & 447.605 & 5:25:25.918 & –69:29:50.48 & A-wf3 & 17.370 (0.015) 42 & 622.056 & 472.000 & 5:25:26.184 & –69:29:51.50 & D-wf3 & 18.517 (0.036) 43 & 612.000 & 379.000 & 5:25:24.704 & –69:29:48.25 & A-wf3 & 17.819 (0.018) 43 & 639.653 & 383.644 & 5:25:24.652 & –69:29:47.48 & D-wf3 & 18.228 (0.028)
44 & 607.375 & 335.463 & 5:25:23.887 & –69:29:47.28 & A-wf3 & 17.379 (0.014) 44 & 628.585 & 344.541 & 5:25:23.886 & –69:29:47.48 & D-wf3 & 17.239 (0.014) 127 & 355.420 & 150.181 & 5:25:23.879 & –69:29:47.56 & B-wf3 & 17.319 (0.014)
45 & 708.733 & 534.000 & 5:25:28.060 & –69:29:44.36 & A-wf3 & 18.843 (0.067) 45 & 713.612 & 546.306 & 5:25:27.988 & –69:29:44.84 & D-wf3 V & 21.214 (0.713) 46 & 793.000 & 300.134 & 5:25:24.429 & –69:29:28.83 & A-wf3 & 18.695 (0.042) 51 & 047.224 & 456.328 & 5:25:12.589 & –69:31:10.06 & A-wf4 & 18.857 (0.064) 52 & 084.929 & 118.391 & 5:25:15.438 & –69:30:39.84 & A-wf4 & 19.605 (0.098) 52 & 053.441 & 080.329 & 5:25:15.422 & –69:30:39.98 & D-wf4 & 18.972 (0.041)
53 & 094.265 & 149.601 & 5:25:15.396 & –69:30:43.01 & A-wf4 & 14.354 (0.008) 53 & 060.674 & 112.426 & 5:25:15.375 & –69:30:43.18 & D-wf4 & 14.151 (0.003) 104 & 792.612 & 429.584 & 5:25:15.426 & –69:30:43.56 & B-pc & 14.212 (0.003)
54 & 098.477 & 754.720 & 5:25:11.608 & –69:31:39.53 & A-wf4 & 19.469 (0.097) 55 & 109.924 & 713.486 & 5:25:12.074 & –69:31:36.14 & A-wf4 & 18.812 (0.067) 56 & 135.128 & 552.157 & 5:25:13.531 & –69:31:21.99 & A-wf4 & 16.911 (0.026) 56 & 077.397 & 516.988 & 5:25:13.519 & –69:31:22.15 & D-wf4 & 16.939 (0.011) 57 & 143.718 & 188.409 & 5:25:16.011 & –69:30:48.25 & A-wf4 & 17.467 (0.034) 57 & 108.329 & 153.795 & 5:25:16.019 & –69:30:48.35 & D-wf4 & 17.254 (0.013) 58 & 206.444 & 734.497 & 5:25:13.637 & –69:31:41.34 & A-wf4 & 19.138 (0.080) 58 & 159.454 & 706.861 & 5:25:14.009 & –69:31:42.41 & D-wf4 & 20.012 (0.093) 59 & 206.333 & 553.424 & 5:25:14.783 & –69:31:24.48 & A-wf4 & 19.166 (0.078) 59 & 148.403 & 522.668 & 5:25:14.773 & –69:31:24.61 & D-wf4 & 18.894 (0.032)
60 & 210.734 & 485.336 & 5:25:15.289 & –69:31:18.25 & A-wf4 & 18.633 (0.059) 60 & 157.354 & 454.904 & 5:25:15.296 & –69:31:18.37 & D-wf4 & 18.395 (0.024) 61 & 227.305 & 219.682 & 5:25:17.303 & –69:30:53.94 & A-wf4 & 14.985 (0.011) 61 & 189.720 & 190.583 & 5:25:17.285 & –69:30:54.08 & D-wf4 & 14.857 (0.004) 62 & 253.328 & 175.909 & 5:25:18.043 & –69:30:50.80 & A-wf4 & 17.814 (0.040) 62 & 218.465 & 148.465 & 5:25:18.035 & –69:30:50.90 & D-wf4 & 17.585 (0.015) 63 & 257.563 & 532.684 & 5:25:15.824 & –69:31:24.24 & A-wf4 & 18.909 (0.068) 63 & 200.553 & 503.562 & 5:25:15.815 & –69:31:24.25 & D-wf4 & 19.600 (0.054) 64 & 342.984 & 796.393 & 5:25:15.676 & –69:31:51.71 & A-wf4 & 16.687 (0.024) 64 & 269.822 & 773.505 & 5:25:15.667 & –69:31:51.87 & D-wf4 & 16.149 (0.008) 65 & 367.672 & 614.385 & 5:25:17.258 & –69:31:35.62 & A-wf4 & 18.945 (0.072) 65 & 305.810 & 593.494 & 5:25:17.244 & –69:31:35.77 & D-wf4 & 18.705 (0.032) 66 & 332.477 & 179.500 & 5:25:19.430 & –69:30:53.73 & A-wf4 & 19.378 (0.086) 66 & 297.485 & 156.963 & 5:25:19.421 & –69:30:53.93 & D-wf4 & 19.060 (0.036) 67 & 380.004 & 751.588 & 5:25:16.617 & –69:31:48.79 & A-wf4 & 18.907 (0.074) 67 & 309.666 & 731.261 & 5:25:16.591 & –69:31:48.99 & D-wf4 & 18.739 (0.039) 68 & 458.448 & 160.663 & 5:25:21.800 & –69:30:56.21 & A-wf4 & 20.619 (0.169) 68 & 424.491 & 145.929 & 5:25:21.791 & –69:30:56.33 & D-wf4 & 20.102 (0.080) 69 & 473.572 & 554.724 & 5:25:19.533 & –69:31:33.58 & A-wf4 & 19.108 (0.078) 69 & 413.697 & 538.809 & 5:25:19.500 & –69:31:33.55 & D-wf4 & 19.334 (0.050) 70 & 499.100 & 538.825 & 5:25:20.100 & –69:31:32.96 & A-wf4 & 20.228 (0.151) 70 & 441.542 & 526.181 & 5:25:20.074 & –69:31:33.18 & D-wf4 & 19.673 (0.068) 71 & 483.147 & 229.535 & 5:25:21.800 & –69:31:03.49 & A-wf4 & 18.764 (0.064) 71 & 444.793 & 216.463 & 5:25:21.779 & –69:31:03.65 & D-wf4 & 18.507 (0.026) 72 & 575.000 & 691.462 & 5:25:20.468 & –69:31:49.74 & A-wf4 & 18.744 (0.076) 72 & 507.595 & 682.304 & 5:25:20.444 & –69:31:49.85 & D-wf4 & 18.388 (0.034) 73 & 637.564 & 553.584 & 5:25:22.458 & –69:31:38.97 & A-wf4 & 16.397 (0.021) 73 & 578.677 & 549.316 & 5:25:22.442 & –69:31:39.18 & D-wf4 & 16.211 (0.008) 74 & 661.381 & 711.638 & 5:25:21.864 & –69:31:54.43 & A-wf4 & 23.046 (5.329) 75 & 690.004 & 709.468 & 5:25:22.390 & –69:31:55.20 & A-wf4 V & 16.453 (0.027) 75 & 629.224 & 724.146 & 5:25:22.429 & –69:31:57.16 & D-wf4 V & 16.275 (0.012) 76 & 735.370 & 686.709 & 5:25:23.333 & –69:31:54.54 & A-wf4 V & 15.700 (0.019) 76 & 668.214 & 687.938 & 5:25:23.332 & –69:31:54.73 & D-wf4 V & 15.748 (0.008) 77 & 798.535 & 543.642 & 5:25:25.366 & –69:31:43.37 & A-wf4 V & INDEF INDEF 77 & 739.637 & 549.094 & 5:25:25.352 & –69:31:43.60 & D-wf4 V & 14.723 (0.004) 78 & 655.113 & 651.461 & 5:25:22.145 & –69:31:48.68 & A-wf4 & 16.992 (0.029) 78 & 590.229 & 647.508 & 5:25:22.136 & –69:31:48.82 & D-wf4 & 16.882 (0.013) 79 & 759.209 & 568.999 & 5:25:25.604 & –69:31:46.04 & D-wf4 V & 16.451 (0.014) 80 & 760.697 & 349.908 & 5:25:26.789 & –69:31:25.25 & D-wf4 & 15.582 (0.006) 81 & 773.092 & 177.880 & 5:25:27.921 & –69:31:09.27 & D-wf4 & 19.485 (0.080) 82 & 673.152 & 602.838 & 5:25:22.780 & –69:31:44.73 & A-wf4 & 19.070 (0.094) 82 & 608.571 & 600.550 & 5:25:22.713 & –69:31:44.86 & D-wf4 & 18.372 (0.036) 83 & 627.839 & 723.154 & 5:25:21.202 & –69:31:54.46 & A-wf4 V & 19.087 (0.134) 83 & 556.981 & 718.108 & 5:25:21.159 & –69:31:54.63 & D-wf4 & 19.264 (0.094)
100 & 168.000 & 446.000 & 5:25:16.766 & –69:30:16.13 & B-pc & 18.408 (0.028) 101 & 215.000 & 513.000 & 5:25:16.097 & –69:30:17.35 & B-pc & 19.122 (0.051) 102 & 635.401 & 631.425 & 5:25:14.121 & –69:30:34.21 & B-pc & 18.717 (0.037) 103 & 647.000 & 632.000 & 5:25:14.085 & –69:30:34.72 & B-pc & 18.640 (0.036)
110 & 299.000 & 701.661 & 5:25:19.213 & –69:29:04.33 & B-wf2 & 17.863 (0.021) 111 & 300.088 & 387.000 & 5:25:17.617 & –69:29:34.38 & B-wf2 & 18.471 (0.028) 112 & 297.306 & 361.398 & 5:25:17.540 & –69:29:36.95 & B-wf2 & 19.023 (0.041) 113 & 347.844 & 719.000 & 5:25:18.430 & –69:29:01.34 & B-wf2 & 18.084 (0.025) 114 & 406.051 & 641.000 & 5:25:16.966 & –69:29:07.22 & B-wf2 & 19.030 (0.050) 115 & 483.812 & 398.539 & 5:25:14.333 & –69:29:28.46 & B-wf2 & 18.852 (0.036) 116 & 622.497 & 323.459 & 5:25:11.424 & –69:29:31.95 & B-wf2 & 18.583 (0.033) 117 & 745.000 & 439.029 & 5:25:09.786 & –69:29:17.60 & B-wf2 & 19.280 (0.077)
123 & 225.288 & 772.000 & 5:25:34.508 & –69:30:16.64 & B-wf3 & 17.930 (0.035) 124 & 241.545 & 725.462 & 5:25:33.743 & –69:30:13.89 & B-wf3 & 18.377 (0.048) 125 & 303.973 & 496.000 & 5:25:29.912 & –69:30:01.78 & B-wf3 & 19.302 (0.065)
128 & 378.000 & 650.865 & 5:25:33.092 & –69:29:58.83 & B-wf3 & 18.508 (0.052) 129 & 412.622 & 530.883 & 5:25:31.087 & –69:29:52.35 & B-wf3 & 18.764 (0.050) 130 & 529.365 & 075.459 & 5:25:23.422 & –69:29:28.92 & B-wf3 & 19.852 (0.090) 131 & 584.646 & 510.429 & 5:25:31.605 & –69:29:35.35 & B-wf3 & 17.379 (0.022) 132 & 653.131 & 348.662 & 5:25:29.013 & –69:29:24.40 & B-wf3 & 18.302 (0.041) 133 & 707.435 & 485.697 & 5:25:31.777 & –69:29:22.96 & B-wf3 V & 17.089 (0.021) 134 & 771.515 & 521.146 & 5:25:32.752 & –69:29:17.87 & B-wf3 V & 19.138 (0.256) 135 & 786.484 & 300.654 & 5:25:28.825 & –69:29:10.50 & B-wf3 & 17.318 (0.025)
140 & 105.858 & 432.353 & 5:25:19.410 & –69:30:54.01 & B-wf4 & 19.175 (0.050) 141 & 253.847 & 491.243 & 5:25:21.776 & –69:31:03.78 & B-wf4 & 18.505 (0.032)
142 & 488.014 & 783.042 & 5:25:24.477 & –69:31:38.04 & B-wf4 & 18.736 (0.078) 143 & 568.832 & 625.478 & 5:25:26.787 & –69:31:25.34 & B-wf4 & 15.517 (0.006) 144 & 601.438 & 272.557 & 5:25:29.249 & –69:30:52.58 & B-wf4 & 19.118 (0.055) 145 & 612.297 & 329.781 & 5:25:29.146 & –69:30:58.32 & B-wf4 & 18.940 (0.057) 146 & 629.694 & 328.539 & 5:25:29.459 & –69:30:58.70 & B-wf4 & 17.674 (0.022) 147 & 704.360 & 415.911 & 5:25:30.346 & –69:31:09.18 & B-wf4 & 19.057 (0.081) 148 & 729.542 & 595.259 & 5:25:29.839 & –69:31:26.88 & B-wf4 V & 15.263 (0.007) 149 & 749.617 & 198.932 & 5:25:32.299 & –69:30:49.79 & B-wf4 & 17.924 (0.032)
150 & 377.802 & 209.461 & 5:25:32.266 & –69:30:50.04 & C-pc & 17.332 (0.018) 151 & 657.185 & 470.329 & 5:25:29.433 & –69:30:58.98 & C-pc & 16.819 (0.012)
160 & 068.322 & 145.245 & 5:25:34.631 & –69:30:25.36 & C-wf2 & 18.601 (0.039) 161 & 088.595 & 469.467 & 5:25:35.909 & –69:29:53.87 & C-wf2 & 17.623 (0.018) 162 & 093.028 & 072.716 & 5:25:33.807 & –69:30:31.64 & C-wf2 & 19.435 (0.073) 163 & 144.503 & 243.762 & 5:25:33.751 & –69:30:14.05 & C-wf2 & 17.623 (0.018) 164 & 142.619 & 086.538 & 5:25:32.990 & –69:30:29.06 & C-wf2 & 19.019 (0.052) 165 & 356.613 & 587.368 & 5:25:31.621 & –69:29:35.47 & C-wf2 & 16.552 (0.011) 166 & 380.868 & 710.416 & 5:25:31.780 & –69:29:23.07 & C-wf2 & 16.193 (0.010) 167 & 476.880 & 190.383 & 5:25:27.413 & –69:30:10.35 & C-wf2 & 18.190 (0.028) 168 & 499.370 & 213.699 & 5:25:27.127 & –69:30:07.56 & C-wf2 & 18.937 (0.049) 169 & 506.226 & 205.546 & 5:25:26.956 & –69:30:08.14 & C-wf2 & 18.945 (0.049) 170 & 475.997 & 458.809 & 5:25:28.786 & –69:29:44.57 & C-wf2 & 18.148 (0.031) 171 & 517.928 & 656.790 & 5:25:29.015 & –69:29:24.61 & C-wf2 & 17.389 (0.027) 172 & 628.318 & 047.514 & 5:25:23.963 & –69:30:19.93 & C-wf2 & 17.052 (0.014) 173 & 718.884 & 359.675 & 5:25:23.866 & –69:29:47.74 & C-wf2 & 16.350 (0.011) 174 & 751.491 & 113.581 & 5:25:22.071 & –69:30:10.36 & C-wf2 & 17.248 (0.019)
180 & 051.431 & 142.692 & 5:25:36.224 & –69:30:38.05 & C-wf3 & 18.946 (0.060) 181 & 065.528 & 505.268 & 5:25:42.880 & –69:30:46.50 & C-wf3 & 17.114 (0.014) 182 & 072.679 & 141.546 & 5:25:36.310 & –69:30:36.03 & C-wf3 & 18.104 (0.028) 183 & 114.264 & 105.729 & 5:25:35.871 & –69:30:31.07 & C-wf3 & 18.040 (0.027) 184 & 148.323 & 181.513 & 5:25:37.413 & –69:30:29.90 & C-wf3 & 18.058 (0.028) 185 & 202.542 & 443.744 & 5:25:42.451 & –69:30:31.82 & C-wf3 & 17.172 (0.015) 186 & 257.723 & 099.560 & 5:25:36.480 & –69:30:17.29 & C-wf3 & 19.778 (0.109) 187 & 281.897 & 420.586 & 5:25:42.442 & –69:30:23.55 & C-wf3 & 17.904 (0.028) 188 & 296.206 & 425.715 & 5:25:42.605 & –69:30:22.35 & C-wf3 & 16.062 (0.008) 189 & 353.159 & 510.794 & 5:25:44.447 & –69:30:19.17 & C-wf3 & 13.448 (0.002) 200 & 374.164 & 500.066 & 5:25:44.372 & –69:30:16.89 & C-wf3 & 17.764 (0.026) 201 & 397.615 & 345.481 & 5:25:41.664 & –69:30:10.52 & C-wf3 & 17.503 (0.019) 202 & 424.650 & 161.750 & 5:25:38.456 & –69:30:02.98 & C-wf3 & 18.740 (0.044) 203 & 535.846 & 358.931 & 5:25:42.610 & –69:29:57.64 & C-wf3 & 13.922 (0.003) 204 & 570.584 & 293.642 & 5:25:41.606 & –69:29:52.55 & C-wf3 & 17.414 (0.019) 205 & 570.775 & 210.666 & 5:25:40.098 & –69:29:50.34 & C-wf3 & 17.940 (0.027) 206 & 584.649 & 525.937 & 5:25:45.901 & –69:29:57.45 & C-wf3 & 16.287 (0.011) 207 & 668.161 & 219.805 & 5:25:40.767 & –69:29:41.24 & C-wf3 & 18.819 (0.063)
210 & 083.143 & 175.582 & 5:25:34.401 & –69:30:50.67 & C-wf4 & 17.310 (0.016) 211 & 089.948 & 147.000 & 5:25:34.660 & –69:30:48.19 & C-wf4 & 18.478 (0.037) 212 & 117.067 & 153.880 & 5:25:35.128 & –69:30:49.61 & C-wf4 & 19.662 (0.095) 213 & 130.968 & 461.566 & 5:25:33.725 & –69:31:19.20 & C-wf4 & 15.590 (0.006) 214 & 132.439 & 438.356 & 5:25:33.882 & –69:31:17.06 & C-wf4 & 17.674 (0.022) 215 & 173.487 & 145.009 & 5:25:36.188 & –69:30:50.29 & C-wf4 & 15.173 (0.005) 216 & 235.989 & 625.341 & 5:25:34.780 & –69:31:37.80 & C-wf4 & 18.937 (0.059) 217 & 279.585 & 274.929 & 5:25:37.422 & –69:31:05.48 & C-wf4 & 17.918 (0.025) 218 & 319.086 & 620.000 & 5:25:36.312 & –69:31:39.65 & C-wf4 & 18.998 (0.064) 219 & 397.579 & 230.891 & 5:25:39.800 & –69:31:04.67 & C-wf4 & 16.305 (0.009) 220 & 423.869 & 126.613 & 5:25:40.826 & –69:30:55.40 & C-wf4 & 14.387 (0.003) 221 & 470.121 & 563.511 & 5:25:39.361 & –69:31:38.42 & C-wf4 & 13.829 (0.002) 222 & 508.547 & 240.440 & 5:25:41.768 & –69:31:08.63 & C-wf4 & 16.744 (0.011) 223 & 581.176 & 315.656 & 5:25:42.697 & –69:31:17.83 & C-wf4 & 18.833 (0.055) 224 & 600.979 & 310.116 & 5:25:43.086 & –69:31:17.91 & C-wf4 & 19.132 (0.073) 225 & 668.361 & 305.426 & 5:25:44.324 & –69:31:19.30 & C-wf4 & 18.497 (0.047) 226 & 732.990 & 129.566 & 5:25:46.412 & –69:31:04.39 & C-wf4 & 15.352 (0.006) 227 & 747.563 & 627.673 & 5:25:44.034 & –69:31:52.10 & C-wf4 V & 16.911 (0.051) 228 & 758.000 & 500.755 & 5:25:44.912 & –69:31:40.36 & C-wf4 V & 18.298 (0.086) 229 & 731.764 & 296.000 & 5:25:45.507 & –69:31:20.20 & C-wf4 & 15.889 (0.008)
230 & 058.090 & 084.796 & 5:25:29.128 & –69:30:58.46 & E-pc & 18.231 (0.042) 231 & 706.834 & 185.571 & 5:25:26.768 & –69:31:25.43 & E-pc & 14.984 (0.004)
240 & 072.732 & 070.708 & 5:25:29.258 & –69:30:53.16 & E-wf2 & 18.284 (0.031) 241 & 156.000 & 326.598 & 5:25:29.033 & –69:30:26.59 & E-wf2 & 18.956 (0.048) 242 & 139.914 & 688.479 & 5:25:31.124 & –69:29:52.35 & E-wf2 & 18.108 (0.031) 243 & 237.294 & 301.000 & 5:25:27.431 & –69:30:26.86 & E-wf2 & 17.470 (0.017) 244 & 247.279 & 281.896 & 5:25:27.153 & –69:30:28.42 & E-wf2 & 17.685 (0.020) 245 & 279.606 & 732.290 & 5:25:28.819 & –69:29:44.47 & E-wf2 & 18.230 (0.046) 246 & 281.311 & 462.645 & 5:25:27.439 & –69:30:10.25 & E-wf2 & 18.078 (0.026) 247 & 300.837 & 466.093 & 5:25:27.112 & –69:30:09.36 & E-wf2 & 19.797 (0.106) 248 & 270.900 & 207.564 & 5:25:26.336 & –69:30:34.98 & E-wf2 & 19.185 (0.060) 249 & 331.142 & 079.421 & 5:25:24.598 & –69:30:45.61 & E-wf2 & 14.776 (0.004) 250 & 379.159 & 474.646 & 5:25:25.710 & –69:30:06.51 & E-wf2 & 20.190 (0.149) 251 & 392.711 & 475.000 & 5:25:25.459 & –69:30:06.04 & E-wf2 & 19.592 (0.086) 252 & 431.162 & 322.179 & 5:25:23.994 & –69:30:19.73 & E-wf2 & 17.086 (0.013) 253 & 461.645 & 303.259 & 5:25:23.350 & –69:30:20.76 & E-wf2 & 19.084 (0.053) 254 & 414.554 & 108.455 & 5:25:23.230 & –69:30:40.66 & E-wf2 & 17.120 (0.013) 255 & 498.102 & 076.538 & 5:25:21.541 & –69:30:41.48 & E-wf2 & 16.527 (0.009) 256 & 522.289 & 632.635 & 5:25:23.896 & –69:29:47.61 & E-wf2 & 16.285 (0.011) 257 & 553.549 & 388.252 & 5:25:22.100 & –69:30:10.17 & E-wf2 & 17.394 (0.018) 258 & 635.320 & 568.437 & 5:25:21.518 & –69:29:50.78 & E-wf2 & 18.085 (0.049) 259 & 662.106 & 702.000 & 5:25:21.709 & –69:29:37.33 & E-wf2 V & 16.176 (0.036) 260 & 668.000 & 759.000 & 5:25:21.890 & –69:29:31.79 & E-wf2 V & 16.881 (0.192)
270 & 054.313 & 503.182 & 5:25:37.873 & –69:31:08.27 & E-wf3 & 17.795 (0.023) 271 & 071.041 & 473.684 & 5:25:37.414 & –69:31:05.86 & E-wf3 & 18.086 (0.028) 272 & 069.799 & 675.279 & 5:25:41.076 & –69:31:11.26 & E-wf3 & 18.917 (0.067) 273 & 079.882 & 699.812 & 5:25:41.579 & –69:31:10.96 & E-wf3 & 17.604 (0.025) 274 & 102.244 & 704.530 & 5:25:41.765 & –69:31:08.99 & E-wf3 & 16.773 (0.013) 275 & 113.652 & 592.643 & 5:25:39.793 & –69:31:05.02 & E-wf3 & 16.423 (0.009) 276 & 149.598 & 165.465 & 5:25:32.211 & –69:30:50.16 & E-wf3 & 17.467 (0.018) 277 & 172.319 & 279.436 & 5:25:34.395 & –69:30:51.05 & E-wf3 & 17.335 (0.016) 278 & 202.425 & 369.526 & 5:25:36.187 & –69:30:50.61 & E-wf3 & 15.284 (0.005) 279 & 217.944 & 620.025 & 5:25:40.840 & –69:30:55.78 & E-wf3 & 14.398 (0.003) 280 & 251.000 & 728.724 & 5:25:42.966 & –69:30:55.47 & E-wf3 & 17.932 (0.041) 281 & 338.737 & 699.430 & 5:25:42.888 & –69:30:46.41 & E-wf3 & 16.822 (0.016) 282 & 325.910 & 472.064 & 5:25:38.700 & –69:30:41.51 & E-wf3 & 18.868 (0.054) 283 & 308.327 & 426.849 & 5:25:37.768 & –69:30:42.00 & E-wf3 & 16.365 (0.009) 284 & 326.544 & 338.481 & 5:25:36.255 & –69:30:37.91 & E-wf3 & 18.975 (0.054) 285 & 347.444 & 336.793 & 5:25:36.326 & –69:30:35.85 & E-wf3 & 17.933 (0.025) 286 & 311.361 & 210.011 & 5:25:33.848 & –69:30:35.91 & E-wf3 & 19.317 (0.074) 287 & 388.582 & 301.690 & 5:25:35.898 & –69:30:30.98 & E-wf3 & 17.887 (0.025) 288 & 421.748 & 376.416 & 5:25:37.435 & –69:30:29.83 & E-wf3 & 17.912 (0.024) 289 & 426.167 & 217.360 & 5:25:34.564 & –69:30:25.08 & E-wf3 & 18.697 (0.044) 290 & 475.195 & 638.307 & 5:25:42.482 & –69:30:31.68 & E-wf3 & 16.914 (0.019) 291 & 523.612 & 141.245 & 5:25:33.684 & –69:30:13.77 & E-wf3 & 17.822 (0.023) 292 & 568.756 & 620.401 & 5:25:42.627 & –69:30:22.31 & E-wf3 & 15.916 (0.010) 293 & 626.353 & 706.461 & 5:25:44.472 & –69:30:19.09 & E-wf3 V & 13.132 (0.002) 294 & 670.698 & 539.728 & 5:25:41.672 & –69:30:10.40 & E-wf3 V & 17.446 (0.038) 295 & 682.269 & 146.000 & 5:25:34.596 & –69:29:58.77 & E-wf3 & 18.327 (0.041) 296 & 750.581 & 194.599 & 5:25:35.816 & –69:29:53.59 & E-wf3 & 17.674 (0.028) 297 & 206.881 & 139.141 & 5:25:32.027 & –69:30:44.03 & E-wf3 & 19.052 (0.058) 298 & 236.359 & 150.349 & 5:25:32.378 & –69:30:41.46 & E-wf3 & 19.903 (0.118)
300 & 117.117 & 139.786 & 5:25:30.292 & –69:31:08.99 & E-wf4 & 18.500 (0.037) 301 & 102.388 & 661.826 & 5:25:27.260 & –69:31:58.27 & E-wf4 & 18.468 (0.040) 302 & 151.012 & 732.273 & 5:25:27.777 & –69:32:06.37 & E-wf4 & 17.760 (0.028) 303 & 141.534 & 319.154 & 5:25:29.764 & –69:31:26.75 & E-wf4 & 14.974 (0.004) 304 & 207.554 & 416.185 & 5:25:30.445 & –69:31:37.84 & E-wf4 & 16.768 (0.011) 305 & 202.375 & 632.783 & 5:25:29.216 & –69:31:58.32 & E-wf4 & 17.329 (0.017) 306 & 234.848 & 587.563 & 5:25:30.031 & –69:31:54.93 & E-wf4 & 19.327 (0.076) 307 & 241.896 & 607.643 & 5:25:30.072 & –69:31:57.06 & E-wf4 & 17.275 (0.016) 308 & 221.568 & 338.525 & 5:25:31.113 & –69:31:30.78 & E-wf4 & 14.791 (0.004) 309 & 251.783 & 170.844 & 5:25:32.572 & –69:31:15.62 & E-wf4 & 19.170 (0.064) 310 & 302.518 & 549.339 & 5:25:31.467 & –69:31:53.20 & E-wf4 & 18.555 (0.041) 311 & 322.960 & 533.000 & 5:25:31.934 & –69:31:52.26 & E-wf4 & 18.695 (0.043) 312 & 322.468 & 186.645 & 5:25:33.759 & –69:31:19.08 & E-wf4 & 15.599 (0.006) 313 & 323.569 & 163.505 & 5:25:33.368 & –69:31:26.45 & E-wf4 & 17.680 (0.020) 314 & 382.990 & 243.649 & 5:25:34.566 & –69:31:26.21 & E-wf4 & 19.120 (0.059) 315 & 472.313 & 492.403 & 5:25:34.864 & –69:31:52.49 & E-wf4 & 19.702 (0.115) 316 & 535.841 & 242.504 & 5:25:37.354 & –69:31:30.38 & E-wf4 & 19.307 (0.073) 317 & 536.663 & 461.382 & 5:25:36.194 & –69:31:51.30 & E-wf4 & 18.316 (0.040) 318 & 560.390 & 277.688 & 5:25:37.605 & –69:31:34.39 & E-wf4 & 18.847 (0.052) 319 & 574.503 & 574.484 & 5:25:36.283 & –69:32:03.13 & E-wf4 & 19.150 (0.116) 320 & 587.925 & 473.000 & 5:25:37.057 & –69:31:53.86 & E-wf4 & 18.696 (0.061) 321 & 596.447 & 264.982 & 5:25:38.327 & –69:31:34.14 & E-wf4 & 18.586 (0.043) 322 & 661.675 & 288.691 & 5:25:39.395 & –69:31:38.29 & E-wf4 & 13.836 (0.002) 323 & 724.334 & 265.986 & 5:25:40.628 & –69:31:37.92 & E-wf4 & 18.924 (0.083) 324 & 731.899 & 529.597 & 5:25:39.381 & –69:32:03.18 & E-wf4 V & 18.139 (0.070) 325 & 749.995 & 672.309 & 5:25:38.940 & –69:32:17.20 & E-wf4 V & 14.710 (0.012) 326 & 783.278 & 591.757 & 5:25:39.966 & –69:32:10.44 & E-wf4 V & 16.052 (0.026)
330 & 694.271 & 083.784 & 5:25:26.769 & –69:31:25.29 & F-pc & 15.792 (0.006) 331 & 303.148 & 733.383 & 5:25:22.607 & –69:30:58.82 & F-pc & 16.861 (0.011)
340 & 306.655 & 070.736 & 5:25:24.597 & –69:30:45.41 & F-wf2 & 15.515 (0.005) 341 & 473.100 & 057.589 & 5:25:21.543 & –69:30:41.28 & F-wf2 & 17.273 (0.013) 342 & 531.532 & 611.436 & 5:25:23.891 & –69:29:47.36 & F-wf2 & 17.103 (0.014)
350 & 130.376 & 721.599 & 5:25:41.565 & –69:31:10.63 & F-wf3 & 18.135 (0.028) 351 & 152.138 & 726.227 & 5:25:41.783 & –69:31:08.75 & F-wf3 & 17.530 (0.018) 352 & 156.710 & 613.613 & 5:25:39.810 & –69:31:04.74 & F-wf3 & 17.203 (0.013) 353 & 231.375 & 385.683 & 5:25:36.202 & –69:30:50.37 & F-wf3 & 16.023 (0.007) 354 & 262.401 & 634.561 & 5:25:40.843 & –69:30:55.54 & F-wf3 & 15.121 (0.005) 355 & 612.542 & 613.529 & 5:25:42.638 & –69:30:22.01 & F-wf3 & 16.714 (0.013) 356 & 675.392 & 695.695 & 5:25:44.477 & –69:30:18.82 & F-wf3 V & 13.843 (0.003) 357 & 766.953 & 176.323 & 5:25:35.808 & –69:29:53.41 & F-wf3 & 16.662 (0.010) 358 & 222.567 & 754.311 & 5:25:42.721 & –69:31:03.09 & F-wf3 & 16.104 (0.008) 359 & 520.155 & 636.918 & 5:25:42.486 & –69:30:31.43 & F-wf3 & 17.808 (0.026) 360 & 387.792 & 706.562 & 5:25:42.904 & –69:30:46.11 & F-wf3 & 17.656 (0.022) 361 & 397.182 & 712.011 & 5:25:43.058 & –69:30:45.40 & F-wf3 & 17.932 (0.027) 362 & 340.460 & 436.454 & 5:25:37.790 & –69:30:41.78 & F-wf3 & 17.162 (0.012)
370 & 190.129 & 372.325 & 5:25:29.419 & –69:31:33.22 & F-wf4 & 13.918 (0.003) 371 & 185.623 & 302.985 & 5:25:29.785 & –69:31:26.58 & F-wf4 & 15.736 (0.006) 372 & 257.507 & 395.704 & 5:25:30.469 & –69:31:37.67 & F-wf4 & 17.499 (0.015) 373 & 266.626 & 317.549 & 5:25:31.134 & –69:31:30.65 & F-wf4 & 15.502 (0.005) 374 & 357.737 & 159.598 & 5:25:33.772 & –69:31:18.95 & F-wf4 & 16.368 (0.008) 375 & 359.546 & 139.348 & 5:25:33.934 & –69:31:17.12 & F-wf4 & 16.820 (0.010) 376 & 703.313 & 240.469 & 5:25:39.399 & –69:31:38.14 & F-wf4 & 14.585 (0.003) 377 & 203.469 & 731.000 & 5:25:27.377 & –69:32:07.17 & F-wf4 & 16.609 (0.010)
[ccccccc]{}
26 & 1 & 5:25: 2.59 & –69:30:12.39 & 16.83 (0.01) 25 & 1 & 5:25: 4.62 & –69:30:20.24 & 15.24 (0.01) 23 & 2 & 5:25: 5.17 & –69:29:30.81 & 19.14 (0.16)
22 & 2 & 5:25: 5.61 & –69:29:33.39 & 18.73 (1.89)
21 & 2 & 5:25: 6.09 & –69:29:27.74 & 17.23 (0.05)
20 & 2 & 5:25: 7.32 & –69:30: 4.32 & 17.44 (0.13)
3 & 1 & 5:25: 8.10 & –69:30:35.52 & 18.69 (0.03) 5 & 2 & 5:25: 8.75 & –69:30:43.54 & 18.47 (0.08)
117 & 1 & 5:25: 9.79 & –69:29:17.60 & 18.71 (0.08) 4 & 2 & 5:25: 9.89 & –69:30:41.32 & 13.20 (0.16)
19 & 2 & 5:25:10.77 & –69:30: 8.47 & 18.34 (0.16)
116 & 1 & 5:25:11.42 & –69:29:31.95 & 18.01 (0.03) 54 & 1 & 5:25:11.61 & –69:31:39.53 & 18.95 (0.10) 15 & 2 & 5:25:11.73 & –69:30:10.14 & 18.40 (0.41)
55 & 1 & 5:25:12.07 & –69:31:36.14 & 18.29 (0.07) 8 & 1 & 5:25:12.19 & –69:31: 5.32 & 19.55 (0.08) 2 & 2 & 5:25:12.46 & –69:30:39.65 & 17.70 (0.04)
6 & 2 & 5:25:12.53 & –69:30:53.72 & 17.04 (0.16)
51 & 1 & 5:25:12.59 & –69:31:10.06 & 18.34 (0.06) 1 & 2 & 5:25:12.88 & –69:30:37.22 & 18.62 (0.18)
12 & 1 & 5:25:13.05 & –69:30:23.56 & 18.44 (0.03) 17 & 2 & 5:25:13.43 & –69:29:32.66 & 16.65 (0.10)
56 & 2 & 5:25:13.52 & –69:31:22.15 & 16.40 (0.01)
58 & 1 & 5:25:13.64 & –69:31:41.34 & 19.05 (0.43)
14 & 2 & 5:25:14.04 & –69:29:44.47 & 14.89 (0.11)
16 & 2 & 5:25:14.08 & –69:29:26.70 & 18.46 (0.14) 103 & 1 & 5:25:14.09 & –69:30:34.72 & 18.23 (0.04) 102 & 1 & 5:25:14.12 & –69:30:34.21 & 18.30 (0.04) 115 & 1 & 5:25:14.33 & –69:29:28.46 & 18.28 (0.04) 7 & 1 & 5:25:14.51 & –69:30:50.67 & 16.19 (0.01) 18 & 1 & 5:25:14.73 & –69:29:10.86 & 16.79 (0.02) 59 & 2 & 5:25:14.77 & –69:31:24.61 & 18.49 (0.14)
13 & 2 & 5:25:14.94 & –69:29:49.07 & 15.07 (0.13)
60 & 2 & 5:25:15.29 & –69:31:18.25 & 17.98 (0.13)
53 & 3 & 5:25:15.38 & –69:30:43.18 & 13.75 (0.11)
52 & 2 & 5:25:15.42 & –69:30:39.98 & 18.71 (0.33)
64 & 2 & 5:25:15.67 & –69:31:51.87 & 15.86 (0.28)
10 & 2 & 5:25:15.81 & –69:30: 7.98 & 17.62 (0.08)
63 & 2 & 5:25:15.82 & –69:31:24.24 & 18.68 (0.34)
12 & 1 & 5:25:15.90 & –69:29:40.17 & 18.67 (0.04) 57 & 2 & 5:25:16.01 & –69:30:48.25 & 16.83 (0.12)
101 & 1 & 5:25:16.10 & –69:30:17.35 & 18.71 (0.05) 11 & 2 & 5:25:16.43 & –69:29:41.14 & 15.81 (0.13)
67 & 2 & 5:25:16.59 & –69:31:48.99 & 18.29 (0.09)
100 & 1 & 5:25:16.77 & –69:30:16.13 & 17.99 (0.03) 114 & 1 & 5:25:16.97 & –69:29: 7.22 & 18.46 (0.05) 24 & 2 & 5:25:17.12 & –69:29:49.75 & 18.86 (0.14)
65 & 2 & 5:25:17.24 & –69:31:35.77 & 18.29 (0.13)
61 & 2 & 5:25:17.28 & –69:30:54.08 & 14.39 (0.07)
112 & 1 & 5:25:17.54 & –69:29:36.95 & 18.45 (0.04) 111 & 1 & 5:25:17.62 & –69:29:34.38 & 17.90 (0.03) 62 & 2 & 5:25:18.03 & –69:30:50.90 & 17.17 (0.12)
113 & 1 & 5:25:18.43 & –69:29: 1.34 & 17.51 (0.03) 110 & 1 & 5:25:19.21 & –69:29: 4.33 & 17.29 (0.02)
66 & 3 & 5:25:19.42 & –69:30:53.93 & 18.65 (0.16)
69 & 2 & 5:25:19.53 & –69:31:33.58 & 18.69 (0.10)
70 & 2 & 5:25:20.07 & –69:31:33.18 & 19.39 (0.29)
72 & 2 & 5:25:20.44 & –69:31:49.85 & 18.02 (0.19)
83 & 2 & 5:25:21.16 & –69:31:54.63 & 18.65 (0.08)
39 & 3 & 5:25:21.51 & –69:29:50.51 & 18.19 (0.69)
31 & 5 & 5:25:21.55 & –69:30:41.39 & 16.62 (0.42)
259 & 1 & 5:25:21.71 & –69:29:37.33 & 15.64 (0.04)
71 & 3 & 5:25:21.78 & –69:31: 3.65 & 18.01 (0.15) 68 & 2 & 5:25:21.79 & –69:30:56.33 & 19.80 (0.27)
74 & 1 & 5:25:21.86 & –69:31:54.43 & 22.53 (5.33) 260 & 1 & 5:25:21.89 & –69:29:31.79 & 16.35 (0.19)
36 & 5 & 5:25:22.08 & –69:30:10.12 & 17.29 (0.61)
78 & 2 & 5:25:22.14 & –69:31:48.82 & 16.41 (0.06)
75 & 1 & 5:25:22.39 & –69:31:55.20 & 15.84 (0.11)
73 & 2 & 5:25:22.44 & –69:31:39.18 & 15.77 (0.10)
331 & 1 & 5:25:22.61 & –69:30:58.82 & 16.48 (0.01) 82 & 2 & 5:25:22.71 & –69:31:44.86 & 18.14 (0.36)
40 & 3 & 5:25:22.79 & –69:29:51.16 & 19.15 (0.15)
33 & 4 & 5:25:23.22 & –69:30:40.48 & 17.22 (0.53)
76 & 2 & 5:25:23.33 & –69:31:54.73 & 15.20 (0.02)
253 & 1 & 5:25:23.35 & –69:30:20.76 & 18.55 (0.05) 130 & 1 & 5:25:23.42 & –69:29:28.92 & 19.28 (0.09)
44 & 6 & 5:25:23.89 & –69:29:47.48 & 16.30 (0.38)
35 & 5 & 5:25:23.99 & –69:30:19.67 & 16.99 (0.53)
46 & 1 & 5:25:24.43 & –69:29:28.83 & 18.18 (0.04) 142 & 1 & 5:25:24.48 & –69:31:38.04 & 18.17 (0.08) 32 & 4 & 5:25:24.59 & –69:30:45.32 & 14.85 (0.49)
43 & 2 & 5:25:24.65 & –69:29:47.48 & 17.50 (0.20)
77 & 2 & 5:25:25.37 & –69:31:43.37 & 14.94 (0.00)
251 & 1 & 5:25:25.46 & –69:30: 6.04 & 19.06 (0.09) 79 & 1 & 5:25:25.60 & –69:31:46.04 & 15.92 (0.01) 250 & 1 & 5:25:25.71 & –69:30: 6.51 & 19.66 (0.15) 42 & 2 & 5:25:25.92 & –69:29:50.48 & 17.42 (0.57)
248 & 1 & 5:25:26.34 & –69:30:34.98 & 18.65 (0.06)
80 & 4 & 5:25:26.79 & –69:31:25.25 & 14.96 (0.34) 38b & 2 & 5:25:26.86 & –69:30: 7.05 & 19.89 (0.52)
38a & 3 & 5:25:26.95 & –69:30: 7.75 & 18.74 (0.33)
37b & 2 & 5:25:27.08 & –69:30: 9.24 & 19.84 (1.02)
168 & 1 & 5:25:27.13 & –69:30: 7.56 & 18.37 (0.05) 34 & 4 & 5:25:27.14 & –69:30:28.18 & 17.73 (0.46)
301 & 1 & 5:25:27.26 & –69:31:58.27 & 17.93 (0.04) 377 & 1 & 5:25:27.38 & –69:32: 7.17 & 16.09 (0.01)
37a & 5 & 5:25:27.43 & –69:30:10.07 & 18.05 (0.55) 243 & 1 & 5:25:27.43 & –69:30:26.86 & 16.93 (0.02)
302 & 1 & 5:25:27.78 & –69:32: 6.37 & 17.23 (0.03) 81 & 1 & 5:25:27.92 & –69:31: 9.27 & 18.95 (0.08) 45 & 1 & 5:25:27.99 & –69:29:44.84 & 20.68 (0.71)
41 & 2 & 5:25:28.27 & –69:29:57.50 & 18.63 (0.69)
170 & 2 & 5:25:28.79 & –69:29:44.57 & 17.64 (0.06)
135 & 1 & 5:25:28.83 & –69:29:10.50 & 16.75 (0.03) 132 & 2 & 5:25:29.01 & –69:29:24.40 & 17.18 (0.46)
241 & 1 & 5:25:29.03 & –69:30:26.59 & 18.42 (0.05)
145 & 2 & 5:25:29.15 & –69:30:58.32 & 18.07 (0.26) 305 & 1 & 5:25:29.22 & –69:31:58.32 & 16.79 (0.02) 144 & 2 & 5:25:29.25 & –69:30:52.58 & 18.15 (0.42)
370 & 1 & 5:25:29.42 & –69:31:33.22 & 13.40 (0.00)
146 & 2 & 5:25:29.46 & –69:30:58.70 & 16.70 (0.35)
148 & 3 & 5:25:29.84 & –69:31:26.88 & 14.74 (0.38) 125 & 1 & 5:25:29.91 & –69:30: 1.78 & 18.73 (0.06) 306 & 1 & 5:25:30.03 & –69:31:54.93 & 18.79 (0.08) 307 & 1 & 5:25:30.07 & –69:31:57.06 & 16.74 (0.02)
147 & 2 & 5:25:30.35 & –69:31: 9.18 & 18.20 (0.26) 304 & 2 & 5:25:30.44 & –69:31:37.84 & 16.54 (0.37)
129 & 2 & 5:25:31.09 & –69:29:52.35 & 17.84 (0.31) 308 & 2 & 5:25:31.11 & –69:31:30.78 & 14.56 (0.36)
310 & 1 & 5:25:31.47 & –69:31:53.20 & 18.02 (0.04) 131 & 2 & 5:25:31.60 & –69:29:35.35 & 16.32 (0.41)
133 & 2 & 5:25:31.78 & –69:29:22.96 & 15.98 (0.45)
311 & 1 & 5:25:31.93 & –69:31:52.26 & 18.16 (0.04) 297 & 1 & 5:25:32.03 & –69:30:44.03 & 18.52 (0.06)
149 & 3 & 5:25:32.30 & –69:30:49.79 & 17.05 (0.31) 298 & 1 & 5:25:32.38 & –69:30:41.46 & 19.37 (0.12) 309 & 1 & 5:25:32.57 & –69:31:15.62 & 18.64 (0.06) 134 & 1 & 5:25:32.75 & –69:29:17.87 & 18.57 (0.26) 164 & 1 & 5:25:32.99 & –69:30:29.06 & 18.45 (0.05) 128 & 1 & 5:25:33.09 & –69:29:58.83 & 17.94 (0.05) 313 & 1 & 5:25:33.37 & –69:31:26.45 & 17.15 (0.02)
213 & 3 & 5:25:33.72 & –69:31:19.20 & 15.25 (0.45) 124 & 3 & 5:25:33.74 & –69:30:13.89 & 17.34 (0.39)
162 & 1 & 5:25:33.81 & –69:30:31.64 & 18.87 (0.07) 286 & 1 & 5:25:33.85 & –69:30:35.91 & 18.78 (0.07) 214 & 2 & 5:25:33.88 & –69:31:17.06 & 16.63 (0.40)
210 & 2 & 5:25:34.40 & –69:30:50.67 & 16.77 (0.03) 123 & 1 & 5:25:34.51 & –69:30:16.64 & 17.36 (0.04)
314 & 1 & 5:25:34.57 & –69:31:26.21 & 18.59 (0.06) 295 & 1 & 5:25:34.60 & –69:29:58.77 & 17.79 (0.04) 160 & 2 & 5:25:34.63 & –69:30:25.36 & 18.09 (0.07) 211 & 1 & 5:25:34.66 & –69:30:48.19 & 17.91 (0.04) 216 & 1 & 5:25:34.78 & –69:31:37.80 & 18.37 (0.06) 315 & 1 & 5:25:34.86 & –69:31:52.49 & 19.17 (0.12) 212 & 1 & 5:25:35.13 & –69:30:49.61 & 19.09 (0.09) 296 & 2 & 5:25:35.82 & –69:29:53.59 & 16.53 (0.50) 183 & 2 & 5:25:35.87 & –69:30:31.07 & 17.41 (0.06)
161 & 1 & 5:25:35.91 & –69:29:53.87 & 17.05 (0.02)
215 & 3 & 5:25:36.19 & –69:30:50.29 & 14.89 (0.46) 317 & 1 & 5:25:36.19 & –69:31:51.30 & 17.78 (0.04)
180 & 2 & 5:25:36.22 & –69:30:38.05 & 18.41 (0.03)
319 & 1 & 5:25:36.28 & –69:32: 3.13 & 18.61 (0.12) 182 & 2 & 5:25:36.31 & –69:30:36.03 & 17.46 (0.07) 218 & 1 & 5:25:36.31 & –69:31:39.65 & 18.43 (0.06)
186 & 1 & 5:25:36.48 & –69:30:17.29 & 19.21 (0.11) 320 & 1 & 5:25:37.06 & –69:31:53.86 & 18.16 (0.06) 316 & 1 & 5:25:37.35 & –69:31:30.38 & 18.77 (0.07) 184 & 2 & 5:25:37.41 & –69:30:29.90 & 17.43 (0.06)
217 & 2 & 5:25:37.42 & –69:31: 5.48 & 17.44 (0.10)
318 & 1 & 5:25:37.60 & –69:31:34.39 & 18.31 (0.05) 283 & 2 & 5:25:37.77 & –69:30:42.00 & 16.16 (0.41)
270 & 1 & 5:25:37.87 & –69:31: 8.27 & 17.26 (0.02) 321 & 1 & 5:25:38.33 & –69:31:34.14 & 18.05 (0.04) 202 & 1 & 5:25:38.46 & –69:30: 2.98 & 18.17 (0.04) 282 & 1 & 5:25:38.70 & –69:30:41.51 & 18.33 (0.05) 325 & 1 & 5:25:38.94 & –69:32:17.20 & 14.17 (0.01) 221 & 3 & 5:25:39.36 & –69:31:38.42 & 13.48 (0.43) 324 & 1 & 5:25:39.38 & –69:32: 3.18 & 17.60 (0.07)
219 & 3 & 5:25:39.80 & –69:31: 4.67 & 16.03 (0.49)
326 & 1 & 5:25:39.97 & –69:32:10.44 & 15.52 (0.03) 205 & 1 & 5:25:40.10 & –69:29:50.34 & 17.37 (0.03) 323 & 1 & 5:25:40.63 & –69:31:37.92 & 18.39 (0.08) 207 & 1 & 5:25:40.77 & –69:29:41.24 & 18.25 (0.06) 220 & 3 & 5:25:40.83 & –69:30:55.40 & 14.04 (0.42)
272 & 1 & 5:25:41.08 & –69:31:11.26 & 18.38 (0.07)
273 & 2 & 5:25:41.58 & –69:31:10.96 & 17.31 (0.27) 204 & 1 & 5:25:41.61 & –69:29:52.55 & 16.84 (0.02) 201 & 2 & 5:25:41.66 & –69:30:10.52 & 16.92 (0.01)
222 & 3 & 5:25:41.77 & –69:31: 8.63 & 16.41 (0.45)
187 & 1 & 5:25:42.44 & –69:30:23.55 & 17.33 (0.03) 185 & 3 & 5:25:42.45 & –69:30:31.82 & 16.69 (0.46)
188 & 3 & 5:25:42.60 & –69:30:22.35 & 15.63 (0.42) 203 & 1 & 5:25:42.61 & –69:29:57.64 & 13.35 (0.00)
358 & 1 & 5:25:42.72 & –69:31: 3.09 & 15.59 (0.01) 181 & 3 & 5:25:42.88 & –69:30:46.50 & 16.60 (0.42)
280 & 1 & 5:25:42.97 & –69:30:55.47 & 17.40 (0.04) 223 & 1 & 5:25:42.70 & –69:31:17.83 & 18.26 (0.05) 361 & 1 & 5:25:43.06 & –69:30:45.40 & 17.41 (0.03) 224 & 1 & 5:25:43.09 & –69:31:17.91 & 18.56 (0.07) 227 & 1 & 5:25:44.03 & –69:31:52.10 & 16.34 (0.05) 225 & 1 & 5:25:44.32 & –69:31:19.30 & 17.93 (0.05) 200 & 1 & 5:25:44.37 & –69:30:16.89 & 17.19 (0.03) 189 & 3 & 5:25:44.45 & –69:30:19.17 & 12.89 (0.36)
228 & 1 & 5:25:44.91 & –69:31:40.36 & 17.73 (0.09) 229 & 1 & 5:25:45.51 & –69:31:20.20 & 15.32 (0.01) 206 & 1 & 5:25:45.90 & –69:29:57.45 & 15.72 (0.01) 226 & 1 & 5:25:46.41 & –69:31: 4.39 & 14.78 (0.01)
[rrrr]{} 15414 & 14.06 (0.04) & 25 & 15.24 (0.01) 15511 & 11.47 (0.08) & 4 & 13.20 (0.16) 15543 & 14.59 (0.06) & 6 & 17.04 (0.16) 15556 & 15.05 (0.05) & 17 & 16.65 (0.10) 15557 & 14.90 (0.03) & 56 & 16.40 (0.01) 15565 & 13.35 (0.04) & 14 & 14.89 (0.11) 15584 & 13.58 (0.04) & 13 & 15.07 (0.13) & & 24 & 18.86 (0.14) 15591 & 12.20 (0.03) & 53 & 13.75 (0.11) 15593 & 14.31 (0.04) & 64 & 15.86 (0.28) & & 67 & 18.29 (0.09) 15598 & 14.25 (0.03) & 11 & 15.81 (0.13) & & 12 & 18.67 (0.04) 15602 & 12.79 (0.03) & 114 & 18.46 (0.05) 15616 & 12.83 (0.02) & 61 & 14.39 (0.07) & & 62 & 17.17 (0.12) & & 66 & 18.65 (0.16) 15681 & 15.35 (0.04) & 31 & 16.62 (0.42) & & 33 & 17.22 (0.53) 15697 & 14.30 (0.04) & 73 & 15.77 (0.10) 15709 & 13.86 (0.05) & 76 & 15.20 (0.02) & & 74 & 22.53 (5.33) & & 75 & 15.84 (0.11) 15721 & 15.32 (0.06) & 44 & 16.30 (0.38) & & 43 & 17.50 (0.20) 15731 & 13.64 (0.04) & 32 & 14.85 (0.49) 15744 & 13.06 (0.03) & 77 & 14.94 (0.00) & & 79 & 15.92 (0.01) 15777 & 13.79 (0.04) & 80 & 14.96 (0.34) 15810 & 15.25 (0.05) & 135 & 16.75 (0.03) 15821 & 15.54 (0.05) & 146 & 16.70 (0.35) & & 145 & 18.07 (0.26) & & 144 & 18.15 (0.42) 15835 & 13.74 (0.04) & 148 & 14.74 (0.38) 15853 & 15.71 (0.09) & 304 & 16.54 (0.37) 15865 & 13.65 (0.04) & 308 & 14.56 (0.36) 15873 & 15.11 (0.05) & 133 & 15.98 (0.45) 15906 & 16.53 (0.16) & 134 & 18.57 (0.26) 15922 & 14.25 (0.03) & 213 & 15.25 (0.45) & & 214 & 16.63 (0.40) 15933 & 16.08 (0.12) & 210 & 16.77 (0.03) & & 211 & 17.91 (0.04) & & 212 & 19.09 (0.09) 15953 & 14.04 (0.06) & 215 & 14.89 (0.46) 15978 & 15.02 (0.04) & 283 & 16.16 (0.41) & & 282 & 18.33 (0.05) 15988 & 14.16 (0.04) & 325 & 14.17 (0.01) 15995 & 12.82 (0.03) & 221 & 13.48 (0.43) & & 323 & 18.39 (0.08) 16004 & 14.99 (0.05) & 219 & 16.03 (0.49) 16005 & 15.39 (0.06) & 326 & 15.52 (0.03) 16019 & 13.23 (0.03) & 220 & 14.04 (0.42) & & 280 & 17.40 (0.04) 16038 & 14.98 (0.05) & 222 & 16.41 (0.45) & & 272 & 18.38 (0.07) & & 273 & 17.31 (0.27) 16049 & 14.38 (0.05) & 188 & 15.63 (0.42) & & 187 & 17.33 (0.03) 16052 & 12.76 (0.03) & 203 & 13.35 (0.00) 16053 & 15.37 (0.05) & 181 & 16.60 (0.42) & & 361 & 17.41 (0.03) 16076 & 14.15 (0.41) & 189 & 12.89 (0.36) & & 225 & 17.93 (0.05) 16087 & 15.48 (0.40) & 200 & 17.19 (0.03) 16096 & 14.74 (0.04) & 206 & 15.72 (0.01)
[crr]{} 26 & 16.83$\pm$0.01 & -2.59$\pm$0.04 25 & 15.24$\pm$0.01 & -5.03$\pm$0.04 20 & 17.44$\pm$0.13 & -1.85$\pm$0.16 4 & 13.20$\pm$0.16 & -5.18$\pm$0.18 6 & 17.04$\pm$0.16 & -2.32$\pm$0.18 12 & 18.44$\pm$0.03 & -0.30$\pm$0.08 56 & 16.40$\pm$0.01 & -4.60$\pm$0.04 53 & 13.75$\pm$0.11 & -5.09$\pm$0.15 64 & 15.86$\pm$0.28 & -4.24$\pm$0.24 11 & 15.81$\pm$0.13 & -5.04$\pm$0.16 114 & 18.46$\pm$0.05 & -1.10$\pm$0.10 61 & 14.39$\pm$0.07 & -4.42$\pm$0.12 69 & 18.69$\pm$0.10 & -0.65$\pm$0.14 68 & 19.80$\pm$0.27 & 1.10$\pm$0.23 36 & 17.29$\pm$0.61 & -2.83$\pm$0.35 78 & 16.41$\pm$0.06 & -2.24$\pm$0.01 73 & 15.77$\pm$0.10 & -3.80$\pm$0.14 331 & 16.48$\pm$0.01 & -2.71$\pm$0.04 76 & 15.20$\pm$0.02 & -5.08$\pm$0.06 46 & 18.18$\pm$0.04 & -2.82$\pm$0.09 32 & 14.85$\pm$0.49 & -6.00$\pm$0.31 77 & 14.94$\pm$0.00 & -3.95$\pm$0.01 80 & 14.96$\pm$0.34 & -4.22$\pm$0.26 3702 & 19.84$\pm$1.02 & 0.90$\pm$0.45 168 & 18.37$\pm$0.05 & 1.80$\pm$0.10 302 & 17.23$\pm$0.03 & -2.09$\pm$0.08 81 & 18.95$\pm$0.08 & 0.50$\pm$0.13 45 & 20.68$\pm$0.71 & 1.83$\pm$0.38 45 & 18.33$\pm$0.07 & -2.20$\pm$0.12 170 & 17.64$\pm$0.06 & -2.85$\pm$0.11 305 & 16.79$\pm$0.02 & -1.34$\pm$0.06 146 & 16.70$\pm$0.35 & -4.30$\pm$0.26 148 & 14.74$\pm$0.38 & -5.10$\pm$0.28 304 & 16.54$\pm$0.37 & -3.28$\pm$0.27 308 & 14.56$\pm$0.36 & -5.05$\pm$0.27 131 & 16.32$\pm$0.41 & -4.65$\pm$0.29 133 & 15.98$\pm$0.45 & -2.70$\pm$0.30 134 & 18.57$\pm$0.26 & 0.44$\pm$0.23 313 & 17.15$\pm$0.02 & -2.88$\pm$0.06 213 & 15.25$\pm$0.45 & -3.80$\pm$0.30 212 & 19.09$\pm$0.09 & -2.05$\pm$0.13 215 & 14.89$\pm$0.46 & -5.00$\pm$0.30 319 & 18.61$\pm$0.12 & -2.25$\pm$0.15 182 & 17.46$\pm$0.07 & -1.63$\pm$0.12 186 & 19.21$\pm$0.11 & -0.10$\pm$0.15 316 & 18.77$\pm$0.07 & -0.01$\pm$0.12 221 & 13.48$\pm$0.43 & -5.40$\pm$0.29 326 & 15.52$\pm$0.03 & -3.95$\pm$0.08 205 & 17.37$\pm$0.03 & -2.00$\pm$0.08 220 & 14.04$\pm$0.42 & -4.65$\pm$0.29
273 & 17.31$\pm$0.27 & -2.88$\pm$0.23 201 & 16.92$\pm$0.01 & -3.40$\pm$0.04 188 & 15.63$\pm$0.42 & -4.90$\pm$0.29 203 & 13.35$\pm$0.00 & -5.00$\pm$0.01 223 & 18.26$\pm$0.05 & -0.60$\pm$0.10 200 & 17.19$\pm$0.03 & -0.90$\pm$0.08 189 & 12.89$\pm$0.36 & -5.70$\pm$0.27 226 & 14.78$\pm$0.01 & -4.34$\pm$0.04
[^1]: Email: noah@wise.tau.ac.il
[^2]: Based on observations with the NASA/ESA [*Hubble Space Telescope*]{}, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555
|
---
abstract: 'In this paper we propose a very special relativity (VSR)-inspired generalization of the Maxwell-Chern-Simons (MCS) electrodynamics. This proposal is based upon the construction of a proper study of the SIM$(1)$–VSR gauge-symmetry. It is shown that the VSR nonlocal effects present a significant and health departure from the usual MCS theory. The classical dynamics is analysed in full detail, by studying the solution for the electric field and static energy for this configuration. Afterwards, the interaction energy between opposite charges are derived and we show that the VSR effects play an important part in obtaining a (novel) finite expression for the static potential.'
author:
- |
R. Bufalo$^{1}$[[^1]]{}\
*$^{1}$ Instituto de Física Teórica (IFT), Universidade Estadual Paulista*\
*Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, 01140-070 São Paulo, SP, Brazil*\
title: 'SIM$(1)$–VSR Maxwell-Chern-Simons electrodynamics'
---
Very special relativity; SIM$(1)$ gauge-symmetry; Maxwell-Chern-Simons theory; Classical solutions
Introduction {#sec1}
============
In recent years we have been scrutinizing Planck scale Physics through many theories, proposals, ideas, etc, all this effort expended in order to improve our understanding of the Nature behaviour at shortest distances (as well as in the beginning our Universe [@ref33]). In particular, it is widely aimed by these proposals achieve a better description of a quantum theory of gravity, or at least to make contact with the phenomenology of quantum gravity and hence gain insights about the fundamental structure of space and time at Planck scale.
Within the class of theories to describe Planck scale Physics, at the quantum realm, we can cite String theory and loop quantum gravity as those most prominent candidates up-to-date. Our main interest is to put at the same level a steady description of both quantum mechanics and general relativity. An interesting outcome of these proposals is the presence of a minimal measurable length scale [@ref22], this can be incorporate at the quantum theory by the so-called Generalized Uncertainty Principle [@ref20; @ref21; @ref23]. Another consequence of known theories of quantum gravity is the breaking of (some) symmetry groups. In particular, a well known scrutinized consequence is the violation of the underlying Lorentz symmetry [@ref37; @ref53], since a definitive description of the space-time is expected not to be in terms of a classical smooth geometry.
Among the broad class of attempts trying to encompass and describe consistently Lorentz violating effects [@ref25; @ref26; @ref27; @ref28], we shall focus in exploring features of very special relativity (VSR) [@ref7; @ref30] in this paper. The cornerstone from this proposal is that the laws of physics are not invariant under the whole Poincaré group but rather are invariant under subgroups of the Poincaré group preserving the basic elements of special relativity, but at the same time enhancing the Lorentz algebra by modifying the dynamics of particles. For instance, conservation laws and the usual relativistic dispersion relation, $E^2=p^2+M^2$ for a particle of mass M, etc, are preserved in this case.
In particular, within this proposal, one can use in the realization of VSR the representations of the full Lorentz group but supplemented by a Lorentz-violating factor, such that the symmetry of the Lagrangian is then reduced to one of the VSR subgroups of the Lorentz group. These effects can then be encoded in the form Lorentz-violating terms in the Lagrangian that are necessarily nonlocal. As an example, one can observe that a VSR-covariant Dirac equation has the form $$\left(i\gamma^\mu \tilde{\partial}_\mu -m \right)\Psi \left(x\right)=0,$$ where the wiggle operator is defined such as $\tilde{\partial}_{\mu}=\partial_{\mu}+\frac{1}{2}\frac{\sigma^{2}}{n.\partial}n_{\mu}$, with the chosen preferred null direction $n_{\mu}=\left(1,0,0,1\right)$ so that it transforms multiplicatively under a VSR transformation. Next, we can square the VSR-covariant Dirac equation, and we find $$\left(\partial^\mu \partial_\mu +\mathcal{M}^2 \right)\Psi \left(x\right)=0, \quad \mathcal{M}^2=M^2+\sigma^2 .$$ We thus immediately realize an interesting observable consequence of VSR that is to provide a novel mechanism for introducing neutrino masses without the need for new particles [@ref30]. Moreover, the VSR parameter $\sigma$ sets the scale for the VSR effects. Among the most interesting analysis involving VSR effects we can cite a realization of VSR via a lightlike noncommutative deformation of Poincaré symmetry [@ref6], studies on Dirac equation [@ref13] and hydrogen atom [@ref3], as well as gauge theories [@ref16] and curved spacetime field theories [@ref9], gravitational and cosmological models [@ref8; @ref36].
As it concerns our interest, VSR-effects have been discussed in the context of $(3+1)$-dims electromagnetic theories: Abelian and non-Abelian Maxwell theories [@ref4; @ref5; @ref51], Chern-Simons theory [@ref2; @ref52] and Born-Infeld electrodynamics [@ref1]. By different reasons, we have seen recently a renewed interest in studying Lorentz-violating modifications of the Maxwell-Chern-Simons theory [@ref2; @ref52; @ref32; @ref12]. Although of the interesting features obtained in those analysis, one may wonder what the (nonlocal) VSR-effects may influence the behaviour of a lower-dimensional electromagnetic theory, for instance in a $(2+1)$-dimensional spacetime [@ref15], where we will work with the SIM$(1)$ subgroup (of the $SO(2,1)$ Lorentz group) that preserve all the aforementioned conditions, in particular that a given null-vector is preserved up to a rescaling. [^2]
It worth notice that different approaches have been used to consider mass effects in $(2+1)$-dims generalized electrodynamic theories [@ref45; @ref46]. It is well-known that the Maxwell-Chern-Simons (MCS) electrodynamics describe a single massive gauge mode of helicity $\pm1$, the so-called Topologically massive electrodynamics [@ref42]. Hence, we expect that the VSR setting will not modify Maxwell-Chern-Simons theory only by adding ’massive’ effects, [^3] since a topological mass is already present, but rather that the VSR-effects will be prominent in changing the theory’s dynamics in a significant and novel manner.
In this letter we will examine the Maxwell-Chern-Simons electrodynamics in a VSR setting. We start Sec.\[sec2\] by reviewing the fundamental aspects from the SIM$(1)$–VSR gauge invariance, which allow us to determine the VSR-modified Abelian field strength to be used in our analysis. In Sec.\[sec3\], we define our SIM$(1)$–VSR Maxwell-Chern-Simons theory. Afterwards, we determine the dispersion relation and discuss the electrostatic solution for the equations of motion in the presence of a pointlike charge. In addition, we compute the field energy and gauge-invariant potential between two opposite charges. Along the analysis we will comment at pertinent points the differences obtained by VSR deformations in view of the usual MCS theory. In Sec.\[sec4\] we summarize the results, and present our final remarks.
SIM$(1)$ gauge symmetry overview {#sec2}
================================
Let us start by discussing the SIM$(1)$ VSR gauge invariance [@ref4; @ref5]. An important remark to bear is that although the VSR subgroups do not admit invariant tensors, they select a preferred null direction. For this matter, the first point in order to develop the gauge invariance is to realize that the gauge transformation of a gauge field in VSR is modified so that $$\delta A_{\mu}=\tilde{\partial}_{\mu}\Lambda,\label{eq:1.1}$$ where the wiggle operator is defined such as $\tilde{\partial}_{\mu}=\partial_{\mu}+\frac{1}{2}\frac{\sigma^{2}}{n.\partial}n_{\mu}$, but now with the chosen preferred null direction given as $n_{\mu}=\left(1,0,1\right)$ and multiplicatively covariant under the SIM$(1)$ subgroup of the $(2+1)$-dims Lorentz group [@ref51].
Next, let us consider a charged scalar field $\varphi$ with an infinitesimal gauge transformation given as usual by $\delta\varphi=i\Lambda\varphi$. Moreover, we know that in general a covariant derivative must satisfy the transformation property $$\delta\left(D_{\mu}\varphi\right)=i\Lambda\left(D_{\mu}\varphi\right).\label{eq:1.2}$$ Hence, it can be showed that the covariant operator defined as the following $$D_{\mu}\varphi=\partial_{\mu}\varphi-iA_{\mu}\varphi+\frac{i\sigma^{2}}{2}n_{\mu}\left(\frac{1}{\left(n.\partial\right)^{2}}\left(n.A\right)\right)\varphi\label{eq:1.3}$$ satisfies the condition . Besides, in the same way as we have defined the wiggle operator $\tilde{\partial}$ from the raw derivative $\partial$, we can generalize the covariant derivative $D$ to a wiggle operator $$\tilde{D}_{\mu}=D_{\mu}+\frac{1}{2}\frac{\sigma^{2}}{n.D}n_{\mu}\label{eq:1.4}$$ so that it reduces to the operator $\tilde{\partial}$ when the limit $A_{\mu}=0$ is taken.
Hence, with the above definitions the field strength associated to the operator $D_{\mu}$ can be computed as usual by the following quantity $\left[D_{\mu},D_{\nu}\right]\varphi=-iF_{\mu\nu}\varphi$. This can be shown to result into $$F_{\mu\nu}=\partial_{\mu}A_{\nu}+\frac{\sigma^{2}}{2}n_{\mu}\left(\frac{1}{\left(n.\partial\right)^{2}}\partial_{\nu}\left(n.A\right)\right)-\mu\leftrightarrow\nu\label{eq:1.5}$$ This field strength can be seen as the raw definition of the $A_{\mu}$ gauge field strength. However, one can easily realize that this field-strength does not coincide with the SIM$(1)$ wiggle operator $$\tilde{F}_{\mu\nu}=\tilde{\partial}_{\mu}A_{\nu}-\tilde{\partial}_{\nu}A_{\mu}\label{eq:1.6}$$ On one hand, the wiggle definition $\tilde{F}_{\mu\nu}$ is gauge invariant and it will be used to describe massive gauge fields. Now, on the other hand, we can realize that the difference between the raw and wiggle field-strength must be gauge invariant as well, so that we can write the following expression for wiggle field strength $$\tilde{F}_{\mu\nu}=F_{\mu\nu}+\frac{\sigma^{2}}{2}\frac{1}{\left(n.\partial\right)^{2}}\left(n_{\nu}n^{\lambda}F_{\mu\lambda}
-n_{\mu}n^{\lambda}F_{\nu\lambda}\right)\label{eq:1.7}$$
Some remarks are now in place. By means of illustration, in showing how to describe massive gauge fields, let us consider a VSR modified Maxwell action, $$S=\int d^{\omega}x\left[-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}\right]\label{eq:1.8}$$ it is interesting to notice that this action can be augmented by further quadratic terms in **$A$** as well as by gauge invariant coupling to matter fields [@ref5]. In particular, this prescription also works to generates mass for the matter fields. The field equations follow straightforwardly as $$\tilde{\partial}_{\mu}\tilde{F}^{\mu\nu}=0,$$ now, by taking a VSR-type Lorenz gauge condition, $\tilde{\partial}_{\mu}A^{\mu}=0$, we find that $$\tilde{\partial}^{2}A^{\mu}=\left(\partial^{2}+m^{2}\right)A^{\mu}=0.$$
With this discussion we see that a massive gauge field, defined in terms of the ordinary derivative, can be described suitably in a gauge-invariant fashion when written in terms of the wiggle operator. Moreover, this may be considered our starting point in defining our model of interest.
VSR Maxwell-Chern-Simons electrodynamics {#sec3}
========================================
Let us now characterize the model under consideration. Based on the points discussed above, but taking into account a SIM$(1)$ VSR setting and the wiggle field strength expression , we are in a position to define the SIM$(1)$–VSR Maxwell-Chern-Simons electrodynamics by the following Lagrangian density $$\mathcal{L}=-\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{m}{4}\varepsilon^{\mu\nu\lambda}A_{\mu}\tilde{F}_{\nu\lambda}.\label{eq:2.1}$$ The usual MCS theory, or topologically massive electrodynamics, describe a single massive gauge mode of helicity $\pm1$. We shall now explore the VSR setting in order to look for modification on the solutions of the MCS classical dynamics. Next, the equations of motion for the SIM$(1)$ MCS theory can be readily determined as $$\tilde{\partial}_{\mu}\tilde{F}^{\mu\alpha}+\frac{m}{2}\varepsilon^{\alpha\nu\lambda}\tilde{F}_{\nu\lambda}=0\label{eq:2.2}$$ In order to solve the above equations, it is convenient to introduce the dual field strength $\tilde{G}_{\mu}$, which is a vector in three dimensions $\tilde{G}^{\mu}=\frac{1}{2}\varepsilon^{\mu\nu\lambda}\tilde{F}_{\nu\lambda}$. Moreover, it follows straightforwardly that the Bianchi identity in this case is written as $\partial_{\mu}\tilde{G}^{\mu}=0$. Hence, we see that the field equations are now written in the form $$\left[\varepsilon^{\mu\nu\lambda}\tilde{\partial}_{\mu}+m\eta^{\nu\lambda}\right]\tilde{G}_{\lambda}=0\label{eq:2.3}$$ From this expression we can identify the (on-shell) projection operators [@ref42] $$\left[P\left(\pm m\right)\right]_{\nu}^{\mu}=\frac{1}{2}\left[\delta_{\nu}^{\mu}\mp\frac{1}{m}\varepsilon_{ ~~~~\nu}^{\mu\lambda}\tilde{\partial}_{\lambda}\right],$$ it is easy to show that, as expected, they satisfy $\left[P\left(\pm m\right)\right]^{2}=\left[P\left(\pm m\right)\right]$. Actually, these operators project onto the Poincaré (irreducible) representations [@ref42]. Hence, in terms of the dual field strength $\tilde{G}_{\mu}$, it finally follows the field equation $$\left(\partial^{2}+M^{2}\right)\tilde{G}_{\mu}=0\label{eq:2.4}$$ where we have defined a new mass parameter $M^{2}=m^{2}+\sigma^{2}$. This shows, nonetheless, that the dispersion relation for the gauge field is only slightly modified, since the dispersion relation $\omega=\pm\sqrt{p^{2}+M^{2}}$ has the same form as the one obtained in the usual theory, being only shifted on the mass parameter.
By means of discussion, let us now add a (electrostatic) source term $A_{0}J^{0}$ into the Lagrangian . Thus, a new set of field equations now read $$\tilde{\partial}_{\mu}\tilde{F}^{\mu\alpha}+\frac{m}{2}\varepsilon^{\alpha\nu\lambda}\tilde{F}_{\nu\lambda}=J^{0}\delta_{0}^{\alpha}. \label{eq:2.5}$$ Hence, for the temporal component of , we find a modified Gauss’s law $$\tilde{\partial}_{i}\tilde{E}^{i}+\frac{m}{2}\tilde{B}=J^{0}\label{eq:2.6}$$ where we have defined the wiggle electric and magnetic fields such as $\tilde{E}^{i}=\tilde{F}^{i0}$ and $\tilde{B}=\frac{1}{2}\varepsilon^{ij}\tilde{F}_{ij}$, respectively. Besides, it follows that for $\alpha=i$ in , we have the relation $$\tilde{E}_{i}=\frac{1}{m}\tilde{\partial}_{i}\tilde{B} .\label{eq:2.7}$$ Finally, we can use the relation to rewrite in the following form, $$\left(-\nabla^{2}+M^{2}\right)\tilde{B}=mJ^{0}. \label{eq:2.8}$$
In particular, we can consider a simple scenario in order to solve (i.e. Eq.), this can be chosen by taking the current density for a pointlike charge $J_{0}\left(t,\mathbf{r}\right)=g\delta^{\left(3\right)}\left(\mathbf{r}\right)$. Hence, one can easily solve to find $$\begin{aligned}
\tilde{B}\left(\mathbf{r}\right) & =\frac{gm}{2\pi}K_{0}\left(Mr\right).\label{eq:2.9}\end{aligned}$$ Finally, we can determine the electric field by replacing back into $$\tilde{{\bf E}}\left(\mathbf{r}\right) =-\frac{gM}{2\pi}K_{1}\left(Mr\right)\left(\tilde{\nabla} r\right). \label{eq:2.15}$$ One can see that the wiggle derivative results into $\tilde{\nabla}r = \hat{{\bf r}}
-\frac{\sigma^{2} \hat{{\bf n}}}{2}\left(\frac{1}{\nabla_{y}}r\right)$, where the unit vector is given as $\hat{{\bf n}}=\left(0,1\right)$. Let us now concentrate in computing the nonlocal term of the above expression. This can be worked out by means of the following representation $$\frac{\sigma}{\nabla_{y}}r=\int_{0}^{\infty}ds\left(\sum_{n=0}\frac{1}{n!}\left(-\frac{s\nabla_{y}}{\sigma}\right)^{n}\right)\sqrt{x^{2}+y^{2}}=\int_{0}^{\infty}ds\sqrt{x^{2}+\left(\frac{s}{\sigma}-y\right)^{2}} . \label{eq:2.16}$$ Besides, the above derivative has been calculated by means of standard manipulations: One can make use of Newton’s binominal to rewrite $\sqrt{x^{2}+y^{2}}$ conveniently as $\sum_{k}\left(\begin{array}{c}
\frac{1}{2} \\ k \end{array} \right)\left(x^{2}\right)^{1/2-k}\left(y^{2}\right)^{k} $, so that we can compute the operation $\frac{1}{n!}\frac{d^{n}}{dy^{n}}\left(y^{2}\right)^{k}=\left(\begin{array}{c}2k\\ n \end{array}\right)y^{2k-n}$. Finally, one can solve the integration in and find that $$\begin{aligned}
\frac{\sigma}{\nabla_{y}}r & =\frac{\sigma}{2}\left[yr-x^{2}\ln\left(\sigma\left[r-y\right]\right)\right]+\lim _{\rho \rightarrow \infty}\Lambda\left(\rho\right). \label{eq:2.17}\end{aligned}$$ We thus see in that as a consequence of the nonlocality of the VSR-effects the distribution $\Lambda\left(\rho\right)\equiv \frac{1}{2}\left[-y \sqrt{\sigma^2x^{2}+\left(\sigma y-\rho\right)^{2}}+\sigma x^{2}\ln\left(\rho-\sigma y+\sqrt{\sigma^2x^{2}+\left(\sigma y-\rho\right)^{2}}\right)\right]$ is not regular, diverging as $\rho\rightarrow\infty$. Nonetheless, in the first term of , we have a finite and well-behaved contribution, which we shall consider in our following analysis while disregarding the non-regular $\Lambda\left(\rho\right)$ contribution. This approximation is valid since the finite part is sufficient to propagate VSR deviations.
Therefore, from the above discussion, we find that the wiggle electric field is then given by $$\left|\tilde{\mathbf{E}}\right|=-\frac{gM}{2\pi}K_{1}\left(Mr\right)\left[1-\frac{\sigma^{2}\left(\hat{\mathbf{r}}.\hat{\mathbf{n}}\right)}{4}\left[yr-x^{2}\ln\left(\sigma\left[r-y\right]\right)\right]\right].\label{eq:2.12}$$ The complete expression for the electric field can be rewritten in polar coordinates, so that $\left(\hat{\mathbf{r}}.\hat{\mathbf{n}}\right)=\sin\theta$. Thus, we find that it now reads $$\left|\tilde{\mathbf{E}}\right|=-\frac{gM}{2\pi}K_{1}\left(Mr\right)\left[1-\frac{\sigma^{2}r^{2}\sin\theta}{4}\left(\sin\theta-\cos^{2}\theta\ln\left[\sigma r\left(1-\sin\theta\right)\right]\right)\right]. \label{eq:2.10}$$
By means of illustration, let us consider a fixed angle $\theta=\pi/2$, so that we can examine the electric field short distance behaviour. With these considerations, we find $$\left|\tilde{\mathbf{E}}\right|=-\frac{gM}{2\pi}K_{1}\left(Mr\right)\left[1-\frac{\sigma^{2}r^{2}}{4}\right]\simeq -\frac{g}{2\pi}\left[\frac{1}{r}-\frac{\sigma^{2}}{4}r\right]. \label{eq:2.18}$$ Hence, we see that the electric field $\left|\tilde{\mathbf{E}}\right|$ at the SIM$(1)$ MCS theory is still non-regular at the origin, as $r\rightarrow0$, due to the usual MCS part. Nonetheless, it is worth of mention that the SIM$(1)$–VSR contribution already gives a well-behaved and regular contribution. We will observe further this positive consequence of VSR acting as a regulator of singular points when computing the interparticle potential (see Sec.\[sec:3.2\]).
Electrostatic energy {#sec:3.1}
--------------------
Is of our interest to proceed and compute the total amount of energy stored in the electrostatic field of a pointlike charge, $U=\int d^{2}xT_{0}^{0}$. The energy-momentum tensor can be evaluated as usual $T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{L}\right)}{\delta g^{\mu\nu}}$. However, notice that the Chern-Simons contribution, $\int dx\varepsilon^{\mu\nu\lambda}A_{\mu}\tilde{F}_{\nu\lambda}$, is already coordinate invariant [@ref42], without additional metric factors; so that the CS mass term does not contribute to $T_{\mu\nu}$ (as expected from a topological term). Hence, we find in our case, that the energy-momentum tensor is simply given as $$T_{\nu}^{\mu}=-\tilde{F}^{\mu\lambda}\tilde{F}_{\nu\lambda}+\frac{\delta_{\nu}^{\mu}}{4}\tilde{F}_{\sigma\lambda}\tilde{F}^{\sigma\lambda}.$$ So, in the electrostatic limit we find $T_{0}^{0}=\frac{1}{2}\left|\tilde{\mathbf{E}}\right|^{2}$. Thus, by using the solution we have that $$U =\frac{g^{2}M^{2}}{8\pi^{2}}\int rdr\left(K_{1}\left(Mr\right)\right)^{2}\int_{0}^{2\pi}d\theta\left[1-\frac{\sigma^{2}r^{2}\sin\theta}{4}\left[\sin\theta-\cos^{2}\theta\ln\left[\sigma r\left(1-\sin\theta\right)\right]\right]\right]^{2} .$$ The angular integration can be computed by means of standard results, so that we get $$\begin{aligned}
U & =\frac{g^{2}M^{2}}{8\pi}\int r^{1-\varepsilon}dr\left(K_{1}\left(Mr\right)\right)^{2}\biggl[2-\frac{2}{3}\sigma^{2}r^{2}\nonumber \\
&+\frac{\sigma^{4}r^{4}}{9216}\biggl( 469-60\ln2 +72\left(\ln \sigma r\right)^{2}+12\left(5-12\ln2\right)\ln \sigma r\biggr)\biggr]\end{aligned}$$ where we have introduced into the numerator a $r^{-\varepsilon}$ factor, as $\varepsilon \rightarrow 0$, so that we can compute the radial integration exactly. A straightforward computation of the remaining integration results into the following expression for the field energy $$\begin{aligned}
U &=\frac{g^{2}}{8\pi}\biggl[-\frac{2}{\varepsilon}-\left(1+2\gamma+2\ln\frac{M}{2}\right) -\frac{4\sigma^{2}}{9M^{2}}\nonumber \\
&+\frac{\sigma^{4}}{72000M^{4}}\biggl(6922+90\gamma\left(-21+10\gamma\right)-900\left(\ln2\right)^{2}+90\ln \left( \frac{M}{\sigma}\right)\left(-21+20\gamma+10\ln \left( \frac{M}{\sigma}\right)\right)\biggr)\biggr].\label{eq:2.11}\end{aligned}$$
We thus find a regularized divergence in the first term of the field energy ; moreover, we clearly see that this divergent term is inherent from the usual MCS theory (a similar fact is also present in the Maxwell theory at $(3+1)$-dims.). So, in order to compare the field energy between the MCS and SIM$(1)$–VSR MCS theories, we shall consider only the finite contribution from the energy expression . First, for the VSR parameter $\sigma=0$, we find the usual MCS (finite) contribution $$\begin{aligned}
U^{MCS} & =-\frac{g^{2}}{8\pi}\biggl[1+2\gamma+2\ln\frac{m}{2}\biggr]\end{aligned}$$ while, the SIM$(1)$–VSR (finite) contribution $U^{VSR}$ follows by taking $m=0$ in , i.e. $M=\sigma$.
We can easily compute and find $U^{MCS}$ has one zero point in $m=0.681085$, while $U^{VSR}$ has one zero points in $m_{1}=0.567385$. So the VSR-modified MCS contribution has a shorter range of positivity than the usual MCS contribution. This is depicted in the Figure \[fig1\].
Static potential {#sec:3.2}
----------------
In this last part of our analysis we will compute the VSR contribution for the static potential energy $V$ between pointlike sources. This study is well motivated since it is usually chosen to describe bound states of particle-antiparticle pairs. Moreover, we will show that the VSR-effects can be chosen conveniently so that the potential is well-behaved and regular. A suitable framework to compute the potential is found to be in terms of physical gauge-invariant objects [@ref40; @ref41]. Let us start by defining the vector gauge-invariant field by $$\mathcal{A}_{\mu}\left(x\right)=A_{\mu}\left(x\right)-\partial_{\mu}\int_{\mathcal{C}_{\xi x}}dz^{\lambda}A_{\lambda}\left(z\right),\label{eq:3.1}$$ where the contour $\mathcal{C}_{\xi x}$ is chosen such as a spacelike path from the point $\xi$ and $x$, on a fixed slice time. Without loss of generality, we consider here a straight path $z_{i}=\xi_{i}+\zeta\left(x-\xi\right)_{i}$ parametrized by $\zeta$ ($0\leq\zeta\leq1$); besides, we can take by simplicity the fixed (reference) point to be $\xi_{i}=0$. This construction for a gauge-invariant variable is, in fact, closely related to the Poincaré gauge conditions $A_{0}\approx0$ and $\int_{\mathcal{C} }dz^{\lambda}A_{\lambda}\approx0$.
Within our interest, we can work out the expression under the above consideration, and after some manipulation, we find that its temporal component reads [@ref40] $$\mathcal{A}_{0}\left(t,\mathbf{r}\right)=\int_{0}^{1}d\zeta x^{i}E_{i}\left(t,\zeta\mathbf{r}\right). \label{eq:3.2}$$
A remark is now in place. On one hand, the interaction energy $V$ of a quantum mechanical system is usually computed by means of a perturbative analysis, i.e. $\left\langle H\right\rangle _{\Omega}=\left\langle H\right\rangle _{0}+V$, where the complete Hamiltonian is obtained by a canonical analysis following Dirac’s procedure. Moreover, in this case one have Dirac’s gauge-invariant fermion–antifermion physical state $\left|\Omega\right\rangle \equiv\left|\overline{\Psi}\left(\mathbf{0}\right)\Psi\left(\mathbf{L}\right)\right\rangle $. On the other hand, instead, we may equally consider the gauge-invariant field in as to provide an equivalent but rather simple framework to compute the expression for the potential $V$ [@ref41].
In particular, we can consider the scenario of a pair of static pointlike (opposite) charges, i.e. $J^{0}\left(t,\mathbf{r}\right)=g\left[\delta^{\left(3\right)}\left(\mathbf{r}\right)-\delta^{\left(3\right)}\left(\mathbf{r}-\mathbf{L}\right)\right]$, where $L=\left|\vec{x}-\vec{y}\right|$. In this case, the potential is then defined as $$V=g\left[\mathcal{A}_{0}\left(\mathbf{0}\right)-\mathcal{A}_{0}\left(\mathbf{L}\right)\right]\label{eq:3.7}$$ Hence, in order to compute first the field $\mathcal{A}_{0}$ from we take the electric field solution Eq.. After some straightforward manipulation, we get the following expression $$\begin{aligned}
\mathcal{A}_{0}\left(t,\mathbf{r}\right)&=\frac{gMr}{2\pi}\int_{0}^{1}d\zeta K_{1}\left(\zeta Mr\right)\left[1-\zeta^{2}\frac{\sigma^{2}r^{2}}{4}\sin\theta\left[\sin\theta-\cos^{2}\theta\ln\left(\zeta \sigma r\left[1-\sin\theta\right]\right)\right]\right], \nonumber \\
& =\frac{g}{2\pi}\int_{0}^{Mr}dwK_{1}\left(w\right)\left[1+w^{2}\left[a^{2}\ln w-b^{2}\right]\right], \label{eq:3.10}\end{aligned}$$ where we have made a change of variables $w=Mr\zeta$ and defined by simplicity $$\begin{aligned}
a^{2} & =\frac{\sigma^{2}}{4M^{2}}\sin\theta\cos^{2}\theta ,\label{eq:3.11}\\
b^{2} & =\frac{\sigma^{2}}{4M^{2}}\sin\theta\left[\sin\theta+\cos^{2}\theta\ln\left(\frac{M}{\sigma}\frac{1}{1-\sin\theta}\right)\right]. \label{eq:3.12}\end{aligned}$$ The integration in can be readily computed, and the complete expression for the gauge-invariant field reads $$\begin{aligned}
\mathcal{A}_{0}\left(t,\mathbf{r}\right) & =-\frac{g}{2\pi}\biggl[\left(1+2a^{2}\right)\left.K_{0}\left(w\right)\right|_{0}^{Mr}+a^{2}\left(Mr\right)K_{1}\left(Mr\right)\nonumber \\
&+\left(Mr\right)\left(\ln\left(Mr\right)a^{2}-b^{2}\right)\left[\left(Mr\right)K_{0}\left(Mr\right)+2K_{1}\left(Mr\right)\right]\biggr]. \label{eq:3.3}\end{aligned}$$ It is worth noticing the singular behaviour of $\left.K_{0}\left(w\right)\right|_{0}^{Mr}$ on . Since the expansion of $K_{0}\left(w\right)$ for $w\rightarrow0$ goes as $K_{0}\left(w\right) \sim -\ln w$, we thus see that the lower limit from the first term is not regular. This is indeed the case in the usual MCS theory, where such a term is usually disregarded. Surprisingly, we see that the novel coefficient (VSR dependent) of this term can be chosen conveniently in such a way that this divergence is removed. Hence, for the case when the identity for coefficient holds $1+2a^{2}=0$, it yields for $$\begin{aligned}
\mathcal{A}_{0}\left(t,\mathbf{r}\right) =\frac{gMr}{4\pi}\biggl[K_{1}\left(Mr\right)+ \left(\ln\left(Mr\right)+2b^{2}\right)\left[\left(Mr\right)K_{0}\left(Mr\right)+2K_{1}\left(Mr\right)\right]\biggr]\label{eq:3.4}\end{aligned}$$
Otherwise, we can conceive this choice for the coefficient as if we are taking the following value for the VSR parameter $$\sigma^{2}=-\frac{2m^{2}}{2+\sin\theta\cos^{2}\theta}, \label{eq:3.5}$$ where we can think that such relation holds for a fixed value of $\theta = \sin^{-1} \left(\hat{\mathbf{r}}.\hat{\mathbf{n}}\right)$. In this case, it also follows that $$\begin{aligned}
b^{2} =-\frac{1}{2}\left[\tan\theta\sec\theta+\ln\left(\frac{M}{\sigma}\frac{1}{1-\sin\theta}\right)\right].\label{eq:3.6}\end{aligned}$$ Besides, the missing piece to evaluate the potential $V$ is obtained by taking the limit $r\rightarrow0$ in , $\mathcal{A}_{0}\left(\mathbf{0}\right) =\frac{g}{4\pi}\biggl[4b^{2}+1\bigg]$. Finally, under the above considerations and by collecting the results and substituting them back into , we find for the potential the following result $$\begin{aligned}
V^{VSR}&=-\frac{g^{2}}{4\pi}\bigg[ \left(ML\right)K_{1}\left(ML\right)-1-4b^{2}\nonumber \\
&+\left(ML\right)\left(\ln\left(ML\right)+2b^{2}\right)\left[\left(ML\right)K_{0}\left(ML\right)+2K_{1}\left(ML\right)\right]\bigg]. \label{eq:3.8}\end{aligned}$$ At last, we see that the VSR deformed expression shows a significant departure from the usual behaviour of the MCS theory, see Figure \[fig2\]. By means of illustration, we can consider the short distance regime of the potential , i.e. $ML\ll 1$, this results into the simple (confining) expression $$\begin{aligned}
V^{VSR} =-\frac{g^{2}}{2\pi}\biggl[\ln\left(M\left|\vec{x}-\vec{y}\right|\right)+\mathcal{O}\left(M^{2}L^{2}\right)\biggr].\label{3.9}\end{aligned}$$
At first sight, this simplified expression might looks exactly the same as the one obtained in the usual MCS theory, since if we consider the short distance regime we have $K_{0}\left(w\right) \sim -\ln w$ (see for $\sigma =0$). However, notice two major differences: one, the VSR-modified potential is completely regular and finite under the condition $1+2a^{2}=0$, i.e., we have removed the term $K_{0}\left(0\right)$; second, the effective mass $M^2=\sigma^2+m^2$ is shifted from the usual MCS parameter $m$. At last, since the VSR deformed potential displays a confining behaviour at short distance (i.e. $V^{VSR} \rightarrow \infty$ as $L \rightarrow 0$), it can used to describe stable bound states of particle-antiparticle pairs.
Concluding remarks {#sec4}
==================
In this paper we have studied a VSR inspired modification of Maxwell-Chern-Simons electrodynamics. The analysis consisted in formulating a SIM$(1)$–VSR topologically electrodynamics, with the expectation that the nonlocal effects would contribute not only as massive contributions but rather in a significant way showing a distinct departure from the usual MCS theory.
We started with a brief construction of the SIM$(1)$ Abelian gauge symmetry. Hence, with a proper definition for the wiggle field strength we proposed a SIM$(1)$–VSR MCS theory. By adding an electrostatic source, we have determined the VSR-modified solution for the electric field. In particular, we showed that at short distances, although the usual MCS contribution is still singular, as $r \rightarrow 0$, the VSR-effects give a finite contribution in this case.
Next, the electrostatic field energy has been computed, and was used in order to compare the VSR contributions in face of the usual MCS result. At last, we have made use of the gauge-invariant formalism in order to compute the static potential between opposite charges. Surprisingly, we found that VSR-effects contribute so that the usual (MCS) singular contribution for the potential can be suitably removed for a particular choice of the VSR parameter. Hence, in addition to its regular form, the complete expression for the VSR modified (confining) potential shows a prominent and health departure from the MCS theory as shown in Fig.\[fig2\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
R.B. thankfully acknowledges FAPESP for support, Project No. 2011/20653-3.
[99]{} \#1\#2[[\#2](http://eudml.org/#1)]{} \#1\#2[[\#2](http://dx.doi.org/#1)]{} \#1\#2[[arXiv:\#1 \[\#2\]](http://arxiv.org/abs/#1)]{} \#1[[arXiv:\#1](http://arxiv.org/abs/#1)]{}
E. W. Kolb and M. S. Turner, “The Early Universe,” Front. Phys. [**69**]{}, Addison-Wesley Publishing (1990). S. Hossenfelder, “Minimal Length Scale Scenarios for Quantum Gravity,” , . D. Amati, M. Ciafaloni and G. Veneziano, “Superstring collisions at planckian energies,” . M. Maggiore, “A Generalized uncertainty principle in quantum gravity,” , . A. F. Ali, S. Das and E. C. Vagenas, “Discreteness of Space from the Generalized Uncertainty Principle,” , . J. Collins, A. Perez, D. Sudarsky, L. Urrutia and H. Vucetich, “Lorentz invariance and quantum gravity: an additional fine-tuning problem?,” , . D. Mattingly, “Modern tests of Lorentz invariance,” , . D. Colladay and V. A. Kostelecky, “Lorentz violating extension of the standard model,” , . M. R. Douglas and N. Nekrasov, “Noncommutative field theory”, , . G. Amelino-Camelia, “Relativity in space-times with short distance structure ...,” , . J. Magueijo and L. Smolin, “Lorentz invariance with an invariant energy scale,” , . A. G. Cohen and S. L. Glashow, “Very special relativity,” , . A. G. Cohen and S. L. Glashow, “A Lorentz-Violating Origin of Neutrino Mass?,” . M. M. Sheikh-Jabbari and A. Tureanu, “Realization of Cohen-Glashow Very Special Relativity on NC Space-Time,” , . J. Fan, W. D. Goldberger and W. Skiba, “Spin dependent masses and Sim(2) symmetry,” , . R. V. Maluf, J. E. G. Silva, W. T. Cruz and C. A. S. Almeida, “Dirac equation in very special relativity for hydrogen atom,” , . S. Upadhyay, “Reducible Gauge Theories in Very Special Relativity,” . E. Alvarez and R. Vidal, “Very Special (de Sitter) Relativity,” , . G. W. Gibbons, J. Gomis and C. N. Pope, “General very special relativity is Finsler geometry,” , . D. V. Ahluwalia and S. P. Horvath, “Very special relativity as relativity of dark matter: The Elko connection,” , . S. Cheon, C. Lee and S. J. Lee, “SIM(2)-invariant Modifications of Electrodynamic Theory,” , . J. Alfaro and V. O. Rivelles, “Non Abelian Fields in Very Special Relativity,” , . J. Vohánka and M. Faizal, “Super-Yang-Mills Theory in SIM(1) Superspace,” , . A. C. Nayak, R. K. Verma and P. Jain, “Effect of VSR invariant Chern-Simon Lagrangian on photon polarization,” , . J. Vohánka and M. Faizal, “Chern-Simons Theory in SIM(1) Superspace,” . R. Bufalo, “Born–Infeld electrodynamics in very special relativity,” , . A. A. Andrianov, P. Giacconi and R. Soldati, “Lorentz and CPT violations from Chern-Simons modifications of QED,” , . A. J. Hariton and R. Lehnert, “Spacetime symmetries of the Lorentz-violating Maxwell-Chern-Simons model,” , . A. O. Barut and C. Fronsdal, “On Non-Compact Groups. II. Representations of the 2 + 1 Lorentz Group,” . P. Gaete and I. Schmidt, “Remarks on screening in a gauge invariant formalism,” , . P. Gaete, “Static potential in a topologically massive Born–Infeld theory,” , . S. Deser, R. Jackiw and S. Templeton, “Topologically Massive Gauge Theories,” , . P. Gaete, “On gauge invariant variables in QED,” . P. Gaete, “Remarks on gauge invariant variables and interaction energy in QED,” , .
[^1]: E-mail: <rbufalo@ift.unesp.br>
[^2]: A detailed account of the SIM$(1)$ subgroup can be found in Ref. [@ref51]
[^3]: Notice however that although VSR engender a nonzero mass, it preserves the number of polarization states [@ref4].
|
---
abstract: |
I show that neutral lithium couples strongly to the cosmic microwave background (CMB) through its 6708Å resonant transition after it recombines at $z\sim 500$. At observed wavelengths of $\la 335\mu$m, the CMB anisotropies are significantly altered since the optical depth for resonant scattering by neutral lithium is substantial, $\tau_{\rm LiI}\sim
0.5$. The scattering would suppress the original anisotropies, but will generate strong new anisotropies in the CMB temperature and polarization on sub-degree scales ($\ell \ga 100$). Observations at different wavelengths in this spectral regime can probe different thin slices of the early universe. The anisotropy noise contributed by continuum radiation from foreground far–infrared sources could be taken out by subtracting maps at slightly different wavelengths. Detection of the above effects can be used to study structure at $z\la 500$ and to constrain the primordial abundance and recombination history of lithium.
author:
- Abraham Loeb
title: Probing the Universe After Cosmological Recombination Through the Effect of Neutral Lithium on the Microwave Background Anisotropies
---
Introduction
============
The dynamical coupling between the baryons and the Cosmic Microwave Background (CMB) during the epoch of cosmological recombination ($z\sim
10^3$) is commonly assumed to be mediated only through Thomson scattering (Peebles & Yu 1970). This coupling becomes ineffective when hydrogen recombines, at around the same time when the universe becomes transparent, and hence it leaves a clear signature on the temperature anisotropies of the CMB (White, Scott, & Silk 1994, and references therein). Forthcoming satellite missions, such as MAP[^1] in 2001 or Planck[^2] in 2007, offer the sensitivity to probe deviations from this simple model at the percent level (Scott 1999; Yu, Spergel, & Ostriker 2001).
In this [*Letter*]{}, I examine the potential significance of the coupling between the cosmic gas and the CMB due to resonant line transitions of neutral atoms. For such transitions, the integral of the absorption cross-section over frequency, $$\int \sigma(\nu) d\nu = f_{12} \left({\pi e^2\over m_e c}\right),
\label{eq:int_s}$$ is typically many orders of magnitude larger than for Thomson scattering, where $f_{12}$ is the absorption oscillator strength (Rybicki & Lightman 1979). For example, the transition of hydrogen provides an average cross-section which is seven orders of magnitude larger than the Thomson value, when averaged over a frequency band as wide as the resonant frequency itself \[$(\Delta \nu/\nu) \sim 1$\]. Since the abundance of atoms in excited states is highly suppressed by large Boltzmann factors at the recombination redshift, we consider only transitions from the ground state. For helium, these transitions correspond to photon frequencies that are too far on the Wien tail of the CMB spectrum and only couple to a negligible fraction of the radiation energy density. Hence, we focus our discussion on hydrogen and lithium.
In §2, we derive the resonant–line opacity and drag force for atoms that move relative to a blackbody radiation field. In §3, we consider the cosmological implications of these results for hydrogen and lithium. Finally, §4 summarizes the main conclusions of this work. Throughout the paper, we adopt the density parameters $\Omega_0\approx 0.3$ for matter and $\Omega_b = 0.04$ for baryons, and a Hubble constant $H_0\approx 70~{\rm km~s^{-1}~Mpc^{-1}}$. The effect of the cosmological constant can be neglected at the redshifts of interest here.
Drag Force and Opacity Due to a Resonant Line
=============================================
Consider a single atom moving at a non-relativistic peculiar velocity ${\bf
v}$ relative to an isotropic blackbody radiation field of temperature $T$. Because of its motion, the atom sees a dipole anisotropy in the radiation temperature, $T_a(\theta)=T [1+(v/c)\cos \theta]$, where $\theta$ is the angle relative to its direction of motion (Peebles 1993). We assume that the atom couples to the radiation through a resonant line of frequency $\nu_0$ with $(h\nu_0/kT)\gg 1$. Hence, the line resonates in the Wien tail of the spectrum, for which the radiation energy density per unit frequency (in erg cm$^{-3}$ Hz$^{-1}$) is given by, $$u_\nu(\theta)={8\pi h\nu^3\over c^3}\exp\left\{-{h\nu \over
kT_a(\theta)}\right\}\approx {8\pi h\nu^3\over c^3}\exp\left\{-{h\nu \over
kT}\right\}\left(1+ {h\nu\over kT}{v\over c}\cos \theta\right),$$ where we have used the fact that $[(h\nu/kT)(v/c)\cos \theta]\ll 1$. Absorption of a resonant photon by the atom is followed by isotropic emission in the atom’s rest frame, and so the net drag force (momentum transfer per unit time) is given by, $$F=-\int_0^{\infty} d\nu ~~{1\over 2} \int_{-1}^{1} d \cos \theta~~
\sigma(\nu)u_\nu(\theta) \times \cos \theta~~~ .
\label{eq:int_F}$$ Since the absorption cross-section is sharply peaked around $\nu_0$, we may use equation (\[eq:int\_s\]) to get, $${\bf F}=-\left({8\pi^2 h^2 f_{12} \nu_0^4\over 3 c^3 kT}{e^2 \over m_e c^2}
\exp\left\{-{h\nu_0\over kT}\right\} \right) {\bf v} ~~~ .
\label{eq:force}$$
For an ensemble of atoms of density $n_{\rm a}$, which are all moving at the same peculiar velocity ${\bf v}$, the force per unit volume is given by $n_{\rm a} {\bf F}$. If the resonant atoms are embedded within a hydrogen plasma of total (free$+$bound) proton density $n_p$ and are strongly coupled to the plasma through binary particle collisions, then the cosmological equation of motion for the gas as a whole reads, $$\left(m_p n_p + m_{\rm a} n_{\rm a}\right) \left({d{\bf v}\over dt} +H{\bf
v}\right) = n_{\rm a} {\bf F} ,
\label{eq:cosm}$$ or equivalently, $${dv\over dt} +Hv= -{v\over t_{\rm a}},
\label{eq:acc}$$ where $H=({\dot{a}/a})$ is the Hubble expansion rate, and $t_{\rm a}$ is the characteristic time over which the peculiar velocity of the gas is damped due to the drag force on the resonant atoms, $$t_{\rm a}(t)\equiv {3 m_p \lambda_0^4 \over 8 \pi^2h^2 c f_{12}} {m_e
c^2\over e^2}{kT \over X_{\rm a}} \exp\left\{{h\nu_0\over kT}\right\} .
\label{eq:t_a}$$ Here, $\lambda_0=c/\nu_0$ is the resonant wavelength, $m_p$ is the proton mass, and $X_{\rm a} \equiv n_{\rm a}/(n_p+ A_{\rm a} n_{\rm a})$, where $A_{\rm a}=m_{\rm a}/m_p$ is the atomic weight of the resonant atoms. In the cosmological context, $T=2.725~{\rm K}\times (1+z)$ is the CMB temperature (Mather et al. 1999). Both $T$ and $X_{\rm a}$ are functions of cosmic time, and so equation (\[eq:acc\]) admits the solution, $v=[{v_0/a(t)}] \exp\left\{-\int {dt /t_{\rm a}(t)}\right\}$, where $v_0=const$ and $a(t)$ is the cosmological scale factor. In general, the right-hand-side of equation (\[eq:acc\]) should include other terms which source the peculiar velocity field, such as gravitational or pressure forces due to density inhomogeneities. The significance of the drag force can be calibrated in terms of the Hubble time, $H^{-1}(z)$, through the product $Ht_{\rm a}$. For the redshifts of interest, $H(z)\approx
\sqrt\Omega_0 H_0 (1+z)^{3/2}$.
The assumption of an isotropic radiation background is only valid if the moving fluid element is [*optically-thin*]{} to resonant scattering. In the opposite regime of high opacity, the background photons are isotropized locally through multiple scatterings and exert a reduced drag force on an atom embedded deep inside the moving fluid element. The Sobolev optical depth for resonant scatterings in an expanding and nearly uniform universe is given by (Sobolev 1946; Dell’Antonio & Rybicki 1993), $$\tau_{\rm a}(z)= f_{12} {\pi e^2\over m_e c} {\lambda_0 n_{\rm a}(z) \over
H(z)}= {A_{21}(g_2/g_1) \lambda_0^3 n_{\rm a}(z)\over 8\pi H(z)}~~,
\label{eq:tau_a}$$ where $A_{21}$ is the transition probability per unit time for spontaneous emission between the two energy levels (in s$^{-1}$), and $(g_2/g_1)$ is the ratio between the statistical weights of the excited and ground states. The optical depth is dominated by the velocity gradient of the Hubble flow, because gravitationally–induced peculiar velocities perturb this flow only slightly at the early cosmic times of interest here.
For $\tau_{\rm a} \gg 1$, the flux of resonant photons inside the moving fluid element is reduced by a factor of $\sim \tau_{\rm a}$ (Rybicki & Hummer 1978), and so the drag force in equation (\[eq:int\_F\]) is lowered by the same factor. This reduction has dramatic consequences for the resonant transition of neutral hydrogen, as we show next.
Implications For Cosmological Recombination
===========================================
Hydrogen
--------
For the $1S$–$2P$ transition of hydrogen, $\lambda_0=1216$Å , $A_{21}=6.262\times 10^8~{\rm s}^{-1}$, $f_{12}=0.4162$, and $(g_2/g_1)=3$, yielding an exceedingly large Sobolev optical–depth, $\tau_{\rm H}=
7.5\times 10^8 X_{\rm H} [(1+z)/10^3]^{3/2}$, where $X_{\rm H}$ denotes the neutral fraction of hydrogen. The thermally–broadened line has an effective cross–section of $\langle\sigma\rangle\sim 3\times 10^{-17}~{\rm
cm^2}$ (Peebles 1993) and a corresponding mean-free-path of $1/n_{\rm
H}\langle \sigma\rangle \sim 10^{14}~{\rm cm}$, at $z\sim 10^3$. Only on smaller scales can viscosity damp velocity gradients effectively. On the much larger scales where the CMB anisotropies are measurable, the drag time in equation (\[eq:t\_a\]) is lengthened by a factor of $\sim
\tau_{\rm H}$, which makes it very much longer than the Hubble time. The situation is similar for deuterium. Balmer line transitions of hydrogen have a small optical depth, but the occupation probability of their lower level is suppressed by an exceedingly large Boltzmann factor, $\sim
\exp\{-4.35\times 10^4/(1+z)\}$.
The resulting drag force is small, even when including the enhancement in the flux beyond the blackbody value, due to the photons released during recombination ($\sim 1$ photon per hydrogen atom; see Dell’Antonio & Rybicki 1993 for the distorted spectrum).
Lithium
-------
### Drag Force and Opacity
Lithium has a closed $n=1$ shell with two electrons, and one electron outside this shell. The transition between the ground state ($2S$) and the first excited state ($2P$) has $\lambda_0=6708$Å and $A_{21}=3.69\times
10^7~{\rm s^{-1}}$, with $f_{12}=0.247$ and $(g_2/g_1)=1$ for the $2^2S$–$2^2P^0_{1/2}$ transition, and $f_{12}=0.494$, $(g_2/g_1)=2$ for the $2^2S$–$2^2P^0_{3/2}$ transition (Radzig & Smirnov 1985; Yan & Drake 1995). We therefore get from equations (\[eq:t\_a\]) and (\[eq:tau\_a\]), $$H t_{\rm LiI}= 4.96\times 10^{-4} \left({1+z\over 10^3}\right)^{5/2}
\left({X_{\rm Li I} \over 3.8\times 10^{-10}}\right)^{-1}
\exp\left\{{7.89\times 10^3\over 1+z}\right\} ,
\label{eq:t_Li}$$ and $$\tau_{\rm LiI}=2.82 \left({X_{\rm Li I} \over 3.8\times 10^{-10}}\right)
\left({1+z\over 10^3}\right)^{3/2} .
\label{eq:tau_Li}$$
Figure 1 depicts the redshift dependence of the drag time in equation (\[eq:t\_Li\]) and compares it to the drag time due to Thomson scattering in the recombining plasma, $$H t_{\rm e}={3m_p c H\over 4\sigma_T a T^4 X_e}=
4.92\times 10^{-3}
X_{\rm e}^{-1} \left({1+z\over 10^3}\right)^{-5/2}~~,$$ where $X_{\rm e}=(n_{\rm e}/n_p)= n_{\rm e}/(n_{\rm e}+n_{\rm H})$ is the ionization fraction of hydrogen. We calculated $X_{\rm e}(z)$ from the updated version of the standard recombination model (Seager, Sasselov, & Scott 2000). Despite the low abundance of lithium, its resonant drag force is not much lower than the Thomson drag force at $z\sim 500$, when roughly half of the lithium atoms are expected to have recombined (Palla, Galli, & Silk 1995; Stancil, Lepp, & Dalgarno 1996, 1998).
The net effect of the lithium drag is highly sensitive to the recombination history of lithium and hydrogen. Previous discussions (e.g., Stancil et al. 1996, 1998) did not include the delaying effect of photons on the recombination history of hydrogen (Peebles 1993), and therefore underestimated the free electron fraction and correspondingly the neutral lithium abundance at $z\la 10^3$. Another process with an opposite sign that was omitted, involves the excitation of neutral lithium from its ground level by the CMB radiation field. For this effect, it is important to take account of the Lyman and Balmer series distortions of the CMB spectrum (Dell’Antonio & Rybicki 1993), which redshift into resonance at lithium recombination. In fact, the lithium chemistry in the early universe might provide a way for inferring the existence of these distortions. Finally, the recombination history of $^7$Be should also be followed since a substantial fraction of the $^7$Li forms through electron-capture in the $^7$Be produced by Big-Bang nucleosynthesis (T. Walker, private communication). Since the half-life of neutral $^7$Be is only 53 days, this conversion can be regarded as instanteneous on the cosmological timescale, as soon as $^7$Be recombines. Careful calculations of the neutral lithium fraction as a function of redshift are required in order to quantify the imprint of the lithium drag on the CMB anisotropies (Dalgarno, Loeb, & Stancil 2001).
The lithium chemistry could also be affected by a large drift velocity between the Li I fluid and the hydrogen plasma. To examine this effect, we consider the equation of motion for the Li I fluid (Burgers 1969), $$m_{\rm LiI}
\left( {d{\bf v}_{\rm LiI}\over dt} + H{\bf v}_{\rm LiI}\right)=
-m_{\rm LiI} {{\bf v}_{\rm LiI}\over X_{\rm LiI} t_{\rm LiI}}
+ m_{\rm p} W_{\rm LiI}
\left({\bf v} - {\bf v}_{\rm LiI}\right)~~.
\label{eq:Li_eom}$$ Here $W_{\rm LiI}=\Sigma_k \mu_{\rm k:Li} W_{\rm k:Li}$ is the collision rate of a Li atom with the gas, where $W_{\rm k:LiI}=n_{\rm k}\langle
\sigma_{\rm k:LiI} v\rangle$, and $\mu_{\rm k:Li}=[m_{\rm k}m_{\rm
Li}/m_{\rm p}(m_{\rm k}+m_{\rm Li})]$. The index ${\rm k} $ runs over all other particle species in the gas which are assumed to move at a common bulk velocity ${\bf v}$ because they are not subject to the resonant drag force. The geometric cross-section of Li I atoms, $\sigma_{\rm LiI} \sim
10^{-15}~{\rm cm^2}$, couples them to the rest of the gas on a time scale $\sim 10^7~{\rm s}$, much shorter than the Hubble time, $H^{-1}\sim 3
\times 10^{13}~{\rm s}$ at $z\sim 10^3$. Hence, we may ignore the terms on the left-hand-side in equation (\[eq:Li\_eom\]), as well as the perturbed gravitational force which is of the same order. The remaining terms give the solution, $${\left(v -v_{\rm LiI}\right)\over v}={1\over 1+ Y}$$ where $Y= W_{\rm LiI} t_{\rm LiI} X_{\rm LiI}/A_{\rm Li}$. Substituting $A_{\rm Li}=7$, we find that $W_{\rm LiI} \approx
10^{-7}[(1+z)/10^3]^{3.5}~{\rm s^{-1}}$, and that $Y\gg 1$ at $z\la 400$, after lithium recombination. At higher redshifts, the assumption that $v_{\rm LiI}=v$ seems not to apply and one may wonder whether the drag time in equation (\[eq:acc\]) should be multiplied by a factor of $(v/v_{\rm LiI})=(1+Y)/Y > 1$, since the drag force acts on ${\bf v}_{\rm Li I}$ and not on ${\bf v}$. In this context it is crucial to realize that the Li I nuclei do not maintain their identity as neutral atoms; in fact, they continuously form and get ionized on a timescale much shorter than a Hubble time. The cross-section for Coulomb collisions of Li$^+$ is far greater than that of Li I, $\sigma_{\rm Li^+} =2\sqrt\pi
e^4\ln \Lambda/(kT)^2 = 2.7 \times 10^{-11} [(1+z)/10^3]^{-2}~{\rm cm^2}$, where $\ln \Lambda\approx 20$ is the Coulomb logarithm. The fraction of time spent by a Li nucleus in an ionized state is the ionization fraction, $f_{+}=X_{\rm Li^+}/(X_{\rm LiI}+X_{\rm Li^+})$, and so $\sigma_{\rm LiI}$ should be replaced by $[f_+\sigma_{\rm Li^+}+ (1-f_+)\sigma_{\rm LiI}]$ and the drag force term should be multiplied by $(1-f_+)$ in equation (\[eq:Li\_eom\]). We consequently find that the substantial ionization fraction of the lithium fluid (Stancil et al. 1996, 1998) diminishes its drift velocity relative to the gas at all relevant redshifts, $z\la
10^3$. Hence, non-thermal relative velocities may be ignored in the Li I chemistry.
At $z\la 10^3$, the mean-free-path for resonant scattering by lithium exceeds the relevant spatial scales for the CMB anisotropies ($10^{23}$–$10^{24}~{\rm cm}$), even if the lithium abundance is taken to be somewhat higher than the value, $X_{\rm SBBN}=3.8\times 10^{-10}$, predicted by the latest deuterium measurements and Standard Big-Bang Nucleosynthesis (Burles, Nollett, & Turner 2001; see also Walker et al. 1991; Smith, Kawano, & Malaney 1993). Note that values as high as $X_{\rm Li I}\sim 10^{-8}$ were suggested by models of inhomogeneous Big-Bang nucleosynthesis (Applegate & Hogan 1985; Sale & Mathews 1986; Mathews et al. 1990), and would result in $Ht_{\rm LiI}\la 10$ for velocity gradients on sufficiently small scales with $\tau_{\rm LiI}\la 1$.
### Effects on CMB Anisotropies
If a substantial fraction of lithium recombines by $z\sim 500$ (as predicted by Palla et al. 1995 or Stancil et al. 1996, 1998), then the flux of the original anisotropies will be suppressed by the absorption factor, $\exp (-\tau_{\rm LiI})$, at observed wavelengths below $(6708$Å $\times
500)=335\mu$m, for which $\tau_{\rm LiI}\approx 0.6 (X_{\rm Li I}/X_{\rm
SBBN})$. The resonant scattering effect is different from that caused by Thomson scattering at the epoch of reionization (Hu & White 1997), in that it has a larger optical depth and it occurs within a much thinner shell of gas (for a sufficiently narrow band of observed wavelengths). Consequently, it should induce new first-order anisotropies \[multiplied by $(1-\exp (-\tau_{\rm LiI})$\] due to the coherence of the velocity field in the thin scattering shell. The dominant contribution to the new anisotropies would come from the Doppler effect. In contrast to reionization, there should be no severe cancellations of the Doppler effects from line-of-sight velocity fluctuations at large wavenumbers, because the last scattering surface (or “visibility function”) for the resonance is extremely thin. The increase in amplitude of sub-horizon fluctuations between $z=10^3$ and $z=500$ would enhance the new anisotropies relative to the original ones and introduce a distortion of order unity in the original anisotropy spectrum on these scales. The anisotropy spectrum would be modified at multipole moments $\ell \ga 100$, around and below the angular scale of the first acoustic peak (which is a factor of $\sim \sqrt{6}=2.4$ smaller than the horizon scale at $z\sim
500$), where the original anisotropy amplitude reaches its maximum ($\ell
\approx 220$). Resonant scattering would also result in enhanced polarization anisotropies on the same angular scales (Chandrasekhar 1960). Since the scattering is done by lithium, the anisotropies will reflect the peculiar velocity of the lithium fluid which, as shown above, is not expected to deviate significantly from the velocity of the gas as a whole.
Since the temperature fluctuations are in the Wien tail of the CMB spectrum, they translate to brightness fluctuations (in ${\rm
erg~s^{-1}cm^{-2}sr^{-1}Hz^{-1}}$) of a much larger contrast, $${\Delta B_\nu\over B_\nu}=
\left({d\ln B_\nu \over d\ln T}\right){\Delta T\over T}=
\left({h\nu\over kT}\right){\Delta T\over T}=
15.78 \times \left({500\over 1+z}\right){\Delta T\over T}~~,$$ where $z$ is the redshift being probed and we have substituted $B_\nu(T)\propto \exp(-h\nu/kT)$. Hence the first acoustic peak would correspond to brightness fluctuations of up to $\sim 5\times 10^{-4}$.
At a given observed wavelength, the redshift thickness of the scattering shell is only determined by two contributions: (i) the possibility that a photon at this wavelength was either absorbed by the $2^2S$–$2^2P^0_{3/2}$ transition with $\lambda_0=6707.76$Å or at a slightly lower redshift by the $2^2S$–$2^2P^0_{1/2}$ transition with $\lambda_0=6707.91$Å (Radzig & Smirnov 1985), yielding $[\Delta z/(1+z)] =2.2\times 10^{-5}$; and (ii) the thermal width of the line[^3], $[\Delta z/(1+z)] = (\Delta
\lambda/\lambda)_{\rm th} \approx 10^{-5}$. In practice, the minimum thickness would be limited by the band–width of the detector. The maximum band–width and minimum angular resolution should be tuned to match the expected coherence length of the peculiar velocity field at $z\sim 500$, which corresponds to $(\Delta \lambda/\lambda)\sim 0.2$ and $\Delta
\theta\sim 6^{\prime}$ or equivalently $\ell\sim 10^3$.
Observations at different far–infrared (FIR) wavelengths would probe different thin slices of structure in the early universe. The anisotropy pattern would vary gradually as a function of wavelength and “3D tomography” is in principle possible, by which one may probe correlations in the structure along the line-of-sight. The feasibility of this measurement might be severely compromised by the anisotropies of the FIR foreground. Unfortunately, $350\mu$m is just the wavelength where the CMB intensity blends into the extragalactic FIR foreground (see Fig. 1 in Scott et al. 2001). However, the contribution of discrete sources (see Fig. 6 in Knox et al. 2001) could be separated out through observations at shorter wavelengths with a higher angular resolution. Furthermore, continuum foreground emission would result in similar anisotropies for all wavelengths within a band width $(\Delta \lambda/\lambda)\sim 1$, while the CMB anistropies would vary considerably across such a band since its width corresponds to $\Delta z/(1+z)\sim 1$. Subtraction of anisotropy maps at slightly different wavelengths within such a band can therefore be used to isolate out the CMB component (as long as these wavelengths do not overlap with strong emission lines from the foreground sources). For $\ell\la
100$, the anisotropies should have the same power-spectrum as measured by MAP or Planck at much longer wavelengths. The predictable correlations between the temperature and polarization patterns of the CMB and their general statistical properties could be used to further enhance the signal-to-noise ratio (Zaldarriaga & Loeb 2001). Obviously, these difficult observations need to be made through holes in the Galactic dust emission at high Galactic latitudes.
Conclusions
===========
The 6708Å resonant transition of neutral lithium provides a previously unexplored coupling between the baryons and the CMB after cosmological recombination. The lithium opacity is substantial at observed wavelengths $\la 335 [(1+z_{\rm LiI})/500]\mu$m, where $z_{\rm LiI}$ is the redshift at which $\sim 50\%$ of the lithium recombines. At these wavelengths, the CMB anisotropies on sub-degree angular scales would be significantly altered, due to the finite optical depth for resonant scattering, $\tau_{\rm
LiI}\approx 0.5[(1+z_{\rm LiI})/500]^{3/2}$. This scattering would generate new temperature and polarization anisotropies at mutipole moments $\ell \ga 100$. The detection of these anisotropies depends on the prospects for cleaning the expected anisotropies of the FIR foreground (Knox et al. 2001), and may be feasible through the subtraction of anisotropy maps for pairs of slightly different wavelengths.
The relevant wavelength range overlaps with the highest frequency channel of the Planck mission ($352\mu$m) and with the proposed baloon–borne Explorer of Diffuse Galactic Emissions[^4] (EDGE) which will survey 1% of the sky in 10 wavelength bands between $230$–$2000\mu$m with a resolution ranging from $6^\prime$ to $14^\prime$ (see Table 1 in Knox et al. 2001). However, a new instrument design with multiple narrow bands ($\Delta \lambda/\lambda\la 0.1$) at various wavelengths in the range $\lambda=250$–$350\mu$m and with angular resolution $\sim 6^\prime$, is necessary in order to optimize the detection of the lithium signature on the CMB anisotropies.
The effects discussed in this [*Letter*]{} depend critically on the recombination history of lithium, and emphasize the need to explore the related lithium chemistry in more detail in the future.
The author thanks Alex Dalgarno, Daniel Eisenstein, George Rybicki, and Matias Zaldarriaga for useful discussions. This work was supported in part by NASA grants NAG 5-7039, 5-7768, and by NSF grants AST-9900877, AST-0071019.
Applegate, J. H.& Hogan, C. J. 1985, , 31, 3037
Burgers, J. M. 1969, Flow Equations For Composite Gases (New York: Academic)
Burles, S., Nollett, K. M., & Turner, M. S. 2001, ApJL, in press; astro-ph/0010171
Chandrasekhar, S. 1960, Radiative Transfer, (New-York: Dover), pp. 50-53
Dalgarno, A., Loeb, A., & Stancil, P. 2001, in preparation
Dell’Antonio, I. P., & Rybicki, G. B. 1993, in ASP Conf. Ser. 51, Observational Cosmology, , ed. G. Chincarini et al. (San Francisco: ASP), 548
Knox, L, Cooray, A., Eisenstein, D., & Haiman, Z. 2001, ApJ, 550, 7
Mather, J. C., Fixsen, D. J., Shafer, R. A., Mosier, C., & Wilkinson, D. T. 1999, , 512, 511
Mathews, G. J., Alcock, C. R., & Fuller, G. M. 1990, , 349, 449
Hu, W., & White, M. 1997, ApJ, 479, 568
Palla, F., Galli, D., & Silk, J. 1995, , 451, 44
Peebles, P. J. E. 1993, Principles of Physical Cosmology, (Princeton: Princeton Univ. Press), pp. 154-157, 168–176
Peebles, P. J. E. & Yu, J. T. 1970, , 162, 815
Radzig, A. A., & Smirnov, B. M. 1985, Reference Data on Atoms, Molecules, and Ions (New York: Springer), p. 224
Rybicki, G. B., & Hummer, D. G. 1978, ApJ, 219, 654
Rybicki, G. B., & Lightman, A. P. 1979, Radiative Processes in Astrophysics, (New York: Wiley), p. 102
Sale, K. E. & Mathews, G. J. 1986, ApJL, 309, L1
Seager, S., Sasselov, D. D., & Scott, D. 2000, ApJS, 128, 407
Scott, D. 1999, to appear in “Beyond the Desert ‘99’’, edited by H.V. Klapdor-Kleingrothaus; astro-ph/9911325
Scott, D. et al. 2001, to appear in UMass/INAOE conference proceedings on Deep Millimeter Surveys, eds. J. Lowenthal and D. Hughes; astro-ph/0010337
Smith, M. S., Kawano, L. H., & Malaney, R. A. 1993, , 85, 219
Sobolev, V. V. 1946, Moving atmospheres of Stars (Leningrad: Leningrad State University \[in Russian\]; English translation: 1960 (Cambridge: Harvard Univ. Press)
Stancil, P. C., Lepp, S., & Dalgarno, A. 1996, ApJ, 458, 401
—————————————————–. 1998, ApJ, 509, 1
Yan, Z. & Drake, G. W. F. 1995, , 52, 4316
Yu, Q., Spergel, D. N., & Ostriker, J. P. 2001, ApJ, submitted, astro-ph/0103149
Walker, T. P., Steigman, G., Kang, H., Schramm, D. M., & Olive, K. A. 1991, , 376, 51
White, M., Scott, D., & Silk, J. 1994, , 32, 319
Zaldarriaga, M., & Loeb, A. 2001, in preparation
\[Fig1\]
[^1]: http://map.gsfc.nasa.gov
[^2]: http://astro.estec.esa.nl/Planck
[^3]: Since $\tau_{\rm LiI}\la 1$, the photon does not need to redshift away from the line center by more than the line width in order to escape to infinity.
[^4]: http://topweb.gsfc.nasa.gov
|
---
abstract: 'We demonstrate a dual-rail optical Raman memory inside a polarization interferometer; this enables us to store polarization-encoded information at GHz bandwidths in a room-temperature atomic ensemble. By performing full process tomography on the system we measure up to $97\pm 1\%$ process fidelity for the storage and retrieval process. At longer storage times, the process fidelity remains high, despite a loss of efficiency. The fidelity is $86\pm 4\%$ for storage time, which is 5,000 times the pulse duration. Hence high fidelity is combined with a large time-bandwidth product. This high performance, with an experimentally simple setup, demonstrates the suitability of the Raman memory for integration into large-scale quantum networks.'
address: |
$^1$Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom\
$^2$ Centre for Quantum Technologies, National University of Singapore, 117543, Singapore\
$^\ast$ Current address: Department of Physics, ETH Zürich, CH-8093, Zürich, Switzerland\
$^\dag$ Current address: Department of Physics, Royal Holloway, University of London, Egham Hill, Egham TW20 0EX, United Kingdom
author:
- 'D. G. England$^1$, P. S. Michelberger$^1$, T. F. M. Champion$^1$, K. F. Reim$^{1, \ast}$, K. C. Lee$^1$, M. R. Sprague$^1$, X.-M. Jin$^{1,2}$, N. K. Langford$^{1,\dagger}$, W. S. Kolthammer$^{1}$, J. Nunn$^{1}$ and I. A. Walmsley$^1$'
bibliography:
- 'QPT\_Bib.bib'
title: 'High-Fidelity Polarization Storage in a Gigahertz Bandwidth Quantum Memory'
---
Introduction
============
Photons are well-established as carriers of quantum information and their transmission through high-bandwidth fibers or free-space opens the possibility of a global quantum network [@Duan:2001p16886; @Kimble:2008p17348]. To compensate for the effects of photon loss in a fiber network and the inherently probabilistic nature of quantum processes, it is necessary to map quantum information from a ‘flying’ photonic qubit to a stationary one and back again, in a controllable manner. This is the essence of a quantum memory: it must faithfully store, and reproduce, the quantum state of a photonic qubit, including polarization. Key performance benchmarks of a quantum memory are high efficiency, long storage times and large bandwidths. Ultimately the clock-rate of a quantum information protocol will depend on the pulse durations used; a high bandwidth is required to store the temporally short pulses which lead to high processing rates. Also, in order to perform operations while the qubit is in storage, the storage time of the memory must be several times larger than the pulse duration. For this reason, the time-bandwidth product (TBP) is an important metric for a quantum memory. Note that the TBP, which represents the number of clock cycles spanned by the storage time, is distinct from the multimode capacity which represents the number of temporal or spectral bins that can be simultaneously stored. A large TBP is critical for synchronisation tasks, while large multimode capacity is advantageous for some protocols that invoke multiplexing.
Quantum memories have been demonstrated in ultracold atoms [@Liu:2001p13466; @Zhao:2008p17809; @Zhao:2008p17808], cryogenically cooled solids [@Longdell:2005p25; @Chaneliere:2010p17807; @Afzelius:2010p17806] and single atoms in high-finesse optical cavities [@Maitre:1997p13519]. However, if quantum memories are to be used as part of a global quantum network, they will eventually have to operate in remote, unmanned locations, so the apparatus must be simple and robust. A promising candidate for such robust operation is a room-temperature atomic ensemble. Storage times of several milliseconds have been achieved using electromagnetically induced transparency (EIT) in simple vapor cells [@Julsgaard:2004p16892]. The gradient-echo memory (GEM) technique has also been used to great effect in atomic gases, achieving up to 87% readout efficiency utilizing a switched magnetic field gradient [@Hosseini:2009p17793; @Hosseini:2011p17792]. Despite these long storage times, and high efficiencies, the bandwidth of atomic memories is often limited by the narrow linewidth of the atomic transitions, hence precluding a large TBP. However this limitation can be overcome by using a controlled read-in and read-out mechanism based on an off-resonant Raman interaction, which has been proposed [@Nunn:2007p13183] and demonstrated [@Reim:2010p13180] by this group.
The Raman memory protocol is based on a two-photon off-resonant process in an atomic $\Lambda$-level system, with a weak signal and strong control pulse, which maps the electric field of the signal pulse onto a collective excitation in an atomic ensemble known as a [*spin wave*]{}, as shown in figure \[fig:Mem\]. Here the strong control pulse produces a virtual excited state, with a linewidth determined only by the bandwidth of the control pulse. Because the control and signal fields must address the two ground-states separately, the bandwidth of this memory is limited, in practice, by the ground state splitting. Based on this interaction, we have implemented a memory in room-temperature Cesium (Cs) vapor [@Reim:2010p13180; @Reim:2011p13181], which is capable of storing pulses of duration, corresponding to a bandwidth with a maximum efficiency of 30%. The maximum storage time, currently limited by residual magnetic fields, is approximately , which is $10^4$ times longer then the pulse duration. These parameters yield the highest TBP of any quantum memory so far. By attenuating the signal field such that it contained, on average, 1.6 photons per pulse, we demonstrated memory operation on the single-photon level. The unconditional noise floor on these measurements was , indicating the functionality of this system in the quantum regime. The origins of this noise are expected to be spontaneous Raman scattering and four-wave mixing [@Reim:2011p13181], the former could be removed by improved frequency filtering, but the latter is intrinsic to the vapor cell memory. Raman quantum memory represents a robust and reliable option for integration in large-scale quantum networks. Its high bandwidth, coupled with an unprecedented TBP, and the abililty to store single photons makes it ideally suited to synchronizing probabilistic quantum events, for example in entanglement swapping or enhancing multi-photon rates.
![[**(a)**]{} The Raman quantum memory scheme: A weak signal pulse (thin green) is mapped into a collective excitation of an atomic ensemble by a strong orthogonally polarized control pulse (thick red). Upon further application of a second control pulse, the signal is read out of the memory due to imperfect memory efficiency some signal light is no t stored; this is transmitted by the memory. [**(b)**]{} Level diagram for the Raman memory operating on the Cesium $D_2$ line. The signal is stored via an off-resonant Raman transition in a $\Lambda$-level system in atomic Cesium vapor. The ground states $6S_{1/2}, F=3$ (denoted by ${|3\rangle}$) and $6S_{1/2}, F=4$ (${|1\rangle}$) are split by and the detuning from the excited state, $6P_{3/2}$ (${|2\rangle}$), is typically around . Prior to the memory operation, the atoms are prepared in ${|1\rangle}$ by optical pumping.[]{data-label="fig:Mem"}](Figure_1.pdf){width="80.00000%"}
To interface effectively with future quantum networks, a memory must also be capable of storing the quantum information encoded in the incoming photons. In fiber-based networks, the polarization degree of freedom is particularly useful, because it allows a single-photon qubit to be transmitted in a single spatial and temporal mode. Furthermore, because photons generally interact only weakly with their environment, polarization-encoded information can be transmitted over long distances without decoherence — for example, radiation from the Big Bang is still partially polarized [@Kogut:2003p17131]. This is a critical requirement for the feasibility of large-scale networks. The ability to store polarization information with high fidelity is, therefore, a key benchmark for a useful quantum memory.
Raman memory may operate in a multimode configuration, but in its simplest form it is effectively single-mode [@Nunn:2008p17074]; hence it cannot store an arbitrary polarization state. However, this problem can be addressed by building a dual-rail memory inside a polarization interferometer with one arm storing the vertical component of the polarization state and the other the horizontal. In this way the polarization state of the input light can be perfectly stored in the two ensembles. This has been successfully demonstrated using the EIT technique in ultracold atoms [@Jin:2010p17638; @Chou:2007p15771; @Choi:2008p17756] and a warm atomic ensemble [@Cho:2010p16987]. Unlike the EIT memory, the novel off-resonant nature of the Raman memory allows the storage of high-bandwidth pulses. Furthermore, a low unconditional noise floor due to the suppressed collisional fluorescence facilitates single-photon level operation at room-temperature [@Reim:2011p13181], which is not possible in many schemes, and technically challenging in others. These advantages motivate the investigation of polarization storage in the Raman memory. In this paper we demonstrate that the Raman memory can store polarization-encoded information using the dual-rail procedure. We perform state tomography on the dual-rail Raman memory to characterize its capability to store polarization. We observe a process fidelity of up to $97\pm1\%$ at storage time. This fidelity remains high for longer storage times, yielding $86\pm4\%$ after — 5000 times longer than the pulse duration. Hence we demonstrate high fidelity polarization storage at large bandwidths with unprecedented time-bandwidth products.
Dual-rail memory \[sec\_TwoModeMemory\]
=======================================
The experimental implementation of the Raman memory in warm Cesium gas is discussed elsewhere [@Reim:2010p13180], so is only briefly described here. The master laser for this experiment is a Spectra Physics Tsunami which produces pulses of duration at a repetition rate. The laser is tuned close to the Cesium $D_2$ line at . A Pockels cell picks two pulses from this 80 MHz pulse train with a variable delay, where the first pulse defines the storage and the second the retrieval time window. From these picked pulses, the orthogonally polarized signal and control pulses are derived by a polarizing beam splitter. The signal is subsequently shifted by the Cs hyperfine ground state splitting of , using an electro-optic modulator (EOM), to obtain two-photon Raman resonance. The EOM is time gated, such that only pulses in the storage time window are frequency modulated, yielding the presence of signal field only in the storage time bin. Subsequently, the signal and control arms are re-overlapped spatially and temporally on a PBS and are focussed into the Cs vapor cell, which is heated to 67using resistive heating coils. The Raman interaction only couples orthogonal polarizations far from resonance, hence the signal and control fields remain orthogonal. In the memory output mode, this orthogonality enables extinction of the strong control field. Frequency filters are used to further extinguish the control field before detecting the signal on a photodiode. Prior to the memory procedure, the atomic ensemble is prepared in the memory ground state ($6 \rm{S}_{1/2}, \rm{F=4}$) by optical pumping with a diode laser. The optical pumping laser is orthogonally polarized to the signal and is in a counter-propagating geometry. The cell is shielded from stray magnetic fields with several layers of $\mu$-metal. By fitting the storage time to our model of magnetic dephasing, we estimate the stray magnetic field to be on the order of [@Reim:2011p13181], which is consistent with residual magnetization generated by our heating coils.
Since, in this regime, the Raman memory is single-mode, polarization-encoded information cannot be stored in a single atomic ensemble. Instead, we construct a passively stable polarization interferometer, employing two polarizing beam displacers (PBD) [@Cho:2010p16987; @Obrien:2003p17856], with the Cs cell positioned inside the interferometer (see fig. \[fig:PBD\_mem\]). The PBDs split the signal pulses into their constituent horizontal (H) and vertical (V) components inside the interferometer with subsequent recombination at the interferometer output. The orthogonally polarized optical pumping laser and control pulses are overlapped with the signal field in the interferometer but are spatially separated outside of it. The phase accumulated by the signal due to unequal path lengths between the H and V arms is compensated behind the interferometer output [@Bhandari:1988p14296]. In this way, we create a two-mode memory by accessing two non-overlapping atomic ensembles in the same Cs cell, with one mode storing horizontally (H) and the other vertically (V) polarized light. By balancing the efficiencies of these two memories, which prevents an artificial rotation of the output polarization, we can accurately store polarization information.
![Layout of the two-memory interferometer. The signal field (red) is prepared in an arbitrary polarization state, $\cos{\theta}{|H\rangle} + e^{i\phi}\sin{\theta}{|V\rangle}$, which is split into two arms in a polarization interferometer using a pair of polarizing beam displacers (PBD). The control field (black) and optical pumping laser (blue dashed) are introduced along each arm with the orthogonal polarization, hence enter and exit the PBDs at different ports. The signal is read in at time $t_1$, and out at time $t_2$, by a strong control pulse. Following the memory, the signal field polarization is analyzed with a polarizer and a pair of calibrated waveplates before Fabry–Perot etalons are used to extinguish stray control field light. Additional phase picked up in the interferometer is compensated by a pair of quarter-wave plates set to $\pm$45$\rm{^\circ}$ sandwiching a half-wave plate. The signal and control field preparation, as well as the focussing lenses are omitted for clarity, for details refer to reference \[14\]. The inset shows a typical memory signal, in this instance the storage time is , the read-out signal is magnified by a factor of 10 for clarity.[]{data-label="fig:PBD_mem"}](Figure_2.pdf){width="80.00000%"}
Quantum process tomography {#sec_QPT}
==========================
In order to benchmark the polarization storage capability of the memory, we use quantum process tomography (QPT) [@Chuang:1997p2455; @OBrien:2004p14369; @Lobino:2008p17804; @RahimiKeshari:2011p17805]. A quantum process is any physical process, unitary or non-unitary, which takes a physical input state $\rho_{\rm{in}}$ and produces a physical output state $\rho_{\rm{out}}$. In the quantum process formalism [@Nielsen:2003p14361], any such process can be written as $$\label{eq:QPT1}
\rho_{\rm out} = \sum_{i,j}\chi_{ij}\Gamma_i\rho_{\rm in}\Gamma_j,
\label{eq_QPTformula}$$ where $\chi_{ij}$ is known as the [*process matrix*]{} and contains the complete information about the dynamics of the system, and the $\Gamma_{i,j}$ are a complete set of orthonormal basis matrices for the density matrix. QPT is a technique for estimating an unknown quantum process by preparing a range of different input states and making measurements on the output state. For our polarization qubit, we prepare and measure six polarization states: $\{ {|H\rangle}, {|V\rangle}, {|D\rangle} = \frac{1}{\sqrt{2}}({|H\rangle} + {|V\rangle}), {|A\rangle} = \frac{1}{\sqrt{2}}({|H\rangle} - {|V\rangle}), {|R\rangle} = \frac{1}{\sqrt{2}}({|H\rangle} -i {|V\rangle}), {|L\rangle} = \frac{1}{\sqrt{2}}({|H\rangle} + i{|V\rangle}) \}$. The 36 resulting input-output measurement settings provide a reliable basis set with which to fully characterize the storage process for the qubit system [@deBurgh:2008p052122; @Adamson:2010p14531]. Here, we reconstruct the measured process matrix using maximum likelihood estimation [@OBrien:2004p14369] (for a detailed recipe, see [@LangfordNK2007phd]). Each measurement was repeated 500 times to determine the measurement statistics, which were in turn used to determine error bars via Monte Carlo simulation.
To characterize our qubit memory, we measure the process with the memory switched “on”, by analyzing the retrieved signal field with the control field present, and “off”, by analyzing the transmitted signal field with the control field blocked. We then quantify the memory performance by calculating the process fidelity between the two resulting process matrices, defined by $F = {\rm tr}\left[\sqrt{\sqrt{\chi_{\rm on}}\chi_{\rm off}\sqrt{\chi_{\rm on}}} \right]^2$ [@Gilchrist:2005p062310], a measure of the similarity of two different quantum processes which in our case is synonymous with the memory’s ability to preserve polarization encoded information. Hence, for an ideal Raman memory, the “on” process is identical to the “off” process, which should just be the identity process, corresponding to $F=1$. Fidelities of less than 1 imply that the state has been altered by the process.
Results
=======
The experiment is performed using classical weak coherent states containing on the order of 1000-10,000 photons per pulse. However, these results will also hold for truly quantum single-photon inputs because the counting statistics of single photons passing through a linear optical system always follow the classical behavior [@Loudon:2004p14489]. An obvious example of this being the interference visibility of light attenuated below the single-photon-level [@Taylor:1909p14511]. In order to run the experiment in the single-photon regime, the interferometer would have to be modified to include small waveplates in each arm to compensate for the birefringence of the cell windows. Currently, this birefringence causes a small rotation of the control field polarization leading to leakage through the polarization filtering; this added control field noise precludes single-photon level measurements. In addition, the long counting times required to build up single-photon statistics require stability of the interferometer on long time scales of several hours, which would require active stabilization of the interferometer.
To assess the coherence of the polarization storage, full process tomography of the memory was performed at a range of storage times and the fidelity is obtained from the reconstructed process matrices $\chi_{ij}$, as described in sec. \[sec\_QPT\]. By running the experiment in its “off” state, with the control field blocked, we also obtain the process matrix of the interferometer. Figure \[fig:QPT\] shows typical reconstructed process matrices for the input (control blocked) and retrieved pulse for an exemplary storage time of . The input process matrix, $\chi_{\rm off}$, can be seen to consist mainly of the identity, $I$, demonstrating that the interferometer is stable throughout the measurement. The retrieved matrix, $\chi_{\rm on}$, is also dominated by the identity showing that the memory replicates the polarization state faithfully. In this case, the process fidelity was calculated to be 85$\pm$4%.
![[**(a)**]{} The process matrix, $\chi_{\rm off}$, as measured with the control field blocked. This evaluates the performance of the polarization interferometer. As can be seen, only the identity transformation appears in the process, proving the stability of the interferometer during the measurement. [**(b)**]{} The process matrix of the retrieved pulse, $\chi_{\rm on}$, after storage time. The fidelity of the memory process is calculated, by comparison of these two matrices (see sec. \[sec\_QPT\]), to be 85$\pm$4%.[]{data-label="fig:QPT"}](Figure_3.pdf){width="70.00000%"}
The individual values for the process fidelity are plotted, alongside the memory efficiency, as a function of storage time in Figure \[fig:Visibility\]. This shows that the fidelity was highest for storage ($97\pm1\%$), but remained constantly above 84% for storage times of up to , beyond which the retrieved signal became too weak for a meaningful reconstruction of the process matrix due to the decreasing memory efficiency (see fig. \[fig:Visibility\]). Notably the fidelity is approximately constant as a function of storage time. Thus it does not degrade despite the decreasing efficiency of the memory, showing that no polarization coherence, and hence no information, is destroyed by memory losses. This is an important feature if this memory is to be integrated in future quantum networks, illustrating the high quality and the robustness of this memory protocol for polarization information with respect to decoherence.
![The process fidelity of the memory (red squares) plotted alongside the efficiency of the memory (blue dots) with increasing storage time. The fidelity remains high even as memory efficiency decreases, implying that the fidelity of stored polarization information is insensitive to loss. The efficiency is lower here than has previously been reported [@Reim:2011p13181] as the control field power is shared between both arms of the polarization interferometer.[]{data-label="fig:Visibility"}](Figure_4.pdf){width="60.00000%"}
From previous studies, we expect the limiting factor in the storage time to be stray magnetic fields [@Reim:2011p13181] as a non-zero magnetic field causes dephasing of the spin wave inside the atomic ensemble and therefore a loss of efficiency. However, a consistent fidelity seems to suggest that the spinwave remains coherent, despite the decrease in efficiency. One way to resolve this apparent discrepancy is to consider the effect of the control pulse. The spin-wave is created, and read-out, by pulses of the same polarization. For this reason the read-out pulse selects only the component of the spin wave which remains coherent with the read-in. This causes the read-out signal to have the same polarization as the input signal, hence the polarization state, and thereby the process fidelity, are maintained.
The off-resonant nature of the Raman memory means that, ordinarily, the ensemble is transparent to the signal field. An advantage of this feature is that unstored signal photons are simply transmitted by the memory (as shown in fig. 1) and can be used in subsequent experiments. To confirm that these transmitted photons are not affected by the action of the control pulse, the process fidelity of the unstored photons was also calculated. Typically, a fidelity of over 99.5% was measured indicating that the transmitted photons are unaffected by the storage process.
Conclusion
==========
In conclusion, we have demonstrated the storage and on-demand retrieval of polarization-encoded information in a room-temperature Raman quantum memory with high fidelity at GHz bandwidths. The high time-bandwidth product and rugged design of this memory make it a promising candidate for integration in scalable quantum networks. The polarization basis represents a reliable and robust option for the transmission of photonic quantum information. Thus the preservation of polarization during storage and retrieval is of paramount importance for a quantum memory. In this paper we have performed process tomography on a dual-rail Raman quantum memory, demonstrating storage of the polarization of a weak coherent state with up to $97\pm1\%$ process fidelity. The fidelity remains above 84% for up to storage time which is around 5000 times longer than the pulse duration, so this high-fidelity storage is coupled with a record time-bandwidth product. Furthermore, the fidelity does not decrease with increasing storage time, despite losses in memory efficiency, showing that the fidelity of the information remaining in storage is insensitive to loss.
The off-resonant operation of the Raman memory suppresses collision-induced fluorescence, making single-photon storage and retrieval possible with low noise. This has already been demonstrated in a single-mode Raman memory. Although the polarization memory was implemented here with weak coherent states, only technical difficulties preclude the storage of single photons in the dual-rail memory. Hence this result represents a key step towards the storage of true single-photon polarization qubits.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by EPSRC (project number EP/C51933/01) and the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement n$^\circ$ 248095 for project Q-ESSENCE, and the Royal Society. KFR was supported by the Marie-Curie Initial Training Network (ITN) EMALI and PSM was supported by the ITN FASTQUAST. XMJ acknowledges support from the Centre for Quantum Technologies at the National University of Singapore. MRS was supported by a Clarendon scholarship.
References {#references .unnumbered}
==========
|
---
abstract: 'We continue our study of the realization problem for prism manifolds. Every prism manifold can be parametrized by a pair of relatively prime integers $p>1$ and $q$. We determine a complete list of prism manifolds $P(p, q)$ that can be realized by positive integral surgeries on knots in $S^3$ when $q>p$. The methodology undertaken to obtain the classification is similar to that of the case $q<0$ in an earlier paper.'
address: 'Department of Mathematics, California Institute of Technology, Pasadena, CA 91125'
author:
- William Ballinger
- Yi Ni
- Tynan Ochse
- Faramarz Vafaee
bibliography:
- 'Reference.bib'
title: The prism manifold realization problem II
---
|
---
abstract: 'Enforcing a non-classical behavior in mesoscopic systems is important for the study of the boundaries between quantum and classical world. Recent experiments have shown that optomechanical devices are promising candidates to pursue such investigations. Here we consider two different setups where the indirect coupling between a three-level atom and the movable mirrors of a cavity is achieved. The resulting dynamics is able to conditionally prepare a non-classical state of the mirrors by means of projective measurements operated over a pure state of the atomic system. The non-classical features are persistent against incoherent thermal preparation of the mechanical systems and their dissipative dynamics.'
address: |
$^1$Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore\
$^2$Centre for Theoretical Atomic, Molecular, and Optical Physics, School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom\
$^3$NEST Istituto Nanoscienze-CNR and Dipartimento di Fisica, Universitá degli Studi di Palermo, via Archirafi 36, I-90123 Palermo, Italy\
$^4$QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, United Kingdom\
$^5$Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom\
$^6$Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore
author:
- 'G. Vacanti$^1$, M. Paternostro$^2$, G. M. Palma$^3$, M. S. Kim$^4$, and V. Vedral$^{1,5,6}$'
title: 'Non-classicality of optomechanical devices in experimentally realistic operating regimes.'
---
Introduction
============
Interesting experimental endeavors have recently challenged the widely-accepted assumption that quantumness is an exclusive prerogative of microscopic and isolated systems. These efforts show that complex and large objects comprising many elementary constituents or endowed with a variety of degrees of freedom can display important non-classical features [@naik; @thompson; @schliesser1; @corbitt; @schliesser2; @schliesser3; @oconnell; @demartini; @sekatski]. In general, quantum control under unfavorable operating conditions is an important milestone in the study of the quantum-to-classical transition and, as such, should be pursued to achieve a better understanding of the conditions enforcing and implying quantum mechanical features in the state of a given system. This topic has recently become the focus of an intense research activity, at all levels, boosted by the ability to experimentally manipulate systems composed of subparts having variegated nature. We can now coherently control the interaction between radiation and Bose-Einstein condensates [@Brennecke; @colombe] while mesoscopic superconducting devices compete with atoms and ions for the realization of cavity quantum electrodynamics [@schoelkopf; @majer; @Wallraff]. Equally remarkable is the progressive entering of purely mechanical systems into the realm of experimental controllability [@gigan; @arcizet1; @kleckner; @arcizet2; @Groblacher1; @Groblacher2; @chan]. The operative conditions and the intrinsic nature of the systems involved in these examples often deviate from the naive requirements for “quantumness”: ultra-low temperatures, full addressability and ideal preparation of the system. The design and exploitation of such interesting setups is giving further emphasis to investigations performed along the lines of the question raised above [@marshall; @ferreira; @mauro].
Here, we prove how non-classical behaviors can be induced in massive mesoscopic systems out of the reach of direct addressability. The indirect interaction with a fully controllable microscopic system enforces non-classical mesoscopic states, robust against adverse operative conditions (such as temperature). Our study is performed in the micro-scale domain and involves two different optomechanical cavity-quantum electrodynamics settings. It proposes a scenario for the observation of induced non-classical features, such as non-local correlations and negative values of the Wigner function, that are truly mesoscopic (thus different from more extensively studied nano-scale setups [@armour; @rabl; @tian; @rodrigues]), well-controllable and, although close to experimental capabilities in the fields of optomechanics and light-matter interaction, yet unexplored.
The paper is organized as follow: in Sec. \[SM\], we discuss a setup in which one mesoscopic object (a movable end-mirror of an optical cavity) interacts with a microscopic system (a three-level atom) through the radiation inside the cavity. In this context, we study the correlations between the two systems as well as the non-classical features induced on the state of the mirror. In Sec. \[TM\] we extend our analysis to a system where both cavity mirrors interact with the atom. This setup allows us to investigate the correlations between two truly mesoscopic systems, revealing how quantum effects can survive to adverse environmental conditions such as dissipation and thermalization.
![[ (Color online) **(a)**]{} Scheme of the system. [**(b)**]{} Energy levels of the atom driven by an off-resonant two-photon Raman transition.[]{data-label="scheme"}](Fig1Scheme.eps){width="42.00000%"}
Single Mirror {#SM}
=============
Here we consider an optomechanical system consisting of a cavity whose end-mirror can oscillate under the action of the radiation-pressure force. A three-level atom is placed inside the cavity and the system parameters are chosen so that an effective atom-mirror coupling is achieved. We show how the state of the system reveals strong non-classical features such as non-local correlations between the atom and the mirror and negative values of the Wigner function of the mirror, even in presence of dissipative processes and non-zero temperature.
The Model {#subsectionOMModel}
---------
The system that we consider involves a three-level atom in a $\Lambda$ configuration, coupled to a single-mode optical cavity pumped by an laser field at frequency $\omega_p$ and with a movable mirror. The atom is driven by a second external field at frequency $\omega_i$ that enters the cavity radially (see Fig. \[scheme\]). We label $\{{\left\vert0\right\rangle},{\left\vert1\right\rangle}\}$ the states belonging to the fundamental atomic doublet and ${\left\verte\right\rangle}$ the excited state. The atomic transition $|0\rangle{\leftrightarrow}|e\rangle$ is guided, at rate $\Omega,$ by the external field at frequency $\omega_i$. On the other hand, the transition $|1\rangle{\leftrightarrow}|e\rangle$ is coupled to the cavity field at frequency $\omega_c$ with coupling constant $g$. We call $\delta$ the detuning between each transition and the respective driving field, while $\Delta{=}\omega_c{-}\omega_p$ is the cavity-pump detuning. The movable mirror is modeled as a harmonic oscillator with frequency $\omega_m$, coupled to the cavity field through radiation-pressure. We assume large single-photon Raman detuning and negligible decay rate $\gamma_e$ from the atomic excited state, so that $\delta\gg{\Omega,g}\gg\gamma_e$ and an off-resonant two-photon Raman transition is realized. Moving to an interaction picture defined by the operator $\omega_p\hat{a}^\dag\hat{a}+\omega_i{\left\verte\right\rangle}_a\!{\left\langlee\right\vert}+\omega_{10}{\left\vert1\right\rangle}_{a}\!{\left\langle1\right\vert},$ the Hamiltonian of the overall system reads \[we set $\hbar{=}1$ throughout the paper\] $\hat{\cal H}_{\rm sys}=\hat{\cal H}_{a}+\hat{\cal H}_{R}+\hat{\cal H}_{m}+\hat{\cal H}_{c}+\hat{\cal H}_{mc}+\hat{\cal H}_{cp}\label{HTot}$, where $$\begin{split}\label{HTot}
&\hat{\cal H}_{a}\!=\!{\delta}{\left\verte\right\rangle}_a\!{\left\langlee\right\vert},\hat{\cal H}_{m}=\omega_m\hat{b}^\dag\hat{b},\\
&\hat{\cal H}_{c}=-\Delta\hat{a}^\dag\hat{a}, \hat{\cal H}_{mc}=\chi\hat{a}^\dag\hat{a}(\hat{b}+\hat{b}^\dag),\\
&\hat{\cal H}_{R}\!=\! \Omega({\left\verte\right\rangle}_a\!{\left\langle0\right\vert}+{\left\vert0\right\rangle}_a\!{\left\langlee\right\vert})+{g}(e^{i\Delta{t}}\hat{a}^\dag{\left\vert1\right\rangle}_a\!{\left\langlee\right\vert}+h.c.)\\
\end{split}$$ Here, $\hat{\cal H}_{a}$ is the atomic energy, $\hat{\cal H}_{R}$ is the Raman coupling, $\hat{\cal H}_m$ ($\hat{\cal H}_c$) is the mirror (cavity) free Hamiltonian and $\hat{\cal H}_{mc}$ is the radiation-pressure term [@law] (with coupling rate $\chi$), where $\hat{a}$ ($\hat{a}^\dag$) is the annihilation (creation) operator of the cavity field and $\hat{b}$ ($\hat{b}^\dag$) is the corresponding operator of the mirror. Finally, $\hat{\cal H}_{cp}$ is the cavity-pump interaction [@walls]. The pumping field ensures that a few photons are always present in the cavity, allowing a mediated interaction between the atom and the mirror. On the other hand, the purpose of the external field with rate $\Omega$ is to trigger the passages between the excited level $|e\rangle$ and the ground level $|0\rangle.$
If we further assume $\Delta\gg{g,\chi}$, both the atomic excited state and the cavity field are virtually populated and they can be eliminated from the dynamics of the system. This leads to the effective interaction Hamiltonian $$\hat{\cal H}_{\rm eff} = \eta{\left\vert0\right\rangle}_a\!{\left\langle0\right\vert}(\hat{b}^\dag+\hat{b})
\label{HEffOneMirr}$$ where $\eta={\chi{g}^2\Omega^2}/{\delta^2\Delta^2}.$ The form of the effective coupling rate $\eta$ shows that all the considered coupling mechanisms are necessary in order to achieve the atom-mirror coupling. Through the two-photon Raman transition, the virtual quanta resulting from the atom-cavity field interaction are transferred (by the bus embodied by the cavity field) to the mechanical system. As a consequence, the state of the latter experiences a displacement (in phase space) conditioned on the state of the effective two-level atomic system resulting from the elimination of the excited state. $\hat{\cal H}_{\rm eff}$ involves the position quadrature operator $\hat{q}\propto\hat{b}+\hat{b}^\dag$ of the movable mirror. It is worth noticing that, if the cavity is driven by a bichromatic pump with frequencies $\omega_{p}$ and $\omega_{p}+\omega_m$ and a relative phase $\phi$, the effective coupling between the atom and the movable mirror can be made [*flexible*]{} in the sense that $\hat{q}$ is replaced by $\hat{b}e^{i\phi}+\hat{b}^\dag{e}^{-i\phi}$, making possible the displacement in any direction of the phase space of the movable mirror [@cam; @noiGP; @phase; @Leibfried].
Atom-Mirror Entanglement {#subsecOMEnt}
------------------------
We now focus on the quantification of microscopic-macroscopic correlations between the atom and the mirror. First, we assume that the initial state of the movable mirror is a coherent state ${\left\vert\alpha\right\rangle}_m$ with amplitude $\alpha\in\mathbb{C},$ while the atom is assumed intially in ${\left\vert+\right\rangle}_a=({\left\vert0\right\rangle}+{\left\vert1\right\rangle})_{a}/\sqrt{2}$. Under the action of the effective Hamiltonian in Eq. (\[HEffOneMirr\]), the initial state evolves into ${\left\vert\psi(t)\right\rangle}=\hat{\cal U}_t{\left\vert+,\alpha\right\rangle}_{am}$, where $${\left\vert\psi(t)\right\rangle}=\frac{1}{\sqrt{2}}({\left\vert1,\alpha\right\rangle}+e^{-i\varPhi(t)}{\left\vert0,\alpha-i\eta t e^{-i\phi}\right\rangle})_{am}
\label{catstate}$$ with $\varPhi(t)=\eta t\text{Re}[\alpha{e}^{i\phi}]$ and $\hat{\cal U}_t\equiv{e}^{-i\hat{\cal H}_{\rm eff} t}={\left\vert1\right\rangle}_a\!{\left\langle1\right\vert}\otimes\openone+{\left\vert0\right\rangle}_a\!{\left\langle0\right\vert}\otimes \hat{D}(-i\eta t e^{i\phi})$, where $\hat{D}(\zeta) = e^{\zeta \hat{b}^\dag- \zeta^*\hat{b}}$ is the single-mode displacement operator [@walls]. Eq. (\[catstate\]) is, in general, an entangled state of a microscopic and a mesoscopic system: its Von Neumann entropy depends on the value of $\eta t$ only. Intuitively, the larger the phase-space distance between ${\left\vert\alpha\right\rangle}$ and ${\left\vert\alpha-i\eta t\right\rangle}$, the closer the evolved state to a balanced superposition of bipartite orthogonal states, thus maximizing the entanglement. To give a figure of merit, for $\eta t=0.82$ the entropy is $\sim0.8,$ while for $\eta t>1.7$ the entropy is $>0.996$. Interestingly, the kind of control over the mirror state reminds of the “quantum switch” protocol for microwave cavities [@davidovicharoche], although here it is achieved over a truly mesoscopic device.
![(Color online) Maximum violation of the Bell-CHSH inequality against the displacement $d$. From top to bottom, the curves correspond to $V=1,3,5$ with $\eta t=2d$ and $\theta_1\simeq{3}\pi/2$ and are optimized with respect to $\theta$. The inset shows, from top to bottom, the logarithmic negativity $E$ against $V$ for projected states with $p=0,1$ and $2$, for $d=2$.[]{data-label="CHSHoneMirr"}](Fig2CHSH.eps){width="42.00000%"}
Although impressive progresses have recently been accomplished in active and passive cooling of micro- and nano-mechanical oscillators [@chan], it is realistic to expect the mirror to be affected by thermal randomness due to its exposure to the driving field and/or to a phononic background at temperature $T$. Exploiting the handiness of Eq. (\[catstate\]), we write the initial state of the mirror at thermal equilibrium (temperature $T$) and displaced by $d$ (due to the external pump) as $$\varrho_{m}^{\rm th}=\int{d}^2\alpha{\cal P}(\alpha,V){\left\vert\alpha\right\rangle}_m\!{\left\langle\alpha\right\vert}
\label{thermalstate}$$ with ${\cal P}(\alpha,V)=\frac{{2}e^{-\frac{2|\alpha-d|^2}{V-1}}}{{\pi(V-1)}}$, $V=\coth (\omega_m/2k_bT)$ and $k_b$ the Boltzmann constant. Under $\hat{\cal U}_t$, the state ${\left\vert+\right\rangle}_a\!{\left\langle+\right\vert}\otimes\varrho_{m}^{\rm th}$ evolves into $$\label{final1}
\hat{\cal U}_t({\left\vert+\right\rangle}_a\!{\left\langle+\right\vert}\varrho_{m}^{\rm th}\,)\hat{\cal U}^\dag_t\!=\!\int\!{d}^2\alpha{\cal P}(\alpha,V){\left\vert\psi(t)\right\rangle}\!{\left\langle\psi(t)\right\vert},$$ which reduces to the pure case of Eq. (\[catstate\]) for $T=0$. We proceed to show that the coupling mechanism described above is characterized by interesting features, at the core of current experimental and theoretical interests [@demartini; @jeong1; @jeongralph]. Let us consider the case of $\phi=\pi/2$, $V=1$ (i.e. $T=0$) and $\alpha\in\mathbb{R}$, which gives ${\left\vert\psi(\tau)\right\rangle}({\left\vert1,\alpha\right\rangle}+{\left\vert0,\alpha-\eta t\right\rangle})/\sqrt2$. This entangled state represents a mesoscopic instance of a pure Schrödinger-cat state. Interestingly, it has been discussed that a faithful implementation of the Schrö[d]{}inger’s cat paradox would use a mesoscopic subsystem initially prepared in a thermal state, rather than a pure one [@demartini; @jeong1; @jeongralph]. The state in Eq. (\[final1\]) is a significant example of such case. Unravelling the entanglement properties of this state is demanding due to the difficulty of finding an analytical tool for its undisputed revelation. In order to gain insight, here we propose to follow two paths.
The first relies on the nonlocality properties of this class of states, induced by the strong entanglement between the subsystems. Following Ref. [@banaszek; @banwod], the microscopic part is projected along the direction ${\bf n}=(\sin\theta,0,\cos\theta)$ of the single-qubit Bloch sphere while the mesoscopic one is probed by using the displaced parity observable $\hat{\Pi}(\beta)=\hat{D}^\dag(\beta)(-1)^{\hat{b}^\dag\hat{b}}\hat{D}(\beta)$, where $\hat{D}(\beta)$ is the displacement operator of amplitude $\beta=\beta_r+i\beta_i$. This approach has been used recently to address the micro-macro non-locality in an all-optical setting [@spagnolo2011]. The correlation function for a joint measurement is thus $${\cal C}(\beta,\theta)=\int{d}^2\alpha{\cal P}(\alpha,V){\left\langle\psi(t)\right\vert}({\bf n}\cdot\hat{{\bm \sigma}})\otimes\hat{\Pi}(\beta){\left\vert\psi (t)\right\rangle}$$ and a Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality is formulated as $|{\cal C}(0,\theta_1)+{\cal C}(0,\theta)+{\cal C}(\beta,\theta_1)-{\cal C}(\beta,\theta)|\le{2}$ [@chsh]. Any state satisfying this constraint can be described by a local-realistic theory. Let us first discuss the pure case of $V=1$, which gives $$\begin{split}
{\cal C}(\beta,\theta)&=\frac{1}{2}e^{-2(d^2+\eta^2 t^2+|\beta|^2+\beta_r\eta t-2\beta_rd)}\\
&\times[\cos\theta(e^{4d\eta t-2\eta t \beta_r}\!-\!e^{2\eta^2 t^2+2\eta t \beta_r})\!\\ &+\!2e^{\eta t(2{d}+\frac{3}{2}\eta t)}\cos({2\eta t \beta_i})\sin\theta].
\end{split}$$ At $\eta t=0$, the microscopic and mesoscopic subsystems are uncorrelated and ${\cal C}(\beta,\theta)$ can indeed be factorized. For a set value of $d$ and a non-zero value of $\eta t$, we observe violation of the Bell-CHSH inequality as illustrated in Fig. \[CHSHoneMirr\]. Moreover, there is a range of values of $\theta$ ($\sim\pi/2$) where, for $d\neq{0}$, the local-realistic bound is violated, symmetrically with respect to $d=0$. When the thermal character of the mesoscopic part is considered, the expression for the correlation function becomes cumbersome and we omit it. However, [*the strong entanglement between microscopic and mesoscopic subsystems allows violation of Bell-CHSH inequality also in the mixed-state case*]{}: the dotted curve in Fig. \[CHSHoneMirr\] corresponds to $V\simeq{5}$. Beyond this value, the inequality is no longer violated.
The second path we follow uses the technique put forward in Ref. [@bosekim] and later reprised by Ferreira [*et al.*]{} in Ref. [@ferreira]. In this approach, Eq. (\[final1\]) is projected onto a bidimensional subspace spanned by the microscopic states $\{{\left\vert0\right\rangle},{\left\vert1\right\rangle}\}_a$ and the phononic ones $\{{\left\vertp\right\rangle},{\left\vertp+1\right\rangle}\}_m$ ($p\in\mathbb{Z}$). The entanglement within Eq. (\[final1\]) cannot be increased by this projection, which is just a local operation. Thus, by quantifying the entanglement for fixed $p$, we provide a lower bound to the overall quantum correlations in the state of the system. As a measure for entanglement in each $2\times{2}$ subspace we use the [*logarithmic negativity*]{}, which accounts for the degree of violation of the positivity of partial transposition criterion [@peres; @horo3; @lee; @plenio05]. An example of the results achieved with this method is given in the inset of Fig. \[CHSHoneMirr\], where we show the case of $d=2$ and $p=0,1,2$. Entanglement is found in each subspace with fixed $p$, up to values of $V\sim{5}$, strengthening our findings about the resilience of non-classical correlations set by the coupling being studied.
Non-classicality of the mirror
------------------------------
We now consider the effects of the microscopic-mesoscopic interaction over the state of the movable mirror. This is a hot topic in the current research of opto and electro-mechanical systems. The grounding of opto/electro-mechanical devices as potential candidates for quantum information processing requires the design of protocols for the preparation of non-classical states of massive mechanical systems. Various attempts have been performed in this direction, mainly at the nano-scale level, where a cantilever can be capacitively coupled to a superconducting two-level system [@armour; @rabl; @tian; @rodrigues].
Let us consider the case of $\phi=0$. The optomechanical evolution encompassed by $\hat{\cal U}_t$ alone is unable to give rise to any non-classicality in the state of the mirror. This is easy to check simply by tracing out the state of the atom in Eq. (\[catstate\]), which would leave us with a statistical mixture of two displaced mirror’s states. On the other hand, a conditional process is able to project the coherence of a quantum mechanical superposition and simultaneously get rid of the atomic degree of freedom [@KochPRL; @RitterNature; @SpechtNature; @MonteiroNJP; @Kiesel; @MauroPRA]. In order to illustrate our claim, we consider an initial state of the system having the form $\rho(0){=}|\varphi\rangle \langle \varphi |{\otimes}\rho_m(0)$ where $|\varphi\rangle{=}c_0 |0\rangle{+}c_1 |1\rangle$ is a pure state of the atom and $\rho_m(0)$ is an arbitrary state of the mechanical mode. We then project the atomic part of the evolved state $\hat{\cal U}_t \rho(0)\hat{\cal U}_t^\dag$ onto $|\varphi\rangle \langle \varphi |$, thus post-selecting the mechanical state $\rho_m(t) = \langle \varphi | \hat{\cal U}_t |\varphi \rangle \rho_m(0)\langle \varphi | \hat{\cal U}_t^\dag |\varphi\rangle$. Therefore, the state of the mirror undergoes an effective evolution driven by the operator $$\langle \varphi | \hat{\cal U}_t | \varphi \rangle = |c_1|^2 \hat\openone + |c_0|^2\hat{D}(-i \eta t).
\label{pUpOneMirr}$$ In the remainder of this paper, we consider again the case where $|{\varphi}\rangle=|{+}\rangle\equiv(|0\rangle + |1\rangle)/\sqrt{2}$, which optimizes the performance of our scheme terms of the degree of non-classicality enforced in the mechanical subsystem. For an initial coherent state of the mirror, i.e. $\rho_m(0)=|\alpha\rangle\langle\alpha|,$ applying the conditional time evolution operator in Eq. (\[pUpOneMirr\]) leads to ${\left\vert\mu_+\right\rangle}_m={\cal N}_+({\left\vert\alpha\right\rangle}+{e}^{-i\varPhi(t)}{\left\vert\alpha-i\eta t\right\rangle})_m$, where ${\cal N}_+$ is the normalization factor. Depending on the value of $\eta t$, such states exhibit quantum coherences. Obviously, the thermal convolution inherent in the preparation of mirror’s state $\varrho_m$ may [*blur*]{} them. In what follows we prove that this is not the case for quite a wide range of values of $V$.
![(Color online) Density plot of fidelity against $V$ and $\eta$. Darker regions correspond to smaller values of $F_{W}$. []{data-label="fidelity"}](Fig4Fidelity.eps){width=".8\columnwidth"}
The figure of merit that we use to estimate non-classicality is the negativity in the Wigner function associated with the mirror state resulting from the measurement performed over the atomic part of the system. The Wigner function for a single bosonic mode is defined as $${W(\mu)=\frac{1}{\pi}\int d^2\nu e^{\mu \nu^* - \mu^* \nu} \chi(\nu)},
\label{WignerDef}$$ where $\mu {\in}\mathbb{C}$ and $\chi(\nu){=}\text{Tr}[\hat{D}(\nu)\rho]$ is the Weyl characteristic function. Considering an initial thermal state of the mirror and applying the conditional unitary evolution operator given in Eq. (\[pUpOneMirr\]), the Wigner function of the mirror after the post-selection process is $$\begin{split}
W_m(\mu)=&{\cal M}^{-1}e^{-\frac{2|\mu|^2+2\eta t \mu_i+\eta^2 t^2}{V}}\\ \times&[\cosh(\frac{\eta^2 t^2+2\eta t\mu_i}{V})+e^{\frac{\eta^2 t^2}{2V}}\cos(2\eta t\mu_r)]
\end{split}$$ with ${\cal M}=(1+e^{-\frac{V\eta^2}{2}})\pi{V}/2$. The behavior of $W_m(\mu)$ in the phase space is shown in Fig. \[Wignerevolution\], where we clearly see the appearance of regions of negativity, witnessing non-classicality of the corresponding state as induced by our microscopic-to-mesoscopic coupling. Interference fringes are created between two positive Gaussian peaks (not shown in the figure) corresponding to the position, in the phase space, of mutually displaced coherent states. This reminds of the Wigner function of a pure Schrödinger cat state although, as we see later, the analogy cannot be pushed. Remarkably, in contrast with the fragility of the nonlocality properties of the microscopic-mesoscopic state, $W_{m}(\mu)$ has a negative peak of $-0.01$ up to $V\sim100$, which implies strong thermal nature of the mirror state. For a mechanical system embodying one of the mirrors of a cavity, $\omega_m/2\pi\sim5$MHz is realistic [@parameters]. For $V=10$ ($100$), this corresponds to an effective temperature of $1$mK ($10$mK), [*i.e.*]{} energies $10$ ($100$) times larger than the ground-state energy of the mirror.
It is interesting to compare the mixed state resulting from the thermal convolution to a pure state in Eq. (\[catstate\]) (with $\phi=0$). As a measure of the closeness of two states, we use quantum fidelity between a mixed and a pure state written as the overlap between the corresponding Wigner functions $F_W=\pi\int{d}^2\mu{W}_P(\mu)W_M(\mu)$, where $W_P(\mu)$ ($W_M(\mu)$) is the Wigner function of the pure (mixed) state. $F_W$ is shown in Fig. \[fidelity\] against $\eta\tau$ and $V$. While the thermal effect reduces the value of the fidelity as $V$ grows, the behavior of $F_W$ against $\eta t$ is, surprisingly, non-monotonic. At a given $V$, there is always a finite value of $\eta t$ associated with a maximum of $F_W$. Remarkably, the values of $\eta t$ maximizing $F_W$ differ from those at which the Wigner function achieves its most negative value.
![(Color online) Wigner function of the mirror under dissipation, for $\gamma\sim{0.1}\eta$ and $V=5$.[]{data-label="WdissOneMirr"}](Fig5Wigner1.eps){width=".9\columnwidth"}
Finite temperature dissipative dynamics {#subsectionOMDissipative}
---------------------------------------
So far, we have assumed a movable mirror of large mechanical quality factor. The progresses recently accomplished in fabrication processes guarantee very small mechanical dissipation rates. However, they are not yet negligible and their effect should be considered in any proposal for quantumness in optomechanical devices. We thus include mechanical losses in our analysis, looking for their effects onto the non-classicality induced in the movable mirror. We concentrate on the finite-temperature dissipative mechanism described by $${\cal L}^{V}(\rho)=\frac{\gamma}{2}\big[(2\hat{b}\rho\hat{b}^\dag-\{\hat{b}^\dag\hat{b},\rho\})+(V-1)(\hat{b}\rho-\rho\hat{b},\hat{b}^\dag)\big],$$ which is the weak-damping limit of the Brownian-motion master equation [@walls]. The density matrix $\rho$ describes the state of the atom-mirror system. The full master equation, including the unitary part $-i[\hat{\cal H}_e,\rho]$, is easily translated into a set of equations of motion for the mirror reduced density matrix obtained by considering the projections onto the relevant atomic states $\rho_{ij}={}_a\langle{i}|\rho|j\rangle_a~(i,j=0,1)$. These can then be recast as Fokker-Planck equations for the Wigner functions $W_{ij}$ of such mirror’s state components. These read $$\partial_t{\bf W}(x,p, t)={\bf M}{\bf W}(x,p, t)+\tilde{\cal L}_d{\bf W}(x,p, t),
\label{FokkerPlanck}$$ where $$\begin{split}
&{\bf W}(x,p, t){=}\left[\begin{matrix}W_{00}(x,p,t)\\W_{01}(x,p,t)\\W_{10}(x,p,t)\\W_{11}(x,p,t)\end{matrix}\right],\\
&{\bf M}{=}\sqrt 2\eta\left[\begin{matrix}\partial_p&0&0&0\\ 0&-\frac{ix+\partial_p}{2}&0&0\\0&0& \frac{ix+\partial_p}{2}&0\\0&0&0& 0\end{matrix}\right],\\
&\tilde{\cal {L}}_d=\Big[\frac{\gamma}{2}(x\partial_x+p\partial_p)+\frac{\gamma}{4}V(\partial^2_{p^2}+\partial^2_{x^2})+\gamma\Big]\openone,
\end{split}$$ where we have introduced the quadrature variables $x=\sqrt 2\text{Re}(\mu), p=\sqrt 2\text{Im}(\mu)$. Each of these equations preserves the Gaussian nature of the corresponding Wigner function’s component, whose time-evolved form is taken from the ansatz $$W_{ij}(x,p,t)\propto[{\text{det}}({\bf D}_{ij})]^{-1/2}{e^{-\frac{1}{2}{{\bf q}^T_{ij}{\bf D}^{-1}_{ij}{\bf q}_{ij}}+i\Theta_{ij}(t)}}
\label{ansatz}$$ with $${\bf q}_{ij}=\left[\begin{matrix}x-\overline{x}_{ij}\\p-\overline{p}_{ij}\end{matrix}\right],~{\bf D}_{ij}=\left[\begin{matrix}\sigma^x_{ij}&\sigma^{xp}_{ij}\\\sigma^{xp}_{ij}&\sigma^{p}_{ij}\end{matrix}\right]
\label{qandcovariance}$$ parameterized by the time-dependent mean values $\overline{x}_{ij},\overline{p}_{ij}$ and variances $\sigma^{x,p,xp}_{ij}$ of the variables $x,p$ and $xp$. We have also introduced the time-dependent phases $\Theta_{ij}$’s which account for the contributions from $\varPhi(t)$ in Eq. (\[catstate\]). The solution is readily found to be $\sum_{i,j=0,1}W_{ij}(x,p,t)$ (apart from the normalization factor), which gives back the non-Gaussian character of the mirror’s state. The negativity of the Wigner function can be studied at set values of $\gamma$ and $T$ and chosing the time at which the ideal case achieves the most negative value. The results are shown in Fig. \[WdissOneMirr\], where we see that non-classicality is found even for quite a large value of $\gamma/\eta$. Clearly, this results from a subtle trade off between temperature and mechanical quality factor. Although small $\gamma$ and $T$ guarantee non-classicality, such a behavior is still present at $\gamma/\eta\sim{0.1}$ and for $T$ well above the ground-state one.
Two Mirrors {#TM}
============
In this section, we will consider a different setup, where both cavity mirrors are free to oscillate around their equilibrium positions and they are both interacting with a three level atom inside the cavity. Using this setup, we can study the correlations between the two mesoscopic systems and their quantum features. In this section, we will only focus on the conditional evolution of the two mirrors after a measurement of the atomic subsystem.
Hamiltonian and conditional unitary evolution {#subsectionTMH}
---------------------------------------------
Let us consider the same Fabry-Perot cavity discussed in Sec \[subsectionOMModel\], pumped by a laser field at frequency $\omega_p$ and with a three-level $\Lambda$-type atom trapped within the mode-volume of the cavity field. The model is very similar to the one describing the single mirror case, with the difference that here the two mirrors of the cavity are both able to oscillate around their equilibrium positions and they are modeled as two harmonic oscillators with frequencies $\omega_1$ and $\omega_2.$ By moving to an interaction picture respect to the same operator considered in the one-mirror scheme, the Hamiltonian of the system can be written in the same form as the one given in Eq. (\[HTot\]), where only the terms involving the mirror’s degrees of freedom are changed to take into account the addition of the second mirror. These terms read $$\begin{split}
&\hat{\cal H}_{\rm m}{=}\sum^2_{j=1}\omega_j \hat{b}^\dag_j \hat{b}_j,~~\hat{\cal H}_{\rm mc}{=}\hat{a}^\dag \hat{a} \sum^2_{j=1}(-1)^{j-1}\chi_j (\hat{b}^\dag_j{+}\hat{b}_j),
\end{split}$$ where the bosonic operators $\hat{a}^\dag,\hat{a}$ and $\hat{b}^\dag_j,\hat{b}_j$ refer to the cavity field and the two mechanical mirrors, respectively. By assuming a large cavity quality factor and a small spontaneous emission rate from ${\left\verte\right\rangle}$, in the limit of $(\Delta,\delta){\gg}(\Omega, g)$ we can eliminate both the cavity field and the excited atomic level, thus arriving at the effective atom-mirrors Hamiltonian $$\hat{\cal H}_{\rm eff} = |0\rangle \langle 0|\otimes \sum^2_{j=1}(-1)^{j-1}\eta_j (\hat{b}^\dag_j{+}\hat{b}_j)$$ with $\eta_j= ( \Omega^2 g^2/ \delta^2 \Delta^2)\chi_j. $ The corresponding time-evolution operator is $\hat{\cal U}_t = |1\rangle \langle 1| + |0\rangle \langle 0|\otimes \hat{D}_1(-i \eta_1 t)\otimes \hat{D}_2(i \eta_2 t)$, where $\hat{D}_j(\zeta) = \exp[\zeta \hat{b}^\dag_j -\zeta^* \hat{b}_j]$ is the displacement operator for mode ${j=1,2}$ [@walls]. In analogy with the one-mirror case, the resulting dynamics of the mechanical systems is thus a conditional displacement controlled by the state of the atomic part: while nothing happens to the mechanical modes when the atom is prepared in ${\left\vert1\right\rangle}$, their state gets displaced in phase space when the atomic state is ${\left\vert0\right\rangle}$. In what follows we generalize the analysis performed in the previous Section and show how this mechanism, complemented with an appropriate post-selective step, results in non-classicality of the mechanical subsystem.
The generalization of the conditional time evolution operator given in Eq. (\[pUpOneMirr\]) to the two-mirrors case is straightforward. The new operator simply reads $$\langle \varphi | \hat{\cal U}_t | \varphi \rangle = |c_1|^2 \hat\openone + |c_0|^2\hat{D}_1(-i \eta_1 t)\hat{D}_2(i \eta_2 t) \label{pUp}$$ with $\hat\openone$ the identity operator. We consider again the case in which $| \varphi \rangle = |+\rangle $ and the initial state of the mirror is $\rho_m(0)=|\alpha_1, \alpha_2\rangle\langle\alpha_1,\alpha_2|$ where $|\alpha_j\rangle$ is a coherent state of mode $j$ having amplitude $\alpha{\in}\mathbb{C}$. The state of the mirrors at time $t$ is $$|\psi_m(t)\rangle =(| \alpha_1, \alpha_2 \rangle + e^{-i \varPhi(t)} | \beta_1(t) , \beta_2(t) \rangle)/{\sqrt2}
\label{psit}$$ where $\varPhi(t)=\sum^2_{j=1}(-1)^{j-1}\eta_j\text{Re}\{\alpha_j\}t$ and $\beta_j(t)=\alpha_j{+}(-1)^{j} i \eta_j t~(j{=}1,2)$. Eq. (\[psit\]) is an Entangled Coherent State (ECS) of modes $1$ and $2$ [@sanders]. Its Von Neumann entropy depends on a delicate trade off among the amplitudes $\alpha_j(t)$ and $\beta_j(t)$. ECSs play an important role in continuous-variable (CV) quantum information processing as a valuable resource for communication and computation [@jeong2].
Mirror-Mirror correlations
--------------------------
![(Color online) Negative volume of $W(\mu_1,\mu_2)$ against $V$ for $\eta{t}=5$. Inset: Wigner function $W_1(\mu_1)$ at $\mu_2{=}-(1+i)$, $\eta{t}{=}2$ and $T{=}0$. []{data-label="WignerT"}](Fig6Negativity.eps){width=".8\columnwidth"}
In Sec \[subsectionTMH\] we have considered the simple case in which the two oscillators are initialized in a pure coherent state. This example is instructive and, as we will see later, mathematically useful. However, as pointed out in Sec. \[subsecOMEnt\], the interaction of the two oscillators with the thermal bath has to be taken into account, and it is realistic to assume a initial thermal state for the two mirros. The thermal state of a single bosonic mode is given by Eq. (\[thermalstate\]) . In the case of two modes, the initial mechanical state is ${\rho_{\rm m}(0){=}\varrho^{\rm th}_1{\otimes}\varrho^{\rm th}_2},$ and it evolves under the action of $\langle+|\hat{\cal U}_t|+\rangle$ so as to give $$\rho_m(t){=}\int d^2 \alpha_1 d^2 \alpha_2 {\cal P}(\alpha_1,V){\cal P}(\alpha_2,V)|\psi_m(t) \rangle \langle \psi_m(t)|.
\label{EvolTherStateTwoMirr}$$ As in the one mirror setup, we now show that, despite the thermal convolution at the basis of the definition of $\rho_{}(t),$ the mechanical state of two mirrors can exhibit strong non-classical features even at non-zero temperature. We will focus on two different signatures of non-classicality: the negative values of the Wigner function associated with the state $\rho_m(t)$ and the non-local correlations between the two mirrors. The Wigner function of a two-modes system is defined as the straightforward generalization of Eq. (\[WignerDef\]), i.e. $${W(\mu_1,\mu_2)=\frac{1}{\pi^{2}}\!\int d^2\nu_1 d^2\nu_2 \prod^2_{j=1}e^{\mu_j \nu^*_j - \mu^*_j \nu_j} \chi(\nu_1,\nu_2)}$$ where $(\mu_j,\nu_j){\in}\mathbb{C}$ and $\chi(\nu_1,\nu_2){=}\text{Tr}[\hat{D}_1(\nu_1)\hat{D}_2(\nu_2)\rho]$ is the two-modes Weyl characteristic function. Together with the study of Wigner function’s negativity, we also investigate the quantum correlations between the two mirrors. To overcome the problem of inferring non-classical correlations in a mixed non-Gaussian state of a CV system, which is a very demanding task due to the lack of appropriate entanglement measures, we use the same approach taken in the previous Section, which relies on the investigation of Bell inequality violations. This route is particularly viable in our case as we can take advantage of the dualism between density matrix and Wigner function for CV states. Here, one can formulate a Bell-CHSH test using the two-mode Wigner function associated to $\rho_{m}(t)$.
![(Color online) Numerically optimized violation of the Bell-CHSH inequality for the two-mirror state against $\eta{t}$ and $V$. []{data-label="WignerTCHSH"}](Fig7CHSH.eps){width=".8\columnwidth"}
To begin with, one can study the behavior of the single-mirror Wigner functions calculated for a fixed point $\mu_0$ in the other mirror phase space, i.e. $W_j(\mu_j){=}W(\mu_j,\mu_i = \mu_0)$ with $i{\neq}{j}{=}1,2$. It is seen from the inset of Fig. \[WignerT\] that, depending on the operating conditions of the system, $W_1(\mu_1)$ \[equivalently $W_2(\mu_2)$\] can be considerably negative, thereby proving its non-classical nature. This is remarkable, especially when compared to the case of a standard optomechanical setting where a mechanical mirror is coupled to the field of an optical resonator. There, in fact, it can be proven that the state of the mechanical subsystem is only [*classically squeezed*]{} and the device cannot be utilized in order to engineer non-classical states of the movable mirror [@mauro2]. Differently, using the mechanism we propose here, we have checked that the negative regions of $W_1(\mu_1)$ persist even at non-zero temperature. These considerations can be strengthened by extending them to the Wigner function of both the mechanical mirrors and studying the [*negative volume*]{} ${\cal V}_-{=}\int{d}^2\mu_1{d}^2\mu_2[|W(\mu_1,\mu_2)|{-}W(\mu_1,\mu_2)]/2$. In Fig. \[WignerT\], ${\cal V}_-$ is plotted against $V$ for ${\eta{t}=5}$, revealing that non-classicality persists up to $V{\sim}{10}$, [*i.e.*]{} well above zero temperature. We give an estimate of actual temperatures corresponding to such order of magnitude for $V$ later on.
We now pass to the study of the Bell-CHSH inequality test [@chsh] to infer non-classical correlations shared by the mechanical systems. For a two-mode bosonic system, the Bell-CHSH inequality can be re-cast in terms of the expectation values $\langle\hat\Pi_1(\mu_1)\otimes\hat\Pi_2(\mu_2) \rangle$, with the displaced parity operator $\hat{\Pi}_i(\mu_j) =\hat{D}_j(\mu_j)(-1)^{\hat{b}^\dag_j\hat{b}_j }\hat{D}_j^\dag(\mu_j)$, as before [@banaszek], in terms of which $W(\mu_1,\mu_2)=(4/\pi^2)\langle\hat\Pi_1(\mu_1){\otimes}\hat\Pi_2(\mu_2) \rangle$. The CHSH function can thus be written as $$\text{CHSH}{=}\frac{\pi^2}{4}[W(\mu_1,\mu_2){+}W(\mu'_1,\mu_2){+}W(\mu_1,\mu'_2){-}W(\mu'_1,\mu'_2)].$$ Any local realistic theory imposes the bound $|\text{CHSH}|\le{2}$. If the mechanical state is such that $|\text{CHSH}|>2$, correlations of non-classical nature are necessarily shared by the two mirrors. In Fig. \[WignerTCHSH\] we show that, although hindered by the thermal nature of the mechanical modes, the two-mirror state violates the local realistic bound up to $V=1.1$, which corresponds to ${T \approx0.1}$mK (5$\mu$K) at ${\omega_m/2\pi\sim6}$MHz (${300}$KHz), a frequency easily achievable by current experimental setups [@Groeblacher]. This shows that the mechanical state remains non-classically correlated even for thermal energies that are 10 times larger than the ground-state energy of each mirror [@commentT0].
The decreasing behavior of the CHSH function at ${T{>}0}$ can be explained by considering that, under such conditions, the coherences in the two-mirror state are suppressed. In fact, let us study the off-diagonal terms of $\rho_m(t)$ in the coherent-state basis. These are given by $\int d^2 \alpha_1 d^2 \alpha_2 P(\alpha_1,V)P(\alpha_2,V)
e^{i\varPhi(t)}|\alpha_1, \alpha_2 \rangle \langle \beta_1(t), \beta_2(t) |$ and Hermitian conjugate. As a function of $\text{Re}(\alpha_j)$, the phase factor $e^{i\varPhi(t)}$ oscillates at frequency $\eta_j t$. At $T{=}0$, $P(\alpha_j,1)$ becomes a bidimensional Dirac delta-function $\delta^2(\alpha_j)$, which sets the phase factor to unity. At the same time, by increasing $\eta_j{t}$, the components of the ECS entering state $\rho_{m}(t)$ become increasingly orthogonal, which optimizes the violation of the CHSH inequality. Differently, at finite temperature $P(\alpha_j,V)$ has a non-null width within which the increasingly oscillating time-dependent phase factor is eventually averaged to zero. This occurs more rapidly as $V$ grows.
Dissipative dynamics
--------------------
We now proceed to include the mechanical damping of the two oscillator in our analysis on the same lines followed in Sec \[subsectionOMDissipative\]. We consider the dynamics of the mirror-atom density matrix $\rho$ as driven by the weak-damping limit of the standard Brownian-motion superoperator, whose generalization to the a two mirrors system reads as $$\hat{\cal L}^V(\rho)=\sum_{j{=}1,2}\frac{\gamma}{2}(2\hat{b}^{}_j\rho\hat{b}^\dag_j{-}\{\hat{b}_j^\dag\hat{b}^{}_j,\rho\}{+}(V{-}1)[\hat{b}^{}_j\rho{-}\rho\hat{b}^{}_j,\hat{b}_j^\dag]).$$ From such master equation one can obtain with standard techniques four Fokker-Planck equations for the Wigner functions $W_{ij}$ of the mechanical state components associated the atomic operator ${\left\verti\right\rangle}{\left\langlej\right\vert}$ ($i,j{=}0,1$). The Foller-Planck equations can be written in the same form given in Eq. (\[FokkerPlanck\]), and each equation is solved by using the Gaussian ansatz in Eq. (\[ansatz\]), which is worth recalling $$W_{ij}(x,p,t){\propto}[{\text{det}}({\bf D}_{ij})]^{-1/2}{e^{{-\frac{1}{2}{{\bf q}^T_{ij}{\bf D}^{-1}_{ij}{\bf q}_{ij}}+i\Theta_{ij}(t)}}}$$ Here the vector ${\bf q}$ and the covariance matrix ${\bf D}_{ij}$ are the generalization of Eq. (\[qandcovariance\]) to the two-mode case and are given by $$\begin{split}
{\bf q}_{ij}=\left[\begin{matrix} x_1{-}\overline{x}_{1,ij}\\p_1{-}\overline{p}_{1,ij}\\x_2{-}\overline{x}_{2,ij}\\p_2{-}\overline{p}_{2,ij}\end{matrix}\right], {\bf D}_{ij}=\left[\begin{matrix}\sigma^{x_1x_1}_{ij}&\sigma^{p_1x_1}_{ij}&\sigma^{x_2x_1}_{ij}&\sigma^{p_2x_1}_{ij}\\ \sigma^{x_1p_1}_{ij}&\sigma^{p_1p_1}_{ij}&\sigma^{x_2p_1}_{ij}&\sigma^{p_2p_1}_{ij}\\ \sigma^{x_1x_2}_{ij}&\sigma^{p_1x_2}_{ij}&\sigma^{x_2x_2}_{ij}&\sigma^{p_2x_2}_{ij}\\ \sigma^{x_1p_2}_{ij}&\sigma^{p_1p_2}_{ij}&\sigma^{x_2p_2}_{ij}&\sigma^{p_2p_2}_{ij}\end{matrix}\right].
\end{split}$$ As explained in the previous section, the sum of the four term $\sum_{i,j=0,1}W_{ij}(x,p,\tau)$ gives the full non-gaussian solution of the Fokker-Planck equations, and the negativity of the Wigner function can be use to witness non-classicality.
\
![(Color online) Wigner function for a mechanical system open to dissipation. [**(a)**]{} Wigner function of a single mirror for $\mu_2{=}1{+}i$, $\eta/\gamma{=}2$, $\gamma{t}{=}V{=}1$. [**(b)**]{} ${\cal V}_-$ against $V$ and $\eta/\gamma$ for $\gamma{t}=1$ (we assume that all the relevant parameter are the same for both mirrors). []{data-label="NoisyWigner"}](Fig8aWigner1.eps "fig:"){width=".45\columnwidth"} ![(Color online) Wigner function for a mechanical system open to dissipation. [**(a)**]{} Wigner function of a single mirror for $\mu_2{=}1{+}i$, $\eta/\gamma{=}2$, $\gamma{t}{=}V{=}1$. [**(b)**]{} ${\cal V}_-$ against $V$ and $\eta/\gamma$ for $\gamma{t}=1$ (we assume that all the relevant parameter are the same for both mirrors). []{data-label="NoisyWigner"}](Fig8bNegativity.eps "fig:"){width=".45\columnwidth"}
![(Color online) Violation of the CHSH inequality as a function of $\gamma t$ for four values of $\eta/\gamma$.[]{data-label="NoisyWignerCHSH"}](Fig9CHSH.eps){width=".8\columnwidth"}
Fig. \[NoisyWigner\][**(a)**]{} reveals that $W(\mu_1,\mu_2)$ exhibits considerable regions of negativity also for $\gamma{\neq}0$. As expected, the negativity of the Wigner function increases when the coupling constant $\eta$ becomes larger than the damping rate. In this situation it is indeed possible to neglect the dissipation of the mirror and recover the purely unitary dynamics treated above. Interestingly, the Wigner function has still negative values when $\eta \sim \gamma,$ which means that in the dissipative regime the state of the two mirrors is non-classical. The decrease of ${\cal V}_-$ as $\eta/\gamma\gg{1}$ shown in Fig. \[NoisyWigner\][**(b)**]{} is simply due to our choice for the interaction time. By adjusting $t$, non-zero values of ${\cal V}_-$ are retrieved. The interplay between $\gamma,\eta$ and $t$ in setting non-classicality in the mechanical state can be best seen by studying non-locality. As shown in Fig. \[NoisyWignerCHSH\], as $\eta/\gamma$ increases for damped mechanical systems at zero-temperature, the interaction-time window has to be set so as to maximize the degree of violation of the CHSH inequality. As expected, the violation increases with the ratio between the coupling constant and the decay rate. However, large values of $\eta/\gamma$ correspond to shorter time-windows for the violation to occur. This point can be understood solving explicitly the open-system dynamics corresponding to a low-temperature bath in an alternative way.
Following the approach used in [@noiGP], we divide the evolution time as $t{=}N\delta{t}$, with $\delta t/t{\ll}{1}$ and approximate the dynamics of the total system as a sequence of the unitary dynamics ruled by $\hat{U}_t$ and a purely dissipative one. After $N$ steps, the evolved state reads $$\rho(N \delta t){=}\big(\hat{\mathcal{D}}^1_{\delta t}\hat{\mathcal{D}}^2_{\delta t}\hat{\mathcal{U}}_{\delta t}\big)^N \rho(0)$$ where we have introduced the superoperators $$\begin{split}
&\hat{\mathcal{D}}_{\delta t}^{j} \rho{=} e^{{\hat{\mathcal{L}}^{V{=}1}_j \delta t} }\rho,~~\hat{\mathcal{U}}_{\delta t} \rho{=}\hat{U}_{\delta t} \rho \hat{U}^\dag_{\delta t}
\end{split}$$ and where ${\rho(0){=}|+\rangle \langle +|{\otimes}|\alpha_1,\alpha_2\rangle \langle \alpha_1,\alpha_2|}$ is the initial state. This approach is particularly useful in treating a damped harmonic oscillator. Indeed, the action of the dissipative superoperator $\hat{\mathcal{D}}_{\delta{t}}^{j}$ on the diadic form $|\lambda\rangle\langle\sigma|$ (with ${\left\vert\lambda\right\rangle}$ and ${\left\vert\sigma\right\rangle}$ two coherent states) is given by [@phoenix] $$\hat{\mathcal{D}}_t^{j} |\lambda\rangle \langle \sigma| = \langle \sigma | \lambda\rangle^{\gamma \delta t} | \lambda e^{-\gamma \delta t} \rangle \langle \sigma e^{-\gamma \delta t}|.$$ In the limit $\delta t{\rightarrow}0$, $N{\rightarrow}\infty$ (so as to keep $t{=}N\delta{t}$ finite), we get an accurate description of the dissipation-affected dynamics[@comment2]. After the projection on the atomic part of the system, the state of the two mirrors is $$\begin{split}
\rho_m(t)&=\frac{1}{2}\big[\sum_{\mu=\alpha,\beta}|\mu_1(t),\mu_2(t) \rangle \langle \mu_1(t),\mu_2(t) |\\
&+ e^{-i \vartheta(t)-\Gamma(t) } |\beta_1(t),\beta_2(t) \rangle \langle \alpha_1(t),\alpha_2(t) |+h.c.\big]
\end{split}$$ where $$\begin{split}
&\alpha_j(t){=}\alpha_j e^{-\gamma t},~\beta_j(t){=}\alpha_j(t){+}(-1)^{j-1}i{\eta_j}(1{-}e^{-\gamma t})/{\gamma},\\
& \vartheta(t) = \sum_{j=1,2}({\eta_j}/{2\gamma})\alpha_j (1-e^{-2\gamma t}),\\
&\Gamma(t) =\sum_{j=1,2}({\eta_j^2}/2{\gamma^2})[\gamma t +\frac{1}{2}(1-e^{-2\gamma t}) - 2(1 - e^{-\gamma t})].
\end{split}$$ The analysis of the CHSH inequality using $\rho_m(t)$ leads to features consistent with the solutions gathered through the Fokker-Planck approach. As the decoherence factor $\Gamma(t)$ grows with $(\eta/\gamma)^2$, the time-window where violation of the local realistic boundary can be observed gets smaller.
Conclusions
===========
We have studied a mediated coupling mechanism between a microscopic and a mesoscopic system in two different setup involving optical cavities with movable mirrors interacting with a three level atom. The resulting dynamics drives the system into states which exhibit strong quantum features in both cases considered. The study of the first setup, involving a single mechanical oscillator and described in Sec. \[SM\], reveals strong non-local correlations between the atom and the movable mirror. Considerable violations of Bell-CHSH inequality are observed even when the thermal nature of the mirror’s initial state is taken into account. Moreover, projective measurements over the atomic system probabilistically create non-classical mixed states of the mirror. Such non-classicality, quantified by the negativity of the Wigner function, is robust against mechanical damping, while the dynamical mechanism we used ensures a good protection from other sources of noise. In the second part of the paper (Sec. \[TM\]) a cavity where both mirrors oscillate around their equilibrium positions is considered. The conditional dynamics obtain by a post-selection process on the microscopic part of the system induces truly mesoscopic quantum correlations between the two mirrors which lead again to a violation of CHSH inequality at finite temperature. In analogy with the one-mirror setup, negative values of the Wigner function are found in the dissipative regime.
Apart from stimulating the experimental achievement of non-classical states of a massive system, which will be the focus of optomechanics at the quantum level, the first part of our proposal triggers the study of microscopic-mesoscopic interplay for mechanical manipulation and control. As a significant example, the bichromatic version of the coupling Hamiltonian opens up the interesting possibility to attach a non-trivial geometric phase to the state of the mechanical system. This can be done by adjusting the amplitude of displacements and the phase $\phi$ in a way so as to realize a cyclic evolution in the mirror’s phase space, along the lines with Refs. [@phase; @Leibfried]. Such possibilities for microscopically-induced control of a mesoscopic device has already been studied elsewhere [@noiGP] and it will be the topic of further investigations. The second part of our study focus on the quantum correlations shared by two massive objects, bringing our analysis to the boundary between the quantum and the classical world. In such operating conditions, the dissipative part of the dynamics induced by damping processes in the mechanical oscillators plays an important role. It is thus clear that the achievement of the condition $\eta\sim\gamma$ is crucial in our scheme, and a comment about the possibility of reaching this regime is unavoidable. For state-of-the-art mechanical systems, typical values of $\gamma$ are in the range of a few Hz. On the other hand, the effective coupling rate $\eta$ is given by $\eta={\chi{g}^2\Omega^2}/{\delta^2\Delta^2},$ where $\chi$ is the radiation pressure interaction constant given by $\chi = (\omega_c /L) \sqrt{\hbar/2 m \omega_m}.$ For mechanical modes having $\omega_m/(2\pi){=}300$KHz and mass $m \sim 50$ng placed to a cavity of $L = 10$mm [@Groblacher1; @Groblacher2; @Groeblacher] and assuming $g^2 \Omega^2 / \delta^2 \Delta^2 \sim 0.1$ and $\omega_c \sim 10^15$Hz, a straightforward calculation shows that $\eta \sim 1.$ This value is indeed comparable to $\gamma$, thus demonstrating the achievability of the conditions required by our proposal. It is remarkable that the state of the two mechanical systems exhibits non-classical features both for one and two mirrors, in contrast with a purely optomechanical coupling between a movable mirror and a cavity field [@mauro].
Our analysis demonstrates the broad validity of our arguments, both at the single and two-mirror level. We stress the full generality of our method. Although we have illustrated it using a specific setup, the same sort of quantum-correlated state can be engineered in settings consisting of a Bose-Einstein condensate in an optomechanical cavity, two nano-mechanical resonators capacitively coupled to a Cooper-pair box or two planar superconducting resonators mutually connected via an off-resonant phase or transmon qubit [@Noi2010; @martinis; @koch]. We hope that the results of our study to trigger experimental endeavors directed towards the achievement of the working conditions discussed here.
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank R. Fazio for valuable comments. We acknowledge financial support from the National Research Foundation and Ministry of Education in Singapore, the UK EPSRC \[through a Career Acceleration Fellowship (MP) and the “New Directions for EPSRC Research Leaders" initiative, the Royal Society and the Wolfson Trust. VV is a fellow of Wolfson College, Oxford.
Adiabatic elimination {#appendix1}
=====================
We start from Eq. (\[HTot\]) and we adiabatic eliminate the excited state of the atom $|e\rangle$ and the electromagnetic field inside the cavity. In order to do so, we assume $\Delta >> \Omega, g$ and $\delta >> \Omega, g$. We notice that the only terms in the Hamiltonian involving the atomic degrees of freedom are ${\cal H}_a$ and ${\cal H}_R$. Hence, we perform first the adiabatic elimination of the exited level $|e\rangle$ of the atom. The Hamiltonian ${\cal H}_a + {\cal H}_R$ can be formally written as a $3\times3$ matrix with respect of the basis $\{|0\rangle,|1\rangle,|e\rangle\}$: $${\cal H}_a + {\cal H}_R= \begin{pmatrix} \label{atomH}
0 & 0& \Omega\\
0&0& g e^{i \Delta t} \hat{a}^\dag \\
\Omega & g e^{-i \Delta t} \hat{a} & \delta \end{pmatrix}.$$ By writing a generic state of the atom as $|\lambda\rangle = c_0 |0\rangle + c_1 |1\rangle + c_e |e\rangle$ and by setting to zero $\dot{c}_e$ in the corresponding Schrodinger equation $i \partial_t |\lambda\rangle = ({\cal H}_a + {\cal H}_R) |\lambda\rangle$, we find the effective Hamiltonian $$\begin{split}
{\cal H}_{1}= &-\frac{\Omega^2}{\delta} |0\rangle \langle 0| - \frac{\Omega g e^{-i \Delta t} }{\delta} \hat{a} |0\rangle \langle 1|\\
&- \frac{\Omega g e^{i \Delta t}}{\delta} \hat{a}^\dag |1\rangle \langle 0| - \frac{g^2}{\delta} \hat{a}^\dag \hat{a} |1\rangle\langle 1|
\end{split}$$ After the adiabatic elimination we substitute the terms ${\cal H}_a + {\cal H}_R$ in Eq. (\[HTot\]) with the expression above and the total Hamiltonian of the system reads now ${\cal H}_{\rm sys} = {\cal H}_{eff}+ {\cal H}_c + {\cal H}_m + {\cal H}_{mc} + {\cal H}_{cp}$
The next step is the elimination of the cavity field operators $\hat{a}$ and $\hat{a}^\dag$. In order to do so, we consider the equations describing the time evolution of those operators $\dot{\hat{a}} = - i[H_{sys},\hat{a}]$ and $\dot{\hat{a}}^\dag = - i[H_{sys},\hat{a}^\dag]$ and we set to zero the time derivative. Considering that $[H_{sys},\hat{a}] = [H_c, \hat{a}] + [H_{mc}, \hat{a}] + [H_{eff}, \hat{a}]$, we find that $$\begin{split}
[H_{sys}, \hat{a}] = &- \hat{a} \big(\Delta + \chi_1 (\hat{b}^\dag_1 + \hat{b}_1) + \chi_2 (\hat{b}^\dag_2 + \hat{b}_2) - \frac{g^2}{\delta} |1\rangle\langle 1| \big)\\
& + \frac{\Omega g }{\delta} e^{i \Delta t} |1\rangle \langle 0|
\end{split}$$ Setting this quantity to zero and considering that $\Delta \gg \chi, g^2/\delta$, we find $$\hat{a} = \frac{\Omega g}{\delta \Delta} e^{i \Delta t} |1\rangle \langle 0|$$ In a similar way we find that $$\hat{a}^\dag = \frac{\Omega g}{\delta \Delta} e^{-i \Delta t} |0\rangle \langle 1|$$ By substituting these equation in the expression for $H_{mc}$ we find the effective atom-mirrors interaction which reads as $$H_{am}^{eff} = \frac{\Omega^2 g^2}{\delta^2 \Delta^2} \chi |0\rangle \langle 0|(\hat{b}^\dag + \hat{b})$$ With $\eta = \frac{\Omega^2 g^2}{\delta^2 \Delta^2} \chi$ we recover the expression in Eq. (\[HEffOneMirr\]).
[99]{}
A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe and K. C. Schwab, Nature (London) [**443**]{}, 193 (2006).
J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin and J. G. E. Harris, Nature (London) [**452**]{}, 72 (2008).
A. Schliesser, P. DelÕHaye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, 243905 (2006).
T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf and N. Mavalvala, , 150802 (2007).
A. Schliesser, R. Riviere, G. Anetsberger, O. Arcizet and T. J. Kippenberg, Nature Phys. [**4**]{}, 415 (2008).
A. Schliesser, O. Arcizet, R. Riviere, G. Anetsberger and T. J. Kippenberg, Nature Phys. [**5**]{}, 509 (2009).
A. D. O’ Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis and A. N. Cleland, Nature (London) [**464**]{}, 697 (2010);
F. De Martini, F. Sciarrino, and C. Vitelli, , 253601 (2008).
P. Sekatski, N. Brunner, C. Branciard, N. Gisin and C. Simon, , 113601 (2009).
F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl and T. Esslinger, Nature (London) [**450**]{}, 268 (2007).
Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger and J. Reichel, Nature (London) [**450**]{}, 272 (2007).
A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.- S. Huang, J. Majer, S. Kumar, S. M. Girvin and R. J. Schoelkopf, Nature (London) [**431**]{}, 161 (2004).
J. Majer, J. M. Chow, J. M. Gambetta, Jens Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf, Nature (London) [**449**]{}, 443 (2007).
A. Wallraff, D. I. Schuster, A. Blais, J. M. Gambetta, J. Schreier, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, , 050501 (2007).
S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer and A. Zeilinger, Nature (London) [**444**]{}, 67 (2006).
O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard and A. Heidmann, Nature (London) [**444**]{}, 71 (2006).
D. Kleckner and D. Bouwmeester, Nature (London) [**444**]{}, 75 (2006).
O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, A. Heidmann, J.-M. Mackowski, C. Michel, L. Pinard, O. Francais and L. Rousseau , 133601 (2006).
S. Gröblacher, K. Hammerer, M. R. Vanner and M. Aspelmeyer, Nature (London) [**460**]{}, 724 (2009).
S. Gröblacher, J. B. Hertzberg, M. R. Vanner, G. D. Cole, S. Gigan, K. C. Schwab and M. Aspelmeyer, Nature Phys. [**5**]{}, 485 (2009).
J. Chan, T. P. Mayer Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, Nature (London) [**478**]{}, 89 (2011).
W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, , 130401 (2003).
A. Ferreira, A. Guerreiro, and V. Vedral, , 060407 (2006).
M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert and M. Aspelmeyer, , 250401 (2007).
A. D. Armour, M. P. Blencowe, and K. C. Schwab, , 148301 (2002).
P. Rabl, A. Shnirman, and P. Zoller, , 205304 (2004).
L. Tian, , 195411 (2005).
D. A. Rodrigues, J. Imbers, and A. D. Armour, , 067204 (2007).
M. Koch, C. Sames, M. Balbach, H. Chibani, A. Kubanek, K. Murr, T. Wilk, and G. Rempe, , 023601 (2011).
S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, Nature [**484**]{}, 195 (2012).
H. P. Specht, C. Nölleke, A. Reiserer, M. Uphoff, E. Figueroa, S. Ritter, and G. Rempe, Nature [ **473**]{}, 190 (2011).
T. S. Monteiro, J. Millen, G. A. T. Pender, F. Marquardt, D. Chang and P. F. Barker, New J. Phys. [**15**]{}, 015001, (2012).
N. Kiesel, F. Blaser, U. Delic, D. Grass, R. Kaltenbaek, and M. Aspelmeyer, arXiv:1304.6679 (2013).
A. Xuereb, and M. Paternostro, , 023830 (2013)
C. K. Law, [Phys. Rev. A]{} [**49**]{}, 433 (1994).
D. F. Walls and G. J. Milburn, [*Quantum Optics*]{} (Springer, Heidelberg, 1994).
S. Mancini, D. Vitali, and P. Tombesi, , 137901 (2003).
G. Vacanti, R. Fazio, M. S. Kim, G. M. Palma, M. Paternostro and V. Vedral, Phys. Rev. A [**85**]{}, 022129 (2012).
G. J. Milburn, S. Schneider, and D. F. V. James, Fortschr. Phys. [**48**]{}, 801 (2000).
D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband and D. J. Wineland, Nature (London) [**422**]{}, 412 (2003).
L. Davidovich, A. Maali, M. Brune, J. M. Raimond and S. Haroche, , 2360 (1993).
H. Jeong and T. C. Ralph, [Phys. Rev. Lett.]{} [**97**]{}, 100401 (2006).
H. Jeong, M. Paternostro, and T. C. Ralph, , 060403 (2009).
K. Banaszek and K. Wódkiewicz, [Phys. Rev. A]{} [**58**]{}, 4345 (1998).
K. Wódkiewicz, New. J. Phys. [**2**]{}, 21 (2000).
N. Spagnolo, C. Vitelli, M. Paternostro, F. De Martini, and F. Sciarrino, Phys. Rev. A [**84**]{}, 032102 (2011).
J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, , 880 (1969).
S. Bose, I. Fuentes-Guridi, P. L. Knight and V. Vedral, , 050401 (2001).
A. Peres, , 1413 (1996).
M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A [**223**]{}, 1 (1996).
J. Lee, M. S. Kim, Y. J. Park and S. Lee, J. Mod. Opt. [**47**]{}, 2151 (2000).
M. B. Plenio, Phys. Rev. Lett. [**95**]{}, 090503 (2005).
For $g\Omega/\delta\Delta\sim0.1$, cavity-length $\sim0.3$mm and a light mirror (mass $\sim15$ng), $\chi\sim{2.8}$KHz, which corresponds to $\zeta\sim0.1$KHz. The elimination of ${\left\verte\right\rangle}_a$ and the cavity field, whose damping rate is quenched by the off-resonant coupling, make the proposed dynamics quite feasible.
B. C. Sanders, [Phys. Rev. A]{} **45**, 6811 (1992).
H. Jeong and M. S. Kim, [Phys. Rev. A]{} [**65**]{}, 042305 (2002).
M. Paternostro, Phys. Rev. Lett. [**106**]{}, 183601 (2011) .
S. Gröblacher (Private communication).
Quite expectedly, the CHSH inequality is increasingly violated in time at $T=0$, asymptotically reaching Tsirelson’s bound.
S. J. D. Phoenix, [Phys. Rev. A]{} [**41**]{}, 5132 (1990).
Indeed, for the particular system considered here the superoperators $\hat{\mathcal{D}}_{\delta t}$ and $\hat{\mathcal{U}}_{\delta t}$ commute when $\delta t \rightarrow 0$ and this approach gives the exact dynamics of the system.
M. Paternostro, G. De Chiara, and G. M. Palma, [Phys. Rev. Lett.]{} [**104**]{}, 243602 (2010); G. De Chiara, M. Paternostro, and G. M. Palma, Phys. Rev. A [**83**]{}, 052324 (2011).
J. M. Martinis, S. Nam, J. Aumentado and C. Urbina, [Phys. Rev. Lett.]{} [**89**]{}, 117901 (2002);
J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf, [Phys. Rev. A]{} [**76**]{}, 042319 (2007).
|
---
abstract: 'We examine relation between neutrino oscillation parameters and prediction of lepton flavor violation, in light of deviations from tri-bimaximal mixing. Our study shows that upcoming experimental searches for lepton flavor violation process can provide useful implications for neutrino mass spectrum and mixing angles. With simple structure of heavy right-handed neutrino and supersymmetry breaking sectors, the discovery of $\tau\to \mu\gamma$ decay determines neutrino mass hierarchy if large (order 0.1) reactor angle is established.'
---
[KYUSHU-HET-116]{}
[**Probing neutrino masses and tri-bimaximality\
with lepton flavor violation searches**]{}
Kentaro Kojima$^{a,}$[^1] and Hideyuki Sawanaka$^{b,}$[^2]
[*$^a$ Center for Research and Advancement in Higher Education,\
Kyushu University, Fukuoka 810-8560, Japan\
$^b$ Department of Physics, Kyushu University, Fukuoka 812-8581, Japan*]{}
[(January, 2009)]{}
Introduction {#sec:int}
============
In the last decades, one of the most striking developments in particle physics beyond the standard model (SM) is the experimental establishment [@SK98] of neutrino masses and the large mixing property, which is quite different from the small mixing in the quark sector. Neutrino oscillation experiments have revealed neutrino mass-squared differences and its flavor mixing angles. Notably, a simple form of mixing matrix, referred to as tri-bimaximal mixing, is well descriptive of the observed mixing structure [@tbm]. Vast numbers of flavor models have been proposed in order to derive the tri-bimaximal mixing [@model]; thus, from a theoretical viewpoint, it is one of the most important subjects to realize difference between tri-bimaximal and observed mixing angles. In addition, recent results of the three-flavor global data analysis [@Gfit; @13mix] indicate non-zero $\theta_{13}$; as the best-fit value, not so small one $\sin\theta_{13}\simeq {\cal O}(0.1)$ is obtained. Therefore, it seems interesting to examine deviations of neutrino mixing angles from the tri-bimaximal pattern [@deviation], which leads to $\sin^2\theta_{12}=1/3$, $\sin^2\theta_{23}=1/2$, and $\sin^2\theta_{13}=0$, for the coming future precise experiments.
The present knowledge of neutrino parameters (i.e. masses, mixing angles and phases) is not only important for low-energy characters of neutrinos, but also a key ingredient of the origin of flavor structure in the SM fermions. It is thus meaningful to make clear the relation between neutrino parameters and high-energy phenomenologies. Among them, charged lepton flavor violation (LFV) process would give an intriguing clue, since supersymmetry (SUSY) and the seesaw mechanism [@seesaw] could enhance LFV as reachable in near future experiments [@lfv]. The seesaw mechanism naturally provides desired neutrino mass scale, and predicted LFV fractions are affected by low-energy neutrino parameters and heavy Majorana masses via SUSY breaking terms [@rdep; @lfvfl].
In this Letter, we examine the relation between neutrino parameters and LFV prediction, in light of the tri-bimaximal mixing and the recent precision oscillation data. We use a particular parametrization [@tmin] of the MNS matrix [@mns], where the tri-bimaximal mixing is taken as its zeroth order approximation, and give a detailed analysis of LFV prediction especially for a simple case of right-handed Majorana mass and SUSY breaking structures. Our study shows that upcoming experimental LFV searches provide useful implications for neutrino mass spectrum and mixing angles. If we possess simple structure of heavy Majorana masses, future discovery of $\tau\to \mu\gamma$ implies that inverted hierarchy (IH) and quasi-degenerate (QD) neutrino mass spectra are inconsistent with a large reactor angle of order 0.1, though normal hierarchy (NH) is still allowed.
The Letter is organized as follows. In Section \[sec:Neu\], we introduce a useful parametrization [@tmin] of the MNS matrix for examining how the mixing angles deviate from the tri-bimaximal ones. In Sections \[sec:LFV\] and \[sec:LFV-TB\], relation between neutrino parameters and LFV prediction in a literature of the minimal supersymmetric standard model (MSSM) with the seesaw mechanism is studied. Detailed analysis is given in Section \[sec:LFV-TB\] by using the parametrization. Section \[sec:clarif\] is devoted to study implications for neutrino parameters with future LFV searches. We summarize our result in Section \[sec:conclusion\].
Neutrino parameters and tri-bimaximal mixing {#sec:Neu}
============================================
Current data obtained by neutrino oscillation experiments is consistent with a simple mixing structure called tri-bimaximal mixing [@tbm]. We adopt a particular parametrization proposed in Ref. [@tmin] to describe the MNS matrix by deviations from exact tri-bimaximal mixing angles. It is helpful to systematically analyze the neutrino tri-bimaximality.
In the basis where the charged lepton mass matrix is diagonal, neutrino mass matrix is given by $M_\nu= U {\cal D}_m U^T$, where ${\cal D}_m=diag(m_1, m_2, m_3)$ with positive neutrino mass eigenvalues $m_{1, 2, 3}$ and $U$ is the unitary lepton mixing matrix. Three mixing angles and phases are involved in the matrix $U$; using the standard parametrization [@PDG08], it can be expressed as $$\begin{aligned}
U &=&
\begin{pmatrix}
c_{12}c_{13}&s_{12}c_{13}&s_{13} e^{-i \delta}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13} e^{i\delta}&
c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&
s_{23}c_{13}\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&
-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&
c_{23}c_{13}
\end{pmatrix}
P_M,
\label{mnspdg}\end{aligned}$$ where $c_{ij}\equiv\cos\theta_{ij}$, $s_{ij}\equiv\sin\theta_{ij}$, and $\delta$ is the Dirac CP violating phase. $P_M$ stands for the diagonal phase matrix which involves two Majorana phases.
Recent progress in neutrino experiments greatly increase data for neutrino masses and mixing angles. The updated result of the three-flavor global data analysis [@Gfit] indicates the following best-fit values with 3$\sigma$ intervals of three (solar, atmospheric, reactor) mixing angles and two (solar, atmospheric) mass squared differences: $$\begin{aligned}
\label{mixex}
\sin^2\theta_{12}\;=\;0.304^{+0.066}_{-0.054}, &&
\sin^2\theta_{23}\;=\;0.50^{+0.17}_{-0.14},\qquad
\sin^2\theta_{13}\;=\;0.010\ (\le 0.056),\\[2mm]\notag\Delta m_{\rm sol}^2&\equiv& \, m_2^2-m_1^2\ \;=\;
(7.65\pm^{0.69}_{0.60})\times 10^{-5}~{\rm eV}^2,\\\label{msqdiff}
\Delta m_{\rm atm}^2&\equiv&|m_3^2-m_1^2|\;=\;
(2.40\pm^{0.35}_{0.33})\times 10^{-3}~{\rm eV}^2. \end{aligned}$$ The Dirac phase $\delta$ has not yet been constrained by the experimental data.
The flavor structure could be determined by profound principles such as flavor symmetry in a high-energy regime. Although the origin of the flavor is unrevealed, the current neutrino mixing angles are known to be consistent with a simple mixing matrix $U_{\rm TB}$, $$\begin{aligned}
U_{\rm TB} &=&
({\cal R}_{\rm TB})_{23}({\cal R}_{\rm TB})_{12}\;=\;
{1\over \sqrt{6}}
\begin{pmatrix}
2&\sqrt{2}&0\\
-1&\sqrt{2}&\sqrt{3}\\
1&-\sqrt{2}&\sqrt{3}
\end{pmatrix},
\label{tbm}\end{aligned}$$ where two (non-diagonal) rotational matrices are given by $$\begin{aligned}
({\cal R}_{\rm TB})_{23}&=&
\begin{pmatrix}
1&0&0\\
0&{1/\sqrt{2}}&{1/\sqrt{2}}\\
0&{-1/\sqrt{2}}&{1/\sqrt{2}}
\end{pmatrix}, \qquad
({\cal R}_{\rm TB})_{12}\;=\;
\begin{pmatrix}
{2/\sqrt{6}}&{1/ \sqrt{3}}&0\\
{-1/ \sqrt{3}}&{2/ \sqrt{6}}&0\\
0&0&1
\end{pmatrix}. \end{aligned}$$ Although the matrix $U_{\rm TB}$ approximately describes the MNS matrix , it is an interesting subject to investigate the difference between them in reality. For this purpose, the following unitary matrix is useful to parametrize the MNS mixing structure: $$\begin{aligned}
U &=&
({\cal R}_{\rm TB})_{23}
\begin{pmatrix}
c_x c_z&s_x c_z&s_z e^{-i \delta}\\
- s_x c_y - c_x s_y s_z e^{i\delta}&
c_x c_y - s_x s_y s_z e^{i\delta}&s_y c_z\\
s_x s_y - c_x c_y s_z e^{i\delta}&
- c_x s_y - s_x c_y s_z e^{i\delta}&c_y c_z
\end{pmatrix}
({\cal R}_{\rm TB})_{12}
P_M,
\label{mnstb}\end{aligned}$$ where $c_w$ and $s_w$ denote $\cos\epsilon_w$ and $\sin\epsilon_w$ ($w=x, y, z$), respectively.[^3] In the limit where $\epsilon_{x,y,z}\to 0$, $U$ in goes back to $U_{\rm TB}$ except $P_M$. With this parametrization, experimental data indicates $$\begin{aligned}
-0.092\le \;\epsilon_{x}\;\le 0.038,\qquad
-0.14\le \;\epsilon_{y}\;\le 0.17,\qquad
|\epsilon_z|\; \le 0.239,
\label{devrange} \end{aligned}$$ in its $3\sigma$ ranges. Since the deviation parameters are much suppressed than ${\cal O}(1)$, expansions around $\epsilon_{x,y,z}=0$ could give a good approximation to physical quantities related to lepton mixing angles.
LFV in MSSM with type-I seesaw {#sec:LFV}
==============================
Let us study LFV prediction in MSSM with heavy right-handed Majorana neutrinos for the type-I seesaw mechanism [@seesaw]. The relevant part of the MSSM superpotential is given by $$\begin{aligned}
W_{\rm lepton}&=&
L_i(Y_e)_{ij}\bar e_jH_d+L_i(Y_\nu)_{ij}\bar \nu_jH_u
+{1\over 2}\bar \nu_i(M_R)_{ij}\bar \nu_j, \end{aligned}$$ where $Y_e$ and $Y_\nu$ are charged lepton and neutrino Yukawa matrices. Superfields $L_i$, $\bar e_i$, $\bar \nu_i$ ($i=1,2,3$) and $H_{u(d)}$ include lepton doublets, charged leptons, right-handed neutrinos and up (down)-type Higgs doublet, respectively. $M_R$ gives Majorana mass matrix of the right-handed neutrinos, whose scale is assumed to be much larger than the electroweak scale ($\sim 10^{2}$ GeV). It is also noted that scale of $M_R$ is required to be smaller than the grand unified theory (GUT) scale in order to reproduce known solar and atmospheric neutrino mass scales unless $Y_\nu$ is much larger than ${\cal O}(1)$. Without loss of generality, we take a basis where $M_R$ is diagonal.
SUSY breaking should be incorporated in realistic models, since it is not exact symmetry in Nature. We thus introduce SUSY breaking terms, which in general could be new sources of the flavor violation. Although several breaking scenarios have been proposed, one of the most economical and predictive ansatz is to assume the universal form of soft terms in a high-energy regime. In this case, universal SUSY breaking parameters are listed as scalar mass $m_0$, trilinear coupling $A_0$, gaugino mass $M_{1/2}$ and the Higgs bilinear coupling. We refer to these SUSY breaking parameters as their GUT scale values.
The flavor violation in the supersymmetric sector is transmitted to SUSY breaking sector thorough renormalization group (RG) evolution between GUT and heavy Majorana mass scales. At a low-energy regime, one-loop diagrams with SUSY particles give leading corrections to the LFV processes; the branching fractions are approximately written as $$\begin{aligned}
{\cal B}(\ell_j\to \ell_i+\gamma)&\simeq& {\alpha^3\over G_F^2m_S^8}
\left[{3m_0^2+A_0^2\over 8\pi^2v_H^2\sin^2\beta}\right]^2\tan^2\beta
|B_{ij}|^2,
\label{brap}\end{aligned}$$ where $\alpha$ and $G_F$ are the fine-structure and the Fermi coupling constants, $v_H\simeq 174~{\rm GeV}$, and $\beta$ parametrizes the ratio between vacuum expectation values of the two Higgs scalars in $H_{u,d}$. The mass parameter $m_S$ is a typical mass scale of SUSY particles. Note that flavor indices only appear in $B_{ij}$ as $$\begin{aligned}
B_{ij}&=&v_H^2\sin^2\beta\sum_{k=1}^3 (Y_\nu)_{ik}(Y_\nu^\dag)_{kj}
\ln{M_G\over {M_R}_k},
\label{fl}\end{aligned}$$ where ${M_R}_k$ denotes the $k$-th eigenvalue of $M_R$, and $M_G\simeq 10^{16}$ GeV is the GUT scale. Thus the flavor dependence in LFV branching fractions is completely involved in , that is SUSY breaking parameters do not affect the flavor structure in the approximate formula of LFV prediction .
Although it is generally difficult to reconstruct the combination of neutrino Yukawa matrix in from low-energy data, if heavy Majorana neutrinos have an approximately degenerate mass $M_U$ and there are no large CP phases except the Dirac phase $\delta$ in the neutrino sector, the branching fractions are tightly connected to neutrino parameters as $$\begin{aligned}
{\cal B}(\ell_j\to \ell_i+\gamma)&\simeq& {\alpha^3\over G_F^2m_S^8}
\left[{3m_0^2+A_0^2\over 8\pi^2v_H^2\sin^2\beta}\right]^2\tan^2\beta
M_U^2\left(\ln{M_G\over M_U}\right)^2
|b_{ij}|^2, \\\label{cij}
&& b_{ij}\;=\;\sum_{k=1}^3m_k(U^*)_{ik}(U^T)_{kj}. \end{aligned}$$ Note that $P_M$ in disappears in the factor $b_{ij}$. Here the low-energy neutrino parameters, namely their masses, mixing angles and the Dirac phase, are completely involved in (\[cij\]); thus the LFV branching fractions depend on SUSY parameters in a flavor independent manner. We will discuss more general cases with non-degenerate Majorana masses in the end of the next section.[^4]
LFV prediction around tri-bimaximal mixing {#sec:LFV-TB}
==========================================
In this section, we discuss the relation between neutrino parameters and prediction of LFV processes. The parametrization allows us to handle deviations from tri-bimaximal mixing in a systematic way. In a definite framework for high-energy theory, we analyze how LFV prediction depends on small deviation parameters with current neutrino oscillation data.
#### Analytical results
Applying the parametrization to describe the LFV prediction, $b_{ij}$ is explicitly written as $$\begin{aligned}
\notag
b_{12}&=&{m_{12}\over 6\sqrt{2}}c_z(c_y-s_y)(2\sqrt{2}c_{2x}+s_{2x})
+{e^{i\delta}\over 6\sqrt{2}}c_zs_z(c_y+s_y)
\left[3m_{123}+m_{12}(c_{2x}-2\sqrt{2}s_{2x})\right],\\[2mm]
\notag
b_{13}&=&-{m_{12}\over 6\sqrt{2}}c_z(c_y+s_y)(2\sqrt{2}c_{2x}+s_{2x})
+{e^{i\delta}\over 6\sqrt{2}}c_zs_z(c_y-s_y)
\left[3m_{123}+m_{12}(c_{2x}-2\sqrt{2}s_{2x})\right],\\[2mm]\notag
b_{23}&=&{m_{123}\over 4}c_z^2(c_y^2-s_y^2)
-{m_{12}\over 12}(1+s_z^2)(c_y^2-s_y^2)
(c_{2x}-2\sqrt{2}s_{2x})
\\&&\hspace{3cm}
-{e^{i\delta}\over 12}m_{12}s_z\left[(c_y-s_y)^2
-(c_y+s_y)^2e^{-2i\delta}\right](2\sqrt{2}c_{2x}+s_{2x}),
\label{cijfull}\end{aligned}$$ where $c_{2x}=\cos2\epsilon_x$ and $s_{2x}=\sin2\epsilon_x$. Note that neutrino masses appear just as particular combinations $$\begin{aligned}
m_{12}\;=\;m_2-m_1,\qquad m_{123}\;=\;2m_3-m_2-m_1. \end{aligned}$$ It is also noted that $b_{12}(\epsilon_y, \epsilon_z; \delta)
=-b_{13}(-\epsilon_y, -\epsilon_z; \delta)
=-b_{13}(-\epsilon_y, \epsilon_z; \delta\pm\pi)$ holds between $b_{12}$ and $b_{13}$; the relation is just the $\mu$-$\tau$ symmetric property in the tri-bimaximal limit.
As mentioned in Section \[sec:Neu\], the observed values of lepton mixing angles are consistent with the tri-bimaximal mixing pattern. Focusing on the LFV prediction with the tri-bimaximal limit $(\epsilon_x,\epsilon_y,\epsilon_z)\to (0,0,0)$ in , we obtain $$\begin{aligned}
\notag
{\cal B}(\mu\to e\gamma)&\propto&|b_{12}|^2\;\to\;
\Bigl({m_{12}\over 3}\Bigr)^2\;=\;{m_{12}^2\over 9},
\\\notag
{\cal B}(\tau\to e\gamma)&\propto&|b_{13}|^2\;\to\;
\Bigl({m_{12}\over 3}\Bigr)^2\;=\;{m_{12}^2\over 9},
\\\label{lfvtbl}
{\cal B}(\tau\to \mu\gamma)&\propto&|b_{23}|^2
\;\to\;
{1\over 16}\left(m_{123}-{m_{12}\over 3}\right)^2\;=\;
{m_{123}^2\over 16}-{m_{123}m_{12}\over 24}+{m_{12}^2\over 144}, $$ where only $|b_{23}|^2$ depends on $m_{123}$ and $|b_{12}|^2=|b_{13}|^2$ holds.
Experimental values of solar and atmospheric neutrino mass differences indicate that $m_{123}$ is much larger than $m_{12}$. The ratio $\hat{m}\equiv m_{12}/m_{123}$ depends on the neutrino mass spectrum. For NH of neutrino masses ($m_1<m_2< m_3$), we obtain $$\begin{aligned}
\label{mhatn}
\hat{m}&=&
{\sqrt{\Delta m_{\rm sol}^2+m_1^2}-m_1\over
2\sqrt{\Delta m_{\rm atm}^2+m_1^2}-\sqrt{\Delta m_{\rm sol}^2+m_1^2}-m_1}
\;\to \;{1\over 2}\sqrt{\Delta m_{\rm sol}^2\over\Delta m_{\rm atm}^2}, \end{aligned}$$ while for the IH case ($m_3< m_1<m_2$), we have $$\begin{aligned}
\label{mhati}
\hat{m}&=&
{\sqrt{\Delta m_{\rm atm}^2+\Delta m_{\rm sol}^2+m_3^2}
-\sqrt{\Delta m_{\rm atm}^2+m_3^2}\over
2m_3-\sqrt{\Delta m_{\rm atm}^2+\Delta m_{\rm sol}^2+m_3^2}-
\sqrt{\Delta m_{\rm atm}^2+m_3^2}}
\;\to \;-{1\over 4}{\Delta m_{\rm sol}^2\over\Delta m_{\rm atm}^2}. \end{aligned}$$ The last expressions in and imply the massless limit of the lightest neutrino, where $\Delta m_{\rm sol}^2/\Delta m_{\rm atm}^2\simeq 0.04$ with the present data. We plot in Fig. \[massdifs\] the ratio $\hat{m}$ as the function of the lightest neutrino mass eigenvalue $m_{\rm ref}$. It shows that the ratio becomes relatively large (small) if the neutrino mass spectrum is NH (IH). With being large value of $m_{\rm ref}$, namely QD mass spectrum limit, $\hat{m}$ takes similar values for normal and inverse ordering cases.
![Absolute values of the ratio between $m_{12}$ and $m_{123}$ are shown as functions of reference neutrino mass scale $m_{\rm ref}$; for the NH (IH) case we take $m_{\rm ref}=m_1(m_3)$. Colored bands indicate the predicted values with 3$\sigma$ ranges of input parameters $\Delta m_{\rm sol}^2$ and $\Delta m_{\rm atm}^2$. Lines in the bands correspond to the plot with the best-fit values in . []{data-label="massdifs"}](mass.eps){width="7cm"}
In the tri-bimaximal limit, ${\cal B}(\mu\to e\gamma)$ and ${\cal B}(\tau \to e\gamma)$ in are much suppressed than ${\cal B}(\tau\to \mu\gamma)$ since they do not involve $m_{123}$. This implies that these processes are sensitive to deviations from the tri-bimaximal mixing. As argued in Section \[sec:Neu\], the deviation parameters in are sufficiently small, so that we can use them as expansion parameters in LFV branching fractions. Up to ${\cal O}(\epsilon_w^2)$, one can obtain the following expressions: $$\begin{aligned}
\notag
|b_{12}|^2&\simeq &\tilde m_{12}^2+\sqrt{2}\tilde m_{12}^2\epsilon_x
-2\tilde m_{12}^2\epsilon_y+{\tilde m_{12}\over\sqrt{2}}(\tilde m_{12}+m_{123})
\epsilon_z\cos\delta -{7\over 2}\tilde m_{12}^2\epsilon_x^2
-2\sqrt{2}\tilde m_{12}^2\epsilon_x\epsilon_y\\\label{c122}
&&
-{1\over 8}(7\tilde m_{12}^2-2\tilde m_{12}m_{123}-m_{123}^2)\epsilon_z^2
-{\tilde m_{12}\over 2}(7\tilde m_{12}-m_{123})\epsilon_x\epsilon_z\cos\delta,\\
\notag
|b_{23}|^2&\simeq &{1\over 16}(\tilde m_{12}-m_{123})^2
-{\tilde m_{12}\over\sqrt{2}}(\tilde m_{12}-m_{123})\epsilon_x
+{\tilde m_{12}\over 4}(7\tilde m_{12}+m_{123})\epsilon_x^2
-{1\over 4}(\tilde m_{12}-m_{123})^2\epsilon_y^2\\\label{c232}
&&
+{1\over 8}(\tilde m_{12}^2-m_{123}^2+16\tilde m_{12}^2\sin^2\delta)\epsilon_z^2
-\sqrt{2}\tilde m_{12}(\tilde m_{12}-m_{123})\epsilon_y\epsilon_z\cos\delta, \end{aligned}$$ where $\tilde m_{12}\equiv m_{12}/3$ and $|b_{13}|^2(\epsilon_y, \epsilon_z\cos\delta)
=|b_{12}|^2(-\epsilon_y, -\epsilon_z\cos\delta)$.
From the fact that $m_{123}$ is much larger than $m_{12}$, the typical correlation between neutrino parameters and the branching fractions can be understood. In , the terms which remain in the tri-bimaximal limit are proportional to $m_{12}^2$, and $m_{123}$ always appears with involving the deviation parameters, especially $\epsilon_z$. Thus $|b_{12(13)}|^2$, namely ${\cal B}(\mu(\tau)\to e\gamma)$, is sensitive to the parameters, while ${\cal B}(\tau\to \mu\gamma)$ does not have so large dependence on them since $m_{123}^2$ is the dominant term in $|b_{23}|^2$. Moreover, the leading contributions in $|b_{12}|^2$ and $|b_{23}|^2$ can be expressed as $$|b_{12}|^2\;\simeq\;
{m_{123}^2\over 9}
\left(
\hat m^2+{3\over \sqrt{2}}\hat m \epsilon_z \cos \delta+{9\over 8}\epsilon_z^2
\right)+\cdots,
\qquad
|b_{23}|^2\;\simeq\;
{m_{123}^2\over 16}+\cdots,
\label{ld}$$ since the ratio $\hat{m}$ is also the small quantity as well as deviation parameters. One can find that $\epsilon_z$ plays an important role to determine ${\cal B}(\mu(\tau)\to e\gamma)$ and that $\epsilon_x$ and $\epsilon_y$ are less effective because they only appear as sub-leading corrections. The $\epsilon_z$ dependence is controlled by the neutrino mass spectrum and the Dirac phase through $\hat{m}$ and $\cos\delta$. For example, $|b_{12}|^2$ is minimized at $$\begin{aligned}
\epsilon_z&=&\theta_{13}\;\simeq\;-{2\sqrt{2}\over 3}\hat m\cos\delta
\;\simeq\;-\hat m\cos\delta.
\label{ez-min}\end{aligned}$$ Note that the value of $\epsilon_z$ in has significant distinction between NH and IH by the magnitude of $\hat m$. We can check these properties with numerical analysis.
#### Numerical searches
We discussed the neutrino parameter dependence of the LFV prediction using the approximate formula in . To make our study more complete, we proceed to numerical examination of the LFV prediction. Here we take SPS1$_a$ [@sps1a] for SUSY particle mass spectrum; SUSY breaking parameters are fixed as $(m_0,M_{1/2},A_0)=(100,250,-100)$ GeV at the GUT scale and $\tan\beta=10$. SUSY parameter dependence is mostly flavor blind, and has been greatly studied [@lfv]. If one takes other types of SUSY mass spectrum, following results do not alter as long as the universality of the SUSY breaking is assumed.
Given set of SUSY parameters, we numerically estimate RG evolutions between GUT and electroweak scale taking heavy right-handed neutrinos into account. Above the right-handed neutrino mass scale, which is taken as $M_U=10^{14}$ GeV in the analysis, the right-handed neutrinos are decoupled with the theory. Two-loop RG equations for gauge and Yukawa couplings, and one-loop ones for the soft SUSY breaking parameters are numerically solved. LFV fractions are estimated with one-loop diagrams in the SUSY particle mass eigenbasis rather than the mass-insertion approximation.
As stressed in the analytical discussion, among the three deviation parameters in , $\epsilon_z$ is crucial for the prediction of $\mu(\tau)\to e\gamma$ branching fractions. To see its dependence, we plot the LFV predictions as the functions of $\epsilon_z$ in Fig. \[ez\].
![Predictions of LFV branching fractions as the functions of $\epsilon_z$ are shown. Black, red and gray plots correspond to ${\cal B}(\mu\to e\gamma)$, ${\cal B}(\tau\to \mu\gamma)$ and ${\cal B}(\tau\to e\gamma)$, respectively. Each figure corresponds to different neutrino mass spectrum: from left to right figures, NH ($m_1=10^{-4}$ eV), IH ($m_3=10^{-4}$ eV), and QD ($m_1=10^{-1}$ eV) mass spectra are taken, respectively. The mass squared differences are fixed to central values in and the Dirac phase is taken as $\delta=0$. $\epsilon_x$ and $\epsilon_y$ are scanned with $3\sigma$ ranges in . []{data-label="ez"}](ez.eps){width="15.cm"}
The prediction of ${\cal B}(\tau\to \mu\gamma)$ is insensitive to size of $\epsilon_z$, by contrast that of ${\cal B}(\mu(\tau)\to e\gamma)$ strongly depends on. The branching fraction ${\cal B}(\mu\to e\gamma)$ is minimized around $\epsilon_z\simeq -0.1 (+0.01)$ for the case with NH (IH and QD) mass spectrum, as shown in . These results are consistent with the previous argument using the analytical expressions. Note that ${\cal B}(\mu(\tau)\to e\gamma)$ is highly suppressed than ${\cal B}(\tau \to \mu\gamma)$ around the minima. This is an important point to extract implications for neutrino parameters from future LFV searches, and we discuss the issue in the next section.
#### Effects of heavy Majorana mass hierarchies
In general, heavy Majorana masses have non-degeneracy and it affects LFV prediction. In order to incorporate the non-degeneracy of $M_R$, it is convenient to focus on $B_{ij}$ in rather than $b_{ij}$ in . $B_{ij}$ can be generally rewritten as follows [@rdep]: $$\begin{aligned}
B_{ij}
&=&
\sum_{k=1}^3
(U^* {\cal D}_{\sqrt{m}}R{\cal D}_{\sqrt{M_R}})_{ik}
({\cal D}_{\sqrt{M_R}}R^\dag{\cal D}_{\sqrt{m}}U^T)_{kj}
\ln{M_G\over M_{Rk}},
\label{yygen}\end{aligned}$$ where ${\cal D}_{\sqrt m}=diag(\sqrt{m_1},\sqrt{m_2},\sqrt{m_3})$, ${\cal D}_{\sqrt {M_R}}=diag(\sqrt{M_{R1}},\sqrt{M_{R2}},\sqrt{M_{R3}})$ and a complex matrix $R$ satisfies $RR^T={\bf 1}$. The additional mixing matrix $R$ appears in $Y_\nu Y_\nu^\dag$ because the right-handed neutrino mixing is not unphysical. As a result, LFV prediction generally depends on the mixing structure of $R$, as minutely studied in [@rdep].
If the right-handed mixing is not important for low-energy neutrino parameters, namely $R\simeq {\bf 1}$, $B_{ij}$ is simplified. Especially $R\to {\bf 1}$ leads to $$\begin{aligned}
B_{ij}&\simeq&\sum_{k=1}^3
{\cal M}_{k}
(U^*)_{ik}
(U^T)_{kj}
, \qquad
{\cal M}_i\;\equiv \;
m_iM_{Ri}\ln{M_G\over M_{Ri}}.
$$ It is obvious that $B_{ij}$ has a similar form to $b_{ij}$, where low-energy neutrino mass $m_i$ in is replaced by a combination of light and heavy neutrino masses. Hence, explicit expression of $B_{ij}$ is easily obtained by . In this case, LFV prediction depends on both neutrino masses through ${\cal M}_{12}={\cal M}_2-{\cal M}_1$ and ${\cal M}_{123}=2{\cal M}_3-{\cal M}_2-{\cal M}_1$. In particular, the new mass ratio $\hat {\cal M}={\cal M}_{12}/{\cal M}_{123}$ is essential to determine the neutrino parameter dependence. With typical mass hierarchies of $M_R$, LFV prediction has examined in Ref. [@MReff]. Corrections to $R={\bf 1}$ also affect LFV prediction when right-handed neutrinos have large non-degeneracy. For example, if $R$ matrix in contains a small mixing angle $\kappa_{ij}$,[^5] then $\mu\to e\gamma$ prediction is modified with including the counterpart of ; up to ${\cal O}(\kappa_{ij})$ contribution except for ${\cal O}(\kappa_{ij}\cdot\epsilon_{x,y,z})$ terms, the prediction is explicitly written as follows: $$\begin{aligned}
|B_{12}|^2\;\simeq \; {{\cal M}_{123}^2\over 9}
\bigg[\hat{\cal M}^2+{3\over \sqrt{2}}\hat{\cal M} \epsilon_z\cos\delta
+{9\over 8}\epsilon_z^2
+{\hat{\cal M}}
{\sqrt{2}\Delta_{12}+\sqrt{6}\Delta_{23}+2\sqrt{3}\Delta_{13}
\over {\cal M}_{123}}
+\cdots
\bigg],
\label{B12sq}\end{aligned}$$ where we use the notation $$\begin{aligned}
R\;=\;
\begin{pmatrix}
1&\kappa_{12}&\kappa_{13}\\
-\kappa_{12}&1&\kappa_{23}\\
-\kappa_{13}&-\kappa_{23}&1
\end{pmatrix},\quad
\Delta_{ij}\;\equiv \;\kappa_{ij}\sqrt{m_im_j}\left(
M_{Rj}\log{M_G\over M_{Rj}}-M_{Ri}\log{M_G\over M_{Ri}}\right). \end{aligned}$$ The contribution due to $R\neq {\bf 1}$ strongly depends on heavy Majorana mass hierarchy. One can similarly see that the leading contribution from $\kappa_{ij}$ appears in $\Delta_{ij}$ for the $\tau\to e\gamma$ and $\tau\to \mu\gamma$ predictions.
The parameter dependence of the branching fractions is modified from the degenerate heavy Majorana case by mainly given difference between $\hat m$ and $\hat {\cal M}$. Nevertheless, a particular Majorana mass spectrum is taken such as $M_{R3}$ is dominantly heavy, then similar discussion to the degenerate case is possible as long as effects from $R$ in is sufficiently small.[^6] Further study on effects of $R$ and Majorana masses from our viewpoint is also important and future task.
Probing neutrino parameters with LFV searches {#sec:clarif}
=============================================
Finally, we investigate possible implications for neutrino parameters from future LFV searches with the analysis obtained in the previous section. In the following we concentrate on a limited scenario where $R={\bf 1}$ is assumed, and show how future LFV searches give constraints for neutrino mass spectrum and $\theta_{13}=\epsilon_z$.
Experimental discovery of lepton rare decay processes $\ell_j\to\ell_i+\gamma$ is one of smoking gun signals of physics beyond the SM; thus several experiments have been developed to detect LFV processes. The present experimental upper bounds are given at 90% C.L. [@muex; @tauex]: $$\begin{aligned}
\notag
{\cal B}(\mu\to e\gamma)&\leq &1.2\times 10^{-11},\quad
{\cal B}(\tau\to \mu\gamma)\;\leq \;4.5\times 10^{-8},\quad
{\cal B}(\tau\to e\gamma)\;\leq \;1.2\times 10^{-7}. \end{aligned}$$ These bounds are to be modified in near future searches. MEG experiment searches $\mu\to e\gamma$; the bound is expected to reach ${\cal B}(\mu\to e\gamma)\leq {\cal O}(10^{-13}\sim 10^{-14})$ [@meg]. Future $B$-factories would also greatly reduce the $\tau$ decay upper bounds [@superb]. In our analysis, we conservatively adopt $$\begin{aligned}
{\cal B}(\mu\to e\gamma)&\lesssim &10^{-13},\quad
{\cal B}(\tau\to \mu\gamma)\;\lesssim \;10^{-9},\quad
{\cal B}(\tau\to e\gamma)\;\lesssim \;10^{-9},
\label{future}\end{aligned}$$ as upcoming upper bounds of LFV fractions. Since the bound for ${\cal B}(\mu\to e\gamma)$ is most severe, ${\cal B}(\mu\to e\gamma)/{\cal B}(\tau\to\mu(e)\gamma)$ must be sufficiently suppressed as $10^{-2}\sim 10^{-5(6)}$ in order to observe both $\mu$ and $\tau$ decay processes. Prediction of ${\cal B}(\tau\to e\gamma)$ is always suppressed than ${\cal B}(\tau\to \mu\gamma)$ in Fig. \[ez\], and we focus on $\tau\to \mu\gamma$ between the tau decay processes in the following.
The LFV fractions depend on $M_R$ and SUSY breaking parameters in a flavor blind way, and on the neutrino parameters in a flavor dependent manner. As seen in the previous section, predictions of $\mu\to e\gamma$ and $\tau\to e\gamma$ highly depend on $\epsilon_z$, but $\tau\to \mu \gamma$ is nearly independent of it. Thus ${\cal B}(\tau\to \mu \gamma)$ is mostly determined by $M_R$ and SUSY breaking parameters; namely, the universal dependence in LFV fractions can be read from ${\cal B}(\tau\to \mu \gamma)$. Hence we can express ${\cal B}(\mu\to e\gamma)$ using $\epsilon_z$ and ${\cal B}(\tau\to\mu\gamma)$. Fig. \[ez2\] shows contour plots of ${\cal B}(\mu\to e\gamma)$ as the function of $\epsilon_z$ and ${\cal B}(\tau\to \mu\gamma)$ for the cases with NH and IH. In the figure, shaded regions have already been excluded by the current experimental bound for $\mu\to e\gamma$, and the current and expected bounds for $\tau\to \mu\gamma$ are shown by the solid and dotted lines, respectively.
![Contour plots of ${\cal B}(\mu\to e\gamma)$ as the functions of $\epsilon_z$ and ${\cal B}(\tau\to \mu\gamma)$ for the cases with NH and IH. The other deviation parameters are set to zero, and the neutrino masses are fixed to their central values in . The Dirac phase is taken as $\delta=0$. Shaded regions have already been excluded by the current experimental bound for $\mu\to e\gamma$, and the current (expected) bounds for $\tau\to \mu\gamma$ is shown by the solid (dotted) lines. []{data-label="ez2"}](cont.eps){width="14.cm"}
From the figure, one can realize that future LFV searches give us implications for neutrino parameters. For instance, if near future experiments discover both ${\cal B}(\mu\to e\gamma)$ and ${\cal B}(\tau\to \mu\gamma)$, then $\epsilon_z$ and neutrino mass spectrum are strongly constrained. In the scenario, on the one hand for NH case $|\theta_{13}|$ is close to 0.1, on the other hands for IH and QD cases such a large value of $\theta_{13}$ is not allowed. It is interesting that the above value of reactor angle for NH is in accord with the best-fit value reported by recent data analyses. Hence, the LFV discovery excludes IH and QD mass spectra when the large $\theta_{13}$ is established in experiments like T2K [@t2k] and Double Chooz [@dch]. Though in the analysis the Dirac phase is taken as zero, the result is preserved if non-zero value of $\delta$ is incorporated. This is because IH and QD spectra are still inconsistent with the LFV discovery and the large $\theta_{13}$ value, as easily realized from .
![Prediction of ${\cal B}(\tau\to \mu\gamma)$ as the functions of $\epsilon_z$ for the cases with NH (black circles) and IH (gray triangles). All the plotted points satisfy ${\cal B}(\mu\to e\gamma)\leq 1.2\times 10^{-11}$. The deviation parameters $\epsilon_x$ and $\epsilon_y$ are scanned with 3$\sigma$ ranges , and the Dirac phase is taken as $\delta=0$. The current (expected) bounds for $\tau\to \mu\gamma$ is shown by the solid (dotted) lines. []{data-label="scan3"}](scan3sig.eps){width="10.cm"}
As another scenario, when future LFV searches discover only ${\cal B}(\tau\to \mu\gamma)$, the above discussion is still valid. Fig. \[scan3\] shows the prediction of ${\cal B}(\tau\to \mu\gamma)$ as the functions of $\epsilon_z$ for the cases with NH and IH. All the plotted points satisfy ${\cal B}(\mu\to e\gamma)\leq 1.2\times 10^{-11}$. One can see that NH and IH require different values of $\epsilon_z$ to predict ${\cal B}(\tau\to \mu\gamma)$ in future discovery range. However, constraints on $\epsilon_z$ and neutrino mass spectrum are weakened, if ${\cal B}(\tau\to \mu\gamma)$ is sufficiently suppressed than the future experimental limit.
Conclusion {#sec:conclusion}
==========
In this Letter, we have examined relation between neutrino parameters and LFV predictions, in light of the tri-bimaximal mixing and the recent precision data. By using a particular parametrization for the lepton mixing matrix, which is useful to study difference between the MNS and tri-bimaximal mixing matrices, we have explicitly showed that the flavor dependence in LFV predictions is controlled by deviation parameters and neutrino mass differences.
In the setup with universal heavy Majorana masses and soft SUSY breaking parameters, we have found that $\epsilon_z$ and the neutrino mass spectrum are important for predictions of ${\cal B}(\mu\to e\gamma)$ and ${\cal B}(\tau\to e\gamma)$, while $\epsilon_x$ and $\epsilon_y$ are less effective to determine the LFV predictions. The branching fraction ${\cal B}(\tau\to \mu\gamma)$ is also shown to be insensitive to neutrino parameters. In addition, we have discussed the effects from heavy Majorana mass structure, namely non-degeneracy in right-handed neutrinos and small mixing angles in $R$ matrix.
We have examined and extracted the possible implications for neutrino parameters from upcoming LFV searches. Future discovery of LFV process can give strong constraints on $\theta_{13}$ with respect to the type of neutrino mass hierarchy as long as effects from $R$ in is sufficiently small. In particular, $\tau\to\mu\gamma$ discovery excludes IH and QD neutrino mass spectra if $|\theta_{13}|\simeq 0.1$ would be established.
In general, inclusive studies of the precision neutrino parameters and LFV could give us implications for unrevealed issues in the lepton flavor structure, such as the neutrino tri-bimaximality. Further investigations of LFV prediction focusing on effects from the phases and the Majorana mass structure are our future works.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors would like to thank M. Tanimoto for helpful discussions, and K. Inoue, K. Harada, H. Kubo, N. Yamatsu and S. Kaneko for fruitful comments. This work was supported by a Grant-in-Aid for Scientific Research on Priority Areas (\#441) “Progress in elementary particle physics of the 21st century through discoveries of Higgs boson and supersymmetry”(No. 16081209) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
[99]{} Y. Fukuda [*et al.*]{} \[Super-Kamiokande Collaboration\], Phys. Rev. Lett. [**81**]{} (1998) 1562. P.F. Harrison, D.H. Perkins and W.G. Scott, Phys. Lett. B [**530**]{} (2002) 167. See for example F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, arXiv:0808.0812 \[hep-ph\], and references therein. T. Schwetz, M. Tortola and J.W.F. Valle, New J. Phys. [**10**]{} (2008) 113011. G.L. Fogli, E. Lisi, A. Marrone, A. Palazzo and A.M. Rotunno, Phys. Rev. Lett. [**101**]{} (2008) 141801. F. Plentinger and W. Rodejohann, Phys. Lett. B [**625**]{} (2005) 264; C.H. Albright and W. Rodejohann, Phys. Lett. B [**665**]{} (2008) 378. P. Minkowski, Phys. Lett. [**B67**]{} (1977) 421; T. Yanagida, in [*Proceedings of the Workshop on Unified Theories and Baryon Number in the Universe (Tsukuba, Japan, 1979)*]{}, eds. O. Sawada and A. Sugamoto, KEK report [**79-18**]{}, 1979; Prog. Theor. Phys. [**64**]{} (1980) 1103; M. Gell-Mann, P. Ramond and R. Slansky, in [*Supergravity*]{}, eds. P. van Nieuwenhuizen and D.Z. Freedman, North Holland, Amsterdam, 1979;
S.L. Glashow, in [*Quarks and Leptons, proceedings of the advanced study institute (Cargèse, Corsica, 1979)*]{}, eds. M. Lévy [*et al*]{}., Plenum, New York, 1980;
R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. [**44**]{} (1980) 912. F. Borzumati and A. Masiero, Phys. Rev. Lett. [**57**]{} (1986) 961; R. Barbieri, L.J. Hall and A. Strumia, Nucl. Phys. B [**445**]{} (1995) 219; J. Hisano, T. Moroi, K. Tobe and M. Yamaguchi, Phys. Rev. D [**53**]{} (1996) 2442. J.A. Casas and A. Ibarra, Nucl. Phys. B [**618**]{} (2001) 171. J. Hisano and D. Nomura, Phys. Rev. D [**59**]{} (1999) 116005; J. Sato, K. Tobe and T. Yanagida, Phys. Lett. B [**498**]{} (2001) 189; A. Kageyama, S. Kaneko, N. Shimoyama and M. Tanimoto, Phys. Lett. B [**527**]{} (2002) 206; Phys. Rev. D [**65**]{} (2002) 096010; S.T. Petcov, T. Shindou and Y. Takanishi, Nucl. Phys. B [**738**]{} (2006) 219; S.T. Petcov and T. Shindou, Phys. Rev. D [**74**]{} (2006) 073006; S. Antusch and S.F. King, Phys. Lett. B [**659**]{} (2008) 640. S. Pakvasa, W. Rodejohann and T.J. Weiler, Phys. Rev. Lett. [**100**]{} (2008) 111801. Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. [**28**]{} (1962) 870. C. Amsler [*et al.*]{} \[Particle Data Group\], Phys. Lett. B [**667**]{} (2008) 1. For a review, see W. Rodejohann, Pramana [**72**]{} (2009) 217. R.N. Mohapatra and G. Senjanovic, Phys. Rev. D [**23**]{} (1981) 165;
J. Schechter and J.W.F. Valle, Phys. Rev. D [**22**]{} (1980) 2227; Phys. Rev. D [**25**]{} (1982) 774.
B.C. Allanach [*et al.*]{}, Eur. Phys. J. C [**25**]{} (2002) 113. M. Hirsch, J.W.F. Valle, W. Porod, J.C. Romao and A. Villanova del Moral, Phys. Rev. D [**78**]{} (2008) 013006. M. Bando and T. Kugo, Prog. Theor. Phys. [**101**]{} (1999) 1313; M. Bando, T. Kugo and K. Yoshioka, Prog. Theor. Phys. [**104**]{} (2000) 211; K.S. Babu and S.M. Barr, Phys. Lett. B [**525**]{} (2002) 289. M.L. Brooks [*et al.*]{} \[MEGA Collaboration\], Phys. Rev. Lett. [**83**]{} (1999) 1521. K. Hayasaka [*et al.*]{} \[Belle Collaboration\], Phys. Lett. B [**666**]{} (2008) 16. T. Mori, Nucl. Phys. Proc. Suppl. [**169**]{} (2007) 166. J.L. Hewett [*et al.*]{}, arXiv:hep-ph/0503261.
I. Kato \[T2K Collaboration\], J. Phys. Conf. Ser. [**136**]{} (2008) 022018. V.I. Kopeikin, I.N. Machulin, L.A. Mikaelyan, V.V. Sinev, M.D. Skorokhvatov, S.V. Sukhotin and A.V. Etenko \[Double Chooz Collaboration\], Phys. Atom. Nucl. [**72**]{}, 279 (2009) \[Yad. Fiz. [**72**]{}, 307 (2009)\].
[^1]: E-mail: kojima@rche.kyushu-u.ac.jp
[^2]: E-mail: sawanaka@higgs.phys.kyushu-u.ac.jp
[^3]: One can easily find that $\epsilon_z=\theta_{13}$ from and , owing to $\theta_{13}=0$ in the tri-bimaximal limit.
[^4]: One can also discuss relation between neutrino parameters and LFV prediction with other seesaw mechanisms than the conventional type-I [@rode]. For instance, in the type-II seesaw scenario [@seesawII], neutrino masses in $b_{ij}$ are replaced by those squared, and mixing parameter dependence of LFV prediction can be studied as our analysis.
[^5]: Here $\kappa_{ij}$ is assumed to be real. Complex phases of $R$ can bring further modification into LFV prediction [@rdep].
[^6]: In a class of models where all the leptonic flavor violation originates in the charged lepton sector correspond to $R={\bf 1}$ as in Ref. [@rdep]. This is equivalent to the case that neutrino Yukawa and heavy Majorana mass matrices can be taken as simultaneously diagonal. It is notified that such the situation is approximately realized in some $E_6$ and $SO(10)$ grand unified models, called lopsided mass structure [@lops].
|
---
abstract: 'We propose a mechanism in which two molecular knots pass through each other and swap positions along a polymer strand. Associated free energy barriers in our simulations only amount to a few $k_{B}T$, which may enable the interchange of knots on a single DNA strand.'
author:
- Benjamin Trefz
- Jonathan Siebert
- Peter Virnau
title: How molecular knots can pass through each other
---
Ever since Kelvin conjectured atoms to be composed of knots in the ether [@1], knots have stimulated the imagination of natural scientists and mathematicians alike. In recent years the field went through a renaissance and progressed considerably, spurred by the realization that topology may not only diversify structure, but can also have a profound impact on the function of biological macromolecules. Knots in proteins have been reported [@2; @3; @4; @5; @6; @33; @7] and even created artificially [@8]. Topoisomerases can remove [@9] or create [@10] knots in DNA, which may otherwise inhibit transcription and replication, and viral DNA is known to be highly knotted in the capsid [@11; @12; @13; @14; @15]. Artificial knots have also been tied in single DNA molecules with optical tweezers and dynamics have been studied both experimentally and with computer simulations [@16; @17]. Knots are also known to weaken strands, which tend to rupture at the entrance to the knot [@32; @34]. Even though most of these examples are not knotted in a strict mathematical sense [@18], which only defines knots in closed curves, they nevertheless raise fundamental questions and challenge our understanding of topics as diverse as DNA ejection [@19] and protein folding [@20]. Knots may also play a role in future technological applications, particularly in the advent of DNA nanopore sequencing [@21]. While the probability of observing a knot in a DNA strand of 10 kilo base pairs in good solvent conditions only amounts to a few percent [@22; @23], knots and even multiple knots will become abundant once strand sizes exceed $100000$ base pairs in the near future. Part of this problem was recently addressed in a simulation study [@24]: a single knot will not necessarily jam the channel once it arrives at the pore, but may slide along its entrance.
In the following we would like to elucidate a fascinating and little-known property of composite knots: Two knots can diffuse through each other. In our simulations we employ a standard bead-spring polymer model [@25], which does not allow for bond crossings if local dynamics are applied. Furthermore we apply an additional angular potential and tune the stiffness of the chain such that in good solvent (high salt) conditions the persistence length of DNA is reproduced (for $\kappa = 20 k_{B}T$). Our polymer consists of $250$ monomers which corresponds to roughly $1875$ base pairs. Details on our coarse-grained model, the mapping onto DNA and determination of knot sizes are given in Materials and Methods section. Note that no bias was applied so that our simulations are solely driven by thermal fluctuations. As Supporting Information a video of one “tunneling” event is provided (Video \[SupVid1\]).
![\[Fig1\] Left: Snapshot pictures taken before (a), while (b) and after (c) $3_1$ (green) and the $4_1$ (red) knots interchange positions along the strand. Right: Simplified representation. ](trefzfig1){width="8.7cm"}
In FIG. \[Fig1\] we have prepared a starting configuration with a trefoil knot ($3_{1}$) on the right hand side (green), which is characterized by three non-reducible crossings in a projection onto a plane, and a figure-eight knot ($4_{1}$) on the left (red), which has four crossings. Both termini are connected to a repulsive wall on each side. The distance between walls was chosen to correspond to the typical end-to-end distance ($R_{ee}$) of an unentangled polymer strand of this size ($d \approx 88 \sigma$). Simulations take place in the NVT ensemble (Langevin thermostat) using the CPU version of the HOOMD package [@26]. Knots are identified using an implementation of the Alexander polynomial as described in [@27]. The location and the size of each knot is determined by successively deleting monomers from both ends [@28] until we detect first the trefoil or the figure-eight knot and finally the unknot or vice-versa. The arithmetic mean of the starting and the end monomer of the knot is called its “center”.
Results
=======
![\[Fig2\] (a) Distance between the respective “knot centers” as a function of simulation time. The positions around $+100$ correspond to configurations in which the two knots are separated. At $-100$ the knots are also separated, but positions along the strand are interchanged. The transition region in which the knots are entangled and pass through each other is located around $0$. (b) Corresponding probability profile (blue) obtained from (a). Interestingly a triple peak forms in the intertwined state. Simulations in which the $4_1$ knot passes through the enlarged $3_1$ knot only contribute a single peak (green), while for the opposite situation two peaks arise (red). (c) “Size” of the trefoil (green) and the figure-eight knot (red) as a function of simulation time. The same section was chosen as in (a). “Swapping events” and attempted events are accompanied by a considerable enlargement of one of the two knots to around the combined equilibrium size of both knots (blue line), while the other knot which diffuses along the big knot only grows a bit. (d) Corresponding probability profile obtained from (c). The data shown in (a) and (c) is smoothed by applying a running average. For details and implications see Materials and Methods. FIG. \[SupFig1\] also shows the raw data. ](trefzfig2){width="8.7cm"}
In FIG. \[Fig2\]a we follow the location of the knot centers with respect to each other, and record their distance (in units of monomers) as a function of simulation time. In this framework, the two knots are separated when the two centers are around $100$ monomers apart. At $-100$ the knots are also separated, but positions along the strand are interchanged. Knots are intertwined when centers coincide. As shown in FIG. \[Fig2\]a knots may pass through each other over and over again via an entangled intermediate state. Can we understand this peculiar diffusion mechanism? In FIG. \[Fig2\]c (which shows the same section as in FIG. \[Fig2\]a) we record the size of each knot. When two knots are separated the trefoil knot occupies around $60$ monomers, whereas the figure-eight knot is slightly larger at around $80$ monomers. In the entangled intermediate state one of the knots suddenly expands to a bit less than the combined size of the two knots in the separated state, whereas the size of the other knot grows only marginally. Intriguingly, it is not always the larger figure-eight knot which expands even though its expansion is a bit more likely as can be seen in the accumulated histogram in FIG. \[Fig2\]d. As the two knot centers more or less coincide in the entangled state we conclude that the smaller knot diffuses along the strand of the enlarged knot (as depicted in FIG. \[Fig1\]b) until the two are separated again. They may then either occupy the same positions as before or have interchanged positions along the strand. At large stiffness, the probability distribution of the intertwined state is split up into a triple peak (FIG. \[Fig2\]b) which emerges from two separate contributions. If the $4_1$ knot passes through the enlarged $3_1$ knot there is only a single peak in the middle (green curve). Vice versa, two slightly shifted peaks arise (red curve) due to the symmetry of the enlarged $4_1$ knot (compare with FIG. \[Fig1\]b).
Topological free energy
-----------------------
We can also derive an estimate for the “topological” free energy barrier which needs to be overcome in a “knot swapping” event. This barrier essentially accounts for the obstruction caused by entanglements. In FIG. \[Fig2\]b we have accumulated data from simulations as shown in FIG. \[Fig2\]a to obtain a histogram of the time series and a corresponding probability distribution. For $\kappa = 20 k_{B}T$ the most likely state is the combined state whereas the separated states are metastable.
![\[Fig3\] (a) Free energy profiles derived from probability distributions as shown in FIG. \[Fig2\]b. The curves correspond to simulations in which the stiffness of the chain was modified at a constant wall distance ($d \approx 44 \sigma$) to apply more or less tension to the string. While the separated states are stable for a flexible chain (blue) the intermediate state becomes stable and more likely at higher stiffnesses (green and yellow). The barrier in the free energy is also reduced by a higher chain stiffness. Note that smaller values for $\kappa$ can also be mapped onto DNA and correspond to larger DNA strands in physiological and lower salt conditions as detailed in the Materials and Methods section. (b) Free energy profiles derived from probability distributions as shown in FIG. \[Fig2\]b. The three curves correspond to three simulations in which the separation of the walls was modified to apply more or less tension to the string while the same angular stiffness ($\kappa = 20 k_{B}T$) was used. For all wall distances the intertwined state is the most likely state. Note that a pronounced triple peak only emerges for large stiffnesses. ](trefzfig3){width="8.7cm"}
From FIG. \[Fig2\]b the “topological” free energy is derived as $F = -k_{B} T \ln(P)$. When the separated states are stable (as for flexible chains with $\kappa = 0 k_{B}T$ in FIG. \[Fig3\]a) the system first needs to overcome a barrier ${\Delta F_{1} = -k_{B} T} \cdot \allowbreak \ln( P(\text{entrance to intertwined state}) / P(\text{separated state}))$ to reach the metastable intertwined state. Then a second barrier ${\Delta F_{2} = -k_{B} T} \cdot \ln( \allowbreak P(\text{entrance to intertwined state}) \allowbreak/ \allowbreak P(\text{intertwined state}))$ needs to be overcome to finally swap positions or go back to the original state. If the intertwined state is stable (as in $\kappa=20 k_{B}T$ in FIG. \[Fig3\]b) the system needs to overcome $\Delta F_{2}$ to escape into the metastable separated state. In all cases the barriers only amount to $2-5 ~ k_{B}T$, which would be accessible in experiments. Can we alter this barrier? FIG. \[Fig3\]a shows free energy profiles from simulations with different angular stiffness at the same wall distance. While in the case of the lowest stiffness the separated states are more likely, the intertwined state is more probable at larger stiffnesses as indicated above. FIG. \[Fig3\]b also shows free energy profiles from simulations in which the walls were placed closer together (to $0.5 R_{ee}$ and $0.75 R_{ee}$). While the free energy barrier decreases only slightly for $0.75 R_{ee}$, the separated states nearly vanish when the two knots are pushed together by the smaller distance of the walls (at $0.5 R_{ee}$).
Discussion and Conclusion
=========================
In conclusion, we present a mechanism which allows for two molecular knots to diffuse through each other and swap positions along a strand. The corresponding free energy barrier in our simulations only amounts to a few $k_{B}T$ and should be attainable in experiments similar to [@16] (with loose composite knots) and potentially in vivo. The barrier can be altered by changing the chain stiffness as well as the wall distance to make the “tunneling” event more or less probable. To which extent this peculiar diffusion mechanism might affect DNA behavior in nano-manipulation experiments will be investigated in future studies.
Materials and Methods
=====================
Model and simulation details
----------------------------
The model we apply is essentially a discrete variant of the well-known worm-like chain model (with excluded volume interactions) which has been used extensively to characterize mechanical properties of DNA [@22; @23; @30; @31]. We start with a standard bead-spring polymer model from reference [@25] which does not allow for bond crossings. All beads interact via a cut and shifted Lennard-Jones potential (eq. \[eq:lj\]). Adjacent monomers interact via the finitely extensible nonlinear elastic (FENE) potential (eq. \[eq:fene\]). Chain stiffness is implemented via a bond angle potential (eq. \[eq:angle\]), where angle $\theta_{i}$ is measured between the beads $i-1$, $i$ and $i+1$. For the interaction with the wall we also apply the repulsive part of the Lennard-Jones potential (eq. \[eq:wall\]), where $d_{i}$ is the orthogonal distance from the respective wall to bead $i$. For simplicity we define the normal vector of the walls to coincide with the x-axis of our system. $$\begin{aligned}
U_{\text{WCA}}(r_{ij}) &= \left \{
\begin{array}{l l}
4 \epsilon \left[ (\sigma / r_{ij})^{12} - (\sigma / r_{ij})^{6} \right] + \epsilon, & r_{ij} \le 2^{1/6} \sigma \\
0, & r_{ij} > 2^{1/6} \sigma
\end{array} \right. \label{eq:lj} \\
U_{\text{fene}}(r_{ij}) &= \left \{
\begin{array}{l l}
- 0.5 k R_{0}^{2} \ln \left[ 1 - (r_{ij} / R_{0})^{2} \right], & r_{ij} < R_{0} \\
\infty, & r_{ij} \ge R_{0}
\end{array} \right. \label{eq:fene} \\
% U_{\text{angle}}(\theta_{i}) &= \quad~ k_{\theta}~ [ 1 - \cos (\theta_{i} - \pi) ] \label{eq:angle} \\
U_{\text{angle}}(\theta_{i})&= \quad~ \frac{1}{2} ~ \kappa~ ( \theta_{i} - \pi)^2 \label{eq:angle} \\
U_{\text{wall}}(d_{i}) &= \left \{
\begin{array}{l l}
4 \epsilon \left[ (\sigma / d_{i})^{12} - (\sigma / d_{i})^{6} \right], & d_{i} \le 2^{1/6} \sigma \\
0, & d_{i} > 2^{1/6} \sigma
\end{array} \right. ,\label{eq:wall}\end{aligned}$$ with $\epsilon = 1 k_{B}T$, $k = 30\ \epsilon / \sigma^{2}$ and $R_{0} =1.5\ \sigma$. The two end beads are grafted to the walls and have the same y and z coordinates. The simulations are run with the CPU version of HOOMD [@26] and use the implemented Langevin dynamics thermostat at $T=1$ and $\gamma = 1$. All simulations take place in a regime where differences of the strain energies along the knots can be measured but are far away from breaking bonds as seen in [@32; @34]. We use a time step of $\Delta t = 0{.}01$ [@25] and evaluate the data each $10^{5}$ MD-steps. Each simulation ran for $4\cdot10^9$ MD steps and each parameter set was simulated in at least $66$ independent simulations. For $\kappa=10 k_{B}T$, $d=44\sigma$, e.g., we have performed 131 independent simulations and observed 973 successful swapping events.
Mapping onto DNA
----------------
In the context of knots a similar model was applied in [@22] where the parameters were obtained from mapping the probability for obtaining trefoil knots in the polymer model onto the experimental probability observed for a DNA strand of $11.6$ kilo bases as a function of NaCl concentration. Our model (which is based on this model) has essentially two parameters, which can be fitted to mimic real DNA: The chain stiffness $\kappa$ and the diameter of the chain ($\sigma$ in Lennard-Jones units). $\sigma$ is taken from [@22]. For high salt concentration ($1$ M NaCl), the effective diameter of the chain is slightly larger then the locus of DNA ($\sigma = 2.5 \text{nm}$). In physiological salt conditions ($0.15$ M NaCl) the effective diameter (according to [@22]) is somewhat larger ($\sigma = 5 \text{nm}$). For all salt conditions we assume a persistence length $l_{p}$ of $50 \text{nm}$ or $150$ base pairs.
The relevant energy scale of our model is defined by $\kappa$ in eq. \[eq:angle\]. For the discrete worm-like chain model $$\begin{aligned}
\kappa \approx \frac{l_{p} k_{B} T}{\sigma} \quad. \label{eq:kappa}\end{aligned}$$ As our model features excluded volume interactions, variable bond lengths and angles, we have verified this relation by measuring the persistence length (from the decay of the bond angle autocorrelation function) as a function of $\kappa$ in simulations of unbound chains. Hence, for high salt conditions ($\sigma = 2.5 \text{nm}$) we obtain $\kappa = 20 k_{B}T$. For physiological conditions ($\sigma = 5 \text{nm}$) $\kappa = 10 k_{B}T$.
From eq.\[eq:kappa\] we also obtain the persistence length in simulation units. For $\kappa = 20 k_{B}T$, $l_{p} \approx 20 \sigma$. Therefore, our chain of $N=250$ monomers contains $12.5$ persistence lengths or $12.5 \cdot 150 \text{bp} = 1875$ base pairs. For physiological conditions $l_{p} = 10 \sigma$ and our chain corresponds to $25 \cdot 150 \text{bp} = 3750$ base pairs. (In this calculation we have neglected that the typical distance between adjacent beads is slightly smaller than $\sigma$.)
To confirm the validity of our model we have undertaken extensive Monte Carlo simulations (with fixed bond lengths). We have obtained the probability of observing trefoil knots in a $11.6$ kilo base DNA strand in high salt concentration ($1$ M NaCl, $\kappa = 20 k_{B}T$, $\sigma = 2.5$ nm, $N=1547$) and physiological salt conditions ($0.15$ M NaCl, $\kappa = 10 k_{B}T$, $\sigma = 5.0$ nm, $N=773$). In both cases the probability of observing trefoil knots is only slightly smaller (just outside the experimental errorbars) than the values for the experimental system [@22].
Detection and localization of knots
-----------------------------------
To be able to detect knots, the chain has to be closed first. This is done by drawing a line outwards and parallel to the walls from the fixed beads. Then we connect these lines with a large half circle. After the closure we calculate the products of the Alexander polynomials $\Delta_{p}(-1{.}1) = |\Delta(-1.1)\cdot\Delta(-1/1.1)|$ as described in reference [@27; @28], which yields the composite knot.
![\[Fig4\] Schematic drawing of the knot size analysis. We start removing beads from the left hand side first ($1.$). When the Alexander polynomial yields neither composite nor $3_1$ knot the beginning of the $3_1$ knot is reached ($t_{\text{start}}$). Restarting this procedure from the right hand side ($2.$) gives us the end of the $3_1$ knot ($t_{\text{end}}$). For the $4_1$ knot we start from the left hand side again ($3.$) and change the criteria to neither composite nor $4_1$ knot which gives us $f_{\text{start}}$. Analogously $f_{\text{end}}$ is obtained.](trefzfig4){width="8.7cm"}
For each configuration we confirm that there was no bond-crossing by computing $\Delta_{p}(-1.1)$. Now we start to reduce the chain from one side by successively removing beads starting with the first bead which is not fixed to the wall. The first remaining bead is connected to the fixed bead on the wall and the chain is thereby closed again. To determine the starting monomer of one knot, e.g., the $3_1$ knot $t_{\text{start}}$, we check after each reduction if the result of the Alexander polynomial is still the composite knot or the knot itself. The end of this knot is determined similarly by starting from the other end and applying the same criteria which leads to $t_{\text{end}}$. Likewise, we determine the start and the end monomer of the $4_1$ knot $f_{\text{start}}$ and $f_{\text{end}}$. A scheme of this process is shown in FIG. \[Fig4\]. With these four values we can calculate both knot centers $m_{1/2}$ on the x-axis by using the arithmetic mean.
Data analysis
-------------
The computational determination of knot sizes as described above typically results in strongly fluctuating data even if underlying structures are similar. This method immanent noise covers up relevant features of the transition such as the triple peak in FIG. \[Fig2\]b and the slightly increased size of the translocating knot in the intertwined state (FIG. \[Fig2\]d, compare with FIG. \[SupFig1\]). It also (artificially) broadens the peaks of the probability distribution at the expense of the transition states. For this reason it is not recommended to apply the data analysis directly to raw data. Instead, we have chosen to smoothen the data by applying a running average over 100 adjacent data points. Note that the length of this interval has a minor influence on the barrier height as shown in FIG. \[SupFig2\].
P.V. would like to thank M. Kardar for pointing out that two knots on a rope may change their position and G. Dietler, C. Micheletti and E. Rawdon for helpful discussions. B.T. and P.V. would like to acknowledge the MAINZ Graduate School of Excellence for financial support. We would also like to thank F. Rieger for performing the MC simulations in the Materials and Methods section and D. Richard for his work on the analysis method in the Supporting Information.
Supporting Information
======================
![\[SupFig1\] Same as FIG. \[Fig2\], but including raw data in panel (a) and (d). If our analysis is applied directly to raw data, important features such as the occurrence of the triple peak in panel (c) and the slightly increased size of the translocating (figure-eight) knot in panel (f) are lost in the immanent noise of the detection method. The noise also results in an artificial broadening of the peaks in panel (c) and (f). ](trefzSupportingFigure1){width="8.7cm"}
![\[SupFig2\] Free energy profiles for $\kappa=20k_{B}T$ and $D=88 \sigma$ as derived from raw data (magenta) and derived after averaging over 50 (orange) and 100 (blue) data points (compare with FIG. \[Fig3\]b). In the raw data profile the noise artificially broadens the minima, which results in a sharper transition state. In addition, information on the triple peak is lost. Note that there is a small difference in the barrier height ($<0.5 k_{B}T$) if we change the interval over which we average from $50$ to $100$ data points. ](trefzSupportingFigure2){width="8.7cm"}
{width="8.7cm"}
[32]{} Thompson W. (1867) On vortex atoms. [*Philos. Mag.*]{} 34:15-24.
Mansfield M. L. (1994) Are there knots in proteins. [*Nature Structural Biology*]{} 1:213-214.
Taylor W. R. (2000) A deeply knotted protein structure and how it might fold. [*Nature*]{} 406:916-919.
Lua R. C., Grosberg A. Y. (2006) Statistics of knots, geometry of conformations, and evolution of proteins. [*Plos Computational Biology*]{} 2:350-357.
Virnau P., Mallam A., S. Jackson (2011) Structures and folding pathways of topologically knotted proteins. [*Journal of Physics-Condensed Matter*]{} 23:033101.
Potestio R., Micheletti C., Orland H. (2010) Knotted vs. unknotted proteins: Evidence of knot-promoting loops. [*Plos Computational Biology*]{} 6:e1000864.
B[ö]{}linger, D. et al. (2010) A stevedore’s protein knot [*PLoS computational biology*]{} 6:4
Nureki O. et al. (2004) Deep knot structure for construction of active site and cofactor binding site of trna modification enzyme. [*Structure*]{} 12:593-602.
King N. P., Jacobitz A. W., Sawaya M. R., Goldschmidt L., Yeates T. O. (2010) Structure and folding of a designed knotted protein. [*Proceedings of the National Academy of Sciences of the United States of America*]{} 107:20732-20737.
Liu L. F., Liu C. C., Alberts B. M. (1980) Type-ii dna topoisomerases - enzymes that can un-knot a topologically knotted dna molecule via a reversible double-strand break. [*Cell*]{} 19:697-707.
Dean F., Stasiak A., Koller T., Cozzarelli N. (1985) Duplex dna knots produced by escherichia-coli topoisomerase-i - structure and requirements for formation. [*Journal of Biological Chemistry*]{} 260:4975-4983.
Arsuaga J., Vazquez M., Trigueros S., Sumners D. W., Roca J. (2002) Knotting probability of dna molecules confined in restricted volumes: Dna knotting in phage capsids. [*Proceedings of the National Academy of Sciences of the United States of America*]{} 99:5373-5377.
Marenduzzo D., Micheletti C., Orlandini E. (2010) Biopolymer organization upon confinement. [*Journal of Physics-Condensed Matter*]{} 22:283102.
Arsuaga J. et al. (2005) Dna knots reveal a chiral organization of dna in phage capsids. [*Proceedings of the National Academy of Sciences of the United States of America*]{} 102:9165-9169.
Marenduzzo D., Micheletti C. (2003) Thermodynamics of dna packaging inside a viral capsid: The role of dna intrinsic thickness. [*Journal of Molecular Biology*]{} 330:485-492.
Reith D., Cifra P., Stasiak A., Virnau P. (2012) Effective stiffening of dna due to nematic ordering causes dna molecules packed in phage capsids to preferentially form torus knots. [*Nucleic Acids Research*]{} 40:5129-5137.
Bao X. R., Lee H. J., Quake S. R. (2003) Behavior of complex knots in single dna molecules. [*Physical Review Letters*]{} 91:265506.
Vologodskii A. (2006) Brownian dynamics simulation of knot diffusion along a stretched dna molecule. [*Biophysical Journal*]{} 90:1594-1597.
Millett K., Rawdon E., Stasiak A., Sulkowska J. (2013) Identifying knots in proteins. [*Biochemical Society Transactions*]{} 41:533-537.
Matthews R., Louis A. A., Yeomans J. M. (2009) Knot-controlled ejection of a polymer from a virus capsid. [*Physical Review Letters*]{} 102:88101.
Sulkowska J. I., Sulkowski P., Onuchic J. (2009) Dodging the crisis of folding proteins with knots. [*Proceedings of the National Academy of Sciences of the United States of America*]{} 106:3119-3124.
Branton D. et al. (2008) The potential and challenges of nanopore sequencing. [*Nature Biotechnology*]{} 26:1146-1153.
Rybenkov V. V., Cozzarelli N. R., Vologodskii A. V. (1993) Probability of dna knotting and the effective diameter of the dna double helix. [*Proceedings of the National Academy of Sciences of the United States of America*]{} 90:5307-5311.
Shaw S. Y., Wang J. C. (1993) Knotting of a dna chain during ring-closure. [*Science*]{} 260:533–536.
Rosa A., Di Ventra M., Micheletti C. (2012) Topological jamming of spontaneously knotted polyelectrolyte chains driven through a nanopore. [*Physical Review Letters*]{} 109:118301.
Halverson J. D., Lee W. B., Grest G. S., Grosberg A. Y., Kremer K. (2011) Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. i. statics. [*Journal of Chemical Physics*]{} 134:204904.
Anderson J. A., Lorenz C. D., Travesset A. (2008) General purpose molecular dynamics simulations fully implemented on graphics processing units. [*Journal of Computational Physics*]{} 227:5342-5359.
Virnau P. (2010) Detection and visualization of physical knots in macromolecules. [*Physics Procedia*]{} 6:117-125.
Virnau P., Kantor Y., Kardar M. (2005) Knots in globule and coll phases of a model polyethylene. [*Journal of the American Chemical Society*]{} 127:15102-15106.
Vologodskii A. V., Anshelevich V. V., Lukashin A. V., Frank-Kamenetskii M. D. (1979) Statistical mechanics of supercoils and the torsional stiffness of the DNA double helix [*Nature*]{} 280:5720
Gross, P. et al. (2011) Quantifying how DNA stretches, melts and changes twist under tension [*Nature Physics*]{} 7:731–736
Saitta, M. A., Soper, P. D., Wasserman, E., Klein, M. L. (1999) Influence of a knot on the strength of a polymer strand [*Nature*]{} 399:6731 pp. 46–48
Saitta M. A., Klein M. L. (1999) Polyethylene under tensil load: Strain energy storage and breaking of linear and knotted alkanes probed by first-principles molecular dynamics calculations [*The Journal of chemical physics*]{} 111:20 pp. 9434–9440
|
---
abstract: 'Synergistic design of communications and radar systems with common spectral and hardware resources is heralding a new era of efficiently utilizing a limited radio-frequency spectrum. Such a joint radar-communications (JRC) model has advantages of low-cost, compact size, less power consumption, spectrum sharing, improved performance, and safety due to enhanced information sharing. Today, millimeter-wave (mm-wave) communications have emerged as the preferred technology for short distance wireless links because they provide transmission bandwidth that is several gigahertz wide. This band is also promising for short-range radar applications, which benefit from the high-range resolution arising from large transmit signal bandwidths. Signal processing techniques are critical in implementation of mmWave JRC systems. Major challenges are joint waveform design and performance criteria that would optimally trade-off between communications and radar functionalities. Novel multiple-input-multiple-output (MIMO) signal processing techniques are required because mmWave JRC systems employ large antenna arrays. There are opportunities to exploit recent advances in cognition, compressed sensing, and machine learning to reduce required resources and dynamically allocate them with low overheads. This article provides a signal processing perspective of mmWave JRC systems with an emphasis on waveform design.'
author:
- |
\
\
\
\
\
\
Email: kumarvijay-mishra@uiowa.edu, bhavani.shankar@uni.lu, visa.koivunen@aalto.fi, bjorn.ottersten@uni.lu, sergiy.vorobyov@aalto.fi
bibliography:
- 'main.bib'
title: 'Toward Millimeter Wave Joint Radar-Communications: A Signal Processing Perspective'
---
Introduction
============
In recent years, sensing systems (radar, lidar or sonar) that share the spectrum with wireless communications (radio-frequency/RF, optical or acoustical) and still operate without any significant performance losses have captured significant research interest [@bliss_RF_convergece_17; @Hassan16a]. The interest in such spectrum sharing systems is largely because the spectrum required by the wireless media is a scarce resource, while performance of both communications and remote sensing systems improves by exploiting wider spectrum. In this article, we focus on RF spectrum sharing between radar and communications.
Several portions of frequency bands - from Very High Frequency (VHF) to Terahertz (THF) - are allocated exclusively for different radar applications [@cohen2017spectrum]. Although a large fraction of these bands remains underutilized, radars need to maintain constant access to these bands for target sensing and detection as well as obtain more spectrum to accomplish missions such as secondary surveillance, multi-function integrated RF operations, communications-enabled autonomous driving and cognitive capabilities. On the other hand, the wireless industry’s demand for spectrum continues to increase for providing new services and accommodating massive number of users with high data rate requirement. The present spectrum is used very inefficiently due to its highly fragmented allocation. Emerging wireless systems such as commercial Long Term-Evolution (LTE) communications technology, fifth-generation (5G), WiFi, Internet-of-Things (IoT), and Citizens Broadband Radio Services (CBRS) already cause spectral interference to legacy military, weather, astronomy, and aircraft surveillance radars [@bliss_RF_convergece_17; @cohen2017spectrum]. Similarly, radar signals in adjacent bands leak to spectrum allocated for communications and deteriorate the service quality. Therefore, it is essential and beneficial for radar and communications to develop strategies to simultaneously and opportunistically operate in the same spectral bands in a mutually beneficial manner.
The spectral overlap of centimeter-wave (cmWave) radars with a number of wireless systems at 3.5 GHz frequency band led to 2012 U. S. President’s Council of Advisors on Science and Technology (PCAST) report on spectrum sharing [@pcast2012realizing] and changes in regulation for this band became a driver for spectrum sharing research programs of multiple agencies [@cohen2017spectrum]. Today, it is the higher end of the RF spectrum, i.e., the millimeter-wave (mmWave), formally defined with the frequency range 30-300 GHz, that requires concerted efforts for spectrum management because its technologies are in an early development stage. Increasingly , the mmWave systems [@RapMacSam:Wideband-Millimeter-Wave-Propagation:15] are the preferred technology for near-field communications since it provides transmission bandwidth that is several GHz wide and currently unlicensed. This enables applications which require huge data rates such as 5G wireless backhaul, uncompressed high definition (HD) video, in-room gaming, intra-large-vehicle communications, inter-vehicular communications, indoor positioning systems, and IoT-enabled wearable technologies [@daniels200760]. There is also a spurt of novel sensing systems in the mmWave band. Although these devices typically have short ranges because of heavy attenuation by physical barriers, weather, and atmospheric absorption, they provide high range resolution resulting from the wide bandwidth. Typical mmWave radar applications include autonomous vehicles [@dokhanchi2019mmwave], gesture recognition [@lien2016soli], cloud observation [@mishra2018deep], RF identification [@decarli2014novel], indoor localization [@mishra2017sub], and health monitoring [@fortino2012bodycloud]. We now explain the distinct features and JRC challenges of mmWave channel.
The mmWave Channel {#sec:mmw_ch}
==================
Compared to cmWave, the channel environment for mmWave is characterized by unique challenges that motivate the ensuing specific design constraints.
#### Strong Attenuation
Compared to sub-6 GHz transmissions envisaged in 5G, mmWave signals encounter a more complex propagation environment characterized by higher scattering, severe penetration losses, and lower diffraction. These losses result in mmWave communications links being near line-of-sight (LOS) with fewer non-line-of-sight (NLOS) clusters and smaller coverage areas. Similarly, lower diffraction results in poorer coverage around corners. High attenuation also implies that mmWave radars are useful only at short ranges and, as a result, multipath is a less severe problem.
#### High Path-Loss and Large Arrays
Quite naturally, the mmWave signals suffer from higher path-loss for fixed transmitter (TX) and receiver (RX) gains. By Friis transmission formula, compensating for these losses while keeping the same effective antenna aperture (or increasing the gain) imposes constraints on the transceiver hardware. Since the received power is contingent on the beams of the transmitter and receiver being oriented towards each other, same aperture is accomplished by using steerable antenna arrays whose elements are spaced by at most half the wavelength ($\lambda/2$) of the transmitted signal to prevent undesirable grating lobes. This inter-element spacing varies between $0.5$-$5$ mm for mmWave carriers. Such narrow spacings impact the choice of RF and intermediate frequency (IF) elements because they should fit in limited space available and precise mounting may be difficult in, for instance, vehicular platforms.
#### Wide bandwidths
The unlicensed, wide mmWave bandwidth enables higher data rates for communications as well as the range resolution in radar. In automotive radar, this ensures detection of distinct, informative micro-motions of targets such as pedestrians and cyclists [@mishra2019doppler]. The mmWave receivers sampling at Nyquist rate require expensive, high-rate analog-to-digital converters (ADCs). Large bandwidths also imply that use of low-complexity algorithms in transmitter and receiver processing is critical [@dokhanchi2019mmwave]. Further, mmWave channels are sparse in both time and angular dimensions - a property exploited for low-complexity, low-rate reconstruction using techniques such as compressed sensing [@mishra2017sub; @mishra2018sub]. It is crucial to consider if relevant narrowband assumptions hold in a mmWave application; otherwise, the signal bandwidth is very broad with respect to the center frequency and the steering vectors become frequency-dependent.
#### Power Consumption
The power consumption of an ADC increases linearly with the sampling frequency. At baseband, each full-resolution ADC consumes $15$-$795$ mW at $36$ MHz-$1.8$ GHz bandwidths. In addition, power consumed by other RF elements such as power amplifiers and data interface circuits in conjunction with the narrow spacing between antenna elements renders it infeasible to utilize a separate RF-IF chain for each element. Thus, a feasible multi-antenna TX/RX structure and beamformers should be analog or hybrid (wherein the potential array gain is exploited without using a dedicated RF chain per antenna and phase shifter) [@mendez2016hybrid] because fully digital beamforming is infeasible.
#### Short Coherence Times
The mmWave environments such as indoor and vehicular communications are highly variable with typical channel coherence times of nanoseconds [@RapMacSam:Wideband-Millimeter-Wave-Propagation:15]. The reliability and coverage of dynamic mmWave vehicular links are severely affected by the use of narrow beams. The intermittent blockage necessitates frequent beam re-alignment to maintain high data rates. Also, mmWave radar requires wide Doppler range to detect both fast vehicles and slow pedestrians [@mishra2019doppler]. Short coherence times impact the use of feedback and waveform adaptation in many JRC designs, where the channel knowledge may be invalid or outdated when transmit waveform optimization takes place.
We now present details of channels models commonly used in mmWave communications and radars.
Communications Channel
----------------------
Consider a transmitter that employs an antenna array or a single directional antenna with carrier frequency $f$ and TX (RX) antenna gain $G_\txm$ ($G_\mrx$). The LOS communications channel with a delay spread comprising $L_\comm-1$ delay taps is $h_\comm(t,f) = G_\comm \sum_{\ell=0}^{L_\comm-1} \alpha_\ell \me^{- j 2 \pi \tau_\ell f}\me^{j 2 \pi \nu_\ell t}$, where $G_\comm$ is the large-scale communications channel gain at the reception, and $\alpha_\ell$ is the path loss coefficient of the $l^{\text{th}}$ path with time delay $\tau_\ell$ and Doppler shift $\nu_\ell$. The free space attenuation model yields $G_\comm = \frac{G_\txm G_\mrx \lambda^2 } {(4 \pi)^2 \rho_\mcom^\gamma}$, where $\gamma$ is path loss (PL) exponent . Further, $\gamma\approx2$ for mmWave LOS outdoor urban [@RapMacSam:Wideband-Millimeter-Wave-Propagation:15] and rural scenarios [@MacSunRap:Millimeter-Wave-Wireless:16].
Radar Channel
-------------
The doubly selective (time- and frequency-selective) mmWave radar channel is modeled after TX/RX beamforming using virtual representation obtained by uniformly sampling in range dimension [@KumChoGon:IEEE-802.11ad-based-Radar::17]. Assume $L$ uniformly sampled range bins and that the $\ell$-th range bin consists of a few, (say) $K_\ell$, virtual scattering centers. Each $(\ell,k)$-th virtual scattering center is characterized by its distance $\rho_\ell$, delay $\tau_\ell$, velocity $v_{\ell,k}$, Doppler shift $\nu_{\ell,k} = 2 v_{\ell,k}/\lambda$, large-scale channel gain $G_{\ell,k}$, and small-scale fading gain $\beta_{\ell,k}$. Then, the multi-target radar channel model is $h_r (t,f) = \sum_{\ell=0}^{L-1} \sum_{k=0}^{K_{\ell}-1} G_{\ell,k} \beta_{\ell,k} \me^{- j 2 \pi \tau_\ell f} \cdot \me^{- \jm 2\pi \nu_{\ell,k} t}$. The large-scale channel gain corresponding to the $(\ell,k)$-th virtual target scattering center is $G_{\ell,k} = \frac{\lambda^2 \sigma_{\ell,k}}{64 \pi^3 \rho_\ell^4}$, where $\sigma_{\ell,k}$ is corresponding scatterer’s radar cross section (RCS). The small scale gain is assumed to be a superposition of a complex Gaussian component and a fixed LOS component leading to Rician fading. Similarly, the corresponding frequency selective models can also include Rician fading. They capture, as a special case, the spiky model used in prior works on mmWave communications/radar. In this case, the corresponding radar target models are approximated by the Swerling III/IV scatterers [@skolnik2008radar].
Further, clustered channel models can be considered to incorporate correlations and extended target scenarios although they remain unexamined in detail. For instance, the conventional mmWave automotive target model assumes a single non-fluctuating (i.e., constant RCS) scatterer based on the Swerling 0 model. This greatly simplifies the development and analysis of receive processing algorithms and tracking filters [@dokhanchi2019mmwave]. However, when the target is located within the close range of a high-resolution radar, the received signal is composed of multiple reflections from different parts of the same object. This *extended* target model is more appropriate for mmWave applications and may also include correlated RCS [@mishra2019doppler].
It is typical to assume a frequency-selective Rayleigh fading model for both communications and radar channels during the dwell time comprising $\Ndwel$ coherent processing intervals (CPI). In radar terminology, this corresponds to Swerling I/II target models. In each CPI with $M$ frames, the channel amplitude of each tap is considered to be constant, i.e., a block fading model is assumed. Moreover, constant velocity and quasi-stationarity conditions are imposed on the target model.
Channel-Sharing Topologies {#subsec:jrc_top}
--------------------------
![(a) Spectral coexistence system where radar and communications subsystems are independently located and access the associated radio channels such as radar target channel $h_r$, communications channel $h_c$, radar-to-communications interference $h_s$, and communications-to-radar interference $h_d$ [@bica_icassp16]. (b) Co-design system where only Rx are shared. In this *joint multiple access* channel, the radar operates in monostatic mode and both systems transmit different waveforms that are orthogonal in spectrum, code or time [@mishra2017auto]. (c) In TX-shared co-design, the monostatic radar functions as a communications transmitter emitting a common JRC waveform [@muns2017beam]. (d) A bi-static broadcast co-design with common TX, RX, and a joint waveform [@dokhanchi2019mmwave]. The joint waveform transmitted by the TX vehicle bounces off from targets such as T1 and T2 and received by the Rx vehicle. A variant is in-band full duplex system with different waveforms but common TX and Rx [@donnet2006combining]. The term ‘BS’ stands for ‘base station’. []{data-label="fig:jrc_top"}](jrc_top.png){width="100.00000%"}
The existing mmWave JRC systems could be classified by the joint use of the channel [@bliss_RF_convergece_17; @geng2018fusion] (Fig. \[fig:jrc\_top\]). In the *spectral coexistence* approach, radar and communications operate as separate entities and focus on devising strategies to adjust transmit parameters and mitigate the interference adaptively for the other [@cohen2017spectrum]. To this end, some information exchange between the two systems, i.e. spectral cooperation, may be allowed but with minimal changes in the standardization, system hardware and processing. In *spectral co-design* [@dokhanchi2019mmwave; @bliss_RF_convergece_17], new *joint* radio-frequency sensing and communications techniques are developed where a single unit is employed for both purposes while also accessing the spectrum in an opportunistic manner. New fully-adaptive, software-defined systems are attempting to integrate these systems into same platform to minimize circuitry and maximize flexibility. Here, each transmitter and receiver may have multiple antennas in a phased array or Multiple-Input Multiple-Output (MIMO) configuration. In the next section, we discuss mmWave systems based on co-existence and follow it by co-design methods in Section \[sec:jrc\].
JRC at mmWave: Coexistence {#sec:individual}
==========================
Interference management is central to spectral coexistence of different radio systems. This, typically requires sensing the state of the shared spectrum and adjusting TX and RX parameters so that the impact of interference is sufficiently reduced and individual system performance is enhanced. We now present the figures of merit qualifying system performance and then discuss methodologies for mmWave coexistence.
Communications Performance Criteria {#subsec:comm_crit}
-----------------------------------
Since the goal of communications systems is to transfer data at a high rate error-free for a given bandwidth, the commonly used performance criteria include quality of service (QoS) indicators such as spectral efficiency, mutual information, channel capacity, pairwise error probability, bit/symbol error rates (BER/SER), and signal-to-interference-and-noise ratio (SINR). Given a communications signal model, the achievable spectral efficiency can be used as a universal communications performance criterion. In practice, the achievable spectral efficiency $r$ is an upper bound, while the effective spectral efficiency $r_\eff$ depends on the implemented receiver (e.g. minimum mean square error or MMSE [@ShiZhoYao:Capacity-of-single-carrier:04], decision feedback [@TakKyrHan:Performance-evaluation-of-60-GHz-radio:12] or time-domain equalizer [@liu2013digital]), and is a fraction of the achievable spectral efficiency. The effective communications rate is then the product of the signal bandwidth $W$ and $r_\eff$.
Radar Performance Criteria {#subsec:rad_crit}
--------------------------
Radar systems, by virtue of their use in both detection and estimation, lend themselves to a plethora of performance criteria depending on the specific task. Target detection performance is characterized by probabilities of correct detection, mis-detection, and false alarm. In parameter estimation task, mean square error (MSE) or variance in comparison to the Cramér-Rao Lower Bound (CRLB) is commonly considered. The CRLB defines the lower bound for estimation error variance for unbiased estimators. There are also several radar design parameters such as range/Doppler/angular resolution/coverage and the number of targets a radar can simultaneously resolve. In particular, the radar’s ability to discriminate in both range and velocity is completely characterized by the *ambiguity function* (AF) of its transmit waveform; it is obtained by correlating the waveform with its Doppler-shifted and delayed replicas.
Interference Mitigation
-----------------------
The mmWave radar and communications TX and RX can use all of their degrees of freedom (DoFs) such as different antennas, frequency, coding, transmission slots, power, or polarization to mitigate or avoid mutual interference. Interference may also be caused by leakage of signals from adjacent channels because of reusing identical frequencies in different locations. In general, higher the frequency in mmWave bands, weaker the multipath effects. The transmitters can adjust their parameters so that the level of interference is reduced at the receiver. To this end, awareness about the dynamic state of the radio spectrum and interference experienced in different locations, subbands and time instances is desired. This may be in the form of feedback provided by the receivers to the transmitter about the channel response and SINR. Both the TX and RX can be optimized such that the SINR is maximized at the receivers for both subsystems.
### Receiver Techniques
Interference mitigation may be performed only at the RX rendering channel state information (CSI) exchange optional. Typically, this requires multiple antenna at RX, a common feature at mmWave, and processing of the received signals in spatial and/or temporal domain. These techniques employ receive array covariance matrix $\bm{\Sigma}$ (or its estimate $\hat{\bm{\Sigma}}$) in certain interference canceling RX structures. Here, the received signal space spanned by eigenvectors of $\bm{\Sigma}$ is divided into two orthogonal subspaces of signal and interference-plus-noise. The received signal is then projected to a subspace orthogonal to the interference-and-noise subspace to enable processing of practically interference-free signals. If the interference impinges the receiver from angles different than the desired signal, RX beamforming is commonly used [@geng2018fusion]. The beampattern design ensures high gains towards the desired signals and steers nulls towards the interference. Common solutions include Minimum Variance Distortion-less Response (MVDR), Linearly Constrained Minimum Variance (LCMV) and diagonal loading [@vorobyov2014adaptive]. Advanced interference cancellation receivers estimate CSI, use feedback about channel response or sense other properties of the state of the radio spectrum. These estimates are later used to cancel the interference contribution from the overall received signal. The coherence time of the channels should be sufficiently long that the feedback or channel estimates are not outdated during the interference cancellation process. These techniques either require knowledge of modulation schemes employed by coexisting radio systems, or are applied to digital modulation methods only. A prime example is the Successive Interference Cancellation (SIC) method that decodes and subtracts the strongest signal first from the overall received signals and the repeats the same procedure by extracting the next weaker signal from the residual signal and so on [@bliss_RF_convergece_17]. In the absence of CSI, non-traditional radar interference models are used for robust communications signal decoders [@ayyar2019robust].
### Transmitter Techniques
Adapting transmitters and optimizing transmit waveforms may be used to minimize the impact of interferences in coexistence systems. In a radar-communications coexistence scenario, for example, the optimization objective could be maximizing the SINR at each receiver while providing desired data rate for each communications user and target Neyman-Pearson detector performance for radar users. Designing a precoder for each transmitter or/and decoders for each receiver achieves this goal by steering the interferences to different space than the desired signals. One such example design in the context of MIMO communications and MIMO radar is the Switched Small Singular Value Space Projection (SSSVSP) method [@Mahal15] in which the interference is steered to space spanned by singular vectors corresponding to zero or negligible singular values. This method requires information exchange between the radar subsystem and communications base-stations. Another example of a precoder-decoder design for interference management in radar-communications coexistence is via Interference Alignment (IA) [@cui_spawc18] where IA coordinates co-existing multiple transmitters such that their mutual interference aligns at the receivers and occupies only a portion of the signal space. The interference-free signal space is then used for radar and communications purposes.
JRC at mmWave: Co-Design {#sec:jrc}
========================
Central towards facilitating the co-design of radar and communications systems are waveform design and their optimization exploiting available DoFs (spatial, temporal, spectral, polarization). The optimization is based on the system performance criteria and availability of channel state information (CSI), awareness about target scene and the levels of unintentional or intentional interference at the receivers.
JRC Performance Criteria
------------------------
In co-design, JRC waveforms are modeled to simultaneously improve the functionalities of both subsystems with some quantifiable trade-off. In [@Bli:Cooperative-radar-and-communications:14], a radar round-trip delay estimation rate is developed and coupled with the communications information rate. This radar estimation, however, is not drawn from the same class of distributions as that of communications data symbols and, therefore, provides only an approximate representation of the radar performance. However, potential invalidity of some assumuptions limits the extension of this to estimation of other target paramters. The mmWave designs in [@KumVorHea:VIRTUAL-PULSE-DESIGN:19; @KumNguHea:Performance-trade-off-in-an-adaptive:17] for single- and multiple-target scenarios suggest an interesting JRC performance criterion which attempts to parallel the radar CRLB performance with a new effective communications symbol MMSE criteria as a function of effective maximum achievable communications spectral efficiency, $r_\eff$. The MMSE communications criteria here is analogous to the mean-squared error distortion in the rate distortion theory. Let $\mmse_\comm$ be the MMSE of a communications system with spectral efficiency $r$. Then $\mmse_\comm$ and $r$ are related to each other through the equation $\frac{1}{N}
\Tr {\log_2 {\mmse_\mcom}} = -r
$, where $N$ is the code length. Therefore, the effective communications distortion MMSE (DMSE) that satisfies $\frac{1}{N}
\Tr {\log_2 {\dmse_\eff}} = {-r_\eff} = {-\delta \cdot r}$ can be defined as $\dmse_\eff \triangleq \mmse_c^\delta$, where $\delta$ is a constant fraction of communications symbols transmitted in a CPI with the channel capacity $C$. The performance trade-off between communications and radar is quantified in terms of a weighted combination of the scalar quantities $\frac{1}{N}
\Tr{\log_2 \dmse_\eff}$ and $\frac{1}{Q}
\Tr{\log_2 \textrm{CRLB}}$, respectively, where the log-scale is used to achieve proportional fairness between the communications distortion and radar CRLB values and $Q$ is the number of detected targets. Pareto-optimal solutions that assign weights to different design goals have also been explored in this context [@ciuonzo2015pareto].
Mutual information (MI) is also a popular waveform optimization criteria. At the radar receiver, depending on whether the communications signal reflected off the target is treated as useful energy or interference or ignored altogether, a different MI-based criterion results [@bica_icassp16]. Although MI maximization enhances the characterizing capacity of a radar system, it does not maximize the probability of detection. The optimal radar signals for target characterization and detection tasks are generally different [@bica_icassp16; @cohen2017spectrum].
Radar-Centric Waveform design
-----------------------------
We first consider the appropriate radar-centric waveforms here. These range from conventional signals to emerging multi-carrier waveforms.
#### Conventional Continuous Wave and Modulated Waveforms
A simple continuous-wave (CW) radar provides information about only Doppler velocity. To extract range information, either the frequency/phase of CW signal is modulated or very short duration pulses are transmitted. In practice, the well-known Frequency Modulated Continuous Wave (FMCW) and Phase Modulated Continuous Wave (PMCW) radars are used. A typical FMCW radar transmits one or multiple chirp signals wherein the frequency increases or decreases linearly in time and then the chirps reflected off the targets are captured at the receiver. Chirp bandwidth of a few GHz may be used to provide a range resolution of a few centimeters, e.g, $4$ GHz chirp achieves a range resolution of $3.75$ cm. For PMCW, binary pseudorandom sequences with desirable autocorrelation/ cross-correlation properties are typically used. The AF of PMCW has lower sidelobes than FMCW and PMCW is also easier to implement in hardware [@dokhanchi2019mmwave].
A general bi-static, uniform linear array (ULA) PMCW-JRC system [@dokhanchi2019mmwave] follows the topology shown in Fig. \[fig:jrc\_top\]d. The transmitter sends $M$ repetitions of the PMCW code of length $L$ from each of its $N_{t}$ transmit antennas. The Doppler shift and flight time for the paths are assumed to be fixed over the CPI. The reflections from $Q$ targets impinge on $N_{r}$ receive antennas. Let $t_{c}$ be chip time (time for transmitting one element of one PMCW code sequence, i.e., fast-time). The Doppler shifts and the flight time for every path are assumed to be fixed over a coherent transmission time $Mt_b$, where $t_{b}=Lt_{c}$ is the time taken to transmit one block of code, i.e., slow-time. The transmit waveform takes the form, $$\begin{aligned}
\label{2}
x_i(t)=\sum_{m=0}^{M-1} \sum_{l=0}^{L-1} a_{m}e^{j\zeta_l} s(t-lt_{c}-mt_{b})e^{j2\pi f_{c}t} e^{j(i-1)k d \sin\beta},
\end{aligned}$$ where $i\in[1,N_{t}]$ and $ a_{m}= e^{j\phi_{m}} $ denote differential PSK symbols (DPSK) over slow time (time for sending one code sequence). The DPSK modulation is robust to constant phase shifts. Further, $s(t)$ is the elementary baseband pulse shape, $\zeta_{l} \in \{0, \pi \} $ is the binary phase code, $ e^{j(n-1)k d \sin\beta}$ is beam-steering weight for $n$th antenna, $k=\frac{2\pi}{\lambda}$ is wave number, and $\beta$ is angle between the radiating beam and the perpendicular to the ULA (for simplicity, we consider only azimuth and ignore common elevation angles). The transmitter steers the beam in multiple transmission from $[\dfrac{-\pi}{2}$, $\dfrac{\pi}{2}]$, each time with angle $\beta$. As shown in Fig. \[fig:jrc\_block\], the communications and radar waveform for PMCW-JRC are combined in analog hardware.
![A simplified block diagram showing major steps of transmit and receive processing for a general mmWave JRC system. In case of PMCW-JRC, the radar and communications waveforms are combined in the analog hardware before the RF stage. On the other hand, the information bits from these two subsystems are mixed digitally in OFDMA-JRC. The multiplexing of radar-only and radar-communications frame for both PMCW- and OFDMA-JRC are depicted in the transmit portion. The receive processing for both systems is largely similar. []{data-label="fig:jrc_block"}](JRC_Block_v2.png){width="100.00000%"}
Let $\Delta V^{(1)}_{q}$ be the radial relative velocity between the transmitter and $q$th path, where superscript $(\cdot)^{(1)}$ refers to transmitter-target path, and the corresponding Doppler shift is $f^{(1)}_{D_{q}}= \frac{\Delta V^{{(1)}}_{q}}{c}f_c$, where $c=3\times 10^8$ m/s is the speed of light. The signal impinging on $q$th scatterer is, $$\begin{aligned}
\label{eq:signal@target}
\hspace*{-0.1in}
z_{q,n}(t)= \sum_{m=0}^{M-1} \sum_{l=0}^{L-1} h^{(1)}_{q,n}a_{m}e^{j\zeta_{l}} s\big( t-lt_{c}-mt_{b}-\tau^{(1)}_{q} \big)
e^{j2\pi f_{c}t- j2\pi f^{(1)}_{D_{q}}t - j 2\pi f_{c} \tau^{(1)}_{q}},\end{aligned}$$ where $\tau^{(1)}_{q}$ and $h^{(1)}_{q,n}$ are $q$th point scatterer time delay and propagation loss for each path, respectively. We exploit the standard narrowband assumption to express the received signal as a phase-Doppler shifted version of the transmit signal. Assume $\tau_{q}=\tau^{(1)}_{q}+\tau^{(2)}_{q}$ be the total flight time corresponding to a bi-static range $R_{q}=c\tau_{q}$, where superscript $(\cdot)^{(2)}$ denotes variable dependency on the target-receiver path. Assume $f_{D_{q}}=f^{(1)}_{D_{q}}+f^{(2)}_{D_{q}}$ to be the bi-static Doppler shift, and $\psi_{q}$ be the angle between the $q$th scatterer and perpendicular line to receive ULA. After TX/RX beamforming and frequency synchronization, the received signal at antenna $p$, obtained as a superposition of these reflections takes the form,
\[4\] & y\_[p]{}(t)= \_[q=1]{}\^[Q]{}\_[n=1]{}\^[N\_[t]{}]{} h\^[(2)]{}\_[q,p]{}z\_[q,,n]{}(t-\^[(2)]{}\_[q]{}) e\^[j2f\^[(2)]{}\_[D\_[q]{}]{}t]{} + N\_p(t)\
& =\_[q=1]{}\^[Q]{}\_[n=1]{}\^[N\_[t]{}]{}\_[m=0]{}\^[M-1]{} \_[l=0]{}\^[L-1]{} h\^[(2)]{}\_[q,p]{} h\^[(1)]{}\_[q,n]{} a\_[m]{} e\^[j\_[l]{}]{} s(t-lt\_[c]{}-mt\_[b]{}-\^[(1)]{}\_[q]{}-\^[(2)]{}\_[q]{}) e\^[j2(f\_[c]{}-f\^[(1)]{}\_[D\_[q]{}]{}-f\^[(2)]{}\_[D\_[q]{}]{})t]{} e\^[j\_[q]{}]{} e\^[-jk d (\_[q]{}) (p-1)]{}+ N\_p(t),
where $ e^{j\eta_{q}}= e^{-j2\pi \big( f_{c}(\tau^{(1)}_{q}+\tau^{(2)}_{q})+f^{(1)}_{D_{q}}\tau^{(2)}_{q} \big)}$ is a static phase shift, $h^{(2)}_{q,p}$ accumulates the effect of $q$th transmitter-target-receiver point scatterer, path-loss and RCS of the target, and $\tilde N_p(t)$ is complex circularly symmetric white Gaussian noise with variance $\sigma^2$. An extended target is modeled as a cluster of points. This combined with the superposition of reflections from independent scatterer renders the model in applicable for extended targets. After downconversion to baseband and ignoring RCS dependency on Tx and Rx antennas, i.e., $\sum\limits_{n=1}^{N_t}h^{(1)}_{q,n}h^{(2)}_{q,p} e^{j\eta_{q}} = \sum_{n=1}^{N_t}\acute d_{q,p,n}=N_t \acute d_q = d_q$, received signal is $$\begin{aligned}
\label{PMCW-JRC_rx}
y_{p}(t)= \sum_{q=1}^{Q}\sum_{ m=0}^{M-1} \sum_{l=0}^{L-1} d_{q} a_{m} e^{-j2\pi f_{D_q}t}c^{p-1}_q e^{j\zeta_{l}}s(t- lt_{c}- mt_{b}-\tau_{q}) + N_p(t) , \ p\in[1,N_{r}],\end{aligned}$$ where $c_{q}= e^{-jk d \sin(\psi_{q})}$. Collecting the Nyquist time samples for the antenna $p$ and rearranging them accordingly to slow/fast-time, we form a matrix, $$\begin{aligned}
\label{eq:sys_mod1}
\boldsymbol Y_p^{\text{PMCW-JRC}} = \sum_{q=1}^{Q} c_q^{p-1} d_q \text{Diag}\{\boldsymbol a\} \big[ ( \boldsymbol b_q^T \odot \boldsymbol s^T \boldsymbol P_{k_q}) \otimes \boldsymbol e_q \big] +\boldsymbol N_p \ \in \mathbb{C}^{M\times L}, \end{aligned}$$ where vectors $ \boldsymbol{e}_q = [e^{j2\pi f_{D_q} m L t_{c}}]_{m=1}^{M}$ and $\boldsymbol{b}_q= [e^{j2\pi f_{D_q} l t_{c}}]_{l=1}^{L}$ collect Doppler samples in slow and fast time, respectively, $\boldsymbol{s}= [e^{j\zeta_l}]_{l=0}^{L}$ contains $L$ chips of code sequence, and $\boldsymbol P_{k_q}$ is a cyclic permutation matrix for a shift of ${k_q}$ as $$\begin{aligned}
\boldsymbol P_{k_q} = \begin{bmatrix}
\mathbf0_{K_q\times L-K_q} & \mathbf I_{K_q\times K_q}\\
\mathbf{I}_{L-K_q\times L-K_q} & \mathbf{0}_{L-K_q\times K_q}
\end{bmatrix} \in \mathbb{C}^{L \times L},\end{aligned}$$ where $k_q \in \{0,\cdots , L-1\}$ is determined by range of the $q$th scatterer. If there is no delay between transmitter and receiver for all paths, then $k_q=0$ for all $q$ and $\boldsymbol P_{k_q}$ becomes identity matrix.
In a PMCW-JRC, the communications symbols and Doppler parameters are coupled thus leading to a non-identifiable model. This is resolved by a multiplexing strategy through which unknown parameters in the received signal are uniquely identified. The PMCW-JRC adopts time-division multiplexing between radar-only ($\bm X_r$) and joint radar-communications ($\bm X_{rc}$) frames which are transmitted for $\mu$ and ($1-\mu$) % of the CPI, respectively. The value of $\mu$ depends on the amount of prior knowledge about the target scene. As a case in point, when the scene is stationary such as driving a straight path on a highway, we may not need full sensing capacity and can scale up the allocated time appropriately for communications. A coarse estimate of radar target parameters (range, angle and Doppler) is obtained from $\boldsymbol Y_p^{\text{PMCW-JRC}}$ of radar-only frames $\bm X_r$ while communications symbols are extracted from the received signal samples of $\bm X_{rc}$ frame. After extracting communications symbols from $\bm X_{rc}$, the residual signal is exploited for further improving the radar target estimates through low-complexity JRC super-resolution algorithms [@dokhanchi2019mmwave].
#### Multi-Carrier Waveforms
Multi-carrier waveform radars provide additional DoFs to deal with dense spectral use and demanding mmWave target scenarios like drones, low-observable objects, and large number of moving vehicles in automotive scenario. Different DoFs can be used in an agile manner to achieve optimal performance depending on the radar task, nature of targets, and state of the radio spectrum. A general drawback of multi-carrier radar waveforms is their time-varying envelope leading to an increased Peak-to-Average-Power-Ratio (PAPR) or Peak-to-Mean-Envelope-Power-Ratio (PMEPR) which makes it difficult to use the amplifiers efficiently when high transmit powers are needed. However, in mmWave radars, the transmit powers tend to be small and surveillance ranges are short. The PAPR reduction is achieved by not allocating all subcarriers or by using appropriate coding/waveform design. Hence, the PAPR issue in mmWave may be less severe. Multi-carrier Complementary Phase Coded (MCPC) waveform [@levanon_mf_radar_2000], wherein each subcarrier is modulated by a pseudorandom code sequence of a specific length, is also a viable mmWave JRC candidate. The MCPC design exploits DoFs in spectral and code domain. In a sense, it is related to OFDM because after each subcarrier is modulated by a code in time-domain, the subcarriers remain orthogonal without intercarrier interference. If the subcarriers are uncoded, the waveform is exactly OFDM. The inter-carrier spacing in MCPC needs to accommodate the spreading of the signals in frequency due to phase codes such as Barker, P3 or P4 polyphase codes [@skolnik2008radar]. This is achieved by choosing the inter-carrier spacing to be inverse of the chip duration. In OFDM, intercarrier spacing is smaller. A Generalized Multi-carrier Radar (GMR) waveform devised in [@bica_ciss_14; @bica_trsp_16] subsumes most of the widely used radar waveforms such as pseudo random frequency hopping (FH), MCPC, OFDM and linear step approximations of linear FM signals, as special cases. A matrix model of transmitter and receiver is developed for GMR that allows for defining the waveforms and codes, spreading in time and frequency domain, power allocations and active subcarriers using a compact notation. Different waveforms are obtained by choosing the dimensions of the matrix model and filling the entries appropriately. This approach allows for relaxing perfect orthogonality requirement; this may lead to a better resolution of target delays and Doppler velocities at mmWave.
#### Spatial DoFs and Multiple Waveforms
A few different solutions use the same waveform for both subsystems but make use of radar’s spatial DoFs for communications symbols. For instance, in [@Hassan16], the radar array beampattern sidelobes are modulated by communications messages along user directions. In [@Hassan18], the communications symbols are represented by different pairing of antennas and waveforms in a MIMO configuration. Spatial DoFs are also useful for adaptively canceling specific users. A joint beamforming method is suggested in [@hassanien_spawc18] for a dual-function radar-communications (DFRC) that comprises MIMO radar and communications systems assuming full-duplex transmission. The downlink communications signal is embedded into the transmit radar waveform and uplink communications takes place when the radar is in listening mode. This necessitates accurate synchronization among the subsystems. The technique utilizes spatial diversity by enforcing the spatial signature of the uplink signals to be orthogonal to the spatial steering vectors associated with the radar target returns. The receiver beamformer employs adaptive and non-adaptive strategies to separate the desired communications signal from echoes of targets, clutter, and noise even if they impinge the array from the same direction. Other solution paths consist of finding spatial filters to mitigate in-band MIMO communications interference through optimization of the sidelobe and cross-correlation levels in MIMO radar systems [@Aittom17; @LiPetr16], exploiting co-array processing with multiple waveforms [@ZhangVor15] and designing precoders/decoders through interference alignment [@CuiKoivu17].
However, for mmWave JRC systems, the full-resolution ADCs at the baseband signal result in an unacceptably high power consumption. This makes it infeasible to utilize an RF chain for each antenna element implying that the prevailing MIMO systems that employ fully digital beamforming are not practical for mmWave systems. Thus, the benefits of using multiple waveforms for spatial mitigation in mmWave JRC systems are yet to be carefully evaluated. Currently, a single data stream model that supports analog beamforming with frequency flat TX/RX beam steering vectors is more common [@KumChoGon:IEEE-802.11ad-based-Radar::17]. Use of large antenna arrays in mmWave suggests that a feasible JRC approach could be to simply partition the arrays for radar and communications functionalities [@mishra2018sub].
Communications-Centric Waveform design
--------------------------------------
The most popular communications signal for mmWave JRC is OFDM because it provides a stable performance in multipath fading and relatively simple synchronization [@donnet2006combining]. Also, frequency division in duplexing has an added advantage; unlike time-division duplexing, the former employs different bands for uplink and downlink so that the impact on the interference in radar systems is less severe. Some solutions [@donnet2006combining; @dokhanchi2019mmwave] also employ the related Orthogonal Frequency Division Multiple Access (OFDMA) waveform for a JRC system. While the OFDM users are allocated on only time domain, the OFDMA users can be differentiated by both time and frequency. The latter, therefore, provides DoFs in both temporal and spectral domains. Although OFDM-JRC offers high dynamic range and efficient receiver processing implementation based on fast Fourier transform (FFT), it requires additional processing to suppress high side-lobes in receiver processing and reduce PAPR. Further, the OFDM cyclic prefix (CP) used to transform frequency selective channel to multiple frequency flat channels leading to a simplified equalizer, may be a nuisance in the radar context. The CP may adversely affect the radar’s ability to resolve ambiguities in radar ranging. Its length depends on number of channels, particularly the maximum excess delay that the radar signal may experience (time difference between first and last received component of the signal). For radar applications, the CP duration should be equal to or longer than the total maximum signal travel time between the radar platform and target. Other communications waveforms proposed for mmWave automotive JRC include spread spectrum, noise-OFDM, and multiple encoded waveforms [@dokhanchi2019mmwave]. We now examine mmWave OFDMA-JRC in detail.
#### OFDMA-JRC
Consider the same bi-static scenario of Fig. \[fig:jrc\_top\]d that we earlier analyzed for the PMCW-JRC system. The OFDMA-JRC transmitter (Fig. \[fig:jrc\_block\]) sends $N_s$ OFDM symbols from $N_{t}$ transmit antennas and reflections from $Q$ targets impinge on $N_{r}$ receive antennas. Assume that $\beta$ is angle of departure. The Doppler shift and flight time for the paths are assumed to be fixed over a CPI, i.e., $N_sT_{sym}$, where $T_{sym}$ is the duration of one OFDM symbol and $a_{n,m}$ are multiplexed communications/radar DPSK on $n$th carrier of $m$th OFDM symbol. Let $N_c$ be the number of subcarriers and $\Delta f$ be the subcarriers spacing, then the joint transmit waveform in baseband neglecting the CP is, $$\begin{aligned}
\label{MODEL_TX}
& x_i(t)= \sum_{m=0}^{N_s-1} \sum_{n=0}^{N_c-1} a_{n,m} e^{j2\pi f_nt}
e^{j k \sin(\beta) (i-1) \frac{\lambda}{2}} s(t-mT_{\text{sym}}),\end{aligned}$$ where $s(t)$ is a rectangular pulse of the width $T_{\text{sym}}$, $i\in[1,N_t]$, $n$ and $m$ are frequency and time indices respectively, and $f_n=n\Delta f=\frac{n}{T_{sym}}$ [@dokhanchi2019mmwave]. The received signal at the $p$th receiver over a CPI is, $$\begin{aligned}
\label{MODEL}
\tilde y_p(t)= \sum_{m=0}^{N_s-1} \sum_{q=1}^{Q} \sum_{n=0}^{N_c-1} \sum_{i=1}^{N_t} d_{q,i,p} a_{n,m}e^{j2\pi f_n(t-\tau_{q})} e^{j2\pi f_{D_{q}} t} e^{j k \sin(\psi_{q}) (p-1) \frac{\lambda}{2}} s(t-m T_{sym}-\tau_{q}) + \tilde{N}_p(t), \end{aligned}$$ where $\tilde{N}_p(t)$ is the additive noise on antenna $p$, Similar to PMCW-JRC, $d_{q,i,p}$ denotes path-loss, phase-shift caused by carrier frequency and RCS of the target; $d_{q,i,p}$ is independent of the subcarrier index due to narrowband assumption. Similarly, the Doppler is assumed to be identical for all subcarriers given a small inter-carrier spacing. For notational convenience, we omit the noise in the following. We sample (\[MODEL\]) at intervals $t_s=\dfrac{1}{N_c \Delta f}$ as, $$\begin{aligned}
\label{ofdm_final0}
&
\tilde y_p[t_s]
= \sum_{m=0}^{N_s-1} \sum_{q=1}^{Q} \sum_{n=0}^{N_c-1} d_{q} s_{n,m}
e^{j2\pi \frac{nl}{N_c}}s(lt_s-m T_{\text{sym}}- \tau_{q}),\end{aligned}$$ where $l\in[1,L]$, $n\in[1,N_c]$ and $L \leq N_c$, $d_{q}=\sum_{i=1}^{N_t} d_{q,i,p}$ as before, and $\tilde s_{n,m}= a_{n,m} e^{-j2\pi n\Delta f \frac{R_{q}}{c}} e^{j2\pi m T_{\text{sym}}f_{D_{q}}} e^{j \pi \sin(\psi_{q}) (p-1)}$ $\tilde s_{n,m}$ contains information about range, Doppler, angle of arrival and communications. We assume the number of inverse Fast Fourier Transform (IFFT) points $N_c$ is equal to the number of fast-time samples $L$ in each OFDM symbol. The received signal samples can be viewed as a radar data cube in spatial, spectral and temporal domains with $N_t$ antennas, $N_c$ subcarriers and $N_s$ OFDM symbols. Let us stack the entire DPSK symbols into a matrix $\bm A \in \mathbb{C}^{N_c\times N_s} $ and $\bm a_m = [\bm A]_m$ be the communications symbols over all subcarriers at $m$th OFDM symbol time. For a given OFDM symbol, say $m$, collecting signals from all subcarriers across different antennas leads to the following slow-time slice of the data cube $$\begin{aligned}
\label{time_slice_ofdm}
\bm Y_{m}^{\text{ OFDMA-JRC}}
= \bm F_{N_c} Diag(\bm{a}_m ) \boldsymbol \Xi( \frac{-\Delta fR_q}{c}) Diag(\boldsymbol d) \boldsymbol C \ \in \mathbb{C}^{N_c\times N_r},
\end{aligned}$$ where $m\in[1,N_s]$, $ \bm \Xi( \frac{-\Delta fR_q}{c}) =[e^{-j2\pi n\Delta f \frac{R_q}{c}}]_{n=1,q=1}^{N_c,Q} \in \mathbb{C}^{N_c\times Q}$, $\bm C =[e^{j k \sin(\psi_q) (p-1) \frac{\lambda}{2}} ]_{q=1,p=1}^{Q,N_r} \in \mathbb{C}^{Q\times N_{r}}$ and $\bm d=\begin{bmatrix} d_1 & \cdots & d_Q \end{bmatrix}$. Further, $\bm F_{N_c} = [e^{j2\pi \frac{nl}{N_c}}]_{l=0,n=0}^{N_c-1,N_c-1}$ denotes $N_c$-point IFFT matrix. To estimate Doppler shifts, we consider subcarrier slice of data cube : $$\begin{aligned}
\label{eq:fd_est}
\bm Z_{n}^{\text{OFDMA-JRC}}= Diag(\bm a_n) \bm \Xi(f_{D_q}T_{sym}) Diag(\bm d) \bm C \in \mathbb{C}^{N_s\times N_r},
\end{aligned}$$ where $\bm a_n=[\bm A]_n\in \mathbb{C}^{N_s}$ are the DPSK symbols over slow-time, $\bm \Xi(f_{D_q}T_{sym})=[e^{j2\pi m T_{sym}f_{D_q}}]_{m=1,q=1}^{N_s,Q}$.
As in PMCW-JRC, the receive processing of OFDMA-JRC is affected by coupling of communications symbols with a radar parameter (range in case of OFDMA-JRC). To ensure that range estimation does not suffer by using all subcarriers, frequency-division multiplexing is employed (\[fig:jrc\_block\]) such that $\mu$% of the OFDMA subcarriers are allocated to radar (with known $a_{n,m}$ on these subcarriers) and the rest to JRC. The rest of the OFDMA-JRC receive processing is similar to PMCW-JRC (Fig. \[fig:jrc\_block\]) [@dokhanchi2019mmwave].
#### Comparison of PMCW- and OFDMA-JRC
While OFDMA encodes radar and communications simultaneously in the entire *time and space*, the PMCW does so in the entire *frequency and space*; hence, their DoFs and design spaces are in different domains. While it turns out that the receive system models of both waveforms are mathematically identical after matched filtering and retrieve all JRC parameters using similar super-resolution algorithms [@dokhanchi2019mmwave; @Dokhanchi2017Joint], their individual performances mimic the respective communications and radar-centric properties. For example, the AF of the bi-static PMCW-JRC inherits the low sidelobes from its parent stand-alone PMCW radar waveform as shown in a comparison with the AF of OFDMA-JRC in Fig. \[fig:amb\_fcns\], given the same bandwidth. On the other hand, the PMCW-JRC is more sensitive to the number of users while the orthogonality of waveforms in OFDMA-JRC makes the latter robust to inter-channel interference. Finally, in a networked vehicle scenario, it requires less complex infrastructure and processing to apply PMCW with predefined or stored sequences rather than using OFDMA to adaptively allocate band to each user [@dokhanchi2019mmwave; @donnet2006combining]. A comparison of estimation errors in the coupled parameter - range for OFDMA-JRC and Doppler for PMCW-JRC - using JRC super-resolution recovery [@dokhanchi2019mmwave] is shown in Fig. \[fig:ofdm\_pmcw\_res\] for $\mu=50$%.
![The AFs of bi-static mmWave JRC using (a) OFDMA (b) PMCW signals with the (c) Doppler and (d) delay cuts [@dokhanchi2019mmwave]. []{data-label="fig:amb_fcns"}](amb_fcns_v2.png){width="100.00000%"}
![The root-mean-square-error (RMSE) of estimated range of a single target using OFDMA-JRC with respect to (a) SNR and (b) BER using half ($\mu$=50%) or all subcarriers (full $N_c$) with perfect and imperfect recovery of communications symbols. The RMSE in Doppler estimate of a single target for PMCW-JRC using all and half frames with respect to (c) SNR and (d) BER. In both cases, JRC super-resolution algorithms [@dokhanchi2019mmwave] have been employed.[]{data-label="fig:ofdm_pmcw_res"}](ofdm_pmcw_res_v2.png){width="100.00000%"}
Joint Coding
------------
Recently, existing mmWave communications protocols that are embedded with codes which exhibit favorable radar ambiguity functions are garnering much attention for JRC. In particular, the 60 GHz IEEE 802.11ad wireless protocol has been employed with time-division multiplexing of radar-only and radar-communications frame. In general, these designs have temporal DoF (for a monostatic radar case). The IEEE 802.11ad single-carrier physical layer (SCPHY) frame consists of a short training field (STF), a channel estimation field (CEF), header, data and beamforming training field. The STF and CEF together form the SCPHY preamble. CEF contains two 512-point sequences $Gu_{512}[n]$ and $Gv_{512}[n]$, each containing a *Golay complementary pair* of length $256$, $\{Gau_{256}, Gbu_{256}\}$ and $\{Gav_{256}, Gbv_{256}\}$, respectively. A Golay pair has two sequences $Ga_N$ and $Gb_N$ each of the same length $N$ with entries $\pm1$, such that the sum of their *aperiodic* autocorrelation functions has a peak of $2N$ and zero sidelobes: $$\begin{aligned}
Ga_N[n]*Ga_N[-n] + Gb_N[n]*Gb_N[-n] = 2N\delta[n], \end{aligned}$$ where $*$ denotes linear convolution. This property is useful for channel estimation and target detection.
By exploiting the preamble of a single SCPHY frame for radar, the existing mmWave 802.11ad waveform simultaneously achieves a cm-level range resolution and a Gbps data rate [@KumChoGon:IEEE-802.11ad-based-Radar::17]. The limited velocity estimation performance of this waveform can be improved by using multiple fixed length frames in which preambles are reserved for radar [@KumChoGon:IEEE-802.11ad-based-Radar::17]. While this increases the radar integration duration leading to more accurate velocity estimation, the total preamble duration is also prolonged causing a significant degradation in the communications data rate [@KumNguHea:Performance-trade-off-in-an-adaptive:17]. A joint coding scheme based on the use of sparsity-based techniques in the time domain can minimize this trade-off between communications and radar [@KumVorHea:VIRTUAL-PULSE-DESIGN:19]. Here, the frame lengths are varied such that their preambles (exploited as radar pulses) are placed in non-uniformly. These non-uniformly pulses in a CPI are then used to construct a virtual block of several pulses increasing the radar pulse integration time and enabling an enhanced velocity estimation performance. If the channel is sparse, the same can be achieved in frequency-domain using sub-Nyquist processing [@mishra2017sub]. In [@mishra2019doppler], the wide bandwidth of mmWave is exploited using a Doppler-resilient 802.11ad link to obtain very high resolution profiles in range and Doppler with the ability to distinguish various automotive targets. Fig. \[fig:dop\_res\_sign\] shows distinct, detailed movements of each wheel of a car and body parts of a pedestrian as detected by an 802.11ad-based Doppler-resilient short range radar.
![Radar signatures generated from animation models of (a) a small car and (b) a pedestrian using Doppler-resilient 802.11ad waveform [@duggal2019micro; @mishra2019doppler]. As the targets move radially in front of the radar on the marked trajectories, the movements of the front right, front left, rear right, and rear left wheels (FRW, FLW, RRW and LLW, respectively) of the car as well as the torso, arms, and legs of the pedestrian are individually observed in (b, e) range-time and (c, f) Doppler-time domains.[]{data-label="fig:dop_res_sign"}](dop_res.png){width="100.00000%"}
Carrier Exploitation
--------------------
![Power allocation solutions for JRC carrier exploitation via (a) water-filling and (b) Neyman-Pearson test [@bica2015opportunistic].[]{data-label="fig:wf"}](wf.pdf){width="100.00000%"}
Selecting active subcarriers and controlling their power levels or PAPR in an adaptive manner is also useful for interference management. Radar systems generally utilize entire bandwidth to achieve high resolution. On the other hand, communications systems often allocate resource blocks of certain number of subcarriers to each user based on channel quality indicator (CQI) to satisfy their rate and system QoS requirements. Through feedback from the receivers, spectrum sensing, databases or other sources, the transmitters of both systems can have information about occupancy of different subcarriers, instantaneous or desired SINR levels, channel gains, and power constraints imposed by other coexisting subsystems. This awareness can be exploited in adaptively optimizing the power allocation among different subcarriers. An example of optimizing subcarrier power ($P_k$) allocations and imposing minimum desired rate constraints on wireless communications users and maximum power constraint $P_\mathrm{T}$ for the radar is as follows: $$\begin{aligned}
\label{eq:c3_opt}
&\underset{P_k,\eta}{\text{maximize}}\;\; p_{\mathrm{D}}\nonumber\\
&\text{subject to} \;p_{\mathrm{FA}} \leq \alpha,\nonumber\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\log \left( {1+ SINR_k} \right) \geq t_k, \; \forall k,\nonumber\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\displaystyle \sum_{k=0}^{N-1}} P_k \leq P_\mathrm{T},\end{aligned}$$ where $\eta$ is detection threshold for likelihood ratio test using Neyman-Pearson detection strategy with false alarm constraint $\alpha$. Two example power allocations from the radar perspective are depicted in Fig. \[fig:wf\]. A water-filling solution (Fig. \[fig:wf\]a) obtained by maximizing Mutual Information between received data and target and channel response allocates the radar power to those parts of spectrum where the signal experiences the least attenuation and interference level is low. The second approach (Fig. \[fig:wf\]b) takes into account channel gains, required SINR values at communications subsystems while maximizing the radar performance in the Neyman-Pearson sense for target detection task.
Cognition and Learning in mmWave JRC
====================================
Some more recent enabling architectures and technologies for mmWave JRC where the system can sense, learn and adapt to the changes in the channel are as follows.
#### Cognitive Systems
Cognitive radars and radios sense the spectrum and exchange information to build and learn their radio environment state. This typically implies channel estimation and feedback on channel quality. Spectrum cartography methods, that generate a map of spectrum access in different locations and frequencies at different time instances, have been developed in this context [@giannakis_cartography_2011]. Based on the obtained awareness, operational parameters of transmitters and receivers in each subsystem are adjusted to optimize their performance [@cohen2017spectrum]. Channel coherence times should be long enough for JRC to apply cognitive actions. Since this duration is in nanoseconds for mmWave environments, compressed sensing-based solutions aid in reducing required samples for cognitive processing [@mishra2017sub; @mishra2017performance].
#### Fast Waveforms
Algorithms that develop cognitive waveforms should have low computational complexity in order to re-design waveforms on-the-fly, typically within a single CPI. This is especially important for mmWave systems where the fast-time radar waveform can easily have a length of tens of thousands samples. In [@LiVor18a], waveform design in spectrally dense environment does not exceed a quadratic complexity. In [@mishra2017auto; @mishra2017sub], the mmWave radar based on sub-Nyquist sampling adaptively transmits in disjoint subbands and the vacant slots are used by vehicular communications.
#### Machine Learning
In order to facilitate fast configuration of mmWave JRC links with low latency and high efficiency, machine learning is useful to acquire situational awareness. This implies learning the evolution of spectrum state over time (including classifying radar target responses or other waveforms occupying the spectrum), acquiring the channel responses, identifying underutilized spectrum and exploiting it in an opportunistic manner. The deep learning methods are widely applied for tasks such as target classification, automatic waveform recognition and determining optimal antennas and RF chains [@elbir2019deep]. Optimal policies for coexisting systems may be learned using reinforcement learning approaches like partially observable Markov decision process (POMDP) and restless multiarm bandit (RMAB) [@lundenSPM].
#### Game Theoretic Solutions
The interaction between radar and communications systems sharing spectrum can be analyzed from a game theory perspective [@mishra2019power]. The two systems or players form an adversarial, non-cooperative game because of conflicting interests in sharing the spectrum. The game is also dynamic due to continuously evolving spectral states over time. The utility function is designed to reflect the possible strategies based on the respective players’ requirements. The solutions result in Nash or Stackelberg equilibrium which are the game states with the property that none or one of the players can do better, respectively. In comparison to sub-6 GHz, the solution space for mmWave is several GHz wide with much lower maximum transmit power.
Summary
=======
We outlined various aspects of implementing JRC systems at mmWave. The sheer number of mmWave antennas and huge bandwidth pose new challenges in waveform design and receiver processing that was not seen in other bands. The dynamic and highly variable environments of mmWave applications require continuous cognition of the mmWave channel by both radar and communications. While there are still many open problems in this area, mmWave JRC is a precursor to an emerging frontier of sub-mmWave or THF JRC where THF communications would coexist with the promising technology of low-THF ($.1$-$1$ THz) automotive and imaging radars.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is partially funded by the European Research Council grant titled Actively Enhanced Cognition based Framework for Design of Complex Systems and Luxembourg National Research Fund project Adaptive mmWave Radar Platform for enhanced Situational Awareness: Design and Implementation.
|
---
author:
- 'Tomoki <span style="font-variant:small-caps;">Endo</span>$^1$, Toshiki <span style="font-variant:small-caps;">Maruyama</span>$^2$, Satoshi <span style="font-variant:small-caps;">Chiba</span>$^2$ and Toshitaka <span style="font-variant:small-caps;">Tatsumi</span>$^1$'
title: 'Charge screening effect in the hadron-quark mixed phase '
---
Introduction {#intro}
============
Deconfinement phase transition is believed to occur in hot and/or high-density matter, while its mechanism has not been well understood yet. Many authors have studied this transition by model calculations or first-principle calculations like lattice QCD[@rev]. Nowadays it is widely accepted that quark matter exists in hot and/or high-density region like inner core of neutron stars. Static and dynamic properties of quark matter have been actively studied theoretically for quark-gluon plasma (QGP), color superconductivity [@alf1; @alf3] or magnetism [@tat1; @tat2; @tat3]. Phenomenologically quark matter has been actively searched for in relativistic heavy-ion collisions (RHIC)[@rhic1], or in early universe and compact stars[@mad3; @chen].
Many theoretical calculations have suggested that the deconfinement phase transition should be of first order in low temperature and high density area[@pisa; @latt]. Therefore we assume first-order phase transition in this study. We, hereafter, consider the phase transition from nuclear matter to three-flavor quark matter in neutron-star matter for simplicity. If the deconfinement transition is of first order, we may expect the [*mixed phase*]{} during the transition. The hadron-quark mixed phase has been considered during the hadronization in RHIC[@rhic21; @rhic22; @rhic23] or the boundary between quark matter and hadron matter in neutron stars[@gle2].
There is an issue about the mixed phase for the first-order phase transitions with more than one chemical potential[@gle1]. We often use the Maxwell construction (MC) to derive the equation of state (EOS) in thermodynamic equilibrium, as in the water-vapor phase transition. In this case both phases consist of single particle species (${\rm H_2O}$). However, if many particle species participate in the phase transition as in neutron-star matter, MC is no more an appropriate method. Before Glendenning first pointed out [@gle1], many people have applied MC to get EOS of the first-order phase transitions[@weis; @migd; @elli; @rose] expected in neutron stars, such as pion or kaon condensation and the deconfinement transition.
For the deconfinement transition in neutron-star matter, we consider quark degrees of freedom as well as hadrons and leptons. Accordingly we must introduce many chemical potentials for particle species, but the independent ones in this case are reduced to two, i.e. baryon-number chemical potential $\mu_\mathrm{B}$ and charge chemical potential $\mu_\mathrm{Q}$, due to beta-equilibrium and total charge neutrality. They are nothing but the neutron and electron chemical potentials, $\mu_n$ and $\mu_e$, respectively. In the mixed-phase these chemical potentials should be spatially constant. When we naively apply MC to get EOS in thermodynamic equilibrium, we immediately notice that $\mu_\mathrm{B}$ is constant in the mixed-phase, while $\mu_\mathrm{e}$ is different in each phase because of the difference of the electron number in these phases. This is because MC uses EOS of bulk matter in each phase, which is of locally charge-neutral and uniform matter; many electrons are needed in hadron matter to cancel the positive charge of protons, while in quark matter total charge neutrality is almost fulfilled without electrons. Thus $$\mu_{\mathrm{B}}^{\mathrm{Q}} = \mu_{\mathrm{B}}^{\mathrm{H}}, \hspace{10pt}
\mu_{\mathrm{e}}^{\mathrm{Q}} \neq \mu_{\mathrm{e}}^{\mathrm{H}},
\label{chemeqMC}$$ in MC, where superscripts “Q” and “H” denote the quark and hadron phase, respectively. Glendenning emphasized that we must use the Gibbs conditions (GC) in this case instead of MC, which relaxes the charge-neutrality condition to be globally satisfied as a whole, not locally in each phase [@gle1]. GC imposes the following conditions, $$\begin{aligned}
\mu_{\mathrm{B}}^{\mathrm{Q}} &= \mu_{\mathrm{B}}^{\mathrm{H}},
\hspace{5pt} \mu_{\mathrm{e}}^{\mathrm{Q}} =
\mu_{\mathrm{e}}^{\mathrm{H}}, \nonumber \\
P^{\mathrm{Q}} &=P^{\mathrm{H}} , \hspace{5pt} T^{\mathrm{Q}} = T^{\mathrm{H}}.\end{aligned}$$ He demonstrated a wide region of the mixed phase, where two phases have a net charge but totally charge-neutral: EOS thus obtained, different from that given by MC, never exhibits a constant-pressure region. He simply considered the mixed phase consisting of two bulk matters separated by a sharp boundary without any surface tension and the Coulomb interaction, which we call “bulk Gibbs” for convenience.
“Bulk Gibbs” requires that each matter can have a net charge but total charge is neutral, $$f_V \rho_{\mathrm{ch}}^\mathrm{Q} + (1-f_V) \rho_{\mathrm{ch}}^\mathrm{H} = 0,$$ where $f_V$ means the volume fraction of quark matter in the mixed phase and “$\rho_\mathrm{ch}^\mathrm{Q,H}$” means charge density in each matter. Figure \[bulk\] shows the phase diagram in the $\mu_\mathrm{B}$ - $\mu_e$ plane. We can see that there is a discontinuous jump in $\mu_e$ for the case of MC, while the curve given by “bulk Gibbs” smoothly connects uniform hadron matter and uniform quark matter; the mixed phase can appear in the wide $\mu_\mathrm{B}$ region in “bulk Gibbs”, in contrast with MC[@alf2].
![ Phase diagram in the $\mu_\mathrm{B}-\mu_\mathrm{e}$ plane. There appears no region of the mixed phase by the calculation with MC, while a wide region of the mixed phase by the “bulk Gibbs” calculation.[]{data-label="bulk"}](mubmue00.eps){width="70mm"}
However, this “bulk Gibbs” should be too simple to study the mixed phase, since we must consider non-uniform structures to be more realistic, instead of two bulk uniform matters; the mixed phase should have various geometrical structures, where both the number and charge densities are no more uniform. Then we have to take into account the finite-size effects like the surface and the Coulomb interaction energies. In this paper we study the structured mixed phase by treating the finite-size effects self-consistently. We show that the mixed phase should be narrow in the $\mu_\mathrm{B}$ space by the charge screening effect, and derive EOS for the deconfinement transition in neutron-star matter. We shall see it results in EOS being similar to the one given by MC. We also discuss the interplay of the Coulomb interaction effect and the surface effect in the context of the hadron-quark mixed phase. Preliminary results for the droplet case has been already reported in Ref.[@end1].
The plan of the paper is as follows. We briefly review the previous works in Sec. \[history\]. Section \[formalism\] is devoted to our formalism and the numerical procedure. We show the results of our calculation and discuss the screening effect in Sec. \[results\]. Finally, a summary and concluding remarks are given in Sec. \[summary\].
Brief review of the previous works {#history}
==================================
Heiselberg et al. [@pet] studied a geometrical structure in the mixed phase: they considered spherical quark droplets embedded in hadron matter by including the surface and the Coulomb energies. They introduced the surface tension and treated its strength as a free parameter because the surface tension at the hadron-quark interface has not been clearly understood. They pointed out that if the surface tension parameter $\sigma$ is large ($\sigma \geq$ 90 MeV/fm$^2$), the region of the mixed phase is largely limited or cannot exist. Subsequently Glendenning and Pei [@gle2] have suggested the “crystalline structures of the mixed phase” which have some geometrical structures, “droplet”, “rod”, “slab”, “tube”, and “bubble”, assuming a small $\sigma$[@gle2; @gle3].
The finite-size effects are obvious in these calculations by observing energies. We may consider only a single cell, by dividing the whole space into equivalent Wigner-Seitz cells: the cell size is denoted by $R_W$ and the size of the lump (droplet, rod, slab, tube or bubble) by $R$. Then the surface energy density is expressed in terms of the surface tension parameter $\sigma$ as $$\begin{aligned}
\epsilon_\mathrm{S} = \frac{f_V \sigma d}{R},\end{aligned}$$ where $d$ denotes the dimensionality of each geometrical structure; $d=3$ for droplet and bubble, $d=2$ for rod and tube, and $d=1$ for slab. The Coulomb energy density reads $$\begin{aligned}
\epsilon_\mathrm{C} &=& 2\pi e^2 \left(\rho_\mathrm{ch}^\mathrm{H}
- \rho_\mathrm{ch}^\mathrm{Q} \right)^2 R^2 \Phi_d(f_V) \hspace{5pt},\\
\Phi_d(f_V) &\equiv& \left[2 (d-2)^{-1} \left( 1 -\frac{1}{2} d f_V^{1-2/d} \right)
+ f_V \right] \left(d+2 \right)^{-1},\end{aligned}$$ where we simply assumed uniform density in each phase. When we minimize the sum of $\epsilon_S$ and $\epsilon_C$ with respect to the size $R$ for a given volume fraction $f_V$, we can get the well known relation, $$\epsilon_S = 2 \epsilon_C.
\label{edensCS}$$ This implies that an optimal size of the lump is determined by the balance of these finite-size effects. Eventually we can express the size-dependent energy density besides the bulk energy density [@pet]: $$\epsilon^{(d)}_\mathrm{C} + \epsilon^{(d)}_\mathrm{S} = 3 f_V d \left(
\frac{\pi
\sigma^2
\left(
\rho_\mathrm{ch}^\mathrm{H} - \rho_\mathrm{ch}^\mathrm{Q} \right)^2 \Phi_d (f_V)}{2d} \right)^{1/3}.
\label{bulk_cs}$$ Thus we can calculate the energy of any geometrically structured mixed phase with (\[bulk\_cs\]) by changing the parameter $d$. Many authors have taken this treatment for the mixed phase[@gle2; @alf2; @pet]. Note that the energy sum in Eq. (\[bulk\_cs\]) becomes larger as the surface tension gets stronger, while the relation Eq.(\[edensCS\]) is always kept.
However, this treatment is not a self-consistent, but a perturbative one, since the charge screening effect for the Coulomb potential or the rearrangement of charged-particles in the presence of the Coulomb interaction is completely discarded. We shall see that the Coulomb potential is never weak in the mixed phase, and thereby this treatment overestimates the Coulomb energy. The charge screening effect is included only if we introduce the Coulomb potential and consistently solve the Poisson equation with other equations of motion for charged particles. Consequently it is a highly non-perturbative effect. Norsen and Reddy[@nors] have studied the Debye screening effect in the context of kaon condensation to see a large change of the charged-particle densities like kaons and protons. Maruyama et al. have numerically studied it in the context of liquid-gas phase transition at subnuclear densities [@maru1], where nuclear pastas can be regarded as geometrical structures in the mixed phase. Subsequently, they have also studied kaon condensation at high-densities [@maru2], where we have seen that kaonic pastas appear in the mixed phase. Through these works we have figured out the role of the Debye screening in the mixed phase. We have also studied the interplay of the Coulomb effect and the surface effect.
Voskresensky et al. [@vos] explicitly studied the Debye screening effect for a few geometrical structures of the hadron-quark mixed phase. They have shown that the optimal value of the size of the structure cannot be obtained due to the charge screening even if the surface tension is not so strong. They called it as mechanical instability. It occurs because the Coulomb energy density is suppressed at larger size than the Debye screening length (cf. Eq. (\[lambda\])). They also suggested that the properties of the mixed phase become very similar to those given by MC, if the charge screening effectively works. They also noted that the apparent violation of the Gibbs condition (Eq. (\[chemeqMC\])) can be remedied by including the Coulomb potential in a gauge-invariant way: the number of the charged particles is given by a gauge invariant combination of the chemical potential and the Coulomb potential, and thereby the number can be different in each phase for a constant charge chemical potential if the the Coulomb potential takes different values in both phases. However, they used a linear approximation to solve the Poisson equation analytically.
If the Coulomb interaction effect is so important, it would be important to study it without recourse to any approximation. In this paper we numerically study the charge screening effect on the structured mixed phase during the deconfinement transition in neutron-star matter in a self-consistent way. Actually we shall see importance of non-linear effects included in the Poisson equation.
Self-consistent calculation {#formalism}
===========================
Thermodynamic potential
-----------------------
We consider the geometrically structured mixed phase (SMP) where one phase is embedded in the other phase with a certain geometrical form.
[l]{}[50mm]{}
{width="30mm" height="60mm"}
We divide the whole space into equivalent charge-neutral Wigner-Seitz cells with a size $R_W$ and a size of embedded phase $R$ as illustrated in Fig. \[wsapp\]. Quark phase consists of [*u*]{}, [*d*]{}, [*s*]{} quarks and electron. Hadron phase consists of proton, neutron and electron. We incorporate the MIT Bag model and assume the sharp boundary at the hadron-quark interface. We use density functional theory (DFT) and incorporate local density approximation (LDA) [@parr; @drez].
We consider total thermodynamic potential ($\Omega_\mathrm{total}$) which consists of the hadron, quark and electron and the Coulomb interaction contributions: $$\Omega_\mathrm{total} = \Omega_\mathrm{hadron} +\Omega_\mathrm{quark} +\Omega_\mathrm{em},
\label{ometot}$$ where we summarize the contributions of electrons and the Coulomb interaction as $\Omega_\mathrm{em}$ because they are present in both phases.
We briefly present the expressions of thermodynamic potentials. The details of the derivation of the expressions are given in Ref.[@end2].
First, the Coulomb interaction energy is expressed in terms of particle densities, $$E_V = \frac{1}{2} \sum_{i,j} \int d^3 r d^3 r^{\prime} \frac{Q_i \rho_i(\vec{r}) Q_j \rho_j(\vec{r}^{\prime})}{\left| \vec{r} - \vec{r}^{\prime} \right|},$$ where $i=u,d,s,p,n,e$ with $Q_i$ being the particle charge ($Q=-e < 0$ for the electron). Accordingly the Coulomb potential is defined as $$V (\vec{r}) = -\sum_i \int d^3 r^{\prime} \frac{e Q_i
\rho_i(\vec{r}^{\prime})}{\left| \vec{r} - \vec{r}^{\prime} \right|}+V_0,
\label{vcoul}$$ where $V_0$ is an arbitrary constant representing the gauge degree of freedom. We fix the gauge by a condition $V(R_W) = 0$ in this paper (see Sec. 2.2). Operating a Laplacian $\nabla^2$ on the Coulomb potential $V(\vec{r})$, we automatically derive the Poisson equation.
Therefore, the electron contribution and the Coulomb interaction energy (in both phases) are expressed as $$\begin{aligned}
\Omega_{\mathrm{em}} \! \! &=& \! \! \int \! d\vec{r} \Biggl[ -\frac{1}{8\pi e^2} \left(
\nabla V
(\vec{r})
\right)^2
+ \epsilon_e (\rho_e (\vec{r})) - \mu_e \rho_e (\vec{r}) + V (\vec{r}) \rho_e (\vec{r}) \Biggr] \nonumber \\
\! \! &=& \! \int \! d\vec{r} \left[ -\frac{1}{8\pi e^2} \left( \nabla V (\vec{r})
\right)^2 \!-\! \frac{\left(V
(\vec{r}) \!- \! \mu_e \right)^4}{12\pi^2} \! \right], \end{aligned}$$ where $\epsilon_e (\rho_e(\vec{r}))=\frac{\left( 3 \pi^2 \rho_e (\vec{r})\right)^\frac{4}{3}}{4\pi^2}$ is the kinetic energy density of electron.
Secondly, in the quark phase, [*u*]{} and [*d*]{} quarks are treated as massless particles and only [*s*]{} quark massive one, $m_s= 150$ MeV. The kinetic energy of quark of flavor $f$ is simply expressed as[@tama] $$\epsilon_{f \mathrm{kin}}= \frac{3}{8 \pi^2} m_f^4 \left[ x_f \eta_f \left( 2x_f^2+1 \right) -\ln\left( x_f +\eta_f \right) \right],$$ where $m_f$ is the quark mass, $x_f = p_{\mathrm{F}f}(\vec{r})/m_f$ with Fermi momentum $p_{\mathrm{F}f}(\vec{r}) = (\pi^2 \rho_f (\vec{r}) )^\frac{1}{3}$ and $\eta_f=\sqrt{1+x^2_f}$.
For the interaction energy, we take into account the leading order contribution coming from the one-gluon exchange interaction. Since the contribution from the Hartree term disappears due to the traceless property of the Gell-Mann matrix ($\lambda$), the leading order contribution only comes from the Fock term, $$\epsilon_{f \mathrm{Fock}} = -\frac{\alpha_{\mathrm{c}}}{\pi^3} m_f^4 \left\{ x_f^4 - \frac{3}{2} \left[ x_f \eta_f -\ln\left( x_f +\eta_f \right) \right]^2 \right\}.
\label{fock}$$ Including this interaction, the quark contribution to the thermodynamic potential is expressed as $$\begin{aligned}
& & \Omega_{\mathrm{quark}} =
\Omega_{\mathrm{u}}+\Omega_{\mathrm{d}}+\Omega_{\mathrm{s}} +\int d\vec{r} B , \label{omeq}\\
\Omega_f &=& \int \! d\vec{r} \left[
\epsilon_f (\rho_f (\vec{r})) - \mu_f \rho_f (\vec{r}) -
N_i V (\vec{r})
\rho_f (\vec{r}) \right] , \hspace{10pt} N_i=\frac{Q_i}{e},
\nonumber \end{aligned}$$ where, the energy density $\epsilon_f (\rho_f
(\vec{r}))$ stands for $\epsilon_{f \mathrm{kin}} + \epsilon_{f
\mathrm{Fock}}$ of $f$ quark, and $B$ is the Bag constant. The Bag constant is taken as 120 MeV/fm$^3$, and the QCD fine structure constant as $\alpha_{\mathrm{c}}=0.4$, which are also used by Heiselberg et al. [@pet] and in the previous work[@vos].
Thirdly, we consider the hadron contribution. The thermodynamic potential for the non-relativistic nucleons becomes $$\begin{aligned}
\Omega_{\mathrm{hadron}} = E_N - \sum_{a=p, n} \mu_a \int d\vec{r} \rho_a
(\vec{r}) - \int d \vec{r} \, V (\vec{r}) \rho_{\mathrm{p}}(\vec{r}) ,
\label{omeh}\end{aligned}$$ where $E_N$ is the energy of the nucleons, $$\begin{aligned}
E_N = \int d \vec{r} \left[ \sum_{a=p, n} \frac{3}{10m}\left( 3 \pi^2 \right)^\frac{2}{3} \rho^\frac{5}{3}_a
(\vec{r}) + \epsilon_\mathrm{pot} \left( \rho_{\mathrm{p}} (\vec{r})
, \rho_{\mathrm{n}} (\vec{r})
\right) \right].\end{aligned}$$ Here we use the effective potential $\epsilon_\mathrm{pot} (\rho_{\mathrm{p}} (\vec{r}), \rho_{\mathrm{n}}
(\vec{r}))$ parametrized by the local densities for simplicity, $$\begin{aligned}
\epsilon_\mathrm{pot} (\vec{r}) &=& S_0 \frac{\left( \rho_\mathrm{n}(\vec{r}) - \rho_\mathrm{p}(\vec{r}) \right)}{\rho_0(\vec{r})} + \left( \rho_\mathrm{n}(\vec{r}) + \rho_\mathrm{p}(\vec{r}) \right) \epsilon_\mathrm{bind} \nonumber \\
&+& K_0 \frac{\left( \rho_\mathrm{n}(\vec{r}) + \rho_\mathrm{p}(\vec{r}) \right)}{18} \left( \frac{\rho_\mathrm{n}(\vec{r}) + \rho_\mathrm{p}}{\rho_0} - 1 \right)^2 \nonumber\\
&+& C_\mathrm{sat} \left( \rho_\mathrm{n}(\vec{r}) +
\rho_\mathrm{p}(\vec{r}) \right) \left(
\frac{\rho_\mathrm{n}(\vec{r}) +
\rho_\mathrm{p}(\vec{r})}{\rho_0} - 1
\right) ,
\label{effpot}\end{aligned}$$ where $S_0$, $K_0$, $\epsilon_\mathrm{bind}$, and $C_\mathrm{sat}$ are adjustable parameters to reproduce the saturation properties of nuclear matter[@vos]. We consider beta equilibrium at the hadron-quark interface as well as in each phase: $$\begin{aligned}
&& \mu_\mathrm{u}+\mu_\mathrm{e} = \mu_\mathrm{d},\nonumber\\
&& \mu_\mathrm{d} = \mu_\mathrm{s}, \nonumber \\
&& \mu_\mathrm{p}+\mu_\mathrm{e} = \mu_\mathrm{n} \equiv \mu_\mathrm{B}, \nonumber \\
&& \mu_\mathrm{n} = \mu_\mathrm{u}+2 \mu_\mathrm{d}, \nonumber\\
&& \mu_\mathrm{p} = 2 \mu_\mathrm{u}+\mu_\mathrm{d}.
\label{chemeq}\end{aligned}$$ The last relation can be derived from other four relations, so that there are left four independent conditions for chemical equilibrium.
We get the equations of motion from $\frac{\delta \Omega_\mathrm{total}}{\delta\phi_i}=0$ ( $\phi_{i}=\rho_u(\vec{r}), \rho_d(\vec{r}), \rho_s(\vec{r}),$ $\rho_p(\vec{r}), \rho_n(\vec{r}), \rho_e(\vec{r}), V(\vec{r})$ ): The Poisson equation then reads $$\begin{aligned}
\nabla^2 V (\vec{r}) \! \! = \! \! 4 \pi e^2 \left[ \left(\frac{2}{3}\rho_u
(\vec{r})- \frac{1}{3}\rho_d
(\vec{r}) - \frac{1}{3} \rho_s
(\vec{r}) \right)
+ \rho_{\mathrm{p}} (\vec{r}) - \rho_e (\vec{r}) \right].
\label{poisson}\end{aligned}$$ Other equations of motion give nothing but the expressions of the chemical potentials, $$\mu_i = \frac{\delta E_\mathrm{kin+str}}{\delta \rho_i (\vec{r})} - N_i V(\vec{r}),
\label{mu_i}$$ where $\displaystyle E_\mathrm{kin+str} = \sum_{i= u, d,
s,e} \int d\vec{r} \epsilon_i + E_N$. Then quark chemical potentials are expressed as $$\begin{aligned}
\mu_{\mathrm{u}} &=& \left( 1 + \frac{2 \alpha_{\mathrm{c}}}{3 \pi} \right) \pi^\frac{2}{3} \rho_{\mathrm{u}}^\frac{1}{3} (\vec{r})- \frac{2}{3} V (\vec{r})\\
\mu_{\mathrm{d}} &=& \left( 1 + \frac{2 \alpha_{\mathrm{c}}}{3 \pi} \right) \pi^\frac{2}{3} \rho_{\mathrm{d}}^\frac{1}{3}(\vec{r}) + \frac{1}{3} V (\vec{r})\\
\mu_{\mathrm{s}} &=& \epsilon_{{\mathrm{Fs}}}(\vec{r}) + \frac{2
\alpha_{\mathrm{c}}}{3 \pi} \left[ p_{\mathrm{Fs}}(\vec{r})- 3
\frac{m_{\mathrm{s}}^2}{\epsilon_{\mathrm{Fs}}(\vec{r})} \ln \left( \frac{\epsilon_{\mathrm{Fs}}(\vec{r})+p_{\mathrm{Fs}}(\vec{r})}{m_{\mathrm{s}}} \right) \right] + \frac{1}{3} V (\vec{r}), \nonumber \\\end{aligned}$$ with $\epsilon_{\mathrm{Fs}} (\vec{r})=
\sqrt{m_{\mathrm{s}}^2+p_{\mathrm{Fs}}^2(\vec{r})}$. On the other hand chemical potentials of nucleons and electrons are $$\begin{aligned}
\mu_{\mathrm{n}} &=& \frac{p_{\mathrm{Fn}}^2}{2m} + \frac{2S_0 \left(\rho_n(\vec{r})-\rho_{\mathrm{p}}(\vec{r})
\right)}{\rho_0}+ \epsilon_{\mathrm{bind}} \nonumber \\
&+& \frac{K_0}{6} \left( \frac{\rho_{\mathrm{n}} (\vec{r}) \!+\!
\rho_{\mathrm{p}} (\vec{r})} {\rho_0} - 1 \right)^2 +
\frac{K_0}{9} \left( \frac{\rho_{\mathrm{n}} (\vec{r})+
\rho_{\mathrm{p}} (\vec{r})}{\rho_0}- 1 \right) \nonumber \\
&+& 2 C_{\mathrm{sat}} \frac{\rho_{\mathrm{n}}(\vec{r}) + \rho_{\mathrm{p}}(\vec{r})}{\rho_0} - C_{\mathrm{sat}} \\
\mu_{\mathrm{p}} &=& \mu_{\mathrm{n}} - \frac{p_{\mathrm{Fn}}^2
(\vec{r})}{2m}+ \frac{p_{\mathrm{Fp}}^2 (\vec{r})}{2m} - \frac{4 S_0
\left( \rho_{\mathrm{B}} - 2 \rho_{\mathrm{p}} (\vec{r})
\right)^2}{\rho_0} - V (\vec{r}) \nonumber \\
\mu_e &=& \left( 3 \pi^2 \rho_e (\vec{r}) \right)^\frac{1}{3} + V (\vec{r}).\end{aligned}$$ We solve these equations of motion under GC. Important point is that the Coulomb potential $V(\vec{r})$ is included in each expression in a proper way. The Coulomb potential is the function of charged-particle densities, and in turn densities are functions of the Coulomb potential. As a result, the Poisson equation becomes highly non-linear. Since it should be difficult to solve them analytically, we numerically solve them without any approximation.
Once the geometrical structure is concerned, we have to take into account the surface tension as well at the interface of the hadron and quark phases. It may be connected with the confining mechanism and unfortunately we have no definite idea about how to incorporate it. Actually many authors have treated its strength as a free parameter and seen how the results are changed by its value[@gle2; @pet; @alf2]. Here we also follow this manner by introducing the surface tension parameter $\sigma$ to simulate the surface effect. One might be afraid that the surface tension will be modified once the Coulomb interaction is explicitly introduced. However, such modification might be rather small, as inferred from the previous result[@maru2].
Note that we have to determine now eight variables, i.e., six chemical potentials, $\mu_\mathrm{u}$, $\mu_\mathrm{d}$, $\mu_\mathrm{s}$, $\mu_\mathrm{p}$, $\mu_\mathrm{n}$, $\mu_\mathrm{e}$, and radii $R$ and $R_W$. First, we fix $R$ and $R_W$. Here we have four conditions due to $\beta$ equilibrium (\[chemeq\]). Therefore, once two chemical potentials $\mu_\mathrm{B}$ and $\mu_\mathrm{e}$ are given, we can determine other four chemical potentials, $\mu_\mathrm{u}$, $\mu_\mathrm{d}$, $\mu_\mathrm{s}$ and $\mu_\mathrm{p}$. Next, we determine $\mu_\mathrm{e}$ by the global charge neutrality condition: $$f_V \rho_{\mathrm{ch}}^{\mathrm{Q}} + (1-f_V) \rho_{\mathrm{ch}}^{\mathrm{H}} = 0,$$ where the volume fraction $f_V=\left(\frac{R}{R_W}\right)^d$, and $d$ denotes the dimensionality of each geometrical structure. At this point $f_V$ is still fixed.
The pressure coming from the surface tension is given by $$P_{\sigma}= \sigma \frac{d S}{d V_{\rm Q}},$$ where $S$ is the area of the surface and $V_{\rm Q}$ is the volume of the quark phase. Then we find the optimal value of $R$ ($R_W$ is fixed and thereby $f_V$ is changed by $R$) by using one of GC; $$P^\mathrm{Q} = P^\mathrm{H} + P_\sigma.
\label{pbalance}$$ The pressure in each phase $P^\mathrm{Q(H)}$ is given by the thermodynamic relation: $
P^\mathrm{Q(H)}=-\Omega_\mathrm{Q(H)}/V_\mathrm{Q(H)}$, where $\Omega_\mathrm{Q(H)}$ is the thermodynamic potential in each phase and given by adding electron and the Coulomb interaction contributions to $\Omega_\mathrm{quak(hadron)}$ in Eqs. (\[omeq\]) and (\[omeh\]). Finally, we determine $R_W$ by minimizing thermodynamic potential. Therefore once $\mu_\mathrm{B}$ is given, all other values $\mu_i$ ($i=u,d,s,p,e$) and $R$, $R_W$ can be obtained.
Note that we keep GC throughout the numerical procedure. We will see later how the mixed phase would be changed by including finite-size effects keeping GC completely. Although MC is not rigidly correct as we have seen in Sec. 1, our results will show a similar behavior to those by MC as a result of including the finite-size effects.
In numerical calculation, every point inside the cell is represented by a grid point (the number of grid points $N_\mathrm{grid} \approx 100 $). Equations of motion are solved by a relaxation method for a given baryon-number chemical potential under constraints of the global charge neutrality.
Proper treatment of the Coulomb interaction
-------------------------------------------
With the Coulomb potential (\[vcoul\]) and thermodynamic potentials (\[ometot\]), the gauge invariance of our treatment can easily be seen as follows: varying the expression of chemical potentials (\[mu\_i\]) with respect to the Coulomb potential $V(\vec{r})$, as is shown in the previous work[@vos], we have $$A_{ij} \frac{\partial \rho_j }{\partial V } = N_i , \hspace{10pt}
\hspace{5pt} A_{ij} B_{jk} = \delta_{ik},$$ where matrices $A$ and $B$ are defined as $$A_{ij} \equiv \frac{ \delta^2 E_{\mathrm{kin+str}}}{\delta \rho_i \delta
\rho_j}
\hspace{10pt} B_{ij} \equiv \frac{\partial \rho_i}{\partial \mu_j}.$$ From these equations, the gauge-invariance relation follows, $$\frac{\partial \rho_i}{\partial V} = N_j \frac{\partial \rho_j}{\partial \mu_i}.$$ We can understand that chemical potential is gauge variant from this relation. When the Coulomb potential is shifted by a constant value, $V(\vec{r}) \Longrightarrow V({\vec{r}}) - V_0$, the charge chemical potential should be also shifted as $\mu_i \Longrightarrow
\mu_i+N_i V_0$. To take into account the Coulomb interaction, we have to include $V(\vec{r})$ in the gauge invariant way like in Eq. (\[mu\_i\]). Note that the phase diagram in the $\mu_\mathrm{B}-\mu_\mathrm{e}$ plane (see ,e.g., Fig. \[bulk\]) is not well-defined, since the charge chemical potential $\mu_e$ is not gauge invariant by itself.
Numerical results {#results}
=================
We show the thermodynamic potential in Figs. \[omed40\] and \[omed60\]. In uniform matter, hadron phase is thermodynamically favorable for $\mu_\mathrm{B} < 1225$ MeV and quark phase for $\mu_\mathrm{B} > 1225$. Therefore we plot $\delta \omega$, difference of the thermodynamic potential density between the mixed phase and each uniform matter: $$\delta \omega = \begin{cases} \omega_\mathrm{total}-\omega^\mathrm{uniform}_\mathrm{H} \quad \mu_\mathrm{B}
\leq 1225 \mathrm{MeV}, \\
\omega_\mathrm{total}-\omega^\mathrm{uniform}_\mathrm{Q} \quad \mu_\mathrm{B} \geq 1225 \mathrm{MeV},
\end{cases}$$ where $\omega_\mathrm{total}=\Omega_\mathrm{total}/V_W$, etc. There we also depict two results for comparison: one is given by the “bulk Gibbs” calculation, where the finite-size effects are completely discarded. The other is the thermodynamic potential given by a perturbative treatment of the Coulomb interaction, which is denoted by “no Coulomb”; discarding the Coulomb potential $V(\vec{r})$, we solve the equations of motion to get density profiles, then evaluate the Coulomb interaction energy by using the density profiles thus determined. We can see the screening effects by comparing this “no Coulomb” calculation with the self-consistent one denoted by “screening”. $\delta \omega$ given by MC appears as a point denoted by a circle in Figs. \[omed40\] and \[omed60\] where two conditions, $P^\mathrm{Q}=P^\mathrm{H}$ and $\mu_\mathrm{B}^\mathrm{Q}=\mu_\mathrm{B}^\mathrm{H}$, are satisfied. On the other hand the mixed phase derived from “bulk Gibbs” appears in a wide region of $\mu_{\mathrm{B}}$. Therefore, if the region of the mixed phase becomes narrower, it signals that the properties of the mixed phase become close to those of MC. One may clearly see that $\omega_{\mathrm{total}}$ becomes close to that given by MC due to the finite-size effects, the effects of the surface tension and the Coulomb interaction.
![ Same as Fig. 3 for $\sigma=60$ MeV/fm$^{2}$. The negative $\delta
\omega$ region is narrower than the $\sigma=40$ MeV/fm$^{2}$ case.[]{data-label="omed60"}](alfo40.eps){width="70mm"}
![ Same as Fig. 3 for $\sigma=60$ MeV/fm$^{2}$. The negative $\delta
\omega$ region is narrower than the $\sigma=40$ MeV/fm$^{2}$ case.[]{data-label="omed60"}](alfo60.eps){width="70mm"}
The large increase of $\delta\omega$ from the “bulk Gibbs” curve comes from the effects of the surface tension and the Coulomb potential. Since the surface tension parameter is introduced by hand, we must carefully study the effects of the surface tension and the Coulomb interaction, separately. From the difference between the result given by “no Coulomb” and that by “bulk Gibbs”, we can roughly say that about $2/3$ of the increase comes from the effect of the surface tension and $1/3$ from the Coulomb interaction (see Eq. (\[edensCS\])).
Comparing the result of self-consistent calculation with that of “no Coulomb”, we can see that the change of energy caused by the screening effect is not so large, but still the same order of magnitude as that given by the surface effect.
![Density profiles and the Coulomb potential given by the self-consistent calculation for the same parameter set as Fig. 5. $R=7.7$ fm and $R_W=18.9$ fm.[]{data-label="densprof-sc"}](densprof-dropno.eps){width="70mm"}
![Density profiles and the Coulomb potential given by the self-consistent calculation for the same parameter set as Fig. 5. $R=7.7$ fm and $R_W=18.9$ fm.[]{data-label="densprof-sc"}](densprof-dropsc.eps){width="70mm"}
If the surface tension is stronger, the relative importance of the screening effect becomes smaller and the effect of the surface tension becomes more dominant, as is seen in Fig. 4.
[r]{}[70mm]{} {width="70mm"}
To be more realistic we have to take into account the modification of the surface tension as the structure size changes. Though we cannot clearly say how the surface tension is affected, we may infer from the previous study that it is not so large. Although the charge screening has not so large effects on bulk properties of the matter, we shall see that it is remarkable for the charged particles to change the properties of the mixed phase.
The screening effect induces the rearrangement of the charged particles. We can see this screening effect by comparing Fig. \[densprof-no\] with Fig. \[densprof-sc\]. The quark phase is negatively charged and the hadron phase is positively charged. The negatively charged particles in the quark phase such as [*d*]{}, [*s*]{}, [*e*]{} and the positively charged particle in the hadron phase [*p*]{} are attracted toward the boundary. On the contrary the positively charged particle in the quark phase [*u*]{} and negatively charged particle in the hadron phase [*e*]{} are repelled from the boundary. The charge screening effect also reduces the net charge in each phase. In Fig. \[chdensprof\], we show the local charge densities of the two cases shown in Figs. \[densprof-no\] and \[densprof-sc\]. The change of the number of charged particles due to the screening is as follows: In the quark phase, the numbers of $d$ and $s$ quarks and electrons decrease, while the number of $u$ quark increases. In the hadron phase, on the other hand, the proton number should decrease and the electron number should increase. Consequently the local charge decreases in the both phases. In Fig. \[chdensprof\] we can see that the core region of the droplet tends to be charge-neutral and near the boundary of the Wigner-Seitz cell is almost charge-neutral.
![ Same as Fig. 8 given by the self-consistent calculation with the screening effect. The size of the structure becomes larger than that given by “no Coulomb”, and consequently exceeds the Debye screening length.[]{data-label="rho-cell40sc"}](rho-cell40no.eps){width="70mm"}
![ Same as Fig. 8 given by the self-consistent calculation with the screening effect. The size of the structure becomes larger than that given by “no Coulomb”, and consequently exceeds the Debye screening length.[]{data-label="rho-cell40sc"}](rho-cell40.eps){width="70mm"}
[r]{}[80mm]{} {width="80mm"}
In Figs. \[rho-cell40no\] and \[rho-cell40sc\] we present the lump and cell radii for each density. As we have shown in the previous paper[@vos], the Coulomb energy is suppressed for larger $R$ by the screening effect. The $R$ dependence of the total thermodynamic potential comes from the contributions of the surface tension and the Coulomb interaction: the optimal radius giving the minimum of the thermodynamic potential is then determined by the balance between two contributions, since the former gives a decreasing function, while the latter an increasing one. If the Coulomb energy is suppressed, the minimum of the thermodynamic potential is shifted to larger radius. As a result the size of the embedded phase ($R$) and the cell size ($R_W$) become large. In Ref. [@vos] they demonstrated that the minimum disappears for a large value of the surface tension parameter: the structure becomes mechanically unstable in this case. We cannot show it directly in our framework because such unstable solutions are automatically excluded during the numerical procedure, while we can see its tendency in Figs. \[rho-cell40no\] and \[rho-cell40sc\]: $R$ and $R_W$ get larger by the screening effect.
We also see the relation between the size of the geometrical structure and the Debye screening length. The Debye screening length appears in the [*linearized*]{} Poisson equation and is then given as $$\left(\lambda^{q}_D\right)^{-2}\!\!=\! 4 \pi \sum_f Q_f \! \left( \frac{\partial \langle
\rho_f^\mathrm{ch}
\rangle}{\partial \mu_f} \right), \hspace{5pt}
\left(\lambda^{p}_D\right)^{-2}\!\!=\! 4 \pi Q_p \! \left( \frac{\partial \langle
\rho_p^\mathrm{ch}
\rangle}{\partial \mu_p} \right), \hspace{5pt}
\left(\lambda^{e}_D\right)^{-2}\!\!=\! 4 \pi Q_e \! \left( \frac{\partial \langle
\rho_e^\mathrm{ch}
\rangle}{\partial \mu_e} \right),
\label{lambda}$$ where $\langle \rho_f^\mathrm{ch} \rangle$ stands for the averaged density in quark phase, $\langle \rho_p^\mathrm{ch}
\rangle$ is proton number averaged density in the hadron phase and $\langle \rho_e^\mathrm{ch} \rangle$ is the electron charge density averaged inside the cell. It gives a rough measure for the screening effect: At a distance larger than the Debye screening length, the Coulomb interaction is effectively suppressed.
In Fig. \[rho-cell40no\] we show sizes of geometrical structure for “no Coulomb” case. If we ignore the screening effect, the size of the embedded phase is comparable or smaller than the corresponding quark Debye screening length $\lambda_D^q$ (Fig. \[lambdadrop\]). This may mean that the Debye screening is not so important. Actually, many authors have neglected the screening effect due to this argument[@pet; @alf2]. In Fig. \[rho-cell40sc\], however, we see that the size of the embedded phase can be larger than $\lambda_D^q$ (Fig. \[lambdadrop\]) in the self-consistent calculation. We can also see the similar situation about $R_W$ and $\lambda_D^e$. This means that the screening has important effects in this mixed phase. We cannot expect such a effect without solving the Poisson equation because of the non-linearity. We show the EOS in Figs. \[pres40no\] and \[pres40sc\]. The pressure of the mixed phase becomes similar to that given by MC due to the screening effect.
![ Same as Fig. \[pres40no\] given by the self-consistent calculation with the screening effect.[]{data-label="pres40sc"}](rho-pres40no.eps){width="70mm"}
![ Same as Fig. \[pres40no\] given by the self-consistent calculation with the screening effect.[]{data-label="pres40sc"}](rho-pres40.eps){width="70mm"}
We have used a fixed surface tension parameter in the present study. Surface tension is a very difficult problem because it should be self-consistent with the two phases of matter, quark and hadron. Lattice QCD, based on the first principle, would be the most reliable theory. It predicts that the surface tension can be 10-100 MeV/fm$^2$[@kaja; @huan]. Although this range is for high temperature, our choice is within it. Moreover, other model calculations of the surface tension[@mad1; @mad2; @berg; @mond] are similar to our choice. Although we cannot conclude that MC is [*perfectly*]{} correct, we can say that the results obtained by the “no Coulomb” calculation, which many authors have used, have to be checked again taking into account the finite-size effects.
Let us consider some implication of these results for neutron star phenomena. Glendenning[@gle2] suggested many SMP appear in the core region by using “bulk Gibbs”: the mixed phase should appear for several kilometers. However we can say that the region of SMP should be narrow in the $\mu_\mathrm{B}$ space and EOS is more similar to that of MC due to the finite-size effects. These results correspond to recent other calculations. Bejger et al. [@bejg] have examined the relation between the mixed phase and glitch phenomena, and shown that the mixed phase should be narrow if the glitch is generated by the mixed phase in the inner core. On the other hand the gravitational wave asks for density discontinuity in the core region[@mini]. These studies support our result.
Summary and concluding remarks {#summary}
==============================
We have numerically studied the charge screening effect in the hadron-quark mixed phase, by fully including the non-linear effects in the Poisson equation. Comparing the results with those given by “no Coulomb” calculation, we have elucidated the screening effect. The density profiles of the charged particles are much modified by the screening effect, while the thermodynamic potential is not so much affected; the charge rearrangement induced by the screening effect tends to make the net charge smaller in each phase. Consequently the system tends to be locally charge-neutral, which suggests that MC is effectively justified even if it is thermodynamically incorrect. In this context, it would be interesting to refer to the work by Heiselberg [@hei], who studied the screening effect on a quark droplet (strangelet) in the vacuum, and suggested the importance of the rearrangement of charged particles.
We have seen that thermodynamic quantities such as thermodynamic potential and pressure become close to those derived from MC by the screening effect, which also suggests that MC is effectively justified due to the screening effect. As another case of more than one chemical potential system, kaon condensation has been also studied[@maru1] and the results are similar to those in the present study. Thus the importance of the screening effect should be a common feature for the first-order phase transitions in high-density matter.
We have included the surface tension at the hadron-quark interface, while its definite value is not clear at present. There are also many estimations for the surface tension at the hadron-quark interface in lattice QCD [@kaja; @huan], in shell-model calculations [@mad1; @mad2; @berg] and in model calculations based on the dual-Ginzburg Landau theory [@mond]. Our parameter is in that reasonable range.
We have considered some implications of our results for neutron star phenomena. The screening effect would restrict the allowed SMP region in neutron stars, in contrast with a wide region given by “bulk Gibbs”[@gle2; @gle3]. It could be said that they should change the bulk property of neutron stars, especially the structure of the core region.
Compact stars have the strong magnetic field and its origin is not well understood. One possibility is that it comes from quark matter in the core[@tat1; @tat2; @tat3]. Therefore it should be interesting to include the magnetic field contribution in our formalism.
We have assumed zero temperature here. It would be much interesting to include the finite-temperature effect. Then it is possible to draw the phase diagram in the $\mu_\mathrm{B}$ - $T$ plane and we can study the properties of the deconfinement phase transition; our study may be extended to treat the mixed phase to appear during the hadronization of QGP in the nucleus-nucleus collisions and supernova explosions.
In this study we have used a simple model for quark matter to figure out the finite-size effects in the SMP. However, it has been suggested that the color superconductivity would be a ground state of quark matter [@alf1; @alf2]. Hence we will include it in a further study. The hadron phase should be also treated more realistically; for example, we should include the hyperons or kaons in hadron matter. In the recent studies the mixed phase has been also studied [@shov; @redd] in the context of various phases in the color superconducting phase.
Acknowledgements {#acknowledgements .unnumbered}
================
We wish to acknowledge Dr. D.N. Voskresensky and Dr. T. Tanigawa for fruitful discussion. T.E. appreciates Dr. H. Suganuma for his encouragement. This work is supported in part by the Grant-in-Aid for the 21st Century COE “Center for the Diversity and Universality in Physics” from the Ministry of Education, Culture, Sports, Science and Technology of Japan. The work of one of the authors (T.T.) is partially supported by the Japanese Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Culture, Sports, Science and Technology (13640282,16540246).
[99]{} For review, D. H. Rischke, Prog. Part. Nucl. Phys. [**52**]{} (2004) 197; J. Macher and J. Schaffner-Bielich, Eur. J. Phys. [**26**]{} (2005) 341 and references therein.
M. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys. B [ **64**]{} (1999) 443, D. Bailin and A. Love, Phys. Rep. [**107**]{} (1984) 325. As resent reviews, M. Alford, hep-ph/0102047; K. Rajagopal and F. Wilczek, hep-ph/0011333
M. Alford and S. Reddy, Phys. Rev. D [**67**]{} (2003) 074024
T. Tatsumi, T. Maruyama and E. Nakano, Prog. Theor. Phys. Suppl. [**153**]{} (2004) 190
E. Nakano, T. Maruyama and T. Tatsumi, Phys. Rev. D [**68**]{} (2003) 105001
T. Tatsumi, Phys. Lett. B [**489**]{} (2000) 280
K. Adcox et al. (PHENIX collaboration), Phys. Rev. Lett. [**88**]{} (2002) 022301; C. Adler et al. (STAR collaboration), Phys. Rev. Lett. [**90**]{} (2003) 082302
J. Madsen, Lect. Notes Phys. [**516**]{} (1999) 162
K. S. Cheng, Z. G. Dai and T. Lu, Int. Mod. Phys. D[**7**]{} (1998) 139
R. D. Pisalski and F. Wilczek, Phys. Rev. Lett. [**29**]{} (1984) 338
R. V. Gavai, J. Potvin and S. Sanielevici, Phys. Rev. Lett. [**58**]{} (1987) 2519
Y. A. Terasov, Phys. Rev. C [**70**]{} (2004) 054904
S. Pratt and S. Pal, Phys. Rev. C [**71**]{} (2005) 014905
J. M. Peters and K. L. Haglin J. Phys. G: Nucl. Part. Phys.31 (2005) 49
N. K. Glendenning and S. Pei, Phys. Rev. C [**D52**]{} (1995) 2250
N. K. Glendenning, Phys. Rev. D [**D46**]{} (1992) 1274; hys. Rep. [**342**]{} (2001) 393.
W. Weise, and G. E. Brown, Phys. Lett. B [**58**]{} (1975) 300
A. B. Migdal, A. I. Chernoustan and I. N. Mishustin, Phys. Lett. B [**83**]{} (1979) 158
J. Ellis, J. Kapusta and K. A. Olive, Nucl. Phys. B [**348**]{} (1991) 345
A. Rosenhauer, E. F. Staubo, L. P. Csernai, T. [Ø]{}verg[å]{}rd, and E. [Ø]{}stgaad, Nucl. Phys. A [**540**]{} (1992) 630
M. Alford, K. Rajagopal, S. Reddy, and F. Wilczek, Phys. Rev. D [**64**]{} (2001) 074017
H. Heiselberg, C. J. Pethick and E. F. Staubo, Phys. Rev. Lett. [**70**]{} (1993) 1355
N. K. Glendenning, *Compact stars*, Springer 2000.
T. Norsen and S. Reddy, Phys. Rev. C. [**63**]{} (2001) 065804
Toshiki Maruyama, T. Tatsumi, D. N. Voskresensky, T. Tanigawa and S. Chiba, Nucl. Phys. A [**749**]{} (2005) 186c; Phys. Rev C [**72**]{} (2005) 015802
Toshiki Maruyama, T. Tatsumi, D. N. Voskresensky, T. Tanigawa, T. Endo and S. Chiba, nucl-th/0505063
D. N. Voskresensky, M. Yasuhira and T. Tatsumi, Phys. Lett. [**B541**]{} (2002) 93, ; Nucl. Phys. [**A723**]{} (2003) 291; T. Tatsumi, M. Yasuhira and D. N. Voskresensky, Nucl. Phys. [**A718**]{} (2003) 359c; T. Tatsumi and D. N. Voskresensky, nucl-th/0312114.
T. Endo, Toshiki Maruyama, S. Chiba and T. Tatsumi, Nucl. Phys. A [**749**]{} (2005) 333c
R. G. Parr and W. Yang, *Density-Functional Theory of atoms and molecules*, Oxford Univ. Press, 1989.
E. K. U. Gross and R. M. Dreizler, *Density functional theory*, Plenum Press (1995)
T. Endo, Toshiki Maruyama, S. Chiba and T. Tatsumi, hep-ph/0502216
R. Tamagaki and T. Tatsumi, Prog. Theor. Phys. Suppl. [ **112**]{} (1993) 277
K. Kajantie, L. Kärkäinen, and K. Rummukainen, Nucl. Phys. [**B357**]{} (1991) 693
S. Huang, J. Potvion, C. Rebbi and S. Sanielevici, Phys. Rev. D [**43**]{} (1991) 2056
J. Madsen, Phys. Rev. D [**46**]{} (1992) 329
J. Madsen, Phys. Rev. Lett. [**70**]{} (1993) 391
M. S. Berger and R. L. Jaffe, Phys. Rev. C [**35**]{} (1987) 213
H. Monden, H. Ichie, H. Suganuma and H. Toki, Phys. Rev. C [**57**]{} (1998) 2564
M. Bejger, P. Haensel and J. L. Zdunik, Mon. Not. Roy. Astron. Soc. [**359**]{} (2005) 699
G. Miniutti, J. A. Pons, E. Berti, L. Gualtieri, V. Ferrari, Mon. Not. Roy. Astron. Soc. [**338**]{} (2003) 389
H. Heiselberg, Phys.Rev.D. [**48**]{} (1993) 1418
I. Shovkovy, M. Hanauske and M. Huang, Phys. Rev. D [**67**]{} (2003) 103004
S. Reddy and G. Rupak, nucl-th/0405054
|
---
abstract: 'In this paper, we study a new type of BSDE, where the distribution of the $Y$-component of the solution is required to satisfy an additional constraint, written in terms of the expectation of a loss function. This constraint is imposed at any deterministic time $t$ and is typically weaker than the classical pointwise one associated to reflected BSDEs. Focusing on solutions $(Y,Z,K)$ with deterministic $K$, we obtain the well-posedness of such equation, in the presence of a natural Skorokhod type condition. Such condition indeed ensures the minimality of the enhanced solution, under an additional structural condition on the driver. Our results extend to the more general framework where the constraint is written in terms of a static risk measure on $Y$. In particular, we provide an application to the super hedging of claims under running risk management constraint.'
author:
- 'Philippe Briand[^1]'
- 'Romuald Elie[^2]'
- 'Ying Hu[^3]'
title: '**BSDEs with mean reflection**'
---
Introduction
============
Backward Stochastic Differential Equations (BSDEs) have been introduced by Pardoux and Peng [@parpen90] and share a strong connection with stochastic control problems. Solving a BSDE typically consists in the obtention of an adapted couple process $(Y,Z)$ with the following dynamics: $$\begin{aligned}
Y_t &=& \xi + \int_t^T f(s,Y_s,Z_s) ds - \int_t^T Z_s \cdot dB_s\;, \qquad 0\le t \le T\;.\end{aligned}$$
In their seminal paper, Pardoux and Peng provide the existence of a unique solution $(Y,Z)$ to this equation for a given square integrable terminal condition $\xi$ and a Lipschitz random driver $f$. Since then, many extensions have been derived in several directions. The regularity of the driver can for example be weakened. The underlying dynamics can be fairly more complex, via for example the addition of jumps. These extensions allow in particular to provide representation of solutions to a large class of stochastic control problems, and to tackle several meaningful applications in mathematical finance.\
More interestingly, the consideration of additional conditions on the stochastic control problems of interest naturally led to the consideration of constrained BSDEs. In such a case, the solution of a constrained BSDE contains an additional adapted increasing process $K$, such that $(Y,Z,K)$ satisfies $$\begin{aligned}
\label{Constrained_BSDE}
Y_t &=& \xi + \int_t^T f(s,Y_s,Z_s) ds - \int_t^T Z_s \cdot dB_s + K_T - K_t \;, \qquad 0\le t \le T,\end{aligned}$$ together with a chosen constraint on the solution. The process $K$ interprets as the extra cost on the value process $Y$, due to the additional constraint. In such a framework, this equation admits an infinite number of solutions, as the roles of $Y$ and $K$ are too closely connected. The underlying stochastic control problem of interest typically indicates that one should look for the minimal solution (in terms of $Y$) of such equation. Motivated by optimal stopping or related obstacle problems, El Karoui et al. [@EKP] introduced the notion of reflected BSDE, where the constraint is of the form $$\begin{aligned}
\label{Pointwise constraint}
Y_t &\ge& L_t \;, \qquad 0\le t \le T.\end{aligned}$$ The obstacle process $L$ is a lower bound on the solution $Y$ and interprets as the reward payoff, if one chooses to stop immediately. It is worth noticing that the minimal solution $(Y,Z,K)$ is fully characterized by the following so-called Skorokhod condition $$\begin{aligned}
\int_0^T (Y_t -L_t)^+ dK_t &=& 0.\end{aligned}$$ This condition intuitively indicates that the process $K$ is only allowed to push the value process $Y$ whenever the constraint is binding.\
The class of constrained BSDEs has been significantly enlarged in the recent literature. The resolution of zero sum Dynkin game problems led Cvitanic and Karatzas [@CK] to the study of doubly reflected BSDE, where the process $Y$ lies in between two processes. Considering super-hedging problems where the admissible portfolios are restricted to belong to a convex set $\textbf{C}$ (e.g. $\textbf{C}={\mathbb{R}}^+$ for no short sell constraints), Buckdahn and Hu [@BuHu1; @BuHu2] and Cvitanic et al. [@CKS] studied the well posedness of BSDE (\[Constrained\_BSDE\]) together with the constraint: $Z_t\in\textbf{C}$, for $t\in[0,T]$. More generally, Peng and Xu [@PenXu] considered pointwise constraints of the form $\varphi(t,Y_t,Z_t)\ge 0$, where $\varphi$ is non-decreasing in $y$. The study of optimal switching problems [@HamJea; @HamZha; @HuTan; @CEK] led to the consideration of multidimensional systems of BSDEs with oblique reflections.\
In contrast to the previously mentioned pointwise constraints on the solution, Bouchard et al. [@BER] introduced the notion of BSDE with weak terminal condition. In their framework, the terminal condition is replaced by a constraint on the distribution of the random variable $Y_T$ and would typically rewrite $$\begin{aligned}
{\mathbb{E}}[\ell(Y_T-\xi)] \ge 0.\end{aligned}$$ The term $\ell(X_T-\xi)$ identifies to the quantification of a loss depending on the distance between $Y_T$ and the target $\xi$. This type of BSDE relates in particular to quantile hedging or related controlled loss control problems.\
The purpose of this paper is to determine the impact of a dynamic version of such type of constraint, by studying the BSDE (\[Constrained\_BSDE\]) together with a running constraint in expectation of the form $$\begin{aligned}
\label{runningconstraint}
{\mathbb{E}}[\ell(t,Y_t)] \ge 0\;, \qquad 0\le t \le T,\end{aligned}$$ where $(\ell(t,.))_{0\le t\le T}$ is a collection of non decreasing possibly random functions. It is worth noticing that the previous running constraint is only imposed on deterministic times $t\in[0,T]$. In the spirit of the above mentioned Skorokhod condition for reflected BSDEs, we look towards so-called flat solutions, i.e. satisfying the extra condition $$\begin{aligned}
\label{skorokhodcondition}
\int_0^T {\mathbb{E}}[\ell(t,Y_t)] dK_t &=& 0. \end{aligned}$$ Whenever $K$ is allowed to be random, we observe that the construction of a minimal continuous (in $Y$) solution to the system (\[Constrained\_BSDE\])-(\[runningconstraint\]) is not possible in general. For such reason, we focus on the derivation of solutions $(Y,Z,K)$ with deterministic $K$ component. Whenever $\ell$ is deterministic linear, we provide an explicit construction of the unique flat solution to the system (\[Constrained\_BSDE\])-(\[runningconstraint\])-(\[skorokhodcondition\]). For general loss functions, we are also able to derive the well-posedness of the system (\[Constrained\_BSDE\])-(\[runningconstraint\])-(\[skorokhodcondition\]), under mild assumptions satisfied for example whenever $\ell$ is bi-Lipschitz. Besides, restricting to drivers with any dependence in $z$ but deterministic linear dependence in $y$, we verify that the condition (\[skorokhodcondition\]) ensures the minimality in $Y$ of the enhanced solution, among all considered solutions of the BSDE (\[Constrained\_BSDE\]) with mean reflexion (\[runningconstraint\]).\
In terms of applications, it is worth noticing that the constraint (\[runningconstraint\]) can easily be replaced by a more general version of the form $$\begin{aligned}
\label{risk_measure_constraint}
\rho(t,Y_t) \le q_t\;, \qquad 0\le t \le T\;,\end{aligned}$$ where $(\rho(t,.))_t$ is a time indexed collection of static risk measures, and $(q_t)_t$ are associated benchmark levels. This framework is in fact the main motivation of this paper, but we chose to present our main argumentation within the constraint (\[runningconstraint\]) for sake of clarity and simplicity. We detail in particular in the last section of the paper an application to the super-replication of claims, restricting to investment portfolio $Y$ satisfying risk management constraint of the form .\
The paper is organized as follows: Section \[sec:setup\] presents the problem of interest, clarifies the assumptions and discusses the main results of the paper. In Section \[sec:llin\], we build the unique solution to the system (\[Constrained\_BSDE\])-(\[runningconstraint\])-(\[skorokhodcondition\]) whenever $\ell$ is linear and deterministic. The general case is treated in Section \[sec:general\_case\], where we derive the well-posedness of the system (\[Constrained\_BSDE\])-(\[runningconstraint\])-(\[skorokhodcondition\]). The minimality of the enhanced solution is discussed in Section \[sec:minimal\], whereas the mathematical finance application is given in Section \[sec:application\].\
#### Notations
Throughout this paper, we are given a finite horizon $T$ and a complete probability space $(\Omega,\Fc,\P)$ endowed with a $d$-dimensional standard Brownian motion $B=(B_t)_{0\geq t\leq T}$. We will work with the usual augmented filtration of $B$, $\{ {\mathcal}F_t\}_{0\leq t\leq T}$. Any element $x \in \R^d$ will be identified to a column vector with $i$-th component $x^i$ and Euclidian norm $|x|$. $\Cc_T$ denotes the set $C([0,T],\R)$ of continuous functions from $[0,T]$ to $\R$. For a given set of parameters $\alpha$, $C(\alpha)$ will denote a constant only depending on these parameters, and which may change from line to line. Finally, we classically denote by:\
- $L^2(\Fc_t)$ the set of real valued $\Fc_t$-measurable square integrable random variables, for any $t\in[0,T]$.
- $\Sc^2$ the set of real valued $\Fc$-adapted continuous processes $Y$ on $[0,T]$ such that\
$
\|Y\|_{\Sc^2} := \Esp{\sup_{0\le r \le T} |Y_r|^2}^{\frac{1}{2}} <\infty
$;
- $\Hc^2$ the set of predictable $\R^d$-valued processes $Z$ s.t. $
\!\|Z\|_{\Hc^2}\! := \! \Esp{\int_0^T |Z_r|^2 dr}^\frac{1}{2}\!\!\!\!~<~\!\!\infty
$;
- ${\Ac^2}$ is the closed subset of $\Sc^2$ consisting of nondecreasing processes $K$ $=$ $(K_t)_{0\leq t\leq T}$ with $K_0$ $=$ $0$;
- ${\Ac^2_D}$ the subset of deterministic elements of ${\Ac^2}$.
Problem set up {#sec:setup}
==============
Presentation of BSDEs with mean reflexion
-----------------------------------------
The main purpose of this paper is to construct solutions $(Y,Z,K)$ to the following BSDE $$\begin{gathered}
Y_t =\xi+\int_t^T f(s,Y_s,Z_s)\, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t,\quad 0\leq t\leq T, \label{eq:main_dyn}\\
{\mathbb{E}}[\ell(t,Y_t)] \geq 0, \quad 0\leq t \leq T, \label{eq:main_constraint}\end{gathered}$$ where the second equation is a running constraint in expectation on the component $Y$ of the solution. In opposition to classical reflected BSDE where would typically be a pointwise constraint, the constraint considered here concerns the distribution of the $Y$-component. We pin this new type of constrained equations as *BSDEs with mean reflexion*.
The non-decreasing function $\ell$ interprets as a loss function and typical examples of interest are
- $\ell(t,x)=x-u_t$ where $u$ is a deterministic continuous benchmark, that the process $Y$ is required to beat in expectation;
- $\ell(t,x) = {\mathbf{1}}_{x \geq u_t} - v_t$ (or any smoothed equivalent), so that the process $Y$ is now required to beat deterministic continuous benchmark $u$ with a probability greater than $v_t$, for any time $t$;
- $\ell(t,x) = U(x,\xi_t) -u_t$ where $U$ is a concave utility function, $(\xi_t)_t$ is a running random benchmark of interest and $(u_t)_t$ a given deterministic confidence level.
Whenever $\ell$ is a strictly increasing function, the corresponding classical reflected BSDE is characterized by the dynamics together with the stronger pointwise constraint $$\begin{aligned}
\ell(t,Y_t) \geq 0, \quad 0\leq t \leq T.
$$ In such a case, the $Y$-component of the solution to the BSDE is reflected on the boundary process $([\ell(t,.)]^{-1}(0))$. Observe that our constrained BSDE of interest weakens the condition imposed on $Y$, by only constraining its distribution.
Observe that the condition is only written on the deterministic dates of $[0,T]$, and not on all the possible stopping times smaller than $T$. In our framework, considering a constraint on all stopping times would strongly strengthen the constraint of interest. On the contrary, both type of pointwise conditions are by construction equivalent for classical reflected BSDEs.
Assumptions on the coefficients
-------------------------------
The parameters of the BSDE with mean reflection are the terminal condition $\xi$, the driver $f$ as well as the loss function $\ell$. These parameters are supposed to satisfy the following standard running assumptions:
- The driver $f:\Omega\times[0,T]\times{\mathbb{R}}\times{\mathbb{R}}^d {\longrightarrow}{\mathbb{R}}$ is a measurable map with respect to ${\mathcal}P\times {\mathcal}B({\mathbb{R}})\times{\mathcal}B\left({\mathbb{R}}^d\right)$ and ${\mathcal}B({\mathbb{R}})$, ${\mathcal}P$ being the sigma algebra of progressive sets of $\Omega\times[0,T]$, and there exists $\lambda\geq 0$ such that, ${\mathbb{P}}$-a.s., for all $t\in[0,T]$, $$\forall y,p,z,q\qquad \left| f(t,y,z) - f(t,p,q)\right| \leq \lambda \left( |y-p| + |z-q|\right),$$ and $${\mathbb{E}}\left[ \int_0^T | f(t,0,0) | ^2\, dt\right] <+\infty.$$
- The terminal condition $\xi$ is a square-integrable ${\cal F}_T$-measurable random variable such that $${\mathbb{E}}[\ell(T,\xi)] \geq 0.$$
- The loss function $\ell : \Omega\times[0,T]\times{\mathbb{R}}{\longrightarrow}{\mathbb{R}}$ is a measurable map with respect to ${\mathcal}F_T\times{\mathcal}B([0,T])\times {\mathcal}B({\mathbb{R}})$ and there exists $C\geq 0$ such that, ${\mathbb{P}}$-a.s.,
1. $(t,y)\longmapsto \ell(t,y)$ is continuous,
2. $\forall t\in[0,T]$, $y\longmapsto \ell(t,y)$ is strictly increasing,
3. $\forall t\in[0,T]$, ${\mathbb{E}}\left[\ell(t,\infty)\right] >0$,
4. $\forall t\in[0,T]$, $\forall y\in{\mathbb{R}}$, $|\ell(t,y)| \leq C(1+|y|)$.
We chose to work in this paper under that seminal Lipschitz and square integrability assumptions on the driver and terminal condition. We restrict here to this simple framework, in order to decrease the amount of technical details and emphasize the novelty induced of the additional constraint .
Observe that Condition ($H_\xi$) ensures that the constraint is automatically satisfied at maturity. This condition implies that no a priori facelift procedure is required on the terminal payoff $\xi$.
Definition of solution, main results and discussion
---------------------------------------------------
We now turn to the definition of a solution to the BSDE with mean reflexion of interest.
A square integrable solution to the BSDE with mean reflection is a triple of processes $(Y,Z,K)\in\Sc^2\times\Hc^2\times\Ac^2$ satisfying together with the constraint . A solution is said to be *flat* if moreover $K$ increases only when expected, i.e. when we have $$\label{main_flat}
\int_0^T {\mathbb{E}}[\ell(t,Y_t)] \, dK_t = 0.$$ By a *deterministic solution*, we mean a solution for which the process $K$ is deterministic., i.e. $K\in\Ac^2_D$.
As detailed in Remark \[rem\_uniqueness\] below, we observe that allowing $K$ to be random leads to the existence of multiple flat solutions. We even verify that it may induce the non-existence of minimal solution for the BSDE with mean reflection , see the example provided at the end of Section \[sec:minimal\]. This is why we chose here to restrict to the consideration of so-called *deterministic solutions*, i.e. solutions $(Y,Z,K)$ with deterministic compensator $K$.
In particular, focusing on deterministic solutions, we verify that the flatness condition can directly imply the minimality property of the solution beyond all the deterministic ones. This is in particular the case for drivers with deterministic linear dependence in $y$, see Condition .\
The main result of this paper is the existence and uniqueness of deterministic flat solution to the BSDE with mean reflection . This is first achieved for the particular case of linear loss function $\ell$, see Proposition \[prop:existence1\_linear\] and Theorem \[thm existence llinear\] in Section \[sec:llin\]. The line of proof follows a constructive approach when the driver does not depend on $Y$ and $Z$, together with a contraction property in order to tackle any Lipschitz driver function. An alternative approach via penalization is also provided in Section \[subsec:penalize\]. When the driver is not linear, the well posedness of the system -- is also established, under an additional assumption on the loss function, denoted ($H_L$) below, see Proposition \[prop:existence\_uniqueness\_2\] and Theorem \[thm: main\] in Section \[sec:general\_case\].\
In a similar fashion, we explain in Section \[sec:application\] below how the constraint in expectation can be replaced by a constraint of the form $\rho(\cdot,Y_\cdot)\le q_\cdot$, where $(\rho(t,\cdot))_t$ is a collection of static risk measures computed at time $0$, and $q$ is a collection of time-indexed benchmarks. In particular, solving this equation allows for example to represent the super-hedging price of a claim $\xi$, whenever any admissible portfolios require to satisfy at any date $t$ a running risk management constraint written in terms of risk measures.\
Since the constraint concerns the distribution of the solution to the BSDE, it is tempting to understand the possible connection between such type of BSDE and corresponding constrained McKean Vlasov BSDEs. This topic seems promising in particular for the mean field game literature and is left for further research.
A priori estimate
-----------------
Let us conclude this section by providing a usefull a priori estimate on any solution to the BSDE - of interest.
\[en:S2\] Let $(Y,Z,K)$ be a square integrable solution to the BSDE with mean reflection . Then $Y$ satisfies the following $${\mathbb{E}}\left[ \sup_{0\leq t \leq T} |Y_t|^2\right] \leq C(\lambda, T) \, {\mathbb{E}}\left[ |Y_0|^2 + K_T^2 + \int_0^T |f(s,0,0)|^2 ds + \int_0^T |Z_s|^2 ds\right].$$
By construction, we have, $$Y_t = Y_0 - \int_0^t f(s,Y_s,Z_s)\, ds + \int_0^t Z_s\cdot dB_s - K_t, \qquad 0\le t \le T\;.$$ Because $K$ is non decreasing, Assumption $(H_f)$ leads to $$|Y_t| \leq |Y_0| + K_T + \int_0^T |f(s,0,0)| ds + \lambda \int_0^T |Z_s| ds + \sup_{0\leq t \leq T} \left | \int_0^t Z_s\cdot dB_s \right| + \lambda \int_0^t |Y_s|\, ds \,,$$ for $t\in[0,T]$. Since $Y$ has continuous paths, Gronwall’s lemma gives $$\sup_{0\leq t \leq T} |Y_t| \leq e^{\lambda T} \left(|Y_0| + K_T + \int_0^T |f(s,0,0)| ds + \lambda \int_0^T |Z_s| ds + \sup_{0\leq t \leq T} \left | \int_0^t Z_s\cdot dB_s \right|\right) \,,$$ and the result follows from the Burkholder-Davis-Gundy inequality.
We deduce from this lemma that, when the generator has linear growth, the process $Y$ belongs to ${\mathcal}S^2$ as soon as $Z$ and $K$ are square integrable.
The particular case of linear mean reflection {#sec:llin}
==============================================
In this section, we consider the simpler particular case where the mean reflection is linear. Namely, $\ell:(t,y)\mapsto y-u_t$ so that the condition is replaced by $$\begin{aligned}
{\mathbb{E}}[Y_t] \geq u_t, \qquad 0\leq t \leq T, \label{eq:main_constraint_linear}
$$ where $u$ is a deterministic continuous map from $[0,T]$ to ${\mathbb{R}}$. Hereby, we impose a running deterministic lower bound $u$ on the expected value of the $Y$-component of the solution. Besides, we recall that Assumption $H_\xi$ ensures that this constraint is already satisfied at maturity so that we have $$\begin{aligned}
{\mathbb{E}}[\xi] \geq u_T \,. \qquad \label{eq:terminal_constraint_linear}
$$
In this linear framework, we are able in Proposition \[prop:existence1\_linear\] to construct an explicit deterministic flat solution $(Y,Z,K)$ to a BSDE with linear mean reflexion , when the driver does not depend on $Y$ nor $Z$. Building modifications on this deterministic flat solution, we exhibit an infinite number of non deterministic flat solutions to the same BSDE. This feature is our main motivation in order to focus solely on deterministic flat solutions in order to ensure the well posedness of BSDEs with mean reflection. Indeed, Proposition \[prop:uniqueness\_linear\] indicates that uniqueness holds within the class of deterministic flat solutions to -.\
Hereafter, we first derive an a priori estimate on the solution, and then tackle respectively the uniqueness and existence issues. In order to handle general drivers, the enhanced demonstration relies on a contraction argument, but an alternative approach via penalization is also provided in Section \[subsec\_penalization\].
A priori estimate {#subsec_estimate}
-----------------
The main mathematical advantage of considering a linear loss function $\ell$ is that it allows to use some of the computational tricks associated to classical reflected BSDEs, in particular when the compensator $K$ is moreover deterministic. As detailed in the proof below, this enables us to derive the following a-priori estimate on the solution to the BSDE with linear mean reflexion.
\[en:est\] Let $(Y,Z,K)$ be a deterministic square integrable flat solution to the BSDE with linear mean reflexion . Then $${\mathbb{E}}\left[\sup_{0\leq t \leq T} |Y_t|^2 + \int_0^T |Z_s|^2 ds\right] + K_T^2 \leq C(\lambda, T) \left({\mathbb{E}}\left[|\xi|^2 + \int_0^T |f(s,0,0)|^2 ds\right] + \| u \|_\infty^2\right).$$
Let us recall that the Lipschitz property of $f$ implies $$2 y\cdot f(t,y,z) \leq |f(t,0,0)|^2 + \frac{1}{2} |z|^2 + \left(1+2\lambda + 2\lambda^2\right) |y|^2, \qquad \forall (y,z)\in\R\times\R^d\;.$$ Setting $\beta := 1 + 2\lambda + 2 \lambda^2$, Itô’s formula together with the previous inequality provides $$\begin{gathered}
e^{\beta t} |Y_t|^2 + \frac{1}{2} \int_t^T e^{\beta s} |Z_s|^2 ds \leq e^{\beta T} |\xi|^2 + \int_0^T e^{\beta s} |f(s,0,0)|^2 ds + 2 \int_t^T e^{\beta s} Y_s dK_s -2 \int_t^T e^{\beta s} Y_s Z_s\cdot dB_s,
\end{gathered}$$ for all $t\in[0,T]$. Since $K$ is deterministic and $\ell$ is linear, we compute $$\begin{aligned}
2 {\mathbb{E}}\left[\int_t^T e^{\beta s} Y_s dK_s\right] = 2 \int_t^T e^{\beta s} {\mathbb{E}}\left[Y_s\right] dK_s & = 2 \int_t^T e^{\beta s} \left({\mathbb{E}}\left[Y_s\right] -u_s\right)dK_s + 2 \int_t^T e^{\beta s} u_s dK_s \;.
\end{aligned}$$ Besides the solution is flat so that condition directly implies $$\begin{aligned}
2 {\mathbb{E}}\left[\int_t^T e^{\beta s} Y_s dK_s\right]
& = 2 \int_t^T e^{\beta s} u_s dK_s \leq 2 e^{\beta T} \| u \|_\infty K_T.
\end{aligned}$$ We deduce that $$\begin{gathered}
\sup_{0\leq t\leq T} {\mathbb{E}}\left[e^{\beta t} |Y_t|^2\right] + {\mathbb{E}}\left[ \int_0^T e^{\beta s} |Z_s|^2 ds \right]
\leq 3 \left( {\mathbb{E}}\left[e^{\beta T} |\xi|^2 + \int_0^T e^{\beta s} |f(s,0,0)|^2 ds\right] + 2 e^{\beta T} \| u \|_\infty K_T \right),
\end{gathered}$$ from which, we get, for any ${\varepsilon}>0$, $$\label{eq:p1}
\sup_{0\leq t\leq T} {\mathbb{E}}\left[ |Y_t|^2\right] + {\mathbb{E}}\left[ \int_0^T |Z_s|^2 ds \right] \leq C(\lambda,T,{\varepsilon}) \left({\mathbb{E}}\left[|\xi|^2 + \int_0^T |f(s,0,0)|^2 ds\right] + \| u \|_\infty^2\right) + {\varepsilon}\, K_T^2.$$ On the other hand, since $K$ is deterministic, we have $$K_T = {\mathbb{E}}\left[K_T\right] = Y_0 - {\mathbb{E}}\left[\xi\right] - {\mathbb{E}}\left[\int_0^T f(s,Y_s,Z_s) ds \right],$$ from which we deduce the inequality $$\label{eq:p2}
K_T^2 \leq C(\lambda,T) \left({\mathbb{E}}\left[ \int_0^T |f(s,0,0)|^2 ds \right] + \sup_{0\leq t\leq T} {\mathbb{E}}\left[|Y_t|^2\right] + {\mathbb{E}}\left[\int_0^T |Z_s|^2 ds \right] \right).$$ Combining this estimate with and ${\varepsilon}$ small enough, we get $$\sup_{0\leq t\leq T} {\mathbb{E}}\left[ |Y_t|^2\right] + {\mathbb{E}}\left[ \int_0^T |Z_s|^2 ds \right] + |K_T|^2\leq C(\lambda,T) \left({\mathbb{E}}\left[|\xi|^2 + \int_0^T |f(s,0,0)|^2 ds\right] + \| u \|_\infty^2\right)$$ and the result follows from Lemma \[en:S2\].
Uniqueness of the deterministic flat solution {#subsec_unique_linear}
----------------------------------------------
The uniqueness of flat deterministic solution for a BSDE with linear mean reflection follows mainly from a similar argumentation as the one used for classical reflected BSDE. This is detailed in the next Proposition.
\[prop:uniqueness\_linear\] The BSDE with linear mean reflexion has at most one square integrable deterministic flat solution.
Let us consider two such solutions $(Y^1, Z^1, K^1)$ and $(Y^2, Z^2, K^2)$ and denote $$\delta Y:=Y^1-Y^2,\quad \delta Z:=Z^1-Z^2 \;\;\mbox{ and } \;\; \delta K:=K^1-K^2.$$ Setting $a:=2\lambda + 2 \lambda^2$ and arguing as in Lemma \[en:est\], Itô’s formula gives easily $$e^{a t}|\delta Y_t |^2 + \frac{1}{2 }\int_t^T e^{as} |\delta Z_s|^2 \, ds \leq - 2 \int_t^T e^{as} \delta Y_s\,\delta Z_s\cdot dB_s + 2 \int_t^T e^{as} \delta Y_s\, d\delta K_s \,,$$ for $t\in[0,T]$. Let us observe that since both solutions are flat and deterministic and $\ell$ is linear, we nicely have $$\begin{aligned}
{\mathbb{E}}\left[\int_t^T e^{as} \delta Y_t \, d \delta K_s\right] & = \int_t^T e^{as} \left[\left({\mathbb{E}}[Y^1_s]-u_s\right) - \left({\mathbb{E}}\left[Y^2_s\right]-u_s\right)\right] d K^1_s \\
& \quad - \int_t^T e^{as} \left[\left({\mathbb{E}}[Y^1_s]-u_s\right) - \left({\mathbb{E}}\left[Y^2_s\right]-u_s\right)\right] d K^2_s \\
& = - \int_t^T e^{as} \left({\mathbb{E}}\left[Y^2_s\right]-u_s\right) d K^1_s - \int_t^T e^{as} \left({\mathbb{E}}\left[Y^1_s\right]-u_s\right) d K^2_s \leq 0,
\end{aligned}$$ for any $t\in[0,T]$. Thus the result follows by taking expectations in the previous inequality.
As detailed in Remark \[rem\_uniqueness\] below, considering deterministic $K$ processes is a key for the obtention of a unique solution to the BSDE of interest. We now turn to the existence property.
Existence of a deterministic flat solution {#subsec_existence_linear}
-------------------------------------------
We first focus on the particular case where the driver $f$ does not depend on $Y$ nor $Z$. In this simple case, we are able to construct explicitly the unique solution to a BSDE with linear mean reflection.
\[prop:existence1\_linear\] Let $C$ be a square integrable progressively measurable stochastic process or more generally in the space ${\mathrm{L}}^2\left(\Omega;{\mathrm{L}}^1(0,T)\right)$. The BSDE with linear mean reflection $$\label{eq:linov}
Y_t =\xi+\int_t^T C_s\, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t,\qquad {\mathbb{E}}[Y_t] \geq u_t, \qquad 0\leq t\leq T,$$ has a unique square integrable deterministic flat solution.
Let us set $\displaystyle x_t = {\mathbb{E}}\left[\xi + \int_t^T C_s\, ds\right]$. By Skorokhod’s lemma, there exists a unique pair of deterministic functions $(y,K):[0,T]\rightarrow{\mathbb{R}}$ such that $K$ is non decreasing and $K_0=0$ and we have $$\label{temp1}
y_t = x_t + K_T-K_t,\qquad y_t \geq u_t,\qquad \int_0^T (y_t-u_t) \, dK_t=0.$$ By construction, observe that $K$ is continuous and $K_t = \sup_{0\leq s \leq T} \left(x_s-u_s\right)_- - \sup_{t\leq s\leq T}\left(x_s-u_s\right)_-$. Now, $K$ being given, we know that the BSDE $$Y_t =\xi+\int_t^T C_s\, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t,\quad 0\leq t\leq T,$$ has a unique square integrable solution $(Y,Z)$. Moreover, we have by construction $y_t = {\mathbb{E}}[Y_t]$. It follows from that $(Y,Z,K)$ is a deterministic flat solution of the BSDE . The uniqueness follows from Proposition \[prop:uniqueness\_linear\].
\[rem\_uniqueness\] Let us observe that the BSDE with mean reflexion has infinite many flat solutions with random $K$. Let us start with $(Y^0,Z^0,K^0)$ the deterministic flat solution to constructed above in the proof of Proposition \[prop:existence1\_linear\]. For any real $\alpha$, let us set $M^\alpha := (e^{\alpha B_t - \alpha^2 t/2})_t$ and define $$K^\alpha_t := \int_0^t M^\alpha_s\, dK^0_s, \qquad 0\le t \le T\;.$$ Being given $K^\alpha$, let $(Y^\alpha,Z^\alpha)$ be the solution to the BSDE $$Y^\alpha_t =\xi+\int_t^T C_s\, ds - \int_t^T Z^{\alpha}_s\cdot dB_s + K^{\alpha}_T-K^{\alpha}_t,\quad 0\leq t\leq T.$$ For all $0\le t \le T$, since ${\mathbb{E}}[M^\alpha_t]=1$ and $K^0$ is deterministic, we have ${\mathbb{E}}\left[K^{\alpha}_t\right] = K^0_t$ so that ${\mathbb{E}}\left[Y^\alpha_t\right] = {\mathbb{E}}\left[Y^0_t\right] \geq u_t$. Moreover, since ${\mathbb{E}}\left[Y^0_t\right]-u_t = 0$ $dK$-a.e., we compute $$\int_0^T \left({\mathbb{E}}\left[Y^{\alpha}_t\right] - u_t\right) dK^{\alpha}_t = \int_0^T \left({\mathbb{E}}\left[Y^0_t\right] - u_t\right) M^\alpha_t \, dK^0_t = 0.$$ Hence, for any real $\alpha$, $\left(Y^{\alpha},Z^{\alpha},K^{\alpha}\right)$ is also a flat solution to .
We are now in position to turn to the general driver case and we will derive the well-posedness of the BSDE of interest via the classical use of a well chosen contraction property.
\[thm existence llinear\] The BSDE with linear mean reflexion has a unique deterministic square integrable flat solution.
For given processes $U\in\Sc^2$ and $V\in\Hc^2$, let $(Y,Z,K)$ be the deterministic square integrable flat solution to the BSDE $$Y_t =\xi+\int_t^T f(s,U_s,V_s)\, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t, \qquad {\mathbb{E}}\left[Y_t\right] \geq u_t,\qquad 0\le t \le T\;,$$ as provided by Proposition \[prop:existence1\_linear\]. Let us show that the mapping $\Phi:(U,V) \longmapsto (Y,Z)$, from ${\mathcal}S^2\times {\mathcal}H^2$ into itself, has a unique fixed point.
For this purpose, let us denote $(Y^1,Z^1,K^1)$ and $(Y^2,Z^2,K^2)$ the two deterministic square integrable flat solutions to the above BSDE with given $(U^1,V^1)$ and $(U^2, V^2)$ respectively. Set $$\delta Y:=Y^1-Y^2,\quad \delta Z:=Z^1-Z^2,\quad \delta K:=K^1-K^2, \quad \delta U:=U^1-U^2, \quad
\delta V:=V^1-V^2.$$ For $a=4\lambda^2 +1$, Itô’s formula leads to $$\begin{gathered}
|\delta Y_0 |^2 + \int_0^T e^{as} \left(|\delta Y_s|^2 + |\delta Z_s|^2\right) \, ds \\
\leq \frac{1}{2} \int_0^T e^{as} \left( |\delta U_s|^2 + |\delta V_s|^2\right) ds - 2\int_0^T e^{as} \delta Y_s\,\delta Z_s\cdot dB_s + 2 \int_0^T e^{as} \delta Y_s\, d\delta K_s.
\end{gathered}$$ As observed in the proof of Proposition \[prop:existence1\_linear\], we compute $$\begin{aligned}
{\mathbb{E}}\left[\int_0^T e^{as} \delta Y_s \, d \delta K_s\right] & = - \int_0^T e^{as} \left({\mathbb{E}}\left[Y^2_s\right]-u_s\right) d K^1_s - \int_0^T e^{as} \left({\mathbb{E}}\left[Y^1_s\right]-u_s\right) d K^2_s \leq 0.
\end{aligned}$$ It follows directly that $${\mathbb{E}}\left[\int_0^T e^{as} \left(|\delta Y_s|^2 + |\delta Z_s|^2\right) \, ds\right] \leq \frac{1}{2} {\mathbb{E}}\left[\int_0^T e^{as} \left( |\delta U_s|^2 + |\delta V_s|^2\right) ds\right].$$ Since we have $$\begin{gathered}
\delta Y_t = {\mathbb{E}}\left[ \int_t^T \left(f(s,U^1_s,V^1_s)-f(s,U^2_s,V^2_s)\right)ds \: \Big| \: {\mathcal}F_t\right] + (K^1_T-K^1_t) - (K^2_T-K^2_t), \\
K^i_T-K^i_t = \sup_{t\leq s\leq T} \left({\mathbb{E}}\left[\xi+\int_s^T f\left(r,U^i_r,V^i_r\right)dr\right]-u_s\right)_- \;,
\end{gathered}$$ we get immediately $${\mathbb{E}}\left[\sup_{0\leq t\leq T} |\delta Y_t|^2\right] \leq C\, {\mathbb{E}}\left[\int_0^T \left( |\delta U_s|^2 + |\delta V_s|^2\right) ds\right].$$ As a byproduct, $\Phi$ is continuous from ${\mathcal}S^2\times{\mathcal}H^2$ into itself.
Moreover, starting from $(Y^0,Z^0) = (0,0)$ and setting for $n\geq 1$, $(Y^n,Z^n) = \Phi\left(Y^{n-1},Z^{n-1}\right)$, we deduce easily from the previous estimates that $${\mathbb{E}}\left[\sup_{0\leq t\leq T} \left|Y^{n+1}_t -Y^n_t\right|^2 + \int_0^T \left|Z^{n+1}_t -Z^n_t\right|^2 dt\right] \leq C\, 2^{-n},$$ and finally that the sequence $\left\{(Y^n,Z^n)\right\}_{n\geq 0}$ converges in ${\mathcal}S^2\times{\mathcal}H^2$ to the unique fixed point of $\Phi$.
Alternative approach via penalization {#subsec_penalization}
--------------------------------------
\[subsec:penalize\]
In order to handle classical reflected BSDE, a very helpful feature is the characterization of the solution as a limit of corresponding penalized classical BSDEs. The idea simply relies on the addition of a strong penalization on the driver of a classical BSDE, which is only active whenever the constraint is not satisfied. As the penalization strength increases, the $Y$ component of the penalized solution also increases and converges at the limit to the minimal solution of the reflected BSDE. In our framework, the constraint only integrates the distribution of $Y$, and not the pointwise value of the process $Y$. For this reason, no comparison argument can ensure that a sequence of penalized BSDEs will be increasing and the classical line of proof falls down. Nevertheless, whenever the benchmark function $u$ is constant, we are able to identify the unique deterministic flat solution of a BSDE with linear mean reflexion as the limit of corresponding penalized BSDEs of McKean-Vlasov type. This is the purpose of the next Proposition.
Suppose that the benchmark $(u_t)_t$ is constant and also denoted $u$. For any positive integer $n$, let us consider $(Y^n,Z^n)$ solution to the BSDE of McKean-Vlasov type $$\begin{aligned}
Y^n_t & = \xi + \int_t^T f\left(s,Y^n_s,Z^n_s\right) ds + \int_t^T n \left(u-{\mathbb{E}}\left[Y^n_s\right]\right)_+ ds - \int_t^T Z^n_s\cdot dB_s \,. \quad 0\le t \le T\;,\\
\end{aligned}$$ and denote $K^n:= \int_0^. n \left(u-{\mathbb{E}}\left[Y^n_s\right]\right)_+ ds$. As $n$ goes to infinity, $(Y^n,Z^n,K^n)$ converges to the unique flat deterministic solution of the BSDE with linear mean reflexion .
Observe first the the solution $(Y^n,Z^n)$ is well and uniquely defined, according to the results of [@Buckdahn] up to slight modifications discussed for example in [@ChaGar].
#### Step 1. Uniform a priori estimate on the sequence $(Y^n,Z^n,K^n)_n$
Since $K^n$ is deterministic, we have $$\begin{aligned}
2 {\mathbb{E}}\left[\int_t^T e^{as} Y^n_s dK^n_s\right] = 2 \int_t^T e^{as} {\mathbb{E}}\left[Y^n_s\right] dK^n_s & = 2 \int_t^T e^{as} \left({\mathbb{E}}\left[Y^n_s\right] -u\right)dK^n_s + 2 \int_t^T e^{as} u dK^n_s \\
& = - 2n \int_t^T e^{as} \left(u-{\mathbb{E}}\left[Y^n_s\right]\right)_+^2 ds + 2 \int_t^T e^{as} u dK^n_s \\
& \leq 2 u \int_t^T e^{as} dK^n_s\;,
\end{aligned}$$ for any constant $a$ and $t\in[0,T]$. Thus, arguing as in the proof of Lemma \[en:est\], we get the following estimate on the solution $(Y^n,Z^n)$ $$\sup_{n\geq 1 }\left({\mathbb{E}}\left[\sup_{0\leq t \leq T} |Y^n_t|^2 + \int_0^T |Z^n_s|^2 ds\right] + \left|K_T^n\right|^2\right) \leq C(\lambda, T) \left({\mathbb{E}}\left[|\xi|^2 + \int_0^T |f(s,0,0)|^2 ds\right] + u ^2\right).$$
#### Step 2. Convergence of the sequence $(Y^n,Z^n,K^n)_n$
Since the constraint is satisfied at maturity, observe also that $(\left(u-{\mathbb{E}}\left[Y^n_0\right]\right)_+)^2$ rewrites $$\begin{aligned}
|\left(u-{\mathbb{E}}\left[Y^n_0\right]\right)_+|^2+2n\int_0^T |\left(u-{\mathbb{E}}\left[Y^n_s\right]\right)_+|^2ds
&=&-2\int_0^T \mathbb E[f(s,Y_s^n,Z_s^n)] \left(u-{\mathbb{E}}\left[Y^n_s\right]\right)_+ ds\\
&\le& n\int_0^T |\left(u-{\mathbb{E}}\left[Y^n_s\right]\right)_+|^2ds+\frac{C(\lambda,T)}{n},\end{aligned}$$ according to the previous estimate. Hence we deduce for later use that $$\label{eq:fonda}
n^2 \int_0^T |\left(u-{\mathbb{E}}\left[Y^n_s\right]\right)_+|^2ds \le C(\lambda,T).$$
We now look towards a contracting property of the sequence $(Y^n,Z^n)$ and denote $\delta X := X^{n+1}-X^n$ for $X=Y,Z$ or $K$. Setting $a := \frac{1}{2} + 2\lambda + 2 \lambda^2$, a standard computation based on Itô’s formula provides $$e^{a t}|\delta Y_t |^2 + \frac{1}{2}\int_t^T e^{as} \left(|\delta Y_s|^2 + |\delta Z_s|^2\right) \, ds
\leq 2 \int_t^T e^{as} \delta Y_s\, d\delta K_s - 2\int_t^T e^{as} \delta Y_s\,\delta Z_s\cdot dB_s, \quad 0\le t \le T,$$ from which we deduce that $$\label{eq:utile}
\sup_{0\leq t\leq T} {\mathbb{E}}\left[\left| \delta Y_t \right|^2 +\int_0^T \left(|\delta Y_s|^2 + |\delta Z_s|^2\right) \, ds \right] \leq 2\, \sup_{0\leq t\leq T} {\mathbb{E}}\left[\int_t^T e^{as}\delta Y_s\, d\delta K_s\right].$$ For any $s\in[0,T]$, denoting $v^n_s:=\left(u-y^n_s \right)_+$ where $y^n_s$ stands for ${\mathbb{E}}\left[Y^n_s\right]$, we have $dK^n_s = nv^n_s ds $ and $${\mathbb{E}}\left[\int_t^T e^{as} \delta Y_s\, d\delta K_s\right] = \int_t^T e^{as} \left[y^{n+1}_s-y^n_s\right] \left[(n+1) v^{n+1}_s - nv^n_s\right] ds, \quad 0\le t \le T.$$ Moreover, we compute $$\begin{aligned}
\left[y^{n+1} -y^n\right] \left[(n+1) v^{n+1} - nv^n\right] &=&
\left[\left(u-y^n\right) - \left(u-y^{n+1})\right)\right] \left[(n+1) v^{n+1} - nv^n\right] \\
&\le& -n |v^n|^2 +(2n+1) v^n v^{n+1} - (n+1) |v^{n+1}|^2 \;.
\end{aligned}$$ But, we have $$-nx^2 + (2n+1) xy -(n+1) y^2 = -n \left(x-\left(1 + \dfrac{1}{2n}\right)y\right)^2 + \dfrac{y^2}{4n} \;, \qquad x,y\in\R\;,$$ so that combining the previous estimates with , we deduce $${\mathbb{E}}\left[\int_0^T e^{as} \delta Y_s\, d\delta K_s\right] \leq \frac{1}{4n} \int_0^T |v^{n+1}_s|^2 ds \leq \frac{C(\lambda,T)}{n^3} \;.$$ Plugging this estimate in , it follows that $$\sup_{0\leq t\leq T} {\mathbb{E}}\left[ |\delta Y_t|^2 \right] + {\mathbb{E}}\left[ \int_0^T \left(|\delta Y_s|^2 + |\delta Z_s|^2\right) ds \right] \leq \frac{C(\lambda,T)}{n^3}.$$ Setting $\Delta_t K^n = K^n_T - K^n_t$ and reminding that $K^n$ is deterministic, observe that $$\Delta_t K^{n+1} - \Delta_t K^{n} = \mathbb E[ \delta Y_t] - \mathbb E\left[\int_t^T (f(s,Y^{n+1}_s,Z^{n+1}_s) - f(s,Y^{n}_s,Z^{n}_s)) \, ds\right] \;,$$ from which we deduce $$\sup_{0\leq t\leq T}|\Delta_t K^{n+1} - \Delta_t K^{n} |\le \frac{C(\lambda,T)}{n^3}.$$ Since we have $$\delta Y_t = {\mathbb{E}}\left(\int_t^T \left( f\left(s,Y^{n+1}_s,Z^{n+1}_s\right) - f(s,Y^{n}_s,Z^{n}_s) \right) \, ds \: \Big|\: {\mathcal}F_t\right) + \Delta_t K^{n+1} - \Delta_t K^n,$$ combining the above and Burkholder-Davis-Gundy inequality, we conclude that $(Y^n,Z^n,K^n)_n$ converges strongly to a limit $(Y, Z, K)$, namely $${\mathbb{E}}\left[ \sup_{0\leq t\leq T} |Y^n_t- Y_t|^2 + \int_0^T | Z^n- Z_s|^2 ds \right] + \sup_{0\leq t\leq T} | K^n_t-K_t|^2 \longrightarrow_{n\rightarrow \infty} 0.$$
#### Step 3. Properties of the limit $(Y,Z,K)$
Passing to the limit the dynamics of $(Y^n,Z^n,K^n)_n$, remark that $(Y,Z,K)$ satisfies . Observe also that, by construction, $K$ is deterministic, nondecreasing with $K_0=0$. Besides, the estimate directly implies that $$\int_0^T |(u-{\mathbb{E}}[Y_t])_+|^2 dt \,=\, \lim_{n\rightarrow\infty} \int_0^T |(u-{\mathbb{E}}[Y^n_t])_+|^2 dt \,=\, 0\;,$$ so that ${\mathbb{E}}[Y_t]\ge u$, for any $t\in[0,T]$. Finally from Lemma \[en:analysis\] below, since $({\mathbb{E}}[Y^n],K^n)$ converges to $({\mathbb{E}}[Y], K)$ in ${\mathcal}C([0,T])$, we have $$\lim_{n\to\infty} \int_0^T ({\mathbb{E}}[Y^n_t]-u)_+ dK^n_t = \int_0^T ({\mathbb{E}}[Y_t]-u)_+ dK_t,$$ and, on the other hand, $$\int_0^T ({\mathbb{E}}[Y^n_t]-u)_+ dK^n_t \,=n\, \int_0^T ({\mathbb{E}}[Y^n_t]-u)_+ (u-{\mathbb{E}}[Y^n_t])_+ dt \;=\; 0.$$ It follows that $(Y,Z,K)$ is the unique flat deterministic solution to the BSDE with linear mean reflection .
We now complete the argumentation by proving a rather elementary lemma, that we just used in the previous proof.
\[en:analysis\] Let $(u^n)_{n\geq 1}$ and $(K^n)_{n\geq 1}$ be two convergent sequences of $\left({\mathcal}C_T, |\cdot |_\infty\right)$. We assume that, for each $n\geq 1$, $K^n$ is nondecreasing and we denote by $u$ and $K$ the corresponding limits of $(u^n)_n$ and $(K^n)_n$. We have $$\lim_{n\to\infty} \int_0^T u^{n}_t dK^n_t = \int_0^T u_tdK_t.$$
For any piecewise constant function $h$, we have $$\begin{aligned}
\int_0^T u^n_s dK^n_s - \int_0^T u_s dK_s & = \int_0^T [u^n_s-u_s] dK^n_s + \int_0^T [u_s-h_s] dK^n_s + \int_0^T h_s dK^n_s \\
& \quad - \int_0^T h_s dK_s + \int_0^T [h_s-u_s] dK_s,
\end{aligned}$$ from which we deduce that $$\begin{aligned}
\left|\int_0^T u^n_s dK^n_s - \int_0^T u_s dK_s\right| & \leq |u^n-u|_\infty |K^n|_\infty + |u-h|_{\infty} \left(|K^n|_\infty + |K|_\infty\right) \\
&\quad + \left|\int_0^T h_s dK^n_s - \int_0^T h_s dK_s\right|.
\end{aligned}$$ Since $h$ is piecewise constant, we have $$\lim_{n\to\infty}\int_0^T h_s dK^n_s = \int_0^T h_s dK_s, \quad\text{and}\quad\limsup \left|\int_0^T u^n_s dK^n_s - \int_0^T u_s dK_s\right| \leq 2 \, |u-h|_\infty\, |K|_\infty,$$ from which we get the result since piecewise constant functions on $[0,T]$ are dense in $\left({\mathcal}C_T, |\cdot |_\infty\right)$.
BSDE with general mean reflection {#sec:general_case}
==================================
We now turn to the general case where $x\longmapsto \ell(t,\omega,x)$ is non necessarily linear. We recall that we still work under Assumptions ($H_\xi$)-($H_f$)-($H_\ell$) presented in Section \[sec:setup\]. In the same spirit as the approach presented in the previous section, we first construct explicitly a solution whenever the driver does not depend on $Y$ nor $Z$, and then tackle the general case via a Picard contraction argument. The construction of an explicit solution in the non linear case is less natural and relies a lot on the use of the following operator: $$\begin{aligned}
L_t : {\mathrm{L}}^2\left(\Fc_T\right) &\rightarrow& [0,\infty) \\
X &\mapsto& \inf\left\{ x\geq 0 : {\mathbb{E}}\left[\ell(t,x+X)\right] \geq 0 \right\}\;,\end{aligned}$$ defined for any $t\in[0,T]$. Since $\ell$ is of linear growth at infinity and ${\mathbb{E}}\left[\ell(t,\infty)\right]>0$, $L_t$ is well defined. Namely, $L_t(X)$ represents the minimal deterministic strength with which the random variable $X$ must be pushed upward in order to satisfy the constraint of interest at time $t$. In the previous linear case where $\ell:(t,x)\mapsto x-u_t$, we simply explicitly have $L_t : X \mapsto \left({\mathbb{E}}\left[X\right]-u_t\right)_-$.\
We first focus on the constant driver case and we then are able to tackle the general case. For this last framework, a Lipschitz property for the operator $L$ will be required.
The constant driver case
-------------------------
In this section, we demonstrate the well posedness of the BSDE of interest in the constant driver case. As explained above, the operator $L$ plays an important role in order to build a solution to such BSDE.
\[prop:existence\_uniqueness\_2\] Let $C$ be a square integrable progressively measurable stochastic process or more generally in the space ${\mathrm{L}}^2\left(\Omega;{\mathrm{L}}^1(0,T)\right)$. Then, the BSDE with mean reflection $$\label{eq:nov}
Y_t =\xi+\int_t^T C_s\, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t,\qquad {\mathbb{E}}[\ell(t,Y_t)] \geq 0, \qquad 0\leq t\leq T,$$ has a unique square integrable deterministic flat solution.
We derive the existence and uniqueness properties separately.
#### Step 1. Existence
In order to solve , let us define $$\Psi_t := L_t\left(X_t\right),\text{ where }X_t = {\mathbb{E}}_t\left(\xi + \int_t^T C_s\, ds\right), \qquad 0\le t \le T\;.$$ Since $\ell$ is continuous in space, observe that $$\label{temp123123}
{\mathbb{E}}\left[\ell\left(t,X_t + \Psi_t\right)\right] \geq 0, \qquad 0\le t \le T\;.$$ Let us now show that $\Psi$ is moreover continuous. Observe first that the map $x\longmapsto {\mathbb{E}}\left[\ell(t,x+X)\right]$ is continuous and strictly increasing. If ${\mathbb{E}}\left[\ell(t,X_t)\right]\leq 0$, since $\ell$ is continuous and has linear growth, for any $x< L_t(X_t)<y $, one has $$\lim_{s\to t} {\mathbb{E}}\left[\ell(s,x+X_{s})\right]={\mathbb{E}}\left[\ell(t,x+X_t)\right] < 0 = {\mathbb{E}}\left[\ell(t,L_t(X_t)+X_t)\right] < {\mathbb{E}}\left[\ell(t,y+X_t)\right]=\lim_{s\to t} {\mathbb{E}}\left[\ell(s,y+X_{s})\right].$$ Then, if $|s-t|$ is small enough, ${\mathbb{E}}\left[\ell(s,x+X_{s})\right]<0$, ${\mathbb{E}}\left[\ell(s,y+X_{s})\right]>0$ and $x\leq L_s(X_s) \leq y$.
If ${\mathbb{E}}\left[\ell(t,X_t)\right] >0 $, $L_t(X_t)=0$, and $\lim_{s\to t} {\mathbb{E}}\left[\ell(s,X_{s})\right]={\mathbb{E}}\left[\ell(t,X_t)\right]>0$. If $|s-t|$ is small enough, ${\mathbb{E}}\left[\ell(s,X_{s})\right]>0$ and $L_s(X_s)=0$.
We are now in position to define the continuous process $K$ by $$K_t := \sup_{0\leq s\leq T} \Psi_s - \sup_{t\leq s\leq T} \Psi_s\;, \; \qquad \mbox{so that } \qquad K_T-K_t = \sup_{t\leq s \leq T} \Psi_s \;, \qquad 0\le t \le T\;.$$ Observe that $K$ is deterministic, non decreasing with $K_0=0$. Given this process $K$, let $(Y,Z)$ be the unique solution to the classical BSDE with the dynamics of . Then, since $x\longmapsto \ell(t,x)$ is non decreasing, we deduce from that $$\label{eq:pos}
{\mathbb{E}}\left[\ell(t,Y_t)\right] = {\mathbb{E}}\left[\ell\left(t,X_t + K_T-K_t\right)\right] = {\mathbb{E}}\left[\ell\left(t,X_t + \sup_{t\leq s\leq T} \Psi_s\right)\right] \geq {\mathbb{E}}\left[\ell\left(t,X_t + \Psi_t\right)\right] \geq 0.$$ Hence, $(Y,Z,K)$ is a deterministic solution to the BSDE with weak reflexion .
Let now verify that it is also flat. By definition of $K$, observe that $\sup_{t\leq s\leq T} \Psi_s = \Psi_t$ $dK_t$-a.e. and ${\mathbf{1}}_{\Psi_t = 0} = 0$ $dK_t$-a.e. Thus, by , we compute $$\int_0^T {\mathbb{E}}\left[\ell(t,Y_t)\right] \, dK_t = \int_0^T {\mathbb{E}}\left[\ell(t,X_t + \Psi_t)\right] \, dK_t = \int_0^T {\mathbb{E}}\left[\ell(t,X_t + \Psi_t)\right]{\mathbf{1}}_{\Psi_t >0} \, dK_t.$$ Besides, since $\ell$ is continuous in space, we have ${\mathbb{E}}\left[\ell(t,X_t + \Psi_t)\right]=0$ as soon as $\Psi_t>0$, so that $$\int_0^T {\mathbb{E}}\left[\ell(t,X_t + \Psi_t)\right]{\mathbf{1}}_{\Psi_t >0} \, dK_t = 0,$$ and $(Y,Z,K)$ is a flat solution.\
#### Step 2. Uniqueness
Let $(Y^1,Z^1,K^1)$ and $(Y^2,Z^2,K^2)$ be two deterministic flat solutions to the BSDE with mean reflexion . We work towards a contradiction and suppose that there exists $t_1<T$ such that $$K^1_T-K^1_{t_1}> K^2_T - K_{t_1}^2.$$ Setting $t_2$ as the first time $t$ after $t_1$ such that $K^1_T-K^1_{t}=K^2_T - K_{t}^2$, we observe that $$K^1_T-K^1_{t}>K^2_T - K_{t}^2,\quad t_1\le t < t_2 \;.$$ Since $\ell$ is strictly increasing, this implies that $${\mathbb{E}}[\ell(t,X_t+K_T^1-K_t^1)]>{\mathbb{E}}[\ell(t,X_t+K_T^2-K_t^2)]\ge 0,\quad t_1\le t < t_2 \;.$$ But $(Y^1,Z^1,K^1)$ is a flat solution and hereby $$\int_{t_1}^{t_2} {\mathbb{E}}[\ell(t,X_t+K_T^1-K_t^1)]dK_t^1 =0\;,$$ so that we must have $dK^1=0$ on the interval $[t_1,t_2]$. We deduce that $$K_T^1-K^1_{t_2}=K_T^1-K_{t_1}^1>K_T^2-K_{t_1}^2\ge K^2_T-K^2_{t_2}\;,$$ which contradicts the definition of $t_2$. Hence $K^1=K^2$ and the uniqueness of solution to classical BSDEs directly implies that $(Y^1,Z^1,K^1)$ coincides with $(Y^2,Z^2,K^2)$.
Existence and uniqueness for the general case
----------------------------------------------
Now that the well posedness for constant driver is established, we can focus on the BSDE with mean reflexion in full generality. In order for the solution to be well defined, we will require a Lipschitz property of the operator $L$, that we present in the following additional Assumption:
- The operator $L_t$ is Lipschitz continuous for the ${\mathrm{L}}^1$-norm, uniformly in time: namely there exists a constant $C\geq 0$ such that $$|L_t(X)-L_t(Y)|\le C\;\mathbb {\mathbb{E}}\left[|X-Y|\right] , \qquad 0\le t \le T\;, \;\; X,Y \in {\mathrm{L}}^2\left(\Fc_t\right)\;.$$
We are now in position to state the main result of the paper, providing the well-posedness of BSDEs with mean reflexion.
\[thm: main\] In addition to the running assumptions ($H_\xi$)-($H_f$)-($H_\ell$), let us moreover assume that ($H_L$) is satisfied. Then, there exists a unique deterministic flat solution $(Y,Z,K)\in\Sc^2\times \Hc^2\times\Ac^2_D$ to the BSDE with mean reflexion .
Let us consider $\sigma$ and $\tau$ in the time interval $[0,T]$ with $\sigma\leq \tau$. Given $Y_\tau \in {\mathrm{L}}^2\left({\mathcal}F_\tau\right)$, $\{U_t\}_{\sigma\leq t\leq \tau}\in{\mathcal}S^2$ and $\{V_t\}_{\sigma\leq t\leq \tau}\in{\mathcal}H^2$, Proposition \[prop:existence\_uniqueness\_2\] ensures the existence of a triple of processes $\{(Y_t,Z_t,R_t)\}_{\sigma\leq t\leq \tau}$ solution to the BSDE with mean reflexion $$\begin{gathered}
Y_t= Y_\tau + \int_t^\tau f(s,U_s,V_s)\, ds - \int_t^\tau Z_s\cdot dB_s + R_t, \qquad \sigma\leq t\leq \tau, \\
{\mathbb{E}}\left[\ell(t,Y_t)\right] \geq 0, \quad \sigma\leq t\leq \tau, \qquad \int_\sigma^\tau {\mathbb{E}}\left[\ell(t,Y_t)\right]\, dR_t = 0\,,
\end{gathered}$$ where we conveniently denoted $R_. = K_\tau-K_.$. In this setting, $R$ is non increasing with $R_\tau=0$ and, for $\sigma\leq t\leq \tau$, $$\label{eq:repR}
R_t = \sup_{t\leq s \leq \tau}\,L_s(X_s), \quad\text{with}\quad X_t = {\mathbb{E}}\left[Y_\tau + \int_t^\tau f(s,U_s,V_s)\, ds \:\Big|\: {\mathcal}F_t\right].$$ Let $(Y',Z',R')$ be the solution associated to $(U',V')$ and the same $Y_\tau$.
We have, with usual notations, $$\delta Y_t = {\mathbb{E}}\left[\int_t^\tau \left[f(s,U_s,V_s)-f(s,U'_s,V'_s)\right]\, ds \:\Big|\: {\mathcal}F_t\right] + \delta R_t, \quad \sigma\leq t\leq \tau,$$ from which we deduce immediately, since $f$ is assumed to be Lipschitz, that $${\mathbb{E}}\left[\sup_{\sigma\leq t\leq \tau}|\delta Y_t|^2 \right] \leq C(\lambda) \, {\mathbb{E}}\left[ \left(\int_\sigma^\tau \left(|\delta U_s|+|\delta V_s|\right)ds\right)^2\right]+ \sup_{\sigma \leq t \leq \tau} \left|\delta R_t\right| ^2.$$ Besides, since ($H_L$) holds, we deduce from the representation together with $\delta Y_\tau=0$ and the Lipschitz property of $f$ that, for $\sigma\leq t\leq\tau$, $$\begin{aligned}
\left|\delta R_t\right| & \leq \left|\sup_{t\leq s\leq\tau} L_s(X_s) - \sup_{t\leq s\leq\tau} L_s(X'_s)\right| \leq \sup_{t\leq s\leq\tau} \left| L_s(X_s)-L_s(X'_s) \right| \leq \sup_{t\leq s\leq\tau} {\mathbb{E}}\left[\left|\delta X_s\right|\right] \\
& \leq C(\lambda)\, {\mathbb{E}}\left[ \int_\sigma^\tau \left(|\delta U_s|+|\delta V_s|\right)ds\right] \;.
\end{aligned}$$ Combining the previous estimates together with the Cauchy Schwartz inequality, we deduce $${\mathbb{E}}\left[\sup_{\sigma\leq t\leq \tau}|\delta Y_t|^2 \right] \leq C(\lambda) \, {\mathbb{E}}\left[ \left(\int_\sigma^\tau \left(|\delta U_s|+|\delta V_s|\right)ds\right)^2\right],$$ and writing $$\int_\sigma^\tau \delta Z_s\cdot dB_s = \delta Y_\tau-\delta Y_\sigma + \delta R_\tau-\delta R_\sigma+\int_\sigma^\tau \left[f(s,U_s,V_s)-f(s,U'_s,V'_s)\right]\, ds$$ we finally have $$\begin{aligned}
\nonumber
{\mathbb{E}}\left[\sup_{\sigma\leq t\leq \tau}|\delta Y_t|^2 + \int_\sigma^\tau |\delta Z_s|^2 \, ds\right] & \leq C(\lambda) \, {\mathbb{E}}\left[ \left(\int_\sigma^\tau \left(|\delta U_s|+|\delta V_s|\right)ds\right)^2\right] , \\
& \leq C(\lambda)\, \left(\tau-\sigma\right)\max\left(1, \tau-\sigma\right) \, {\mathbb{E}}\left[\sup_{\sigma\leq t\leq \tau}|\delta U_t|^2 + \int_\sigma^\tau |\delta V_s|^2 \, ds\right].
\label{eq:smalltime}
\end{aligned}$$ Of course, this inequality shows that the BSDE with mean reflexion has a unique solution whenever $T$ is small enough.
To cover the general case, let us pick $n\geq 1$ such that $C(\lambda) \min(T,T^2) / n^2 < 1$. For $i=0,\ldots, n$, let us set $T_i:=iT/n$. Starting from the interval $[T_{n-1},T_n]$ and $Y_{T_n}=\xi$, let, for $i=n,\ldots,1$, $(Y^i,Z^i,R^i)$ the unique solution to the BSDE with mean reflexion $$\begin{gathered}
Y^i_t= Y^{i+1}_{T_i}+ \int_t^{T_i} f\left(s,Y^i_s,Z^i_s\right)\, ds - \int_t^{T_i} Z^i_s\cdot dB_s + R^i_t, \qquad {\mathbb{E}}\left[\ell\left(t,Y^i_t\right)\right] \geq 0 \qquad T_{i-1}\leq t\leq T_i, \\
\int_{T_{i-1}}^{T_i} {\mathbb{E}}\left[\ell\left(t,Y^i_t\right)\right]\, dR^i_t = 0, \qquad R^i \text{ continuous and non increasing on } [T_{i-1}, T_{i}] \text{ with }R^i_{T_i}=0.
\end{gathered}$$ Let us define $(Y,Z,R)$ on $[0,T]$ by setting $$Y_t = Y^1_0 {\mathbf{1}}_{{0}}(t) + \sum_{i=1}^n Y^i_t {\mathbf{1}}_{]T_{i-1},T_i]}(t), \qquad Z_t = \sum_{i=1}^n Z^i_t {\mathbf{1}}_{]T_{i-1},T_i[}(t),$$ and $R_t=R^n_t$ on $[T_{n-1},T_{n}]$ and, for $i=n-1,\ldots 1$, $R_t=R^{i}_t + R_{T_i}$ on $[T_{i-1},T_i]$. Since $R^i_{T_i}=0$, $R$ is continuous and non increasing. Finally, let us define $K_t = R_0-R_t$ to get a non decreasing continuous function with $K_0=0$. Since $R_T=0$, $K_T=R_0$ and $R_t=K_T-K_t$.
It is plain to check that $(Y,Z,K)$ is a solution to the BSDE with mean reflexion . Uniqueness follows from the uniqueness on each small interval.
It is worth noticing that the previous assumption $(H_L)$ is automatically satisfied as soon as $\ell$ is a bi-Lipschitz function in $x$. More precisely, we consider the following alternative assumption on $\ell$:
- The loss function $\ell : \Omega\times[0,T]\times{\mathbb{R}}{\longrightarrow}{\mathbb{R}}$ is a measurable map with respect to ${\mathcal}F_T\times{\mathcal}B([0,T])\times {\mathcal}B({\mathbb{R}})$ and there exists $0< c_l\leq C_l$ such that, ${\mathbb{P}}$-a.s.,
1. $\forall y\in{\mathbb{R}}$, $t\longmapsto \ell(t,y)$ is continuous,
2. $\forall t\in[0,T]$, $y\longmapsto \ell(t,y)$ is strictly increasing,
3. $\forall t\in[0,T]$, $\forall y\in{\mathbb{R}}$, $|\ell(t,y)| \leq C_l(1+|y|)$.
4. $\forall t\in[0,T]$, $$\label{bilip}
{c_{\ell}} |x-y| \leq |\ell(t,x)-\ell(t,y) | \leq C_{\ell} |x-y|\;, \qquad x,y\in{\mathbb{R}}\;,$$
Assume $(H_{b\ell})$. Then both Assumptions ($H_\ell$) and ($H_L$) hold.
Observe first that $(H_{b\ell})$ implies directly that $(H_\ell)$ holds. Fix now $t\in[0,T]$ and let $X$ and $Y$ be two random variables in ${\mathrm{L}}^2\left(\Fc_T\right)$.
Since $\ell$ is non decreasing, the lower bound of gives $$\begin{aligned}
\ell\left(t,L_t(X)+\frac{C_{\ell}}{c_{\ell}}{\mathbb{E}}\left[|X-Y|\right]+Y\right) &\ge& c_{\ell} \frac{C_{\ell}}{c_{\ell}}{\mathbb{E}}\left[|X-Y|\right] + \ell(t,L_t(X)+Y),\end{aligned}$$ and using the upper bound we get $$\ell(t,L_t(X)+Y) \geq \ell(t,L_t(X)+X) -C_l |X-Y|,$$ from which it follows $$\ell\left(t,L_t(X)+\frac{C_{\ell}}{c_{\ell}}{\mathbb{E}}\left[|X-Y|\right]+Y\right) \geq \ell(t,L_t(X)+X) - C_l\, |X-Y| +C_l\, {\mathbb{E}}\left[|X-Y|\right]$$ Since ${\mathbb{E}}\left[\ell(t,X+L_t(X))\right]\geq 0$, we obtain by taking the expectation of the previous inequality $${\mathbb{E}}\left[\ell\left(t,L_t(X)+\frac{C_{\ell}}{c_{\ell}}{\mathbb{E}}\left[|X-Y|\right]+Y\right)\right] \geq 0.$$ By definition of $L_t(Y)$, this directly implies that $$L_t(Y)\le L_t(X)+\frac{C_{\ell}}{c_{\ell}}{\mathbb{E}}\left[|X-Y|\right] \;.$$ By symmetry of $X$ and $Y$, we conclude that $$|L_t(X)-L_t(Y)|\le \frac{C_{\ell}}{c_{\ell}}\,{\mathbb{E}}\left[|X-Y|\right].$$
As a byproduct, we have the following result.
\[en:TheMain\] Let ($H_\xi$), ($H_f$) and ($H_{b\ell}$) hold.
Then, there exists a unique deterministic flat solution $(Y,Z,K)\in\Sc^2\times \Hc^2\times\Ac^2_D$ to the BSDE with mean reflexion .
Minimality of the deterministic flat solution {#sec:minimal}
=============================================
Let us recall that for classical reflected BSDE, the Skorokhod condition ensures the minimality of the enhanced solution in the class of all supersolutions to the reflected BSDE. By minimality, we refer to minimality in terms of the $Y$-component of the solution. The Skorokhod condition indicates that the compensator $K$ only pushes the solution when the condition is binding, i.e. only when it is really necessary. In this spirit, we chose in this paper to look towards solutions to BSDEs with mean reflection which satisfy the corresponding flatness condition .\
Now, that the existence of a unique deterministic flat solution to the BSDE with mean reflexion has been established, it is natural to wonder if this flatness condition also implies the minimality among all the deterministic solutions. Since the constraint is given in expectation instead of pointwisely, it is not obvious that only the condition at time $t$ determines the minimal upward kick to apply on the solution at time $t$. Under additional assumption on the structure of the driver function $f$, we are able to verify that such minimality property is indeed satisfied.
\[minimal\] Suppose that the driver function $f$ is of the form $$\label{struct_f}
f : (t,y,z) \mapsto a_t y+h(t,z) \;,$$ where $a$ is a deterministic and bounded measurable function. If $\ell$ is strictly increasing, a deterministic flat solution $(Y,Z,K)$ is minimal among all the deterministic solutions.
Let $(Y,Z,K)$ be a deterministic flat solution, and $(Y',Z',K')$ be any deterministic solution. We want to prove that $Y\le Y'$. We first focus on the particular case where the driver does not depend on $y$ and then tackle the general case where $f$ is given by .\
[**Step 1. Driver of the form $f(t,z)$.**]{}\
Since the driver function $f$ does not depend on $y$, the processes $(Y-(K_T-K),Z)$ and $(Y'-(K'_T-K'),Z')$ are both solutions of the same classical BSDE, and we deduce that $$\label{tmptmp}
Y_t-(K_T-K_t)=Y'_t-(K'_T-K'_t), \qquad 0\le t\le T \;.$$ Hereby, proving that $Y\le Y'$ boils down to showing that $K_T-K \le K'_T-K'$. We work towards a contradiction and suppose the existence of $t_1<T$ such that $$K_T-K_{t_1}>K'_T-K'_{t_1}.$$ Let $t_2$ be the first time such that $K_T-K_. \ge K_T'-K'_.$. Obviously $t_2$ is a deterministic time smaller than $T$ and by continuity of $K$ and $K'$, we get $K_T-K_{t_2} = K'_T-K'_{t_2}$ and $$K_T-K_{t} > K'_T-K'_{t}\;, \qquad t_1\le t < t_2\;.$$ We deduce from that $Y> Y'$, on $[t_1,t_2)$, and the strict monotony of $\ell$ implies $$\mathbb E[\ell(t,Y_t)]>\mathbb E[\ell(t,Y'_t)]\ge 0\;, \qquad t_1\le t \le t_2 \,.$$ Since $Y$ is a flat solution, we have $\int_0^T \mathbb E[\ell(Y_s)]dK_s=0$ and we deduce that $dK_t=0$, for $t\in [t_1,t_2)$. Therefore, $$K_T'-K'_{t_1}<K_T-K_{t_1}=K_T-K_{t_2}=K'_T-K'_{t_2}$$ which is a contradiction since $K'$ must be non decreasing.
[**Step 2. Driver of the form .**]{}\
Let us denote $A_t:= \int_0^t a_sds$ for $0\le t \le T$. Making the following transformation $$\tilde{Y}_t=e^{A_t}Y_t,\quad \tilde{Z}_t=e^{A_t}Z_t, \quad \tilde{K}_t=e^{A_t}K_t,$$ we verify easily that $(\tilde{Y},\tilde{Z},\tilde{K})$ is a flat deterministic solution to the BSDE with mean reflection associated to the parameters $$\tilde \xi = e^{A_T} \xi \;, \qquad \tilde f(t,z)=e^{A_t} f(t,e^{-{A_t}}z) \; \quad \mbox{and}\quad \tilde \ell(t,y)=\ell(t,e^{-{A_t}}y) \;.$$ According to the previous step $\tilde Y$ is minimal within the class of deterministic solutions, and $Y$ inherits this property by a straightforward argument.
As a by-product, this proof provides an alternative argument in order to derive the uniqueness of the flat deterministic solution of BSDEs with mean reflexion and driver of the form . It is in fact a generalization of the proof presented in Proposition \[prop:existence\_uniqueness\_2\] for the constant driver case.\
We now exhibit an example to show that if we allow $K$ to be random, then there exists no minimal flat solution to the BSDE with mean reflection. This argument strengthens our choice to focus solely in this paper on so-called deterministic solutions.
For this purpose, let consider BSDE with mean reflection $$\begin{gathered}
Y_t = \xi - \int_t^T \gamma \, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t, \quad 0\leq t\leq T ,\\
{\mathbb{E}}[Y_t] \geq u,\quad 0\leq t \leq T, \qquad \int_0^T \left({\mathbb{E}}[Y_t]-u\right) dK_t = 0,
\end{gathered}$$ with $\gamma>0$, and the terminal condition $\xi$ such that $u < {\mathbb{E}}[\xi] < u+\gamma T$.
As detailed in Section \[sec:llin\], the deterministic flat solution to the BSDE is given by $$\begin{gathered}
Y_t = {\mathbb{E}}\left(\xi\:|\: {\mathcal}F_t\right) - \gamma (T-t) + {\left({\mathbb{E}}[\xi]-\gamma (T-t)-u\right)^-},
\end{gathered}$$ and $K_t = \gamma (t\wedge t^*)$, where we pick $t^*$ to verify $$\qquad {\mathbb{E}}[\xi] - \gamma (T-t^*) = u.$$ Starting from the previous solution, for $\alpha \in{\mathbb{R}}$, we set $$M^\alpha_t := \exp\left(\alpha B_t - \alpha^2 t/2\right)\quad \mbox{ and }\quad \quad K^\alpha_t := \int_0^t M^\alpha_s\, dK_s \;, \qquad 0\le t \le T\,.$$ Given $K^\alpha$, let $\left(Y^\alpha,Z^\alpha\right)$ be the solution to the classical BSDE $$Y^\alpha_t = \xi - \int_t^T \gamma \, ds - \int_t^T Z^\alpha_s dB_s + K^\alpha_T - K^\alpha_t, \quad 0\leq t\leq T.$$ Then $\left(Y^\alpha,Z^\alpha,K^\alpha\right)$ is still a flat solution to the reflected BSDE, see Remark \[rem\_uniqueness\] in Section \[sec:llin\].\
Let us suppose the existence of a minimal solution $(\bar Y,\bar Z,\bar K)$ and look towards a contradiction. We have $$\begin{aligned}
\bar Y_t & \leq & Y^\alpha_t = {\mathbb{E}}_t(\xi) - \gamma (T-t) +{\mathbb{E}}_t\left( \int_t^T M^\alpha_s dK_s\right)
\; = \; {\mathbb{E}}_t(\xi) - \gamma (T-t) + M^\alpha_t (K_T-K_t)\,,
\end{aligned}$$ for $t>0$. As a byproduct, sending $\alpha$ to $+\infty$, we deduce $\bar Y_t \leq {\mathbb{E}}_t(\xi)-\gamma(T-t)$ for $t>0$, and in particular $$\forall t>0, \qquad {\mathbb{E}}\left[\bar Y_t\right] \leq {\mathbb{E}}\left[\xi\right] -\gamma (T-t).$$ Since ${\mathbb{E}}\left[\xi\right]-\gamma T < u$, for $t>0$ small enough, ${\mathbb{E}}\left[\bar Y_t\right]<u$. The constraint is not satisfied and we get a contradiction.
Extension and application {#sec:application}
=========================
Interpreting $Y$ as the value of a portfolio, the constraint imposes at any date $t$ a constraint on the distribution of $Y_t$, seen from time $0$. The form of constraint that we considered so far is the expectation of a loss function. From a financial point of view, an investor may be required to control the risk of any admissible portfolio. In order to measure the underlying risk of a portfolio, the natural tool in the mathematical finance literature are the so-called risk measures, see e.g. [@ADEH]. We emphasize in this section how our framework of study allows to encompass such type of running static risk measure constraint. Then, we present an application for the problem of super hedging a claim under a given running risk measure constraint.\
BSDE with risk measure reflection {#subsec_risk}
----------------------------------
For a fixed $t$, a static risk measure is a map $\rho(t,.) : L^2({\mathcal}F_t) {\longrightarrow}{\mathbb{R}}$ satisfying $\rho(t,0)=0$ together with
- Monotonicity: $X\leq Y \Longrightarrow \rho(t,X) \geq \rho(t,Y)$, for $X,Y\in L^2({\mathcal}F_t)$ ;
- Translation invariance: $\rho(t,X+m) = \rho(t,X) - m$, for $X\in L^2({\mathcal}F_t)$ and $m\in\R$ .
Hereby, for a given $t\in[0,T]$, $\rho(t,X)$ is a real number which measures the risk associated to the wealth random variable $X$. Risk measures can similarly be characterized by their so-called acceptance set, which defines as $$\mathcal{A}_\rho^t = \{ X\in L^2({\mathcal}F_t) : \rho(t,X) \leq 0 \}.$$ Similarly, given a set $\mathcal{A}^t$, one can define a static risk measure by setting $$\rho(t,X) = \inf\{ m\in{\mathbb{R}}: m+X \in \mathcal{A}^t \},$$ so that the acceptance set $\mathcal{A}^t$ and the risk measure $\rho(t,.)$ share a one to one correspondence. For a given collection of static risk measures $(\rho(t,.))_t$, a wealth process $Y$ will be considered admissible in our framework as soon as it satisfies $$\begin{aligned}
\label{constraint_riskmeasure}
\rho(t,Y_t) \le q_t\;, \qquad 0\le t \le T\;,
\end{aligned}$$ where $q$ is a given time indexed deterministic benchmark. For example, the risk measuring tool of $\rho$ could simply not depend on time, but be compared to the deterministic benchmark $q$, which evolves with time, by either tightening or relaxing the constraint. We now look towards solutions of BSDEs subject to the additional constraint . In the same spirit as above, a flat solution to such type of BSDE will be required to satisfy $$\begin{aligned}
\label{flatness_riskmeasure}
\int_0^T [q_t-\rho(t,Y_t)] dK_t=0 \;.
\end{aligned}$$
The next theorem indicates that we are able to consider BSDEs under risk measure constraint of the form , in a similar fashion as the one developed in the previous sections.
Let $\rho(t,.) : [0,T]\times{\mathrm{L}}^2{\longrightarrow}{\mathbb{R}}$ be a collection of monotonic and translation invariant risk measures, which are continuous with time and Lipschitz in space, i.e. $$\begin{aligned}
|\rho(t,X)-\rho(t,Y)|\le C {\mathbb{E}}[|X-Y|] \;, \qquad 0\le t \le T\;, \;\; X,Y \in {\mathrm{L}}^2\left(\Fc_t\right)\;.
\end{aligned}$$ If we are moreover given a continuous deterministic benchmark $q$ and $\xi$ satisfies $\rho(T,\xi)\le q_T$, then the “*BSDE with risk measure reflection*” $$\begin{gathered}
Y_t = \xi + \int_t^T f(s,Y_s,Z_s)\, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t, \quad 0\leq t\leq T \\
\rho(t,Y_t)\leq q_t,\quad 0\leq t \leq T, \qquad \int_0^T [q_t-\rho(t,Y_t)] dK_t = 0.
\end{gathered}$$ admits a unique deterministic flat solution.\
Besides, if $f$ satisfies , the deterministic flat solution is minimal among all deterministic solutions.
The reasoning simply follows the arguments of Proposition \[prop:existence\_uniqueness\_2\], Theorem \[thm: main\] and Theorem \[minimal\]. The main distinction is that the map $L_t$ is replaced by the risk measure $\rho(t,.)-q_t$, for any $t\in[0,T]$. Besides, the translation invariance property conveniently replaces the strict monotonicity of $\ell$ in the proofs.
Typical examples considered in the literature are coherent risk measures of the form $$\rho(t,X) = \sup\{ {\mathbb{E}}^{\mathbb{Q}}\left[-X\right] : \mathbb{Q} \in {\mathcal}Q_t\} \;,$$ where ${\mathcal}Q_t$ is a set of probabilities absolutely continuous w.r.t. ${\mathbb{P}}$. As soon as the set of probability change densities is bounded, $\rho(t,)$ is Lipschitz. This is particular the case for the classical Expected Shortfall risk measure, defined as $$\begin{aligned}
\rho^{ES}_\alpha(t,X) := \frac{1}{\alpha_t} \int_0^{\alpha_t} VaR_{s}(X) ds\;,
\end{aligned}$$ where $\alpha_t\in(0,1)$ denotes a given precision level and $VaR_s$ is the Value at Risk of level $s$. Indeed, the Expected Shortfall (or AVaR) rewrites also this way $$\rho^{ES}_\alpha(t,X) = \sup\left\{ {\mathbb{E}}^{\mathbb{Q}}\left[-X\right] : \frac{d\mathbb{Q}}{d{\mathbb{P}}} \leq \frac{1}{\alpha_t} \right\} \;.$$
Application to super hedging under risk constraint {#subsec_application}
--------------------------------------------------
We now turn to an application in mathematical finance and consider a stock market endowed with a Bond with deterministic interest rate $r$ and a vector of $d$ stocks with dynamics $$d S_t \,=\, S_t \left( \mu_t dt + \sigma_t dB_t \right)\;, \qquad 0\le t \le T\;,$$ where the drift $\mu$ and the volatility $\sigma$ are square integrable predictable processes. We assume that $\sigma_t\sigma_t'-\varepsilon I\succeq 0$ for some $\varepsilon>0$, in order to ensure the completeness of the market. For a given initial capital $x$, we consider portfolios $X^{x,\pi,K}$ driven by a consumption-investment strategy $(\pi,K)$, and whose dynamics are given by $$\begin{aligned}
dX^{x,\pi,K}_t &=& X^{x,\pi,K}_t \left(r_t dt+ (\mu_t-r_t{\bf 1}) '\pi_t \frac{d S_t}{S_t} \right) - dK_t \;, \\
&=& r_t X^{x,\pi,K}_t dt + (\mu_t-r_t{\bf 1})'\pi_tdt+\pi_t'\sigma_t dB_t-dK_t\,,\qquad 0\le t \le T \,.\end{aligned}$$ Using such portfolios, a financial engineer is willing to hedge a possibly non Markovian claim $\xi\in L^2(\Fc_T)$. For regulatory purposes, the risk management department of his financial institution imposes him restrictions on the class of admissible investment strategies. Namely, a portfolio wealth process $X^{x,\pi,K}$ is considered admissible if and only if it satisfies the following constraint : $$\rho^{ES}_\alpha(t,X^{x,\pi,K}_t) \le q_t \;,\qquad 0\le t \le T\;,$$ where $(\alpha,q)$ are a time indexed collection of deterministic quantile and level benchmarks. These benchmarks can for example be chosen in such a way that the constraint becomes either tighter or weaker, as we approach the maturity $T$. In such a case, the careful investor is looking for the super hedging price $$Y_0 = \inf\{ x\in\R \;, \quad \exists (\pi,K)\in\Ac\,, \quad s.t. \;\; X^{x,\pi,K}_T\ge \xi \;\;\;\; \mbox{ and } \;\;\;\; \rho^{ES}_\alpha(t,X_t) \le q_t \;,\quad \forall \;\; t\in[0,T] \} \;,$$ and associated consumption-investment strategy. Applying the results of this paper, we deduce that, if the investor restricts to deterministic consumption strategies, $Y_0$ is well defined as the starting point of the unique deterministic flat solution to the following BSDE with risk measure reflection $$\begin{aligned}
Y_t = \xi + \int_t^T r_t X^{x,\pi,K}_t dt + (\mu_t-r_t{\bf 1})' \sigma_t^{-1} Z_t^\top \, ds - \int_t^T Z_s\cdot dB_s + K_T-K_t, \quad 0\leq t\leq T, \\
\rho^{ES}_\alpha(t,Y_t)\leq q_t,\quad 0\leq t \leq T, \qquad \int_0^T [q_t-\rho^{ES}_\alpha(t,Y_t)] dK_t = 0.\end{aligned}$$ Indeed, the driver function satisfies , so that the flat solution is minimal among all deterministic ones.
[99]{}
(1999), Coherent measures of risk, [*Math. Finance*]{}, [**9(3)**]{}, 203-228.
(2015), BSDEs with weak terminal condition, [*Ann. Probab.*]{}, [**43(2)**]{}, 572-604.
(1998), Pricing of American contingent claims with jump stock price and constrained portfolios, [*Math. Oper. Res.*]{}, [**23(1)**]{}, 177-203.
(1998), Hedging contingent claims for a large investor in an incomplete market, [*Adv. in Appl. Probab.*]{} [**30(1)**]{}, 239-255.
(2009), Mean-field backward stochastic differential equations and related partial differential equations, [*Stochastic Process. Appl.*]{}, [**119(10)**]{}, 3133-3154.
(2011), A note on existence and uniqueness for solutions of multidimensional reflected BSDEs, [*Electron. Comm. Probab.*]{}, [**16**]{}, 120-128.
(2015), A cubature based algorithm to solve decoupled McKean-Vlasov forward-backward stochastic differential equations, [*Stochastic Process. Appl.*]{}, [**125(6)**]{}, 2206-2255.
(1996), Backward stochastic differential equations with reflection and Dynkin games, [*Ann. Probab.*]{}, [**24(4)**]{}, 2024-2056.
(1998), Backward stochastic differential equations with constraints on the gains-process, [*Ann. Probab.*]{}, [**26(4)**]{}, 1522-1551.
(1997), Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, [*Ann. Probab.*]{}, [**25(2)**]{}, 702-737.
(1997), Backward stochastic differential equations in finance, [*Math. Finance*]{}, [**7(1)**]{}, 1-71.
(2007), On the starting and stopping problem: application in reversible investments, [*Math. Oper. Res.*]{}, [**32(1)**]{}, 182-192.
(2010), Switching problem and related system of reflected backward SDEs, [*Stochastic Process. Appl.*]{}, [**120(4)**]{}, 403-426.
(2010), Multi-dimensional BSDE with oblique reflection and optimal switching, [*Probab. Theory Related Fields*]{}, [**147(1-2)**]{}, 89-121.
(1990), Adapted solution of a backward stochastic differential equation, [*Systems Control Lett.*]{}, [**14(1)**]{}, 55-61.
(2010), Reflected BSDE with a constraint and its applications in an incomplete market, [*Bernoulli*]{}, [**16(3)**]{}, 614-640.
[^1]: Laboratoire de Mathématiques CNRS UMR 5127, Université de Savoie, Campus Scientifique, 73376 Le Bourget du Lac, France (philippe.briand@univ-savoie.fr)
[^2]: LAMA UMR CNRS 8050, Université Paris-Est $\&$ Projet MathRisk ENPC-INRIA-UMLV, Champs-sur-Marne, France (romuald.elie@univ-mlv.fr). Research partially supported by the ANR grant LIQUIRISK and the Chair Finance and Sustainable Development.
[^3]: IRMAR UMR CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France (ying.hu@univ-rennes1.fr), partially supported by Lebesgue center of mathematics (“Investissements d’avenir” program - ANR-11-LABX-0020-01 and by ANR-15-CE05-0024-02.
|
---
abstract: 'STAR collected data in proton-proton collisions at $\sqrt{s}=200$ GeV with transverse and longitudinal beam polarizations during the initial running periods in 2002–2004 at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory. Results on the single transverse spin asymmetries in the production of high energy forward neutral pions and of forward charged hadrons will be presented. Data have been obtained for double longitudinal asymmetries in inclusive jet production in 2003 and 2004. These data provide sensitivity to the polarization of gluons in the proton. In the future, we aim to determine the gluon polarization over a wide kinematic range using coincidences of direct photons and jets. Furthermore, we aim to determine the polarizations of the $u$, $\bar{u}$, $d$ and $\bar{d}$ quarks in the proton by measuring single longitudinal spin asymmetries in the production of weak bosons at $\sqrt{s} = 500$ GeV.'
---
[Spin Physics with STAR at RHIC]{}
J. Kiryluk for the STAR Collaboration\
\
STAR is one of two large experimental facilities at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory. One of the goals of the STAR physics program is to study the internal spin structure of the proton in polarized proton-proton collisions. In particular we aim to determine the gluon polarization in the proton and the flavor decomposition of the quark helicity densities in the nucleon sea.
STAR is capable of tracking charged particles and measuring their momenta in a high multiplicity environment. The experimental setup[@nim] provides tracking, particle identification and electromagnetic calorimetry covering a large acceptance. The identification and measurement of jets, electrons, photons, and neutral pions are of particular importance to the spin program.
During the first three polarized proton running periods at RHIC crucial machine components, including spin rotators and polarimeters[@nim], were successfully commissioned. Progress has been made towards the projected design luminosities and polarizations, $\cal{L}_{\rm{max}}$$=0.8(2.0)\times10^{32}\rm{cm^{-2}s^{-1}}$ and P=$0.7$ at $\sqrt{s}=200 (500)$ GeV. Specifically, polarization development has resulted in an increase in beam polarization from about $P= 0.15$ in the first running period in 2002 to $P = 0.40$ in 2004, and peak luminosities have reached $0.5
\times 10^{31}\rm{cm^{-2}s^{-1}}$.
STAR has measured the single transverse spin cross-section asymmetry $A_N=$\
$(\sigma_{\downarrow}-\sigma_{\uparrow})/(\sigma_{\downarrow}+\sigma_{\uparrow})$ at $\sqrt{s}=200$ GeV in neutral pion and charged particle production in the forward region during the initial beam periods at lower luminosities and polarizations. In addition, STAR has developed a sensitive local polarimeter to measure the radial and transverse polarization components at the STAR Interaction Region (IR). We discuss the instrumentation and results below.
A prototype of the Forward Pion Detector (FPD) was installed in STAR during the first pp run at RHIC. Its main component was a Pb-scintillator sampling calorimeter with its associated shower maximum detector for $\pi^0$ identification placed about $30$ cm left of the polarized beam direction and $750$ cm from the interaction point. Data were collected when the energy deposited in the calorimeter exceeded $15$ GeV. Pions were reconstructed for total energies of $15 < E < 80$ GeV with a mass resolution of $20$ MeV.
\[fig:fpd\]
Figure \[fig:fpd\] shows the published[@fpd] analyzing power for inclusive neutral pions production at $\sqrt{s}=200$ GeV as a function of Feynman-$x$, where $x_F\sim 2E/\sqrt{s}$. The filled points are for reconstructed $\pi^0$ mesons. The curves are predictions from different $A_N(x_F,p_T=1.5 {\rm{GeV/c}})$ models. The model predictions[@fpd] are in qualitative agreement with the measured analyzing power, which is found to increase with $x_F$. Higher precision measurements of $A_N$ will map the dependencies in $x_F$ and $p_T$ and, together with complementary measurements, may help to unravel the physics mechanism. STAR aims to further study $A_N$ for forward $\pi^0$ and aims to measure the transverse spin dependence of di-jet back-to-back correlations related to the Sivers function[@boer].
The Beam-Beam Counters (BBC) are scintillator annuli mounted around the beam pipe beyond the east and west poletips of the STAR magnet at 370cm from the interaction region. The small tiles of the BBC have full azimuthal coverage in the pseudorapidity range of $3.4 < |\eta| < 5.0$, cf. Fig. \[fig:bbc\](a). A signal from any of the 18 tiles on the east side and any of the 18 tiles on the west side of the interaction region constitutes a BBC coincidence. The number of BBC coincidences is a measure of the luminosity, with the BBC acceptance covering about 50% of the total proton-proton cross section. The BBC are used in STAR during proton runs as a trigger detector, to monitor the overall luminosity, and to measure the relative luminosities for different proton spin orientations[@kiryluk] with an accuracy of $10^{-3}$.
In an experiment with a transversely polarized beam and a left-right symmetric detector, such as the BBC, the single spin asymmetry can be determined by measuring the beam polarization and the asymmetry of yields, $$\epsilon_{\rm{BBC}} = \frac{ \sqrt{ N_L^{\uparrow} N_R^{\downarrow} }
- \sqrt{ N_L^{\downarrow} N_R^{\uparrow} }}{
\sqrt{ N_L^{\uparrow} N_R^{\downarrow} } + \sqrt{ N_L^{\downarrow} N_R^{\uparrow} }}
\simeq A_N^{\rm{BBC}} \times P,$$ in which $N_{L}^{i}$ and $N_{R}^{i}$ are the spin dependent yields from the detector on the left ($N_L$) and right ($N_R$) side of the beam, and $i = \uparrow, \downarrow$ denote the different spin orientations of the polarized beam. The beam polarization $P$ was measured by the Coulomb-Nuclear Interference (CNI) polarimeter at RHIC[@cni]. STAR has no tracking and particle identification in the BBC acceptance, however, the BBC segmentation allows the classification of the counted occurrences by pseudorapidity (2 bins: inner $3.9 < \eta < 5.0$ and outer $3.4 < \eta < 3.9$ BBC rings) and azimuth. The group of the 4 small tiles labeled 1, 7 and 8 in Fig. \[fig:bbc\](a) is referred to as [*[Up]{}*]{}, whereas the group of tiles 4, 12 and 13 is called [*[Down]{}*]{}. The remaining small tiles are labeled [*[Left (Right) ]{}*]{} for the groups of tiles on the left (right). An example of the hit topology for the ‘inner-right’ BBC event is shown in Fig. \[fig:bbc\](a).
\[fig:bbc\]
Figure \[fig:bbc\](b) shows the time variation of the charged particle asymmetries determined with the BBC for $3.9 < \eta < 5.0$ (filled points) and the asymmetry measured with the RHIC CNI polarimeter (open points). Each data point corresponds to one STAR run, which typically lasts for 30-60 min. The indicated uncertainties on the CNI and BBC asymmetries are statistical only. The dashed line indicates when the spin rotators at STAR were turned on and the transverse polarization direction in RHIC was made longitudinal at the STAR interaction region.
From the data with transverse beam polarization at STAR we find that $A_{\rm{N}}^{\rm{BBC}} = 0.67(8)\times A_{\rm{N}}^{\rm{CNI}} \sim 1 \% $ for $ 3.9 < \eta < 5.0 $, while for smaller pseudorapidities, $ 3.4 < \eta < 3.9 $, the BBC asymmetries are found to be $A_{\rm{N}}^{\rm{BBC}} = 0.02(9)\times A_{\rm{N}}^{\rm{CNI}}$ consistent with zero. The [*[Left-Right]{}*]{} asymmetries in the BBC are sensitive to the transverse polarization vector. Their numerical values when the rotator magnets were on, that is, when the beam polarization was longitudinal at the STAR IR, were significantly smaller. The currents in the rotator magnets were adjusted to make these asymmetries consistent with zero, while at the same time the CNI polarimeter - located at a different IR at RHIC - continued to measure non-zero beam polarization. The asymmetries have also been evaluated with the [*[Up]{}*]{} and [*[Down]{}*]{} groups of tiles in the BBC. They were found to be close to zero for both transverse and longitudinal beam polarizations, as expected.
STAR has thus far collected about $1$ pb$^{-1}$ of data at $\sqrt{s}=200$ GeV with longitudinally polarized beams. These data will allow an exploratory measurement of the double longitudinal spin asymmetry $A_{LL}$ in inclusive pion and jet production, which is sensitive to the magnitude of the gluon polarization in the proton[@jager]. At the partonic level, the cross section receives contributions from the (a) $g+g\rightarrow g+g$, (b) $g+q \rightarrow g+q$ and (c) $q+q \rightarrow q+q$ processes. Their relative contribution varies with $p_T$ and is dominated by gluon-gluon and quark-gluon scattering at low $p_T$. Preliminary precisions and the status of the ongoing jet analysis of the existing data have been reported in Ref [@trentalange]. The ongoing completion of the Barrel Electromagnetic Calorimeter, and the successful installation of an Endcap Electromagnetic Calorimeter, will expand STAR’s acceptance and triggering capabilities for pions and jets. Figure \[fig:jets\] shows prospects for the upcoming running period in 2005.
In the longer term, when the STAR calorimeter upgrades have been completed and design polarizations and beam luminosities have been reached, we will measure the double longitudinal spin asymmetry $A_{LL}$ for coincident photon jet production $\vec{p}+\vec{p}\rightarrow \gamma + {\rm{jet + X}}$ at both 200 and 500 GeV center-of-mass energy to determine the gluon polarization over a wide range in $x$. At leading order QCD, the prompt photon production in $pp$ collisions is dominated by the gluon Compton process $q+g \rightarrow \gamma+q$ and $A_{LL}$ can be written as: $$A_{LL} \simeq {\frac{\Delta G (x_g,Q^2)}{G(x_g,Q^2)}} \times A_1^p (x_q,Q^2)
\times \hat{a}_{LL}^{\rm Compton}.$$ The proton asymmetry $A_1^p$ is known from inclusive DIS measurements and the partonic asymmetry for the Compton process $\hat{a}_{LL}^{\rm Compton}$ can be calculated in perturbative QCD.
\[fig:jets\]
The QCD scale $Q^2$ is on the order of the $p_T^2$ of the prompt photon, and the fraction $x_q (x_g)$ of the hadron momentum carried by the quark (gluon) can be reconstructed on an event-by-event basis from the observed photon and jet pseudorapidities and the transverse momentum of the photon[@bland]. The measurement of $A_{LL}$ thus forms a direct determination of $\Delta G(x,Q^2)/G(x,Q^2)$. Theoretical description of $A_{LL}$ is also known at next-to-leading order [@frixione]. The measurement for two values of $\sqrt{s}$ is essential to cover a relatively wide region in $x_g$, $0.01 < x_g < 0.3$. The integral $\Delta G = \int_{0}^{1} \Delta G(x)$ d$x$ from these measurements is expected to be determined to a precision better than $\pm 0.5$ assuming integrated luminosities of at and at [@bland]. The upcoming proton run (2005-2006) will be the first run for which all of the STAR detector components essential for this measurement are commissioned.
In addition to the measurements of the gluon polarization, STAR aims to decompose the quark spin densities in the nucleon sea by measuring the parity violating single spin asymmetries for $W$ production in $\vec{p} + p \rightarrow W + X \rightarrow e + X$ collisions at $\sqrt{s} = 500\,\mathrm{GeV}$. At these energies, $W$ bosons are produced predominantly through $u + \bar{d} \rightarrow W^+ $ and $d + \bar{u} \rightarrow W^-$, valence-sea processes in $pp$ collisions. The measurements, which will require the operation of RHIC at high luminosities and an upgrade of the tracking capability in the forward region at STAR, are expected to aid the flavor decomposition of the quark spin densities in the nucleon sea[@soffer] and to distinguish various symmetry scenarios. Their theoretical description is known at next-to-leading order [@nadolsky].
[99]{}
Special Issue: RHIC and Its Detectors, Nucl. Instrum. Meth.A [**[499]{}**]{} (2003).
STAR, J. Adams et al., Phys. Rev. Lett. [**[92]{}**]{} (2004) 171801 and references therein.
D. Boer and W. Vogelsang, Phys. Rev. D [**[69]{}**]{} (2004) 094025.
J. Kiryluk, AIP Conf. Proc. [**[675]{}**]{} (2003) 424.
O. Jinnouchi et al., RHIC/CAD Accelerator Physics Note 171 (2004).
B. Jäger, M. Stratmann and W. Vogelsang, Phys. Rev. D [**[70]{}**]{} (2004) 034010.
S. Trentalange, to appear in DIS2004 Proceedings (2004).
L.C. Bland, in EPIC99 Proceedings; hep-ex/9907058.
S. Frixione and W. Vogelsang, Nucl. Phys.B [**[568]{}**]{} (2000) 60.
M. Glück, E. Reya, M. Stratmann and W. Vogelsang, Phys. Rev. D [**[63]{}**]{} (2001) 094005.
C. Bourrely and J. Soffer, Phys. Lett.B [**[314]{}**]{} (1993) 132.
P.M. Nadolsky and C.P. Yuan, Nucl. Phys.B [**[666]{}**]{} (2003) 31.
|
---
author:
- |
Benjamin D. Haeffele bhaeffele@jhu.edu\
René Vidal rvidal@jhu.edu\
Department of Biomedial Engineering\
Johns Hopkins University\
Baltimore, MD 21218, USA
bibliography:
- 'posfactor.bib'
- '../biblio/sparse.bib'
- '../biblio/vidal.bib'
title: 'Global Optimality in Tensor Factorization, Deep Learning, and Beyond'
---
|
---
abstract: 'Transfer learning enhances learning across tasks, by leveraging previously learned representations–if they are properly chosen. We describe an efficient method to accurately estimate the appropriateness of a previously trained model for use in a new learning task. We use this measure, which we call “Predict To Learn” (“P2L”), in the two very different domains of images and semantic relations, where it predicts, from a set of “source” models, the one model most likely to produce effective transfer for training a given “target” model. We validate our approach thoroughly, by assembling a collection of candidate source models, then fine-tuning each candidate to perform each of a collection of target tasks, and finally measuring how well transfer has been enhanced. Across 95 tasks within multiple domains (images classification and semantic relations), the P2L approach was able to select the best transfer learning model on average, while the heuristic of choosing model trained with the largest data set selected the best model in only 55 cases. These results suggest that P2L captures important information in common between source and target tasks, and that this shared informational structure contributes to successful transfer learning more than simple data size.'
author:
- |
Bishwaranjan Bhattacharjee, Noel Codella, Siyu Huo, Michael R. Glass,\
Parijat Dube, Matthew Hill, Brian Belgodere\
IBM Research\
[{bhatta,nccodell,mrglass,pdube,mh,bmbelgod}@us.ibm.com, siyu.huo@ibm.com]{}
- |
John R. Kender[^1]\
Columbia University\
[jrk@cs.columbia.edu]{}
- |
Patrick Watson[^2]\
Minerva University\
[pwatson@minerva.kgi.edu]{}
bibliography:
- 'references.bib'
title: 'P2L: Predicting Transfer Learning for Images and Semantic Relations '
---
Introduction
============
Good machine learning quality often benefits from a large number of examples to capture a robust representation of the unknown input distribution [@caffe1] [@Kavzoglu2012]. Small data sets may not sufficiently sample the input space. However, in practice, small training jobs are common and labeled data is scarce in many domains. In a survey of industry visual recognition workloads the average number of images submitted by customers was only 250, and the average number of labels was 5 (see Section \[sec:realWorld\]).
To be clear, our goal is not cross-task transfer: our aim is to devise a heuristic for domain adaptation, for intra-task (such as image classification, or relationship prediction) cross-domain transfer, such as transfer from a classification model trained on a subset of ImageNet to a classification model for some unknown image classes.
*Inductive transfer* learning methods [@QiangYang2010], [@Weiss2016] have been identified as a possible solution to this problem. These methods use knowledge acquired in a “source” task to enhance the learning of a new “target” task. However, these methods commonly assume that there is a “best” transfer model, usually the model trained with the largest data set [@DBLP:journals/corr/RazavianASC14]. Yet this assumption stands in tension with results showing that while a well chosen source can improve performance significantly, a poorly chosen one can be worse than random initialization [@QiangYang2010] [@rosenstein2005transfer]. An open challenge remains: for fine-tuning of neural nets, how to predict the degree of transfer between different source and target domains prior to training.
In this work, we describe a method for identifying good transfer models [*prior*]{} to training that we validate in both the semantic relations and image domains. This is valuable since a general learning service must be prepared to train accurate models from widely varied target tasks automatically. Such a service must balance efficient training time and classification accuracy, precluding the exhaustive approach of fine-tuning all existing source models. At target task training time, P2L requires only a single forward pass of the target data set through a single reference model to identify the most likely candidate for fine-tuning.
Beginning with a single reference model (for images, VGG16 trained on ImageNet-1k, and for semantic relations PCNN), we generate feature vectors for each source dataset. We then use these models to characterize the similarity between source domain features and each target domain to select the source most likely to “donate” useful features, independent of the reference model. Using this metric, we estimate the similarity between a conceptual category of inputs, and each member of our family of classifiers. We then fine-tune a network for each combination of source and target to assess the degree to which each of the source models enhanced learning.
Related Work {#csofa}
============
The transfer learning literature explores a number of different topics and strategies such as few-shot learning [@bollegala2011relation] [@Socher2013], domain adaptation [@Patricia2014], weight synthesis [@Sussillo2012], and multi-task learning [@jiang2009multi] [@nguyen2016] [@torrey2010transfer]. Some works propose novel combinations of these approaches, yielding new training architectures and optimization objectives to improve transfer performance under conditions of domain transfer with limited or incomplete annotations [@transfer3].
**Representation transfer:** Several approaches have been tried to transfer robust representations based on large numbers of examples to new tasks. These transfer learning approaches share a common intuition [@Bengio2011]: that networks which have learned compact representations of a “source” task, can reuse these representations to achieve higher performance on a related “target” task. Different approaches use different techniques to transfer previous representations. *Instance-based* approaches attempt to identify appropriate data used in the source task to supplement target task training, *feature-representation* approaches attempt to leverage source task weight matrices, and *parameter-transfer* approaches involve re-using the architecture or hyper-parameters of the source network [@Dai2007] [@QiangYang2010]. These approaches, often supplemented by related small-data techniques such as bootstrapping, can yield improvements in performance [@Azizpour2016] [@Mou2016] [@nguyen2016] [@vgg16] [@transfer4]. One approach to transfer learning is to leverage existing deep nets trained on a large dataset, for example VGG16 [@DBLP:journals/corr/RazavianASC14] [@vgg16] for images, or PCNN [@zeng2015distant] for relation prediction. The trained weights in these networks have captured a representation of the input that can be transferred by fine-tuning the weights or retraining the final dense layer of the network on the new task.
While all these methods seek to improve performance on the target task by transfer from the source task, most assume there is only one source model, usually trained from ImageNet. [@ImageNet22k] Additionally, this approach involves a number of meta-learning decisions, although in general each change from the original source architecture tends to decrease resulting classification performance [@transfer4]. Meta-learning [@Lemke2015] is another approach for representation transfer. While meta-learning typically deals with training a base model on a variety of different learning tasks, transfer learning is about learning from multiple related learning tasks [@FinnAL17]. Efficiency of transfer learning depends on the right source data selection, whereas meta-learning models could suffer from ’negative transfer’ [@QiangYang2010] of knowledge if source and target domains are unrelated. Surprisingly, in image classification performance gains are commonly observed even in cases where initialization data appears visually and semantically different from the target dataset (such as ImageNet and Medical Imaging datasets).
The Learning to Transfer [@ying2018transfer] framework learns a *reflection function* that transforms feature vector representations to be more effectively classified by a kNN approach. Although it uses a model trained on ImageNet to produce the initial feature vectors, it is not a parameter-transfer method, since this model is not fine tuned on the target domains.
In contrast, for relation prediction, semantic dissimilarity between source and target task typically prevents effective transfer learning [@Mou2016], and so the semantic-relations transfer is more poorly explored. However, semantic relations can contain information that can support transfer, one approach used vectorized representations of semantic relations as an added source of information to support image-segmentation [@Myeong2013] [@Rohrbach2010].
**Fine-tuning with co-training:** Our approach is most similar to that of fine-tuning with co-training [@transfer4]. That method begins by using low-level features to identify images within a source dataset having similar textures to a target dataset, and concludes by using a multi-task objective to fine-tune the target task using these images. A related approach has been used to enhance performance and reduce training time in document classification [@Das2018] and to identify examples to supplement training data [@Ge] [@transfer4]. Our goal is to extend this approach to high-level features, and to domains outside computer vision to construct a more complete map of the feature space of a trained network. In this way our approach has some parallels with “learning to transfer” approaches [@DBLP:journals/corr/abs-1708-05629], which attempt to train a source model optimized for transfer rather than target accuracy.
**Taskonomy and Model-Recommendation:** When transferring information captured by previous task-learning for a new task, it is important to take into account the nature of both tasks. One promising approach involves use recommender systems which identify models with similar latent-space representations of labeled data. In an object-detection context [@wang2015model], this approach has been used to select likely candidates for inclusion in an ensemble model for object recognition. In multi-task visual learning, a model learned to estimate the similarity space of various visual tasks, to estimate the degree to which models trained to perform these tasks might contribute to transfer on a novel task [@DBLP:journals/corr/abs-1804-08328]. The current paper aims, in part, to combine the low computational cost of the former technique with the enhanced transfer performance of the latter by learning a novel method for selection among previously trained source models. However, our goal is not cross-task transfer - for example, we are not trying to transfer a model learned for a depth estimation task to a classification task. Although within a task type, (such as image classification) we do sometimes refer to a source task and target task, our aim is to devise a heuristic for domain adaptation, for intra-task, cross-domain transfer, such as transfer from a classification model trained on ImageNet to a classification model for Oxford Flowers. We show that P2L works for multiple domains and intra-task for 2 types of tasks: image classification and semantic relation prediction.
Methods
=======
Theoretical Framework {#methods:theory}
---------------------
This work addresses the problem of how to make an optimal choice of pre-trained network weights learned from source tasks $s_j \in S = (s_1, ..., s_M)$ for some target task $t_i \in T= (t_1, ..., t_N)$. Given a target task and dataset $t_i$, a model $M(t_i,s_j)$ is generated by first training on a source task and dataset $s_j$ , and then transferring this information to $t_i$, through mechanisms such as fine-tuning. For each pair $(t_i,s_j)$, performance improvement by transfer in each scenario can be measured:
$$I(t_i, s_j) = P(M(t_i,s_j)) - P(M(t_i, \phi))$$
where $P()$ is some defined performance evaluation (such as accuracy), $\phi$ represents the nil dataset (randomly initialized weights), and $I(t_i, s_j)$ is the measured performance improvement. Selecting the optimal $s_i$ would then trivially be achieved by:
$$\theta (t_i,S) = argmax_{s_j} I(t_i,s_j)$$
However, since training a model $M$ for all combinations of $(t_i,s_j)$ and $(t_i, \phi)$ is computationally expensive, we build $E(t_i,s_i)$ as an estimate of $I$, which could be used in its place to predict $\theta (t_i,S)$ more efficiently. In this work, we demonstrate the approach by simply defining $E(t_i,s_i)$ as
$$E(t_i,s_j) = G \left(A(F(t_i)),A(F(s_j)),|s_j| \right)$$
where $G()$ is an empirically-derived monotonically increasing goodness measure, and $A()$ is a statistical aggregation technique to combine sets of individual data instances $F()$ into vectors representing the entire dataset. As an example, $F(t_i)$ are a set of feature vectors over images contained in $t_i$, and $A(F(t_i))$ is simply the average over those feature vectors. As another example, $F(t_i)$ could be a set of SIFT features over images in the dataset, and $A(F(t_i))$ is a corresponding codebook histogram. Noting that performance of $G()$ should increase as the cardinality of the datasets increase, $z=|s_j|$ is the number of elements, or size, of dataset $s_j$. Specifically,
$$\begin{split}
G(x,y,z) = \left ( 1 - sig(\alpha_{d} (d(x,y)-\mu_{d} )/\sigma_{d}) \right ) \\
\cdot \left ( sig(\alpha_{z} (z -\mu_{z})/\sigma_{z}) \right )
\label{eq:predictionFunc}
\end{split}$$
![Image Deep Learning Pipeline[]{data-label="fig.vision_pipeline"}](figures/vision_impl.jpg){width="3in"}
where $d(x,y)$ represents the statistical dissimilarities in the datasets as measured by standard methods, such as KL or Jensen-Shannon divergence ([@js_distance] [@kullback1951]), or Chi-square or Euclidean distance. $(\mu_{d,z},\sigma_{d,z}$) are the mean and standard deviations of the dissimilarities and the source dataset size, respectively, and $\alpha_{d,z}$ are learned parameters (from $T_{tr},S_{tr}$) that change how quickly each term reaches saturation. $sig(x)$ is the logistic sigmoid function $sig(x) = 1 / (1 + exp(-x))$. Intuitively, the first term captures the negative effect of dissimilarity of the average feature vectors, while the second term captures the positive effect of dataset size. Both effects are normalized and bounded, and the central multiplication effectively “ANDs” them together. In practice, we have found that the KL-divergence measure works best, possibly because of its asymmetry. In order to evaluate the performance of a given approximation function $E(t_i,s_i)$ in comparison to some ground truth $I(t_i,s_i)$, $E(t_i,s_j)$ is learned by experimenting on a collection of target and source datasets $(T_{tr},S_{tr})$, and then evaluating on a held-out set of datasets $(T_{te},S_{te})$. In order to evaluate the performance of $E(t_i,s_j)$ in comparison to exhaustive ground truth, we measure both the Spearman’s $\rho$ of its choices, as well as the accuracy of its Top-1 choice.
While this work takes an engineering design approach to an approximation function $E(t_i,s_i)$, this framework paves the way for future works which may explicitly learn linear or non-linear functions to approximate $I$.
Implementation Details for Images {#impl_image}
---------------------------------
As described in Figure \[fig.vision\_pipeline\], We use the VGG16 model pre-trained on ImageNet-1k. For $F(\cdot)$, we extract the response of the penultimate full connection layer, a 4096-dimensional vector. For $A(\cdot)$ in a learning task with $k$ images, we extract $k$ such vectors $v_i$, compute their mean, $v_\mu$, and then L1-normalize this mean, giving $\overline{v_\mu}$ as the summary feature vector for this task. For $d(\cdot, \cdot)$, we compute one of several possible distance measures, smoothing any zero components by adding an appropriate $\epsilon$ value.
Implementation Details for Semantic Relations {#gen_inst}
---------------------------------------------
The task of relation prediction provides a second benchmark for source domain selection. In this task, a semantic relations base is extended with information extracted from text. We use the CC-DBP [@ccdbp] dataset: the text of Common Crawl[^3] and the semantic relations schema and training data from DBpedia [@dbpedia]. DBpedia is a knowledge graph extracted from the infoboxes from Wikipedia. An example edge in the DBpedia knowledge graph is [$\langle {\textsc{Larry McCray}}\ {\textit{genre}}\ {\textsc{Blues}} \rangle$]{}, meaning Larry McCray is a blues musician. This relationship is expressed through the DBpedia [*genre*]{} relation, a sub-relation of the high level relation [*isClassifiedBy*]{}. The relation prediction task is to predict the relations (if any) between two nodes in the knowledge graph from the entire set of textual evidence, rather than each sentence separately as in mention level relation extraction.
Figure \[fig.rkiarch\] shows the relation prediction neural architecture. The feature representations are taken from the penultimate layer, the max-pooled network-in-network. All models have the same architecture and hyperparameters, shown in Table \[tbl.rkihypers\]
![Deep Learning Architecture for Relation Prediction[]{data-label="fig.rkiarch"}](figures/knowledgeArchBasic.pdf){width="0.98\linewidth"}
\[tbl.rkihypers\]
----------------------------- --
**Hyperparameter & **Value\
word dim. & 50\
position dim. & 5\
sentence vector & 400\
NiN filters & 400\
CNN filters & 1000\
CNN filter width & 3\
dropout & 0.5\
****
----------------------------- --
Experimental Results and Analysis {#gen_inst}
=================================
Experimental Approach Images
----------------------------
ImageNet22k contains 21841 categories spread across hierarchical categories such as person, animal, fungus. We extracted some of the major hierarchies from ImageNet22k (Table \[fig.ImageNet22k\]) to form multiple source and target domains image sets for our evaluation. As the figure indicates, approx 9 million images were used. Some of the domains like animal, plant, person and food contained substantially more images (and labels) than categories such as weapon, tool, or sport. This skew is reflective of real world situations and provides a natural test bed for our method when comparing training sets of different size.
Each of these domains was then split into four equal partitions. One was used to train the source model, two were used to validate the source and target models, and the last was used for the transfer learning task. One-tenth of the fourth partition was used to create a transfer learning target. For example, the person hierarchy has more than one million images. This was split into four equal partitions of more than 250K each. The source model was trained with data of that size, whereas the target model was fine-tuned with one-tenth of that data size taken from one of the partitions. The smaller target datasets is reflective of real transfer learning tasks.
In this way, we generated 15 source workloads and 15 target training workloads. These source and target workloads were divided into two groups. One group, ($S_{tr}$,$T_{tr}$), consisting of sport, garment, fungus, weapon, plant, and animal as $S_{tr}$ and $T_{tr}$ was used to generate parameters for approximation function E in equation \[eq:predictionFunc\], as well as to determine which dissimilarity measure to use. A second held-out group, ($S_{te}$,$T_{te}$) consisting of furniture, food, person, nature, music, fruit, fabric, tool, and building as $S_{te}$ and $S_{te}$ were used to validate these parameters. The same identical parameters was also validated on 71 real world image classification tasks, Oxford Flowers dataset as well as for Semantic Relations.
The training of the source and target models was done using Caffe [@caffe1] using a ResNet-27 model [@resnet27]. The source models were trained using SGD [@SGD] for 900,000 iterations with a step size of 300,000 iterations and an initial learning rate of 0.01. The target models were trained with an identical network architecture, but with a training method with one-tenth of both iterations and step size. A fixed random seed was used throughout all training.
Experimental Approach Semantic Relations
----------------------------------------
We split the task of relation prediction into seven subtasks composed of the high-level relations with the most positive examples in the CC-DBP (other relations were discarded). This was intended to be mirror the partitions of ImageNet by high-level class. The seven source domains are shown in Table \[tbl.ccdbpSourceBinary\]. A model is trained for each of these domains on the full training data for the relevant relation types.
-------------------- ----------- --------------------
Division Name Number of Positives in Train
Relations
coparticipatesWith 227 78598
hasLocation 85 72065
sameSettingAs 169 40359
isClassifiedBy 34 22743
hasPart 64 12319
hasMember 45 36706
hasRole 4 7320
-------------------- ----------- --------------------
: Source domains division of CC-DBP for binary relation extraction[]{data-label="tbl.ccdbpSourceBinary"}
Our approach to transfer learning was the same as in images: a deep neural network trained on the source domain was fine-tuned on the target domain. Fine-tuning involves re-initializing and re-sizing the final layer, since different domains have different numbers of relations. The final layer is updated at the full learn rate $\alpha$ while the previous layers are updated at $f \cdot \alpha$, with $f < 1$. We used a fine-tune multiplier of $f = 0.1$.
A new, small training set is built for each target task. For each split of CC-DBP we take 20 positive examples for each relation from the full training set or all the training examples if there are fewer than 20. We then sample ten times as many negatives (unrelated pairs of entities). These form the target training sets. The model trained from the full training data of each of the different subtasks is then fine-tuned on the target domain. We measure the area under the precision/recall curve for each trained model. We also measure the area under the precision/recall curve for a model trained without transfer learning.
Results
-------
When training a model, a user commonly may (1) choose the source model trained with the largest amount of data (LTD), or (2) randomly choose a model from the basket of available models as a source for transfer learning, or (3) not use transfer learning at all but instead initialize the weights of the network randomly. We have used this to compare P2L across two domains : Images (ImageNet22k in 4.3.1, Oxford Flowers in 4.3.3, Real World Tasks in 4.3.5) and Semantic Relations (DBpedia in 4.3.2).
In summary, across 95 tasks in the above 4 contexts, P2L was able to pick a better source model on average. In contrast, the heuristic LTD (to choose the source model trained with the largest amount of data) was able to pick the best source in 55 cases only.
Table \[fig.gain\] shows the relative increase in final performance for our proposed method in comparison to each of these three methods, across ImageNet22k (in 4.3.1) and DBpedia (in 4.3.2). Our method selects the best dataset for transfer learning in all but one case. On average, we improve the accuracy over the next best method by 2 percent. While it is fair to say that the gain shown Table \[fig.gain\] is consistent but modest, we found it encouraging, and sought to test it further in 2 more independent experiments: the Oxford Flower dataset and on tasks sampled from a real-world, commercial classifier training service. For the validation on real world tasks (in 4.3.5), the largest source, Imagenet1K, was the best in only 35 of 71 cases. P2L picked better source models on average, boosting mean top-1 accuracy across the 71 real-world tasks from the public cloud service. We feel this result is the most significant of this work, since the real, “in-the-wild” classification tasks from the service had no guaranteed relationship to the ImageNet classification images used in Table \[fig.gain\]. Similarly for the Oxford Flower dataset (in 4.3.3), Imagenet1K was not optimal, and P2L identified a better data source.
### Validation on subsets of ImageNet22K
![Spearman’s $\rho$ for various measures[]{data-label="fig.spearmans_average"}](figures/average_spearman_rho.pdf){width="0.9\linewidth"}
\[fig.ImageNet22k\]
------------------------------------- --
**Domain & **% of Evaluated Images\
animal & 32%\
plant & 24%\
person & 13%\
food & 11%\
tool, building & 2% each\
sport, garment & 2% each\
nature, music & 2% each\
furniture & 2%\
fruit, fabric & 2% each\
fungus, weapon & 1% each\
****
------------------------------------- --
We tested distance measures based on Kullback-Leibler Divergence (KLD), Jensen-Shannon Divergence (JSD), Chi-square (Chi2), and Euclidean distance (ED). For each training task ($S_{tr}$,$T_{tr}$), we calculated the rank-correlation (Spearman’s $\rho$) between the predictions of each of these measures, and the ground-truth transfer performance based on top-1 classification accuracy.
Figure \[fig.spearmans\_average\] shows the average Spearman’s $\rho$ of the top-1 ground truth rank and our prediction rank as they varied with various $\alpha$ values of equation (\[eq:predictionFunc\]). For $\alpha$ in this interval, the KLD based measure is most sensitive, and we use it exclusively for evaluations with $\alpha_d$ = 1 and $\alpha_z$ = 4. The parameter $\alpha_d$ was fixed at 1, since what is important is the ratio of $\alpha_z$ to $\alpha_d$. For these parameters, the average Spearman’s $\rho$ for the transfer learning task is 0.83. The gains from our prediction method are shown in Table \[fig.gain\].
The parameters $\alpha_d$ and $\alpha_z$ which were formulated as described above on the 6x6 Imagenet22K training set ($S_{tr}, T_{tr}$), were subsequently used for validation on the 9x9 Imagenet22K (in 4.3.1), real world image classification tasks (in 4.3.4), Oxford flowers (in 4.3.3), as well as dbpedia (in 4.3.2) datasets. The set size of ($S_{tr}, T_{tr}$) can be increased from the current 6x6. But even with the current size it shows the potential of P2L.
This parameter selection of $\alpha_d$ and $\alpha_z$ is essentially offline, and only needs to be done once to pick the parameter values. It required 30 custom training jobs. All subsequent predictions for the 9x9 Imagenet22K, real work image classification tasks, Oxford flowers, as well as dbpedia, did not require further training.
---------- ------------ -- ----------- ------------
Target Spearman’s Target Spearman’s
Dataset $\rho$ Dataset $\rho$
Fabric 0.976 Food 0.833
Nature 0.952 Fruit 0.762
Person 0.929 Tool 0.667
Music 0.929 Furniture 0.595
Building 0.905
---------- ------------ -- ----------- ------------
: Spearman’s $\rho$ for predictions vs ground truth for transfer learning on images[]{data-label="fig.spearmans_for_measures"}
### Validation on Common Crawl - DBpedia
Figure \[fig.ccdbpBinaryCS\] shows the correlation of the prediction $E(t_i,s_j)$ with the improvement $I(t_i,s_j)$, when using KLD in addition to size of the source domains’ training set in $E$. Figure \[fig.ccdbpBinaryS\] shows the same when only size is used. Using the estimator produced better predictions, that is, $E(t_i,s_j)$ and $I(t_i,s_j)$ were then better correlated (Spearman’s $\rho$ = .763, Pearson’s $R$ = .823).
-------- ---------------- ------------------------------------------------------ ------------------ ---------------- ---------------------- --
Domain Target Dataset P2L Picked Largest Training Random Dataset No Transfer
$t_i$ Best Dataset Selection Learning
Dataset ? (OP-LTD)/LTD (OP-RDS)/RDS (OP-A$\phi$)/A$\phi$
Images Fruit **Yes & 0.18 & 0.59 & 1.00\
&Fabric & **Yes & 0.00 & 0.32 & 0.67\
& Building &**Yes & 0.00 & 0.36 & 0.63\
&Music &**Yes & 0.00 & 0.25 & 0.53\
&Nature & **Yes & 0.00 & 0.21 & 0.42\
&Food & **Yes & 0.00 & 0.37 & 0.31\
&Tool & **Yes & 0.00 & 0.12 & 0.25\
&Furniture & **Yes & 0.00 & 0.22 & 0.22\
&Person & **Yes & 0.00 & 0.13 & 0.25\
Semantic & hasPart & **Yes & 0.15& 0.13 & 0.40\
Relations & copartWith & **Yes & 0.00 & 0.14 & 0.34\
& sameSettingAs & **Yes & 0.00& 0.17 & 0.30\
& hasLocation & **Yes & 0.00& 0.09 & 0.14\
& hasMember & **Yes & 0.00 & 0.07 & 0.14\
& hasRole & **Yes & 0.00 & 0.01 & 0.13\
& isClassifiedBy & No & -0.01& 0.08 & 0.08\
******************************
-------- ---------------- ------------------------------------------------------ ------------------ ---------------- ---------------------- --
### Validation on Oxford Flower 102 Dataset
We also evaluated fine tuning the Oxford Flower 102 [@oflower] dataset using P2L and compared it to other methods on the ResNet27 architecture using the same training regime as in section 4.1. The dataset contains 102 commonly occurring flower types each with only 10 training images per class. Of the 16 source candidates, including ImageNet1k, P2L predicted plants. Intuitively, this is because of the strong visual resemblance of many plants and flowers. Experimental evaluation validated this prediction: fine tuning with plants as the source produced a top-1 accuracy of 91.6% accuracy in comparison to 85.12% accuracy for ImageNet1k.
### Validation on Real World Image Classification Tasks {#sec:realWorld}
To provide a practical test of P2L, we obtained data for 71 training tasks that were submitted to a commercial machine learning service, by users of the service who had allowed their data to be used for research. This service takes images with labels as input, and produces a classifier via supervised learning. Our goal was to validate the prediction made by equation \[eq:predictionFunc\] on real world-data, by selecting the single most appropriate source model from the collection of 16 candidates generated in our transfer learning experiment, and then fine-tuning that candidate for each of the 71 target jobs. We validated our prediction method by exhaustively fine-tuning for each of the 1,136 possible source-target pairs. We assume that for efficiency at classification time it is necessary to select the [*single*]{} best source model instead of using an ensemble of multiple source models.
For our experiments, we randomly split each set of images with labels into 80% for fine-tuning and 20% for validation. For these 71 training sets, we had a total of approximately 18,000 images: an average of 204 training images and 50 held-out validation images each. There were 5.2 classes per classifier on average, with a range of 2 to 60 classes per classifier. We used 14 models trained from sub-domains of ImageNet as possible source models (listed in table \[fig.ImageNet22k\], excluding “music”), plus a variant of the “animal” source model which was trained with twice as many examples. We also used a “standard” model trained on all of the ImageNet-1K training data. We used the same ResNet-27 architecture [@resnet27] described above. Due to the small size of target domain data, we set the learning rate to 0 for the convolutional layers, and otherwise used a learning rate of 0.01 for 40 epochs. We ranked the performance of each of these models by top-1 accuracy using the 20% held out data.
Based on a manual inspection of the classifier labels for the 71 target jbos, we found a wide variety of domains, with the largest (animals) representing only about 14% of the total. This high level of variety appears common in real-world learning service scenarios, since users are training custom classifiers to address problems for which ready-made models don’t exist.
---------------- ---------------------------------- -- ----- --
Domain
P2L (ours) LTD
Oxford Flowers **91.6 & & 85.1\
Real World & **79.3 & & 78.1****
---------------- ---------------------------------- -- ----- --
: Results on Non-ImageNet Data []{data-label="fig.customoxford"}
### Results of Using P2L for Real World Tasks
Compared to the most robust baseline we identified in our experimental results (i.e., using ImageNet-1k as the source model for every target), our method was able to enhance the performance of target learning jobs. For our sample of 71 tasks, the P2L method provided a top-1 image classification accuracy of 79.3% compared to an ImageNet-1k baseline result of 78.1%. ImageNet-1k was the optimal baseline in 35 out of the 71 cases, but in the 36 remaining tasks P2L was able on average to identify a more effective source model. This increase was most often driven by the selection of the [*food*]{} and [*person*]{} source models. We speculate that variation within these domains may not be well captured by ImageNet-1k (which contains relatively few labeled examples of people or food), or that the target task may rely on a very specific feature domain. These findings are summarized in table \[fig.customoxford\].
### Comparing against merged source datasets
We have investigated how a merged dataset of various source domains could do in comparison to its individual components. While it may seem that a single merged dataset would perform as well or better than individual sources, in reality we have noticed results to the contrary. For example, we trained a custom learning workload for “car”, using the real world image classification dataset (in 4.3.4). Using the “weapon” dataset as a source delivered 87.5% accuracy. But combining the “music” and “tool” datasets with “weapon” actually reduced final accuracy to 73%. This combination was chosen since “music”, “tool”, and “weapon” are the three most convergent datasets.
The likely reason for this is that merging datasets, without consideration for the semantic similarity of labels results in confusion for the neural network. For example, images of “knife” are part of both the weapon and tool datasets labels. By merging these two, the training process has to differentiate between two labels which are similar and does not learn much.
Future Work
===========
The current P2L approach estimates transfer performance at the level of large conceptual categories (e.g., “animal”, or “location”). However, large labeled data sets, such as those used in ImageNet-1k, contain deep hierarchies (e.g., animal $\rightarrow$ mammal $\rightarrow$ cat $\rightarrow$ cheetah) that may help to characterize finer resolution maps of the feature space. Identifying crucial sub-features can assist further in selecting more specific source categories, and in developing more efficient source models and transfer techniques.
We currently use one modality in isolation for determining which source model to use. However,there are a lot of information besides the image (or semantic relation) which could additionally aid in determining a good source model. Like accompanying text or audio feed etc. Bringing in these multi modal aspects could enhance the accuracy of prediction. For example, blight is a crop disease and crops are likely to occur in a plant dataset than any other dataset. If one can determine these links from external datasets, it would help zero in on a good dataset and choose the best especially when there are two or more close candidates. Extracting tags from the images or using other available information and using them to find out semantically closest source categories from a large knowledge graph can yield substantial improvement in image recognition.
Additionally,we have currently proven our methods with images and knowledge. We propose to enhance it for temporal domains like machine translation and video. We also propose to investigate refining and simplify our method and improving its understand-ability.
Conclusion
==========
We described an efficient method for using a small data sample to select and fine-tune a candidate from a family of pre-trained models, applicable to both the image and semantic relations. We conducted an empirical test of the method using models trained on specific conceptual categories across images and semantic relations, demonstrating improved transfer learning results, outperforming baselines such as picking the model trained with the largest data set, or using a common industry standards such as a model based based on ImageNet-1k. These findings suggest that a learned representation from previous tasks can be used to select the best transfer candidate, and to provide greater transfer learning.
Despite order of magnitude differences in training set sizes, we were able to obtain transfer gains by computing an estimate of conceptual closeness. Although prior work has described a saturating curve for training set size contributions to accuracy [@Kavzoglu2012]–which we also observed in our data–we showed that feature similarity provided transfer benefits not predicted by set-size alone. Our method is efficient at training and classification time, and has been shown to improve accuracy versus the baseline, on publicly available image and semantic relations datasets as well as on real-world datasets, and across a wide range of task sizes. These results help to explain the tension in the literature between results showing that larger datasets usually outperform smaller [@DBLP:journals/corr/RazavianASC14], but that ill-selected transfer models can degrade performance [@rosenstein2005transfer]. We suggest that rather than there being a single “best” transfer model, transfer performance critically depends upon the similarity between the source and target models. Further, methods such as P2L can map the degree of overlap between disparate tasks to select more optimal models and and enhance transfer learning performance. Exploring these “maps” of feature space similarities could be a valuable future direction for machine learning research.
[^1]: Author is also affiliated with IBM Research.
[^2]: Work done when author was with IBM Research.
[^3]: <http://commoncrawl.org>
|
---
abstract: 'We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as the Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral theory of operators in Hilbert space, to optimization, to large sparse systems, to iterated function systems (IFS), and to fractal harmonic analysis. We present a new recursive iteration scheme involving as input a prescribed sequence of selfadjoint projections. Applications include random Kaczmarz recursions, their limits, and their error-estimates.'
address:
- '(Palle E.T. Jorgensen) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A.'
- '(Myung-Sin Song) Department of Mathematics, Southern Illinois University Edwardsville, Edwardsville, IL 62026, U.S.A.'
- '(Feng Tian) Department of Mathematics, Hampton University, Hampton, VA 23668, U.S.A.'
author:
- Palle Jorgensen
- 'Myung-Sin Song'
- Feng Tian
bibliography:
- 'ref.bib'
title: 'A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spaces'
---
\[sec:Intro\]Introduction
=========================
In this paper, we consider certain infinite products of projections. Our framework is motivated by problems in approximation theory, in harmonic analysis, in frame theory, and the context of the classical Kaczmarz algorithm [@K-1937]. Traditionally, the infinite-dimensional Kaczmarz algorithm is stated for sequences of vectors in a specified Hilbert space $\mathscr{H}$, (typically, $\mathscr{H}$ is an $L^{2}$-space.) We shall here formulate it instead for sequences of projections. As a corollary, we get explicit and algorithmic criteria for convergence of certain *infinite products of projections* in $\mathscr{H}$.
***Organization and main results.***
Our first two sections outline a certain frame-harmonic analysis. This is the immediate focus of our present applications, but our main results, dealing with general projection valued Kaczmarz algorithms, we believe, are of independent interest. They include (products of projections,) and its related results, Corollaries \[cor:fi\], \[cor:gf\], \[cor:kg\], and \[cor:ke\]. The connection between infinite products of projections, on the one hand, and more classical Kaczmarz recursions (for frames), on the other, is spelled out in Corollaries \[cor:ke\] and \[cor:ke2\]. Our main result for *random* Kaczmarz algorithms is , combined with . In the remaining three sections, we return to applications, iterated function system, fractals, and random power series.
Our extension of the Kaczmarz algorithm to sequences of projections is highly nontrivial: While in general convergence questions for infinite products of projections (in Hilbert space) is difficult (see e.g., [@MR0051437; @MR647807; @MR2129258; @MR3796644]), we show that our projection-valued formulation of Kaczmarz’ algorithm yields an answer to this convergence question; as well as a number of applications to stochastic analysis, and to frame-approximation questions in the Hilbert space $L^{2}\left(\mu\right)$, where $\mu$ is in a class of *iterated function system* (IFS) measures (see [@MR625600; @MR1656855; @MR2319756; @2016arXiv160308852H; @MR3800275]). The latter refers to a precise multivariable setting, and the class of measures $\mu$ we consider are fractal measures. (The notion of fractal is defined here relative to the rank $d$ of the ambient Euclidean space $\mathbb{R}^{d}$ for the particular IFS measure $\mu$ under consideration.) Indeed, our measures $\mu$ will be singular relative to the Lebesgue measure on $\mathbb{R}^{d}$. In addition to singularity questions for $\mu$ itself, one must also consider properties of the marginal measures for $\mu$, and the corresponding slice-direct integral decompositions. Our first two applications will be the IFS-measures for the Sierpinski gasket and the Sierpinski carpet, so $d=2$.
In the next section, we introduce this family of measures $\mu$, called slice-singular measures. We then turn to our Kaczmarz algorithm for sequences of projections, and its applications.
\[sec:SSM\]Slice-singular measures
==================================
The purpose of the current paper is to perform a systematic analysis of fractal measures embedded in higher dimensions $d$, such as Sierpinski triangles ($d=2$), and higher dimensional analogues, $d>2$. The analysis for $d=1$ begins with the following variant of the F&M Riesz theorem:
Consider a choice of period interval, $\left[0,1\right]$, or $\left[-\pi,\pi\right]$, a positive finite measure $\mu$ with support in the chosen period interval; and the usual Fourier frequencies realized as complex exponentials $e_{n}$, $n\in\mathbb{Z}$. Set $\mathbb{N}_{0}=\left\{ 0\right\} \cup\mathbb{N}$.
\[thm:FM\]The subset $\left\{ e_{n}\mid n\in\mathbb{N}_{0}\right\} $ is total in $L^{2}\left(\mu\right)$ if and only if $\mu$ is singular with respect to Lebesgue measure.
The corresponding result is false when $d>1$, and the question is: What is a natural extension of F&M Riesz’ theorem to higher dimensions, modeling the above formulation? One of the motivations for this is a certain construction of frame algorithms in $L^{2}\left(\mu\right)$; in the form started for $d=1$ in [@MR2319756; @2016arXiv160308852H; @MR3796641; @HERR2018]. For general frame theory, including projection valued frames, see e.g., [@MR2147063; @MR2367342; @MR2362796; @MR3423689; @MR3611473; @MR3817340; @MR3894265; @MR3906284; @MR3910931; @MR3778680; @MR3896128].
does *not* extend to 2D, or higher dimensions. In 1D, the standard F&M Riesz theorem is used at a crucial point; but there is *not* a direct extension of the theorem in one variable. To get a harmonic analysis of $L^{2}\left(\mu\right)$, with $supp\left(\mu\right)\subset\mathbb{R}^{d}$, $d\geq2$, one must assume instead that $\mu$ is *slice singular*; see . It is possible to view the result as an extension of F&M Riesz’ theorem to higher dimensions.
For the sake of stressing the idea, we shall consider the case $d=2$ in most detail.
Let $\left(X,\mathscr{F}\right)$ be a measurable space. $\mathcal{M}\left(X\right)$ denotes all Borel measures on $\mathscr{F}$. The set $\mathcal{M}^{+}\left(X\right)$ consists of all positive measures in $\mathcal{M}\left(X\right)$, and $\mathcal{M}_{1}^{+}\left(X\right)$ the subset of probability measures. We shall also use standard multi-index notations.
Let $\left(X\times Y,\mathscr{B}_{X}\times\mathscr{B}_{Y},\mu\right)$ be a measure space, where $X$, $Y$ are equipped with $\sigma$-algebras $\mathscr{B}_{X}$, $\mathscr{B}_{Y}$ respectively, and $\mu$ is defined on the *product $\sigma$-algebra*.
Every positive measure $\mu$ on $X\times Y$ w.r.t. the product $\sigma$-algebra yields a unique representation as follows:
1. $\xi:=\mu\circ\pi_{X}^{-1}$ is a measure on $\left(X,\mathscr{B}_{X}\right)$;
2. \[enu:dis21\]There exists a conditional measure $\sigma^{x}\left(dy\right):=\sigma\left(x,dy\right)$ on $\left(Y,\mathscr{B}_{Y}\right)$, defined for a.a. $x\in X$, such that $$d\mu=\int\sigma^{x}\left(dy\right)d\xi\left(x\right).\label{eq:A1}$$
The precise meaning of (\[eq:A1\]) is as follows: For all measurable functions $F$ on $X\times Y$, we have $$\iint_{X\times Y}Fd\mu=\int_{X}\left(\int_{Y}F\left(x,y\right)\sigma^{x}\left(dy\right)\right)d\xi\left(x\right).\label{eq:A2}$$ The decomposition (\[eq:A2\]) is often referred to as a *Rohlin disintegration formula*.
\[def:SS\]A Borel measure $\mu$ on $J^{2}:=\left[0,1\right]\times\left[0,1\right]$ is called *slice singular* iff (Def.)
1. $\xi=\mu\circ\pi_{1}^{-1}$ is singular; and
2. for a.a. $x$ w.r.t. $\xi$, the measure $\sigma^{x}\left(\cdot\right)$ is singular.
“Singular” is defined relative to Lebesgue measure.
\[thm:SM\]If $\mu$ is slice singular on $J^{2}$, then $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}^{2}}$ has dense span in $L^{2}\left(\mu\right)$, where $e_{n}\left(x\right)=e^{i2\pi\left(n_{1}x_{1}+n_{2}x_{2}\right)}$, for all $n=\left(n_{1},n_{2}\right)\in\mathbb{N}_{0}^{2}$, and $x=\left(x_{1},x_{2}\right)\in J^{2}$.
We shall show that, if $\left\langle F,e_{n}\right\rangle _{L^{2}\left(\mu\right)}=0$, $\forall n\in\mathbb{N}_{0}^{2}$, then $F=0$ $\mu$-a.e. But $$\begin{aligned}
\left\langle F,e_{n}\right\rangle _{L^{2}\left(\mu\right)} & =\int_{0}^{1}e_{n_{1}}\left(x\right)\left(\int_{0}^{1}e_{n_{2}}\left(y\right)\overline{F\left(x,y\right)}\sigma^{x}\left(dy\right)\right)d\xi\left(x\right)\nonumber \\
& =0,\;\forall n=\left(n_{1},n_{2}\right)\in\mathbb{N}_{0}^{2}\nonumber \\
& \Downarrow\quad\left(\text{since \ensuremath{\xi} is singular}\right)\label{eq:fm1}\\
\int_{0}^{1}e_{n_{2}}\left(y\right) & \overline{F\left(x,y\right)}\sigma^{x}\left(dy\right)=0,\;a.a.\;x,\;\forall n_{2}\in\mathbb{N}_{0}\nonumber \\
& \Downarrow\quad\left(\text{since \ensuremath{\sigma^{x}\left(\cdot\right)} is singular a.a. \ensuremath{x}}\right)\label{eq:fm2}\\
F\left(x,y\right) & =0,\;a.a.\;\left(x,y\right)\;\text{w.r.t. }\mu.\nonumber \end{aligned}$$
This gives the desired conclusion that $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}^{2}}$ is total in $L^{2}\left(\mu\right)$.
$\mu\in\mathcal{M}^{+}\left(\mathbb{T}^{2}\right)$, $W=$ Sierpinski gasket/carpet ().
Note that, for a.a. $x$ w.r.t. $\xi$, the measure $\sigma^{x}$ on $A\left(x\right)=\left\{ y\mid\left(x,y\right)\in W\right\} $ is a fractal measure with variable gap size; and by Kakutani’s theorem, for a.a. $x$, $\sigma^{x}\left(dy\right)$ is singular relative to the Lebesgue measure. Hence we can apply F&M Riesz as in (\[eq:fm1\]), and (\[eq:fm2\]).
The detailed properties of the fractals from (A) and (B) will be derived in below.
------------------------------------------------------------------------------------------ -- -- -- ------------------------------------------------------------------------------------------
![\[fig:2df\]Examples of slice singular measures.](sp1 "fig:"){width="0.25\columnwidth"} ![\[fig:2df\]Examples of slice singular measures.](sp2 "fig:"){width="0.25\columnwidth"}
\(A) Sierpinski gasket \(B) Sierpinski carpet
------------------------------------------------------------------------------------------ -- -- -- ------------------------------------------------------------------------------------------
While the Sierpinski constructions in are better known as self-affine planar sets, it is in fact the corresponding *measures* which are important for algorithms and for frame-harmonic analysis. As it turns out, the particular affine maps (see (\[eq:F1\]), (\[eq:F2\]), and below) going into the Sierpinski constructions are in fact special cases of a more general family of iterated function systems (IFS.) They are discussed in detail in sections \[sec:gifs\] and \[sec:Sie\] below. Brief preview: Given a system of contractive mappings, affine or conformal, there are then two associated fixed-point problems, one for compact sets, and the other for probability measures: The case of the sets $W$ is discussed in (\[eq:d9\]), and the measures $\mu$ in (\[eq:d7\]). For a fixed IFS, the set in question arises as the support of an associated IFS-measure $\mu$. Probabilistic features of these constructions are outlined in sect \[sec:gifs\], and their fractal properties, in sect \[sec:Sie\], below. In particular, we show that these planar Sierpinski measures $\mu$ are slice-singular.
Frames, projections, and Kaczmarz algorithms
============================================
While earlier approaches to the Kaczmarz algorithm in Hilbert space have dealt with recursive constructions of vectors, as needed in optimization problems, or in harmonic analysis, we present here an extension of the algorithm to the context of countable systems of *selfadjoint projections* in a Hilbert space. As outlined in subsequent sections of our paper, the projection setting is motivated directly by applications; the *randomized* Kaczmarz algorithms, just one of them.
For the benefit of readers, and for later reference, we include below a brief review of fundamentals for the classical Kaczmarz algorithm, and its variants. This also gives us a suitable framework for our present results: An operator theoretic extension of Kaczmarz, with applications to multivariable fractal measures.
*Literature guide*: In addition to Kaczmarz’ pioneering paper [@K-1937], there are also the following more recent developments of relevance to our present discussion [@MR1898684; @MR2208766; @MR2140451; @MR2263965; @MR2311862; @MR2721177; @MR2835851; @MR3117886; @MR3159297; @MR3450541; @MR3439812; @MR3796634; @MR3846956; @MR3896982], as well as [@2016arXiv160308852H; @MR3796641; @HERR2018].\
The classical Kaczmarz algorithm is an iterative method for solving systems of linear equations, for example, $Ax=b$, where $A$ is an $m\times n$ matrix.
Assume the system is consistent. Let $x_{0}$ be an arbitrary vector in $\text{\ensuremath{\mathbb{R}^{n}}}$, and set $$x_{k}:=\operatorname*{\textit{argmin}}_{\left\langle a_{j},x\right\rangle =b_{j}}\left\Vert x-x_{k-1}\right\Vert ^{2},\;k\in\mathbb{N};\label{eq:C1}$$ where $j=k\mod m$, and $a_{j}$ denotes the $j^{th}$ row of $A$. At each iteration, the minimizer is given by $$x_{k}=x_{k-1}+\frac{b_{j}-\left\langle a_{j},x_{k-1}\right\rangle }{\left\Vert a_{j}\right\Vert ^{2}}a_{j}.\label{eq:C2}$$ That is, the algorithm recursively projects the current state onto the hyperplane determined by the next row vector of $A$.
There is a stochastic version of (\[eq:C2\]), where the row vectors of $A$ are selected randomly [@MR2500924]. Also see Sections \[subsec:rkac\] and \[subsec:axy\] below.
Following standard conventions in approximation theory, we use the notation *argmin* for denoting the vector which realizes a specified optimization; in this case (see ), we refer to the minimum problem on the right hand side in eq (\[eq:C1\]). So in the particular instance of the Kaczmarz algorithm (\[eq:C2\]), we are in finite dimensions, and there is then an easy, geometric, and explicit formula for the argmin vector occurring in each step of the algorithm, see .
The Kaczmarz algorithm can be formulated in the Hilbert space setting as follows:
Let $\left\{ e_{j}\right\} _{j\in\mathbb{N}_{0}}$ be a spanning set of unit vectors in a Hilbert space $\mathscr{H}$, i.e., $span\left\{ e_{j}\right\} $ is dense in $\mathscr{H}$. For all $x\in\mathscr{H}$, let $x_{0}=e_{0}$, and set $$x_{k}:=x_{k-1}+e_{k}\left\langle e_{k},x-x_{k-1}\right\rangle .\label{eq:C3}$$ We say the sequence $\left\{ e_{j}\right\} _{j\in\mathbb{N}_{0}}$ is *effective* if $\left\Vert x_{k}-x\right\Vert \rightarrow0$ as $k\rightarrow\infty$, for all $x\in\mathscr{H}$.
A key motivation for our present analysis is an important result by Stanisław Kwapień and Jan Mycielski [@MR2263965], giving a criterion for stationary sequences (referring to a suitable $L^{2}\left(\mu\right)$) to be effective.
**Observation.** Equation (\[eq:C3\]) yields, by forward induction: $$\begin{aligned}
x-x_{k} & = & \left(1-P_{k}\right)\left(x-x_{k-1}\right)\\
& = & \left(1-P_{k}\right)\left(1-P_{k-1}\right)\left(x-x_{k-2}\right)\\
& \vdots\\
& = & \left(1-P_{k}\right)\left(1-P_{k-1}\right)\cdots\left(1-P_{0}\right)x,\end{aligned}$$ where $P_{j}$ is the orthogonal projection onto $e_{j}$.
[>p[0.45]{}>p[0.45]{}]{} ![\[fig:kap\] Solution to $Ax=b$ by the Kaczmarz algorithm, with $a_{1}=\left(\cos\left(\pi/3\right),\sin\left(\pi/3\right)\right)$, $a_{2}=\left(\cos\left(0.1\right),\sin\left(0.1\right)\right)$, and $b=\left(1,2\right)$.](kac1 "fig:"){width="0.35\columnwidth"} & ![\[fig:kap\] Solution to $Ax=b$ by the Kaczmarz algorithm, with $a_{1}=\left(\cos\left(\pi/3\right),\sin\left(\pi/3\right)\right)$, $a_{2}=\left(\cos\left(0.1\right),\sin\left(0.1\right)\right)$, and $b=\left(1,2\right)$.](kac2 "fig:"){width="0.35\columnwidth"}[\
]{}(A) Approximate solution; random starting point $x_{0}$ & (B) orthogonality relation
$\left\Vert x_{k-1}-x\right\Vert ^{2}=\left\Vert x_{k-1}-x_{k}\right\Vert ^{2}+\left\Vert x_{k}-x\right\Vert ^{2}$[\
]{}
Algorithms, and products of projections
---------------------------------------
We now present an extension of the Kaczmarz algorithm; an extension to a setting of an infinite sequence of selfadjoint projections, as opposed to the classical case of sequences of vectors in Hilbert space. There are more general results on limits of iterated products of selfadjoint projections. See [@MR3796644] and also [@MR0051437; @MR647807; @MR2129258]. For applications of infinite products of operators to central problems in mathematical physics, see e.g., papers by D. Ruelle et al [@MR0284067; @MR534172; @MR647807].
**Preliminaries**
Let $\mathscr{H}$ be a Hilbert space. An operator $P:\mathscr{H}\rightarrow\mathscr{H}$ is said to be a *selfadjoint projection* iff (Def.) $P=P^{*}=P^{2}$. It is known that there is a bijective correspondence between:
1. all closed subspaces $\mathscr{M}\subset\mathscr{H}$; and
2. the set of all selfadjoint projections $P$.
If $\mathscr{M}$ is as in (i), then $P$ may be obtained from the axioms for $\mathscr{H}$; and we have $$P\mathscr{H}=\mathscr{M}=\left\{ x\in\mathscr{H}\mathrel{;}Px=x\right\} .\label{eq:cp1}$$ Conversely, if $P$ is given as in (ii), then $\mathscr{M}$ (see (\[eq:cp1\])) is a closed subspace in $\mathscr{H}$.
The ortho-complement $$\mathscr{M}^{\perp}:=\mathscr{H}\ominus\mathscr{M}=\left\{ x\in\mathscr{H}\mathrel{;}Px=0\right\}$$ is the closed subspace corresponding to the selfadjoint projection $P^{\perp}:=1-P$. (Here, we denote the identity operator in $\mathscr{H}$ by $1$, as it is the unit in the $C^{*}$-algebra $\mathscr{B}\left(\mathscr{H}\right)$.)
For our present purpose, all projections will be assumed selfadjoint. On occasion, to save space, we shall simply say projection when selfadjointness is implicit. (We note that selfadjoint projections yield orthogonal sum-splittings, and are therefore often, equivalently, referred to as orthogonal projections.)
We shall further make use of the *lattice operations* corresponding to the correspondence (i)$\leftrightarrow$(ii) above:
If $\mathscr{M}_{i}$, $i=1,2$, are closed subspaces with corresponding projections $P_{i}$, $i=1,2$; then TFAE: $$\begin{aligned}
\mathscr{M}_{1} & \subseteq\mathscr{M}_{2},\;\text{and}\\
P_{1} & =P_{1}P_{2}.\end{aligned}$$ Moreover, for a pair of projections $\left\{ P_{i}\right\} _{i=1,2}$, TFAE: $$\begin{aligned}
P_{1} & = & P_{1}P_{2}\\
& \Updownarrow\\
P_{1} & = & P_{2}P_{1}\\
& \Updownarrow\\
\left\Vert P_{1}x\right\Vert & \leq & \left\Vert P_{2}x\right\Vert ,\;\forall x\in\mathscr{H}\\
& \Updownarrow\\
\left\langle x,P_{1}x\right\rangle & \leq & \left\langle x,P_{2}x\right\rangle ,\;\forall x\in\mathscr{H}.\end{aligned}$$
**Caution:** In general, the class of selfadjoint projections is *not* closed under products, under sums, or under differences.
\[thm:kac\]Let $\left\{ P_{j}\right\} _{j\in\mathbb{N}_{0}}$ be a system of selfadjoint projections in a Hilbert space $\mathscr{H}$. For all $n\in\mathbb{N}_{0}$, set $$\begin{aligned}
T_{n} & =\left(1-P_{n}\right)\left(1-P_{n-1}\right)\cdots\left(1-P_{0}\right),\;\text{and}\label{eq:f8}\\
Q_{n} & =P_{n}\left(1-P_{n-1}\right)\cdots\left(1-P_{0}\right),\quad Q_{0}=P_{0}.\label{eq:f2}\end{aligned}$$ Then, $$\begin{aligned}
1-T_{n}^{*}T_{n} & =\sum_{j=0}^{n}Q_{j}^{*}Q_{j},\;\text{and}\label{eq:cc6}\\
1-T_{n} & =\sum_{j=0}^{n}Q_{j}.\label{eq:cc7}\end{aligned}$$
The operator products introduced in formulas (\[eq:f8\]) and (\[eq:f2\]) above will play an important role in our subsequent considerations. Hence, when we refer to $Q_{n}$, and $T_{n}$, we shall mean the particular operator products in (\[eq:f8\]) and (\[eq:f2\]). The input in our algorithm will be a fixed system of selfadjoint projections, $P_{n}$.
Note that the factors making up the operator products in (\[eq:f8\]) and (\[eq:f2\]) are non-commuting. We stress that non-comutativity is an important (and subtle) feature of the theory of operator frames; see e.g., [@MR3642406].
One checks that $T_{n}=T_{n-1}-Q_{n}$, so that $$\begin{aligned}
T_{n}^{*}T_{n} & = & \left(T_{n-1}^{*}-Q_{n}^{*}\right)\left(T_{n-1}-Q_{n}\right)\\
& = & T_{n-1}^{*}T_{n-1}-T_{n-1}^{*}Q_{n}-Q_{n}^{*}T_{n-1}+Q_{n}^{*}Q_{n}\\
& = & T_{n-1}^{*}T_{n-1}-Q_{n}^{*}Q_{n}-Q_{n}^{*}Q_{n}+Q_{n}^{*}Q_{n}\\
& = & T_{n-1}^{*}T_{n-1}-Q_{n}^{*}Q_{n}\\
& = & T_{n-2}^{*}T_{n-2}-Q_{n-1}^{*}Q_{n-1}-Q_{n}^{*}Q_{n}\\
& \vdots\\
& = & 1-P_{0}-\sum_{j=1}^{n}Q_{j}^{*}Q_{j}\\
& = & 1-\sum_{j=0}^{n}Q_{j}^{*}Q_{j}.\end{aligned}$$ Since $Q_{n}=T_{n-1}-T_{n}$, so $$\begin{aligned}
\sum_{j=0}^{n}Q_{j} & =Q_{0}+\left(T_{0}-T_{1}\right)+\left(T_{1}-T_{2}\right)+\cdots+\left(T_{n-1}-T_{n}\right)\\
& =P_{0}+1-P_{0}-T_{n}=1-T_{n}.\end{aligned}$$
Let $\mathscr{H}$ be a Hilbert space, and let $\left\{ A_{n}\right\} _{n\in\mathbb{N}}$ be a sequence of bounded operators in $\mathscr{H}$, i.e., $A_{n}\in\mathscr{B}\left(\mathscr{H}\right)$, $\forall n\in\mathbb{N}$. We shall need the following two notions of convergence in $\mathscr{B}\left(\mathscr{H}\right)$.
1. We say that $A_{n}\rightarrow0$ in the *strong operator topology* (SOT) iff (Def.) $\lim_{n\rightarrow\infty}\left\Vert A_{n}x\right\Vert =0$ for all vectors $x\in\mathscr{H}$.
2. We say that $A_{n}\rightarrow0$ in the *weak operator topology* (WOT) iff (Def.) $\lim_{n\rightarrow\infty}\left\langle x,A_{n}y\right\rangle =0$ for all pairs of vectors $x,y\in\mathscr{H}$. Here $\left\langle \cdot,\cdot\right\rangle $ refers to the inner product in $\mathscr{H}$.
\[cor:fi\]The following are equivalent:
1. \[enu:mf1\]$1=\sum_{j\in\mathbb{N}_{0}}Q_{j}^{*}Q_{j}$ in the weak operator topology.
2. $1=\sum_{j\in\mathbb{N}_{0}}Q_{j}$ in the strong operator topology.
3. \[enu:mf3\]$T_{n}\rightarrow0$ in the strong operator topology.
Under suitable conditions on $Q_{n}$ one can show that the convergence in part (\[enu:mf1\]) of the corollary also holds in the strong operator topology.
The system $\left\{ P_{j}\right\} _{j\in\mathbb{N}_{0}}$ is called *effective* if $T_{n}\rightarrow0$ in the strong operator topology.
\[cor:gf\]Suppose the system $\left\{ P_{j}\right\} _{j\in\mathbb{N}_{0}}$ is effective. Then, for all $x\in\mathscr{H}$, $$x=\sum_{j\in\mathbb{N}_{0}}Q_{j}x.\label{eq:C5}$$ Moreover, for all $x,y\in\mathscr{H}$, $$\left\langle x,y\right\rangle =\sum_{j\in\mathbb{N}_{0}}\left\langle Q_{j}x,Q_{j}y\right\rangle ;$$ and in particular, $$\left\Vert x\right\Vert ^{2}=\sum_{j\in\mathbb{N}_{0}}\left\Vert Q_{j}x\right\Vert ^{2}.\label{eq:C7}$$
The system of operators $\left\{ Q_{j}\right\} _{j\in\mathbb{N}_{0}}$ in has frame-like properties. Specifically, the mapping $$\mathscr{H}\ni x\xmapsto{\;V\;}\left(Q_{j}x\right)\in l^{2}\left(\mathbb{N}_{0}\right)\otimes\mathscr{H}$$ plays the role of an analysis operator, and the synthesis operator $V^{*}$ is given by $$l^{2}\left(\mathbb{N}_{0}\right)\otimes\mathscr{H}\ni\xi\xmapsto{\;V^{*}\;}\sum_{j\in\mathbb{N}_{0}}Q_{j}^{*}\xi_{j}.$$ Note that $1=V^{*}V$, by part (\[enu:mf1\]) of ; and eq. (\[eq:C7\]) is the generalized Parseval identity. Also see below.
\[prop:fd\]Let $\left\{ P_{j}\right\} _{j\in\mathbb{N}_{0}}$ be an effective system. Then there exits a Hilbert space $\mathscr{K}$, an isometry $V:\mathscr{H}\rightarrow\mathscr{K}$, and selfadjoint projections $E_{j}$ in $\mathscr{K}$, such that $Q_{j}^{*}Q_{j}=V^{*}E_{j}V$, for all $j\in\mathbb{N}_{0}$. Thus, $$1=\sum_{j\in\mathbb{N}_{0}}Q_{j}^{*}Q_{j}=\sum_{j\in\mathbb{N}_{0}}V^{*}E_{j}V.\label{eq:d1}$$
Let $\mathscr{K}=l^{2}\left(\mathbb{N}_{0}\right)\otimes\mathscr{H}\left(=\oplus_{\mathbb{N}_{0}}\mathscr{H}\right)$, and set $V:\mathscr{H}\rightarrow\mathscr{K}$ by $$Vx=\left(Q_{j}x\right)_{j\in\mathbb{N}_{0}}.$$ Then, for all $x\in\mathscr{H}$ and $y=\left(y_{j}\right)\in\mathscr{K}$, $$\left\langle Vx,y\right\rangle _{\mathscr{K}}=\sum\left\langle Q_{j}x,y_{j}\right\rangle _{\mathscr{H}}=\left\langle x,\sum Q_{j}^{*}y_{j}\right\rangle _{\mathscr{H}}.$$ Hence the adjoint operator $V^{*}$ is given by $$V^{*}y=\sum_{j\in\mathbb{N}_{0}}Q_{j}^{*}y_{j}.$$
For all $j\in\mathbb{N}_{0}$, let $E_{j}:\mathscr{K}\rightarrow\mathscr{K}$ be the projection, $$E_{j}y=\left(0,\cdots,0,y_{j},0,\cdots\right),\;\forall y=\left(y_{j}\right)\in\mathscr{K}.$$ Then $Q_{j}^{*}Q_{j}=V^{*}E_{j}V$, and (\[eq:d1\]) follows from this.
Let $\mathscr{H}$ be a fixed Hilbert space. We shall have occasion to use Dirac’s notation for rank-one operators in $\mathscr{H}$: If $u,v\in\mathscr{H}$, we set $\left|u\left\rangle \right\langle v\right|$ the operator, which is defined by $$\left|u\left\rangle \right\langle v\right|\left(x\right)=\left\langle v,x\right\rangle _{\mathscr{H}}u;$$ or in physics terminology, $$\left|u\left\rangle \right\langle v\right|\left.x\right\rangle =\left|u\right\rangle \left\langle v,x\right\rangle _{\mathscr{H}}.$$ Note the following: For vectors $u_{i},v_{i}$, $=1,2$, we have: $$\left(\left|u_{1}\left\rangle \right\langle v_{1}\right|\right)\left(\left|u_{2}\left\rangle \right\langle v_{2}\right|\right)=\left\langle v_{1},u_{2}\right\rangle _{\mathscr{H}}\left|u_{1}\left\rangle \right\langle v_{2}\right|.$$ For the adjoint operators, we have: $$\left|u\left\rangle \right\langle v\right|^{*}=\left|v\left\rangle \right\langle u\right|.$$ If $B\in\mathscr{B}\left(\mathscr{H}\right)$, we have $$B\left|u\left\rangle \right\langle v\right|=\left|Bu\left\rangle \right\langle v\right|;$$ and $$\left|u\left\rangle \right\langle v\right|B=\left|u\left\rangle \right\langle B^{*}v\right|.$$
The case of rank-1 projections in Hilbert space
-----------------------------------------------
Let $\left\{ P_{j}\right\} _{j\in\mathbb{N}_{0}}$ be a system of rank-1 projections, i.e., $P_{j}=\left|e_{j}\left\rangle \right\langle e_{j}\right|$, where $\left\{ e_{j}\right\} _{j\in\mathbb{N}_{0}}$ is a set of unit vectors in $\mathscr{H}$. When the system $\left\{ e_{j}\right\} $ is independent, then the corresponding family of projections $P_{j}=\left|e_{j}\left\rangle \right\langle e_{j}\right|$ is non-commutative.
It follows from (\[eq:f2\]) that every $Q_{j}$ is a rank-1 operator with range in $span\left\{ e_{j}\right\} $. Thus there exists a unique $g_{j}\in\mathscr{H}$ such that $$Q_{j}=\left|e_{j}\left\rangle \right\langle g_{j}\right|,\;j\in\mathbb{N}_{0}.\label{eq:C16}$$
\[lem:qn\]Given $\left\{ P_{j}\right\} _{j\in\mathbb{N}_{0}}$ a sequence of s.a. projections in $\mathscr{H}$; set $$Q_{n}:=P_{n}P_{n-1}^{\perp}\cdots P_{1}^{\perp}P_{0}^{\perp},\label{eq:C17}$$ where $P_{j}^{\perp}:=1-P_{j}$; then $$Q_{n}=P_{n}\left(1-\sum\nolimits _{j=0}^{n-1}Q_{j}\right).$$
By definition, we have $$\begin{aligned}
Q_{n} & = & P_{n}\left(1-P_{n-1}\right)P_{n-2}^{\perp}\cdots P_{0}^{\perp}\\
& = & P_{n}P_{n-2}^{\perp}\cdots P_{0}^{\perp}-P_{n}P_{n-1}P_{n-2}^{\perp}\cdots P_{0}^{\perp}\\
& = & P_{n}P_{n-2}^{\perp}\cdots P_{0}^{\perp}-P_{n}Q_{n-1}\\
& = & P_{n}P_{n-3}^{\perp}\cdots P_{0}^{\perp}-P_{n}Q_{n-2}-P_{n}Q_{n-1}\\
& \vdots\\
& = & P_{n}P_{0}^{\perp}-P_{n}Q_{1}-P_{n}Q_{2}-\cdots-P_{n}Q_{n-1}\\
& = & P_{n}-\sum\nolimits _{j=0}^{n-1}P_{n}Q_{j}\\
& = & P_{n}\left(1-\sum\nolimits _{j=0}^{n-1}Q_{j}\right).\end{aligned}$$
\[cor:kg\]The vectors $\left\{ g_{j}\right\} $ in (\[eq:C16\]) are determined recursively by $$\begin{aligned}
g_{0} & =e_{0}\\
g_{n} & =e_{n}-\sum_{j=0}^{n-1}\left\langle e_{j},e_{n}\right\rangle g_{j}.\label{eq:C20}\end{aligned}$$
For all $x\in\mathscr{H}$, it follows from , that $$\begin{aligned}
Q_{n}x & = & P_{n}x-\sum_{j=0}^{n-1}P_{n}Q_{j}x\\
& \Updownarrow\\
e_{n}\left\langle g_{n},x\right\rangle & = & e_{n}\left\langle e_{n},x\right\rangle -\sum_{j=0}^{n-1}e_{n}\left\langle e_{n},e_{j}\right\rangle \left\langle g_{j},x\right\rangle .\end{aligned}$$ That is, $g_{n}=e_{n}-\sum_{j=0}^{n-1}\left\langle e_{j},e_{n}\right\rangle g_{j}$.
\[cor:ke\]Assume $\left\{ \left|e_{j}\left\rangle \right\langle e_{j}\right|\right\} _{j\in\mathbb{N}_{0}}$ is effective, and let $Q_{j}=\left|e_{j}\left\rangle \right\langle g_{j}\right|$ be as above. Then, for all $x\in\mathscr{H}$, we have $$x=\sum_{j\in\mathbb{N}_{0}}\left\langle g_{j},x\right\rangle e_{j}.\label{eq:C21}$$ In particular, for all $A\in\mathscr{B}\left(\mathscr{H}\right)$, then $$Ax=\sum_{j\in\mathbb{N}_{0}}\left\langle A^{*}g_{j},x\right\rangle e_{j}.$$
Moreover, for all $x,y\in\mathscr{H}$, $$\begin{aligned}
\left\langle x,y\right\rangle & =\sum_{j\in\mathbb{N}_{0}}\left\langle x,g_{j}\right\rangle \left\langle g_{j},y\right\rangle ,\;\text{and}\\
\left\Vert x\right\Vert ^{2} & =\sum_{j\in\mathbb{N}_{0}}\left|\left\langle g_{j},x\right\rangle \right|^{2}.\end{aligned}$$
By assumption, $Q_{j}^{*}Q_{j}=\left|g_{j}\left\rangle \right\langle g_{j}\right|$, hence $$\left\langle x,y\right\rangle =\sum_{j\in\mathbb{N}_{0}}\left\langle x,Q_{j}^{*}Q_{j}y\right\rangle =\sum_{j\in\mathbb{N}_{0}}\left\langle x,g_{j}\right\rangle \left\langle g_{j},y\right\rangle .$$
\[cor:ke2\]The system $\left\{ \left|e_{j}\left\rangle \right\langle e_{j}\right|\right\} _{j\in\mathbb{N}_{0}}$ is effective iff $\left\{ g_{j}\right\} _{j\in\mathbb{N}_{0}}$ is a Parseval frame in $\mathscr{H}$.
\[rem:nf\]We note that when $\mu$ is slice singular, then the Fourier frequencies $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}}$ is effective in $L^{2}\left(\mu\right)$, and every $f\in L^{2}\left(\mu\right)$ has Fourier series expansion.
This conclusion is based on (\[eq:C20\]) and (\[eq:C21\]) from Corollaries \[cor:kg\] & \[cor:ke\]. In more detail: Assume $\mu$ is slice singular, and take $\mathscr{H}=L^{2}\left(\mu\right)$. We may then think of as a (generalized) Fourier expansion result since every $f$ in the specified $L^{2}\left(\mu\right)$ space admits a non-orthogonal Fourier expansion in terms of explicit coefficients and the standard Fourier functions $e_{n}$. Indeed, the corresponding generalized Fourier coefficients are computed with the use of the functions $g_{n}$ of the Kaczmarz algorithm, see eq. (\[eq:C21\]) and .
We stress that while the coefficients in the expansion for $f$ are explicitly given in (\[eq:C21\]), this is nonetheless a non-orthogonal expansion; see also [@2016arXiv160308852H; @MR3796641].
\[subsec:rkac\]Random Kaczmarz constructions and sequences of projections
-------------------------------------------------------------------------
In the discussion below, the word “*random*” will refer to a fixed probability space $\left(\Omega,\mathscr{F},\mathbb{P}\right)$, where $\Omega$ is a set (sample space), $\mathscr{F}$ is a $\sigma$-algebra (specified events), and $\mathbb{P}$ is a probability measure defined on $\mathscr{F}$. *Random variables* will then be measurable functions on $\left(\Omega,\mathscr{F}\right)$. For example, if $\xi:\Omega\rightarrow\mathscr{B}\left(\mathscr{H}\right)$ is an operator valued random variable, measurability will then refer to the $\sigma$-algebra of subsets in $\mathscr{B}\left(\mathscr{H}\right)$ which are w.r.t. the usual operator topology.
Equivalently, $\xi:\Omega\rightarrow\mathscr{B}\left(\mathscr{H}\right)$ is a random variable iff (Def.) for all pairs of vectors $x,y\in\mathscr{H}$, then the functions $$\Omega\longrightarrow\mathbb{C},\quad\omega\longmapsto\left\langle x,\xi\left(\omega\right)y\right\rangle _{\mathscr{H}}$$ are measurable w.r.t. the standard Borel $\sigma$-algebra $\mathscr{B}_{\mathbb{C}}$ of subsets of $\mathbb{C}$.
Given a probability space $\left(\Omega,\mathscr{F},\mathbb{P}\right)$ we shall denote the corresponding expectation $\mathbb{E}$, i.e., $$\mathbb{E}\left(\cdots\right)\overset{\text{Def.}}{=}\int_{\Omega}\left(\cdots\right)d\mathbb{P}.$$\
below is a stochastic variant of the classical Kaczmarz algorithm; also see . For recent development and applications, we refer to [@MR2113344; @MR2721177; @MR3210983; @MR3450541; @MR3345342; @MR3424852; @MR3439812; @MR3847751; @MR3796634].
Let $\mathscr{H}$ be a Hilbert space. Given a family of selfadjoint projections $\left\{ P_{j}\right\} _{j\in\mathbb{N}_{0}}$ in $\mathscr{H}$, let $\xi:\Omega\rightarrow\mathscr{B}\left(\mathscr{H}\right)$ be a random variable, such that $$\mathbb{P}\left(\xi=P_{j}\right)=p_{j},\;j\in\mathbb{N}_{0},\label{eq:rm1}$$ where $p_{j}>0$, and $\sum_{j\in\mathbb{N}_{0}}p_{j}=1$.
Suppose further that there exists a constant $C$, $0<C<1$, such that $$\mathbb{E}\left[\left\Vert \xi x\right\Vert ^{2}\right]:=\sum\nolimits _{j\in\mathbb{N}_{0}}p_{j}\left\Vert P_{j}x\right\Vert ^{2}\geq C\left\Vert x\right\Vert ^{2},\;\forall x\in\mathscr{H}.\label{eq:rm2}$$
Let $\xi$, $\eta:\Omega\rightarrow\mathscr{B}\left(\mathscr{H}\right)$ be two operator-valued random variables. We say $\xi$ and $\eta$ are *independent* iff (Def.) for all $x,y\in\mathscr{H}$, the scalar valued random variables $\left\langle x,\xi y\right\rangle $, and $\left\langle x,\eta y\right\rangle $ are independent.
We shall use the standard abbreviation i.i.d. for independent, identically distributed; also in the case of an indexed family of operator valued random variables. In the present case, the common distribution is specified by fixing the data in (\[eq:rm1\]).
The key feature of our present *randomized* Kaczmarz algorithm is that it outputs a recursively generated sequence of operator valued random variables; see (\[eq:rm3\]) and (\[eq:rm4\]). Each output, in turn, will be a product of a specified i.i.d. system of projection valued random variables. The latter i.i.d. system serves as input into the algorithm.
\[thm:rkac\]Let $\left\{ \xi_{j}\right\} _{j\in\mathbb{N}_{0}}$ be an i.i.d. realization of $\xi$ from (\[eq:rm1\]). Fix $\xi_{0}=P_{0}$, and set $$\begin{aligned}
T_{n} & =\left(1-\xi_{n}\right)\left(1-\xi_{n-1}\right)\cdots\left(1-\xi_{0}\right),\;\text{and}\label{eq:rm3}\\
Q_{n} & =\xi_{n}\left(1-\xi_{n-1}\right)\cdots\left(1-\xi_{0}\right),\;Q_{0}=\xi_{0}.\label{eq:rm4}\end{aligned}$$ Note that each product in (\[eq:rm3\]) and (\[eq:rm4\]) is an operator-valued random variable.
Then, for all $x\in\mathscr{H}$, we have: $$\lim_{n\rightarrow\infty}\mathbb{E}\text{\ensuremath{\left[\left\Vert T_{n}x\right\Vert ^{2}\right]}}=0.\label{eq:rm5}$$
For all $x\in\mathscr{H}$, we have $$T_{n}x=T_{n-1}x-\xi_{n}T_{n-1}x.$$ But each $\xi_{n}$ is a random variable taking values in the set of selfadjoint projections, as specified in (\[eq:rm1\]), and so $$\left\Vert T_{n}x\right\Vert ^{2}=\left\Vert T_{n-1}x\right\Vert ^{2}-\left\Vert \xi_{n}T_{n-1}x\right\Vert ^{2}.$$ It follows from (\[eq:rm2\]) that $$\begin{aligned}
\mathbb{E}_{\xi_{1},\cdots,\xi_{n-1}}\left[\left\Vert T_{n}x\right\Vert ^{2}\right] & =\mathbb{E}_{\xi_{1},\cdots,\xi_{n-1}}\left[\left\Vert T_{n-1}x\right\Vert ^{2}\right]-\mathbb{E}_{\xi_{1},\cdots,\xi_{n-1}}\left[\left\Vert \xi_{n}T_{n-1}x\right\Vert ^{2}\right]\\
& \leq\left\Vert T_{n-1}x\right\Vert ^{2}\left(1-C\right).\end{aligned}$$ Therefore, by taking expectation again, we get $$\begin{aligned}
\mathbb{E}\left[\left\Vert T_{n}x\right\Vert ^{2}\right] & \leq & \mathbb{E}\left[\left\Vert T_{n-1}x\right\Vert ^{2}\right]\left(1-C\right)\\
& \leq & \mathbb{E}\left[\left\Vert T_{n-2}x\right\Vert ^{2}\right]\left(1-C\right)^{2}\\
& \vdots\\
& \leq & \mathbb{E}\left[\left\Vert T_{0}x\right\Vert ^{2}\right]\left(1-C\right)^{n}\\
& = & \left\Vert x_{0}-x\right\Vert ^{2}\left(1-C\right)^{n}\rightarrow0,\;n\rightarrow\infty.\end{aligned}$$
\[cor:rkac\]Let $T_{n}$ and $Q_{n}$ be as in (\[eq:rm3\])–(\[eq:rm4\]), then the following hold.
1. For all $x\in\mathscr{H}$, $$\lim_{n\rightarrow\infty}\mathbb{E}\left[\left\Vert x-\sum\nolimits _{j=0}^{n}Q_{j}x\right\Vert ^{2}\right]=0.\label{eq:rm6}$$
2. For all $x,y\in\mathscr{H}$, $$\lim_{n\rightarrow\infty}\mathbb{E}\left[\left|\left\langle x,y\right\rangle -\sum\nolimits _{j=0}^{n}\left\langle x,Q_{j}^{*}Q_{j}y\right\rangle \right|^{2}\right]=0.\label{eq:rm7}$$
The assertion (\[eq:rm6\]) follows from (\[eq:rm5\]) and (\[eq:cc7\]).
By (\[eq:cc6\]), we have $\left\Vert T_{n}x\right\Vert ^{2}=\left\langle x,T_{n}^{*}T_{n}x\right\rangle =\left\langle x,x\right\rangle -\sum_{j=0}^{n}\left\langle x,Q_{j}^{*}Q_{j}x\right\rangle $, and so $$\mathbb{E}\left[\left\langle x,T_{n}^{*}T_{n}x\right\rangle \right]\rightarrow0,\;n\rightarrow\infty.$$ Now (\[eq:rm7\]) follows from this and the polarization identity.
Our present equation (\[eq:rm2\]) may be viewed as an instance of what is now called *fusion frames*, and developed extensively by Casazza et al. [@MR2066823; @MR2419707; @MR2440135], and by others. In addition, we note that our present (\[eq:rm7\]) is closely related to a formulation a certain notion of *measure frames*, see e.g., [@MR2147063; @MR2964017; @MR3526434; @MR3688637], and its extensions in [@MR3800275].
\[subsec:axy\]Solutions to $Ax=y$ in finite, and in infinite, dimensional spaces
--------------------------------------------------------------------------------
A natural extension of the classical Kaczmarz algorithm is to solve the equation $$Ax=y,$$ when $x,y$ are vectors in an infinite-dimensional Hilbert space $\mathscr{H}$, and $A,A^{-1}$ are both bounded operators in $\mathscr{H}$; see .
Equivalently, when $\left\{ \varphi_{j}\right\} _{j\in\mathbb{N}}$ is an ONB (or a Parseval frame) in $\mathscr{H}$, we shall consider the system of equations $$\begin{aligned}
\left\langle \varphi_{j},Ax\right\rangle & = & \left\langle \varphi_{j},y\right\rangle \\
& \Updownarrow\\
\left\langle A^{*}\varphi_{j},x\right\rangle & = & \left\langle \varphi_{j},y\right\rangle .\end{aligned}$$
\[que:Axy\]Given the complex numbers $\left\langle A^{*}\varphi_{j},x\right\rangle $, $j\in\mathbb{N}$, is it possible to recover $x$ using the Kaczmarz method?
The closest analog to the finite-dimensional setting is the class of Hilbert-Schmidt operators, and we shall recall the basics below.
Assume $\mathscr{H}$ is separable. $A:\mathscr{H}\rightarrow\mathscr{H}$ is *Hilbert-Schmidt* iff (Def.) $\exists$ an ONB $\left\{ e_{i}\right\} _{i\in\mathbb{N}_{0}}$, such that $$\sum\nolimits _{i}\left\Vert Ae_{i}\right\Vert ^{2}<\infty.$$ We denote the set of all Hilbert-Schmidt operators in $\mathscr{H}$ by $HS\left(\mathscr{H}\right)$.
Note that $A\in HS\left(\mathscr{H}\right)$ iff $A^{*}A$ is *trace class*, and for an ONB $\left\{ e_{i}\right\} $ , we have $$\sum\left\Vert Ae_{i}\right\Vert ^{2}=\sum\left\langle e_{i},A^{*}Ae_{i}\right\rangle =tr\left(A^{*}A\right).$$
$HS\left(\mathscr{H}\right)$ $\simeq\mathscr{H}\otimes\overline{\mathscr{H}}$, where $\overline{\mathscr{H}}$ denotes the conjugate Hilbert space.
If $\left\{ e_{i}\right\} $ is an ONB, set $\left|e_{i}\left\rangle \right\langle e_{j}\right|$ w.r.t. the inner product $$\left(A,B\right)\longmapsto tr\left(A^{*}B\right),$$ for all $A,B\in HS\left(\mathscr{H}\right)$. Hence, $$\left\langle A,B\right\rangle _{HS}=\sum_{i}\left\langle e_{i},A^{*}Be_{i}\right\rangle _{\mathscr{H}}\left(=tr\left(A^{*}B\right)\right).$$
We shall show that $$HS\left(\mathscr{H}\right)\ominus\left\{ \left|e_{i}\left\rangle \right\langle e_{j}\right|\right\} _{\mathbb{N}_{0}\times\mathbb{N}_{0}}=0,\label{eq:hs4}$$ i.e., $\left\{ A_{ij}:=\left|e_{i}\left\rangle \right\langle e_{j}\right|\right\} $ is *total* in $HS\left(\mathscr{H}\right)$. To see this, note that $$tr\left(\left|u\left\rangle \right\langle v\right|\right)=\left\langle v,u\right\rangle _{\mathscr{H}},\;u,v\in\mathscr{H}.$$ In fact, one checks that $$\begin{aligned}
tr\left(\left|u\left\rangle \right\langle v\right|\right) & =\sum_{i}\left\langle e_{i},u\right\rangle \left\langle v,e_{i}\right\rangle \\
& =\sum\left\langle v,u\right\rangle ,\;\text{by Parseval.}\end{aligned}$$ Now, if $B\in HS\left(\mathscr{H}\right)$, then $$\begin{aligned}
\left\langle B,\left|e_{i}\left\rangle \right\langle e_{j}\right|\right\rangle _{HS} & =tr\left(B^{*}\left|e_{i}\left\rangle \right\langle e_{j}\right|\right)\\
& =tr\left(\left|B^{*}e_{i}\left\rangle \right\langle e_{j}\right|\right)\\
& =tr\left(\left|e_{i}\left\rangle \right\langle Be_{j}\right|\right)\\
& =\left\langle Be_{j},e_{i}\right\rangle _{\mathscr{H}},\;\text{by \ensuremath{\left(\ref{eq:hs4}\right)}.}\end{aligned}$$ Therefore, if $\left\langle B,\left|e_{i}\left\rangle \right\langle e_{j}\right|\right\rangle _{HS}=0$, for all $i,j\in\mathbb{N}_{0}$, then $B=0$; since $$Be_{j}=\sum_{i}\left\langle e_{i},Be_{j}\right\rangle _{\mathscr{H}}e_{i}.$$
Now, back to . From earlier discussion, the answer depends on whether the sequence $\left\{ A^{*}\varphi_{j}\right\} $ is effective. In general, we do not get an effective sequence, even if $A$ is assumed Hilbert-Schmidt. However, under certain conditions (see (\[eq:hp0\])) the *random* Kaczmarz algorithm applies, and we get an approximate sequence that converges to $x$ in expectation. See details below.
\[lem:rax\]Suppose $A$ is a bounded operator in $\mathscr{H}$ with bounded inverse. Fix a Parseval frame $\left\{ \varphi_{j}\right\} _{j\in\mathbb{N}}$ in $\mathscr{H}$, let $P_{j}$ be the projection onto $A^{*}\varphi_{j}$, $j\in\mathbb{N}$.
Assume further that $$1/\left\Vert A^{-1}\right\Vert ^{2}<\sum\nolimits _{k}\left\Vert A^{*}\varphi_{k}\right\Vert ^{2}<\infty.\label{eq:hp0}$$ Then, there exists a probability distribution $\left\{ p_{j}\right\} $ on $\left\{ P_{j}\right\} $, given by $$p_{j}=\left\Vert A^{*}\varphi_{j}\right\Vert ^{2}/\sum\nolimits _{k}\left\Vert A^{*}\varphi_{k}\right\Vert ^{2},\label{eq:hp}$$ such that, for all $h\in\mathscr{H}$, $$\sum\nolimits _{j\in\mathbb{N}}p_{j}\left\Vert P_{j}h\right\Vert ^{2}\geq C\left\Vert h\right\Vert ^{2},\label{eq:hp1}$$ where $C$ is a constant, $0<C<1$.
For all $h\in\mathscr{H}$, we have: $$\begin{aligned}
\left\Vert h\right\Vert ^{2} & =\left\Vert A^{-1}Ah\right\Vert ^{2}\\
& \leq\left\Vert A^{-1}\right\Vert ^{2}\left\Vert Ah\right\Vert ^{2}\\
& =\left\Vert A^{-1}\right\Vert ^{2}\sum\left|\left\langle \varphi_{j},Ah\right\rangle \right|^{2}=\left\Vert A^{-1}\right\Vert ^{2}\sum\left|\left\langle A^{*}\varphi_{j},h\right\rangle \right|^{2}\\
& =\left\Vert A^{-1}\right\Vert ^{2}\sum\nolimits _{k}\left\Vert A^{*}\varphi_{k}\right\Vert ^{2}\sum\nolimits _{j}\underset{=p_{j}}{\underbrace{\frac{\left\Vert A^{*}\varphi_{j}\right\Vert ^{2}}{\sum_{k}\left\Vert A^{*}\varphi_{k}\right\Vert ^{2}}}}\left|\left\langle \frac{A^{*}\varphi_{j}}{\left\Vert A^{*}\varphi_{j}\right\Vert },h\right\rangle \right|^{2}\\
& =\underset{=C^{-1}}{\underbrace{\left\Vert A^{-1}\right\Vert ^{2}\sum\nolimits _{k}\left\Vert A^{*}\varphi_{k}\right\Vert ^{2}}}\cdot\sum\nolimits _{j}p_{j}\left\Vert P_{j}h\right\Vert ^{2}.\end{aligned}$$ The desired conclusion follows from this.
Let the setting be as in . An approximate solution to $Ax=y$ is obtained recursively as follows:
Let $\xi:\Omega\rightarrow\mathscr{B}\left(\mathscr{H}\right)$ be a random projection, s.t. $\mathbb{P}\left(\xi=P_{j}\right)=p_{j}$ (see (\[eq:hp\])), and $\left\{ \xi_{j}\right\} $ be an i.i.d. realization of $\xi$. Then, with $x_{0}\neq0$ fixed, and $$x_{j}:=x_{j-1}+\xi_{j}\left(x-x_{j-1}\right),\;j\in\mathbb{N},\label{eq:p1}$$ we have: $$\lim_{j\rightarrow\infty}\mathbb{E}\left[\left\Vert x_{j}-x\right\Vert ^{2}\right]=0.\label{eq:p2}$$ Note that, in (\[eq:p1\]) if $\xi=P_{k}$, then $$\xi x=\frac{\left\langle A^{*}\varphi_{k},x\right\rangle }{\left\Vert A^{*}\varphi_{k}\right\Vert ^{2}}A^{*}\varphi_{k}=\frac{\left\langle \varphi_{k},y\right\rangle }{\left\Vert A^{*}\varphi_{k}\right\Vert ^{2}}A^{*}\varphi_{k}.$$
By , the estimate (\[eq:hp1\]) holds with the probabilities specified in (\[eq:hp\]). See also condition (\[eq:rm2\]). Moreover, it follows from (\[eq:p1\]) that $$x-x_{j}=\left(1-\xi_{j}\right)\left(1-\xi_{j-1}\right)\cdots\left(1-\xi_{1}\right)x_{0}.$$ Therefore, by , the assertion in (\[eq:p2\]) holds (with a suitable choice of index $j$).
System of isometries
====================
Below we discuss a particular aspect of our problem where the polydisk $\mathbb{D}^{d}$ will play an important role. As outlined below, the polydisk is a natural part of our harmonic analysis of frame-approximation questions in the Hilbert space $L^{2}(\mu)$, where $\mu$ is in a suitable class of IFS-measures, i.e., the multivariable setting for fractal measures.
\[lem:viso\]Fix $d>1$, and let $\mathbb{D}^{d}$ be the polydisk. Let $H_{2}\left(\mathbb{D}^{d}\right)$ be the corresponding Hardy space. Let $\mu$ be a Borel probability measure on $\mathbb{T}^{d}\simeq\left[0,1\right]^{d}$. Then there is a bijective correspondence between:
1. \[enu:pfs1\]isometries $V:L^{2}\left(\mu\right)\rightarrow H_{2}\left(\mathbb{D}^{d}\right)$; and
2. \[enu:pfs2\]Parseval frames $\left\{ g_{n}\right\} $ in $L^{2}\left(\mu\right)$.
The correspondence is as follows:
(\[enu:pfs1\])$\rightarrow$(\[enu:pfs2\]). Given $V$, isometric; set $g_{n}:=V^{*}\left(z^{n}\right)$, where $n\in\mathbb{N}_{0}^{d}$.
(\[enu:pfs2\])$\rightarrow$(\[enu:pfs1\]). Given $\left\{ g_{n}\right\} $ a fixed Parseval frame in $L^{2}\left(\mu\right)$, set $$\left(Vf\right)\left(z\right)=\sum_{n\in\mathbb{N}_{0}^{d}}\left\langle g_{n},f\right\rangle _{L^{2}\left(\mu\right)}z^{n},\;z\in\mathbb{D}^{d}.$$
The fact that there is a correspondence between isometries and Parseval frames is general. Let $\mathscr{H}_{1}$ be a separable Hilbert space, then there is a bijective correspondence between the following two:
1. \[enu:pf1\]A Parseval frame $\left(g_{n}\right)_{n\in\mathbb{N}}$ in $\mathscr{H}_{1}$ (with a suitable choice of index);
2. \[enu:pf2\]A pair $\left(\mathscr{H}_{2},V\right)$, where $\mathscr{H}_{2}$ is a Hilbert space, and $V:\mathscr{H}_{1}\rightarrow\mathscr{H}_{2}$ is isometric.
(Note that there is a similar result for Bessel frames as well.) The correspondence is as follows.
Given a Parseval frame $\left(g_{n}\right)_{n\in\mathbb{N}}$ in $\mathscr{H}_{1}$, take $\mathscr{H}_{2}:=l^{2}\left(\mathbb{N}\right)$, and set $Vf=\sum_{n}\left\langle g_{n},f\right\rangle _{\mathscr{H}_{1}}\delta_{n}$, where $\left\{ \delta_{n}\right\} _{n\in\mathbb{N}}$ is the standard ONB in $l^{2}\left(\mathbb{N}\right)$.
Conversely, let $\left(\mathscr{H}_{2},V\right)$ be such that $\mathscr{H}_{1}\xrightarrow{\;V\;}\mathscr{H}_{2}$ is isometric. Choose an ONB $\left\{ \beta_{n}\right\} _{n\in\mathbb{N}}$ in $\mathscr{H}_{2}$, and set $g_{n}=V^{*}\beta_{n}$. Then $\left\{ g_{n}\right\} $ is a Parseval frame in $\mathscr{H}_{1}$. Indeed, for all $h\in\mathscr{H}_{1}$, one checks that, $$\begin{aligned}
\sum_{n}\left|\left\langle g_{n},h\right\rangle _{\mathscr{H}_{1}}\right|^{2} & =\sum_{n}\left|\left\langle V^{*}\beta_{n},h\right\rangle _{\mathscr{H}_{1}}\right|^{2}\\
& =\sum_{n}\left|\left\langle \beta_{n},Vh\right\rangle _{\mathscr{H}_{2}}\right|^{2}\\
& =\left\Vert Vh\right\Vert _{\mathscr{H}_{2}}^{2}=\left\Vert h\right\Vert _{\mathscr{H}_{1}}^{2}.\end{aligned}$$
The lemma follows by setting $\mathscr{H}_{1}=L^{2}\left(\mu\right)$, and $\mathscr{H}_{2}=H_{2}\left(\mathbb{D}^{d}\right)$.
Fix $d>1$. For all $x\in\mathbb{T}^{d}$, and all $z\in\mathbb{D}^{d}$, let $$K^{\ast}\left(z,x\right)=\prod_{j=1}^{d}\frac{1}{1-z_{j}\overline{e\left(x_{j}\right)}}.$$ Let $\mu\in\mathcal{M}\left(\mathbb{T}^{d}\right)$, and set $$\begin{aligned}
\left(C_{\mu}f\right)\left(z\right) & =\int_{\mathbb{T}^{d}}f\left(x\right)K^{\ast}\left(z,x\right)d\mu\left(x\right)\label{eq:nc1}\\
& =\sum_{n\in\mathbb{N}^{d}}\widehat{fd\mu}\left(n\right)z^{n}.\nonumber \end{aligned}$$ In particular, $$\left(C_{\mu}1\right)\left(z\right)=\sum_{n\in\mathbb{N}_{0}^{d}}\widehat{\mu}\left(n\right)z^{n},$$ where $\widehat{\mu}\left(n\right)=\int_{\mathbb{T}^{d}}\overline{e_{n}\left(x\right)}d\mu\left(x\right)$, $n\in\mathbb{N}_{0}^{d}$.
Let $L^{2}\left(\mu\right)\left(=L^{2}\left(\mathbb{T}^{2},\mu\right)\right)$ be as above, where $\mu\in\mathcal{M}^{+}\left(\mathbb{T}^{2}\right)$, $\xi=\mu\circ\pi_{1}^{-1}$, and $\mu$ assumes a disintegration $d\mu=\int\sigma^{x}\left(dy\right)d\xi\left(x\right)$.
\[thm:ssiso\] Assume $\mu$ is slice singular. There are then two associated isometries: $$L^{2}\left(\xi\right)\xrightarrow{\;V_{\xi}\;}H_{2}\left(\mathbb{D}\right),\quad\left(V_{\xi}f\right)\left(z\right)=\frac{\left(C_{\xi}f\right)\left(z\right)}{\left(C_{\xi}1\right)\left(z\right)},\label{eq:vxi}$$ and $$L^{2}\left(\sigma^{x}\right)\xrightarrow{\;V_{\sigma^{x}}\;}H_{2}\left(\mathbb{D}\right),\quad\left(V_{\sigma^{x}}f\right)\left(z\right)=\frac{\left(C_{\sigma^{x}}f\right)\left(z\right)}{\left(C_{\sigma^{x}}1\right)\left(z\right)}.\label{eq:vsig}$$
Let $\nu$ be a positive Borel measure on $\left[0,1\right]$, and $C_{\mu}$ be the Cauchy transform from (\[eq:nc1\]). Assume $\nu$ is singular.
Then, by F.M Riesz (see ), the set $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}}$ is total in $L^{2}\left(\nu\right)$. Moreover, it follows from [@MR2263965], that $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}}$ is effective. Thus, every $f\in L^{2}\left(\nu\right)$ has (non-orthogonal) Fourier expansion $$f=\sum_{n\in\mathbb{N}_{0}}\left\langle g_{n},f\right\rangle _{L^{2}\left(\nu\right)}e_{n},$$ where $\left\{ g_{n}\right\} $ is the Parseval frame in $L^{2}\left(\nu\right)$ constructed from Kaczmarz’ algorithm. See also . One may verify that $$\left(V_{\nu}f\right)\left(z\right)=\frac{C_{\nu}f}{C_{\nu}1}=\sum_{n\in\mathbb{N}_{0}}\left\langle g_{n},f\right\rangle _{L^{2}\left(\nu\right)}z^{n},$$ and so $V_{\nu}:L^{2}\left(\nu\right)\rightarrow H_{2}\left(\mathbb{D}\right)$ is isometric.
The theorem follows from this, and the assumption that $\mu$ is slice singular.
The mapping $$V_{\mu}:L^{2}\left(\mu\right)\longrightarrow H_{2}\left(\mathbb{D}^{2}\right)\left(=H_{2}\left(\mathbb{D}\right)\otimes H_{2}\left(\mathbb{D}\right)\right)$$ given by $$\left(V_{\mu}F\right)\left(z_{1},z_{2}\right)=V_{\xi}\left(\left(V_{\sigma^{x}\left(\cdot\right)}F\left(x,\cdot\right)\right)\left(z_{2}\right)\right)\left(z_{1}\right)$$ is isometric. It follows that $\left\{ g_{n}:=V_{\mu}^{*}\left(z^{n}\right)\right\} _{n\in\mathbb{N}_{0}^{2}}$ is a Parseval frame in $L^{2}\left(\mu\right)$.
Follows from and .
From the above discussion, we see that if $V:L^{2}\left(\mu\right)\rightarrow H_{2}\left(\mathbb{D}^{2}\right)$ is an isometry, then $\left\{ g_{n}:=V^{*}\left(z^{n}\right)\right\} _{n\in\mathbb{N}_{0}^{2}}$ is a Parseval frame in $L^{2}\left(\mu\right)$. This implication holds in general. Since there are “many” such isometries, it follows that there are “many” Parseval frames. For more details, see [@2016arXiv160308852H; @MR3796641; @HERR2018] and the reference therein.
Let $K$ be a kernel on $\mathbb{D}^{d}$, and $\mu$ be a measure on $\mathbb{T}^{d}$. Then for all $z\in\mathbb{D}^{d}$, we have $\lim_{r\rightarrow1}K\left(z,re\left(x\right)\right)=K^{\ast}\left(z,x\right)$, a.a. $x\in\mathbb{T}^{d}$; and $$V_{\mu}^{*}\left(K\left(\cdot,z\right)\right)=K^{\ast}\left(z,x\right),$$ a.a. $x$ w.r.t. $\mu$.
\[sec:gifs\]General iterated function system (IFS)-theory
=========================================================
In this section we turn to an analysis of the IFS measures (see e.g., [@MR625600; @MR1656855; @MR2319756; @MR2431670; @2016arXiv160308852H; @MR3882025]), as introduced in Sections \[sec:Intro\] and \[sec:SSM\] (see (\[eq:E3\]) below). The notion of iterated function systems (IFS) for the case of measures fits the following general idea of patterns with self-similarity across different scales. Also here, the IFS-measures are created by recursive repetition of a simple process in an ongoing feedback loop.
Recall that an IFS measure is obtained from a recursive algorithm involving successive iteration of a finite system of maps in a metric space. IFS systems are self-similar because the same fixed choice of scaling mappings is used in each step of the algorithm. (The simplest IFS measures arise from the standard Cantor construction applied to a finite interval. But the idea works much more generally.) Then the chosen finite index-set for the mappings is called an alphabet, denoted $B$. We shall analyze here the IFS measures with the aid of symbolic dynamics on a probability space $\Omega$, made up of infinite words in $B$. Then a fixed choice of probability weights on $B$ leads to an associated infinite product measure, called $\mathbb{P}$, on $\Omega$, see (\[eq:d2\]). By Kakutani’s theorem, distinct weights yield mutually singular infinite product measures.
We shall construct a random variable $X$ on $\Omega$ such that the IFS then arises as the image under $X$, and the IFS measure $\mu$ becomes the distribution of $X$. Intuitively, $X$ is an infinite address map; see also eq (\[eq:d2\]) and . While the choice of such system of maps could be rather general, we shall restrict attention here to the case of a finite number of contractive affine mappings in $\mathbb{R}^{d}$, $d$ fixed; see e.g., (\[eq:F2\]) for the case of the standard Sierpinski gasket, where $d=2$. In this case, the associated maximal entropy measure $\mu$ (see (\[eq:si3\])) is a probability measure prescribed by the uniform distribution on $B$.\
Let $\left(M,d\right)$ be a complete metric space. Fix an alphabet $B=\left\{ b_{1},\cdots,b_{N}\right\} $, $N\geq2$, and let $\left\{ \tau_{b}\right\} _{b\in B}$ be a contractive IFS with attractor $W\subset M$, i.e., $$W=\bigcup_{b}\tau_{b}\left(W\right).\label{eq:e1}$$ In fact, $W$ is uniquely determined by (\[eq:e1\]).
Let $\left\{ p_{b}\right\} _{b\in B}$, $p_{b}>0$, $\sum_{b\in B}p_{b}=1$, be fixed. Set $\Omega=B^{\mathbb{N}}$, equipped with the product topology. Let $$\mathbb{P}=\vartimes{}_{1}^{\infty}p=\underset{\aleph_{0}\text{ product measure}}{\underbrace{p\times p\times p\cdots\cdots}}\label{eq:d2}$$ be the infinite-product measure on $\Omega$ (see [@MR0014404; @MR562914]).
In this section, we construct a random variable $X:\Omega\rightarrow M$ with value in $M$ (a measure space $\left(M,\mathscr{B}_{M}\right)$), such that the distribution $\mu:=\mathbb{P}\circ X^{-1}$ is a Borel probability measure supported on $W$, satisfying $$\mu=\sum_{b\in B}p_{b}\,\mu\circ\tau_{b}^{-1}.\label{eq:E3}$$ That is, $\mu$ is the IFS measure.
\[thm:pp\]For points $\omega=\left(b_{i_{1}},b_{i_{2}},b_{i_{3}},\cdots\right)\in\Omega$ and $k\in\mathbb{N}$, set $$\begin{aligned}
\omega\big|_{k} & =\left(b_{i_{1}},b_{i_{2}},\cdots,b_{i_{k}}\right),\;\text{and}\label{eq:d3}\\
\tau_{\omega|_{k}} & =\tau_{b_{i_{k}}}\circ\cdots\circ\tau_{b_{i_{2}}}\circ\tau_{b_{i_{1}}}.\label{eq:d4}\end{aligned}$$ Then $\bigcap_{k=1}^{\infty}\tau_{\omega|_{k}}$$\left(M\right)$ is a singleton, say $\left\{ x\left(\omega\right)\right\} $. Set $X\left(\omega\right)=x\left(\omega\right)$, i.e., $$\left\{ X\left(\omega\right)\right\} =\bigcap_{k=1}^{\infty}\tau_{\omega|_{k}}\left(M\right);\label{eq:d5}$$ then:
1. $X:\Omega\rightarrow M$ is an $\left(M,d\right)$-valued random variable.
2. \[enu:pp2\]The distribution of $X$, i.e., the measure $$\mu=\mathbb{P}\circ X^{-1}\label{eq:d6}$$ is the unique Borel probability measure on $\left(M,d\right)$ satisfying: $$\mu=\sum_{b\in B}p_{b}\,\mu\circ\tau_{b}^{-1};\label{eq:d7}$$ equivalently, $$\int_{M}fd\mu=\sum_{b\in B}p_{b}\int_{M}\left(f\circ\tau_{b}\right)d\mu,\label{eq:d8}$$ holds for all Borel functions $f$ on $M$.
3. The support $W_{\mu}=supp\left(\mu\right)$ is the minimal closed set (IFS), $\neq\emptyset$, satisfying $$W_{\mu}=\bigcup_{b\in B}\tau_{b}\left(W_{\mu}\right).\label{eq:d9}$$
We shall make use of standard facts from the theory of iterated function systems (IFS), and their measures; see e.g., [@MR625600; @MR1656855; @MR2431670].
*Monotonicity*: When $\omega\in\Omega$ is fixed, then $\tau_{\omega|_{k}}\left(M\right)$ is a monotone family of compact subsets in $M$ s.t. $$\tau_{\omega|_{k+1}}\left(M\right)\subset\tau_{\omega|_{k}}\left(M\right).\label{eq:d10}$$ Since $\tau_{b}$ is strictly contractive for all $b\in B$, we get $$\lim_{k\rightarrow\infty}diameter\left(\tau_{\omega|_{k}}\left(M\right)\right)=0,\label{eq:d11}$$ and so the intersection in (\[eq:d5\]) is a singleton depending only on $\omega$.
*The $\sigma$-algebras on $\left(\Omega,\mathbb{P}\right)$ and $\left(X,d\right)$*: The $\sigma$-algebra of subsets of $\Omega$ is generated by cylinder sets. Specifically, if $f=\left(b_{i_{1}},b_{i_{2}},\cdots,b_{i_{k}}\right)$ is a *finite word*, the corresponding cylinder set is $$E\left(f\right)=\left\{ \omega\in\Omega\mid\omega_{j}=b_{i_{j}},\;1\leq j\leq k\right\} \subset\Omega.\label{eq:d12}$$ The Borel $\sigma$-algebra on $M$ is determined from the fixed metric $d$ on $M$.
The measure $\mathbb{P}\left(=\mathbb{P}_{p}\right)$ is specified by its values on cylinder sets; i.e, set $$\mathbb{P}\left(E\left(f\right)\right)=p_{b_{i_{1}}}p_{b_{i_{2}}}\cdots p_{b_{i_{k}}}=:p_{f}.\label{eq:d13}$$ Also see e.g., [@MR735967].
*Proof of (\[eq:d7\])*. The argument is based on the following: On $\Omega$, introduce the *shifts* $\widetilde{\tau}_{b}\left(b_{i_{1}},b_{i_{2}},b_{i_{3}},\cdots\right)=\left(b,b_{i_{1}},b_{i_{2}},b_{i_{3}},\cdots\right)$, $b\in B$. Let $X$ be as in (\[eq:d5\])-(\[eq:d6\]), then $$\tau_{b}\circ X=X\circ\widetilde{\tau}_{b},\label{eq:d14}$$ which is immediate from (\[eq:d5\]).
$$\xymatrix{\Omega\ar[rr]^{X}\ar[d]_{\widetilde{\tau}_{b}} & & M\ar[d]^{\tau_{b}}\\
\Omega\ar[rr]_{X} & & M
}$$
We now show (\[eq:d8\]), equivalently (\[eq:d7\]). Let $f$ be a Borel function on $M$, then $$\begin{aligned}
{2}
\int_{M}f\,d\mu & =\int_{\Omega}\left(f\circ X\right)d\mathbb{P} & \quad & \text{\ensuremath{\left(\text{by }\left(\ref{eq:d6}\right)\right)}}\\
& =\sum_{b\in B}p_{b}\int_{\Omega}f\circ X\circ\widetilde{\tau}_{b}\:d\mathbb{P} & & \left(\begin{matrix}\text{since \ensuremath{\mathbb{P}} is the product}\\
\text{ measure \ensuremath{\vartimes_{1}^{\infty}p}, see \ensuremath{\left(\ref{eq:d13}\right)}}
\end{matrix}\right)\\
& =\sum_{b\in B}p_{b}\int_{\Omega}f\circ\tau_{b}\circ X\:d\mathbb{P} & & \text{\ensuremath{\left(\text{by }\left(\ref{eq:d14}\right)\right)}}\\
& =\sum_{b\in B}p_{b}\int_{M}f\circ\tau_{b}\:d\mu & & \text{\ensuremath{\left(\text{by }\left(\ref{eq:d6}\right)\right)}}\end{aligned}$$ which is the desired conclusion.
In general, the random variable $X:\Omega\rightarrow W$ (see (\[eq:d5\])) is not 1-1, but it is always onto. It is 1-1 when the IFS is non-overlap; see below.
\[def:no\]We say that $\left(\tau_{b},W\right)$ is “non-overlap” iff for all $b,b'\in B$, with $b\neq b'$, we have $\tau_{b}\left(W\right)\cap\tau_{b'}\left(W\right)=\emptyset$.
\[cor:kuk\]Assume $p\neq p'$, i.e., $p_{b}\neq p_{b}'$, for some $b\in B$. (Recall that $\sum_{b\in B}p_{b}=\sum_{b\in B}p_{b}'=1$, $p_{b}$, $p_{b}'>0$.) Let $\mathbb{P}=\vartimes_{1}^{\infty}p$, and $\mathbb{P}'=\vartimes_{1}^{\infty}p'$ be the corresponding infinite product measures; and let $\mu=\mathbb{P}\circ X^{-1}$, $\mu'=\mathbb{P}'\circ X^{-1}$ be the respective distributions. Then $\mu$ and $\mu'$ are mutually singular.
This is an application of Kakutani’s theorem on infinite product measures. See [@MR0014404; @MR0023331].
Let $B=\left\{ b_{1},\cdots,b_{N}\right\} $ be a subset of $\mathbb{R}^{d}$, and fix a $d\times d$ matrix $M$. Assume $M$ is expansive, i.e., $\left|\lambda\right|>1$, for all eigenvalues $\lambda$ of $M$. Then the mapping $\Omega=\left\{ 1,\cdots,N\right\} ^{\mathbb{N}}\xrightarrow{\;X\;}W_{B}$ from (\[eq:d5\]) is given by $$\omega=\left(i_{1},i_{2},i_{3}\cdots\right)\longmapsto x:=\sum_{j=1}^{\infty}M^{-j}b_{i_{j}}.$$ Note that $x$ has a random expansion, with the alphabets $b_{i}\in B$, as a sequence of i.i.d. random variables with distribution $p=\left(p_{1},\cdots,p_{N}\right)$.
\[sec:Sie\]Sierpinski and random power series
=============================================
Given a probability measure $\mu$ on $I^{d}$ where $I=\left[0,1\right]$, a key property that $\mu$ may, or may not, have is that the Fourier frequencies $\left\{ e_{n}\right\} _{n\in\mathbb{N}^{d}}$ are *total* in $L^{2}\left(\mu\right)$, i.e., that the closed span of $\left\{ e_{n}\right\} _{n\in\mathbb{N}^{d}}$ is $L^{2}\left(\mu\right)$.
The result in $d=1$, that, if $\nu$ on $I$ is singular, then the set $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}}$ is total in $L^{2}\left(\nu\right)$, fails for $d=2$. There are examples when $\mu$ on $I^{2}$ is positive, singular w.r.t. the 2D Lebesgue measure, but $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}^{2}}$ is *not* total in $L^{2}\left(\mu\right)$.
Take $\mu=\lambda_{1}\times\nu$ (see ), where $\lambda_{1}$ is Lebesgue measure and $\nu$ is a singular measure in $I$, then $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}^{2}}$ is *not* total in $L^{2}\left(\mu\right)$.
![\[fig:pm\]$\lambda_{1}=$ Lebesgue, $\nu\perp\lambda_{1}$](pm){width="0.2\columnwidth"}
For the Sierpinski case (affine IFS), with the Sierpinski measure $\mu$, total does hold in $L^{2}\left(\mu\right)$. See details below.
Let the alphabets be $$B=\left\{ b_{0},b_{1},b_{2}\right\} :=\left\{ \begin{bmatrix}0\\
0
\end{bmatrix},\begin{bmatrix}1\\
0
\end{bmatrix},\begin{bmatrix}0\\
1
\end{bmatrix}\right\} .\label{eq:F1}$$ Set $$M=\begin{bmatrix}2 & 0\\
0 & 2
\end{bmatrix},\;\text{and}\quad\tau_{j}\left(x\right)=M^{-1}\left(x+b_{j}\right).\label{eq:F2}$$ The Sierpinski gasket () is the IFS attractor $W$ satisfying $$W=\bigcup_{j=0}^{2}\tau_{j}\left(W\right).$$
-------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------
![\[fig:isp\]Construction of the Sierpinski gasket.](isp0 "fig:"){width="0.2\columnwidth"} ![\[fig:isp\]Construction of the Sierpinski gasket.](isp1 "fig:"){width="0.2\columnwidth"} ![\[fig:isp\]Construction of the Sierpinski gasket.](isp2 "fig:"){width="0.2\columnwidth"}
![\[fig:isp\]Construction of the Sierpinski gasket.](isp3 "fig:"){width="0.2\columnwidth"} ![\[fig:isp\]Construction of the Sierpinski gasket.](isp4 "fig:"){width="0.2\columnwidth"} ![\[fig:isp\]Construction of the Sierpinski gasket.](isp5 "fig:"){width="0.2\columnwidth"}
-------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------
We have the random variable $B^{\mathbb{N}}\xrightarrow{\;X\;}W$, given by $$\omega=\left(b_{i_{1}},b_{i_{2}},b_{i_{3}},\cdots\right)\longmapsto x=\sum_{k=1}^{\infty}M^{-k}b_{i_{k}}.\label{eq:F3}$$
As a Cantor set, $W$ (the Sierpinski gasket) is the boundary of the tree symbol representation; see .
![\[fig:wtree\]Symbol representations of infinite words.](wtree){width="0.7\columnwidth"}
Recall that every $\omega\in B^{\mathbb{N}}$ is an infinite word $\omega=\left(b_{i_{1}},b_{i_{2}},b_{i_{3}},\cdots\right)$, with $i_{k}\in\left\{ 0,1,2\right\} $. Setting $\omega\big|_{n}=\left(b_{i_{1}},\cdots,b_{i_{n}}\right)$, a finite truncated word, and $\tau_{\omega|_{n}}=\tau_{i_{n}}\circ\cdots\circ\tau_{i_{1}}$; then $\bigcap_{n}\tau_{\omega|_{n}}\left(W\right)=\left\{ x\right\} $, i.e., the intersection is a singleton. And we set $X\left(\omega\right)=x$.
Let $p$ be the probability distribution on $B$, where $$p=\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right).\label{eq:F4}$$ Let $\mathbb{P}=\vartimes_{1}^{\infty}p$, and $\mu=\mathbb{P}\circ X^{-1}$ be the corresponding IFS measure, i.e., $\mu$ is the unique Borel probability measure on $W$, s.t. $$d\mu=\frac{1}{3}\sum_{j=0}^{2}\mu\circ\tau_{j}^{-1}.\label{eq:si3}$$ See for details.
1. The Hausdorff dimension of $W$ is $\ln3/\ln2$, where $3=\#\left\{ B\right\} $ and $2=$ scaling number.
2. Let $O_{j}$ be the triangles removed from the $j^{th}$ iteration (), and let $O=\bigcup_{j=1}^{\infty}O_{j}$. Then, $$\lambda_{2}\left(O\right)=\lambda_{2}\left(\bigcup\nolimits _{j=1}^{\infty}O_{j}\right)=\frac{1}{2}\left[\frac{1}{4}+\frac{3}{4^{2}}+\frac{3^{2}}{4^{3}}+\cdots\right]=\frac{1}{2};$$ and so $\lambda_{2}\left(W\right)=0$, where $\lambda_{2}$ denotes the 2D Lebesgue measure. Note that $\mu\left(W\right)=1$.
Let $W$ be the Sierpinski gasket, and $\mu$ be the corresponding IFS measure. Let $\widehat{\mu}$ be the Fourier transform of $\mu$, i.e., $\widehat{\mu}\left(\lambda\right):=\int_{W}e^{i2\pi\lambda\cdot x}d\mu\left(x\right)$. Then $$\widehat{\mu}\left(\lambda\right)=\frac{1}{3}\left[1+e^{i\pi\lambda_{1}}+e^{i\pi\lambda_{2}}\right]\widehat{\mu}\left(\lambda/2\right),\label{eq:sm}$$ where $\lambda=\left(\lambda_{1},\lambda_{2}\right)\in\mathbb{R}^{2}$.
Immediate from (\[eq:si3\]). More specifically, we have $$\begin{aligned}
\widehat{\mu}\left(\lambda\right) & =\frac{1}{3}\sum_{j=0}^{2}\int_{W}e^{i2\pi\lambda\cdot\tau_{j}\left(x\right)}d\mu\left(x\right)\\
& =\frac{1}{3}\Big(\int_{W}e^{i2\pi\lambda\cdot x/2}d\mu\left(x\right)+\int_{W}e^{i2\pi\lambda/2\cdot\left(x+\left(1,0\right)\right)}d\mu\left(x\right)\\
& \qquad+\int_{W}e^{i2\pi\lambda/2\cdot\left(x+\left(0,1\right)\right)}d\mu\left(x\right)\Big)\\
& =\frac{1}{3}\left(1+e^{i\pi\lambda_{1}}+e^{i\pi\lambda_{2}}\right)\widehat{\mu}\left(\lambda/2\right),\end{aligned}$$ which is the assertion (\[eq:sm\]).
By general theory (see ), the IFS measure $\mu$ as in (\[eq:si3\]) has a disintegration $$d\mu=\int_{0}^{1}\sigma^{x}\left(dy\right)d\xi\left(x\right),\label{eq:si4}$$ where $$\xi=\mu\circ\pi_{1}^{-1}\label{eq:si5}$$ with $supp\left(\xi\right)\subset\left[0,1\right]$. Note, if $S\subset\left[0,1\right]$ is a measurable subset, then $$\xi\left(S\right)=\mu\left(\left\{ \left(x,y\right)\mid x\in S\right\} \right).\label{eq:si6}$$
\[lem:sW\]Let $W$ be the Sierpinski gasket. Then points in $W$ are represented as random power series $$\begin{bmatrix}x\\
y
\end{bmatrix}\in W\Longleftrightarrow\left\{ \begin{matrix}x=\sum_{k=1}^{\infty}\varepsilon_{k}2^{-k}\\
y=\sum_{k=1}^{\infty}\eta_{k}2^{-k}
\end{matrix}\right.\label{eq:F9}$$ where $\left(\varepsilon_{k}\right),\left(\eta_{k}\right)$ are defined on $\Omega=\left\{ 0,1\right\} ^{\mathbb{N}}$, i.e., the binary probability space.
Moreover, $\varepsilon_{k}$ is i.i.d. on $\left\{ 0,1\right\} $, $k\in\mathbb{N}$, with distribution $\left(2/3,1/3\right)$. That is, $Prob\left(\varepsilon_{k}=0\right)=2/3$, and $Prob\left(\varepsilon_{k}=1\right)=1/3$. The same conclusion holds for $\eta_{k}$ as well.
This follows from (\[eq:F3\]) and (\[eq:F4\]).
In detail, let $X:B^{\mathbb{N}}\rightarrow W$ be the random variable from (\[eq:F3\]), $X\left(\omega\right)=\sum_{k=1}^{\infty}M^{-k}b_{i_{k}}$, for all $\omega\in B^{\mathbb{N}}$; then $$\begin{aligned}
W\ni\begin{bmatrix}x\\
y
\end{bmatrix} & =X\left(\omega\right)=\sum_{k=1}^{\infty}\begin{bmatrix}2^{-k} & 0\\
0 & 2^{-k}
\end{bmatrix}b_{i_{k}}\\
& =\sum_{k=1}^{\infty}\begin{bmatrix}2^{-k} & 0\\
0 & 2^{-k}
\end{bmatrix}\left\{ \begin{bmatrix}0\\
0
\end{bmatrix},\begin{bmatrix}1\\
0
\end{bmatrix},\begin{bmatrix}0\\
1
\end{bmatrix}\right\} \\
& =\begin{bmatrix}\sum_{k=1}^{\infty}2^{-k}\varepsilon_{k}\left(x\right)\\
\sum_{k=1}^{\infty}2^{-k}\eta_{k}\left(x\right)
\end{bmatrix},\end{aligned}$$ where $$\begin{aligned}
Pr\left(\varepsilon_{k}=0\right) & =Pr\left(\eta_{k}=0\right)=2/3,\\
Pr\left(\varepsilon_{k}=1\right) & =Pr\left(\eta_{k}=1\right)=1/3.\end{aligned}$$ Also we have the following conditional probabilities: $$\begin{aligned}
Pr\left(\eta_{k}=0\mid\varepsilon_{k}=0\right) & =1/2,\\
Pr\left(\eta_{k}=1\mid\varepsilon_{k}=0\right) & =1/2,\\
Pr\left(\eta_{k}=0\mid\varepsilon_{k}=1\right) & =1.\end{aligned}$$ One checks that $$\begin{aligned}
Pr\left(\eta_{k}=0\right) & =Pr\left(\eta_{k}=0\mid\varepsilon_{k}=0\right)Pr\left(\varepsilon_{k}=0\right)\nonumber \\
& \qquad+Pr\left(\eta_{k}=0\mid\varepsilon_{k}=1\right)Pr\left(\varepsilon_{k}=1\right)=\frac{1}{2}\cdot\frac{2}{3}+1\cdot\frac{1}{3}=\frac{2}{3},\label{eq:F10}\\
Pr\left(\eta_{k}=1\right) & =Pr\left(\eta_{k}=1\mid\varepsilon_{k}=0\right)Pr\left(\varepsilon_{k}=0\right)\nonumber \\
& \qquad+Pr\left(\eta_{k}=1\mid\varepsilon_{k}=1\right)Pr\left(\varepsilon_{k}=1\right)=\frac{1}{2}\cdot\frac{2}{3}+0\cdot\frac{1}{3}=\frac{1}{3}.\label{eq:F11}\end{aligned}$$ See the diagram in .
$$\xymatrix{ & \cdot\ar@/^{1pc}/[rd]^{1/3}\ar@/_{1pc}/[ld]_{2/3}\\
0\ar[d]_{1/2}\ar[rrd]_{1/2} & & 1\ar[d]^{1}\\
1 & & 0
}$$
\[lem:sW2\]Let $\mu$ be the IFS measure of the Sierpinski gasket as above, and $\xi=\mu\circ\pi_{1}^{-1}$ be as in (\[eq:si4\])–(\[eq:si6\]), so that $\mu$ has the disintegration in (\[eq:si4\]).
1. Then the measure $\xi$ is singular and non-atomic. More precisely, $\xi$ is the product measure $\vartimes_{1}^{\infty}\left\{ 2/3,1/3\right\} $ defined on $\left\{ 0,1\right\} ^{\mathbb{N}}$.
2. For a.a. $x$ w.r.t $\xi$, the measure $\sigma^{x}\left(dy\right)$ (in the $y$-variable) is singular. Hence $\mu$ is slice singular (see ), and $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}^{2}}$ is total in $L^{2}\left(\mu\right)$.
For all points $\left(x,y\right)\in W$, let $x=\sum_{k=1}^{\infty}\varepsilon_{k}2^{-k}$, $y=\sum_{k=1}^{\infty}\eta_{k}2^{-k}$ be as in (\[eq:F9\]). Then (\[eq:F10\]) & (\[eq:F11\]) hold for all $x\in I$.
Therefore, we get the product measure $\xi=\vartimes_{1}^{\infty}\left\{ 2/3,1/3\right\} $ on the space $\Omega=\vartimes_{1}^{\infty}\left\{ 0,1\right\} $; see . By contrast, $\lambda=\vartimes_{1}^{\infty}\left\{ 1/2,1/2\right\} $ is Lebesgue measure; hence $\xi$ and $\lambda$ are mutually singular by Kakutani’s thoerem. (See above.)
Note that, for a.a. $x$, the measure $\sigma^{x}\left(dy\right)$ is the middle interval gap supported on $A\left(x\right)=\left\{ y\mid\left(x,y\right)\in W\right\} $, and we conclude that $\sigma^{x}\left(dy\right)$ is singular w.r.t. Lebesgue measure for a.a. $x$. By , it follows that $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}^{2}}$ is total in $L^{2}\left(\mu\right)$.
[cccc]{} ![\[fig:spt\]The measure $\xi$, or $d\xi\left(x\right)$ as an infinite product measure.](xi1 "fig:"){width="0.2\columnwidth"} & ![\[fig:spt\]The measure $\xi$, or $d\xi\left(x\right)$ as an infinite product measure.](xi2 "fig:"){width="0.2\columnwidth"} & ![\[fig:spt\]The measure $\xi$, or $d\xi\left(x\right)$ as an infinite product measure.](xi3 "fig:"){width="0.2\columnwidth"} & ![\[fig:spt\]The measure $\xi$, or $d\xi\left(x\right)$ as an infinite product measure.](xi4 "fig:"){width="0.2\columnwidth"}[\
]{}step 1 & step 2 & step 3 & step 4[\
]{} & & & [\
]{}[\
]{}[\
]{}
\[rem:mca\]There is a Markov chain associated with the transition probabilities (see ). Note that $$\begin{bmatrix}2/3 & 1/3\end{bmatrix}\begin{bmatrix}1/2 & 1/2\\
1 & 0
\end{bmatrix}=\begin{bmatrix}2/3 & 1/3\end{bmatrix},$$ so the conditional expectation can be expressed as a Perron-Frobenius problem with the row vector $\begin{bmatrix}2/3 & 1/3\end{bmatrix}$ as a left Perron-Frobenius vector.
As another example, consider the fractal Eiffel Tower $W_{Ei}$ (see ). In this case, we have $$M=\begin{bmatrix}2 & 0 & 0\\
0 & 2 & 0\\
0 & 0 & 2
\end{bmatrix},\quad B=\left\{ \begin{bmatrix}0\\
0\\
0
\end{bmatrix},\begin{bmatrix}1\\
0\\
0
\end{bmatrix},\begin{bmatrix}0\\
1\\
0
\end{bmatrix},\begin{bmatrix}0\\
0\\
1
\end{bmatrix}\right\} ,$$ and $p=\left(1/4,1/4,1/4,1/4\right)$. It follows that each coordinate of points in $W_{Ei}$ has representation $\sum_{k=1}^{\infty}\varepsilon_{k}2^{-k}$, where $\left\{ \varepsilon_{k}\right\} $ is i.i.d. with $Pr\left(\varepsilon_{k}=0\right)=3/4$, and $Pr\left(\varepsilon_{k}=1\right)=1/4$. The transition probabilities are given by the diagram below. $$\xymatrix{ & \cdot\ar@/_{1pc}/[dl]_{3/4}\ar@/^{1pc}/[dr]^{1/4}\\
0\ar[d]_{1/3}\ar[rrd]^{2/3} & & 1\ar[d]^{1}\\
1 & & 0
}$$ One checks that $$\begin{bmatrix}3/4 & 1/4\end{bmatrix}\begin{bmatrix}2/3 & 1/3\\
1 & 0
\end{bmatrix}=\begin{bmatrix}3/4 & 1/4\end{bmatrix}.$$
[cccc]{} ![\[fig:fei\]Construction of the fractal Eiffel Tower.](eif0 "fig:"){width="0.2\columnwidth"} & ![\[fig:fei\]Construction of the fractal Eiffel Tower.](eif1 "fig:"){width="0.2\columnwidth"} & ![\[fig:fei\]Construction of the fractal Eiffel Tower.](eif2 "fig:"){width="0.2\columnwidth"} & ![\[fig:fei\]Construction of the fractal Eiffel Tower.](eif3 "fig:"){width="0.2\columnwidth"}[\
]{}step 0 & step 1 & step 2 & step 3[\
]{} & & & [\
]{}[\
]{}[\
]{}
Given an affine contractive IFS measure $\mu$ supported in $\left[0,1\right]^{d}$, let $T=\left(T_{ij}\right)$ be the corresponding Markov transition matrix. Then the following are equivalent:
1. The Fourier frequencies $\left\{ e_{n}\right\} _{n\in\mathbb{N}_{0}^{d}}$ are total in $L^{2}\left(\mu\right)$.
2. The Perron-Frobenius vector $v$ ($vT=v$, or $\sum_{j}v_{j}T_{ji}=v_{i}$) is non-constant, i.e., not proportional to $\left(1,1,\cdots,1\right)$.
In the above, we carried out all the detailed computation justifying our conclusions for the case of the Sierpinski gasket, (A). Recall that (B) represents the Sierpinski carpet, a close cousin; and the reader will be able to fill in details from inside the section, spelling out the changes from (A) to (B). In case (B), naturally, the particular affine transformations (\[eq:F1\])–(\[eq:F2\]) are a bit different (i.e., for case (B)), but they are of the same nature. In particular, it follows that the maximal entropy (IFS) measure for the Sierpinski carpet is also slice-singular. Moreover, the other conclusions from Lemmas \[lem:sW\], and \[lem:sW2\], and , carry over from case (A) to case (B), *mutatis mutandis*. As the underlying ideas and methods involved are the same, interested readers will be able to fill in details.
Moreover the above remarks, regarding extension of the conclusions for case (A) to that of (B), also apply *mutatis mutandis*, to the case of , the fractal Eiffel Tower. There again, we conclude that the associated maximal entropy (IFS) measure is also slice-singular.
The co-authors thank the following colleagues for helpful and enlightening discussions: Professors Daniel Alpay, Sergii Bezuglyi, Ilwoo Cho, Wayne Polyzou, Eric S. Weber, and members in the Math Physics seminar at The University of Iowa.
|
---
abstract: 'We study betweenness preserving mappings (we call them *monotone*) on subsets of the plane. We show, in particular, that an open planar set cannot be mapped in a one-to-one monotone way into the real line.'
author:
- |
[Wies[ł]{}aw Kubiś]{}\
[Institute of Mathematics,]{} [Czech Academy of Sciences]{}\
- |
[Janusz Morawiec]{}\
[Institute of Mathematics, University of Silesia in Katowice, Poland]{}\
- |
[Thomas Z" urcher]{}\
[Institute of Mathematics, University of Silesia in Katowice, Poland]{}
date:
title: Monotone mappings and lines
---
Introduction
============
In the abstract setting, a *betweenness* is a ternary relation ${\flat}$ on a set $X$, satisfying some natural axioms. There is quite a lot of literature on this topic, see e.g. the survey [@Pamb]. We are actually interested in *euclidean* and *linear* betweenness only. Euclidean betweenness in a real vector space is the standard one: $x$ is between $a$ and $b$ (denoted by ${\flat}(a,x,b)$) if and only if $x = (1-{{\lambda}})a + {{\lambda}}b$ for some $0{\leqslant}{{\lambda}}{\leqslant}1$, namely, $x$ belongs to the line segment joining $a$ and $b$. This generalizes to vector spaces over arbitrary ordered fields, however we are not going to explore this direction. Linear betweenness is the one induced by a fixed linear ordering: $x$ is between $a$ and $b$ if and only if $a {\leqslant}x {\leqslant}b$ or $b {\leqslant}x {\leqslant}a$. In the real line, the euclidean and the linear betweenness are of course the same.
Given two betweenness structures $(X,{\flat}_X)$, $(Y,{\flat}_Y)$, a mapping ${f\colon X \to Y}$ will be called *monotone* if it preserves the betweenness, that is, $${\flat}_X(a,x,b) \implies {\flat}_Y(f(a),f(x),f(b))$$ for every $a,x,b \in X$. Note that if ${\flat}_X$, ${\flat}_Y$ are linear then this notion coincides with the usual one: A mapping is monotone if and only if it is increasing or decreasing (in the non-strict sense). A monotone *isomorphism* is a monotone bijection whose inverse is monotone. Every monotone bijection between linearly ordered sets is an isomorphism, but this is not the case for euclidean betweenness: Take $X = \{a,b,c\} {\subseteq}{{{\mathbb}{R}}}^2$, where $a,b,c$ are not on a single line, and $Y = \{a',b',c'\}$, where $a',b',c'$ are collinear; then any bijection from $X$ to $Y$ is monotone, while its inverse is not monotone. In fact, among subsets of the plane one can easily find those in which the betweenness is *discrete*, that is, ${\flat}(x,y,z)$ holds if and only if $y \in {\{x,z\}}$. A typical example is the circle $S^1$. Another example is any set $S$ with the property that $S \cap L$ contains exactly two points for every line $L {\subseteq}{{{\mathbb}{R}}}^2$. Such a set can be easily constructed by transfinite induction, knowing that each line has cardinality continuum. In particular, $S$ is isomorphic to $S^1$ and any bijection between these sets is a monotone isomorphism. This is a clear evidence that one should look at sets that are either convex or at least “resemble” convex sets, having sufficiently many collinear triples and perhaps satisfying some axioms involving intersections of line segments.
Affine mappings are obvious examples of monotone mappings between real vector spaces. Perhaps lesser known are projective transformations, defined on halfspaces only. We shall discuss them in Section \[SectGoodDobri\] below.
When it comes to arbitrary subsets of, say, the real plane, one has to note that the induced betweenness may easily become trivial. Again, a good example is any circle. In particular, every mapping defined on a circle is automatically monotone. When it comes to sets of small cardinality, one can concentrate on the plane, due to the following observation.
\[PropReducn2\] Let $X$ be a subset of a real vector space $V$, let $V_0$ be a $2$-dimensional linear subspace of $V$. If the cardinality of $X$ is strictly less than the continuum, then there exists a linear projection ${P\colon V \to V}$ such that ${P V} = V_0$ and $P {\restriction}X$ is a monotone isomorphism onto its image.
We address the question of classifying monotone mappings on “reasonable" (e.g. convex or with nonempty interior) subsets of euclidean spaces. As it happens, again the plane is crucial here.
In the literature one can find several works studying mappings which preserve collinearity, typically assuming injectivity and aiming to show that the mapping is affine or projective. This is closely related to the fundamental theorem on affine geometry: Every bijection of a real vector space mapping lines onto lines is affine. Removing the assumption of being one-to-one, one can easily find examples of (discontinuous) monotone mappings, see Section \[SectZlosciBad\] below.
The role of homographies {#SectGoodDobri}
========================
What are the “nice” monotone isomorphisms between convex sets? The first thing that comes to our mind is: affine transformations. These are precisely those monotone mappings that preserve not only the betweenness but also the one-dimensional barycentric coordinates of the point in an interval. Specifically, if $x = (1-{{\lambda}})a + {{\lambda}}b$ with ${{\lambda}}\in [0,1]$ and $f$ is affine, then $f(x) = (1-{{\lambda}})f(a) + {{\lambda}}f(b)$. No other monotone mappings have this property. Namely, suppose ${f\colon A \to B}$ preserves the one-dimensional coordinates and assume $A$ is convex. Then $f$ extends uniquely to an affine transformation ${\tilde f\colon V_A \to V_B}$, where $V_A$, $V_B$ denote the linear spans of $A$ and $B$, respectively.
Recall that a *homography* is an isomorphism of real projective spaces, that is, a bijection mapping lines onto lines. Note that in the real projective space the betweenness becomes ambiguous, since lines are actually circles. Betweenness can only be defined locally. In any case, every affine isomorphism extends to the projective space, however there are other projective transformations. Specifically, if $n>1$ is a natural number, then the general form of a projective transformation of $V = {{{\mathbb}{R}}}^n$ is $$f(x) = \frac 1 {\alpha + (v | x)} T(x),$$ where ${\alpha}\in {{{\mathbb}{R}}}$, $v \in V$, $T$ is an affine transformation, and $(v | x)$ denotes the usual (real) scalar product of $v$ and $x$. We can actually treat the formula above as the definition of a projective transformation[^1]. Note that it is defined on the union of the two halfspaces: $$H^+ = {\{x \in V\colon (v|x) > \alpha\}} {\qquad\text{and}\qquad}H^- = {\{x \in V\colon (v|x) < \alpha\}}.$$ It is easy to check that both $f {\restriction}H^+$ and $f {\restriction}H^-$ are monotone and $H^-$, $H^+$ are maximal convex sets on which $f$ is monotone. A mapping $f$ defined on a subset of ${{{\mathbb}{R}}}^n$ will be called a *partial projective transformation* (*partial homography*) if there is a projective transformation (homography) extending $f$.
\[THMhomographies\] Let $G {\subseteq}{{{\mathbb}{R}}}^2$ be a convex set, let ${f\colon G \to {{{\mathbb}{R}}}^2}$ be a monotone mapping with the following property:
1. There exist $a,b,c, d \in G$ such that $f(d) \in {\operatorname{int}}[f(a), f(b), f(c)]$.
Then $f$ is the restriction of a unique homography. In particular, $f$ is a homeomorphism and, at the same time, a monotone isomorphism onto its image.
First, note that there exists a unique homography $g$ defined on a half-plane containing the triangle $\Delta := [a,b,c]$, satisfying $f {\restriction}\{a,b,c\} = g {\restriction}\{a,b,c\}$. This is a general fact in projective geometry, however a self-contained proof can be found in [@ArtFloMil].
Now, using induction, we can see that $f$ and $g$ actually coincide on a dense subset $D$ of $\Delta$. Namely, $D$ is the minimal subset of $\Delta$ containing $a,b,c,d$, that is closed under intersections of line segments. More precisely, for every $x_0,x_1, y_0, y_1 \in D$, if $[x_0, x_1] \cap [y_0 , y_1]$ is a single point then this point belongs to $D$. Moreover, if there is a unique $z \in [x_0,x_1]$ such that $y_1 \in [y_0,z]$ then $z \in D$. It is easy to check that $D$ is indeed dense in $\Delta$. Here the fact that $d$ is in the interior of $\Delta$ is essential. It is clear that $f$ are uniquely determined on $D$ (being monotone mappings), therefore also $f {\restriction}D = g {\restriction}D$. What is perhaps even more important, $D$ is also dense in every side of $\Delta$.
Next, $f {\restriction}D = g {\restriction}D$ has a unique monotone extension to the boundary of $\Delta$. Finally, $f=g$ on the boundary of $\Delta$, therefore $f = g$ on $\Delta$. A monotone mapping is totally determined on an open set, therefore $f=g$ on $G$.
As a consequence, we see that the image of a monotone isomorphism of a triangle is again a triangle. This easily extends to higher dimensions:
Let $\Delta_m$ be an $m$-dimensional simplex, where $m {\geqslant}2$. Let ${f\colon \Delta_m \to V}$ be a one-to-one monotone mapping, where $V$ is a real vector space. Then ${f \Delta_m}$ is an $m$-dimensional simplex and $f$ is a partial homography.
First of all, note that $f$ is an isomorphism onto its image. Indeed, if $x,y,z$ are not collinear and $f(x), f(y), f(z)$ are collinear, then $f$ restricted to the triangle $[x,y,z]$ would be a one-to-one monotone mapping into a line segment, which contradicts Theorem \[THMtrojkatNieProsty\] below (its proof is totally self-contained, not using any results preceding it).
Let us start with the case $m=2$. Replacing $f$ by the composition with a suitable linear transformation, we may assume that ${f \Delta_2} {\subseteq}{{{\mathbb}{R}}}^2$. Now, since we know that $f$ is an isomorphism, the assumptions of Theorem \[THMhomographies\] are fulfilled, therefore $f$ is a partial homography. In particular, its image is the triangle spanned by the images of the vertices of $\Delta_2$. Clearly, the same conclusion is true if ${{{\mathbb}{R}}}^2$ is replaced by any other (not necessarily $2$-dimensional) real vector space.
Now assume $m>2$. By the inductive hypothesis, ${f F}$ is an $(m-1)$-dimensional simplex for each $(m-1)$-dimensional face $F {\subseteq}\Delta_m$. Thus, supposing ${f \Delta_m}$ is contained in an $(m-1)$-dimensional space, the Helly number contradicts the fact that $f$ is a monotone isomorphism. Namely, all $(m-1)$-dimensional faces of $\Delta_m$ have empty intersection, while their images have nonempty intersection, because every $m$ of them have nonempty intersection.
Recall that the Helly number is the minimal $k$ such that every finite family of convex sets with empty intersection contains at most $k$ sets with empty intersection. A classical result of Helly says that the Helly number of ${{{\mathbb}{R}}}^m$ is $m+1$.
Bad monotone mappings {#SectZlosciBad}
=====================
One can say that monotone mappings between linearly ordered sets are relatively well understood, at least when it comes to subsets of the real line. Concerning euclidean betweenness, projective transformations provide the class of “nice" monotone maps of convex sets. In fact, these are the continuous ones, when the domain is not contained in a line. On the other hand, there are many “bad" monotone mappings, namely, those that are neither continuous nor one-to-one. Our aim in this section is to classify “bad” monotone mappings of convex planar sets.
Below are some basic examples of discontinuous monotone mappings of a triangle.
\[ExmplDwaJedne\]
Let $\Delta = [a,b,c]$ be a triangle. Define ${f\colon \Delta \to \Delta}$ by setting $f(x)=x$, if $x \in [a,b]$ and $f(x) = c$, otherwise. It is rather clear that $f$ is monotone.
\[ExmplDwwaDwwaa\]
Let $\Delta = [a,b,c]$ be a triangle. Fix $v \in [b, c] \setminus \{b,c\}$. Define ${g\colon \Delta \to \Delta}$ by setting $g(x) = a$ if $x = a$; $g(x) = b$ if $x \ne a$ is in the triangle $[a,b,v]$; and $g(x) = c$, otherwise. It is easy to see that $g$ is monotone.
In the last example, one can take any planar convex set with at least one extreme point and with nonempty interior. Such a set admits a monotone mapping whose image consists of three non-collinear points.
\[FivePoints\] There exists a monotone mapping ${f\colon [0,1]^2 \to {{{\mathbb}{R}}}^2}$ whose image is $${{\mathbb{X}}}= \{ (0,0), (1,0), (0,1), (1,1), (1/2, 1/2) \}.$$ We define $f$ as follows: $$f(x_0,x_1) = \begin{cases}
(0,1) \quad & \text{if } x_0 < x_1,\\
(1,0) & \text{if } x_0 > x_1,\\
(1/2,1/2) & \text{if } 0 < x_0 = x_1 < 1,\\
(0,0) & \text{if } 0 = x_0 = x_1,\\
(1,1) & \text{if } x_0 = x_1 = 1.
\end{cases}$$ We check that $f$ is monotone. Fix $a,b \in [0,1]^2$ and $s \in [a,b]$. We may assume that $a \ne s \ne b$, which also rules out that $s$ is an extreme point of $[0,1]^2$. If $s$ is not in the diagonal $D = {\{(x,x)\colon 0 {\leqslant}x {\leqslant}1\}}$ then one of the points $a$, $b$ is on the same side of $D$ as $s$ (above or below) and hence $f(s) = f(a)$ or $f(s) = f(b)$. Finally, if $s \in D$ then, knowing that $s \ne (0,0)$ and $s \ne (1,1)$, we conclude that $f(s) = (1/2,1/2)$ and either ${f {\{a,b\}}} = {\{(0,1),(1,0)\}}$ or ${f {\{a,b\}}} = {\{(0,0),(1,1)\}}$.
Below is the announced classification.
\[THMgdyifgi\] Assume $G {\subseteq}{{{\mathbb}{R}}}^2$ is a convex set and ${f\colon G \to {{{\mathbb}{R}}}^2}$ is a monotone mapping. Then exactly one of the following possibilities occur.
1. ${f G} {\subseteq}L \cup {\{q\}}$, where $L {\subseteq}{{{\mathbb}{R}}}^2$ is a line and $q \in {{{\mathbb}{R}}}^2$.
2. ${\operatorname{int}}fG {\ne\emptyset}$.
3. ${f G} = \{a,b,c,d,e\}$, where $a,b,c,d$ are convex independent and ${\{e\}} = [a,c] \cap [b,d]$.
We already know from Theorem \[THMhomographies\] that (2) implies $f$ is a continuous monotone isomorphism, namely, restriction of a homography.
Obviously, (1) – (3) are mutually exclusive. Suppose $f$ satisfies neither (1) nor (2). By Theorem \[THMhomographies\], for every $x,y,z \in G$ either the points $f(x), f(y), f(z)$ are collinear or else $${f G} \cap {\operatorname{int}}[f(x), f(y), f(z)] = \emptyset.$$ We claim that ${f G}$ contains four convex independent points $a,b,c,d$. Suppose otherwise. Since ${f G}$ is not contained in a single line, there are non-collinear points $x,y,z \in {f G}$. Let $A = L_{x,y} \cup L_{x,z} \cup L_{y,z}$, where $L_{s,t}$ denotes the line passing through $s,t$. If there is $u \in {f G} \setminus A$ then the set $\{x,y,z,u\}$ is convex independent, because by the remarks above, none of these points is in the interior of the triangle spanned by the other three and we have found our pair of points. Now we assume ${f G} {\subseteq}A$. Again using the fact that $f$ does not satisfy (1), we find $u,v \in {f G} \setminus \{x,y,z\}$ such that $u$ and $v$ are on different lines forming the set $A$. Permuting $x,y,z$ if necessary, we may assume that $u \in L_{x,y}$ and $v \in L_{x,z}$. Checking all possible configurations, we conclude that there are always four convex independent points. Indeed, if $x \in [u,y]$ then necessarily $x \in [v,z]$ since otherwise either $v \in {\operatorname{int}}[u,y,z]$ or $z \in {\operatorname{int}}[u,y,v]$. Thus, in this case the set $\{u,v,y,z\}$ is convex independent. We get the same conclusion if $x \in [v,z]$ (this is symmetric). In the remaining cases we conclude that the set $\{u,v,y,z\}$ is convex independent.
We have proved that there is a convex independent set $\{a,b,c,d\} {\subseteq}{f G}$ consisting of four elements. Let $\{a',b',c',d'\} {\subseteq}G$ be such that $a = f(a')$, $b = f(b')$, $c = f(c')$, and $d = f(d')$. Then the four-element set $\{a',b',c',d'\}$ is convex independent, therefore, possibly permuting the points, we may assume that $[a',c'] \cap [b',d'] = {\{e'\}}$. Let $e = f(e')$. We claim that ${f G} = \{a,b,c,d,e\}$.
Clearly, ${\{e\}} = [a,c] \cap [b,d]$. Suppose $w \in {f G} \setminus \{a,b,c,d,e\}$. If $w$ is in the interior of the angle $\measuredangle(e,a,b)$ then $e$ is in the interior of the triangle $[c,d,w]$, a contradiction. The same arguments show that necessarily $w \in L_{a,c} \cup L_{b,d}$. However, in this case we find, checking all possible configurations (by symmetry, it suffices to check only the cases where $w\in [a,e]$ or $a\in [w,e]$) that the set $\{a,b,c,d,w\}$ contains a point that is in the interior of the triangle spanned by other three points, again obtaining a contradiction. This completes the proof.
One-to-one mappings into lines
==============================
In this section we address the question which planar sets admit a monotone one-to-one mapping into the real line. The first result exhibits a natural influence of the cardinality continuum.
Let $S$ be a subset of a real vector space $V$ and let $L$ be a one-dimensional linear subspace of $V$. If the cardinality of $S$ is strictly less than the continuum then there exists a linear projection ${P\colon V \to L}$ such that $P {\restriction}S$ is one-to-one. In particular, $S$ admits a one-to-one monotone mapping into the real line.
By Proposition \[PropReducn2\], we may assume that $V$ is 2-dimensional. Let ${{{\mathcal}{L}}}$ be the family of lines passing through at least two points of $S$. Then ${{{\mathcal}{L}}}$ has cardinality strictly less than the continuum, therefore there is a line $K$ passing through the origin, not parallel to any line from the family ${{{\mathcal}{L}}}$ and different from $L$. Let $P$ be the projection onto $L$ along $K$. Then $P{\restriction}S$ is obviously one-to-one.
The next result says that convex sets never have the above property, unless they are contained in a single line.
\[THMtrojkatNieProsty\] Let $G {\subseteq}{{{\mathbb}{R}}}^2$ be a set with nonempty interior, let ${f\colon G \to Y}$ be a one-to-one monotone mapping, where $Y$ is a linearly ordered set. Then $Y$ contains continuum many pairwise disjoint intervals, each of them of cardinality at least the continuum.
We may assume $G$ is open and convex, replacing it by a small enough open ball. We may also assume that $Y = {f G}$. Given $t \in Y$, define $$H_t = {{f}^{-1}{[f(a), t]}}.$$ Then $H_t$ is a nontrivial halfspace in $G$, therefore it is determined by a unique line $L_t$. Specifically, $H_t = U_t \cup I_t$, where $U_t$ is an open halfspace disjoint from $L_t$ and $I_t {\subseteq}L_t \cap G$ is an interval. Note that $I_t {\ne\emptyset}$, because, denoting by $s$ the unique point such that $f(s) = t$, we have that $H_t \setminus {\{s\}} = {{f}^{-1}{[f(a), t)}}$ is a hafspace, too, therefore it contains $U_t$. Now let $${{{\mathcal}{L}}}= {\{L_t\colon t \in Y\}}.$$ Note that $G {\subseteq}\bigcup {{{\mathcal}{L}}}$. Indeed, fix $x \in G$ and let $t = f(x)$. Then $L_t \in {{{\mathcal}{L}}}$ and $x \in L_t$. It follows that ${{{\mathcal}{L}}}$ has cardinality continuum.
Now suppose that $L_t \cap G = (c_t,d_t)$, where $f(c_t) < f(d_t)$. Let $J_t = {f (c_t,d_t)}$. We claim that if $s < t$ are such that $L_s \ne L_t$ then $J_s \cap J_t = \emptyset$. Indeed, in this case $(c_t, d_t)$ is is disjoint from $H_s$, therefore $f(d_s) < f(c_t)$. Finally, ${\{J_t\colon f(a) < t < f(b)\}}$ is a family of pairwise disjoint open intervals in $Y$ and it has cardinality $|{{{\mathcal}{L}}}|$, which is the continuum.
Note that a planar convex set with empty interior consists of collinear points, therefore the theorem above says precisely which planar convex sets admit a one-to-one monotone mapping into ${{{\mathbb}{R}}}$.
Concerning arbitrary planar sets, we do not know any characterization (it might be the case that no reasonable one exists). Let us first look at sets consisting of (or containing) finitely (or even countably) many lines.
Assume $S {\subseteq}{{{\mathbb}{R}}}^2$ consists of countably many pairwise parallel lines. Then $S$ admits a one-to-one mapping into the real line.
Assume $S = \bigcup_{i \in J}L_i$, where each $L_i$ is a horizontal line. We may assume that $J {\subseteq}{{{\mathbb}{R}}}$ and $L_i = {{{\mathbb}{R}}}+ (0,i)$ for $i \in J$ (we identify ${{{\mathbb}{R}}}$ with ${{{\mathbb}{R}}}\times {\{0\}}$). Let ${\{U_i\}_{i \in J}}$ be a family of pairwise disjoint open intervals in ${{{\mathbb}{R}}}$ satisfying $U_i < U_j$ whenever $i<j$, $i,j \in J$. This is possible, because $J$ is countable. More precisely, each $U_i$ may be chosen among open intervals removed during the construction of the standard Cantor set, since these intervals are ordered as the rational numbers.
Now let ${f\colon S \to {{{\mathbb}{R}}}}$ be such that $f {\restriction}L_i$ is an increasing mapping onto $U_i$. Then $f$ is monotone, because if $a,b,c$ are such that $B(a,c,b)$ and $a \in L_i$, $c \in L_j$, $b \in L_k$ then either $i {\leqslant}j {\leqslant}k$ or $k {\leqslant}j {\leqslant}i$.
Let $L_0, L_1, L_2$ be different lines in the plane ${{{\mathbb}{R}}}^2$ such that each two of them intersect, let $S = (L_0 \cup L_1 \cup L_2) \setminus B$, where $B$ is a bounded set. Assume $Y$ is a linearly ordered set and ${f\colon S \to Y}$ is monotone and one-to-one. Then $Y$ contains continuum many pairwise disjoint non-degenerate closed intervals. In particular, $S$ does not admit any one-to-one monotone mapping into ${{{\mathbb}{R}}}$.
An interval is *non-degenerate* if it contains at least two points.
We may assume that $B$ is convex, closed, and contains all the intersections of our three lines (replacing $B$ by a sufficiently big rectangle or a closed ball). Now each $L_i$ is divided into two half-lines $L_i^-$, $L_i^+$, where ${f L_i^-} < {f L_i^+}$. This is because $f$ is monotone therefore either preserves or reverses the natural ordering of $L_i \setminus B$.
Rotating the plane and possibly re-enumerating the lines, we may assume that $L_1$ is a horizontal line, $L_0^-$ is above $L_1$ and $L_2^-$ is below $L_1$. By this way, $L_0^+$ is below $L_1$ and $L_2^+$ is above $L_1$. It is easy to check that other configurations (assuming $L_1$ is horizontal) would violate the fact $f$ is a monotone injection. We may also assume that neither $L_0$ nor $L_2$ is vertical. By this way, each of our three lines has a natural *horizontal* linear ordering (namely, the one induced by the first coordinate) and $f$ preserves this ordering.
Given $x \in L_1^-$, denote by $p_2(x)$ the unique point in the intersection of $L_2$ with the line passing through $x$ parallel to $L_0$. Note that if $x$ is far enough from $B$ then $p_2(x) \in L_2^-$. Let us define $p_0(x) \in L_0$ in the same manner. Let us call $x$ *relevant* if both $p_2(x) \in L_2^-$ and $p_0(x) \in L_0^-$. Clearly, all points of $L_1^-$ that are far enough from $B$ are relevant.
Fix a relevant point $x \in L_1^-$. We claim that $f(y) > f(x)$ for every $y \in L_2^-$ above $p_2(x)$ in the horizontal ordering. Suppose otherwise, and fix a witness $y$. Let $z \in L_0^+$ be such that $x, y, z$ are collinear. Then $f(z) > f(y) > f(x)$. Now take $y'$ strictly between $p_2(x)$ and $y$ and let $z' \in L_0^+$ be such that $x, y', z'$ are collinear. Then $f(z') < f(z)$, although it should be the opposite. We have used the fact $y$ was close enough to $p_2(x)$, so that $z \notin B$, but this is not a problem, as one can always replace $y$ by a point closer to $p_2(x)$. We have also used the fact that $S \cap L_2$ is open.
Now fix a relevant point $x \in L_1^-$. Let $$H_x = {\{s \in S\colon f(s) {\leqslant}f(x)\}}.$$ Then $H_x$ is a halfspace in $S$ and by the remarks above, it is determined by a line $K_x$ either parallel to $L_0$ or $L_2$ (and then passing through either $p_0(x)$ or $p_2(x)$) or intersecting both $L_0^-$ and $L_2^-$. In particular, $K_x$ is not horizontal and $H_x$ contains all points of $S$ that are “on the left side" of the line $K_x$.
Let us call a relevant point $x \in L_1^-$ *tame* if $K_x$ is parallel to $L_0$ or $L_2$, otherwise let us call it *wild*.
Suppose that among the relevant points of $L_1^-$ continuum many of them are tame. Then there is a set $T {\subseteq}L_1^-$ of cardinality continuum such that for every $x \in T$ the line $K_x$ passes through $p_i(x)$ where $i\in {\{0,2\}}$ is fixed. Then ${\{[f(x), f(p_i(x))]\}_{x \in T}}$ is a family of pairwise disjoint non-degenerate closed intervals of $Y$.
Now suppose that there is a set $W {\subseteq}L_1^-$ of cardinality continuum, consisting of wild points. If there are distinct $x, x' \in W$ such that $K_x, K_{x'}$ passes through the same point of some $L_i$ ($i \in {\{0,2\}}$) then we proceed as follows. Reverting the picture, if necessary, assume $i=0$. Let $J$ be the interval between $x$ and $x'$. Then ${\{[y, p_2(y)]\}_{y\in J}}$ consists of pairwise disjoint non-degenerate closed intervals.
Finally, let us assume that the lines $K_x$, $x \in W$, do not intersect in $S$. Then ${\{[f(p_0(x)), f(p_1(x))]\}_{x \in W}}$ is a pairwise disjoint family of closed non-degenerate intervals. In the last case, we have that $f(x) \in [f(p_0(x)), f(p_2(x))]$, therefore these intervals have at least three distinct points.
The proof above seems to indicate that only the left-hand side ($L_0^- \cup L_1^- \cup L_2^-$) is significant. On the other hand, we have essentially used the fact that the right-hand side of $S$ is unbounded.
We shall now argue that the assertion of the theorem above cannot be improved much and moreover there may be no wild points. Namely, let ${{{\mathbb}{R}}}\cdot {{{\mathbb}{R}}}$ denote the lexicographic square of two copies of ${{{\mathbb}{R}}}$. Specifically, $(x,y) < (x',y')$ if and only if either $x < x'$ or else $x = x'$ and $y < y'$. Then the identity mapping from the euclidean plane ${{{\mathbb}{R}}}^2$ onto ${{{\mathbb}{R}}}\cdot {{{\mathbb}{R}}}$ is monotone, showing that the plane can actually be mapped to a sufficiently large linearly ordered set.
Now, consider again a set $S {\subseteq}{{{\mathbb}{R}}}^2$ consisting of three pairwise non-parallel lines. Rotating the plane, if necessary, we may assume that one of these lines is vertical. Then the identity mapping is monotone and onto the lexicographic sum $$({{{\mathbb}{R}}}\cdot 2) + {{{\mathbb}{R}}}+ ({{{\mathbb}{R}}}\cdot 2).$$ This in turn embeds into $D = {{{\mathbb}{R}}}\cdot 2$, the linearly ordered set often called the *double arrow*. Note that $D$ is separable in the interval topology, therefore it has no uncountable family of pairwise disjoint open intervals. In particular, no open convex planar set admits a one-to-one mapping into $D$.
It should be possible to show that a set consisting of $n$ lines always admits a one-to-one monotone mapping into ${{{\mathbb}{R}}}\cdot (n-1)$. How about the case where some of the lines are parallel?
\*\*\*
Asssume $S {\subseteq}{{{\mathbb}{R}}}^2$ consists of three bounded line segments intersecting in exactly one point. Then there exists a one-to-one monotone mapping ${f\colon S \to {{{\mathbb}{R}}}}$.
The construction is visualized in Figure \[FigTrziOdcinkis\].
(v1) at (0,10) ; (v2) at (2,-9) ; (v1.center) edge (v2.center); (v3) at (-8,2) ; (v4) at (8,0) ; (v3.center) edge (v4.center); (v6) at (8,4) ; (v5) at (-8,-4) ; (v5.center) edge (v6.center); (v3.center) edge (v2.center); (v5.center) edge (v1.center); (v3)–(v4); (v2)–(v3); (v5)–(v6); (v1)–(v5);
; ; (v2)–(C); ;
(v1)–(B);
;
(v2)–(E);
;
(v2.center) edge (C.center); (v2.center) edge (E.center);
(v1.center) edge (B.center); (v3)–(C) node\[midway,above\] [$1$]{}; (v5)–(B) node\[midway,below right\][$2$]{}; (C) – (E) node\[midway, above\][$3$]{}; (B) – (D) node\[midway, below\][$4$]{};
(v1)–(D); ; (v1)–(D);
\(E) – (G) node\[midway, above\][$5$]{}; (D)– (F) node\[midway, below\][$6$]{};
[AAFM12]{}
Shiri Artstein-Avidan, Dan Florentin, and Vitali Milman. Order isomorphisms on convex functions in windows. In [*Geometric aspects of functional analysis*]{}, volume 2050 of [*Lecture Notes in Math.*]{}, pages 61–122. Springer, Heidelberg, 2012.
Hanfried Lenz. Einige [A]{}nwendungen der projektiven [G]{}eometrie auf [F]{}ragen der [F]{}lächentheorie. , 18:346–359, 1958.
Victor Pambuccian. The axiomatics of ordered geometry [I]{}. [O]{}rdered incidence spaces. , 29(1):24–66, 2011.
[^1]: Typically, *projective transformation* is synonymous to *homography*, but there seems to be no special name for the more general, not necessarily one-to-one, variant. That is why we have decided to call them projective transformations, while only the bijective ones will be called homographies.
|
---
abstract: '[We present [*XMM-Newton*]{} spectral analysis of all 38 Seyfert galaxies from the Palomar spectroscopic sample of galaxies. These are found at distances of up to 67 Mpc and cover the absorbed 2-10 keV luminosity range $\sim 10^{38}-10^{43}$ . Our aim is to determine the distribution of the X-ray absorption in the local Universe. Three of these are Compton-thick with column densities just above $ 10^{24}$ and high equivalent width FeK$_{\alpha}$ lines ($>700$ eV). Five more sources have low values of the X-ray to \[OIII\] flux ratio suggesting that they could be associated with obscured nuclei. Their individual spectra show neither high absorbing columns nor flat spectral indices. However, their stacked spectrum reveals an absorbing column density of $\rm N_H\sim 10^{23}$ . Therefore the fraction of absorbed sources ($>10^{22}$ ) could be as high as $55\pm12$ %. A number of Seyfert-2 appear to host unabsorbed nuclei. These are associated with low-luminosity sources $\rm L_X < 3\times 10^{41}$ . Their stacked spectrum again shows no absorption while inspection of the images, where available, shows that contamination from nearby sources does not affect the [*XMM-Newton*]{} spectra in most cases. Nevertheless, such low luminosity sources are not contributing significantly to the X-ray background flux. When we consider only the brighter, $>10^{41}$ , 21 sources, we find that the fraction of absorbed nuclei rises to $75\pm19 $ % while that of Compton-thick sources to 15-20%. The fraction of Compton-thick AGN is lower than that predicted by the X-ray background synthesis model in the same luminosity and redshift range.]{}'
author:
- 'A. Akylas and I. Georgantopoulos'
title: ' XMM-Newton observations of Seyfert galaxies from the Palomar spectroscopic survey: the X-ray absorption distribution'
---
2[$\chi^{2}$]{}
Introduction
============
The moderate to high redshift Universe has been probed at unparallelled depth with the most sensitive observations performed at X-ray wavelengths in the [*Chandra*]{} Deep fields. The [*Chandra*]{} 2Ms observations (Alexander et al. 2003, Luo et al. 2008) resolved about 80 per cent of the extragalactic X-ray light in the hard 2-10 keV band (see Brandt & Hasinger 2005 for a review). These deep surveys find a sky density of 5000 sources per square degree, the vast majority of which are found to be AGN through optical spectroscopy (e.g. Barger et al. 2003). In contrast, the optical surveys for QSOs (e.g. the COMBO-17 survey) reach only a surface density about an order of magnitude lower (e.g. Wolf et al. 2003). This clearly demonstrates the power of X-ray surveys for detecting AGN. This is because hard X-rays can penetrate large amounts of gas without suffering from significant absorption. Indeed detailed spectral analysis on X-ray selected AGN reveals large amount of obscuration (e.g. Akylas et al. 2006, Tozzi et al. 2006, Georgantopoulos et al. 2007). In particular, about two thirds of the X-ray sources, over all luminosities, present column densities higher than $10^{22}$ . These high absorbing columns are believed to originate in a molecular torus surrounding the nucleus.
However, even the efficient 2-10 keV X-ray surveys may be missing a fraction of highly obscured sources. This is because at very high obscuring column densities ($>10^{24}$ , corresponding to an optical reddening of $\rm A_V>100$), the X-ray photons with energies between 2 and 10 keV are absorbed. These are the Compton-thick AGN (see Comastri 2004 for a review) where the Compton scattering on the bound electrons becomes significant. Despite the fact that Compton-thick AGN are abundant in our vicinity (e.g. NGC1068, Circinus), only a few tens of Compton-thick sources have been identified from X-ray data (Comastri 2004). Although the population of Compton-thick sources remains elusive there is concrete evidence for its presence. The X-ray background synthesis models can explain the peak of the X-ray background at 30-40 keV, where most of its energy density lies, (Frontera et al. 2007, Churazov et al. 2007) only by invoking a large number of Compton-thick AGN (Gilli, Comastri & Hasinger 2007). Additional evidence for the presence of a Compton-thick population comes from the directly measured space density of black holes in the local Universe. It is found that this space density is a factor of two higher than that predicted from the X-ray luminosity function (Marconi et al. 2004). This immediately suggests that the X-ray luminosity function is missing an appreciable number of obscured AGN.
In recent years there have been many efforts to uncover heavily obscured and in particular Compton-thick AGN in the local Universe by examining IR or optically selected, \[OIII\], AGN samples. This is because both the IR and the narrow-line region originate beyond the obscuring region and thus represent an isotropic property of the AGN. Risaliti et al. (1999) examine the X-ray properties of a large sample of \[OIII\] selected Seyfert-2 galaxies whose X-ray spectra were available in the literature. They find a large fraction of Compton-thick sources (over half of their sample). Their estimates are complemented by more recent [**]{} observations of local AGN samples (Cappi et al. 2006, Panessa et al. 2006, Guainazzi et al. 2005). All these authors also claim a large Compton-thick AGN fraction exceeding 30 per cent of the Seyfert-2 population. The advent of the [*SWIFT*]{} and [*INTEGRAL*]{} missions which carry X-ray detectors with imaging capabilities (e.g. Barthelmy et al. 2005, Ubertini et al. 2003) in ultra-hard X-rays (15-200 keV) try to shed new light on the absorption properties of AGN in the local Universe. In principle, at these ultra-hard X-rays obscuration should play a negligible role, at least up to column densities as high as $10^{25}$ . However, because of the limited effective area the above surveys can provide X-ray samples, down to very bright fluxes $10^{-11}$ , with limited quality spectra. Again observations are often required to determine the column density in each source. Interestingly, these surveys find only a limited number of Compton-thick sources (Markwardt et al. 2005, Bassani et al. 2006, Malizia et al. 2007, Ajello 2008, Winter et al. 2008, Tueller et al. 2008, Sazonov et al. 2008).
Here, we present observations of [*all*]{} 38 Seyfert galaxies in the Palomar spectroscopic sample of nearby galaxies (Ho et al. 1997). This is the largest complete optically selected AGN sample in the local Universe analyzed so far. 23 of the Seyfert galaxies presented here have already been discussed in previous works (e.g. Cappi et al. 2006). For 5 of them newer observations are available and are presented here. The current work should provide the most unbiased census of the AGN column density distribution at low redshifts and luminosities.
The sample
==========
[\[thesample\]]{}
The Seyfert sample used in this study is derived from the Palomar optical spectroscopic survey of nearby galaxies (Ho, Fillipenko, & Sargent 1995). This survey has taken high quality spectra of 486 bright ($B_T<12.5$ mag), northern ($\delta>0^{\circ}$) galaxies selected from the Revised Shapley-Ames Catalogue of Bright Galaxies (RSAC, Sandage & Tammann 1979) and produced a comprehensive and homogeneous catalogue of nearby Seyfert galaxies. The catalogue is 100% complete to $B_T<12.0$ mag and 80% complete to $B_T<12.5$ mag (Sandage, Temmann & Yahil 1981).
For the purpose of this work we consider all the Seyfert galaxies from the Palomar survey. Sources lying in-between the Seyfert-Liner or the Seyfert-Transient boundary have been excluded. Furthermore seven Seyfert galaxies (i.e. NGC1068, NGC1358, NGC1667, NGC2639, NGC3185, NGC4235, NGC5548), which have been included in the Palomar survey for various reasons (see Ho et al. 1995), even though they did not satisfy the survey selection criteria, are also excluded.
There are 40 Seyfert galaxies comprising the optical sample. 9 sources are classified as type-1 (contains types 1, 1.2, 1.5) and 31 as type-2 (contains types 1.8,1.9,2) Seyfert galaxies. However NGC4051, NGC4395 and NGC4639 which have been initially classified as Seyfert 1.2, 1.8 and 1 by Ho et al. (1997) has been re-classified as type-1.5, 1 and 1.5 respectively (see Cappi et al. 2006, Panessa et al. 2006, Baskin & Laor 2008). Moreover NGC185 which is classified as a Seyfert-2 may not contain an active nucleus since it presents line intensity ratios possibly produced by stellar processes (Ho & Ulvestad 2001).
The main characteristics of these sources, taken from Ho et al. (1997), are listed in Table \[optical\]. Some galaxies listed here present $B_T$ fainter than the formal limit of the Palomar survey. According to Ho et al. (1995) this discrepancy can be attributed to errors in the apparent magnitudes given in the RSAC.
---------- ----------------- ----------------- ------------- -------- -------
Name $\alpha$(J2000) $\delta$(J2000) $B_T$ (mag) D(Mpc) Class
(1) (2) (3) (4) (5) (6)
NGC 0185 00 38 57.40 +48 20 14.4 10.10 0.7 S2
NGC 0676 01 48 57.38 +05 54 25.70 10.50 19.5 S2:
NGC 1058 02 43 30.24 +37 20 27.20 11.83 9.1 S2
NGC 1167 03 01 42.40 +35 12 21.00 13.38 65.3 S2
NGC 1275 03 19 48.16 +41 30 42.38 12.64 70.1 S1.5
NGC 2273 06 50 08.71 +60 50 45.01 12.55 28.4 S2
NGC 2655 08 55 38.84 +78 13 25.20 10.96 24.4 S2
NGC 3031 09 55 33.17 +69 03 55.06 7.89 1.4 S1.5
NGC 3079 10 01 58.53 +55 40 50.10 11.54 20.4 S2
NGC 3147 10 16 53.27 +73 24 02.40 11.43 40.9 S2
NGC 3227 10 23 30.58 +19 51 53.99 11.10 20.6 S1.5
NGC 3254 10 29 19.96 +29 29 29.60 12.41 23.6 S2
NGC 3486 11 00 24.10 +28 58 31.60 11.05 7.4 S2
NGC 3516 11 06 47.49 +72 34 06.80 12.50 38.9 S1.2
NGC 3735 11 35 57.49 +70 32 07.70 12.50 41.0 S2:
NGC 3941 11 52 55.42 +36 59 10.50 11.25 18.9 S2:
NGC 3976 11 55 57.35 +06 44 57.00 12.30 37.7 S2:
NGC 3982 11 56 28.10 +55 07 30.50 11.78 17.0 S1.9
NGC 4051 12 03 09.61 +44 31 52.80 11.88 17.0 S1.2
NGC 4138 12 09 29.87 +43 41 06.00 12.16 17.0 S1.9
NGC 4151 12 10 32.57 +39 24 20.63 11.50 20.3 S1.5
NGC 4168 12 12 17.30 +13 12 17.9 12.11 16.8 S1.9:
NGC 4169 12 12 18.93 +29 10 44.00 13.15 50.4 S2
NGC 4258 12 18 57.54 +47 18 14.30 9.10 6.8 S1.9
NGC 4378 12 25 18.14 +04 55 31.60 12.63 35.1 S2
NGC 4388 12 25 46.70 +12 39 40.92 11.76 16.8 S1.9
NGC 4395 12 25 48.93 +33 32 47.80 10.64 3.6 S1.8
NGC 4472 12 29 46.76 +07 59 59.90 9.37 16.8 S2::
NGC 4477 12 30 02.22 +13 38 11.30 11.38 16.8 S2
NGC 4501 12 31 59.34 +14 25 13.40 10.36 16.8 S2
NGC 4565 12 36 21.07 +25 59 13.50 10.42 9.7 S1.9
NGC 4639 12 42 52.51 +13 15 24.10 12.24 16.8 S1
NGC 4698 12 48 22.98 +08 29 14.80 11.46 16.8 S2
NGC 4725 12 50 26.69 +25 30 02.30 10.11 12.4 S2:
NGC 5033 13 13 27.52 +36 35 37.78 10.75 18.7 S1.5
NGC 5194 13 29 52.37 +47 11 40.80 8.96 7.7 S2
NGC 5273 13 42 08.33 +35 39 15.17 12.44 21.3 S1.5
NGC 6951 20 37 14.41 +66 06 19.70 11.64 24.1 S2
NGC 7479 23 04 56.69 +12 19 23.20 11.60 32.4 S1.9
NGC 7743 23 44 21.44 +09 56 03.60 12.38 24.4 S2
---------- ----------------- ----------------- ------------- -------- -------
[Column 1: Galaxy name]{}
[Columns 2 & 3: Optical coordinates]{}
[Column 4: Total apparent $B$ magnitude taken from Ho et al. 1997]{}
[Column 5: Source distance in Mpc from Ho et al. 1997]{}
[Column 6: Optical classification from Ho et al. 1997. Quality ratings are given by “:” and “::” for uncertain and highly uncertain classification.]{}
X-ray Observations
==================
The X-ray data have been obtained with the EPIC (European Photon Imaging Cameras; Strüder et al. 2001, Turner et al. 2001) on board . Thirty sources have been recovered from the archive while the remaining ten objects (marked with a ”$\star$” in Table \[xray\]) have been observed by us during the Guest Observer program.
The log of all the observations is shown in Table \[xray\]. The data have been analysed using the Scientific Analysis Software ([*SAS v.7.1*]{}). We produce event files for both the PN and the MOS observations using the [*EPCHAIN*]{} and [*EMCHAIN*]{} tasks of [*SAS*]{} respectively. The event files are screened for high particle background periods. In our analysis we deal only with events corresponding to patterns 0-4 for the PN and 0-12 for the MOS instruments.
The source spectra are extracted from circular regions with radius of 20 arcsec. This area encircles at least the 70 per cent of the all the X-ray photons at off-axis angles less than 10 arcmin. A ten times larger source-free area is used for the background estimation. The response and ancillary files are also produced using [*SAS*]{} tasks [*RMFGEN*]{} and [*ARFGEN*]{} respectively.
We note that 18 of the observations presented here, coincide with these presented in Cappi et al. (2006). However we choose to re-analyze these common data-sets in order to present a uniform treatment of the sample.
-------------------- ------------ ------------ -------- -------- -------- -------- -------- --------
Name Obs. Date Obs. ID
PN MOS1 MOS2 PN MOS1 MOS2
NGC 185 2004-01-09 0204790301 - 11393 11334 closed Medium Medium
NGC 676 2002-07-14 0112551501 17754 21127 21127 Thick Thin Thin
NGC 1058 2002-02-01 0112550201 12902 17019 17019 Medium Thin Thin
NGC 1167$^{\star}$ 2005-08-04 0301650101 9937 11448 11448 Thin Thin Thin
NGC 1275 2006-01-29 0305780101 119697 124801 124832 Medium Medium Medium
NGC 2273 2003-09-05 0140951001 11076 12709 12714 Medium Medium Medium
NGC 2655$^{\star}$ 2005-09-04 0301650301 9850 11564 11570 Thin Thin Thin
NGC 3031 2001-04-22 0111800101 129550 82790 83150 Medium Medium Medium
NGC 3079 2001-04-13 0110930201 20023 24661 24663 Thin Medium Medium
NGC 3147 2006-10-06 0405020601 14963 16923 16912 Thin Thin Thin
NGC 3227 2000-11-28 0101040301 34734 37198 37201 Medium Medium Medium
NGC 3254$^{\star}$ 2005-10-31 0301650401 9869 11489 11481 Thin Thin Thin
NGC 3486 2001-05-09 0112550101 9057 6398 6385 Medium Thin Thin
NGC 3516 2001-11-09 0107460701 12829 12901 12900 Thin Thin Thin
NGC 3735$^{\star}$ 2005-09-27 0301650501 9312 16466 16471 Thin Thin Thin
NGC 3941 2001-05-09 0112551401 9389 14635 14331 Medium Thin Thin
NGC 3976$^{\star}$ 2006-06-16 0301651801 11313 13483 13598 Thin Thin Thin
NGC 3982 2004-06-15 0204651201 10197 11674 11679 Thin Thin Thin
NGC 4051 2002-11-22 0157560101 49808 51510 51520 Medium Medium Medium
NGC 4138 2001-11-26 0112551201 9999 14365 14365 Medium Thin Thin
NGC 4151 2003-05-26 0143500201 18454 18602 18607 Medium Medium Medium
NGC 4168 2001-12-04 0112550501 18498 22864 22849 Medium Thin Thin
NGC 4169$^{\star}$ 2006-06-20 0301651701 11068 12695 12701 Thin Thin Thin
NGC 4258 2006-11-17 0400560301 62607 64179 64184 Medium Medium Medium
NGC 4378$^{\star}$ 2006-01-08 0301650801 10963 12602 12604 Thin Thin Thin
NGC 4388 2002-12-12 0110930701 8292 11666 11666 Thin Medium Medium
NGC 4395 2003-11-30 0142830101 10596 10942 10940 Medium Medium Medium
NGC 4472 2004-01-01 0200130101 89503 94179 94185 Thin Thin Thin
NGC 4477 2002-06-08 0112552101 9500 13501 13527 Medium Thin Thin
NGC 4501 2002-06-08 0112550801 2885 13387 13385 Medium Thin Thin
NGC 4565 2001-07-01 0112550301 10010 14261 14263 Medium Thin Thin
NGC 4639 2001-12-16 0112551001 10000 14365 14265 Medium Thin Thin
NGC 4698 2001-12-17 0112551101 11755 16112 16112 Medium Thin Thin
NGC 4725 2002-06-14 0112550401 13369 17244 17244 Medium Thin Thin
NGC 5033 2002-12-18 0094360501 9999 11616 11614 Medium Medium Medium
NGC 5194 2003-01-15 0303420101 19047 49944 49351 Thin Thin Thin
NGC 5273 2002-06-14 0112551701 10392 16065 16094 Medium Thin Thin
NGC 6951$^{\star}$ 2005-06-05 0301651401 7951 9664 9669 Thin Thin Thin
NGC 7479$^{\star}$ 2005-06-05 0301651201 12315 15740 15750 Thin Thin Thin
NGC 7743$^{\star}$ 2005-06-15 0301651001 11847 13283 13348 Thin Thin Thin
-------------------- ------------ ------------ -------- -------- -------- -------- -------- --------
[Column 1: Name of the Galaxy]{}
[Column 2: Start Observation date (UTC)]{}
[Column 3: Observation identifier]{}
[Columns 3, 4 & 5: Net exposure time for the EPIC instruments]{}
[Columns 6, 7 & 8: Applied filter]{}
[$^{\star}$ Denotes sources observed during our Guest Observer program]{}
X-ray Spectral Analysis
=======================
We investigate the X-ray properties of the sources in our sample by performing spectral fittings with [*XSPEC v.12.4*]{} software package. 2 sources are excluded from the X-ray spectral analysis: the Seyfert-2 galaxy NGC185 for being undetected in the X-rays (see also section \[thesample\]), and the Seyfert-1.5 galaxy NGC1275 which belongs to the Perseus cluster and whose X-ray image shows that its flux is heavily contaminated by diffuse emission.
The X-ray spectra are binned to give a minimum of 15 counts so Gaussian statistics can be applied. We fit the PN and the MOS data simultaneously in the 0.3-10 keV range. However in some cases where a very complex behaviour is present we perform the spectral fits only in the 2-10 keV band. These latter cases are denoted with an asterisk ($\star$) in Table \[fit\].
The normalization parameters for each instrument are left free to vary within 5 per cent in respect to each other to account for the remaining calibration uncertainties.
We assume a standard power-law model with two absorption components ([*wa\*wa\*po*]{} in [*XSPEC*]{} notation) to account for the source continuum emission. The first absorption column models the Galactic absorption. Its fixed values are obtained from Dickey & Lockman (1990) and are listed in Table \[fit\]. The second absorption component represents the AGN intrinsic absorption and is left as a free parameter during the model fitting procedure. A Gaussian component has also been included to describe the FeK$_{\alpha}$ emission line.
When the fitting procedure gives a rejection probability less than 90 per cent we accept the above “standard model”. However when this simple parametrization is not sufficient to model the whole spectrum additional components are included. For example soft-excess emission and reflection are common features in the X-ray spectra of Seyfert galaxies and can be modeled using additional [*XSPEC*]{} models.
In particular we fit a second power-law model, with $\Gamma$ fixed to the direct component value, to account for the scattered X-ray radiation and/or a [*Raymond-Smith*]{} to model the contribution from diffuse emission in the host galaxy. A flattening of the spectrum is usually indicative of reflected radiation from the backside of the torus. The reflected radiation is modelled using the [*PEXRAV*]{} model (Magdziarz & Zdziardski, 1995). In order to accept the new component we apply the F-test criterion. If the addition of the new component significantly improves the fit at the 90 per cent confidence level, then it is accepted. Other characteristics such as ionized features could also be considered however once a reasonable fit is obtained (i.e. with rejection probability less than 90 per cent) we do not include additional components.
The best fit parameters for all the sources are reported in Table \[fit\]. The errors quoted correspond to the 90 per cent confidence level for one interesting parameter. We note here that some of the sources listed show a rather steep photon index. In many cases this happens because of the fixed value of the continuum power-law photon index to the photon index of the soft component (e.g. NGC1358, NGC3079, NGC3735). When these parameters are untied the continuum power-law photon index becomes harder.
18 of the X-ray observations presented here have already been shown in Cappi et al. 2006. In most of these the results are in agreement. However some deviations also appear and are discussed below. In the cases of NGC3486, NGC3079, NGC4051 and NGC4388 the comparison is not straightforward since we use of a different spectral fitting model. When the same model is applied as a test, there is no significant difference in the results. In the cases of NGC1058 and NGC4725 our results show a steeper power-low photon index than that presented in Cappi et al. 2006. However we point out that the results are consistent within the 90 per cent confidence level.
------------------- --------------- --------------------------- ------------------------ ------------------------ ---------------------- --------- -------- ------------ ------------ ----------------
Name $N_{H_{GAL}}$ $N_H$ [$\Gamma_{cont}$]{} $kT$ EW$_{FeK}$ $F_X$ $L_X$ $f_{scat}$ $f_{refl}$ $\chi^2_{\nu}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
NGC676 4.7 $<0.13$ $2.10^{+0.50}_{-0.27}$ - - 1.12 0.05 - - 22.5/17
NGC1058 5.4 $<0.53$ $3.40^{+2.65}_{-1.69}$ - - 0.24 0.02 - - 31.1/34
NGC1167 9.8 $0.32^{+0.08}_{-0.41}$ $2.10^{+0.50}_{-0.27}$ $0.33^{+0.08}_{-0.04}$ - 0.9 0.4 - - 22.52/17
NGC2273 6.4 $104.70^{+15.90}_{-7.36}$ $1.84^{+0.12}_{-0.05}$ $0.77^{+0.08}_{-0.13}$ $1500^{+303}_{-240}$ 89.4 8.5 0.01 1.35 125.1/114
NGC2655 2.2 $42.69^{+5.68}_{-5.30}$ $2.61^{+0.30}_{-0.26}$ $0.73^{+0.30}_{-0.26}$ 110.4 0.01 7.8 - - 240.1/194
NGC3031$^{\star}$ 5.6 $0.05^{+0.02}_{-0.01}$ $1.91^{+0.02}_{-0.08}$ $1.10^{+0.02}_{-0.02}$ $53^{+11}_{-14}$ 1164.7 0.3 - - 1439.9/1350
NGC3079 0.9 $201^{+28.5}_{-89.8}$ $2.56^{+0.22}_{-0.23}$ $0.84^{+0.02}_{-0.03}$ $700^{+200}_{-210}$ 38.4 1.8 0.01 0.14 301.9/245
NGC3147 2.9 $0.07^{+0.01}_{-0.01}$ $1.56^{+0.04}_{-0.03}$ - $144^{+101}_{-55}$ 155.8 31.1 - - 562.1/585
NGC3227 2.0 $6.42^{+0.15}_{-0.17}$ $1.44^{+0.04}_{-0.04}$ - $152^{+22}_{-41}$ 866.3 43.9 0.06 10.12 2089.4/2082
NGC3254 1.8 $<0.12$ 1.9 (fixed) - - 0.7 0.05 - - 4.23/5
NGC3486 1.7 $8.42^{+28.9}_{-4.17}$ 1.9 (fixed) $0.37^{+0.06}_{-0.05}$ - 8.5 0.06 - - 30.29/44
NGC3516$^{\star}$ 3.5 $1.04^{+0.32}_{-0.49}$ $1.58^{+0.02}_{-0.04}$ - $180^{+13}_{-16}$ 1460.3 264.1 - - 2483.9/2056
NGC3735 1.3 $15.23^{+55.43}_{-10.17}$ $2.85^{+0.67}_{-0.49}$ - - 17.21 3.4 0.11 - 139.7/131
NGC3941 1.9 $<0.09$ $2.02^{+0.35}_{-0.24}$ - - 4.19 0.2 87.97/84
NGC3976 1.1 $0.12^{+0.07}_{-0.06}$ $2.01^{+0.04}_{-0.03}$ - - 7.3 1.2 - - 109.42/109
NGC3982 1.0 $43.23^{+28.11}_{-16.96}$ $2.53^{+0.44}_{-0.42}$ $0.28^{+0.04}_{-0.04}$ $802^{+678}_{-420}$ 16.60 0.6 0.025 - 109.42/109
NGC4051$^{\star}$ 1.2 $0.53^{+0.21}_{-0.35}$ $2.08^{+0.11}_{-0.18}$ $155^{+14}_{-18}$ 650.32 22.4 - 9.7 2315.34/2225
NGC4138 1.3 $8.85^{+0.41}_{-0.44}$ $1.63^{+0.05}_{-0.05}$ - $80^{+41}_{-34}$ 554.3 19.1 0.009 - 473.04/449
NGC4151$^{\star}$ 2.3 $5.14^{+0.13}_{-0.14}$ $1.55^{+0.03}_{-0.03}$ - $69^{+11}_{-10}$ 21000.5 1034.4 - 2.54 109.42/109
NGC4168 2.4 $<0.06$ $2.02^{+0.14}_{-0.12}$ - - 0.6 0.2 - - 72.2/73
NGC4169 1.7 $13.47^{+6.07}_{-3.31}$ $2.01^{+0.79}_{-0.44}$ - 23.5 6.9 0.017 - 22.87/30
NGC4258$^{\star}$ 1.6 $6.8^{+0.35}_{-0.28}$ $1.62^{+0.07}_{-0.05}$ - $41^{+17}_{-19}$ 380.9 2.1 - - 1227.17/1258
NGC4378 1.7 $0.18^{+0.06}_{-0.06}$ $1.55^{+0.19}_{-0.15}$ - 14.7 2.1 31.84/38
NGC4388 2.6 $32.14^{+1.19}_{-1.05}$ $1.86^{+0.09}_{-0.08}$ - $173^{+15}_{-32}$ 2007.3 70.8 0.006 1050.7/923
NGC4395$^{\star}$ 1.9 $1.02^{+0.11}_{-0.10}$ $1.18^{+0.03}_{-0.02}$ - $80^{+16}_{-9}$ 590.5 1.2 - - 2298.85/2008
NGC4472$^{\star}$ 1.5 $<0.82$ $1.65^{+0.26}_{-0.14}$ $0.87^{+0.15}_{-0.11}$ - 21.5 0.7 - - 283.25/292
NGC4477 2.4 $<0.02$ $2.12^{+0.25}_{-0.17}$ $0.41^{+0.04}_{-0.16}$ - 3.7 0.1 - - 62.9/58
NGC4501 2.6 $<0.06$ $1.95^{+0.19}_{-0.18}$ $0.66^{+0.01}_{-0.01}$ - 0.6 0.2 - - 40.31/32
NGC4565 1.2 $0.16^{+0.03}_{-0.03}$ $1.87^{+0.14}_{-0.09}$ - 20.7 0.2 - - 102.5/88
NGC4639 2.2 $<0.04$ $1.79^{+0.06}_{-0.05}$ - - 48.2 1.6 - - 268.42/250
NGC4698 1.8 $<0.07$ $1.73^{+0.22}_{-0.27}$ - - 4.8 0.1 - - 33.82/33
NGC4725 0.8 $<0.03$ $2.68^{+0.27}_{-0.23}$ $0.23^{+0.03}_{-0.03}$ - 2.3 0.04 - - 64.4/71
NGC5033 1.1 $<0.04$ $1.72^{+0.02}_{-0.02}$ - $286^{+81}_{-71}$ - 440.6 18.3 - 999.1/974
NGC5194$^{\star}$ 1.8 $<0.65$ $1.16^{+0.14}_{-0.23}$ - $1730^{+422}_{-275}$ 25.5 0.2 - - 89.33/91
NGC5273$^{\star}$ 0.9 $0.72^{+0.07}_{-0.09}$ $1.44^{+0.07}_{-0.09}$ - $191^{+52}_{-72}$ 706.7 38.2 - - 612.24/635
NGC6951 12.4 $0.40^{+0.31}_{-0.15}$ $2.59^{+0.53}_{-0.30}$ $0.67^{+0.16}_{-0.18}$ - 4.8 0.3 - - 32.33/55
NGC7479 5.3 $40.2^{+11.44}_{-8.75}$ $2.56^{+0.15}_{-0.16}$ $0.27^{+0.07}_{-0.07}$ $480^{+210}_{-390}$ 22.5 2.8 0.004 - 137.3/133
NGC7743 4.8 $0.3^{+0.18}_{-0.14}$ $3.63^{+1.14}_{-1.03}$ $0.23^{+0.06}_{-0.06}$ - 1.0 0.07 - - 16.7/18
------------------- --------------- --------------------------- ------------------------ ------------------------ ---------------------- --------- -------- ------------ ------------ ----------------
[Column 1: Galaxy name]{}
[Column 2: Galactic column density in units of $10^{20}$ cm$^{-2}$]{}
[Column 3: Observed column density in units of $10^{22}$ cm$^{-2}$]{}
[Column 4: Power-law photon index of the continuum emission]{}
[Column 5: Temperature of the Raymond-Smith model]{}
[Column 6: Equivalent width of the FeK$_{\alpha}$ emission]{}
[Column 7: Observed 2-10 keV flux in units of $10^{-14}$ ergs s$^{-1}$ cm$^{-2}$]{}
[Column 8: Observed 2-10 keV luminosity in units of $10^{40}$ ergs s$^{-1}$]{}
[Column 9: Ratio of the normalizations of the scattered to the continuum emission]{}
[Column 10: Ratio of the normalizations of the reflected to the continuum emission]{}
[Column 11: Reduced $\chi^2$]{}
[$^\star$ Indicates source with very complex spectra for which only a rough parametrization in the 2-10 keV band is presented here.]{}
The X-ray spectra of our sources are presented in Fig \[xspectra\]. For each object the upper panel shows the X-ray spectrum along with the model presented in Table \[fit\] while the lower panel shows the residuals.
X-ray absorption
================
The spectral fitting results are presented in Table \[fit\]. There are 8 type-1 Seyferts in our sample. Five of them show small amounts of absorption ($<10^{21}$ cm$^{-2}$) while the 3 Seyfert-1.5 sources (NGC3227, NGC3516, and NGC4151) present a considerable amount of $N_H$ ($>10^{22}$ cm$^{-2}$). Our sample contains 30 Seyfert-2 galaxies. The column densities in this population vary from the Galactic to the Compton-thick limit ($N_H>$10$^{24}$ cm$^{-2}$). However, the apparent number of significantly obscured sources is rather small. Only 12 out of 30 type-2 sources present absorption greater than $10^{22}$ cm$^{-2}$.
Compton-thick sources
---------------------
The fraction of Compton-thick sources is more difficult to estimate. This is because the effective area sharply decreases at energies higher than 6 keV. Given the limited bandpass, which extends up to about 10 keV, we are not able to measure the absorption turnover for highly absorbed sources. A column density of $\sim$10$^{24}$ cm$^{-2}$ suppresses 90 % of the flux in the 2-10 keV band. Therefore, we can obtain a direct measurement of the obscuration only up to column densities reaching at most a few times $10^{24}$ . In the case of Compton-thick AGN the X-ray spectrum is dominated by scattered components from cold or warm material as well as an FeK$_\alpha$ with high equivalent width (Matt et al. 2000). Then to unveil the presence of a Compton-thick nucleus we apply the following diagnostics.
- [ Flat X-ray spectrum ($\Gamma<1$). This implies the presence of a strong reflection component, which intrinsically flattens the X-ray spectrum at higher energies (e.g. Matt et al. 2000)]{}
- [ High Equivalent Width of the FeK$_{\alpha}$ line ($\sim$1 keV). This characteristic is consistent with a Compton-thick nucleus since then the line is measured against a much depressed continuum (Leahy & Creighton 1993) or a pure reflected component.]{}
- Low X-ray to optical flux ratio. Bassani et al. (1999) have showed that the 2-10 KeV to the \[OIII\] $\lambda$5007 flux ratio is very effective in the identification of Compton-thick sources. This is because the \[OIII\] $\lambda$5007 (hereafter \[OIII\]) flux which comes from large (usually kpc) scales, remains unabsorbed while the X-ray flux is diminished because of absorption.
These criteria however should be considered with caution. For example high Equivalent Width (EW) lines may also appear in the case of anisotropic distribution of the scattering medium (Ghisellini et al. 1991), or in the case where there is a time lag between the reprocessed and the direct component (e.g. NGC2992, Weaver et al. 1996). Also there have been reports of Compton-thick sources where the value of FeK$_{\alpha}$ line EW is well below 1 keV (e.g. Awaki et al. 2000 for Mkn1210).
In Fig. \[oiii\] we plot the column density obtained from the spectral fittings as a function of the X-ray to optical flux ratio, $F_{2-10~\rm keV}/F_{[OIII]}$. The \[OIII\] fluxes are corrected for the optical reddening using the formula described in Basanni et al. (1999): $F_{[OIII]_{\rm COR}}=
F_{[OIII]_{\rm OBS}}[(H_{\alpha}/H_{\beta})/(H_{\alpha}/H_{\beta})_{o}]^{2.94}$, where the intrinsic Balmer decrement $(H_{\alpha}/H_{\beta})_{o}$ equals 3.
The solid lines in Fig. \[oiii\] show the expected correlation between these quantities, assuming a photon index of 1.8 and $1\%$ reflected radiation (see also Maiolino et al 1998, Cappi et al 2006). The starting point in the x-axis for the middle solid line is determined by averaging the $F_{2-10~\rm keV}/F_{[OIII]}$ values of the Seyfert-1 population only, while the lines at right and left show the $3\sigma$ dispersion. The sources occupying the low ($F_{2-10~\rm keV}/F_{[OIII]}$, $ N_H$) region in this plot could be possibly highly obscured or Compton-thick AGN.
In two cases (NGC2273, NGC3079) we can immediately tell the presence of a Compton-thick nucleus through the presence of an absorption turnover in the spectral fittings. Both sources present high values of the FeK$_{\alpha}$ line EW ($>700$ eV). One more source (NGC5194), despite the fact that it presents the highest value of FeK$_{\alpha}$ ($\sim 1700$ eV), shows no absorption at all. However, the very flat X-ray spectrum and the very low $F_{2-10~\rm keV}/F_{[OIII]}$ value further suggest that this is a highly obscured or a Compton-thick source. According to the $N_H$-$F_{2-10~\rm keV}/F_{[OIII]}$ relation a minimum value for the $N_H$ is $4\times10^{23}$ cm$^{-2}$ (see Fig. \[oiii\].
There are also 5 Seyfert-2 galaxies (NGC676, NGC1167, NGC3254, NGC6951 and NGC7743) occupying the low $F_{2-10~\rm keV}/F_{[OIII]}$ regime. We do not consider NGC4169 because of the large error in the estimation of the \[OIII\] flux (see Ho et al. 1997). These, according to the expected $N_H$-$F_{2-10~\rm keV}/F_{[OIII]}$ relation, should present high values of $N_H$. According to Fig. \[oiii\], the minimum $N_H$ value is $\sim2\times10^{23}$ cm$^{-2}$ for NGC676, $\sim6\times10^{23}$ cm$^{-2}$ for NGC6951 and $\sim10^{24}$ cm$^{-2}$ for NGC1167 and NGC7743. However, the X-ray spectral fittings show low absorption ($<10^{22}$ cm$^{-2}$) while in addition there is no indication for a flat photon index or strong FeK$_\alpha$ line. This may be due to the limited photon statistics in the hard ($>2$ keV) band, which does not allow us to examine in detail the spectral characteristics. Note however, that in the spectrum of at least two sources (NGC1167 and NGC7743) there is some indication for a flattening at hard energies which could suggest a heavily buried or reflected component. We investigate further this issue by deriving the mean, stacked X-ray spectrum. We use the [*MATHPHA*]{} tasks of [*FTOOLS*]{} to derive the weighted stacked X-ray spectrum of the five EPIC-PN observations. The corresponding ancillary files are also produced using [*ADDRMF*]{} and [*ADDARF*]{} tasks of [*FTOOLS*]{}. We perform no correction for the rest-frame energy because the differences in the redshifts are negligible. An absorbed power-law model plus a Gaussian line and a soft excess component (Raymond-Smith model) reproduce well the mean spectrum (Table \[stack\_table\]). In Fig. \[stacked\] we present the data along with the best-fit. The average spectrum shows significant absorption consistent with the measured value of the FeK$_{\alpha}$ line EW.
$N_H$ (cm$^{-2}$) [$\Gamma$]{} kT (keV) EW$_{\rm FeK}$ (eV) $\chi^2_{\nu}$
------------------------- ------------------------ ------------------------ --------------------- ---------------- -- -- -- -- --
$10.30^{+2.01}_{-1.91}$ $1.71^{+0.08}_{-0.17}$ $0.73^{+0.13}_{-0.12}$ $255^{+243}_{-146}$ 72.41/91
Our results above can be summarised as follows. The number of absorbed nuclei ($N_H>10^{22}$ cm$^{-2}$) are 21 out of 38 or $55\pm12$%. The number of Compton-thick sources is three i.e. $8\pm5$% although, if we adopt the extreme case where all the low $F_{2-10~\rm keV}/F_{[OIII]}$ host Compton-thick nuclei this number would rise to 8 or $21\pm 7$ %. Our estimates on the amount of $N_H$ in the local universe are illustrated in Fig. \[nh\]. The solid line describes the $N_H$ distribution. The vertical arrows in the highest $N_H$ bin show the upper and lower limits for the number of Compton-thick sources.
The absorption in the bright sub-sample
---------------------------------------
Our findings should play an important role to the XRB synthesis models (Comastri et al. 1995, Gilli et al. 2007). Gilli et al. 2007 assume in their models a lower luminosity of 10$^{41}$ ergs s$^{-1}$. However, the intrinsic 2-10 keV luminosity of our sources starts from as low as a few times 10$^{38}$ ergs s$^{-1}$ which is about 3 orders of magnitude lower. Therefore it is useful to present our results separately for the fainter ($L_{2-10~\rm keV}<10^{41}$ ergs s$^{-1}$) and the brighter ($L_{2-10~\rm keV}>10^{41}$ ergs s$^{-1}$) sub-sample containing 21 and 17 sources respectively (see Fig. \[nh\_lum\]). The intrinsic $L_X$ values are determined using the best fitting results. For the three Compton-thick sources the intrinsic $L_X$ has been estimated assuming that 1$\%$ of the intrinsic luminosity is actually observed below 10 keV due to scattering and or reflection (e.g. Comastri 2004). In the bright sample the fraction of the highly absorbed sources is $\sim$75 % and the Compton-thick sources most probably account for 15-20 $\%$ of the total population. This fraction can reach a maximum of 29 $\%$ in the unlikely case where all the low $F_{2-10~\rm keV}/F_{[OIII]}$ sources host a Compton-thick nucleus.
Unabsorbed Seyfert-2 Galaxies
-----------------------------
The X-ray spectral analysis reveals several Seyfert-2 galaxies with very little or no X-ray absorption. As we have already discussed some of these, i.e. the five with low X-ray to \[OIII\] flux ratio are most probably associated with a highly obscured or even a Compton-thick nucleus. In Fig. \[oiii\] there are 12 additional Seyfert-2 galaxies (NGC1058, NGC3147, NGCC3941, NGC3976, NGC4168, NGC4378, NGC4472, NGC4477, NGC4501,NGC4565,NGC4698 and NGC4725) with $N_H$ less than 10$^{22}$ cm$^{-2}$ but an average $F_{2-10~\rm keV}/F_{[OIII]}$ value. This behaviour is not unknown (e.g. Pappa et al 2001, Gliozzi, Sambruna & Foschini 2007). In particular NGC3147 is a well established example, through simultaneous optical and X-ray observations, of a spectroscopically classified Seyfert-2 galaxy with very little or no absorption (Bianchi et al. 2008). NGC4698 and NGC4565 have also been discussed to be good candidates, (see Georgantopoulos & Zezas 2003, Panessa & Bassani 2002).
It is possible that some of our new unabsorbed candidates are contaminated by nearby luminous X-ray sources that we are unable to resolve owing to the X-ray telescope’s angular resolution. An inspection of the available images which have a superior resolution (0.5 arcsec FWHM) could be very helpful towards this direction. All but three sources (NGC3941, NGC3976 and NGC4378) present archival data. Although a detailed analysis of the properties of the unabsorbed Seyfert-2 galaxies is the scope of a forthcoming paper, we briefly report on whether there is any evidence for contamination. NGC1058 and NGC4168 are significantly contaminated from nearby luminous X-ray sources (see also Foschini et al. 2002, Cappi et al 2006) while NGC4472 suffers from very strong diffuse emission. Finally, inspection of images show that NGC3941 and NGC4501 are contaminated (less than 30% of the counts) by nearby sources (see also Foschini et al. 2002, Cappi et al 2006).
We further try to examine the X-ray properties of unobscured Seyfert-2 galaxies by deriving their stacked spectrum. We use [*MATHPHA*]{} task of [*FTOOLS*]{} software to create the weighted mean X-ray spectrum of the EPIC-PN observations. Weighted mean ancillary files are produced using the [*ADDRMF*]{} and [*ADDARF*]{} tasks of [*FTOOLS*]{}. NGC3147, NGC4565 and NGC4698 are not considered in the mean spectrum since there is already evidence that they do not present any absorption. We also exclude the five contaminated sources leaving the cases of NGC3976, NGC4725, NGC4378 and NGC4477 to be considered.
We try to detect any spectral feature, such as the FeK$_{\alpha}$ line, that could give away the presence of a hidden nucleus in this population as marginally suggested in some cases (e.g. Brightman & Nandra 2008). We fit the average spectrum with an absorbed power-law model plus a Raymond-Smith model. The spectral fitting results are listed in table \[unabsorbed\]. There is no significant evidence for the presence of an FeK$_\alpha$ emission line. Nevertheless, if we choose to include a Gaussian component, the upper limit of the EW is $\sim$600 eV at the 90 per cent confidence level. In Fig. \[stacked1\] we present the mean spectrum along with the best fit model and the residuals. Assuming that all these sources are truly unabsorbed Seyfert-2 galaxies then their total fraction accounts for $\sim$20 per cent of the total population.
$N_H$ (cm$^{-2}$) [$\Gamma$]{} $\rm kT$ (keV) EW$_{\rm FeK}$ (eV) $\chi^2_{\nu}$
------------------- ------------------------ ------------------------ --------------------- ---------------- -- -- -- -- --
$<0.1$ $2.02^{+0.18}_{-0.15}$ $0.37^{+0.07}_{-0.08}$ $<600$ 118.6/125
It has been proposed that the unabsorbed Seyfert-2 galaxies are ’naked’ nuclei i.e. they lack a Broad-Line-Region, BLR, (see Ho 2008 for a review). Various theoretical models could explain this behaviour. Nicastro (2000) presented a model which relates the width of the Broad Emission Lines of AGN to the Keplerian velocity of an accretion disk at a critical distance from the central black hole. Under this scheme the Broad Line Region is linked to the accretion rate of the AGN i.e. below a minimum accretion rate the BLR cannot form. Recently Elitzur & Shlosman (2006) presented an alternative model which depicts the torus as the inner region of clumpy wind outflowing from the accretion disc. According to this model the torus and the BLR disappear when the bolometric luminosity decreases below $\sim 10^{42}$ ergs s$^{-1}$ because the accretion onto the central black hole can no longer sustain the required cloud outflow rate.
In Table \[model\] we try to compare our results with the above model predictions. In Col. 1 we give an estimate for the mass of the central Black Hole, taken from Panessa et al. (2006) and McElroy (1995). The mass estimation is inferred from the mass-velocity dispersion correlation. In the case of NGC3976 there is no information available in the literature. In Col. 2 we calculate the bolometric luminosities using the corrections determined by Elvis et al. (1994) i.e., $L_{\rm BOL}$ = 35 $\times L_{2-10~\rm keV}$ ergs s$^{-1}$. In Col. 3 we give the accretion rate estimator given by $L_{\rm BOL}$/$L_{\rm EDD}\simeq1.3 \times 10^{38} M/M_{\odot}$. The Eddington Luminosity ($L_{\rm EDD}$) is given by $L_{\rm EDD}$=$4\pi G M m_p c / \sigma_T$ where M is the black hole mass, $m_p$ is the proton mass, $\sigma_T$ is the Thomson scattering cross section.
All the sources present very low accretion rates, well below the threshold of 1-4 $\times10^{-3}$ proposed by Nicastro (2000) and Nicastro et al. (2003). Furthermore all these sources (but NGC3147) present very low bolometric luminosities also below the critical value of $10^{42}$ predicted by Elitzur & Shlosman (2006). This supports the idea that the key parameter is not the orientation but an intrinsic parameter (low accretion rate or luminosity), which prevents the formation of the BLR.
---------- -------------------------- -------------------- -------------------------------- -- --
Name Log($M_{BH}/ M_{\odot})$ $Log(L_{\rm BOL})$ $L_{\rm BOL}$ / $ L_{\rm EDD}$
$\times 10^{-4}$
NGC 1058 4.9 39.8 5.5
NGC 3147 8.8 43.0 1.1
NGC 4168 7.9 41.3 0.15
NGC 4378 7.9 41.8 0.50
NGC 4472 8.8 41.4 0.026
NGC 4477 7.9 40.5 0.025
NGC 4501 7.9 41.3 0.17
NGC 4725 7.5 40.1 0.027
NGC 4565 7.7 40.8 0.084
NGC 4698 7.8 40.5 0.030
---------- -------------------------- -------------------- -------------------------------- -- --
[Column 1: Name]{}
[Column 2: Black Hole mass in units of Solar Masses]{}
[Column 3: Bolometric luminosity in units of ergs s$^{-1}$]{}
[Column 4: Accretion rate $L_{\rm BOL}/L_{\rm EDD}$]{}
Discussion
==========
Comparison with other optically selected samples
------------------------------------------------
In this work we present observations of all the Seyfert galaxies from the Palomar survey (Ho et al. 1995). We find that $\sim$50 per cent of the Seyfert population is absorbed by $N_H>10^{22}$ cm$^{-2}$. In this sample we have identified 3 Compton-thick sources which translates to a fraction of $\sim$8 per cent. Five more sources possibly host a highly absorbed or a Compton thick nucleus. In the very extreme, and rather unlikely case were all these candidates are true Compton-thick sources their fraction reaches 20 per cent of the total population.
Cappi et al. (2006) and Pannesa et al. (2006), also using data from the Palomar survey, provide estimates for the fraction of obscured AGN in the local universe. These authors find that about 50% of their sources are obscured ($>10^{22}$ ). Their estimates on the fraction of Compton thick sources suggest an absolute minimum of 20 per cent of the total population. This result comes in contradiction with our findings. However their sample includes 2 objects not fulfilling the Palomar Survey selection criteria (see also Section \[thesample\]). These are the 2 Compton-thick AGN NGC1068 and NGC3185. When we exclude these an agreement is found.
Risaliti et al. (1999) study the X-ray absorption in a sample of 45 Seyfert-2 galaxies finding that a considerable fraction of these are associated with Compton-thick nuclei. A direct comparison with our results is not straightforward since these authors exclude all the sources with $F_{[OIII]}>4\times 10^{-13}$ . However we think that only a luminosity cutoff could reveal column density distribution of the population that contributes to the XRB (see Section 5.2).
X-ray background synthesis models
---------------------------------
The XRB synthesis models can provide tight constraints on the number density of Compton-thick sources. These models attempt to fit the spectrum of the X-ray background roughly in the 1-100 keV range. It is well established that a large number of Compton-thick sources is needed (Gilli et al. 2007) to reproduce the hump of the X-ray background spectrum at 30-40 keV where most of its energy density lies (Churazov et al. 2007, Frontera et al. 2007). Here, we compare the fraction of the Compton-thick sources predicted by the model of Gilli et al. 2007 with our results. We use the publicly available POMPA software [^1]. This predicts the number counts at a given redshift, flux and luminosity range using the best-fit results for the fraction of obscured, Compton-thick sources, of the X-ray background synthesis model of Gilli et al. (2007).
We restrict the comparison to low redshifts $z<0.017$ and X-ray luminosities in the bin $\rm 10^{41}<L_{2-10~\rm keV}<10^{44}$. As our sample is not flux limited in the X-rays, we have to choose a flux limit deep enough to ensure that all sources in this luminosity and redshift bin are detected. A flux limit of $10^{-14}$ satisfies this constraint. The X-ray background synthesis models predict a fraction of Compton-thick sources of about 40 % which is higher compared with our results $15\pm8$%. Only if all the low $F_X/F_{[OIII]}$ Seyfert-2 galaxies are associated with a Compton-thick nuclei the discrepancy would become less pronounced.
Optically selected samples can still miss a fraction of Compton-thick AGN. For example, NGC6240 is classified as a LINER in the optical while [*BeppoSAX*]{} observations show the presence of a Compton-thick nucleus (Vignati et al. 1999). Moreover [*SUZAKU*]{} observations (Ueda et al. 2007 and Comastri et al. 2007) have demonstrated that a small fraction of AGN may have a 4$\pi$ coverage, instead of the usually assumed toroidal structure. These sources will not exhibit the usual high excitation narrow emission lines and therefore will not be classified as AGN on the basis of their optical spectrum.
Recent results based on [*INTEGRAL*]{} and [*SWIFT*]{} observations reveal a small fraction of Compton-thick sources (e.g. Sazonov et al. 2008, Sazonov et al. 2007, Ajello et al. 2008). In particular at the flux limit of $\sim 10^{-11}$ in the 17-60 keV energy band, observations find $10-15\%$ Compton-thick sources. The [*SWIFT*]{}/BAT hard X-ray survey failed to identify any Compton-thick AGN. This non detection discards the hypothesis that their fraction accounts for the 20 per cent of the total AGN at $>2\sigma$ confidence level. It is true however that some heavily obscured Compton-thick sources with $N_H\sim 10^{25-26}$ would be missed even by these ultra hard X-ray surveys.
Less absorption at very low luminosities
----------------------------------------
In the low-luminosity sub-sample (intrinsic $L_{2-10~\rm keV}<10^{41}$ erg s$^{-1}$) the fraction of obscured sources diminishes to 30%. This result comes in apparent contradiction with recent findings suggesting an increasing fraction of obscuration with decreasing luminosity (e.g. Akylas et al. 2006, La Franca et al. 2005). This behaviour may reflect a physical dependence of the column density with intrinsic luminosity as suggested by Elitzur & Shlosman (2006). These authors present a model where the torus and the BLR disappear when the bolometric luminosity decreases below $\sim 10^{42}$ ergs s$^{-1}$ because the accretion onto the central black hole can no longer sustain the required cloud outflow rate. It is interesting to note that the corresponding luminosity in the 2-10 keV band is about several $\times 10^{40}$ , assuming the Spectral Energy Distribution of Elvis et al. (1994). Interestingly, almost all of our Seyfert-2 sources with no absorption present luminosities below this limit (with the exception of NGC3147). We note however, there are sources (NGC3486, NGC3982) with low luminosity, which present column densities around $10^{22}$-$10^{23}$ cm$^{-2}$. Alternatively, it is possible that at least in a few cases, the large XMM-Newton Point Spread Function results in contamination by nearby sources. Thus the nuclear X-ray emission could be out-shined giving the impression that there is no obscuration (e.g. Brightman & Nandra 2008). However, both the inspection of the images as well as the stacked spectrum of the unabsorbed sources do not favour such a scenario.
Conclusions
===========
observations are available for all 38 Seyfert galaxies from the Palomar spectroscopic sample of galaxies of Ho et al. (1995, 1997). Our goal is to determine the distribution of the X-ray absorption in the local Universe through X-ray spectroscopy. Our sample consists of 30 Seyfert-2 and 8 Seyfert-1 galaxies. The results can be summarised as follows:
- [We find a high fraction of obscured sources ($>10^{22}$ ) of about 50 %. ]{}
- [A number of sources present low $F_X/F_{[OIII]}$ ratio. Their individual spectra show no evidence of high absorbing column densities. However, their stacked spectrum shows significant amount of absorption ($\sim 3\times 10^{23} $ )]{}
- [ Considering only the bright sub-sample ($L_{2-10~\rm keV}>10^{41}$ ) , i.e. only these sources which contribute a significant amount to the X-ray background flux, we find that 75 % of our sources are obscured.]{}
- [ In the bright sub-sample there are at least 3 Compton-thick AGN translating to a fraction of 15% which is lower than the predictions of the X-ray background synthesis models at this luminosity and redshift range. Only if we consider, the rather unlikely scenario, where all Seyfert-2 galaxies with a low $F_X/F_{[OIII]}$ ratio are associated with Compton-thick sources we would alleviate this discrepancy.]{}
- [ We find a large number of unobscured Seyfert-2 galaxies. All these have low luminosities $L_{2-10~\rm keV } <3\times 10^{41}$ . Inspection of the images, where available, demonstrates that in most cases these are not contaminated by nearby sources. Furthermore, their stacked spectrum reveals no absorption. It is most likely that these are genuinely unobscured sources in accordance with the predictions of the models of Elitzur & Sloshman (2006)]{}.
[References]{} Ajello M., Rau A., Greiner J. et al., 2008, ApJ, 673, 96 Akylas A., Georgantopoulos I., Georgakakis A., Kitsionas S., Hatziminaoglou E., 2006 , A&A , 459 , 693 Alexander D. M., Bauer F. E., Brandt W. N., Garmire G. P., Hornschemeier A. E., Schneider D. P., Vignali C., 2003, AN, 324, 8 Awaki H., Ueno S., Taniguchi Y., 2000 , AdSpR , 25 , 797 Barger A. J., Cowie L. L.,Capak, D. M., et al., 2003 , AJ , 126 , 632 Barthelmy S. D., Barbier L. M., Cummings J. R., et al., 2005 , SSRv , 120 , 143 Baskin A., Laor A, 2008, ApJ, 682, 110 Bassani L., Dadina M., Maiolino R., Salvati M., Risaliti G., della Ceca R., Matt G., Zamorani G., 1999 , ApJS , 121 , 473 Bassani L., Molina M., Malizia A., et al., 2006 , ApJ , 636 , L65 Bianchi S., Corral A., Panessa F., Barcons X., Matt G., Bassani L., Carrera F. J., Jim$\acute{e}$nez-Bail$\acute{o}$n E., 2008 , MNRAS , 385 , 195 Brandt W. N., Hasinger G., 2005 , ARA&A , 43 , 827 Brightman M., Nandra K., 2008, MNRAS, 390, 1241 Cappi M., Panessa F., Bassani L., et al., 2006 , A&A , 446 , 459 Churazov E., Sunyaev R., Revnivtsev M., et al., 2007 , A&A , 467 , 529 Comastri A., 2004 , ASSL , 308 , 245 Comastri A., Setti G., Zamorani G., Hasinger G., 1995 , A&A, 296, 1 Dickey J. M., Lockman F. J., 1990 , ARA&A , 28 , 215 Elitzur M., Shlosman I., 2006, ApJ, 648, L101 Elvis M., Wilkes B. J., McDowell J. C., et al., 1994 , ApJS , 95 , 1 Foschini L., Di Cocco G., Ho L. C., et al., 2002 , A&A , 392 , 817 Frontera F., Orlandini M., Landi R., et al., 2007, ApJ, 666 , 86 Georgantopoulos I., Zezas A., 2003, ApJ, 594 , 704 Georgantopoulos I., Georgakakis A., Akylas A. 2007, A&A, 466, 823 Ghisellini G., George I. M., Fabian A. C., & Done C. 1991, MNRAS,248, 14 Gilli R., Comastri A., Hasinger G., 2007 , A&A , 463 , 79 Gliozzi M., Sambruna R. M., Foschini L., 2007, ApJ, 662, 878 Guainazzi M., Matt G., Perola G. C., 2005 , A&A , 444 , 119 Heckman T. M., Ptak A., Hornschemeier A., Kauffmann G., 2005 , ApJ , 634 , 161 Ho L. C., 2008, ARA&A, 46, 475 Ho L. C., Ulvestad J. S., 2001, ApJS, 133, 77 Ho L. C., Filippenko A. V., & Sargent W. L.W. 1997, ApJS,112, 315 Ho L. C., Filippenko A. V., & Sargent W. L. 1995, ApJS, 98, 477 La Franca F., Fiore F., Comastri A., et al., 2005, ApJ , 635 , 864 Leahy D. A., Creighton J., 1993 , MNRAS , 263 , 314 Luo B., Bauer F. E., Brandt W. N., et al., 2008, ApJS, 179, 19L Maiolino R., Salvati M., Bassani L., Dadina M., della Ceca R., Matt G., Risaliti G., Zamorani G., 1998, A&A, 338, 781 Magdziarz P., Zdziarski A. A., 1995 , MNRAS , 273 , 837 Malizia A., Landi R., Bassani L., et al., 2007 , ApJ , 668 , 81 Marconi A., Risaliti G., Gilli R., Hunt L. K., Maiolino R., Salvati M., 2004 , MNRAS , 351 , 169 Markwardt C. B., Barbier L., Barthelmy S., et al., 2005 , ApJ , 37 , 1222 Matt G., Fabian A. C., Guainazzi M., Iwasawa K., Bassani L., Malaguti G., 2000 , MNRAS , 318 , 173 McElroy D. B., 1995 , ApJS , 100 , 105 Nicastro F., 2000 , ApJ , 530 , L65 Nicastro F., Martocchia A., Matt G., 2003 , ApJ , 589 , L13 Panessa F., Bassani L., Cappi M., Dadina M., Barcons X., Carrera F. J., Ho L. C., Iwasawa K., 2006 , A&A , 455 , 173 Panessa F., Bassani L., 2002, A&A, 394, 435 Pappa A., Georgantopoulos I., Stewart G. C., Zezas A. L., 2001, MNRAS, 326, 995 Risaliti G., Maiolino R., Salvati M., 1999 , ApJ , 522 , 157 Sandage A., Tammann G. A., & Yahil, A. 1979, ApJ, 232, 352 Sazonov S., Revnivtsev M., Burenin R., Churazov E., Sunyaev R., Forman W. R., Murray S. S., 2008 , A&A , 487 , 509 Sazonov S., Revnivtsev M., Krivonos R., Churazov E., Sunyaev R., 2007, A&A, 462, 57 Strüder L., Briel U., Dennerl K., et al. 2001, A&A 365, L18 Tozzi P., Gilli R., Mainieri V., et al., 2006 , A&A , 451 , 457 Tueller J., Mushotzky R. F., Barthelmy S., Cannizzo J. K., Gehrels N., Markwardt C. B., Skinner G. K., Winter L. M., 2008 , ApJ , 681 , 113 Turner M. J. L., Abbey A., Arnaud M., Turner M. J. L., et al. 2001, A&A 365, L27 Ubertini P., Lebrun F., Di Cocco G., et al., 2003 , A&A , 411 , L131 Ueda Y., Eguchi S., Terashima Y., et al., 2007, ApJ, 664, 79 Vignati P., Molendi S., Matt, G., et al., 1999, A&A, 349, 57 Weaver K. A., Nousek J., Yaqoob T., Mushotzky R. F., Makino F., & Otani C. 1996, ApJ, 458, 160 Winter L. M., Mushotzky R. F., Tueller J., Markwardt C., 2008 , ApJ , 674 , 686 Wolf C., Wisotzki L., Borch A., Dye S., Kleinheinrich M., Meisenheimer K.,2003, A&A, 408, 499
\[xspectra\]
\[xspectra\]
\[xspectra\]
\[xspectra\]
[^1]: www.bo.astro.it/$\sim$gilli/counts.html
|
---
abstract: 'Let $W$, $X$, $Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\in L^{r}(Y, Z)$ and $B\in L^{r}(W, X)$ let $M_{A,\,B}$ be the two-sided multiplication operator from $L^{r}(X, Y)$ into $L^r(W,\,Z)$ defined by $M_{A,\,B}(T)=ATB$. We show that for every $0\leq A_0\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A,\, B}|=M_{|A|, \,|B|}$ for all $ A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$. Our results generalize the related results of Synnatzschke and Wickstead, respectively.'
address:
- 'Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China'
- 'Department of Mathematics, University of South Carolina, Columbia, SC 29208'
author:
- Jin Xi Chen
- 'Anton R. Schep'
title: 'Two-sided multiplication operators on the space of regular operators'
---
[^1]
4.5mm
Introduction
============
For an algebra $\mathcal{A}$ an operator of the form $T\mapsto\sum_{i=1}^{n}A_{i}TB_{i}$, where $A_i$, $B_i$ are fixed in $\mathcal{A}$, is referred to as an *elementary operator* on $\mathcal{A}$. If $A,\,B\in\mathcal{A}$, we denote by $M_{A,\,B}$ the operator $T\mapsto ATB$. The operator $M_{A,\,B}$ is called a *basic elementary operator* or a *two-sided multiplication operator*. The literature related to (basic) elementary operators is by now very large, much of it in the setting of $C^*$-algebras or in the Banach space setting. In this direction there are many excellent surveys and expositions. See, e.g., [@Ara; @Curto; @Mathieu; @Saksman].
For the study of two-sided multiplication operators in the setting of Riesz spaces (i.e., vector lattices) we would like to mention the work of Synnatzschke [@Synnatz]. The set of all regular operators (order continuous regular operators, resp.) from a Riesz space $X$ into a Dedekind complete Riesz space $Y$ will be denoted by $L^{r}(X, Y)$ ($L^{r}_n(X, Y)$, resp.). When $Y=\mathbb{R}$, we write $X^{\sim}$ and $X^{\sim}_{n}$ respectively instead of $L^{r}(X, \mathbb{R})$ and $L^{r}_n(X, \mathbb{R})$. They are likewise Dedekind complete Riesz spaces. Let $W$, $X$, $Y$ and $Z$ be Dedekind complete Riesz spaces. For all $A\in L^r(Y,\,Z)$ and $B\in L^r(W,\,X)$, $M_{A,\,B}:T\in L^r(X,\,Y)\mapsto ATB\in L^r(W,\,Z)$ is a regular operator, and hence the modulus $|M_{A,\,B}|$ of $M_{A,\,B}$ exists in $ L^{r}\big(L^{r}(X, Y), L^{r}(W, Z)\big)$. It is interesting to know about the relationship of $|M_{A,\,B}|$ with $|A|$ and $|B|$. Among other things, Synnatzschke [@Synnatz Satz 3.1] proved the following result:\
a) If $0\leq B_0\in L^r(W,\,X)$, then $|M_{A,\,B_0}|=M_{|A|,\,B_0}$, $M_{A,\,B_0}\vee\, M_{C,\,B_0}=M_{A\vee C,\,B_0}$ hold for all $A,\,C\in L^r(Y,\,Z)$.\
b) If $0\leq A_0\in L^{r}_n(Y,\,Z)$ and $Y^{\sim}_{n},\,Z^{\,\sim}_{n}$ are total, then we have $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ and $(M_{A_0,\,B}\vee M_{A_0,\,D}) (T)=M_{A_0,\,B\vee D}(T)$ for all $B,\,D\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$.
Hereby $Y^{\sim}_{n}$ is *total* if $Y^{\sim}_{n}$ separates the points of $Y$. Synnatzschke uesd (a) to establish (b) by taking adjoints of operators. For his purpose, the hypothesis of both $Y^{\sim}_{n}$ and $Z^{\,\sim}_{n}$ being total is essential.
Recently, Wickstead [@Wickstead] showed that if $E$ is an atomic Banach lattice with order continuous norm and $A,\,B\in L^r(E)$, then $|M_{A,\,B}|=M_{|A|,\,|B|}$ and $\|M_{A,\, B}\|_{r}=\|A\|_{r} \|B\|_r$. In his proofs he depended heavily upon the ‘atomic’ condition.
In this note, we generalize the related results of Synnatzschke and Wickstead, respectively. We remove the condition of order continuous duals being total in [@Synnatz Satz 3.1(b)] and show that for every $0\leq A_0\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms (not necessarily atomic), then $|M_{A,\, B}|=M_{|A|, \,|B|}$ and $\|M_{A,\, B}\|_{r}=\|A\|_{r} \|B\|_r$ hold for all $ A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$.
Our notions are standard. For the theory of Riesz spaces and regular operators, we refer the reader to the monographs [@AB; @M; @Za].
The modulus of the two-sided multiplication operator
====================================================
We start with two examples which serve to illustrate that the order continuous dual $X^{\sim}_{n}$ of a Dedekind complete Riesz space $X$ need not be total. This justifies our effort to generalize the result of Synnatzschke [@Synnatz Satz 3.1 b)].
\(1) Let $(\Omega,\,\Sigma,\,\mu)$ be a nonatomic finite measure space. Then the Dedekind complete Riesz space $X=L_p(\Omega,\,\Sigma,\,\mu)$ ($0<p<1$) satisfies $X^{\sim}_{n}=X^{\sim}=\{0\}$. This result is due to M. M. Day. (cf. [@AB2 Theorem 5.24, p. 128]).
\(2) Let $K$ be a compact Hausdorff space. It is well known that $C(K)$ is Dedekind complete if and only if $K$ is Stonian (i.e., extremally disconnected). A Hausdorff compact Stonian space $K$ such that $C(K)^{\sim}_{n}$ is total is called hyper-Stonian. Dixmier gave a characterization of hyper-Stonian spaces: $K$ is hyper-Stonian if and only if $C(K)$ is isomorphic to a dual Banach lattice (cf. [@M Theorem 2.1.7]). He also gave a Dedekind complete $C(K)$-space which is not isomorphic to a dual space (see, e.g., [@Alb p. 99, Problems 4.8 and 4.9] for details). That is, such a $C(K)$ is Dedekind complete, but $C(K)^{\sim}_{n}$ is not total.
\[Prop 2.1\] Let $W$, $X$, $Y$ and $Z$ be Riesz spaces with $X$, $Y$ and $Z$ Dedekind complete. Let $0\leq A_0\in L^{r}_{n}(Y, Z)$. Then we have $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ and, equivalently, $M_{A_0,\,B}\vee M_{A_0,\,D} (T)=M_{A_0,\,B\vee D}(T)$ for all $B,\,D\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$.
For $B\in L^{r}(W, X)$ and $0\leq T\in L^{r}_n(X, Y)$, we have to prove that $|M_{A_0, B}|(T)= M_{A_0, |B|}(T)$. Clearly we have $|M_{A_0, B}|(T)\leq M_{A_0, |B|}(T)$, since $|M_{A_0, B}|\leq M_{A_0, |B|}$ holds in $L^{r}\big(L^{r}(X, Y), L^{r}(W, Z)\big)$. For the reverse inequality, let $w\in W^{+}$. By a formula for the modulus of regular operators in [@AB Theorem 1.21(3)] or [@Za Theorem 20.10(i)] we have$$\left(\sum_{i=1}^{n}|Bw_i|:n\in\mathbb{N},\, 0\leq w_{i}\in W,\, \sum_{i}w_i=w\right)\uparrow|B|w.$$Since $A_0$ and $T$ are both positive order continuous operators, $A_0T$ is likewise an order continuous positive operator from $X$ into $Z$. It follows that $$\begin{aligned}
M_{A_{0},\,|B|}(T)(w)&=&A_{0}T|B|w\\
&=&\sup\left(\sum_{i=1}^{n}A_0T|Bw_i|:n\in\mathbb{N},\, 0\leq w_{i}\in W,\, \sum_{i}w_i=w\right)\end{aligned}$$ For each $1\leq i\leq n$, let $P_i$ be the order projection from $X$ onto the band generated by $(Bw_{i})^+$ in $X$ and let $Q_i=P_i-I$, where $I$ is the identity operator on $X$. Clearly, $$P_i\perp Q_i,\quad |P_i|+|Q_i|=I,\quad P_{i}Bw_i=(Bw_{i})^+,\quad (P_i+Q_i)Bw_i=|Bw_i|,$$and $$|TP_i|+|TQ_i|=T.$$Therefore, for each $i$ we have $$\begin{aligned}
A_0T|Bw_i|&=&(A_0TP_i+A_0TQ_i)Bw_i\\
&\leq&\big(|A_0(TP_i)B|+|A_0(TQ_i)B|\big)w_i\\
&\leq&\left(\sup\Bigg\{\sum_{j=1}^{m}|A_0T_jB|:m\in\mathbb{N},\, T_j\in L^{r}(X,Y),\, \sum_{j}|T_j|=T\Bigg\}\right)w_i\\
&=&\left(\sup\Bigg\{\sum_{j=1}^{m}|M_{A_{0},\,B}(T_j)|:m\in\mathbb{N},\, T_j\in L^{r}(X,Y),\, \sum_{j}|T_j|=T\Bigg\}\right)w_i\\
&=&|M_{A_{0},B}|(T)(w_i).\end{aligned}$$ Hence, from this it follows that $$\begin{aligned}
M_{A_{0},\,|B|}(T)(w)&=&\sup\left(\sum_{i=1}^{n}A_0T|Bw_i|:n\in\mathbb{N},\, 0\leq w_{i}\in W,\, \sum_{i}w_i=w\right)\\
&\leq&\sup\left(\sum_{i=1}^{n}|M_{A_{0},B}|(T)(w_i):n\in\mathbb{N},\, 0\leq w_{i}\in W,\, \sum_{i}w_i=w\right)\\
&=&|M_{A_{0},B}|(T)(w),\end{aligned}$$ which implies that $|M_{A_0, B}|(T)\leq M_{A_0, |B|}(T)$ for all $B\in L^{r}(W, X)$ and all $0\leq T\in L^{r}_{n}(X, Y)$.
In general we can not expect that $|M_{A_0, B}|=M_{A_0, |B|}$ holds for all $B\in L^{r}(W, X)$. That is, the linear operator $M_{A_0,\, \cdot}:B\in L^{r}(W, X)\rightarrow L^{r}\big(L^{r}(X, Y), L^{r}(W, Z)\big)$ is not necessarily a Riesz homomorphism. In the last section we give a counterexample to illustrate this. However, for Banach lattices with order continuous norms the situation is quite different. The next result is a consequence of the above proposition and the earlier result of Synnatzschke [@Synnatz Satz 3.1], which generalizes Theorem 3.1 of Wickstead [@Wickstead] recently obtained for atomic Banach lattices with order continuous norms.
\[Corollary 2.2\] Let $W$, $X$, $Y$ and $Z$ be Banach lattices such that $X$, $Y$ have order continuous norms and $Z$ is Dedekind complete. Then we have $|M_{A,\, B}|=M_{|A|, \,|B|}$ for all $ A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$.
Let $\mathcal{M}:L^{r}(Y, Z)\times L^{r}(W, X)\rightarrow L^{r}\big(L^{r}(X, Y), L^{r}(W, Z)\big)$ be the bilinear operator defined via $\mathcal{M}(A, B)=M_{A,\,B}$. Clearly, $\mathcal{M}$ is positive. Since $X$ and $Y$ are Banach lattices with order continuous norms, we have $L^{r}_{n}(X,\, Y)=L^{r}(X,\, Y)$ and $L^{r}_{n}(Y,\, Z)=L^{r}(Y,\, Z)$. From Proposition \[Prop 2.1\] above and the result of Synnatzschke [@Synnatz Satz 3.1] it follows that for every $0\leq A_0\in L^{r}(Y, Z)$ and every $0\leq B_0\in L^{r}(W, X) $, $\mathcal{M}(A_0,\,\cdot)$ and $\mathcal{M}(\cdot\,,B_0)$ are both Riesz homomorphisms. Hence, for all $A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$ we have $$\begin{aligned}
|M_{A, B}|&=&|\mathcal{M}(A, B)|\\&=&|\mathcal{M}(A^{+}-A^{-}, B^{+}-B^{-})|\\
&=&|\mathcal{M}(A^+,\,B^+)-\mathcal{M}(A^+,\,B^-)-\mathcal{M}(A^-,\,B^+)+\mathcal{M}(A^-,\,B^-)|\\
&=&\mathcal{M}(A^+,\,B^+)+\mathcal{M}(A^+,\,B^-)+\mathcal{M}(A^-,\,B^+)+\mathcal{M}(A^-,\,B^-)\\
&=&\mathcal{M}(|A|,\,|B|)=M_{|A|,\,|B|}.\end{aligned}$$ Here we are using the fact that the terms $\mathcal{M}(A^+,\,B^+)$, $\mathcal{M}(A^+,\,B^-)$, $\mathcal{M}(A^-,\,B^+)$ and $\mathcal{M}(A^-,\,B^-)$ are pairwise disjoint.
Let $W$ and $X$ be Banach lattices with $X$ Dedekind complete. Recall that $L^{r}(W,\,X)$ is a Dedekind complete Banach lattice under the regular norm $\|B\|_r:=\||B|\|$ for every $B\in L^{r}(W,\,X)$. Note that $M_{A,\,B}$ is a regular operator from $L^{r}(X, Y)$ into $L^{r}(W, Z)$. The following result deals with the regular norms of two-sided multiplication operators. Its proof is based on Corollary \[Corollary 2.2\]
\[Corollary 2.3\] If $W$, $X$, $Y$ and $Z$ be Banach lattices such that $X$, $Y$ have order continuous norms and $Z$ is Dedekind complete, then $\|M_{A,\, B}\|_{r}=\|A\|_{r} \|B\|_r$ for all $ A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$.
We first assume that $ 0\leq A\in L^{r}(Y, Z)$ and $0\leq B\in L^{r}(W, X)$. Since $M_{A,\,B}\geq0$, we have $\|M_{A,\,B}\|_r=\|M_{A,\,B}\|\leq\|A\|\|B\|=\|A\|_r\|B\|_r$. On the other hand, for every $0\leq x^{\prime}\in X^{\prime}$ and every $0\leq y\in Y$ satisfying $\|x^{\prime}\|\leq1$ and $\|y\|\leq1$, $x^{\prime}\otimes y\in L^{r}(X,\,Y)$ and $\|x^{\prime}\otimes y\|_r=\|x^{\prime}\otimes y\|\leq1$. Then it follows that $$\begin{aligned}
\|M_{A,\,B}\|_r=\|M_{A,\,B}\|&\geq&\sup\Big(\|M_{A,\,B}(x^{\prime}\otimes y)\|:0\leq x^{\prime}\in B_{X^{\prime}}, 0\leq y\in B_Y\Big)\\
&=&\sup\Big(\|(B^{\prime}x^\prime)\otimes Ay\|:0\leq x^{\prime}\in B_{X^{\prime}}, 0\leq y\in B_Y\Big)\\
&=&\|A\|\|B\|=\|A\|_r\|B\|_r.\end{aligned}$$ This implies that $\|M_{A,\,B}\|_r=\|A\|_r\|B\|_r$ holds for all $ 0\leq A\in L^{r}(Y, Z)$ and $0\leq B\in L^{r}(W, X)$.
Now, for the general case let $A\in L^{r}(Y, Z)$ and $B\in L^{r}(W, X)$ be arbitrary. Then by Corollary \[Corollary 2.2\] we have $$\begin{aligned}
\|M_{A,\,B}\|_r=\||M_{A,\,B}|\|=\|M_{|A|,\,|B|}\|=\|M_{|A|,\,|B|}\|_r=\|A\|_r\|B\|_r.\end{aligned}$$
Wickstead [@Wickstead] establishes that even in the case of atomic Banach lattices with order continuous norms the operator norm of two-sided multiplication operators need not be equivalent to the regular norm.
A Counterexample
================
Let $X$ and $Y$ be Riesz spaces with $Y$ Dedekind complete. The set of all $\sigma$-order continuous operators in $L^{r}(X, Y)$ will be denoted by $L^{r}_c(X, Y)$. The disjoint complement $(L^{r}_c(X, Y))^{d}$ of $L^{r}_c(X, Y)$ is denoted by $L^{r}_s(X, Y)$. Every element of $L^{r}_s(X, Y)$ is called a singular operator. When $Y=\mathbb{R}$, we write $X^{\sim}$ and $X^{\sim}_{s}$ respectively instead of $L^{r}(X, \mathbb{R})$ and $L^{r}_s(X, \mathbb{R})$. The following example illustrates that $|M_{A_0, B}|=M_{A_0, |B|}$ does not necessarily hold for all $B\in L^{r}(W, X)$, that is, the linear operator $M_{A_0,\, \cdot}:B\in L^{r}(W, X)\rightarrow L^{r}\big(L^{r}(X, Y), L^{r}(W, Z)\big)$ is not necessarily a Riesz homomorphism in general.
Let $W=X=Y=Z=\ell_{\infty}$ and let $e$ denote the strong unit $(1, 1,\cdots)$ of $\ell_{\infty}$. Let $0\le f\in (\ell_{\infty})^{\sim}_{s}$ be a singular Riesz homomorphism with $f(e)=1$ (one can take, e.g., $f$ equal to a limit over a free ultrafilter). Let $B\in L^{r}(\ell_{\infty})$ be the rank one operator $B=f \otimes e$. Then it is clear that $B\in L^{r}_s(\ell_{\infty})$ and $I\wedge B=0$, where $I$ is the identity operator on $\ell_{\infty}$ (and hence order continuous). We claim that $M_{I,\,I}\wedge M_{I,\,B}\neq M_{I,\,I\wedge B}=0$. To this end, let $0\le T\in L^{r}(\ell_{\infty})$. Then, by [@AB Theorem 1.21(2)] we have $$\begin{aligned}
\Bigg\{\sum\limits_{i=1}^{n}(T_{i}\wedge T_{i}B):n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T \Bigg\}\downarrow (M_{I,\,I}\wedge M_{I,\,B})(T).\end{aligned}$$ From this and [@AB Theorem 1.51(2)] it follows that $$\begin{array}{lc}
(M_{I,\,I}\wedge M_{I,\,B})(T)(e)\\[12pt]
\qquad\quad=\inf\Bigg\{\sum\limits_{i=1}^{n}(T_{i}\wedge T_{i}B)(e): n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T \Bigg\}\\[12pt]
\qquad\quad=\inf\Bigg\{\sum\limits_{i=1}^{n}(T_{i}\wedge (f\otimes T_{i}e))(e): n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T\Bigg\}\\[12pt]
\qquad\quad=\inf\Bigg\{\sum\limits_{i=1}^{n}\inf\Bigg(\sum\limits_{j=1}^{m_i}T_{i}x^{(i)}_{j}\wedge f(x^{(i)}_{j})T_{i}e:x^{(i)}_{j}\wedge x^{(i)}_{k}=0, j\neq k,\sum\limits_{j=1}^{m_i}x^{(i)}_{j}=e\Bigg): \\[12pt] \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T\Bigg\}.
\end{array}$$ Let us put $$\begin{array}{l}
G^{\,\prime}=\Bigg\{\sum\limits_{i=1}^{n}\inf\Bigg(\sum\limits_{j=1}^{m_i}T_{i}x^{(i)}_{j}\wedge f(x^{(i)}_{j})T_{i}e:x^{(i)}_{j}\wedge x^{(i)}_{k}=0, j\neq k,\sum\limits_{j=1}^{m_i}x^{(i)}_{j}=e\Bigg): \\[12pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T \Bigg\},
\end{array}$$ $$\begin{array}{l}
G^{\,\prime\prime}=\Bigg\{\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}T_{i}x_{j}\wedge f(x_{j})T_{i}e:m\in\mathbb{N}, x_{j}\wedge x_{k}=0, j\neq k,\sum\limits_{j=1}^{m}x_{j}=e, \\[12pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T \Bigg\}.
\end{array}$$ We claim that $\inf G^{\,\prime}=\inf G^{\,\prime\prime}$. Indeed, it is clear that $\inf G^{\,\prime}\leq\inf G^{\,\prime\prime}$. For the reverse inequality, let $(T_i)^{n}_{1}$ be a fixed positive partition of $T$ (i.e., $ T_{i}\ge 0$, $\sum_{i} T_{i}=T$). For each $i$ let $(x^{(i)}_j)_{j=1}^{m_i}$ be an arbitrary positive disjoint partition of $e$ (i.e., $x^{(i)}_{j}\wedge x^{(i)}_{k}=0, j\neq k,\sum_{j=1}^{m_i}x^{(i)}_{j}=e$) corresponding to $T_i$. Following the proof of [@AB Theorem 1.51] we can find a positive disjoint partition $(x_j)^{m}_1$ of $e$ such that $$\sum\limits_{j=1}^{m}T_{i}x_{j}\wedge f(x_{j})T_{i}e\leq\sum\limits_{j=1}^{m_i}T_{i}x^{(i)}_{j}\wedge f(x^{(i)}_{j})T_{i}e \qquad (i=1, 2,\cdot\cdot\cdot, n).$$ From this it follows that $$\inf G^{\,\prime\prime}\leq \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}T_{i}x_{j}\wedge f(x_{j})T_{i}e\leq\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m_i}T_{i}x^{(i)}_{j}\wedge f(x^{(i)}_{j})T_{i}e$$Therefore, $$\inf G^{\,\prime\prime}\leq \sum\limits_{i=1}^{n}\inf\Bigg(\sum\limits_{j=1}^{m_i}T_{i}x^{(i)}_{j}\wedge f(x^{(i)}_{j})T_{i}e:x^{(i)}_{j}\wedge x^{(i)}_{k}=0, j\neq k,\sum\limits_{j=1}^{m_i}x^{(i)}_{j}=e\Bigg),$$which implies that $\inf G^{\,\prime\prime}\leq\inf G^{\,\prime}$. Hence, we have $(M_{I,\,I}\wedge M_{I,\,B})(T)(e)=\inf G^{\,\prime\prime}$.
Since $f$ is a Riesz homomorphism, for every positive disjoint partition $(x_j)^{m}_1$ of $e$ appearing in $G^{\,\prime\prime}$ there exists only one, say $x_{j_0}$, such that $$f(x_j)=0, \quad j\neq j_0, \quad f(x_{j_0})=f(e)=1$$ $$\quad\quad x_{j_0}\wedge\sum_{j\neq j_{0}}x_j=x_{j_0}\wedge(e-x_{j_0})=0.$$ It follows that $\sum_{i=1}^{n}\sum_{j=1}^{m}T_{i}x_{j}\wedge f(x_{j})T_{i}e= \sum_{i=1}^{n}T_{i}x_{j_0}.$ On the other hand, for any $x\in E^+$ satisfying $x\wedge(e-x)=0$ and $f(x)=1$, we must have $f(e-x)=0$, and hence $$\sum\limits_{i=1}^{n}T_{i}x=\sum_{i=1}^{n}\Big(T_{i}x\wedge f(x)T_{i}e+T_{i}(e-x)\wedge f(e-x)T_{i}e\Big)$$
Thus, we have $$\begin{array}{l}
(M_{I,\,I}\wedge M_{I,\,B})(T)(e)\\[10pt]
\qquad\qquad=\inf\Bigg\{\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}T_{i}x_{j}\wedge f(x_{j})T_{i}e:m\in\mathbb{N}, x_{j}\wedge x_{k}=0, j\neq k,\sum\limits_{j=1}^{m}x_{j}=e, \\[12pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T \Bigg\}\\[12pt]
\qquad\qquad=\inf\Bigg\{\sum\limits_{i=1}^{n}T_{i}x:x\wedge (e-x)=0, f(x)=1, n\in\mathbb{N}, T_{i}\ge 0, \sum\limits_{i} T_{i}=T\Bigg\}\\[12pt]
\qquad\qquad=\inf\Bigg\{Tx:0\le x\le e, x\wedge(e-x)=0, f(x)=1\Bigg\}.
\end{array}$$ If we now take $T=B=f\otimes e$, then $(M_{I,\,I}\wedge M_{I,\,B})(f\otimes e)(e)=e$. So, $M_{I,\,I}\wedge M_{I,\,B}\neq 0$.
[99]{} F. Albiac and N. J. Kalton, *Topics in Banach Space Theory*, Graduate Texts in Mathematics **233**, Springer, New York, 2006. MR2006h:46005 C. D. Aliprantis and O. Burkinshaw, *Positive Operators* (reprint of the 1985 original), Springer, Dordrecht, 2006. MR2262133 C. D. Aliprantis and O. Burkinshaw, *Locally Solid Riesz Spaces with Applications to Economics*, Math Surveys and Monographs, Volume $\sharp$105, Amer. Math. Soc., 2003. MR2005b:46010. P. Ara and M. Mathieu, *Local Multipliers of $C^{\ast}$-Algebras*, Springer Monographs in Mathematics, Springer-Verlag, London, 2003. MR2008i:47076 R. E. Curto, Spectral theory of elementary operators, *Elementary operators and applications* (M. Mathieu, ed.)(Proc. Int. Conf. Blaubeuren, 1991), 3-52, World Sci. Publ., River Edge, NJ, 1992. MR93i:47041 M. Mathieu, The norm problem for elementary operators, *Recent progress in functional analysis (Valencia, 2000)*, 363-368, North-Holland Math. Stud., **189**, North-Holland, Amsterdam, 2001. MR2002g:47071 P. Meyer-Nieberg, *Banach Lattices*, Universitext, Springer-Verlag, Berlin, 1991. MR 93f:46025 E. Saksman and H.-O. Tylli, Multiplications and elementary operators in the Banach space setting, *Methods in Banach space theory*, 253-292, London Math. Soc. Lecture Note Ser., **337**, Cambridge Univ. Press, Cambridge, 2006. MR2008i:47076 J. Synnatzschke, Über einige verbandstheoretische Eigenschaften der Multiplikation von Operatoren in Vektorverbänden, *Math. Nachr.* **95** (1980), 273-292. MR82b:47048 A. W. Wickstead, Norms of basic elementary operators on algebras of regular operators, *Proc. Amer. Math. Soc.*, to appear. A. C. Zaanen, *Introduction to Operator Theory in Riesz Spaces*, Springer, Berlin, 1997.
MR2000c:47074
[^1]: The first author was supported in part by China Scholarship Council (CSC) and was visiting the University of South Carolina when this work was completed.
|
---
author:
- 'K. Benakli[^1] and M. D. Goodsell[^2]'
title: 'Two-Point Functions of Chiral Fields at One Loop in Type II'
---
*Laboratoire de Physique Théorique et Hautes Energies[^3], Tour 24-25, 5eme étage, Boite 126, 4 place Jussieu, F-75252 Paris Cedex 05 France*
Introduction
============
Today’s high energy physics faces open questions such as the construction of a quantum theory of gravity, explaining baryogenesis, the rotation curves of galaxies, the expansion of the universe, or understanding the gauge hierarchy problem... These motivate investigations of many extensions of the Standard Model of particle physics, including, for instance, supersymmetric and higher dimensional ones. The corresponding effective field theories find in string theory a unique framework for an ultraviolet completion. There, the quantum field theory peturbative expansion is replaced by a sum over world-sheet surfaces, each order being ultraviolet finite in supersymmetric vacua.
Because they characterise the effective low energy theory, the lowest dimension correlation functions are of special interest. It is important, for instance, to understand their dependence on the different data of the string compactification. Of these, the two-point function of scalars $\phi_i$ belonging to chiral multiplets $\Phi^i$ gives information about the Kähler metric: \^[i]{}(k) \^[i]{} (-k)= k\^2 G\_[\^[i]{},\^[i]{}]{}. The computation of these two point functions is the subject of this work.
The chiral fields have a cubic superpotential $W \supset \frac{\lambda_{ijk}}{3!} \Phi_i \Phi_j \Phi_k$. In type $IIB$ string theory, $\lambda_{ijk}$ depend only on the complex structure and open string moduli, while in type $IIA$ it may only depend upon the Kähler and open string moduli. The one loop correlator can be split into field theory part and threshold correction \^[(1)]{} G\_[\^[I]{},\^[I]{}]{} = - \_[\_i]{} k\^2/M\_s\^2 + \_[\_I]{} where $\tilde{\gamma}_{\Phi_i}$ is given by \_[\_i]{} = . \[deltaG\] and is related to the one-loop anomalous dimension $\gamma_{\Phi_i}^{(1)}$ by \_[\_i]{}\^[(1)]{} = = , where $Y_{ijk}$ is the physical Yukawa coupling. Just as threshold corrections modify the tree-level gauge couplings, so corrections to the Kähler potential change the Yukawa couplings (see for example [@Antoniadis:1992pm]) of which the one loop corrections may be significant. Moreover, they can teach us about the expected effects of supersymmetry breaking when mediated by moduli fields.
Below, we will carry out these computations for the case of toroidal orbifold / orientifold compactification factorisable on three two tori $T^2$ of type II strings (see [@IIB] for reviews of IIB model building, and [@IIA:Reviews] for IIA, and [@IIA:Models] for recent work in this area) . These models provide an exceptional laboratory in this respect as they have a simple geometrical picture and they allow an explicit computation through conformal field theory techniques [@Friedan:1985ge; @Burgess:1986ah; @Garousi:1996ad]. The chiral states are identified with the massless modes of two types of open strings: (i) open strings with both ends on the same stack of branes; and (ii) open strings with one end on a brane stack $a$ and the second end on a different one $b$, the two stacks intersecting at non-vanishing angles $\theta^{\kappa}_{ab}$ on each torus $\kappa$. To our knowledge the one-loop two point functions have been computed only for the former case (i) in [@Antoniadis:1996vw; @Bain:2000fb; @Berg:2005ja]. This was related to the calculation of corrections to closed string moduli, as also in [@Antoniadis:2003sw; @Bachas:1997mc; @Berg:2004ek]. Moreover, because of the diverse application of moduli fields for the determination of coupling “constants”, supersymmetry breaking or inflationary models, previous works focused on the contributions from the $N=2$ sectors of orbifolds where the moduli dependence appears. However the states of type (ii) localised at brane intersections play an important role, as for example, usually the matter and the Higgs fields in phenomenological considerations are most often identified with them. We believe it is useful to provide the tools and expressions to compute explicitly the one-loop two point functions for these states. The computation proceeds in a different manner as it involves now computing two point functions for boundary changing operators. Using the techniques of [@Dixon:1986qv] for twist operators, the computation of the tree-level correlations of boundary changing operators have been introduced for open strings in [@Gava:1997jt], successively used, first with four insertions with orthogonal brane intersections in [@Antoniadis:2000jv], then for generic angles, the worldsheet instanton contributions was found in [@Cremades:2003qj] before the full CFT computation was performed in [@Cvetic:2003ch; @Abel:2003vv; @Klebanov:2003my], and finally generic N-point functions were calculated in [@Abel:2003yx]. Later, building on the general methods of [@Atick:1987kd], one-loop diagrams with boundary changing operators have been constructed [@Abel:2004ue; @Abel:2005qn].
Amplitudes involving boundary changing operators are sensitive to the compactification space data (as the vacuum expectation values of the moduli) as they are suppressed by contributions of the world-sheet instantons. For tree-level amplitudes this dependence appear explicitely for three and higher points correlations. However, at higher loops, it is already present at the two-point function. Part of this dependence is due to the Yukawa couplings in equation (\[deltaG\]), but it is interesting to investigate the remaining moduli-dependent parts. For the case of the annulus amplitude, some steps in computing the two-point function have been taken in this direction in [@Abel:2005qn]. Although the method was outlined, the computation in the case where the amplitude involved three boundaries was not performed. Moreover, we did not find a similar computation for the Möbius strip available in the literature. We will present here the relevant techniques and apply them to get explicit results.
The paper is organised as follows. In section 2, we will start by computing the one-loop two point function in the case of open strings with both ends on the same stack of $D$-branes when an orbifold action leads to an $N=1$ massless theory. This will allow us to compare the results with the case of open strings on brane intersections. The two point function for the latter is given in the case of annulus in section 3, and in the case of Moebius strip in section 4. Some useful formulae are listed in the appendix.
Warm Up: Orbifolds
==================
In this section, we are interested by the two point functions for the massless chiral modes of open strings propagating on $D3$ branes in orbifold models. For simplicity, the target space is taken as $\mathbb{R}^4 \times (\mathbb{T}^2 \times \mathbb{T}^2 \times \mathbb{T}^2)/\Gamma$, where $\Gamma \subset SU(3)$ is an abelian orbifold group. The compact space is parametrised by complex pairs of coordinates $X^{I},\ov{X}^{I}$ ($I=1,2,3$) with the torus identifications: X\^[I]{} \~X\^[I]{} + 2R\_1\^[I]{}, X\^I \~X\^[I]{} + 2i R\_2\^[I]{} \^I \[ORBIFOLD:torus\] where the angle $\alpha^I$ parametrises the complex structure of the torus; alternatively the torus data is encoded in the Kähler and complex structure moduli respectively T\^I T\_1\^I + i T\_2\^I = iR\^I\_1 R\^I\_2 \^I,U\^I e\^[i\^I]{}. The action of an element $g$ of the orbifold on the three compact dimensions is specified by the twist vector $(g_1,g_2,g_3) \in SU(3)$ (i.e. $\sum_{I} g_I = 0$ to preserve at least $N=1$ supersymmetry) by g X\^I = e\^[2i g\_I]{} X\^I. This projection acts on the open strings modes leading to chiral massless states $\Phi^I$ (where $I=1,2,3$ refers to an internal dimension and group indices are suppressed)
We will not discuss the brane and orientifold content of the model which depends for instance on $N$, but and suppose a set of $D3$ branes can be located at the fixed points, intersecting with (a possibly vanishing number) of $D7$ branes. On each of the $D3$ brane, we assume a set of chiral states $\Phi^I$ (where $I=1,2,3$ refers to an internal dimension and group indices are suppressed).
Depending on the orbifold, there can be twist vectors $(g_1,g_2,g_3)$ lying in $SU(2)$ or $U(1)$ instead of $SU(3)$. They lead respectively to sectors with states in $N=2$ or $N=4$ supersymmetry representations. Their contribution at one-loop to the Kähler potential has been calculated. For the $N=4$ case, it is found to vanish. In contrast, the $N=2$ sectors have attracted a lot of attention as they give moduli dependent results. It was computed in full in [@Berg:2005ja]. Their result for the correction to the Kähler metric at zero expectation value for the chiral matter fields (analagous to the one that we shall give below for $N=1$ sectors) is \^[I]{}(k) \^[I]{} (-k)= (8\^3 (k\^2) T\_2 U\_2|(U)|\^4) \_ \_g ( (\_[,g]{} \^ Q\_[,g]{}) ) where $\mu(k^2)$ is an infra-red regulator. The parameter $\sigma$ is used to indicate the boundary conditions: $\{\sigma\} = \{\mathcal{A}^{33},\mathcal{A}^{37_I},\mathcal{M}^{33}\}$ denoting annulus diagrams between $D3-D3$, $D3-D7_I$ branes and Möbius diagrams between $D3$ branes respectively. $d_\sigma = 1$ for annulus diagrams, $d_\sigma=2$ for Möbius strip diagrams. Also \_[, g]{} {
[cc]{} \_[g]{}\^3 (\_g\^3)\^[-1]{} & = \^[33]{}\
\_[g]{}\^3 (\_g\^[7\_I]{})\^[-1]{} & = \^[37\_I]{}\
-(\_[g]{}\^3)\^[T]{} (\_[g]{}\^3)\^[-1]{} & = \^[33]{}
. and Q\_[, g]{} \_[J | g\_J 0, h\_J = 0]{} (2g\_J) where $h_I = 0 \, \forall I$ for a $D3-D3$ partition function, and $h_I =0, h_{J\ne I} = \pm 1/2$ for $D3-D7^I$. $\sum_I h_I = 0$ is required to preserve supersymmetry.
The field theory behaviour of the $N=1$ sectors was studied on the $Z_3$ orientifold with purely $D3$ branes in [@Bain:2000fb]; in the following we provide a general analysis with the inclusion of the contribution of $D7$ branes, and give a closed form expression for the moduli-independent constants. We also regularise using the off-shell extension of the amplitude, which allows a direct identification of the infra-red cutoff with the physical momentum; this amplitude proves to be a good example where this technique can be easily applied, rather than, for example, zeta-function regularisation. However, it is worth mentioning that it can be shown that the same techniques apply to the calculations of [@Akerblom:2007np] and give the same result.
In the internal compact space, the total $D3$ and $D7$-brane Ramond-Ramond charges must vanish. The global cancellation of the corresponding tadpoles reads \_i tr (\_[1]{}\^[3\_i]{}) - N\_[O3]{}/2 = \_j \_[7\_j]{}- 8 \_[O7]{}= 0, where the sum is over untwisted sectors and we have included the contribution of the orientifold planes $O3$ and $O7$; $\Pi_A$ denotes the homology element corresponding to the cycle wrapped by the $D7$ brane or orientifold. We find, however, that the global considerations are irrelevant for the calculation of the two point corrections. More important is the twisted tadpole cancellation condition, enforced at each fixed point. Since \_[g]{}\^[-1]{} \^[T]{}\_[g]{} = \_[2g]{} where $\rho = \pm 1$, we cancel Möbius diagrams at twist $g$ with annulus diagrams at twist $2g$ (\_[\^[33]{},2g]{} Q\_[\^[33]{},2g]{}) + \_[J]{} (\_[\^[37\_J]{},2g]{} Q\_[\^[37\_J]{},2g]{}) + 4 (\_[2g]{} Q\_[\^[33]{},g]{}) = 0 . This then factorises: (\_[2g]{}\^3) Q\_[\^[33]{},2g]{} + \_[J]{} (\^[7\_J]{}\_[2g]{}) Q\_[\^[37\_J]{},2g]{} + 4 Q\_[\^[33]{},g]{} = 0 . \[ORBIFOLD:twistRR\]Note that for certain orbifolds (such as $\mathbb{Z}_N$ with $N$ non-prime) there is a separate condition \_[\^[33]{}]{} (\_[,g]{} Q\_[,g]{} )= 0 | g 2g’ for elements of the orbifold not generated by $2g'$.
In the spin structures $\left[\begin{array}{c}\alpha\\ \beta \end{array}\right]$, the partition function for annulus or Möbius diagrams can be written (see, for example, [@Berg:2005ja]): Z\_g\^h()= (0,) \_[I=1]{}\^3 Z\_[int]{}\^I() Our conventions for $\eta_{\alpha\beta}$ and for the theta-functions given in appendix \[App:A\] and Z\_[int]{}\^I ()= f(h\_I) with f(h\_I) {
[cc]{} 2(g\_I) & h\_I = 0\
1 & h\_I = 1/2
. .
The zero ghost picture vertex operators $V^{0}_{\Phi^I}$, $ V^{0}_{\ov{\Phi}^I}$ corresponding to the complex scalars in the chiral $\Phi^I $ and anti-chiral $\ov{\Phi}^I $ multiplets, respectively, are given by $$\begin{aligned}
V^{0}_{\Phi^I} &=& \lambda \bigg[ 2\ap (k \cdot \psi )\Psi^{I} + \dot{X}^{I} \bigg] e^{ik \cdot X} \nonumber \\
V^{0}_{\ov{\Phi}^I} &=& \lambda^{\dagger} \bigg[ 2\ap (k \cdot \psi ) \ov{\Psi}^{I} + \dot{\bar{X}}^{I} \bigg] e^{ik \cdot X}.\end{aligned}$$ without any factors of the string coupling. The worldsheet fields appearing in the above are the compact coordinates $X^I (z_1), \ov{X}^I (z_2)$, their fermionic superpartners $\Psi^I (z_1) , \ov{\Psi}^I (z_2)$ and the non-compact fermionic fields $\psi^{\mu} (z_i)$. The Chan-Paton factors $\lambda, \lambda^{\dagger}$ are determined by the orbifold projections by requiring \_g \_g\^[-1]{} = e\^[2i v\_g\^[I]{}]{} ,
To calculate annulus and Möbius strip diagrams, we insert the vertex operators on the imaginary axis taking $z_1=0$ and $z_2=iq$. The one-loop worldsheets are mapped to complex plane domains defined to be $[0,1/2] \times [0,it]$ for the annulus and $[0,1/2]\times[0,2it]$ for the Möbius strip. However, in order to sum over the diagrams, it is necessary to rescale the modular parameter for the Möbius strip by $t \rightarrow t/4$, and so we shall in this section use $[0,1/2]\times[0,it/d_\sigma]$ as the domain. We can then write all of the diagrams in a unified way, using $\tau \equiv it$ for the annulus, and $\tau \equiv 1/2 + it/4$ for the Möbius strip. Using the elementary Green functions \^[I]{} (iq) \^[I]{} (0) \^[,]{}\_[g\_I,h\_I]{} = we obtain: $$\begin{aligned}
\mathcal{A}^{\sigma}_I &\equiv& 4(\ap)^2 k^2 \int_0^{\infty} dt \int_0^{t/d_{\sigma}} dq \chi(q) \nonumber \\
&&\times \frac{\eta_{\alpha\beta} }{(8\pi^2 \ap t/d_{\sigma}^2)^2} 2\pi \frac{\theta_1^{\prime}(0){\theta\left[\begin{array}{c}\alpha \\ \beta \end{array}\right]} (iq,\tau) }{(\theta_1 (iq,\tau))^2} \frac{{\theta\left[\begin{array}{c}\alpha+h_I \\ \beta+g_I \end{array}\right]}(iq,\tau)}{{\theta\left[\begin{array}{c}1/2+h_I \\ 1/2+g_I \end{array}\right]}(0,\tau)} f(h_I) \prod_{J \ne I} Z_{int}^J \nonumber\end{aligned}$$ where (q) e\^[i k X (iq)]{} e\^[-i k X (0)]{} = (e\^[- q\^2]{})\^[-2k\^2]{}. \[ORBIFOLD:defchi\]The two-point function of interest then reads: \^I (k)\^I (-k) = \_[g]{} \^\_I (\_[,g]{} \^ Q\_[, g]{} ) Note that by using equation \[ORBIFOLD:twistRR\] we can cancel any function that is universal to the annulus and Möbius diagrams; we find (\_[\^[33]{},2g]{} \^ Q\_[\^[33]{}, g]{} ) (2g) + \_[\^[33]{}]{} (\_[,2g]{} \^ Q\_[, 2g]{} ) (2g) = 0 for any $\mathcal{F}(2g)$.
The identity (\[A:thetapluscd\]) and the supersymmetry conditions $\sum_I h_I = \sum_I g_I = 0$ are used to write $\mathcal{A}^{\sigma}$ as (c.f. [@Bain:2000fb]) \^\_I = 8i ()\^2 \_0\^ \_0\^[t/d\_]{} q e\^[-2h\_I q]{} (iq) which, in the closed string channel, i.e. expressed in terms of $l = 1/t$ takes the form: $$\begin{aligned}
\mathcal{A}^{3s}_I &=&
\int_0^{\infty} \frac{i \d l}{8\pi^3} \int_0^{1} \d x \, i\, e^{-2\pi i g_I x}\frac{\theta_1^{\prime} (0,il) \theta_1 (x + h_I - g_I il,il)}{\theta_1 (x,il) \theta_1 (h_I - g_I il ,il)} \chi(ix/l) \\
\mathcal{M}^{33}_I &=& d_{\sigma}^2 \int_0^{\infty} \frac{i \d l}{8\pi^3} \int_0^{1} \d x \, i\, e^{-4\pi i g_I x}\frac{\theta_1^{\prime} (0,il - 1/2) \theta_1 (x - 2 g_I il,il-1/2)}{\theta_1 (x,il-1/2) \theta_1 (- 2g_I il ,il-1/2)} \chi(ix/l) \nonumber\end{aligned}$$ where $s = \{3,7_J\}$. The expansion = (a) + (b) + 4\_[m, n = 1]{}\^ e\^[2m n i ]{} (2m a + 2 n b ) allows the identification of two sources of infrared divergences. The first in the open string channel, proportional to $\log k^2$, corresponds to the usual beta-function running. The second in the closed string channel, ultra-violet in the open string one, which instead appear as a $1/k^2$ pole preceding a divergent integral, indicating a fundamental inconsistency of the theory arising from uncancelled $RR$ charges. If we expand in the closed string channel, then the UV divergence can be simply subtracted; it comes entirely from the closed string zero mode. However, in this channel regulating the infra-red divergence is more subtle. Consider the behaviour of the momentum dependent part in the two regimes {
[cc]{} (d\_l)\^[2k\^2]{} (2 x)\^[-2k\^2]{} & l\
e\^[-]{} & t
. \[ORBIFOLD:chilims\] Since we are interested in the divergent and finite terms, but not those $O(k^2)$, we can split the integral into two regions, $l$ greater or less than $C \epsilon$, where $\epsilon \equiv 2\pi\ap k^2$, and $C$ is some constant. Employing (x) = (x) \[1+ 2 \_[n=1]{}\^ (-2|x|n)\] we find as $\epsilon \rightarrow 0$ $$\begin{aligned}
-8\pi^3 \mathcal{A}^{33}_I &=& -i\int_0^1 \d x \int_0^{C \epsilon} \frac{\d l}{l} \ \pi (-i + \cot \pi g) e^{-\epsilon x(1-x)/l} \nonumber\\
&& + \int_0^1 \d x \int_{C\epsilon}^{\infty} \d l e^{-2\pi i g_I x} 4\pi \sum_{m, n = 1}^{\infty} e^{-2\pi m n l} \sin (2\pi m x - 2 \pi n g i l ) \nonumber \\
&& + \int_0^1 \d x \int_{C\epsilon}^{\infty} \d l e^{-2\pi i g_I x} \pi (\cot \pi g_I il +i) ,\end{aligned}$$ where we have subtracted the zero mode terms without affecting the finite part of the amplitude.
After some algebra, taking $\epsilon \rightarrow 0$ and $C \rightarrow \infty$ such that $C \epsilon \rightarrow 0$, leads to $$\mathcal{A}^{33}_I = \frac{i}{8\pi^2} \frac{e^{-\pi i g_I}}{\sin \pi g_I} \bigg[ \log 2 \pi \epsilon - 2 + \gamma_E \bigg] \nonumber \\
+ \frac{i}{8\pi^4} e^{-\pi i g_I} \sin \pi g_I \bigg[ \zeta' (2,1-g_I) + \zeta' (2,g_I) \bigg].
\label{ppp}$$ In the above, the derivatives of the Hurwitz zeta function are on the *first* argument, so that ’(s,a) \_[m=0]{}\^ .
The contribution from $D3-D7_I$ states is identical to the above. However, for $D3-D7_J$ with $J \ne I$ we have $h_I = 1/2$ whose contribution can be seen to be infra-red finite. Hence we can expand in the closed string channel and set $k^2=0$ directly. Expanding $$\begin{aligned}
8\pi^3 \mathcal{A}^{37_J}_I \! \! &=& \! \! \int_0^1 \! \!\d x \int_0^{\infty} dl \pi e^{-2\pi i g_I x}\nonumber \\ & \bigg[& \cot \pi x + i \tanh \pi g l
+ \! \!4 \! \!\sum_{m,n =1} \! \!(-1)^n e^{-2\pi m n l} \sin (2\pi m x - 2\pi n g i l) \bigg], \nonumber\end{aligned}$$ then subtracting the pole parts and integrating we obtain \^[37\_J]{}\_I = 2
The contribution from Möbius amplitudes does contain an infra-red portion; we obtain $$\begin{aligned}
-8\pi^3 \mathcal{M}^{33}_I &=& -i\int_0^1 \d x \int_0^{C \epsilon} \frac{\d l}{l} \ \pi (-i + \cot \pi g) e^{-\epsilon x(1-x)/2l} \\
&&+ \int_0^1 \d x \int_{C\epsilon}^{\infty}\d l \pi e^{-4\pi i g_I x} \bigg[ \cot \pi x \nonumber \\
&&+ i \coth \pi 2 g_I l + 4 \sum_{m,n=1} (-1)^{mn} e^{-2\pi m n l} \sin (2 \pi m x - 4 \pi n g_I i l ) \bigg] \nonumber \end{aligned}$$ which then becomes $$\begin{aligned}
\mathcal{M}^{33}_I &=& \frac{i}{8\pi^2} \frac{e^{-\pi i g_I}}{\sin \pi g_I} \bigg[ \log 2\pi \epsilon - 2 + \gamma_E \bigg] \nonumber \\
&& +\frac{i}{32\pi^4} \sin 2 \pi g_I e^{-2\pi i g_I} \bigg[ \zeta^{\prime} (2, 1-|g_I|) + \zeta' (2,|g_I|) + \frac{\log 2}{\cos^2 \pi g_I} \bigg]\end{aligned}$$
We have thus computed all of the contributions to the one-loop Kähler metric for the states $\Phi^I$ on such orientifolds. These can be split into beta-function and threshold contributions according to the choice of renormalisation scheme that one wishes to match in the field theory. However, note that, since $M_s^{-2} = 4\pi^2 \ap$, we can rewrite in each contributino $\log 2\pi \epsilon = \log k^2/M_s^2$. It was shown in [@Bain:2000fb] for the $Z_3$ orientifold that the field theory result was reproduced; here we have generalised the approach slightly, included the contribution from $D7$-branes (which do not contribute to the field theory running, only the Kähler metric corrections) and computed the numerical corrections. It is hoped that these may have useful applications as they have a certain universal quality: since they do not depend upon the moduli, we expect them to be unaffected by implanting the singularity in a different geometry.
At the end of this section, we would like to notice that exchanging the internal direction $I$ with one of the non-compact directions takes us to the two-point function for gauge bosons which allows to compute gauge thresholds corrections contribution from $N=1$ sector. The result is moduli independent[^4] and is found to be: -8\^3 \_[GT]{} = \_0\^1 x \_[0]{}\^ \_[I]{} ( e\^[-x\^2 t]{})\^[-2k\^2]{} which becomes for the case of $D3$-branes: $$\begin{aligned}
-i8\pi^2 \mathcal{A}_{GT}^{33} &=& \cot \pi g_I (2 - \gamma_E + \log C) -i \bigg[ -2i \mathrm{sign}(g_I) \frac{\log 2\pi |g_I| C \epsilon}{2\pi |g_I|} \nonumber\\
&& -2i \sum_n \frac{\log 2\pi (n+g_I) C \epsilon}{2\pi (n+g_I)} - \frac{\log 2\pi (n-g_I) C \epsilon}{2\pi (n-g_I)} \bigg] \end{aligned}$$ Noting the identity \_[n=1]{}\^ = we observe that the $C$-dependent parts cancel, and we obtain $$\begin{aligned}
\mathcal{A}_{GT}^{33} = \frac{i}{8\pi^2} \cot \pi g_I (2 - \gamma_E -\log 2 \pi \epsilon) + \frac{2g_I}{\pi} \bigg[ \zeta^{\prime} (1,|g_I|) - \zeta^{\prime} (1,1-|g_I|) \bigg] \nonumber \end{aligned}$$
Annulus Diagrams in IIA
=======================
In this section we compute the related amplitudes to the previous section but in type IIA string backgrounds. Here we take $D6$-branes intersecting at angles $\pi \theta^{{\kappa}}_{ab}$ in the torus ${\kappa}$ with ${\kappa}=1,2,3$, which are the analogues of branes at blown-up orbifolds.
In the orientifold model considered here, there are many one-loop diagrams that could conribute. They can be graphically visualised as cylinders with two boundaries: the first fixed to some brane $a$, and the second to either brane $a$ or another brane. We place the vertex operators for our chiral states both on one boundary (the amplitude vanishes if they are on opposite boundaries) just as in the previous section, but now the vertex operators differ due to the presence of boundary changing operators. We shall suppose that our chiral states are trapped at the intersection $ab$ with angles $\pi \th$. There are then three classes of two-point diagrams that can be constructed, which correspond to the three types of partition function that are possible:
1. Annulus diagrams with the second boundary on brane $a$ or $b$.
2. Annulus diagrams with the second boundary on a third brane $c$ not parallel to $a$ or $b$.
3. Möbius strip diagrams; since there is only one boundary, there is an insertion of an orientifold operator $\Omega R$ which changes the boundary from brane $a$ to its orientifold image $a'$ (or $b$ to $b'$).
The diagrams of type 1 were calculated in [@Abel:2005qn], where it was found that there were poles corresponding to $RR$ tadpoles just as in the orbifold case; these must cancel against similar poles in the diagrams of type 2 and 3, as we shall show. The techniques required to perform the calculation in this section - the diagrams of type 2 - were also developed there for general $N$-point correlators, but an analysis of the two-point function was lacking and is provided here. In the next section we shall compute the third type of diagram.
Correlators of Boundary-Changing Operators
------------------------------------------
The most non-trivial part of the calculation is that involving the boundary-changing operators; these are operators inserted into the worldsheet at a boundary that interpolate between D-branes. To understand their appearance, consider that the target space fields obey Dirichlet boundary conditions perpendicular to the branes, but Neumann along them, and once we have applied the doubling trick we have a periodic boundary condition very much like for orbifolds. On the infinite strip $[-i\infty,i\infty]\times[0,1/2]$ we extend to $[-i\infty,i\infty]\times[-1/2,1/2]$ by $$\partial X (w) = \left\{ \begin{array}{cl} \partial X (w) & \Re(w) > 0 \\ -\bar{\partial} \bar{X} (-w) & \Re(w) < 0 \end{array} \right.$$ to obtain $$\partial X (w+1) = e^{2\pi i \theta} \partial X (w).$$ This global periodicity on the strip for an intersecting state is then mapped to a local periodicity on a worldsheet for fields in the presence of a boundary-changing operator, which represents the bosonic ground state: $$\begin{aligned}
\partial X (w) \sigma_{\theta} (z) &\sim& (w-z)^{\theta-1} \tau_{\theta} (z) \nonumber \\
\partial \bar{X} (w) \sigma_{\theta} (z) &\sim& (w-z)^{-\theta} \tau_{\theta}^{\prime} (z).\end{aligned}$$ They are primary operators in the theory with conformal weight $\frac{\theta}{2} (1-\theta)$.
Using these boundary changing operators we form vertex operators for the massless scalars at the intersection between branes $a$ and $b$, which we shall denote $C_{ab}$, in the $-1$ ghost picture (with $e^{-\phi}$ the bosonised ghost operators) as $$\begin{aligned}
V^{-1}_{C_{ab}} (z_1) &=& \sqrt{2\ap G_{C_{ab},\ov{C}_{ab}}}
e^{-\phi} e^{ik\cdot X} \prod_{{\kappa}=1}^3 e^{i(\theta^{{\kappa}}_{ab} - 1) H^{{\kappa}}} \sigma_{\theta^{{\kappa}}_{ab}} \nonumber \\
V^{-1}_{\bar{C}_{ab}} (z_2) &=& \sqrt{2\ap G_{C_{ab},\ov{C}_{ab}}}
e^{-\phi} e^{-ik\cdot X} \prod_{{\kappa}=1}^3 e^{-i(\theta^{{\kappa}}_{ab} - 1) H^{{\kappa}}} \sigma_{1-\theta^{{\kappa}}_{ab}}
\label{ANNULUS:Vertices}\end{aligned}$$ where the intersection is specified by three angles $\th$, $\kappa=1,2,3$, and where to preserve supersymmetry $\sum_{\kappa=1}^3 \th = 0 \mod 2$. The Kähler metric for these models is [@Lust:2004cx; @Bertolini:2005qh; @Akerblom:2007np; @Blumenhagen:2007ip; @Russo:2007tc]: G\_[C\_[ab]{},\_[ab]{}]{} = \^ . In the following we shall assume $\th \ge 0$ and thus $\sum_{\kappa=1}^3 \th = 2$, and to perform the below calculations with negative angles we can take $\th \rightarrow 1 + \th$. This is a requirement of the formalism rather than merely a choice of convenience.
To calculate the diagrams of type (2) above we must calculate the correlator on an annulus of two boundary changing operators $\sigma_{\th}, \sigma_{1-\th}$ fixed to one boundary of the worldsheet. In the target space this boundary is attached to branes $a$ and $b$, interpolating between them by absorption of an open string state. The other worldsheet boundary is fixed in the target space to a brane $c$ not parallel to $a$ or $b$ (in the parallel case the calculation is that of [@Abel:2005qn]). We take brane $c$ to lie at an angle $\ph$ to brane $a$ in each torus, (where to preserve supersymmetry we take $\sum_{\kappa=1}^3 \ph = 2$, although the techniques below apply also for summing to zero mod 2), and the result is a worldsheet periodicity on the annulus (taken to be $[0,1/2]\times[0,it]$ doubled to $[-1/2,1/2]\times[0,it]$ as above on the infinite strip) of X (w+1) = e\^[2i ]{} X (w).
Correlators are split into quantum and classical parts. The correlator \_ (z\_1) \_[1-]{} (z\_2) = N(t) (i|| W\^||)\^[-1/2]{}\
e\^[2i P\^]{} ()\^[-(1-)]{} e\^[-S\_[cl]{}]{} is determined by the following quantities: P\^ \_i (1/2 - \^i ) z\_i = (1/2 - ) qi, where we have placed z\_1 = 1/2 + iq, z\_2 = 1/2, and || W\^|| = A\_1\^ B\_2\^ + A\_2\^ B\_1\^ where $$\begin{aligned}
A_i &\equiv& \int_{it}^0 dz \omega_i (z) \nonumber \\
B_i &\equiv& \int_{-1/2}^{1/2} dz \omega_i (z)\end{aligned}$$ in addition to $$\begin{aligned}
\omega_1^{{\kappa}} (z) &=& e^{2\pi i \ph z} \frac{\theta_2 ( z- (1-\th) q i + \ph i t)}{\theta_2 (z-iq)} \left(\frac{\theta_2 (z-iq)}{\theta_2 (z)}\right)^{\th} \nonumber \\
\omega_2^{{\kappa}} (z) &=& e^{-2\pi i \ph z} \frac{\theta_2 ( z -\th q i - \ph i t)}{\theta_2 (z-iq)} \left(\frac{\theta_2 (z-iq)}{\theta_2 (z)}\right)^{1-\th}.\end{aligned}$$
We also require these for the classical action, which is given by S\^ = \[ANNULUS:action\]where M\^[AA]{}\_ = -2i = 2i 2i F(,) and $$\begin{aligned}
v_A^{{\kappa}} &=& -\frac{1}{\sqrt{2}} [ n_A^{{\kappa}} L_c^{{\kappa}} ] \nonumber \\
v_B^{{\kappa}} &=& i\sqrt{2} [ h^{{\kappa}} + n_B^{{\kappa}} \frac{4\pi^2 T_2^{{\kappa}}}{L_c^{{\kappa}}} ] = i\sqrt{2} F(\th,\ph) [ b^{{\kappa}} + n_B^{{\kappa}} \frac{4\pi^2 T_2^{{\kappa}}}{ F(\th,\ph) L_c^{{\kappa}}}] .\end{aligned}$$ where $h^{{\kappa}},b^{{\kappa}}$ are the height and base of the smallest triangle $abc$, $L_c^{{\kappa}}$ is the wrapping length of brane $c$, and $h^{{\kappa}} = F(\th,\ph) b$. The classical contribution is then e\^[-S\_[cl]{}]{} \_[n\_A\^, n\_B\^]{} e\^[-S\^]{}
Note that we have written $N(t)$ for the normalisation. This shall be determined by considering the factorisation on the partition function. In doing this and in the following, we note that the integrals $A_i, B_i$ control much of the information about the amplitude, and we can use the following to help determine the $A_i$: $$\begin{aligned}
A_{i}^{{\kappa}} &=& -\frac{\sin \pi \th}{\sin \pi (\th + \ph)}e^{-i \pi \ph} \int_{z_1}^{it+1/2} \omega_{i}^{{\kappa}} \equiv \frac{\sin \pi \th}{\sin \pi \ps}e^{-i \pi \ph} D_{i} \nonumber \\
&=& \frac{\sin \pi \th}{\sin \pi \ph}e^{i \pi \ps} \int_{1/2}^{z_1} \omega_{i}^{{\kappa}} \equiv \frac{\sin \pi \th}{\sin \pi \ph}e^{i \pi \ps} C_{i}.\end{aligned}$$ (see fig. \[2ptannulus\] ).
\[2ptannulus\]
### Normalisation
To normalise, we consider $q\rightarrow 0$. Note that using the above and the integral over the $C$ contour for the $A$ integrals (and the property of the theta-functions that $\theta_1 (x) = x \theta_1^{\prime}(0) + \frac{x^3}{3!} \theta_1^{\prime \prime \prime}(0) + ...$) we determine $$\begin{aligned}
A_{1}^{{\kappa}} &\rightarrow& \frac{\theta_1 (\ph it)}{\theta_1^{\prime} (0)} \frac{\pi}{\sin \pi \ph} \nonumber \\
A_{2}^{{\kappa}} &\rightarrow& A_{1}^{{\kappa}} \nonumber \\
B_{1}^{{\kappa}} &\rightarrow& \infty \nonumber \\
B_{2}^{{\kappa}} &\rightarrow& B_{1}^{{\kappa}} \end{aligned}$$ and thus ||W|| 2 B\_1\^ .
The classical action reduces to S . where we have omitted the subscripts since they become redundant in this limit. We also have $||W|| \rightarrow 2AB$. Note that we have to Poisson-resum on $n_B$ since the second term above vanishes, giving a divergent contribution after summing over $n_B$. The coefficient of $n_B^2$ is then $\frac{ -iA(4\pi^2 T_2^{{\kappa}})^2}{B 2\pi \ap (L_c^{{\kappa}})^2}$, and so the classical part of the boundary changing operator amplitude, plus the determinant factor, becomes ||W||\^[-1/2]{} e\^[-S\_[cl]{}]{} = . \[ANNULUS:factorise\]Note that it can be shown that there is no zero mode contribution to the action ($n_A = n_B = 0$) as required; this can be used to show that there can be no zero mode contribution to $v_A$. We must compare this with the partition function *and the OPE coefficient* $C^{(aba)}_{\th,1-\th}$.
First consider the disk normalisation 1 \_a = g\_a\^[-2]{} where $g_a$ is the Yang-Mills coupling on the brane, given by g\_a\^[-2]{} = where $l_s = 2\pi \sqrt{\ap}$ and $V_a$ is the compact volume of the $p$-brane $a$. We require an expression for a given, internal, complex dimension, and can therefore write 1 \_a = 1\_4 \_[=1]{}\^3 1 \_where for $D6$-branes $$\begin{aligned}
\bra 1\ket_4 &=& \frac{1}{(2\pi\ap)^2} \frac{1}{2\pi g_s} \nonumber \\
\bra 1 \ket_\kappa &=& \frac{L_a^\kappa}{2\pi \sqrt{\ap}}.\end{aligned}$$ Here $L_a^{\kappa}$ is the length of the brane wrapping a one-cycle in complex dimension $\kappa$. Then since we have the freedom to normalise the wavefunctions of the vertex operators, we can use \_\^[ab]{} \_[1-]{}\^[ba]{} \_a = 1 = C\^[(aba)]{}\_[,1-]{} 1 \_a to determine C\^[(aba)]{}\_[,1-]{} = \[ANNULUS:OPE\]
Now we wish to normalise the boundary changing operator amplitudes at one loop, so we consider \_\^[ab]{} (z\_1) \_[1-]{}\^[ba]{}(z\_2) \_[ac]{} \~(z\_1 - z\_2)\^[-(1-)]{} C\^[(aba)]{}\_[,1-]{} Z\_[ac]{}\^[X\^]{} where Z\_[ac]{}\^[X\^]{} = -i I\_[ac]{}\^ . Here, $I_{ac}^{{\kappa}}$ is the number of intersections between branes $a$ and $c$ in the torus $\kappa$. Then, with the aid of the identity ([@Lust:2003ky]) = \[ANNULUS:LcLa\]we can write
N(t) =
### Field Theory Limit {#ANNULUS:FTL}
The field theory limit of the above amplitude is found by considering $t\rightarrow \infty$. In this regime we may expand theta-functions as $$\begin{aligned}
\theta_1 (z) &\rightarrow& i e^{-\pi t/4} (e^{-\pi i z} - e^{\pi i z} - e^{-2\pi t} (e^{-3 \pi i z} - e^{3 \pi i z})) + O(e^{-\pi t}) \nonumber \\
\theta_2 (z) &\rightarrow& e^{-\pi t/4} (e^{-\pi i z} + e^{\pi i z} + e^{-2\pi t} (e^{-3 \pi i z} + e^{3 \pi i z})) + O(e^{-\pi t})\end{aligned}$$ We neglect terms $O(e^{-\pi t})$ and $O(e^{-2\pi q})$ (although retain fractional powers). We use this to determine the integrals $A_i$ and $B_i$, with the aid of the following: \_0\^q dy e\^[2y]{} (1-e\^[-2y]{})\^ (1-e\^[-2q]{} e\^[2y]{})\^ = B(, 1+)+ B(-,1+) + O(e\^[-2q]{}) and \_[-1/2]{}\^[1/2]{} dx e\^[i x]{} (2x)\^ = = . This allows us to determine (for $\ph > 0$) $$\begin{aligned}
A_{1}^{{\kappa}}\!\! \!&\rightarrow& \!\!\!\! -\frac{i}{2\pi} e^{\pi \ph t} \bigg[ e^{-2\pi (1-\th) q } B(\th\! + \!\ph \!-1, 1-\!\ph) + e^{-2\pi \ph q } B(1\!-\!\th \!- \!\ph,\ph) \bigg]\nonumber \\
&& \nonumber \\
A_{2}^{{\kappa}}\!\! \!&\rightarrow& \!\!\!\! -\frac{i}{2\pi} e^{\pi \ph t}\bigg[ e^{2 \pi q (\th + \ph - 1)} B (\ph + \th - 1,1-\ph) + B(1-\th-\ph,\ph) \bigg]\nonumber \\
&& \nonumber \\
B_1^{{\kappa}} \!\! \!&\rightarrow& \!\!\!\!e^{-2\pi q (1-\th) +\pi \ph t } \frac{\Gamma (1-\th)}{\Gamma (\ph) \Gamma (2- \th - \ph)} + e^{-\pi \ph t} \frac{\Gamma(1-\th)}{\Gamma (1+ \ph) \Gamma (1 - \th - \ph)} \nonumber \\
B_2^{{\kappa}} \!\! \!&\rightarrow& \!\!\!\!e^{\pi \ph t} \frac{\sin \pi \phi}{\pi} \bigg[ B(\ph, \th) - e^{2\pi (q\th+(\ph-1)t)} B(\ph-1,\th)\bigg] \nonumber \\
&\rightarrow& \!\!\!\! \frac{e^{\pi \ph t}}{B(1-\ps,1-\ph)} \bigg[ \frac{1}{1-\ps} + \frac{e^{2\pi (q\th+(\ph-1)t)}}{1-\ph} \bigg]
\label{PrecisionABs}\end{aligned}$$
The leading behaviour of $||W||$ (where for simplicity in the following we shall take $\th, \ph > 1/2$) is given by ||W|| - \^ where we have defined \^ for later use; note that $G_{C_{ab},\ov{C}_{ab}}G_{C_{bc},\ov{C}_{bc}}G_{C_{ca},\ov{C}_{ca}} = \prod_{{\kappa}}
(\Gamma^{{\kappa}})^{1/2}$.
In this limit, $A_2 B_1$ dominates over $A_1 B_2$ and we obtain for the classical action S\^ Noting that $\frac{A_1^{{\kappa}}}{B_1^{{\kappa}}} \rightarrow 1/M_{AA} $, we obtain S\^ which gives us $$\begin{aligned}
S \!\!\!\!&\rightarrow&\!\! \!\frac{1}{2\pi \ap} \frac{1}{2} F(\th,\ph) \bigg[\! (b^{{\kappa}} + n_A^{{\kappa}} L_c^{{\kappa}} + n_B^{{\kappa}} \frac{4\pi^2 T_2^{{\kappa}}}{ F(\th,\ph) L_c^{{\kappa}}})^2 \!+ \!(b^{{\kappa}} + n_B^{{\kappa}} \frac{4\pi^2 T_2^{{\kappa}}}{ F(\th,\ph) L_c^{{\kappa}}})^2\bigg]\nonumber \\
\!\!&\rightarrow& \!\!\frac{1}{2\pi \ap} [ A(n_A,n_B) + A(n_B) ]\end{aligned}$$ This is just two sums over areas of triangles $abc$ wrapping the torus, and gives the expected field theory factor as the product of two Yukawa couplings. Note the similarity to the tree level expression as given for instance by equation (A.17) of [@Abel:2005qn].
### Fermionic Part
Accompanying the bosonic amplitude is the fermionic one. The correlators are given by \_[j]{} e\^[ia\_i H(z\_j)]{} \_ = e\^[2i Q]{} \_ (i t + Q) \_[i < j]{} ()\^[a\_i a\_j]{} where Q \_i a\_i z\_i. This gives for us Q = (- 1) (z\_1 - z\_2) = (- 1) qi for the operator $e^{i (\th - 1)H } e^{-i (\th - 1) H}$, while for $e^{i \th H} e^{-i \th H}$ we have Q\^ = q i . Note that the fermionic partition function is Z\^[\^]{}\_[ac]{} = -i. and thus we require a normalisation factor of $i \eta(it)^{-1} \exp [-\pi (\ph)^2 t]$.
Full $N=1$ Amplitude
--------------------
We have now assembled all of the ingredients to write down the full amplitude. This is $$\begin{gathered}
\mathcal{A} \equiv \bra C_{ab} (k)\bar{C}_{ab}(-k) \ket_c = G_{C_{ab},\ov{C}_{ab}} N_c tr(\lambda_{ab} \lambda_{ab}^{\dagger}) 4(\ap)^2 k^2\\
\int_0^\infty \frac{dt}{(8\pi^2\ap t)^2} \frac{1}{\eta^3 (it)}\int_0^t d q \chi(qi) e^{2\pi q} \left(\frac{\theta_1(iq)}{\theta_1^{\prime}(0)}\right)^{-2} \sum_{\nu} \delta_\nu \frac{1}{2}\bigg[ \theta_\nu (qi) + \theta_{\nu} (-qi)\bigg] \\
\prod_{{\kappa}= 1}^3 \frac{\theta_\nu (qi(\th - 1) + \ph i t) }{ \eta^3 (it)}|W^{{\kappa}}|^{-1/2} \sum_{n^{{\kappa}}_A,n^{{\kappa}}_B} e^{-S(n_A^{{\kappa}},n_B^{{\kappa}})}\end{gathered}$$ where $\chi$ is as defined in (\[ORBIFOLD:defchi\]).
After summing over spin structures $\nu$ we find $$\begin{gathered}
\mathcal{A} = \frac{ k^2}{16\pi^2} G_{C_{ab},\ov{C}_{ab}} N_c tr(\lambda_{ab} \lambda_{ab}^{\dagger}) \int_0^\infty \frac{dt}{t^2} \frac{1}{\eta(it)^{6}} \int_0^t d q e^{2\pi q} \chi(qi) \theta_1(iq)^{-1} \\
\prod_{{\kappa}= 1}^3 \theta_1 (qi(\th - 1) + \ph i t) |W^{{\kappa}}|^{-1/2} \sum_{n^{{\kappa}}_A,n^{{\kappa}}_B} e^{-S(n_A^{{\kappa}},n_B^{{\kappa}})}.
\label{annulus2pt}\end{gathered}$$ This, with the expression (\[ANNULUS:action\]) is the main result of this section. We see that all of the moduli dependence is contained in the classical action.
Note that $\chi \sim (qi)^{-2\ap k^2}$ as $q \rightarrow 0$ and $\chi \sim (t-q)^{-2\ap k^2}$ as $q \rightarrow t$ and thus the above amplitude has poles at $q=0,t$, as predicted in [@Abel:2005qn]. Using equation (\[ANNULUS:factorise\]) we can see that the prediction there is exactly correct, and we find $$\begin{aligned}
\mathcal{A} = \frac{G_{C_{ab},\ov{C}_{ab}}}{32\pi^3} \tr(\lambda_{ab} \lambda_{ab}^{\dagger})\bigg[ \frac{(2\pi \sqrt{\ap})^3}{2\ap} \bigg( \frac{N_c I_{ac}}{L_a} + \frac{N_c I_{bc}}{L_b}\bigg) \int_0^{\infty} \frac{dt}{t^2} + \mathrm{finite}\bigg]\end{aligned}$$
### Field Theory Limit {#field-theory-limit}
If we now wish to take the field theory limit of the expression (\[annulus2pt\]) we must consider $t \rightarrow \infty$. Using the expressions from section \[ANNULUS:FTL\] and equation (\[ORBIFOLD:chilims\]), we easily derive $$\begin{aligned}
\mathcal{A} = tr(\lambda_{ab} \lambda_{ab}^{\dagger}) \bigg\{&& G_{C_{ab},\ov{C}_{ab}} \frac{ N_c k^2}{16\pi^2} \bigg [\prod_{{\kappa}=1}^3 (\Gamma^{{\kappa}})^{-1/2} |\lambda|^2 \int_{1/2\pi \ap\Lambda^2}^\infty \frac{dt}{t} e^{-2\pi \ap t k^2 x(1-x)} \bigg] \nonumber \\ &+& k^2 \Delta G_{C_{ab},\ov{C}_{ab}} \bigg\}\end{aligned}$$ where ||\^2 \_[=1]{}\^3 \_[n\_A\^,n\_B\^]{} e\^[- - ]{} is the square of the coupling appearing in the superpotential. $\Delta G_{C_{ab},\ov{C}_{ab}}$ is the correction to the Kähler metric from integrating out the massive string modes. Note that we have used a different cutoff scheme here to section 1; here we cannot claim that there is no contribution from massive modes in the region $[1/2\pi \ap \Lambda^2,\infty]$ of $t$, but instead these give finite contributions to $\Delta G_{C_{ab},\ov{C}_{ab}}$. It would be very interesting to compute this correction, but it is complicated by, among other issues, the explicit summation over worldsheet instantons. Note that the classical action is only a constant in the field theory limit; in general it is a function of the worldsheet coordinates and the modular parameter, and so should give interesting dependence on the Kähler moduli to the full amplitude.
Performing the integration in the above we obtain $$\begin{aligned}
\mathcal{A} &=& -N_c tr(\lambda_{ab} \lambda_{ab}^{\dagger})\frac{ |\lambda|^2 }{16\pi^2} \frac{1}{ G_{C_{bc},\ov{C}_{bc}}G_{C_{ca},\ov{C}_{ca}}} k^2 \bigg( \log k^2/\Lambda^2 - 2 +\gamma_E \nonumber \\
&& - \frac{k^2}{\Lambda^2} {}_2 F_2 (1,2;2,5/2;-k^2/4\Lambda^2)\bigg)+ k^2 \Delta G_{C_{ab},\ov{C}_{ab}}.\end{aligned}$$ This reproduces exactly the field theory result for the anomalous dimension of the superfields.
The Möbius Strip Amplitude in IIA
=================================
In this section we provide new techniques to calculate Möbius strip amplitudes for states at intersections between branes. This further generalises the techniques that were developed for periodic closed string amplitudes in [@Atick:1987kd], were first applied to the case of intersecting branes in [@Abel:2004ue] and we generalised to the case of generic annulus diagrams (i.e. with no restrictions upon the angles of the branes) in [@Abel:2005qn].
Worldsheet Periodicity
----------------------
A Möbius strip can be considered to be a strip closed under an orientation reversal: consider $$X(w+it,\bar{w}-it) = \Omega X(w,\bar{w}) = X(1/2 - \bar{w},1/2-w).$$ If we now combine this with a reflection to make an orientifold model $$X(w+it,\bar{w}-it) = \Omega R X(w,\bar{w}) = \bar{X}(1/2 - \bar{w},1/2-w).$$ we see that we can consistently combine this with the doubling trick for intersecting brane models. We align the coordinate system along the orientifold plane, so that for worldsheet the strip $[0,it]\times[0,1/2]$, on the imaginary axis we have Neumann boundary conditions along $X + e^{2\pi i \phi_{a,O6}} \bar{X} - c ( 1 + e^{2\pi i \phi_{a O6}})$ with Dirichlet conditions perpendicular, and along the axis $\Re(w) = 1/2$ we have Neumann conditions along $X + e^{-2\pi i \phi_{a,O6}}\bar{X} - c ( 1 + e^{-2\pi i \phi_{a O6}})$, where $c$ is the position of the intersection along the $O6$ plane. We then have boundary conditions $\partial X = -e^{\pm2\pi i \phi_{a, O6}} \bar{\partial} \bar{X}, \partial \bar{X} = -e^{\mp2\pi i \phi_{a, O6}} \bar{\partial} X$ where the upper (lower) sign is for $\Re(w) = 0 (1/2)$. Using the doubling trick $$\partial X (w) = \left\{ \begin{array}{cl} \partial X (w) & \Re(w) > 0 \\ -e^{2\pi i \phi_{a,O6}}\bar{\partial} \bar{X} (-\bar{w}) & \Re(w) < 0 \end{array} \right.$$ and similarly for $\partial \bar{X}, \bar{\partial} X$, we arrive at the new periodicity conditions $$\begin{aligned}
\partial X (w+1) &=& e^{-4\pi i \phi_{a,O6}} \partial X (w)\nonumber \\
\partial X (w+1/2+it) &=& e^{-2\pi i \phi_{a,O6}}\partial X (w) \nonumber \\
\partial \bar{X} (w+1) &=& e^{4\pi i \phi_{a,O6}} \partial \bar{X} (w)\nonumber \\
\partial \bar{X} (w+1/2+it) &=& e^{2\pi i \phi_{a,O6}}\partial \bar{X} (w).
\label{Periodicities}\end{aligned}$$ This provides a convenient way to obtain the holomorphic differentials with given boundary conditions. Note that these lead to X (w + 2it) = X (w), |[X]{} (w + 2 i t) = |[X]{} To compute the worldsheet instanton contribution, we integrate the (doubly periodic) function $\partial X\bar{\partial}\bar{X} (w, \bar{w})$ over the fundamental domain - but it is more convenient to extend this to the domain $[0,2it]\times[0,1/2]$, and take half of the resulting action.
If there is also an orbifold projection, we may combine the action with the orientifold as above and adapt the doubling trick accordingly, or we can simply align our coordinate system relative to the new fixed planes.
Cut Differentials
-----------------
The cut differentials with the periodicities (\[Periodicities\]) are given by using the theta-function $$\begin{aligned}
\theta \left[ \begin{array}{c} 1/2 - 2a \\ 1/2 + a \end{array} \right]\! (z+m;\tau) \!\!&=& \!\!\exp (2\pi i (1/2-2a)m)\ \ \theta \left[ \begin{array}{c} 1/2 - 2a \\ 1/2 +a\end{array} \right] (z;\tau) \\
\theta \left[ \begin{array}{c} 1/2 - 2a\\ 1/2 + a\end{array} \right] \!(z+m\tau;\tau) \!\!&=& \!\!\exp(-2\pi i a) \exp (-2\pi i m/2) \nonumber \\ && \times \exp (-\pi i m^2 \tau - 2 \pi i mz) \ \ \theta \left[ \begin{array}{c} 1/2 -2a \\ 1/2 +a \end{array} \right] (z;\tau) \nonumber.\end{aligned}$$ We thus define $$\begin{aligned}
\theta_{\phi_{a,O6}} (z) &\equiv& \theta \left[ \begin{array}{c} 1/2 - 2\phi_{a,O6} \\ 1/2 + \phi_{a,O6} \end{array} \right] (z;\frac{1}{2} + it) \nonumber \\
\theta_1 (z) &\equiv& \theta \left[ \begin{array}{c} 1/2 \\ 1/2 \end{array} \right] (z;\frac{1}{2} + it).\end{aligned}$$ For a correlator of $L$ vertex operators at coordinates $z_i$ (all lying on the imaginary axis), each with angles $\theta_i$ we may take $\sum_{i=1}^L \theta_i = M$.We then have $L-M$ cut differentials as a basis for $\partial X$, with $\{i'\} = \{1,...,L-M\}$: $$\tilde{\omega}_{i'} (z) = \gamma_X (z) \theta_{+\phi_{a,O6}} (z-z_{i'} - Y) \prod_{j \in \{\alpha\}\ne \alpha}^{L-M} \theta_1 (z-z_{j})
\label{TwistOmegaa}$$ and we have the set of $M$ differentials for $\partial \bar{X}$ with $\{i''\} = \{L-M+1,...,L\}$: $$\tilde{\omega}_{i''} (z) = \gamma_{\bar{X}} (z) \theta_{-\phi_{a,O6}} (z-z_{i''} + Y) \prod_{j \in \{\beta\}\ne \beta}^{L} \theta_1 (z-z_{j}).
\label{TwistOmegab}$$ Here $$\begin{aligned}
Y &=& -\sum_{i'} \theta_{i'} \ z_{i'} + \sum_{i''} (1-\theta_{i''} ) z_{i''} \nonumber \\
\gamma_X (z) &=& \prod_{i=1}^L \theta_1 (z-z_i)^{\theta_i - 1} \nonumber \\
\gamma_{\bar{X}} (z) &=& \prod_{i=1}^L \theta_1 (z-z_i)^{-\theta_i}.\end{aligned}$$
These cut differentials are a natural basis which is convenient for deriving the quantum part of the amplitude, but for performing calculations it is convenient to express the above only in usual theta functions. We replace \_[\_[a,O6]{}]{} (z-z\_ Y) \_1 (z - z\_ Y 2 i \_[a,O6]{} t). where we use $z_\alpha$ to denote a member of $z_{i'}$ or $z_{i''}$. We shall denote the new basis $\{\omega_{i'},\omega_{i''}\}$. To convert between the two bases, we have $$\begin{aligned}
\tilde{\omega}_{i'} &=& e^{-2\pi i\phi_{a,O6} ( 1 + \phi_{a, O6} (1-2it))} e^{ 4 \pi i \phi_{a,O6} (z_\alpha + Y )} \omega_{i'} \nonumber \\
\tilde{\omega}_{i''} &=& e^{-2\pi i\phi_{a,O6} ( - 1 + \phi_{a, O6} (1-2it))} e^{- 4 \pi i \phi_{a,O6} (z_\alpha - Y )} \omega_{i''}
\label{convertbasesmob}\end{aligned}$$ In this basis, the cut differentials for a two-point function with vertices at $z_1 = 0, z_2 = iq$ and angles $\theta, 1-\theta$ are $$\begin{aligned}
\omega_1 (z) &=& e^{-4\pi i \phi_{a,O6} z} \frac{\theta_1 ( z - \theta q i - 2\phi_{a,O6} i t)}{\theta_1 (z-iq)} \left(\frac{\theta_1 (z-iq)}{\theta_1 (z)}\right)^{1-\theta} \nonumber \\
\omega_2 (z) &=& e^{4\pi i \phi_{a,06} z} \frac{\theta_1 ( z - (1-\theta) q i + 2\phi_{a,O6} i t)}{\theta_1 (z-iq)} \left(\frac{\theta_1 (z-iq)}{\theta_1 (z)}\right)^{\theta}.\end{aligned}$$ In each of the above, the modulus of the theta functions is $\tau = 1/2 + it$.
Classical Solutions
-------------------
The classical solutions $X_{cl}, \bar{X_{cl}}$ satisfy the boundary conditions $$\begin{aligned}
\int_{\gamma_a} \d z \partial X + \d \bar{z} \bar{\partial} X &=& v_a \nonumber \\
\int_{\gamma_a} \d z \partial \bar{X} + \d \bar{z} \bar{\partial} \bar{X} &=& \bar{v}_a\end{aligned}$$ where the $v_a$ are $L$ displacements corresponding to the independent paths $\gamma_a$ on the worldsheet. We can use these to determine the classical solutions in terms of the basis of cut differentials: $$\begin{aligned}
\partial X_{cl} (z)&=& v_a (W^{-1})^a_{i^{\prime}} \omega_{i^{\prime}} (z) \nonumber \\
\bar{\partial} X_{cl}(\bar{z}) &=& v_a (\ov{W}^{-1})^a_{i^{\prime\prime}} \bar{\omega}_{i^{\prime\prime}} (\bar{z}), \end{aligned}$$ where we have defined the matrix $W$ as $$\begin{aligned}
W^{i^{\prime}}_a &= \int_{\gamma_a} \d z \omega^{i^{\prime}} (z), \qquad &i^{\prime} = \{1..L-M\} \nonumber \\
W^{i^{\prime\prime}}_a &= \int_{\gamma_a} \d \bar{z} \bar{\omega}^{i^{\prime\prime}} (\bar{z}), \qquad &i^{\prime\prime} = \{L-M+1..L\}. \end{aligned}$$ We can then use these to determine $\bar{\partial} \bar{X}_{cl}, \partial \bar{X}_{cl}$ via the doubling trick: $$\begin{aligned}
\bar{\partial} \bar{X}_{cl} (\bar{z})&=& -e^{-2\pi i \phi_{a, O6}} v_a (W^{-1})^a_{i^{\prime}} \omega_{i^{\prime}} (-z) \nonumber \\
\partial \bar{X}_{cl}(z) &=& -e^{2\pi i \phi_{a, O6}} v_a (W^{-1})^a_{i^{\prime\prime}} \bar{\omega}_{i^{\prime\prime}} (-\bar{z}). \end{aligned}$$ However, we may also note that the complex conjugates of the cut differentials are a good basis for $\bar{\partial} \bar{X}_{cl}, \partial \bar{X}_{cl}$ if we extend those fields to $\Re(z) = [-1/2,1/2]$ via $$\bar{\partial} \bar{X} (\bar{w}) = \left\{ \begin{array}{cl} \bar{\partial} \bar{X} (\bar{w}) & \Re(\bar{w}) > 0 \\ -e^{-2\pi i \phi_{a,O6}}\partial X (-w) & \Re(w) < 0 \end{array} \right.$$ to obtain $$\begin{aligned}
\bar{\partial} \bar{X}_{cl} (\bar{z})&=& \bar{v}_a (\ov{W}^{-1})^a_{i^{\prime}} \bar{\omega}_{i^{\prime}} (\bar{z}) \nonumber \\
\partial \bar{X}_{cl}(z) &=& \bar{v}_a (W^{-1})^a_{i^{\prime\prime}} \omega_{i^{\prime\prime}} (z). \end{aligned}$$ These are entirely consistent provided that we choose the $v_a$ correctly. This is a crucial point: we are not at liberty to choose arbitrary cycles for the $\gamma_a$, but must match them to displacements with the correct phase. We can write |\_[i’]{} (|[z]{}) = e\^[i\_[i’]{}]{} \_[i’]{} (-z), \_[i”]{} (z) = e\^[i\_[i”]{}]{} |\_[i”]{} (-|[z]{}) and $$\begin{aligned}
\int_{\gamma_a} \d \bar{z} \bar{\omega}_{i'} (\bar{z}) &=& - e^{i \eta_a} e^{i\xi_{i'}} \int_{\gamma_a} \d z \omega_{i'} (z) \nonumber \\
\int_{\gamma_a} \d z \omega_{i''} (z) &=& - e^{i \eta_a} e^{i\xi_{i''}} \int_{\gamma_a} \d \bar{z} \bar{\omega}_{i''} (\bar{z})\end{aligned}$$ where $\eta_a$ is a phase, constant across $i$ for each $a$, and such that we can write \^[a]{}\_i = - e\^[i \_a]{} e\^[i\_[i]{}]{} W\^a\_i . Then (\^[-1]{})\^a\_[i]{} = - e\^[-i \_a]{} e\^[-i\_[i]{}]{} (W\^[-1]{})\^a\_[i]{} and thus we require for consistency |[v]{}\_a = e\^[-2i \_[a,O6]{}]{} e\^[i \_a]{} v\_a . Note that the phases $e^{\xi_i}$ are always removed from amplitudes (corresponding to normalisation of the basis functions), and indeed, when we take the basis $\{\omega_{i'},\omega_{i''}\}$ they are equal to $1$ anyway. However, as mentioned the phases $\eta_a$ are crucial. A consequence of the above is that $$\partial X (-z) = -e^{2\pi i \phi_{a, O6}} \bar{\partial} \ov{X} (\bar{z}),$$ and therefore $$\int_{it+1/2}^{it} \d x \ \partial X (x) = \int_{0}^{1/2} \d x \ \bar{\partial} \ov{X} (x).$$ Using this, we define \_[\_B]{} z \_[i’]{} (z) . which we denote \_B (\[0,1/2\] + e\^[ i ]{} \[it,it+1/2\] ). The above then results in \_B = + 2 \_[a, O6]{} - , which gives us the phase of $v_B$, and thus v\_B = i e\^[i /2]{} \_B . Here $\hat{v}_B $ is a real number corresponding to the distance traversed by the cycle, and thus it may be negative. However, we have an apparent freedom in choosing $\psi$. This freedom is fixed by the requirement that the action not depend upon the linearly independent combination \_ (\[0,1/2\] - e\^[ i ]{} \[it,it+1/2\] ), as shall be seen in the next subsection.
To fix $\hat{v}_B$, however, we must consider from the above that v\_B = ( e\^[-i/2]{} \_B X - e\^[i/2]{} \_B |[X]{} ).
Classical Action
----------------
The classical action is determined by integrating the classical solutions over the surface: $$\begin{aligned}
S_{cl} &=& \frac{1}{4\pi\ap} \int_R \d^2 z (\partial X \bar{\partial} \bar{X} + \bar{\partial} X \partial \bar{X}) \nonumber \\
&\equiv& \frac{1}{4\pi\ap} v_a \bar{v}_b \bigg[ (W^{-1})^a_{i'} (\ov{W}^{-1})^{b}_{j'} (\omega_{i'},\omega_{j'} ) + (W^{-1})^a_{i''} (\ov{W}^{-1})^{b}_{j''} (\omega_{j''},\omega_{i''} )\bigg] \nonumber \\
&\equiv& \frac{1}{4\pi \ap} (v^{\dagger})_b S^{ba}_{cl} v_a\end{aligned}$$ where the region $R$ is the doubled Möbius strip $[0,1/2] \times [0,2it]$, and we have divided by two; the functions $\partial X \bar{\partial} \bar{X}$ and $\bar{\partial} X \partial \bar{X}$ are even under $z \rightarrow -\bar{z}$ and $z \rightarrow z + 1/2 + it$. It remains to determine the inner products $(\omega_{i'},\omega_{j'})$. To do this we perform a canonical dissection by writing $\omega_i (z) = \d f_i (z)$ and integrate using Green’s Theorem, as in [@Atick:1987kd; @Abel:2005qn]. We split the worldsheet up into paths and use Cauchy’s theorem to express these in terms of the same cycles $\gamma_a$. Two paths are eliminated; the most expedient to eliminate depend upon the precise configuration, and hence we shall provide the procedure and the expressions for the two point function, rather than the general case. In the two point function, we have one vertex fixed at $z=0$, and one at $z=qi$. The range of $q$ is $[0,2t]$. The appropriate contours to take depend upon whether the initial brane is parallel to the orientifold plane or intersects with it, and whether $q > t$.
Suppose that the first $N_D$ vertex operators have $\Im(z_i) < t$, and the following $N_C$ have $\Im(z_i) > t$, ordered in increasing $\Im(z_i)$; they all lie upon the imaginary axis, and so we define the contours $$\begin{aligned}
C_{N_C+1} &\equiv& [2it,z_{N_C+N_D}] \nonumber \\
C_{i} &\equiv& [z_{i+1},z_i], \qquad N_D + 1 < i < N_C + N_D \nonumber \\
C_{1} &\equiv& [z_{N_D+1}, it] \nonumber \\
D_{N_D+1} &\equiv& [t,z_{N_D}] \nonumber \\
D_{i} &\equiv& [z_{i+1},z_i], \qquad i < N_D \nonumber \\
\hat{B} &\equiv& [0,1/2] \nonumber \\
\tilde{\hat{B}} &\equiv& [it,it+1/2]\end{aligned}$$ noting that $z_1 = 0$.
We also require the conjugate contours $$\begin{aligned}
C_i^{\prime} &\equiv& C_i + 1/2 - it \nonumber \\
D_i^{\prime} &\equiv& D_i + 1/2 + it .\end{aligned}$$ and the phases $$\begin{aligned}
C_i^{\prime} &=& e^{-2\pi i \phi_{a, O6}} e^{ic_i} C_i \nonumber \\
D_i^{\prime} &=& e^{-2\pi i \phi_{a, O6}} e^{id_i} D_i\end{aligned}$$ where $$\begin{aligned}
d_i - d_{i-1} =& - 2\pi \theta_i \qquad &\mod 2\pi \nonumber \\
c_i - c_{i-1} =& - 2\pi \theta_{i+ N_D} \qquad &\mod 2\pi\end{aligned}$$ and similarly for $d_i$. We have $ c_{N_C + 1} = 0$ and thus $$\begin{aligned}
d_i &=& -2\pi \sum_{j=1}^i \theta_j \nonumber \\
c_i &=& 2\pi \sum_{j=i+1}^{N_C+i+1} \theta_{j+N_D} \end{aligned}$$ The configuration is illustrated in figure \[MOEBIUS:dissection\].
\[MOEBIUS:dissection\]
The conditions for eliminating paths are $$\begin{aligned}
\sum_{i} C_i + \sum_j D_j &=& \sum_{i} C_i^{\prime} + \sum_j D_j^{\prime} \label{CCPrimeSum} \\
\sum_{i} C_i - \sum_j D_j^{\prime} &=& \hat{B} - \tilde{\hat{B}} \end{aligned}$$
Once all spurious degrees of freedom have been eliminated, we can finally write $$\begin{aligned}
(\omega_{i'},\omega_{j'}) &=& i W^{i'}_a \ov{W}^{j'}_b M^{ab} \nonumber \\
(\omega_{i''},\omega_{j''}) &=& i \ov{W}^{i''}_a W^{j''}_b \ov{M}^{ab}\end{aligned}$$ where $M^{ab}$ is anti-hermitian. We can simplify by using the matrix P\_[ij]{} {
[cc]{} \_[i’j’]{} & i {i’}\
0 & i {i”}
. and defining $\hat{W}_{a i} $ by W\^a\_i = i e\^[- i \_a/2]{} \_[a i]{} (which factors out the phases for the individual cut differentials; as we argued they disappear from the action anyway - although note that it does not exclude elements $\hat{W}_{a i }$ from being negative) to then write the action in matrix form as S\^[ab]{}\_[cl]{} = i \[ M\^T P \^[-1]{} - (M\^T P \^[-1]{})\^ - M\^T \]\^[ab]{}
Classical Action: Two-Point case
--------------------------------
It is possible to deal with the two point functions quite generally; initially we have five paths $C_i, D_i, \hat{B}, \tilde{\hat{B}}$ where $N_C=1,N_D=2$ when $q < t$ or $N_C=2, N_D=1$ when $q < t$, and in both cases we can eliminate all but two: $B$ and $A \equiv \sum_i C_i + \sum_j D_j$. That $A$ is a valid path and has a well-defined phase is straightforward to show using $$\begin{aligned}
\bar{C}_i &=& -e^{ic_i} C_i = -e^{2\pi i \phi_{a, O6}} C_i^{\prime} \nonumber \\
\bar{D}_i &=& -e^{id_i} D_i = -e^{2\pi i \phi_{a, O6}} D_i^{\prime}\end{aligned}$$ and \[CCPrimeSum\]; we find |[A]{} = -e\^[2i \_[a, O6]{}]{} A. The displacement associated with this is then v\_A = (v\_a + v\_[a’]{})= n\_A L\_a \_[a, O6]{}
### $q<t$
In this case, the matrix $M^{ab}$ is given by $$\begin{aligned}
M^{AA} &=& i \frac{ \sin \pi (2\phi_{a, O6} + \theta) \tan \pi \phi_{a, O6}} {\sin \pi \theta} \nonumber \\
M^{AB} &=& e^{-i\pi \phi_{a, O6}}/2 \nonumber \\
M^{BA} &=& - e^{i\pi \phi_{a, O6}}/2 \nonumber \\
M^{BB} &=& 0 .\end{aligned}$$ We also find = -2\_[a, O6]{} and thus $v_B$ is perpendicular to brane $a'$; we find v\_B = i ( n\_B + y\_B ) where $y_B$ is the height of the smallest triangle $abO6$. The action is $$\begin{gathered}
S^{ab}_{cl} = \frac{1}{\Wh^{2}_A \Wh^{1}_B - \Wh^{1}_A \Wh^{2}_B } \times\\
\left(\begin{array}{cc} \bigg[\Wh^{1}_B \Wh^{2}_B + iM^{AA} (\Wh^{1}_A \Wh^{2}_B + \Wh^{2}_A \Wh^{1}_B) \bigg] & -e^{i\pi \phi_{a, O6}} M^{AA} \Wh^{1}_A \Wh^{2}_A \\ e^{-i\pi \phi_{a, O6}} M^{AA} \Wh^{1}_A \Wh^{2}_A & -\Wh^{1}_A \Wh^{2}_A \end{array} \right) \end{gathered}$$ which gives $$\begin{aligned}
\label{Actionqltt}
S_{cl} &=& \frac{1}{4\pi \ap} \frac{2}{\Wh^{2}_A \Wh^{1}_B - \Wh^{1}_A \Wh^{2}_B } \times \\
&&\bigg[ \bigg( \Wh^{1}_B \Wh^{2}_B + iM^{AA} (\Wh^{1}_A \Wh^{2}_B + \Wh^{2}_A \Wh^{1}_B)\bigg)(n_A L_a \cos \pi \phi_{a,O6})^2
\nonumber\\
&& -\frac{\Wh^{1}_A \Wh^{2}_A}{\cos^2 \pi \phi} ( n_B \frac{4\pi^2 T_2}{L_{a}} + y_B ) \bigg( n_B \frac{4\pi^2 T_2}{L_{a}} + y_B + 2iM^{AA} n_A L_a \cos^3 \pi \phi_{a,O6}\bigg) \bigg] \nonumber .\end{aligned}$$ We also have the determinant |W| = i e\^[-3 i \_[a, O6]{}]{} (\^[1]{}\_A \^[2]{}\_B - \^[2]{}\_A \^[1]{}\_B).
For calculating the integrals $W^i_A$ when $\phi_{a, O6} \ne 0$ it is most expedient to use the identity W\^i\_A = - W\^i\_[D\_1]{} e\^[-i(+ \_[a, O6]{})]{} from which one deduces that, in the limit $q\rightarrow 0$, that $$\begin{aligned}
W^1_D &\rightarrow& \frac{\theta_1 (2\phi_{a, O6} i t)}{\theta_1^{\prime} (0)} e^{\pi i \theta} B(\theta, 1-\theta) \nonumber \\
W^1_A &\rightarrow& - \frac{\theta_1 (2\phi_{a, O6} i t)}{\theta_1^{\prime} (0)} e^{-\pi i \phi_{a, O6} } \frac{\pi}{\sin \pi ( \phi_{a, O6})} \nonumber \\
W^2_A &\rightarrow& - W^1_A \equiv -A\end{aligned}$$ while it is also clear that $W^1_B \rightarrow W^2_B \equiv B \rightarrow \infty$.
In this limit we find that the coefficient of the term quadratic in $n_A$ diverges, and thus the sum over $n_A$ is reduced to the zero mode: $\sum_{n_A} e^{-S(n_A,n_B)} \rightarrow e^{-S(0,n_B)}$. However, the coefficient of the quadratic term in $n_B$ reduces to zero, and we must Poisson resum, upon which the sum collapses to a single contribution: \_[n\_a, n\_b]{} e\^[-S(n\_A,n\_B)]{} \[ClassicalLimit\]
### $q>t$
In this case, the matrix $M^{ab}$ is given by $$\begin{aligned}
\tilde{M}^{AA} &=& i \frac{ \sin \pi (2\phi_{a, O6} + \theta) \tan \pi (\theta + \phi_{a, O6}) } {\sin \pi \theta} \nonumber \\
\tilde{M}^{AB} &=& e^{-i\pi (\theta + \phi_{a, O6})}/2 \nonumber \\
\tilde{M}^{BA} &=& - e^{i\pi (\theta + \phi_{a, O6})}/2 \nonumber \\
\tilde{M}^{BB} &=& 0 .\end{aligned}$$ We also find = -2\_[a, O6]{} - 2 , and thus $v_B$ is perpendicular to brane $b'$; we find v\_B = i ( n\_B + y\_B ). The action is $$\begin{gathered}
S^{ab}_{cl} = \frac{1}{\Wh^{2}_A \Wh^{1}_B - \Wh^{1}_A \Wh^{2}_B} \times\\
\left(\begin{array}{cc} \bigg[ \Wh^{1}_B \Wh^{2}_B +i\tilde{M}^{AA} (\Wh^{1}_A \Wh^{2}_B + \Wh^{2}_A \Wh^{1}_B) \bigg] & -e^{i\pi (\theta +\phi_{a, O6} )} \tilde{M}^{AA} \Wh^{1}_A \Wh^{2}_A \\ e^{-i\pi (\theta + \phi_{a, O6})} \tilde{M}^{AA} \Wh^{1}_A \Wh^{2}_A & -\Wh^{1}_A \Wh^{2}_A \end{array} \right) \end{gathered}$$ which gives the same action as equation (\[Actionqltt\]) but with $M^{AA}$ replaced with $\tilde{M}^{AA}$ and $v_B$ modified. The determinant is |W| = i e\^[-3 i \_[a, O6]{} - i ]{} (\^[1]{}\_A \^[2]{}\_B - \^[2]{}\_A \^[1]{}\_B).
For calculating the integrals $W^i_A$ when $\phi_{a, O6} \ne 0$ it is most expedient to use the identity W\^i\_A = W\^i\_[C\_2]{} e\^[-i\_[a, O6]{}/2]{} from which one deduces that, in the limit $q\rightarrow 0$, that $$\begin{aligned}
W^1_{C_2} &\rightarrow& - \frac{\theta_1 (2 i (\theta + \phi_{a, O6}) t)}{\theta_1^{\prime} (0)} e^{-4\pi \theta t} B(\theta, 1-\theta) \nonumber \\
W^1_A &\rightarrow& - e^{-i\pi\phi_{a, O6}} \frac{\theta_1 (2 i (\theta + \phi_{a, O6}) t)}{\theta_1^{\prime} (0)} e^{-4\pi \theta t} \frac{\pi}{\sin \pi (\theta + \phi_{a, O6})} \nonumber \\
W^2_A &\rightarrow& - W^1_A\end{aligned}$$ while it is also clear that $W^1_B \rightarrow W^2_B \rightarrow \infty$.
Quantum Part
------------
The quantum contribution for the Möbius strip can be derived exactly as in [@Abel:2005qn] but with the new basis of cut differentials. The result is $$\begin{gathered}
\bra \prod_{i=1}^L {\sigma}_{\theta_i} (z_i) \ket_{\phi_{a, O6}} = |\tilde{W}|^{-1/2} \theta_{\phi_{a, O6}} (-Y)^{(L-M-1)/2} \ov{\theta_{-\phi_{a, O6}} (Y)}^{(M-1)/2} \\
\prod_{0< i< j}^{L-M} \theta_1 (z_i - z_j)^{1/2} \prod_{L-M< i< j}^{L} \theta_1 (z_i - z_j)^{1/2} \prod_{0< i< j}^{L} \theta_1 (z_i - z_j)^{-\frac{1}{2}[ 1 - \theta_i - \theta_j + 2\theta_i \theta_j]}\end{gathered}$$ written in the basis $\tilde{\omega}_i$. To transform to the basis $\omega_i$, we use \[convertbasesmob\] and note that $|\tilde{W}| \rightarrow e^{4\pi i \phi_{a, O6} Y L} \prod_{i=1}^{L-M} e^{4\pi i \phi_{a, O6} z_i} \prod_{j=L-M+1}^{L} e^{-4\pi i \phi_{a, O6} z_j} |W|$ and $\theta_{\phi_{a, O6}} (-Y) \rightarrow e^{4\pi i \phi_{a, O6} Y} \theta_1 (Y + 2 \phi_{a, O6} i t)$ to give $$\begin{gathered}
\bra \prod_{i=1}^L {\sigma}_{\theta_i} (z_i) \ket_{\phi_{a, O6}} = |W|^{-1/2} \theta_1 (Y+2\phi_{a, O6} i t)^{(L-2)/2} e^{-4\pi i \phi_{a, O6} P} \\
\prod_{0< i< j}^{L-M} \theta_1 (z_i - z_j)^{1/2} \prod_{L-M< i< j}^{L} \theta_1 (z_i - z_j)^{1/2} \prod_{0< i< j}^{L} \theta_1 (z_i - z_j)^{-\frac{1}{2}[ 1 - \theta_i - \theta_j + 2\theta_i \theta_j]}\end{gathered}$$ where $P$ is as defined in [@Abel:2005qn]: P \_[i=1]{}\^L (1/2 - \_i) z\_i .
In the case of a two-point function with $z_1 = 0, z_2 = iq, \theta_1 = \theta, \theta_2 = 1-\theta, Y=\theta q i$ we have \_(0) \_[1-]{} (qi) \_[\_[a, O6]{}]{}\^[qu]{} = |W|\^[-1/2]{} e\^[4\_[a, O6]{} (- 1/2) q]{} ()\^[-(1-)]{} \[Zqumob\]
Normalisation
-------------
To normalise the two-point function we use the OPE \_\^[ab]{} (z\_1) \_[1-]{}\^[ba]{}(z\_2) \~(z\_1 - z\_2)\^[-(1-)]{} C\^[(aba)]{}\_[,1-]{} M\_[aa’]{}\^X , with the same OPE coefficients (\[ANNULUS:OPE\]) as before, but now the partition function is M\_[aa’]{}\^X = -i I\_[aO6]{} . Using equations (\[Zqumob\]) and (\[ClassicalLimit\]) we find for $q < t, q \rightarrow 0$ N (qi)\^[-(1-)]{} e\^[-i /4]{} e\^[3i \_[a, O6]{}/2]{} = (qi)\^[-(1-)]{} M\_[aa’]{}\^X and hence, using the identity (\[ANNULUS:LcLa\]) but for the intersection between $a$ and $a'$ (with angle $2 \pi \phi_{a,O6}$) we find N = e\^[i /4 -3i \_[a, O6]{}/2]{}
Following the same procedure for $q > t$ we factorise onto M\_[bb’]{}\^[X]{} = -i I\_[bO6]{} to obtain = N e\^[-i /2]{}
### $\phi_{a, O6}=0, -\theta$
If either brane $a$ or $b$ is parallel to the orientifold plane ($\phi_{a, O6} = 0, -\theta$ respectively), then the partition function that we factorise onto contains worldsheet instantons. For $a$ parallel, for $q \rightarrow 0$ we factorise onto ([@Lust:2003ky]): M\_[aa’]{}\^X = \_[r,s]{} e\^[- t | r + s |\^2]{} or the same for $M_{bb'}^X$ should that be the parallel brane, at the pole $q \rightarrow 2t$. To see this, note that for $\phi_{a, O6}=0$ we have $M^{AA}=0$; but in addition, as in the case of an annulus diagram with a parallel brane, we find that $W^1_B = W^2_B \equiv B$ and $W^1_A = -W^2_A \equiv -i\hat{A}$ and thus the action (\[Actionqltt\]) becomes S\_[cl]{} = . and $|W| = 2i\hat{A}B$. In the limit $q \rightarrow 0$ we find $B \rightarrow 1$, $\hat{A} \rightarrow 2t$, and so to show equivalence to the above we must perform a Poisson resummation on $n_A$ to obtain $$\begin{aligned}
\bra \sigma_{\theta} (0) \sigma_{1-\theta} (iq) \ket &{}_{\quad\longrightarrow}^{\phi_{a, O6}\rightarrow 0}& \left(\frac{\theta_1 (iq)}{\theta_1^{\prime}(0)} \right)^{-\theta (1-\theta)} \frac{2\pi \sqrt{\ap}}{\eta(1/2+it)^2 L_{a}} \frac{1}{B} \nonumber \\
&&\times \sum_{\tilde{n}_A,n_B}e^{-\frac{4\pi^3 \ap}{L_{a}^2} \frac{\hat{A}}{B} [ (n_A )^2 + (n_B \frac{T_2}{\ap} + \frac{y_B L_{a}}{4\pi^2 \ap} )^2 ] } \end{aligned}$$ which clearly reduces to the expected form in the limit.
Fermionic Correlators
---------------------
Calculation of fermionic correlators is straightforward using bosonised operators: \_[i=1]{}\^L e\^[i a\_i H\_i (z\_i)]{} \_[\_[a, O6]{}, ]{} = e\^[-4i \_[a, O6]{} Q]{} \_ ( Q - 2 \_[a, O6]{} i t) \_[i < j]{} \_1 (z\_i - z\_j)\^[a\_i a\_j]{} where $\nu$ indicates the spin structure and Q \_i a\_i z\_i . To normalise, we require the fermionic partition function M\_[aa’]{}\^ = -i . and thus we must multiply by $i \eta^{-1} (it+1/2) e^{-4\pi \phi_{a, O6}^2 t}$.
For the two-point function with $a_1 = \theta - 1, a_2 = 1 - \theta$ we have $$\begin{aligned}
\bra e^{i(\theta-1)H (0)} e^{-i(\theta-1)H(qi)} \ket = e^{4\pi \phi_{a, O6} (1-\theta)q}i \theta_1 (qi)^{-(1-\theta)^2} \frac{\theta_{\nu} ( 2 \phi_{a, O6} i t - (1-\theta) q i ) }{e^{4\pi \phi_{a, O6}^2 t}\eta (it+1/2)} \end{aligned}$$
Full Two-Point Möbius Amplitude
-------------------------------
We now assemble the above machinery to compute the two-point function for the Möbius strip in $N=1$ supersymmetric sectors. The contribution to sectors with more supersymmetry can be obtained from the below by setting some angles to zero. We use the same vertex operators (\[ANNULUS:Vertices\]) as the previous section, for states at an intersection with angles $\theta^{\kappa}$ where $\sum \theta^{\kappa} = 2$, but we define z\_1 = 0, z\_2 = iq in accordance with the method outlined in this section (but in contrast to that used in the previous one). Here we also have the angle between brane $a$ and the orientifold plane $\phi_{a, O6}^{\kappa}$. Here we take $\sum_{\kappa} \phi_{a, O6}^{\kappa} = 2$, although summing to zero is entirely equivalent for these.
We thus write $$\begin{aligned}
\mathcal{M} &\equiv& \bra C_{ab} (k) \ov{C}_{ab} (-k)\ket \nonumber \\ &=& 4(\ap)^2 k^2G_{C_{ab},\ov{C}_{ab}}\int_0^{\infty} \frac{dt}{(8\pi^2 \ap t)^2} \frac{1}{\eta (it+1/2)^3} \int_0^{2t} dq (2\sqrt{2}) \nonumber \\&& \times
\chi(qi) \left(\frac{\theta_1(iq)}{\theta_1^{\prime}(0)}\right)^{-2}
e^{4\pi q} \nonumber \\ && \times \sum_{\nu} \delta_{\nu} \theta_\nu (qi)
\prod_{\kappa}\frac{\theta_\nu (2\phi_{a, O6}^{{\kappa}} it + (\theta^{{\kappa}} -1)qi )}{\eta(it+1/2)^3}
|W^{{\kappa}}|^{-1/2} \sum_{n^{{\kappa}}_A,n_B^{{\kappa}}} e^{-S^{{\kappa}}} \end{aligned}$$ which after summation over spin structures becomes $$\begin{gathered}
\mathcal{M} = \frac{k^2}{16\pi^2} G_{C_{ab},\ov{C}_{ab}}\int_0^{\infty} \frac{dt}{ t^2} \frac{1}{\eta(it+1/2)^6} \int_0^{2t} dq \theta_1(iq)^{-1} e^{4\pi q}\chi(qi) \\
\times \prod_{\kappa}\theta_1(2\phi_{a, O6}^{{\kappa}} it + (\theta^{{\kappa}} -1)qi )
|W^{{\kappa}}/2|^{-1/2} \sum_{n^{{\kappa}}_A,n_B^{{\kappa}}} e^{-S^{{\kappa}}} .\end{gathered}$$ In the same way as the previous section, but after rescaling $t \rightarrow t/4$ to match the modular parameter of the Möbius strip to the annulus we can find the pole behaviour = -4 \_[R]{} tr(\_[ab]{} \_[ab]{}\^)where, in the same way as in section 2, we have $\gamma_{\Omega R}^{-1} \gamma_{\Omega R}^{T}= \rho_{\Omega R} {\bf 1} = \pm {\bf 1}$. This is then exactly the correct contribution to cancel the poles in the annulus diagrams.
Conclusions
===========
We have calculated the one loop Kähler metric for chiral fields on branes in both branes at orbifold fixed points and intersecting brane models, and in so doing completed the set of techniques for calculating $D6$-brane boundary-changing operator amplitudes in toroidal orientifold models. The two types of calculations are in stark contrast, due to the presence of the boundary changing operators in the second case, although they both reproduce the field theory expectations and both contain closed string tadpoles that must be subtracted. In addition, the first computation involves no moduli dependence, and so we expect that the corrections given have a universal quality independent of the geometry. The intersecting branes, on the other hand, have Kähler and brane modulus dependence through the worldsheet instantons, and so are sensitive to the whole compact space. It is nevertheless possible to use these latter computations in various limiting cases; for example intersecting branes can be used as a toy model for D-term supersymmetry breaking, but we postpone such calculations for further work.
Acknowledgments
===============
Work supported in part by the French ANR contracts BLAN05-0079-01 and\
PHYS@COL&COS. M. D. G. is supported by a CNRS contract.
Theta Functions {#App:A}
===============
The theta functions are defined as (z,) = \_[n = -]{}\^ and have periodicies $$\begin{aligned}
\theta \left[ \begin{array}{c} a \\ b \end{array} \right] (z+m;\tau) &=& \exp (2\pi i am) \theta \left[ \begin{array}{c} a \\ b \end{array} \right] (z;\tau) \\
\theta \left[ \begin{array}{c} a \\ b \end{array} \right] (z+m\tau;\tau) &=& \exp (-2\pi i bm) \exp (-\pi i m^2 \tau - 2 \pi i mz)\theta \left[ \begin{array}{c} a \\ b \end{array} \right] (z;\tau) \nonumber \end{aligned}$$
The modular transformation of $\theta_1(z,\tau) \equiv {\theta\left[\begin{array}{c}\frac{1}{2} \\ \frac{1}{2} \end{array}\right]} (z,\tau)$ is $$\begin{aligned}
\theta_1(z, \tau) &=& i (-i\tau)^{-1/2} \exp(-\pi i z^2/\tau) \theta_1 (z/\tau,-1/\tau) \nonumber \\
&=& \exp(-\pi i /4) \theta_1 (z,\tau + 1) \nonumber \\
\theta_1 (z,1/2 + it) &=& \sqrt{\frac{i}{2t}} \exp ( -\pi z^2 / t )\theta_1 ( \frac{i z}{2t},\frac{i}{4t} -1/2)\end{aligned}$$
Some additional identities used in the text are presented below. (z,) = (z+ c + d,) \[A:thetapluscd\]
= (a) + (b) + 4\_[m, n = 1]{}\^ e\^[2m n i ]{} (2m a + 2 n b ) \[A:thetaaplusb\]
= z + 4 \_[m,n=1]{}\^ e\^[2m n i ]{} (2m z) \[A:thetaprimeovertheta\]
[10]{}
I. Antoniadis, E. Gava, K. S. Narain and T. R. Taylor, Nucl. Phys. B [**407**]{} (1993) 706 \[arXiv:hep-th/9212045\].
C. Angelantonj and A. Sagnotti, Phys. Rept. [**371**]{}, 1 (2002) \[Erratum-ibid. [**376**]{}, 339 (2003)\] \[arXiv:hep-th/0204089\]. G. Aldazabal, A. Font, L. E. Ibanez and G. Violero, Nucl. Phys. B [**536**]{} (1998) 29 \[arXiv:hep-th/9804026\].
E. Kiritsis, Fortsch. Phys. [**52**]{} (2004) 200 \[Phys. Rept. [**421**]{} (2005 ERRAT,429,121-122.2006) 105\] \[arXiv:hep-th/0310001\].
A. M. Uranga, Class. Quant. Grav. [**22**]{} (2005) S41. R. Blumenhagen, B. Kors, D. Lust and S. Stieberger, Phys. Rept. [**445**]{} (2007) 1 \[arXiv:hep-th/0610327\]. F. Marchesano, Fortsch. Phys. [**55**]{} (2007) 491 \[arXiv:hep-th/0702094\].
R. Blumenhagen, L. Gorlich and B. Kors, JHEP [**0001**]{} (2000) 040 \[arXiv:hep-th/9912204\]. M. Cvetic, G. Shiu and A. M. Uranga, Nucl. Phys. B [**615**]{} (2001) 3 \[arXiv:hep-th/0107166\]. M. Cvetic and I. Papadimitriou, Phys. Rev. D [**67**]{} (2003) 126006 \[arXiv:hep-th/0303197\]. R. Blumenhagen, M. Cvetic, F. Marchesano and G. Shiu, JHEP [**0503**]{} (2005) 050 \[arXiv:hep-th/0502095\].
C. M. Chen, T. Li, V. E. Mayes and D. V. Nanopoulos, arXiv:0711.0396 \[hep-ph\]. C. Kokorelis, Nucl. Phys. B [**732**]{} (2006) 341 \[arXiv:hep-th/0412035\]. G. Honecker and T. Ott, Phys. Rev. D [**70**]{}, 126010 (2004) \[Erratum-ibid. D [**71**]{}, 069902 (2005)\] \[arXiv:hep-th/0404055\].
D. Friedan, E. J. Martinec and S. H. Shenker, Nucl. Phys. B [**271**]{}, 93 (1986). V. A. Kostelecky, O. Lechtenfeld, W. Lerche, S. Samuel and S. Watamura, Nucl. Phys. B [**288**]{}, 173 (1987). C. P. Burgess and T. R. Morris, Nucl. Phys. B [**291**]{}, 256 (1987). C. P. Burgess and T. R. Morris, Nucl. Phys. B [**291**]{}, 285 (1987). M. R. Garousi and R. C. Myers, Nucl. Phys. B [**475**]{}, 193 (1996) \[arXiv:hep-th/9603194\]. A. Hashimoto and I. R. Klebanov, Phys. Lett. B [**381**]{}, 437 (1996) \[arXiv:hep-th/9604065\].
I. Antoniadis, C. Bachas, C. Fabre, H. Partouche and T. R. Taylor, Nucl. Phys. B [**489**]{}, 160 (1997) \[arXiv:hep-th/9608012\].
P. Bain and M. Berg, JHEP [**0004**]{} (2000) 013 \[arXiv:hep-th/0003185\]. M. Berg, M. Haack and B. Kors, JHEP [**0511**]{} (2005) 030 \[arXiv:hep-th/0508043\].
I. Antoniadis, R. Minasian, S. Theisen and P. Vanhove, Class. Quant. Grav. [**20**]{} (2003) 5079 \[arXiv:hep-th/0307268\].
C. Bachas, C. Fabre, E. Kiritsis, N. A. Obers and P. Vanhove, Nucl. Phys. B [**509**]{} (1998) 33 \[arXiv:hep-th/9707126\]. M. Berg, M. Haack and B. Kors, Phys. Rev. D [**71**]{} (2005) 026005 \[arXiv:hep-th/0404087\].
L. J. Dixon, D. Friedan, E. J. Martinec and S. H. Shenker, Nucl. Phys. B [**282**]{}, 13 (1987). S. Hamidi and C. Vafa, Nucl. Phys. B [**279**]{}, 465 (1987). M. Bershadsky and A. Radul, Int. J. Mod. Phys. A [**2**]{}, 165 (1987).
E. Gava, K. S. Narain and M. H. Sarmadi, Nucl. Phys. B [**504**]{}, 214 (1997) \[arXiv:hep-th/9704006\]. A. Hashimoto, Nucl. Phys. B [**496**]{}, 243 (1997) \[arXiv:hep-th/9608127\]. B. Chen, H. Itoyama, T. Matsuo and K. Murakami, Nucl. Phys. B [**576**]{}, 177 (2000) \[arXiv:hep-th/9910263\]. J. Frohlich, O. Grandjean, A. Recknagel and V. Schomerus, Nucl. Phys. B [**583**]{}, 381 (2000) \[arXiv:hep-th/9912079\].
I. Antoniadis, K. Benakli and A. Laugier, JHEP [**0105**]{} (2001) 044 \[arXiv:hep-th/0011281\]. D. Cremades, L. E. Ibanez and F. Marchesano, JHEP [**0307**]{} (2003) 038 \[arXiv:hep-th/0302105\]. M. Cvetic and I. Papadimitriou, Phys. Rev. D [**68**]{} (2003) 046001 \[Erratum-ibid. D [**70**]{} (2004) 029903\] \[arXiv:hep-th/0303083\]. S. A. Abel and A. W. Owen, Nucl. Phys. B [**663**]{} (2003) 197 \[arXiv:hep-th/0303124\]. I. R. Klebanov and E. Witten, Nucl. Phys. B [**664**]{} (2003) 3 \[arXiv:hep-th/0304079\].
S. A. Abel and A. W. Owen, Nucl. Phys. B [**682**]{} (2004) 183 \[arXiv:hep-th/0310257\]. J. J. Atick, L. J. Dixon, P. A. Griffin and D. Nemeschansky, Nucl. Phys. B [**298**]{} (1988) 1.
S. A. Abel and B. W. Schofield, JHEP [**0506**]{} (2005) 072 \[arXiv:hep-th/0412206\].
S. A. Abel and M. D. Goodsell, JHEP [**0602**]{} (2006) 049 \[arXiv:hep-th/0512072\].
C. Bachas and C. Fabre, Nucl. Phys. B [**476**]{}, 418 (1996) \[arXiv:hep-th/9605028\]. I. Antoniadis, C. Bachas and E. Dudas, Nucl. Phys. B [**560**]{}, 93 (1999) \[arXiv:hep-th/9906039\]. D. Lust and S. Stieberger, Fortsch. Phys. [**55**]{} (2007) 427 \[arXiv:hep-th/0302221\]. N. Akerblom, R. Blumenhagen, D. Lust and M. Schmidt-Sommerfeld, Phys. Lett. B [**652**]{} (2007) 53 \[arXiv:0705.2150 \[hep-th\]\]. M. Bianchi and E. Trevigne, JHEP [**0601**]{} (2006) 092 \[arXiv:hep-th/0506080\].
D. Lust, P. Mayr, R. Richter and S. Stieberger, Nucl. Phys. B [**696**]{} (2004) 205 \[arXiv:hep-th/0404134\]. M. Bertolini, M. Billo, A. Lerda, J. F. Morales and R. Russo, Nucl. Phys. B [**743**]{}, 1 (2006) \[arXiv:hep-th/0512067\]. R. Blumenhagen and M. Schmidt-Sommerfeld, Models,” JHEP [**0712**]{} (2007) 072 \[arXiv:0711.0866 \[hep-th\]\]. R. Russo and S. Sciuto, JHEP [**0704**]{} (2007) 030 \[arXiv:hep-th/0701292\].
[^1]: kbenakli@lpthe.jussieu.fr
[^2]: goodsell@lpthe.jussieu.fr
[^3]: Unité Mixte de Recherche du CNRS UMR 7589
[^4]: The moduli-dependent parts which arises from $N=2$ sectors, have been explicitely computed for instance in [@Bachas:1996zt; @Antoniadis:1999ge; @Lust:2003ky; @Akerblom:2007np; @Bianchi:2005sa].
|
---
abstract: 'Restricted numerical shadow $P^X_A(z)$ of an operator $A$ of order $N$ is a probability distribution supported on the numerical range $W_X(A)$ restricted to a certain subset $X$ of the set of all pure states – normalized, one–dimensional vectors in ${\mathbb C}^N$. Its value at point $z \in {\mathbb C}$ equals to the probability that the inner product $\langle u |A| u \rangle$ is equal to $z$, where $u$ stands for a random complex vector from the set $X$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. For a Hermitian operator $A$ of order $N$ we derive an explicit formula for its shadow restricted to real states, $P^{\mathbb R}_A(x)$, show relation of this density to the Dirichlet distribution and demonstrate that it forms a generalization of the $B$–spline. Furthermore, for operators acting on a space with tensor product structure, ${\cal H}_A \otimes {\cal H}_B$, we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order $N=4$.'
address:
- 'Department of Mathematics Kerchof Hall, Room 223, P.O. Box 400137 University of Virginia, Charlottesville, VA 22904-4137'
- 'Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Ba[ł]{}tycka 5, 44-100 Gliwice, Poland'
- 'Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krak[ó]{}w, Poland'
- 'Center for Theoretical Physics, Polish Academy of Sciences, al. Lotnik[ó]{}w 32/46, 02-668 Warszawa, Poland'
author:
- 'Charles F. Dunkl'
- Piotr Gawron
- '[Ł]{}ukasz Pawela'
- 'Zbigniew Pucha[ł]{}a'
- 'Karol [Ż]{}yczkowski'
bibliography:
- 'shadow\_V.bib'
title: 'Real numerical shadow and generalized B-splines'
---
=1
numerical range ,probability measures ,numerical shadow ,B–splines
Introduction
============
Consider a complex square matrix $A$ or order $N$. Its standard [*numerical range*]{} is defined as the following subset of the complex plane, $$W(A)=\{{\ensuremath{\langleu|}}A{\ensuremath{|u\rangle}}:u \in {\mathbb{C}}^N, \|u\|=1\},$$ where $u$ denotes a normalized complex vector in ${\cal H}_N$. Due to the Toeplitz–Hausdorff theorem this set is convex, while for a Hermitian $A$ it forms an interval belonging to the real axis – see e.g. [@Dav; @GR1977; @Gu04].
Among numerous generalizations of this notion we will be concerned with the [*restricted numerical range*]{}, $$W_X(A)=\{{\ensuremath{\langleu|}}A{\ensuremath{|u\rangle}}:u \in \omega_X \},$$ where $\omega_X$ forms a certain subset of the set $\omega$ of normalized complex vectors of size $N$. For instance, one can choose $\omega_X$ as the set of all real vectors, and analyze the ’real shadow’ of $A$, denoted by $W_{\mathbb R}(A)$. For an operator $A$ acting on a composed space, one studies also numerical range restricted to tensor product states, $W_{\otimes}(A)$, and the range $W_{E}(A)$ restricted to maximally entangled states [@rrange2010; @rrange2011]. It is worth to emphasize a crucial difference with respect to the standard notion: the resticted numerical range needs not to be convex.
In order to define a probability measure supported on numerical range of $W(A)$ it is sufficient to consider the uniform measure on the sphere $S^{2N-1}$ and the measure induced by the map $u\to {\ensuremath{\langleu|}}A{\ensuremath{|u\rangle}}\in W(A)$ [@Shd1; @GS2012]. Alternatively, one considers the space of quantum states – equivalence classes of normalized vectors in ${\mathbb{C}}^N$, which differ by a complex phase, $u \sim e^{i \alpha}u$, and works with the Haar measure invariant under the action of the unitary group [@shadow1]. For any matrix $A$ one defines in this way a probability measure $P_A(z)$ supported on $W(A)$ and called [*numerical shadow*]{} [@Shd1] or [*numerical measure*]{} [@GS2012]. The former name is inspired by the fact that for a normal matrix this measure can be interpreted as a shadow of an uniformly covered $(N-1)$ dimensional regular simplex projected on a plane [@shadow1; @gutkin2013joint]. In a similar fashion, one can consider numerical shadow of matrices over the quaternion field, defined as the pushforward measure of the uniform measure on the sphere $S^{4N-1}$.
Even though several papers on numerical shadow were published during the last five years [@Shd1; @GS2012; @shadow1], the idea to associate with the numerical range a probability measure is much older: as described in a recent review by Holbrook [@Hol14] it goes back to the early papers of Davis [@Dav].
Another variant of the numerical shadow of $A$ can be obtained by taking random points from the subset $\omega_X$ of the set of pure states. The corresponding probability measure $P_A^X(z)$, called [*restricted numerical shadow*]{} [@shadow3], is by definition supported in restricted numerical range $W_X(A)$. More generally, one may take an arbitrary probability measure $\mu$ on the set of all pure states (or on the hypersphere $S^{2N-1}$) and study the measure induced in the numerical range of $A$.
Let $A$ denote a Hermitian matrix of size $N$, so its numerical range is an interval on the real axis. The probability distribution generated by the map $\eta\mapsto\left\langle \eta|A|\eta\right\rangle $, where $\eta$ is a random point on the unit sphere $\left\{ \eta\in\mathbb{C}^{N}:\sum_{j=1}^{N}\left\vert \eta_{j}\right\vert ^{2}=1\right\} $ equipped with the unitary-invariant surface measure, is then equal to the *shadow* of $A$.
It will be convenient to introduce the set ${\Omega}_N$ containing density matrices of order $N$, i.e. Hermitian positive definite operators, normalized by the trace condition, $\rho^*=\rho\ge 0$ with ${\rm Tr}\rho=1$. The set ${\Omega}_N$ is convex as it can be considered as the convex hull of the set of projectors on the pure states of dimension $N$ – see e.g. [@bengtsson2006geometry]. Specifying a measure $\mu$ on the set of density matrices allows us to propose a more general definition of numerical shadow.
For a given $N \times N$ matrix $A$ and a probability measure $\mu$ on the space ${\Omega}_N$ of density matrices of order $N$ we define the numerical shadow of matrix $A$ with respect to $\mu$ as function on complex numbers $$\mathcal{P}^{\mu}_A (z) = \int_{\Omega_N} d \mu (\rho) \delta (z - {{\mathrm{Tr}}}A \rho).$$
The standard numerical shadow, defined in [@shadow1] and denoted by $\mathcal{P}_A (z)$, fulfills the above definition with $\mu$ supported on a pure states invariant to unitary transformations. In fact all restricted numerical shadow presented in [@shadow3] can be written in the above form.
The main goal of this work is to describe restricted numerical shadow for several relevant cases. For any symmetric real matrix $A$ we derive its real numerical shadow. To this end we use Dirichlet distributions, the properties of which are reviewed in Sec. \[sec:dirichlet\]. We demonstrate that in this case the real shadow has the same distribution as a linear combination of components of a random vector generated by the Dirichlet distribution.
In Sec. \[sec:splines\] we briefly discuss $B$–splines, which correspond to complex shadows of Hermitian matrices, and show their link to generalized Dirichlet distributions Complex and real shadows of illustrative normal matrices are compared in Sec. \[sec:symmetric\], in which some results are obtained for the case of Hermitian matrices.
Main result of this work — Theorem \[th:main-theorem\], which characterizes the real shadow of real symmetric matrices, is presented in Sec. \[sec:real-shadow\]. Continuity of the shadow at knots is discussed in Sec. \[sec:knots\], while formulae for the shadow with respect of real maximally entangled states for any matrix of size $N=4$ are derived in Sec. \[sec:entangled-shadow\].
The Dirichlet Distribution {#sec:dirichlet}
==========================
Let $\mathbb{T}_{N-1}$ in $\mathbb{R}^{N-1}$ denotes the unit simplex of $N$–point probability distributions, $$\mathbb{T}_{N-1}:=\left\{ \left( t_{1},\ldots,t_{N-1}\right) \in
\mathbb{R}^{N-1}:t_{i}\geq0\forall i,\ \sum_{i=1}^{N-1}t_{i}\leq1\right\} .$$ The [*Dirichlet*]{} distribution is a measure $\mu_{\mathbf{k}}$ on the simplex $\mathbb{T}_{N-1}$ parameterized by a vector ${\mathbf{k}}$ of $N$ real numbers $k_{1},\ldots,k_{N}>0$, $$d\mu_{\mathbf{k}}=\frac{\Gamma\left( \sum_{i=1}^{N}k_{i}\right) }{\prod_{i=1}^{N}\Gamma\left( k_{i}\right) }\prod\limits_{i=1}^{N-1}t_{i}^{k_{i}-1}\left( 1-\sum_{i=1}^{N-1}t_{i}\right) ^{k_{N}-1}dt_{1}\ldots
dt_{N-1}.$$ Note that the choice ${\mathbf{k}}=\{1,1,\dots,1\}$ gives the flat, Lebesgue measure on the simplex, while the case ${\mathbf{k}}=\{1/2,1/2,\dots,1/2\}$ corresponds to the statistical distribution – see e.g. [@bengtsson2006geometry].
Set $\widetilde{k}:=\sum_{i=1}^{N}k_{i}$. For $\alpha\in\mathbb{N}_{0}^{N}$ let $\alpha!:=\prod_{i=1}^{N}\alpha_{i}!,\left\vert \alpha\right\vert
:=\sum_{i=1}^{N}\alpha_{i}$ and $t^{\alpha}:={\textstyle\prod_{i=1}^{N-1}}
t_{i}^{\alpha_{i}}\left( 1-\sum_{i=1}^{N-1}t_{i}\right) ^{\alpha_{N}}$. It follows from the Dirichlet integral that$$\label{dirinteg}
\int_{\mathbb{T}_{N-1}}t^{\alpha}d\mu_{\mathbf{k}}=\frac{1}{\left(
\widetilde{k}\right) _{\left\vert \alpha\right\vert }}{\textstyle\prod_{i=1}^{N}}
\left( k_{i}\right) _{\alpha_{i}},$$ where $\left( x\right) _{n}:={\textstyle\prod_{i=1}^{n}} \left( x+i-1\right)$ denotes the Pochhammer product. It satisfies an important asymptotic relationship: $\left( z\right) _{n}=\frac{\Gamma\left( z+n\right) }{\Gamma\left(
z\right) }\sim z^{n}$ as $x\rightarrow\infty$ in the complex half-plane $\left\{ z:\operatorname{Re}z>0\right\}$.
Consider the random vector corresponding to choosing a point in $\mathbb{T}_{N-1}$ according to $d\mu_{\mathbf{k}}$ with components $\left( T_{1},\ldots,T_{N}\right) $ with $T_{N}:=1-\sum_{i=1}^{N-1}T_{i}$. We select an arbitrary vector of $N$ real numbers ordered increasingly, $a_{1} \leq a_{2}\leq\ldots\leq a_{N}$, and will be concerned with the probability distribution of their weighted average, $$\begin{aligned}
X & =\sum_{i=1}^{N}a_{i}T_{i}.\end{aligned}$$
The distribution of random variable $X$ will be denoted as $$\mathcal{D}\left( a_{1},\ldots,a_{N};k_{1},\ldots,k_{N}\right) .\label{DakN}$$
In the case some values of $a_{i}$ are repeated some formulae have to be modified. It is clear that $a_{1}\leq X\leq a_{N}$. There is a moment generating function for $X$.
Let $F\left( x\right) $ denote the cumulative distribution function of $X$, that is,$$F\left( x\right) :=\Pr\left\{ X\leq x\right\} .$$
\[lemma:exp-value\] \[mgenF\]Suppose $\left\vert r\right\vert <\min_{i}\frac{1}{\left\vert
a_{i}\right\vert }$ then$$\mathcal{E}\left[ \left( 1-rX\right) ^{-\widetilde{k}}\right] ={\textstyle\prod_{i=1}^{N}}
\left(1-ra_{i}\right) ^{-k_{i}}.$$
This Lemma, proof of which is provided in \[sec:proofs\] , implicitly gives an expression for the moments, $\mathcal{E}\left[ X^{n}\right]$, because $\left( 1-rX\right) ^{-\widetilde{k}}=\sum_{n=0}^{\infty}\frac{\left( \widetilde{k}\right) _{n}}{n!}r^{n}X^{n}$.
\[mean&var\]The mean $\mu:=\mathcal{E}\left[ X\right] =\frac
{1}{\widetilde{k}}\sum_{i=1}^{N}k_{i}a_{i}$ and the variance $\mathcal{E}\left[ \left( X-\mu\right) ^{2}\right] =\frac{1}{\widetilde{k}\left(
\widetilde{k}+1\right) }\sum_{i=1}^{N}k_{i}\left( a_{i}-\mu\right) ^{2}$.
In the case of $N=2$ it is straightforward to find the density for $\mathcal{D}\left( a_{1},a_{2};k_{1},k_{2}\right) $,$$f\left( x\right) =\frac{1}{B\left( k_{1},k_{2}\right) \left( a_{2}-a_{1}\right) ^{k_{1}+k_{2}-1}}\left( a_{2}-x\right) ^{k_{1}-1}\left(
x-a_{1}\right) ^{k_{2}-1},$$ where $B(a,b)$ denotes the beta function.
Let us now return to the general case of an arbitrary dimension $N$ and consider the behavior of $F\left( x\right) $ for $x\notin\left\{ a_{1},a_{2},\ldots,a_{N}\right\} $. Here we require no repeated values in $\left\{ a_{i}\right\} $. This involves the intersection of the hyperplane $\sum_{i=1}^{N}a_{i}t_{i}=x$ with $\mathbb{T}_{N-1}$, which is a convex polytope whose faces are subsets of $\pi_{i}:=\left\{ t:t_{i}=0\right\} $ for $1\leq i\leq N-1$, $\pi_{N}:=\left\{ t:{\textstyle\sum_{i=1}^{N-1}}
t_{i}=1\right\} $, and $\pi_{x}:=\left\{ t:\sum_{i=1}^{N}a_{i}t_{i}=x\right\} $. Note that $\sum_{i=1}^{N}a_{i}t_{i}=x$ is equivalent to $\sum_{i=1}^{N-1}\left( a_{N}-a_{i}\right) t_{i}=a_{N}-x$. The vertices of this polytope come from the intersection of $N-2$ hyperplanes drawn from $\left\{ \pi_{i}:1\leq i\leq N\right\} $ with $\pi_{x}$. Introduce the unit basis vectors $\varepsilon_{i}$ ($1\leq i\leq N-1$) with components $\left(
\delta_{ij}\right) $. There are two types of vertices:$$\begin{aligned}
\xi_{i}\left( x\right) & ={\textstyle\bigcap_{j=1,j\neq i}^{N-1}}
\pi_{j}\cap\pi_{x}=\frac{a_{N}-x}{a_{N}-a_{i}}\varepsilon_{i},1\leq i\leq
N-1;\\
\xi_{ij}\left( x\right) & ={\textstyle\bigcap_{\ell=1,\ell\neq i,j}^{N-1}}
\pi_{\ell}\cap\pi_{N}\cap\pi_{x}=\frac{a_{j}-x}{a_{j}-a_{i}}\varepsilon
_{i}+\frac{x-a_{i}}{a_{j}-a_{i}}\varepsilon_{j},1\leq i<j\leq N-1.\end{aligned}$$ For any given $x$ some of these vertices are in $\mathbb{T}_{N-1}$ and some are not. Suppose $a_{M}<x<a_{M+1}$ for some $M$ with $1\leq M<N$, then $\xi_{i}\left( x\right) \in\mathbb{T}_{N-1}$ exactly when $1\leq i\leq M$ since the condition is $0<\frac{a_{N}-x}{a_{N}-a_{i}}<1$, that is, $x>a_{i}$. Similarly $\xi_{ij}\left( x\right) \in\mathbb{T}_{N-1}$ exactly when $a_{i}<x<a_{j}$, that is, $1\leq i\leq M$ and $M+1\leq j\leq N-1$. Thus the number of vertices is $M\left( N-M\right) $. Each vertex is an extreme point: to show this one exhibits a linear function $c_{0}+\sum_{i=1}^{N-1}c_{i}t_{i}$ which vanishes at the point and is positive at all other vertices. For $\xi_{i}\left( x\right) $ the function $\sum_{j\neq i}t_{j}$ accomplishes this, and for $\xi_{ij}\left( x\right) $ use $1-t_{i}-t_{j}$ (this applies to the vertices contained in $\mathbb{T}_{N-1}$, by inspection).
\[polyt\]Suppose $a_{M}<x_{1}<x_{2}<a_{M+1}$ then $F\left( x_{2}\right)
-F\left( x_{1}\right) $ is given by the integral of $d\mu_{\mathbf{k}}$ over a convex polytope with $2M\left( N-M\right) $ vertices lying between parallel hyperplanes. The vertices of the polytope are analytic functions of $x$ and so $F\left( x_{2}\right) -F\left( x_{1}\right) $ is analytic in $x_{2}$ and in the parameters $k_{1},k_{2},\ldots,k_{N}$ (in broad terms, decompose the integral as a sum of iterated $\left( N-1\right) $-fold integrals each of which has an analytic expression).
It is straightforward to find the following infinite series expression for the complementary distribution function $1-F\left( x\right)$ for $x \in ( a_{N-1}, a_{N}]$ – see \[sec:proofs\] . We assumed here that $a_{N-1}<a_{N}$, but other repetitions are allowed.
\[topF\] For $a_{N-1}<x\leq a_{N}$$$\begin{aligned}
1-F\left( x\right) & =\frac{\Gamma\left( \widetilde{k}\right) \left(
a_{N}-x\right) ^{\widetilde{k}-k_{N}}}{\Gamma\left( k_{N}\right)
\Gamma\left( \widetilde{k}-k_{N}\right) }\prod_{i=1}^{N-1}\left(
a_{N}-a_{i}\right) ^{-k_{i}}\\
& \times\sum_{\alpha\in\mathbb{N}_{0}^{N-1}}\frac{\left( 1-k_{N}\right)
_{\left\vert \alpha\right\vert }}{\left( \widetilde{k}-k_{N}\right)
_{\left\vert \alpha\right\vert +1}}\left( a_{N}-x\right) ^{\left\vert
\alpha\right\vert }\prod_{i=1}^{N-1}\frac{\left( k_{i}\right) _{\alpha_{i}}}{\alpha_{i}!\left( a_{N}-a_{i}\right) ^{\alpha_{i}}}.\end{aligned}$$
For $x$ near $a_{N}$ (and $x<a_{N}$) $1-F\left( x\right) $ behaves like $\left( a_{N}-x\right) ^{\widetilde{k}-k_{N}}$ and the density $f\left(
x\right) =\frac{d}{dx}F\left( x\right) $ behaves like $\left(
a_{N}-x\right) ^{\widetilde{k}-k_{N}-1}$.
The Dirichlet distribution has a special additivity property which allows us to restrict to the situation where the $a_{i}$’s are mutually distinct. If two numbers $a_{i}$’s are equal, say $a_{N-1}=a_{N}$ then $\sum_{i=1}^{N}a_{i}t_{i}$ is has the same distribution as $\mathcal{D}\left( a_{1},\ldots,a_{N-1};k_{1},\ldots,k_{N-1}+k_{N}\right) $ (see \[DakN\]). In other words if $a_{\ell}=a_{\ell+1}=\ldots=a_{\ell+m-1}$ then the distribution is the same as$$\mathcal{D}\left( a_{1},\ldots,a_{\ell},a_{\ell+m},\ldots,a_{N};k_{1},\ldots,\sum_{i=\ell}^{\ell+m-1}k_{i},k_{\ell+m},\ldots,k_{N}\right) .$$ When each $k_{i}$ is an integer ($\geq1)$ there is a finite sum expression for the density in terms of piecewise polynomials (splines). This theorem is from [@Shd1 p.2070]. For simplicity we state the result for the case $0\leq
a_{1}<a_{2}<\ldots<a_{N}$. Let $x_{+}:=\max\left( 0,x\right) $, with the convention that $x_{+}^{0}=1$ for $x\geq0$ and $=0$ for $x<0$.
\[cxshad\]Suppose $0\leq a_{1}<a_{2}<\ldots<a_{N}$, $k_{i}\in\mathbb{N}$ for each $i$, then $$f\left( x\right) =\sum_{i=1}^{N}\sum_{j=1}^{k_{i}}\frac{\beta_{ij}}{a_{i}B\left( j,\widetilde{k}-j\right) }\left( \frac{x}{a_{i}}\right)
_{+}^{j-1}\left( 1-\frac{x}{a_{i}}\right) _{+}^{\widetilde{k}-j-1},$$ where $$\prod_{i=1}^{N}\left( 1-ra_{i}\right) ^{-k_{i}}=\sum_{i=1}^{N}\sum
_{j=1}^{k_{i}}\frac{\beta_{ij}}{\left( 1-ra_{i}\right) ^{j}}\label{parfrac1}$$ is the partial fraction decomposition (the term with $i=1$ is omitted if $a_{1}=0$).
Observe that each term $\frac{1}{a_{i}B\left( j,\widetilde{k}-j\right)
}\left( \frac{x}{a_{i}}\right) _{+}^{j-1}\left( 1-\frac{x}{a_{i}}\right)
_{+}^{\widetilde{k}-j-1}$ is itself a probability density supported on $0\leq
x\leq a_{i}$. (In the present context $N$ is the number of distinct values, differing from the statement in [@Shd1] where each $k_{i}=1$ and some values are repeated.) The Theorem shows that the density is a piecewise polynomial of degree $\widetilde{k}-2$ with discontinuities (in some order derivative) at the points $\left\{ a_{i}\right\} $. Because of this spline interpretation the quantities $a_{i}$ will henceforth be called *knots*.
B–splines and their generalization {#sec:splines}
==================================
The Dirichlet distribution is closely related to the notion of an $s$–dimensional $B$–spline introduced by de Boor [@dB1976].
Let $\sigma$ be a non-trivial simplex in $\mathbb{R}^{s+k}$. On $\mathbb{R}^s$ we define the B–spline of order $k$ from $\sigma$ by $$\mathcal{M}_{k,\sigma}(x_1,\dots,x_s)=\mathrm{vol}(\sigma\cap\{v\in\mathbb{R}^{s+k}:v_j=x_j\,\,(j=1,2,\dots,s)\}).$$
A measure version of the above definition is more useful, thus we define the normalized measure on $\mathbb{R}^s$ $$\mathcal{M}_{k,\sigma}(B)= \mathrm{vol}(\sigma\cap\{v\in\mathbb{R}^{s+k}: \{v_j\}_{j=1}^s \in B\}) / \mathrm{vol}(\sigma).$$ A non-trivial simplex $\sigma \in \mathbb{R}^{s+k}$ can be written as $W \mathbb{T}_{s+k}$ where $\mathbb{T}_{s+k}$ is a regular simplex and $W$ is an invertible matrix of order $s+k$. The simplex is possibly translated if 0 is not a vertex of $\sigma$. We will use the notation
$$\mathcal{M}_{k,W}(B)= \mathrm{vol}(y \in \mathbb{T}_{s+k}: W y \in B\oplus \mathbb{R}^{k}) / \mathrm{vol}(\mathbb{T}_{s+k}).$$
Instead of calculating the volume with respect to the flat Lebesgue measure one can use instead the Dirichlet measure $\mu_{\mathbf{k}}$ with parameters $\mathbf{k}$ instead. In this way one obtains a generalized notion of $B$-splines. $$\begin{split}
\mathcal{M}_{k,W}^{(\mathbf{k})}(B)
&= \mu_{(\mathbf{k})}(y \in \mathbb{T}_{s+k}: W y \in B\oplus \mathbb{R}^{k}) / \mu_{(\mathbf{k})}(\mathbb{T}_{s+k}).
\end{split}$$
Therefore, the distribution $\mathcal{D}$ can be viewed as a generalized $B$-spline. If we take any $N \times N$ invertible matrix $W$ with the first row given by $\lambda_1\dots\lambda_N$, then a generalized $B$-spline is equal to the distribution $\mathcal{D}$ $$\mathcal{M}_{N-1,W}^{(\mathbf{k})} = \mathcal{D}(\lambda_1,\dots\lambda_N; \mathbf{k}).$$
Shadows of Hermitian and real symmetric matrices {#sec:symmetric}
================================================
Among several probability measures defined on the set of density matrices it is convenient to distinguish a class of measures induced by the partial trace performed on a pure state on the extended system.
We say, that a density matrix $\rho$ of size $N$ is distributed according to the induced measure $\mu^{\mathrm{tr}}_{N,K}$ [@bengtsson2006geometry] if $$\rho = {{\mathrm{Tr}}}_2 {\ensuremath{{\ensuremath{|\psi\rangle}}{\ensuremath{\langle\psi|}}}},$$ where ${\ensuremath{|\psi\rangle}}$ being a uniformly distributed, normalized random vector in $\mathcal{H}_1 \otimes \mathcal{H}_2 = \mathbb{C}^{N} \otimes
\mathbb{C}^{K}$ and the operation of partial trace is defined for product matrices as ${{\mathrm{Tr}}}_2 A \otimes B = A {{\mathrm{Tr}}}B$ and extended to general case by linearity. In the case of $K=1$ we obtain a measure on pure states and in the case of $K=N$ we get a Hilbert-Schmidt measure [@bengtsson2006geometry].
![Numerical shadows of an illustrative normal matrix $A$ of order $N=4$. Upper figure represents the complex shadow, while the lower one the real shadow The tetrahedrons on the left are covered with respect to the uniform and the Dirichlet distribution, respectively. If a parallel beam of light is shined upon them they cast shadows on a plane which coincide with complex and real numerical shadows of $A$. Both marginal distributions above and on the right hand side of each shadow, correspond to complex/real numerical shadows of Hermitian matrices formed from real and imaginary parts of $A$.[]{data-label="fig:tetrahedrons"}](tetrahedrons.pdf){width="1\linewidth"}
In paper [@Shd1] we showed that the (complex) shadow of a Hermitian matrix $A$ with eigenvalues $(\lambda_{1},\ldots,\lambda_{N})$ (counted with multiplicity) has the distribution $$\mathcal{P}_{A} = \mathcal{D}\left(
\lambda_{1},\ldots,\lambda_{N};1,\ldots,1\right).
\label{compl}$$ From Corollary \[mean&var\] the mean is $\mu=\frac{1}{N}\sum_{j=1}^{N}\lambda_{j}=\frac
{1}{N}\mathrm{Tr}A$ and the variance is $\frac{1}{N\left( N+1\right) }\sum_{j=1}^{N}\left( \lambda_{j}-\mu\right) ^{2}$.
In analogy to the standard shadow (\[compl\]) one can introduce the [*mixed states shadow*]{} [@shadow1]. For a Hermitian matrix $A$ the mixed shadow induced by a distribution $\mu^{\mathrm{tr}}_{N,K}$ reads $$\mathcal{P}^K_{A} =
\mathcal{D}(\lambda_1,\dots,\lambda_N; K, \dots, K).
\label{complmixed}$$ This follows directly from the definition of a partial trace and the additivity property of a Dirichlet distribution. As a special case we obtain, that the mixed numerical shadow with respect to flat Hilbert Schmidt distribution is given by $\mathcal{P}^N_{A} =\mathcal{D}(\lambda_1,\dots,\lambda_N;N, \dots, N)$. We can calculate mean and variance for mixed numerical shadow induced by $\mu^{\mathrm{tr}}_{N,K}$, using Corollary \[mean&var\] we have $\mu=\frac{1}{N}\sum_{j=1}^{N}\lambda_{j}=\frac{1}{N}\mathrm{Tr}A$ and the variance is $\frac{1}{N\left( N K + 1\right) } \sum_{j=1}^{N}
\left( \lambda_{j}-\mu\right) ^{2}$.
Let us now return to the main subject of the paper - the shadow $\mathcal{P}_A^{\mathbb{R}}$ of a matrix $A$ of order $N$ with respect to the set of real pure states in $\mathbb{R}^{N}$. It is briefly called the [*real shadow*]{} [@shadow3], and for a real symmetric matrix $A$ it can be related to the Dirichlet distribution,
$$\mathcal{P}_A^{\mathbb{R}} =
\mathcal{D}
\left( \lambda_{1},\ldots,\lambda_{N};\frac{1}{2},\ldots,\frac{1}{2}\right)
\label{realshad}$$
where $\lambda_{1},\ldots,\lambda_{N}$ denotes the eigenvalues of $A$ counted with multiplicity. The mean value is $\mu=\frac{1}{N}\sum_{j=1}^{N}\lambda_{j}$ and the variance is $\frac{2}{N\left( N+2\right) }\sum
_{j=1}^{N}\left( \lambda_{j}-\mu\right) ^{2}$.
In a close analogy to the complex case, one can also consider the shadow with respect to real mixed states obtained by an induced measure $\mu_{N,K}^{\rm tr}$. For any real symmetric matrix $A$ this leads to the distribution $\mathcal{D}$, with all indices equal to $K/2$. Thus the real shadow is obtained for $K=1$, as required.
Henceforth we will concentrate on the distributions $\mathcal{D}\left(
a_{1},\ldots,a_{N};k,\ldots,k\right) $ with pairwise distinct knots $a_{i}$. For integer $k$ we have the interpretation as the shadow of the $Nk\times Nk$ Hermitian matrix $A\oplus\ldots\oplus A$ ($k$ summands) where the eigenvalues of $A$ are $a_{1},\ldots,a_{N}$, or the mixed numerical shadow induced by the measure $\mu^{\mathrm{tr}}_{N,k}$. We consider the distribution as an analytic function of $k$, for $\operatorname{Re}k>0$, and will find more information by extrapolating from the known formulas for integer $k$. Start with finding explicit values of the coefficients $\left\{ \beta_{ij}\right\} $ in Theorem \[cxshad\].
\[kparfrac\] Suppose $\left\{ a_{1},a_{2},\ldots,a_{N}\right\} $ consists of pairwise distinct nonzero real numbers and $k=1,2,3,\ldots$ then$$\prod_{i=1}^{N}\left( 1-ra_{i}\right) ^{-k}=\sum_{i=1}^{N}\sum_{m=0}^{k-1}\frac{\left( -1\right) ^{m}a_{i}^{\left( N-1\right) k}}{\left(
1-ra_{i}\right) ^{k-m}}\sum_{\alpha\in\mathbb{N}_{0}^{N},\left\vert
\alpha\right\vert =m,\alpha_{i}=0}\frac{1}{\alpha!}\prod_{j=1,j\neq i}^{N}\frac{\left( k\right) _{\alpha_{j}}a_{j}^{\alpha_{j}}}{\left(
a_{i}-a_{j}\right) ^{k+\alpha_{j}}}$$
The proof is provided in \[sec:proofs\].
Thus the formula in Theorem \[cxshad\] is completely symmetric in $\left(
a_{1},a_{2},\ldots,a_{N}\right) $, independent of the ordering. This is an ingredient in the derivation of the differential equation satisfied by the density.
Consider the case of a symmetric matrix of size $N=3$. Then the density for its real shadow has an expression in terms of a $_{2}F_{1}$-hypergeometric function which solves a certain second-order differential equation. Suppose $a_{1}=0$, so formulas (\[parfrac1\]) and (\[kparfrac\]) (change $j$ to $k-m$) read for $x\in (a_{2}, x\leq a_{3}]$ $$f\left( x\right) =\frac{x^{k-1}\left( a_{3}-x\right) ^{2k-1}}{B\left(
k,2k\right) a_{3}^{2k-1}\left( a_{3}-a_{2}\right) ^{k}}~_{2}F_{1}\left(
\genfrac{}{}{0pt}{}{k,1-k}{2k};\frac{a_{2}\left( a_{3}-x\right) }{x\left( a_{3}-a_{2}\right) }\right)
;\label{N3&2F1}$$ and the series converges for any $k>0$.
Let us now return to the generalized case of an arbitrary matrix order $N$, for which condition $a_{1}<a_{2}<\ldots<a_{N}$ holds. Basing on computational experiments we are in position to formulate a generalization valid for small $N$ and integers $k$. Set $P_{N}\left( x\right) =\prod_{i=1}^{N}\left( x-a_{i}\right) $ and define a differential operator $\mathcal{T}_{k}$ of order $N-1$ (with $\partial:=\frac{d}{dx}$) by$$\mathcal{T}_{k}:=P_{N}\left( x\right) \partial^{N-1}+\sum_{j=1}^{N-1}\left(
-1\right) ^{j}\frac{N-j}{N}\frac{\left( N\left( k-1\right) \right) _{j}}{j!}\partial^{j}P_{N}\left( x\right) \partial^{N-1-j}.$$ The differential equation $\mathcal{T}_{k}f\left( x\right) =0$ has regular singular points at the knots. We will show that the density function of $\mathcal{D}\left( a_{1},\ldots,a_{N};k,\ldots,k\right) $ satisfies this equation at all $x\notin\left\{ a_{1},\ldots,a_{N}\right\} $, first for integer $k$ then for $k>0$. The idea is to verify the equation for the interval $\left( a_{N-1},a_{N}\right) $ by use of Proposition \[topF\] and then use the symmetry property of Theorem \[cxshad\] to extend the result to all intervals $\left( a_{i},a_{i+1}\right) $.
For arbitrary $a,b,c$ and $n=1,2,\ldots$$$\sum_{j=0}^{n}\frac{a+j}{a}\frac{\left( -n\right) _{j}\left( b\right)
_{j}}{\left( c\right) _{j}~j!}=\frac{\left( c-b\right) _{n-1}}{a~\left(
c\right) _{n}}\left( a\left( c-b+n-1\right) -nb\right) .$$
Expand the sum as $$\begin{aligned}
\sum_{j=0}^{n}\frac{\left( -n\right) _{j}\left( b\right) _{j}}{\left(
c\right) _{j}~j!}+\frac{1}{a}\sum_{j=1}^{n}\frac{j\left( -n\right)
_{j}\left( b\right) _{j}}{\left( c\right) _{j}~j!} & =\sum_{j=0}^{n}\frac{\left( -n\right) _{j}\left( b\right) _{j}}{\left( c\right)
_{j}~j!}-\frac{nb}{ac}\sum_{i=0}^{n-1}\frac{\left( 1-n\right) _{i}\left(
b+1\right) _{i}}{\left( c+1\right) _{i}~i!}\\ \nonumber
& =\frac{\left( c-b\right) _{n}}{\left( c\right) _{n}}-\frac{nb}{ac}\frac{\left( c-b\right) _{n-1}}{\left( c+1\right) _{n-1}},\end{aligned}$$ by the Chu-Vandermonde sum.
Since we intend to work with polynomials in $x-a_{N}$ we set $y:=x-a_{N}$. Start the verification by replacing $P_{N}\left( x\right) $ by $y^{n}$ and apply the resulting operator to $\left( -y\right) ^{c}$ (for $0\leq n\leq
N-1$ and generic $c$ (leaving open the possibility of $c$ being a noninteger and $y<0)$. At times we use the Pochhammer symbol with a negative index: for $m=1,2,3,\ldots$ let $\left( c\right) _{-m}=1/\left( c-m\right) _{m}$, so that $\left( c\right) _{-m}\left( c-m\right) _{m}=\left( c\right)
_{0}=1$. Note that $\partial^{j}\left( -y\right) ^{c}=\left(
-c\right) _{j}\left( -y\right) ^{c-j}$, so the result follows $$\begin{aligned}
\nonumber
& \sum_{j=0}^{N-1}\left( -1\right) ^{j}\frac{N-j}{N}\frac{\left( N\left(
k-1\right) \right) _{j}}{j!}\left( -1\right) ^{j}\left( -n\right)
_{j}y^{n-j}\left( -c\right) _{N-1-j}\left( -y\right) ^{c-N+1+j}\\
& =\left( -1\right) ^{n}\left( -y\right) ^{c+n-N+1}\left( -c\right)
_{N-1}\sum_{j=0}^{n}\frac{j-N}{-N}\frac{\left( -n\right) _{j}\left(
N\left( k-1\right) \right) _{j}}{j!\left( c+2-N\right) _{j}}\\
\nonumber
& =\left( -1\right) ^{n}\left( -y\right) ^{c+n-N+1}\frac{\left(
-c\right) _{N-1}}{\left( c+2-N\right) _{n}}\left( c+2-Nk\right)
_{n-1}\left( c+1-\left( N-n\right) k\right) \\
\nonumber
& =\left( -y\right) ^{c+n-N+1}\left( -c\right) _{N-1-n}\left(
c+2-Nk\right) _{n-1}\left( c+1-\left( N-n\right) k\right) .\end{aligned}$$ In the special case $n=N$ we obtain $-\left( -y\right) ^{c+1}\left(
c+2-Nk\right) _{N-1}$ since $\left( -c\right) _{-1}=-1/\left( 1+c\right)
$. For $n=0$ the result is zero. The calculations used the reversal $\left(
a\right) _{m-j}=\left( -1\right) ^{j}\dfrac{\left( a\right) _{m}}{\left(
1-m-a\right) _{j}}$ and the Lemma with $a,b,c$ replaced by $-N,Nk-N$ and $c+2-N$, respectively. The upper limit of summation is $n$ because $n\leq N$. To proceed further we introduce: $$\begin{aligned}
A\left( N,k,n,c\right) & :=\left( -c\right) _{N-1-n}\left(
c+2-Nk\right) _{n-1}\left( c+1-\left( N-n\right) k\right) ,1\leq
n<N;\label{defAN}\\
A\left( N,k,N,c\right) & :=-\left( c+2-Nk\right) _{N-1}.\nonumber\end{aligned}$$ Next $P_{N}\left( x\right) =y\prod_{i=1}^{N-1}\left( y-\left( a_{i}-a_{N}\right) \right) =\sum_{j=0}^{N-1}\left( -1\right) ^{N-1-j}e_{N-1-j}y^{j+1}$ where $e_{m}$ denotes the elementary symmetric polynomial of degree $m$ in $\left\{ a_{1}-a_{N},\ldots,a_{N-1}-a_{N}\right\} $, $0\leq
m\leq N-1$. Thus $$\mathcal{T}_{k}\left( \left( -y\right) ^{c}\right) =\sum_{j=1}^{N}\left(
-1\right) ^{N-j}e_{N-j}A\left( N,k,j,c\right) \left( -y\right)
^{c-N+j+1}.\label{Tkxc}$$ Up to a multiplicative constant, not relevant in this homogeneous equation, the density in $a_{N-1}<x<a_{N}$ is given by $$f_{0}\left( x\right) =-\partial\sum_{\alpha\in\mathbb{N}_{0}^{N-1}}\frac{\left( 1-k\right) _{\left\vert \alpha\right\vert }}{\left( \left(
N-1\right) k\right) _{\left\vert \alpha\right\vert +1}}\left( -y\right)
^{\left\vert \alpha\right\vert +\left( N-1\right) k}\prod_{i=1}^{N-1}\frac{\left( k\right) _{\alpha_{i}}}{\alpha_{i}!\left( a_{N}-a_{i}\right)
^{\alpha_{i}}}.$$ The series terminates at $\left\vert \alpha\right\vert =k-1$. Define symmetric polynomials $S_{m}\left( k;a\right) $ in $\left\{ a_{1}-a_{N},\cdots,a_{N-1}-a_{N}\right\} $ (note the reversal to $a_{i}-a_{N}$) by $$\sum_{m=0}^{\infty}S_{m}\left( k;a\right) r^{m}=\prod_{i=1}^{N-1}\left(
1-\frac{r}{a_{i}-a_{N}}\right) ^{-k},$$ convergent for $\left\vert r\right\vert <a_{N}-a_{N-1}$, then$$f_{0}\left( x\right) =\sum_{m=0}^{k-1}\frac{\left( 1-k\right) _{m}}{\left( \left( N-1\right) k\right) _{m}}\left( -y\right) ^{\left(
N-1\right) k+m-1}\left( -1\right) ^{m}S_{m}\left( k;a\right) ,$$ and$$\begin{aligned}
\nonumber
\mathcal{T}_{k}f_{0}\left( x\right) & =\sum_{m=0}^{k-1}\sum_{j=1}^{N}\left( -1\right) ^{N-j}e_{N-j}\frac{\left( 1-k\right) _{m}}{\left(
\left( N-1\right) k\right) _{m}}S_{m}\left( k;a\right) \left( -1\right)
^{m}\\
& \times A\left( N,k,j,\left( N-1\right) k+m-1\right) \left( -y\right)
^{\left( N-1\right) k+m-N+j}\\
\nonumber
& =\sum_{\ell=1}^{N+k-1}\left( -y\right) ^{\left( N-1\right) k-N+\ell}\sum_{j=1}^{\min\left( N,\ell\right) }\left( -1\right) ^{N-j}e_{N-j}\frac{\left( 1-k\right) _{\ell-j}}{\left( \left( N-1\right) k\right)
_{\ell-j}}\\
\nonumber
& \times\left( -1\right) ^{\ell-j}S_{\ell-j}\left( k;a\right) A\left(
N,k,j,\left( N-1\right) k+\ell-j-1\right) .\end{aligned}$$ It is required to show that the $j$-sum vanishes for each $\ell$. At $\ell=1,j=1$ there is only one term and $A\left( N,k,1,\left( N-1\right)
k-1\right) =0$. Replace $A\left( \cdot\right) $ by its definition (\[defAN\]) and simplify$$\begin{aligned}
\nonumber
& \frac{\left( 1-k\right) _{\ell-j}}{\left( \left( N-1\right) k\right)
_{\ell-j}}\left( 1+j-\ell-\left( N-1\right) k\right) _{N-1-j}\left(
1-k+\ell-j\right) _{j-1}\left( \ell-j+k\left( j-1\right) \right) \\
& =\left( -1\right) ^{\ell-j}\left( 1-k\right) _{\ell-1}\left(
\ell-j+k\left( j-1\right) \right) \frac{\left( 1+j-\ell-\left(
N-1\right) k\right) _{N-1-j}}{\left( 1+j-\ell-\left( N-1\right) k\right)
_{\ell-j}}\\
\nonumber
& =\left( -1\right) ^{\ell-j}\left( 1-k\right) _{\ell-1}\left(
\ell-j+k\left( j-1\right) \right) \frac{\left( 1-\ell-\left( N-1\right)
k\right) _{N-1}}{\left( 1-\ell-\left( N-1\right) k\right) _{\ell}}.\end{aligned}$$ The denominator does not vanish because $\left( N-1\right) k>0$. Taking out the factors depending only on $\ell$ the $j$-sum becomes $$\sum_{j=1}^{\min\left( N,\ell\right) }\left( -1\right) ^{j}\left(
\ell-j+k\left( j-1\right) \right) e_{N-j}S_{\ell-j}\left( k;a\right)
.\label{jsum}$$ There is a recurrence relation for $S_{m}\left( k;a\right) $; the elementary symmetric function of degree $m$ in $\left\{ \frac{1}{a_{1}-a_{N}},\ldots,\frac{1}{a_{N-1}-a_{N}}\right\} $ equals $\frac{e_{N-1-m}}{e_{N-1}}$ for $0\leq m\leq N-1$. The generating function of $\left\{ S_{m}\left(
k;a\right) \right\} $ is $g\left( r\right) ^{-k}$ where$$g\left( r\right) :=\prod_{j=1}^{N-1}\left( 1-\frac{r}{a_{j}-a_{N}}\right)
=\sum_{i=0}^{N-1}\left( -1\right) ^{i}\frac{e_{N-1-i}}{e_{N-1}}r^{i}.$$ Extract the coefficient of $r^{m}$ in the following equation$$\begin{aligned}
g\left( r\right) \frac{\partial}{\partial r}\left[ g\left( r\right)
^{-k}\right] & =-k\left( \frac{\partial}{\partial r}g\left( r\right)
\right) g\left( r\right) ^{-k}\\
\nonumber
\sum_{i=0}^{N-1}\left( -1\right) ^{i}\frac{e_{N-1-i}}{e_{N-1}}r^{i}\sum_{j=0}^{\infty}jS_{j}\left( k;a\right) r^{j-1} & =-k\sum_{i=0}^{N-1}i\left( -1\right) ^{i}\frac{e_{N-1-i}}{e_{N-1}}r^{i-1}\sum
_{j=0}^{\infty}S_{j}\left( k;a\right) r^{j},\end{aligned}$$ to obtain$$\sum_{i=0}^{\min\left( m+1,N-1\right) }\left( -1\right) ^{i}\left(
m-i+1+ki\right) \frac{e_{N-1-i}}{e_{N-1}}S_{m-i+1}\left( k;a\right) =0.$$ Now set $i=j-1$ and $m=\ell-2$ (recall the case $\ell=1$ was already done) to show that the expression in (\[jsum\]) vanishes.
\[shadeq1\]Suppose $k=1,2,3,\ldots$ then the density $f\left( x\right) $ of $\mathcal{D}\left( a_{1},\ldots,a_{N};k,\ldots,k\right) $ satisfies the linear differential equation $\mathcal{T}_{k}f\left( x\right) =0$ at all $x\notin\left\{ a_{1},\ldots,a_{N}\right\} $.
Assume first that $a_{1}>0$. The above argument showed that $\mathcal{T}_{k}f\left( x\right) =0$ for $a_{N-1}<x<a_{N}$. On this interval $f\left(
x\right) $ is a constant multiple of$$p_{N}\left( x\right) :=\sum\limits_{j=1}^{k}\beta_{Nj}\dfrac{1}{B\left(
Nk-j,j\right) a_{N}}\left( \frac{x}{a_{N}}\right) _{+}^{j-1}\left(
1-\frac{x}{a_{N}}\right) _{+}^{Nk-j-1},$$ – see Theorem \[cxshad\]. Because $\mathcal{T}_{k}p_{N}\left( x\right) =0$ is a polynomial equation it holds for all $x\neq0,a_{N}$. The piecewise polynomial $p_{N}$ has coefficients which are symmetric in $a_{1}
,\ldots,a_{N-1}$ – see equation (\[kparfrac\]). Hence the differential equation is symmetric in $\left( a_{1},\ldots,a_{N}\right)$ and each piece $$p_{i}\left( x\right) :=\sum\limits_{j=1}^{k}\beta_{ij}\dfrac{1}{B\left(
Nk-j,j\right) a_{i}}\left( \frac{x}{a_{i}}\right) _{+}^{j-1}\left(
1-\frac{x}{a_{i}}\right) _{+}^{Nk-j-1}$$ satisfies the differential equation for $x\neq0,a_{i}$. The density is the sum $\sum_{i=1}^{N}p_{i}$ thus $\mathcal{T}_{k}f\left( x\right) =0$ at each $x\notin\left\{ a_{1},\ldots,a_{N}\right\} $. The density $f_{c}\left(
x\right) $ of $\mathcal{D}\left(
a_{1}+c,\ldots,a_{N}+c;k,\ldots,k\right) $ equals the translate $f\left( x-c\right) $ and the differential operator $\mathcal{T}_{k}$ has a corresponding translation property and thus the restriction $a_{1}>0$ can be removed.
If $k>0$ then the density $f\left( x\right) $ of $\mathcal{D}\left(
a_{1},\ldots,a_{N};k,\ldots,k\right) $ satisfies the linear differential equation $\mathcal{T}_{k}f\left( x\right) =0$ at all $x\notin\left\{
a_{1},\ldots,a_{N}\right\} $.
Suppose $a_{M}<x_{1}<x_{2}<a_{M+1}$. The probability $\Pr\left\{
x_{1}<X<x_{2}\right\} $ is given by a definite integral with respect to an integrand which is analytic for $\operatorname{Re}k>0$ over a polytope in $\mathbb{T}_{N-1}$ whose vertices are independent of $k$ and analytic in $x_{1},x_{2}$ – see Remark \[polyt\]. Thus the distribution function $F\left( x\right) $ at $x$ is analytic for $\operatorname{Re}k>0$ and extends to an analytic function in $x$ for $x_{1}<\operatorname{Re} x <x_{2},\left\vert \operatorname{Im}x\right\vert <\varepsilon$ for some $\varepsilon>0$. The differential equation $\mathcal{T}_{k}\frac{\partial
}{\partial x}F\left( x\right) =0$ is satisfied for each $k=1,2,3,\ldots$, this is an analytic relation and extends to all $\operatorname{Re}k>0$ by Carlson’s theorem (see Henrici [@H vol.2,p.334]).
We can now assert the validity of the equation for $k=\frac{1}{2},\frac{3}{2},\ldots$ which applies to real shadows or the repeated eigenvalue case (each is repeated 3 times, or 5 times, etc.). It is not clear what happens if just one eigenvalue is repeated, note that the main result used symmetric functions of $\frac{1}{a_{i}-a_{N}}$. It is plausible that the equation applies in intervals adjacent to simple (non-repeated) eigenvalues.
The case $k=\frac{1}{2}$ is of special interest since it applies to the real shadow when the eigenvalues are pairwise distinct. The equation $\mathcal{T}_{1/2}f\left( x\right) =0$ is$$P_{N}\left( x\right) \partial^{N-1}f\left( x\right) +\sum_{j=1}^{N-1}\left( -1\right) ^{j}\frac{N-j}{N}\frac{\left( -N/2\right) _{j}}{j!}\partial^{j}P_{N}\left( x\right) \partial^{N-1-j}f\left( x\right) =0.$$ When $N$ is even then the terms $\partial^{m}f\left( x\right) $ for $0\leq
m\leq\frac{N}{2}-2$ drop out, that is any polynomial of degree $\frac{N}{2}-2$ satisfies the equation. This property will be made precise in the next section.
The *indicial equation* is important because it provides information about the solutions in neighborhoods of the knots, that is, the solutions have the form$$\sum_{n=0}^{\infty}\gamma_{n}\left( x-a_{j}\right) ^{n+c},\sum_{n=0}^{\infty}\gamma_{n}\left( a_{j}-x\right) ^{n+c},$$ (depending on whether the solution is valid for $x>a_{j}$ or $x<a_{j}$) where $c$ is a solution of the indicial equation: this comes from the coefficient of the lowest power in $\mathcal{T}_{k}\left( a_{N}-x\right) ^{c}$ from equation (\[Tkxc\]), namely$$e_{N-1}A\left( N,k,1,c\right) =\left( -1\right) ^{N-1}e_{N-1}\left(
-c\right) _{N-2}\left( c+1-\left( N-1\right) k\right) =0.$$ The solutions, called *critical exponents*, are $c=0,1,\ldots
,N-3,\left( N-1\right) k-1$. In the real shadow situation with $k=\frac
{1}{2}$ we see there are two different types: when $N=2m+1$ the critical exponent $c=m-1$ is repeated which leads to a logarithmic solution: $\sum_{n=0}^{\infty}$ $\gamma_{n}\left( x-a_{j}\right) ^{m-1+n}$ and $\log\left\vert x-a_{j}\right\vert \sum_{n=0}^{\infty}$ $\gamma_{n}^{\prime
}\left( x-a_{j}\right) ^{m-1+n}$. This actually occurs, as will be shown in the sequel.
The real shadow {#sec:real-shadow}
===============
We will use heuristic extrapolation to postulate a set of formulas for the real shadow (\[realshad\]) – the density of $\mathcal{D}\left(
a_{1},\ldots,a_{N};\frac{1}{2},\ldots,\frac{1}{2}\right)$. In the notation of Theorem \[shadeq1\] there is a set of functions $p_{j}\left( x\right)
$, with a symmetry property, such that the density $f\left( x\right)
=\sum_{j=m}^{N}p_{j}\left( x\right) $ in the interval $\left( a_{m-1},a_{m}\right) $. It is straightforward to do this in the top interval $\left( a_{N-1},a_{N}\right)$ but the expression involves square roots of quantities that become negative for $x<a_{N-1}$. The idea is to adopt certain branches of the complex square roots which give plausible results and then to prove the validity of the postulated formulas. This will be done by using complex contour integration to verify the known moment generating function.
We begin by pointing out that the expression for the density in $\left(
a_{N-1},a_{N}\right) $ found in Proposition \[topF\] is a multiple infinite series which diverges for $\left\vert x-a_{N}\right\vert >a_{N}-a_{N-1}$, not an easy expression to evaluate. We can replace it by a one-variable (definite) integral when $k=\frac{1}{2}$. Suppose the series $g\left( r\right)
=\sum_{n=0}^{\infty}\gamma_{n}r^{n}$ converges for $\left\vert r\right\vert
\leq1$ then$$\frac{1}{B\left( \frac{1}{2},\frac{N}{2}-1\right) }\int_{0}^{1}\sum
_{n=0}^{\infty}\gamma_{n}t^{n}t^{-1/2}\left( 1-t\right) ^{N/2-2}dt=\sum_{n=0}^{\infty}\frac{\left( \frac{1}{2}\right) _{n}}{\left(
\frac{N-1}{2}\right) _{n}}\gamma_{n}.$$ Apply this to $$g\left( r\right) ={\textstyle\prod_{j=1}^{N-1}}
\left( 1-\frac{a_{N}-x}{a_{N}-a_{j}}r\right) ^{-\frac{1}{2}}=\sum_{\alpha
\in\mathbb{N}_{0}^{N-1}}\prod_{j=1}^{N-1}\frac{\left( k_{i}\right)
_{\alpha_{j}}}{\alpha_{j}!}\left( \frac{a_{N}-x}{a_{N}-a_{j}}\right)
^{a_{j}}r^{\left\vert \alpha\right\vert }$$ and use the formula for the density from Proposition \[topF\] and act with $-\frac{\partial}{\partial x}$ on $1-F\left( x\right)$ to obtain the density for $x \in (a_{N-1},a_{N})$, $$\begin{split}
f\left( x\right) = &\frac{N-2}{2\pi\left( a_{N}-x\right) }\prod_{j=1}^{N-1}\left( \frac{a_{N}-x}{a_{N}-a_{j}}\right) ^{\frac{1}{2}} \times \\
& \times \int_{0}^{1}\prod_{j=1}^{N-1}\left( 1-\frac{a_{N}-x}{a_{N}-a_{j}}t\right)
^{-\frac{1}{2}}t^{-1/2}\left( 1-t\right) ^{N/2-2}dt.
\end{split}$$ Note that $B\left( \frac{1}{2},\frac{N}{2}-1\right) B\left( \frac{1}{2},\frac{N-1}{2}\right) =\frac{\Gamma\left( 1/2\right) ^{2}\Gamma\left(
N/2-1\right) }{\Gamma\left( N/2\right) }=\frac{\pi}{N/2-1}$. Make the change of variable $s=a_{N}-t\left( a_{N}-x\right) $, then$$f\left( x\right) =\frac{N-2}{2\pi}
\int_{x}^{a_{N}}\left( a_{N}-s\right)
^{-\frac{1}{2}}\prod_{j=1}^{N-1}\left( s-a_{j}\right) ^{-\frac{1}{2}}\left(
s-x\right) ^{\frac{N}{2}-2}ds.$$ Suppose we want to interpret this integral for $a_{N-2}<x<a_{N-1}$ then we must pick a branch of $\left( s-a_{N-1}\right) ^{-\frac{1}{2}}$, that is we need to choose the sign in $\left( s-a_{N-1}\right) ^{-\frac{1}{2}}=\pm\mathrm{i}\left( a_{N-1}-s\right) ^{-\frac{1}{2}}$ , where $\mathrm{i}=\sqrt{-1}$. Denote the integral by $f_{N}\left( x\right) $. Using the symmetry heuristics we define $$f_{N-1}\left( x\right) =\frac{N-2}{2\pi}\int_{x}^{a_{N-1}}\left(
a_{N-1}-s\right) ^{-\frac{1}{2}}\prod_{j=1,j\neq N-1}^{N}\left(
s-a_{j}\right) ^{-\frac{1}{2}}\left( s-x\right) ^{\frac{N}{2}-2}ds,$$ now we need to pick a branch for $\left( s-a_{N}\right) ^{-\frac{1}{2}}$ for $s<a_{N}$. The requirement that $f_{N}\left( x\right) +f_{N-1}\left(
x\right) $ be real for $a_{N-2}<x<a_{N-1}$ motivates the following:
1. For $0\leq j<N$ and $a_{1}<x\leq a_{N-j}$ let$$\begin{split}
f_{N-j}\left( x\right) = & \frac{N-2}{2\pi}\mathrm{i}^{j}\int_{x}^{a_{N-j}}\bigg( \prod\limits_{m=0}^{j} \left( a_{N-m}-s\right) ^{-\frac{1}{2}}
\times
\\& \times \prod\limits_{m=j+1}^{N-1}\left( s-a_{N-m}\right)
^{-\frac{1}{2}}\left(s-x\right) ^{\frac{N}{2}-2}\bigg)ds,
\end{split}$$
2. for $0\leq j\leq N-2$ and $a_{N-j-1}\leq x<a_{N-j}$ the density is$$f\left( x\right) =\sum_{m=0}^{j}f_{N-m}\left( x\right) ,$$
3. if $s<a_{m}$ then $\left( s-a_{m}\right) ^{-\frac{1}{2}}=-\mathrm{i}\left( a_{m}-s\right) ^{-\frac{1}{2}}$ for $2\leq m\leq N-1.$
Suppose $a_{1}<x<a_{N-j-1}$ for some $j\geq 0$. As a consequence we obtain then $$\begin{split}
& f_{N-j}\left( x\right) +f_{N-j-1}\left( x\right) = \\
& \frac{N-2}{2\pi
}\mathrm{i}^{j}\int_{a_{N-j-1}}^{a_{N-j}}\prod\limits_{m=0}^{j}\left(
a_{N-m}-s\right) ^{-\frac{1}{2}}\prod\limits_{m=j+1}^{N-1}\left(
s-a_{N-m}\right) ^{-\frac{1}{2}}\left( s-x\right) ^{\frac{N}{2}-2}ds\\
& +\frac{N-2}{2\pi}\mathrm{i}^{j}\int_{x}^{a_{N-j-1}}\prod\limits_{m=0}^{j}\left( a_{N-m}-s\right) ^{-\frac{1}{2}}\prod\limits_{m=j+1}^{N-1}\left(
s-a_{N-m}\right) ^{-\frac{1}{2}}\left( s-x\right) ^{\frac{N}{2}-2}ds\\
& +\frac{N-2}{2\pi}\mathrm{i}^{j+1}\int_{x}^{a_{N-j-1}}\prod\limits_{m=0}^{j+1}\left( a_{N-m}-s\right) ^{-\frac{1}{2}}\prod\limits_{m=j+2}^{N-1}\left( s-a_{N-m}\right) ^{-\frac{1}{2}}\left( s-x\right) ^{\frac
{N}{2}-2}ds.
\end{split}$$ Due to equation (\[dirinteg\]) the factor $\left( s-a_{N-j-1}\right) ^{-\frac{1}{2}}$ in the second integral is replaced by $-\mathrm{i}\left( a_{N-j-1}-s\right) ^{-\frac{1}{2}}$. Therefore the second and third integrals cancel out as $\left(-\mathrm{i}\right) \mathrm{i}^{j}+\mathrm{i}^{j+1}=0$. Hence there are two different types of expressions for the density, depending on whether $a_{N-2M}<x<a_{N-2M+1}$ or $a_{N-2M-1}<x<a_{N-2M}$. For $0\leq j\leq
\left\lfloor \frac{N-2}{2}\right\rfloor $ let$$g_{j}\left( s\right) :=\prod\limits_{m=0}^{2j}\left( a_{N-m}-s\right)
^{-\frac{1}{2}}\prod\limits_{m=2j+1}^{N-1}\left( s-a_{N-m}\right)
^{-\frac{1}{2}},$$ then for $a_{N-2M}\leq x<a_{N-2M+1}$ (with $1\leq M\leq\frac{N-1}{2}$) $$f\left( x\right) =\frac{N-2}{2\pi}\sum_{j=0}^{M-1}\left( -1\right)
^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left( s-x\right)
^{\frac{N}{2}-2}ds,\label{evenF}$$ and for $a_{N-2M-1}\leq x<a_{N-2M}$ (with $0\leq M\leq\frac{N-2}{2}$)$$\begin{aligned}
f\left( x\right) & =\frac{N-2}{2\pi}\sum_{j=0}^{M-1}\left( -1\right)
^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left( s-x\right)
^{\frac{N}{2}-2}ds\label{oddF}\\
& +\left( -1\right) ^{M}\frac{N-2}{2\pi}\int_{x}^{a_{N-2M}}g_{M}\left(
s\right) \left( s-x\right) ^{\frac{N}{2}-2}ds.\nonumber\end{aligned}$$ An important consequence of this formulation is that for even $N$ the density is a polynomial of degree $\frac{N}{2}-2$ on the *even* intervals $\left( a_{N-2M},a_{N-2M+1}\right)$, which means that the parity by counting intervals from the top down is even, so that $\left( a_{N-1},a_{N}\right) $ is \#1.
Now we are in position to formulate the main result of this work.
\[th:main-theorem\] For $N>2$ the formulas (\[evenF\]) and (\[oddF\]) give the real shadow of a real symmetric matrix with spectrum $\{a_i\}_{i=1}^N$ – the density of $\mathcal{P}^{\mathbb{R}}_{\mathrm{diag}(a_{1},\ldots,a_{N})} = \mathcal{D}\left( a_{1},\ldots,a_{N};\frac{1}{2},\ldots,\frac
{1}{2}\right) $.
We prove the validity of the above theorem by showing that$$\int_{a_{1}}^{a_{N}}\left( 1-r\left( x-a_{1}\right) \right) ^{-\frac{N}{2}}f\left( x\right) dx=\prod_{j=2}^{N}\left( 1-r\left( a_{j}-a_{1}\right) \right) ^{-\frac{1}{2}},\left\vert r\right\vert <\frac
{1}{a_{N}-a_{1}},$$ this is the moment generating function, see Lemma \[mgenF\]. Start by expressing $\int_{a_{1}}^{a_{N}}\left( x-a_{1}\right) ^{n}f\left( x\right)
dx$ as a sum of integrals, for $n=0,1,2,\ldots$. The contribution of an even interval $a_{N-2M}\leq x\leq
a_{N-2M+1}$ to the moment is$$\frac{N-2}{2\pi}\sum_{j=0}^{M-1}\left( -1\right) ^{j}\int_{a_{N-2M}}^{a_{N-2M+1}}\left( x-a_{1}\right) ^{n}dx\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left( s-x\right) ^{\frac{N}{2}-2}ds,$$ and the contribution of an odd interval $a_{N-2M-1}\leq x\leq a_{N-2M}$ is$$\begin{split}
& \frac{N-2}{2\pi}\sum_{j=0}^{M-1}\left( -1\right) ^{j}\int_{a_{N-2M-1}}^{a_{N-2M}}\left( x-a_{1}\right) ^{n}dx\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left( s-x\right) ^{\frac{N}{2}-2}ds\\
& +\left( -1\right) ^{M}\frac{N-2}{2\pi}\int_{a_{N-2M-1}}^{a_{N-2M}}\left(
x-a_{1}\right) ^{n}dx\int_{x}^{a_{N-2M}}g_{M}\left( s\right) \left(
s-x\right) ^{\frac{N}{2}-2}ds.
\end{split}$$ The term $g_{j}\left( s\right) \left( s-x\right) ^{\frac{N}{2}-2}$ appears in the intervals $a_{N-2M}\leq x\leq a_{N-2M+1}$ for $M\geq j+1$ and in $a_{N-2M-1}\leq x\leq a_{N-2M}$ for $M\geq j$. Collect these terms:$$\begin{split}
&
\!\!\!
\frac{N-2}{2\pi}\left( -1\right)^{j}
\!\!\!
\int_{a_{N-2j-1}}^{a_{N-2j}}
\!\!\!\!\!\!\!\!\!\!\!\!
g_{2j}
\!
\left( s\right)
\!
\left\{ \sum_{M=j+1}^{\left\lfloor \frac{N-1}{2}\right\rfloor }
\!
\int_{a_{N-2M}}^{a_{N-2M+1}}
\!\!\!\!
+
\!\!
\sum_{M=j+1}^{\left\lfloor
\frac{N-2}{2}\right\rfloor}
\!
\int_{a_{N-2M-1}}^{a_{N-2M}}\right\}
\!\!
\left(x-a_{1}\right)^{n}
\!
\left(s-x\right) ^{\frac{N}{2}-2}dxds\\
& +\frac{N-2}{2\pi}\left( -1\right) ^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}\left(
x-a_{1}\right) ^{n}dx\int_{x}^{a_{N-2j}}g_{j}\left( s\right) \left(
s-x\right) ^{\frac{N}{2}-2}ds.
\end{split}$$ The terms in the first line add up to just one interval of integration $a_{1}\leq x\leq a_{N-2j-1}$. In the second line reverse the order of integration (note the region for the double integral is $a_{N-2j-1}\leq x\leq
s\leq a_{N-2j}$) to obtain$$\frac{N-2}{2\pi}\left( -1\right) ^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) ds\int_{a_{N-2j-1}}^{s}\left( x-a_{1}\right)
^{n}\left( s-x\right) ^{\frac{N}{2}-2}dx.$$ The terms with $g_{j}$ add up to $$\begin{split}
& \frac{N-2}{2\pi}\left( -1\right) ^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) ds\int_{a_{1}}^{s}\left( x-a_{1}\right) ^{n}\left(
s-x\right) ^{\frac{N}{2}-2}dx\\
& =\frac{\left( -1\right) ^{j}}{\pi}\left( \frac{N-2}{2}\right) B\left(
\frac{N}{2}-1,n+1\right) \int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right)
\left( s-a_{1}\right) ^{\frac{N}{2}+n-1}ds,
\end{split}$$ from the Beta integral $\int_{a}^{b}\left( b-x\right) ^{\alpha-1}\left(
x-a\right) ^{\beta-1}dx=\left( b-a\right) ^{\alpha+\beta-1}B\left(
\alpha,\beta\right)$ with $a=a_{1},b=s,\alpha=\frac{N}{2}-1,\beta=n+1$. Furthermore $\left( \frac{N-2}{2}\right) B\left( \frac{N}{2}-1,n+1\right)
=\dfrac{n!}{\left( \frac{N}{2}\right) _{n}}$. Therefore we have $$\int_{a_{1}}^{a_{N}}\left( x-a_{1}\right) ^{n}f\left( x\right) dx=\frac
{1}{\pi}\frac{n!}{\left( \frac{N}{2}\right) _{n}}\sum_{j=0}^{\left\lfloor
\frac{N-2}{2}\right\rfloor }\left( -1\right) ^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left( s-a_{1}\right) ^{\frac{N}{2}+n-1}ds,$$ and$$\begin{gathered}
\nonumber
\int_{a_{1}}^{a_{N}}\left( 1-r\left( x-a_{1}\right) \right) ^{-\frac{N}{2}}f\left( x\right) dx=\sum_{n=0}^{\infty}\frac{\left( \frac{N}{2}\right)
_{n}}{n!}r^{n}\int_{a_{1}}^{a_{N}}\left( x-a_{1}\right) ^{n}f\left(
x\right) dx\\
=\frac{1}{\pi}\sum_{j=0}^{\left\lfloor \frac{N-2}{2}\right\rfloor }\left(
-1\right) ^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left(
s-a_{1}\right) ^{\frac{N}{2}-1}\left( 1-r\left( s-a_{1}\right) \right)
^{-1}ds,\end{gathered}$$ where the infinite sum converges for $\left\vert r\right\vert <\frac{1}{a_{N}-a_{1}}$. We will evaluate the integral by residue calculus applied to the analytic function $$\label{eqn:func-G}
G\left( z\right) :=\prod\limits_{j=1}^{N}\left( z-a_{j}\right) ^{-\frac
{1}{2}}
\left( z-a_{1}\right) ^{\frac{N}{2}-1}\left( 1-r\left(
z-a_{1}\right) \right) ^{-1}$$ for fixed small $r>0$ with suitable determination of the square roots. For real $a,b$ with $a<b$ consider the analytic function $\left( z-a\right)
^{-\frac{1}{2}}\left( z-b\right) ^{-\frac{1}{2}}$ defined on $\mathbb{C}\backslash\left[ a,b\right] $, that is, the complex plane with the interval $\left[ a,b\right] $ removed. Set $z=a+r_{1}e^{\mathrm{i}\theta_{1}}=b+r_{2}e^{\mathrm{i}\theta_{2}}$, $r_{1},r_{2}>0$ and $\theta_{1}=\theta
_{2}=0$ for $z$ real and $z>b$, then let$$\left( z-a\right) ^{-\frac{1}{2}}\left( z-b\right) ^{-\frac{1}{2}}:=\left( r_{1}r_{2}\right) ^{-\frac{1}{2}}e^{-\mathrm{i}\left( \theta
_{1}+\theta_{2}\right) /2}$$ and let $\theta_{1},\theta_{2}$ vary continuously (from $0$) to determine the values in the rest of the domain. This is well-defined: suppose $z$ is real and $z<a$; approaching $z$ from the upper half-plane $\theta_{1},\theta_{2}$ change from $0$ to $\pi$ and $e^{-\mathrm{i}\left( \theta_{1}+\theta
_{2}\right) /2}$ changes from $1$ to $e^{-\mathrm{i}\pi}=-1$, and approaching $z$ from the lower half-plane $\theta_{1},\theta_{2}$ change from $0$ to $-\pi$ and $e^{-\mathrm{i}\left( \theta_{1}+\theta_{2}\right) /2}$ changes from $1$ to $e^{\mathrm{i}\pi}=-1$.
Suppose $h$ is analytic in a complex neighborhood of $\left[ a,b\right] $ and $\gamma_{\varepsilon}$ is a closed contour oriented clockwise (negatively) made up of the segments $\left\{ x+\mathrm{i}\varepsilon:a\leq x\leq
b\right\} $, $\left\{ x-\mathrm{i}\varepsilon:a\leq x\leq b\right\} $ and semicircles $\left\{ a+\varepsilon e^{\mathrm{i}\theta}:\frac{\pi}{2}\leq\theta\leq\frac{3\pi}{2}\right\} $, $\left\{ b+\varepsilon
e^{\mathrm{i}\theta}:-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}\right\} $ (for sufficiently small $\varepsilon>0$) then$$\lim_{\varepsilon\rightarrow0_{+}}\oint_{\gamma_\epsilon} h\left( z\right)
\left( z-a\right)
^{-\frac{1}{2}}\left( z-b\right) ^{-\frac{1}{2}}dz=-2\mathrm{i}\int_{a}^{b}h\left( x\right) \left( \left( b-x\right) \left( a-x\right)
\right) ^{-\frac{1}{2}}dx.$$
On the semicircles the integrand is bounded by $M\varepsilon^{-\frac{1}{2}}$ for some $M<\infty$ and the length of the arc is $\pi\varepsilon$ so this part of the integral tends to zero as $\varepsilon\rightarrow0_{+}$. Along $\left\{ z=x+\mathrm{i}\varepsilon:a\leq x\leq b\right\} $ the arguments are $\theta_{1}\approx\pi$ and $\theta_{2}\approx0$ so $\left( z-a\right)
^{-\frac{1}{2}}\left( z-b\right) ^{-\frac{1}{2}}\approx e^{-\mathrm{i}\pi
/2}\left( r_{1}r_{2}\right) ^{-\frac{1}{2}}$ and this part of the integral $\approx-\mathrm{i}\int_{a}^{b}h\left( x+\mathrm{i}\varepsilon\right)
\left( \left( b-x\right) \left( a-x\right) \right) ^{-\frac{1}{2}}dx$. Along $\left\{ z=x-\mathrm{i}\varepsilon:a\leq x\leq b\right\} $ the arguments are $\theta_{1}\approx-\pi$ and $\theta_{2}\approx0$ so $\left(
z-a\right) ^{-\frac{1}{2}}\left( z-b\right) ^{-\frac{1}{2}}\approx
e^{\mathrm{i}\pi/2}\left( r_{1}r_{2}\right) ^{-\frac{1}{2}}$ and this part of the integral $\approx\mathrm{i}\int_{b}^{a}h\left( x-\mathrm{i}\varepsilon\right) \left( \left( b-x\right) \left( a-x\right) \right)
^{-\frac{1}{2}}dx$. Adding the two pieces and letting $\varepsilon
\rightarrow0_{+}$ proves the claim.
Now fix $r>0$ with $\frac{1}{r}>\max\left( \left\vert a_{1}\right\vert
,\left\vert a_{N}\right\vert ,a_{N}-a_{1}\right) $. Define a positively oriented closed contour $\Gamma$ consisting of a large circle $\gamma=\left\{
z=\operatorname{Re}^{\mathrm{i}\theta}:0\leq\theta\leq2\pi\right\} $ with $R>\frac{1}{r}$ and $\left\{ \gamma_{j,\varepsilon}:0\leq j\leq\left\lfloor
\frac{N-2}{2}\right\rfloor \right\} $ where $\gamma_{j,\varepsilon}$ is a closed negatively oriented contour around the interval $\left[ a_{N-2j-1},a_{N-2j}\right] $ as in the Lemma, with $\varepsilon>0$ sufficiently small so that the contours do not intersect – see Fig. \[fig:poles\]. The function $G$ is meromorphic on $\mathbb{C}\backslash\cup_{j=0}^{\left\lfloor \left( N-2\right)
/2\right\rfloor }\left[ a_{N-2j-1},a_{N-2j}\right] $ and has one simple pole at $z=a_{1}+\frac{1}{r}$. By the (generalized) residue theorem$$\label{eqn:complex-integral}
\frac{1}{2\pi\mathrm{i}}\oint\limits_{\Gamma}G\left( z\right)
dz=\mathrm{res}_{z=a_{1}+\frac{1}{r}}G\left( z\right) .$$ Using the determinations of roots described above let $z=a_{j}+r_{j}e^{\mathrm{i}\theta_{j}}$ for $1\leq j\leq N$ with $r_{j}>0$. For large $\left\vert z\right\vert $ we see $\left\vert G\left( z\right) \right\vert
<M\left\vert z\right\vert ^{-2}$ so the integral around $\gamma$ (circle with radius $R)$ tends to zero as $R\rightarrow\infty$. Consider $N$ even or odd separately.
\[fig:int-even\] \[fig:int-odd\]
Case of odd $N$:
----------------
The interval with the lowest index is $\left[ a_{2},a_{3}\right]$ and the analytic function $G\left( z\right) =\prod\limits_{j=2}^{N}\left( z-a_{j}\right) ^{-\frac
{1}{2}}\left( z-a_{1}\right) ^{\frac{N-3}{2}}\left( 1-r\left(
z-a_{1}\right) \right) ^{-1}$. Here $\frac{N-3}{2}$ is an integer thus $\left( z-a_{1}\right) ^{\frac{N-3}{2}}$ is entire. Applying the Lemma to $\gamma_{j,\varepsilon}$ put$$h\left( z\right) =\prod\limits_{m=0}^{2j-1}\left( z-a_{N-m}\right)
^{-\frac{1}{2}}\prod\limits_{m=2j+2}^{N-2}\left( z-a_{N-m}\right)
^{-\frac{1}{2}}\left( z-a_{1}\right) ^{\frac{N-3}{2}}\left( 1-r\left(
z-a_{1}\right) \right) ^{-1}.$$ In this case $\theta_{m}=\pi$ for $N-2j+1\leq m\leq N$ and $\theta_{m}=0$ for $1\leq m\leq N-2j-2$ so for $a_{N-2j-1}\leq x\leq a_{N-2j}$ we have $$\begin{split}
h\left( x\right) = & e^{-\mathrm{i}\left( 2j\pi\right) /2}\prod
\limits_{m=0}^{2j-1}\left( a_{N-m}-x\right) ^{-\frac{1}{2}} \times \\
& \times\prod
\limits_{m=2j+2}^{N-2}\left( x-a_{N-m}\right) ^{-\frac{1}{2}}\left(
x-a_{1}\right) ^{\frac{N-3}{2}}\left( 1-r\left( x-a_{1}\right) \right)
^{-1}\end{split}$$ and$$\begin{split}
& \lim_{\varepsilon\rightarrow0_{+}}\oint\limits_{\gamma_{j,\varepsilon}}G\left( z\right) dz
=-2\mathrm{i}\left( -1\right) ^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}\bigg( \prod\limits_{m=0}^{2j}\left( a_{N-m}-x\right) ^{-\frac{1}{2}} \\& \prod\limits_{m=2j+1}^{N-2}\left( x-a_{N-m}\right) ^{-\frac{1}{2}}\left(
x-a_{1}\right) ^{\frac{N-3}{2}}\left( 1-r\left( x-a_{1}\right) \right)
^{-1} \bigg) dx.
\end{split}$$ The residue at $z=a_{1}+\frac{1}{r}$ is straightforward: $$\begin{split}
\lim_{z\rightarrow a_{1}+\frac{1}{r}}\left( z-a_{1}-\frac{1}{r}\right)
G\left( z\right) & =-\frac{1}{r}\prod\limits_{j=2}^{N}\left( \frac{1}{r}-\left( a_{j}-a_{1}\right) \right) ^{-\frac{1}{2}}\left( \frac{1}{r}\right) ^{\frac{N-3}{2}}\\
& =-\prod\limits_{j=2}^{N}\left( 1-r\left( a_{j}-a_{1}\right) \right)
^{-\frac{1}{2}},
\end{split}$$ because $\frac{1}{r}>a_{N}$ and the determination of the roots gives positive values. Thus in the limit as $\varepsilon\rightarrow0_{+},R\rightarrow\infty$ we obtain$$\begin{aligned}
\frac{1}{2\pi\mathrm{i}}\sum_{j=0}^{\frac{N-3}{2}}\left( -1\right)
^{j}\left( -2\mathrm{i}\right) \int_{a_{N-2j-1}}^{a_{N-2j}}g_{2j}\left(
x\right) \left( x-a_{1}\right) ^{\frac{N}{2}-1}\left( 1-r\left(
x-a_{1}\right) \right) ^{-1}dx= \nonumber \\
-\prod\limits_{j=2}^{N}\left( 1-r\left(
a_{j}-a_{1}\right) \right) ^{-\frac{1}{2}},\end{aligned}$$ and this is the required result.
Case of $N$ even
----------------
The interval with the lowest index is $\left[ a_{1},a_{2}\right] $ and the function $G\left( z\right) =\prod\limits_{j=1}^{N}\left( z-a_{j}\right) ^{-\frac
{1}{2}}\left( z-a_{1}\right) ^{\frac{N-2}{2}}\left( 1-r\left(
z-a_{1}\right) \right) ^{-1}$. Here $\frac{N-2}{2}$ is an integer so $\left( z-a_{1}\right) ^{\frac{N-2}{2}}$ is entire. Applying the Lemma to $\gamma_{j,\varepsilon}$ put$$h\left( z\right) =\prod\limits_{m=0}^{2j-1}\left( z-a_{N-m}\right)
^{-\frac{1}{2}}\prod\limits_{m=2j+2}^{N-1}\left( z-a_{N-m}\right)
^{-\frac{1}{2}}\left( z-a_{1}\right) ^{\frac{N-2}{2}}\left( 1-r\left(
z-a_{1}\right) \right) ^{-1}.$$ In this case $\theta_{m}=\pi$ for $N-2j+1\leq m\leq N$ and $\theta_{m}=0$ for $1\leq m\leq N-2j-2$ so for $a_{N-2j-1}\leq x\leq a_{N-2j}$ we have $$\begin{split}
h\left( x\right) = & e^{-\mathrm{i}\left( 2j\pi\right) /2}\prod
\limits_{m=0}^{2j-1}\left( a_{N-m}-x\right) ^{-\frac{1}{2}} \times \\
& \times \prod \limits_{m=2j+2}^{N-1}\left( x-a_{N-m}\right)
^{-\frac{1}{2}}\left( x-a_{1}\right) ^{\frac{N-2}{2}}\left( 1-r\left(
x-a_{1}\right) \right)^{-1}\end{split}$$ and$$\begin{split}
& \lim_{\varepsilon\rightarrow0_{+}}\oint\limits_{\gamma_{j,\varepsilon}}G\left( z\right) dz
=-2\mathrm{i}\left( -1\right) ^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}\bigg( \prod\limits_{m=0}^{2j}\left( a_{N-m}-x\right) ^{-\frac{1}{2}} \times
\\& \times \prod\limits_{m=2j+1}^{N-1}\left( x-a_{N-m}\right)
^{-\frac{1}{2}}\left(
x-a_{1}\right) ^{\frac{N-2}{2}}\left( 1-r\left( x-a_{1}\right) \right)
^{-1} \bigg) dx.
\end{split}$$ The residue at $z=a_{1}+\frac{1}{r}$ is: $$\begin{split}
\lim_{z\rightarrow a_{1}+\frac{1}{r}}\left( z-a_{1}-\frac{1}{r}\right)
G\left( z\right) & =-\frac{1}{r}\prod\limits_{j=1}^{N}\left( \frac{1}{r}-\left( a_{j}-a_{1}\right) \right) ^{-\frac{1}{2}}\left( \frac{1}{r}\right) ^{\frac{N-2}{2}}\\
& =-\prod\limits_{j=2}^{N}
\big( 1-r\left( a_{j}-a_{1}\right) \big)^{-\frac{1}{2}},
\end{split}$$ because $\frac{1}{r}>a_{N}$ and the determination of the roots gives positive values. Thus in the limit as $\varepsilon\rightarrow0_{+},R\rightarrow\infty$ we obtain the final result $$\begin{aligned}
\frac{1}{2\pi\mathrm{i}}\sum_{j=0}^{\frac{N-2}{2}}\left( -1\right)
^{j}\left( -2\mathrm{i}\right) \int_{a_{N-2j-1}}^{a_{N-2j}}g_{2j}\left(
x\right) \left( x-a_{1}\right) ^{\frac{N}{2}-1}
\big( 1-r\left(x-a_{1}\right)
\big)^{-1}ds= \nonumber \\
-\prod\limits_{j=2}^{N}\big( 1-r\left(
a_{j}-a_{1}\right) \big) ^{-\frac{1}{2}}.\end{aligned}$$
For distributions supported by bounded intervals the moment generating function determines the distribution uniquely. Thus we have established the Theorem \[th:main-theorem\].
Examples
--------
There is a somewhat disguised complete elliptic integral of the first kind which appears in $N=3,4,5$. For $b_{1}<b_{2}<b_{3}<b_{4}$ let$$E\left( b_{1},b_{2};b_{3},b_{4}\right) :=\frac{1}{\pi}\int_{b_{3}}^{b_{4}}\left\{ \left( b_{4}-s\right) \left( s-b_{3}\right) \left(
s-b_{2}\right) \left( s-b_{1}\right) \right\} ^{-\frac{1}{2}}ds.$$ There is a hypergeometric formulation (see formula (\[N3&2F1\]) with $k=\frac{1}{2}$):$$E\left( b_{1},b_{2};b_{3},b_{4}\right) :=\frac{1}{\sqrt{\left( b_{3}-b_{1}\right) \left( b_{4}-b_{2}\right) }}~_{2}F_{1}\left(
\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};\dfrac{\left( b_{4}-b_{3}\right) \left( b_{2}-b_{1}\right) }{\left(
b_{3}-b_{1}\right) \left( b_{4}-b_{2}\right) }\right) .$$ Consider the density for $N=3$ and $a_{1}<a_{2}<x<a_{3}$; by formula (\[oddF\])$$\begin{aligned}
f\left( x\right) & =\frac{1}{2\pi}\int_{x}^{a_{3}}\left( a_{3}-s\right)
^{-\frac{1}{2}}\prod_{j=1}^{2}\left( s-a_{j}\right) ^{-\frac{1}{2}}\left(
s-x\right) ^{-\frac{1}{2}}ds \\
& =\frac{1}{2}E\left( a_{1},a_{2};x,a_{3}\right) .\nonumber \end{aligned}$$ Similarly formula (\[evenF\]) shows that $f\left( x\right) =\frac{1}{2}E\left( a_{1},x;a_{2},a_{3}\right) $ for $a_{1}<x<a_{2}$.
Suppose $N=4$ and $a_{2}<x\leq a_{3}$ then by formula (\[evenF\])$$\begin{aligned}
f\left( x\right) & =\frac{1}{\pi}\int_{a_{3}}^{a_{4}}\left(
a_{4}-s\right) ^{-\frac{1}{2}}\prod\limits_{j=1}^{3}\left( s-a_{j}\right)
^{-\frac{1}{2}}ds=f\left( a_{3}\right) \\
& =E\left( a_{1},a_{2};a_{3},a_{4}\right) \nonumber\end{aligned}$$ and the density is constant on this interval.
Suppose $N=5$ and $a_{3}\leq x<a_{4}$ then by formula (\[evenF\])$$\begin{aligned}
f\left( x\right) & =\frac{3}{2\pi}\int_{a_{4}}^{a_{5}}\left(
a_{5}-s\right) ^{-\frac{1}{2}}\prod\limits_{j=1}^{4}\left( s-a_{j}\right)
^{-\frac{1}{2}}\left( s-x\right) ^{\frac{1}{2}}ds,\\
f\left( a_{3}\right) & =\frac{3}{2\pi}\int_{a_{4}}^{a_{5}}\left(
a_{5}-s\right) ^{-\frac{1}{2}}\left( s-a_{4}\right) ^{-\frac{1}{2}}\prod\limits_{j=1}^{2}\left( s-a_{j}\right) ^{-\frac{1}{2}}ds\\\nonumber
& =\frac{3}{2}E\left( a_{1},a_{2};a_{4},a_{5}\right),\end{aligned}$$ which is independent of $a_{3}$.
The integrals in the density formula have the form $\frac{1}{\pi}\int_{a}^{b}h\left(s\right) \big( \left( b-s\right)
\left( s-a\right) \big) ^{-\frac{1}{2}}ds$, where $h$ is differentiable in a neighborhood of $\left[ a,b\right] $. The technique of Gauss-Chebyshev quadrature is well suited for the numerical evaluation of the desired densities: Set $\phi\left( t\right)
=\frac{1}{2}\left( a+b\right) +\frac{1}{2}\left( b-a\right) t$ then the sums $\frac{1}{n}\sum_{j=0}^{n-1}h\left( \phi\left( \cos\frac{\left(
2j+1\right) \pi}{2n}\right) \right) $ converge rapidly to the integral (as $n\rightarrow\infty$); typically $n=20$ suffices for reasonable accuracy.
Another way of numerical approximation of a real numerical shadow can be done by direct numerical integration of a formula for a cumulative distribution function given in [@provost2000distribution].
\
\
Continuity at the knots {#sec:knots}
=======================
In this section we examine the behavior of the shadow density at the knots, where the curve pieces meet, that is, the regular singular points of the shadow differential equation. The even and odd $N$ cases are quite different. For even $N$ there are even and odd segments based on counting from $a_{N}$, so $\left[ a_{N-1},a_{N}\right] $ is \#1, and this parity is the same if one counts up from $a_{1}$. For odd $N$ there are even and odd knots (the parity of $j$ for the knot $a_{j}$; this remains the same under the transformation $x\mapsto-x$). In the neighborhood of each knot $a_{j}$ there is the analytic part, expandable in a power series $\sum_{n=0}^{\infty}c_{n}\left(
x-a_{j}\right) ^{n}$, and a part with discontinuous derivative of order $\left\lfloor \frac{N}{2}\right\rfloor -1$, as will be shown. For even $N=2M$ the density is polynomial of degree $M-2$ in $x$ in the even intervals $\left[ a_{N-2j},a_{N-2j+1}\right] $, and has a jump of the form $\left\vert
x-a_{i}\right\vert ^{M-\frac{3}{2}}$ at each end on the odd intervals $\left[
a_{N-2j-1},a_{N-2j}\right] $. Recall that the critical exponents are $\frac{N-1}{2}-1=M-\frac{3}{2}$ and $0,1,\ldots,N-3$.
For odd $N=2M+1$ there is just one type of curve piece: behavior like $\left\vert x-a_{i}\right\vert ^{M-1}$ at the odd end-point and $\left\vert
x-a_{i}\right\vert ^{M-1}\log\left\vert x-a_{i}\right\vert $ at the even end-point. The critical exponent $M-1=\frac{N-1}{2}-1$ is repeated, accounting for the logarithmic term. In this case each interval can be considered as even or odd by starting from $a_{N}$ or from $a_{1}$ and using the transformation $x\mapsto -x$.
For $a_{N-2m-1}\leq x<a_{N-2m}$ we have $$\begin{aligned}
f\left( x\right) & =\frac{N-2}{2\pi}\sum_{j=0}^{m-1}\left( -1\right)
^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left( s-x\right)
^{\frac{N}{2}-2}ds\\
& +\frac{N-2}{2\pi}\left( -1\right) ^{m}\int_{x}^{a_{N-2m}}g_{m}\left(
s\right) \left( s-x\right) ^{\frac{N}{2}-2}ds. \nonumber\end{aligned}$$ The integral indexed by $j$ is analytic in $x<a_{N-2j-1}$. Furthermore if $N=2M$ then the sum defines a polynomial in $x$ without any further restrictions on $x$. Consider the integral in the second line for $a_{N-2m}-\varepsilon<x<a_{N-2m}$ for some small $\varepsilon>0$ (and $<a_{N-2m}-a_{N-2m-1}$). Set $h_{m}\left( s\right) =g_{m}\left( s\right)
\left( a_{N-2m}-s\right) ^{\frac{1}{2}}$ so that $h_{m}\left( s\right) $ has a power series expansion $\sum_{n=0}^{\infty}c_{n}\left( a_{N-2m}-s\right) ^{n}$ valid in a neighborhood of $\left[ a_{N-2m}-r,a_{N-2m}\right] $ for small enough $r$. Then$$\begin{gathered}
\int_{x}^{a_{N-2m}}g_{m}\left( s\right) \left( s-x\right) ^{\frac{N}{2}-2}ds=\int_{x}^{a_{N-2m}}\left( a_{N-2m}-s\right) ^{-\frac{1}{2}}\left(
s-x\right) ^{\frac{N}{2}-2}h_{m}\left( s\right) ds\\ \nonumber
=\sum_{n=0}^{\infty}c_{n}\int_{x}^{a_{N-2m}}\left( a_{N-2m}-s\right)
^{-\frac{1}{2}+n}\left( s-x\right) ^{\frac{N}{2}-2}ds\\ \nonumber
=\left( a_{N-2m}-x\right) ^{\frac{N-3}{2}}\sum_{n=0}^{\infty}B\left(
n+\frac{1}{2},\frac{N-2}{2}\right) c_{n}\left( a_{N-2m}-x\right) ^{n}\\
=B\left( \frac{1}{2},\frac{N-2}{2}\right) \left( a_{N-2m}-x\right)
^{\frac{N-3}{2}}\sum_{n=0}^{\infty}\frac{\left( \frac{1}{2}\right) _{n}}{\left( \frac{N-1}{2}\right) _{n}}c_{n}\left( a_{N-2m}-x\right) ^{n}; \nonumber\end{gathered}$$ this is the solution of the shadow equation for the critical exponent $\frac{N-3}{2}$ at the regular singular point $a_{N-2m}$. The leading term is$$\frac{\left( -1\right) ^{m}}{B\left( \frac{N-1}{2},\frac{1}{2}\right)
}\prod\limits_{j=0}^{2m-1}\left( a_{N-j}-a_{N-2m}\right) ^{-\frac{1}{2}}\prod\limits_{j=2m+1}^{N-1}\left( a_{N-2m}-a_{N-j}\right) ^{-\frac{1}{2}}\left( a_{N-2m}-x\right) ^{\frac{N-3}{2}},$$ in analogy to the leading term in formula (\[topF\]). The computation uses an identity, $\frac{1}{\pi}\left( \frac{N}{2}-1\right) \frac{\Gamma\left( \frac{1}{2}\right) \Gamma\left( \frac{N-2}{2}\right) }{\Gamma\left( \frac{N-1}{2}\right) }=\frac{\Gamma\left( \frac{N}{2}\right) }{\Gamma\left(
\frac{N-1}{2}\right) \Gamma\left( \frac{1}{2}\right) }$.
Even $N$
--------
Set $N=2M$. Near a knot $a_{2M-2m}$ ($0\leq m<M$) the density $f\left(
x\right) $ is polynomial for $a_{2M-2m}<x$ and given by the sum of the polynomial and a series $\left( a_{2M-2m}-x\right) ^{M-\frac{3}{2}}\sum_{n=0}^{\infty}c_{n}\left( a_{2M-2m}-x\right) ^{n}$ for $x<a_{2M-2m}$. Thus $f^{\left( j\right) }\left( x\right) $ is continuous in a neighborhood of $a_{2M-2m}$ for $0\leq j\leq M-2$. By applying this result to the reversed knots $b_{1}<\ldots<b_{N}$ where $b_{j}=-a_{2M+1-j}$ and $x$ replaced by $-x$ we find that near a knot $a_{2M-2m+1}=-b_{2M-2\left(
M-m\right) }$ for $1\leq m\leq M$ the density $f\left( x\right)$ is polynomial for $x<a_{2M-2m+1}$ and is given by the sum of the polynomial and a series $\left( x-a_{2M-2m+1}\right) ^{M-\frac{3}{2}}\sum_{n=0}^{\infty}c_{n}\left(
x-a_{2M-2m+1}\right) ^{n}$ for $x>a_{2M-2m+1}$. Thus the lowest order discontinuity of the density is in $f^{\left( M-1\right) }\left( x\right)
$ at each knot, that is, $f^{\left( j\right) }\left( x\right) $ is continuous everywhere for all $j\leq\frac{N}{2}-2$.
Odd $N$
-------
Set $N=2M+1$. Consider the even knot $a_{N-2m-1}$ with $0\leq m<M-1$. Pick $r$ with $0<r<\min\left( a_{N-2m}-a_{N-2m-1},a_{N-2m-1}-a_{N-2m-2}\right) $, then in the interval $\left[ a_{N-2m-1}-r,a_{N-2m-1}+r\right] $ the function $\prod\limits_{j=0}^{2m}\left( a_{N-j}-s\right) ^{-\frac{1}{2}}\prod\limits_{j=2m-2}^{N-1}\left( s-a_{N-j}\right) ^{-\frac{1}{2}}$ can be expanded as a power series $\sum_{n=0}^{\infty}c_{n}\left( s-a_{N-2m-1}\right) ^{n}$. Here the coefficients can be found by using the negative binomial theorem for each factor in the product. Since all the difficulty happens at the knot set$$\begin{aligned}
\phi\left( x\right) & =\frac{N-2}{2\pi}\sum_{j=0}^{m-1}\left( -1\right)
^{j}\int_{a_{N-2j-1}}^{a_{N-2j}}g_{j}\left( s\right) \left( s-x\right)
^{\frac{N}{2}-2}ds\\
& +\frac{N-2}{2\pi}\left( -1\right) ^{m}\int_{a_{N-2m-1}+r}^{a_{N-2m}}g_{m}\left( s\right) \left( s-x\right) ^{\frac{N}{2}-2}ds. \nonumber\end{aligned}$$ Thus $\phi\left( x\right) $ is analytic for $x<a_{N-2m-1}+r$; for $a_{N-2m-1}<x<a_{N-2m-1}+r$$$f\left( x\right) =\phi\left( x\right) +\frac{N-2}{2\pi}\left( -1\right)
^{m}\int_{x}^{a_{N-2m-1}+r}g_{m}\left( s\right) \left( s-x\right)
^{M-\frac{3}{2}}ds,$$ and for $a_{N-2m-1}-r<x<a_{N-2m-1}$$$f\left( x\right) =\phi\left( x\right) +\frac{N-2}{2\pi}\left( -1\right)
^{m}\int_{a_{N-2m-1}}^{a_{N-2m-1}+r}g_{m}\left( s\right) \left( s-x\right)
^{M-\frac{3}{2}}ds.$$ By using the power series and the change of variable $x=a_{N-2m-1}+y$ and $s=t+a_{N-2m-1}$the first integral becomes$$\sum_{n=0}^{\infty}c_{n}\int_{y}^{r}t^{n-\frac{1}{2}}\left( t-y\right)
^{M-\frac{3}{2}}dt,0<y<r,$$ and the second integral becomes$$\sum_{n=0}^{\infty}c_{n}\int_{0}^{r}t^{n-\frac{1}{2}}\left( t-y\right)
^{M-\frac{3}{2}}dt,-r<y<0.$$ We want to analyze the behavior of the integrals in the limit $y\rightarrow0$. In each integral change the variable $t=\frac{y}{1-u^{2}}$, so $dt=\frac
{2yu}{\left( 1-u^{2}\right) ^{2}}du$. Furthermore $t\left( t-y\right)
=\frac{y^{2}u^{2}}{\left( 1-u^{2}\right) ^{2}}$ thus $\frac{1}{\sqrt{t\left( t-y\right) }}=\frac{1-u^{2}}{yu}$ (if $y<0$ then $u^{2}>1$, and if $y>0$ then $u^{2}<1$ so that this is the positive root). Set $u_{r}=\sqrt{\frac{r-y}{r}}$. For $0<y<r$ the integral is $$2y^{n+M-1}\int_{0}^{u_{r}}\frac{u^{2M-2}}{\left( 1-u^{2}\right) ^{n+M}}du,~0<u_{r}<1,$$ and for $-r<y<0$ the integral is $$2y^{n+M-1}\int_{\infty}^{u_{r}}\frac{u^{2M-2}}{\left( 1-u^{2}\right) ^{n+M}}du,~u_{r}>1.$$ Because the integrand is even we deduce that the partial fraction expansion is of the form$$\frac{u^{2M-2}}{\left( 1-u^{2}\right) ^{n+M}}=\sum_{j=1}^{n+M}\beta
_{j}\left( M,n\right) \left\{ \frac{1}{\left( 1-u\right) ^{j}}+\frac
{1}{\left( 1+u\right) ^{j}}\right\} ,$$ for certain constants $\beta_{j}\left( M,n\right) $. Thus$$\begin{split}
I_{M,n}\left( u\right) := & \int\frac{u^{2M-2}}{\left( 1-u^{2}\right)
^{n+M}}du=\beta_{1}\left( M,n\right) \log\left\vert \frac{1+u}{1-u}\right\vert + \\
& +\sum_{j=2}^{n+M}\frac{\beta_{j}\left( M,n\right) }{j-1}\left\{ \frac
{1}{\left( 1-u\right) ^{j-1}}-\frac{1}{\left( 1+u\right) ^{j-1}}\right\}
\end{split}.$$ This antiderivative $I_{M,n}$ vanishes at $u=0$ and at $u=\infty$, thus both integrals have the same value $2y^{n+M-1}I_{M,n}\left( \sqrt{\frac{r-y}{r}}\right) $. The terms for $2\leq j\leq n+M$ contribute$$\frac{4y^{n+M-1}u_{r}}{\left( 1-u_{r}^{2}\right) ^{j-1}}\sum_{i=0}^{\left\lfloor \left( j-2\right) /2\right\rfloor }\binom{j-1}{2i+1}u_{r}^{2i}=4y^{n+M-j}\sqrt{\frac{r-y}{r}}\sum_{i=0}^{\left\lfloor \left(
j-2\right) /2\right\rfloor }\binom{j-1}{2i+1}\left( r-y\right)
^{i}r^{j-1-i},$$ which is analytic in $y$ for $-r<y<r$. So all the singular behavior stems from the logarithmic term$$2y^{n+M-1}\beta_{1}\left( M,n\right) \log\left\vert \frac{1+u_{r}}{1-u_{r}}\right\vert ,$$ and$$\begin{aligned}
\log\left\vert \frac{1+u_{r}}{1-u_{r}}\right\vert & =-\log\left\vert
1-u_{r}^{2}\right\vert +2\log\left\vert 1+u_{r}\right\vert \\
& =-\log\left\vert y\right\vert +\log r+2\log(1+\sqrt{\frac{r-y}{r}}). \nonumber\end{aligned}$$ Collecting the relevant terms we see that for $a_{N-2m-1}-r<x<a_{N-2m-1}+r$ the density $f\left( x\right) $ is the sum of an analytic part and$$\begin{split}
& \frac{2}{\pi}\left( -1\right) ^{m-1}\left( 2M-1\right) \log\left\vert
x-a_{N-2m-1}\right\vert \times \\
& \times \left( x-a_{N-2m-1}\right) ^{M-1}\sum_{n=0}^{\infty
}\beta_{1}\left( M,n\right) c_{n}\left( x-a_{N-2m-1}\right) ^{n}.
\end{split}$$
The coefficients $\beta_{j}\left( M,n\right) $ can be found explicitly as sums but we are only concerned with $\beta_{1}\left( M,n\right) $. Indeed (proof left for reader)$$\beta_{1}\left( M,n\right) =\frac{1}{2}\left( -1\right) ^{M-1}\frac{\left( \frac{1}{2}\right) _{M-1}\left( \frac{1}{2}\right) _{n}}{\left( n+M-1\right) !}.$$ Thus we analyzed the behavior at the even knots and showed that $f^{\left(
j\right) }\left( x\right) $ is continuous everywhere for all $j\leq
M-2=\frac{N-1}{2}-2$.
Entangled shadow {#sec:entangled-shadow}
================
In previous sections we investigated the shadow with respect to real states. Here we discuss another example of the restricted shadow – the shadow with respect to maximally entangled states, briefly called [*entangled shadow*]{}.
Entangled shadow of $4 \times 4$ matrices with direct sum structure
-------------------------------------------------------------------
We shall start recalling the definition of the entangled shadow introduced in [@shadow3].
Maximally entangled numerical shadow of a matrix $A$ of size $N=N_1\times N_2$ is defined as a probability distribution $\mathcal{P}^{\mathrm{ent}}_A(z)$ on the complex plane. $$\mathcal{P}_A^{\mathrm{ent}}(z) := \int {\mathrm d} \mu(\psi) \delta\Bigl( z-\langle
\psi|A|\psi\rangle\Bigr),$$ where $\mu(\psi)$ denotes the unique unitarily invariant (Fubini-Study) measure on the set of complex pure states restricted to the set of bi-partite maximally entangled states $$\left\{{\ensuremath{|\psi\rangle}} \in \mathbb{C}^{N_1\times N_2}:
{\ensuremath{|\psi\rangle}}=\frac{1}{\sqrt{N_{\min}}} (U_1\otimes U_2)\sum_{i=1}^{N_{\min}}
{\ensuremath{|\psi_i^1\rangle}}\otimes {\ensuremath{|\psi_i^2\rangle}} \right\}.$$ Here $N_{\min}=\min(N_1,N_2)$, while ${\ensuremath{|\psi_i^1\rangle}}$, ${\ensuremath{|\psi_i^2\rangle}}$ form orthonormal bases in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$ respectively, while $U_1\in SU(N_1)$ and $U_2\in SU(N_2)$.
Pauli matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$ are defined as $$\sigma_x =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\;
\sigma_y =
\begin{pmatrix}
0 & -{\mathrm{i}}\\
{\mathrm{i}}& 0
\end{pmatrix},
\;
\sigma_z =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.$$
For a $2 \times 2$ unitary matrix $U$ and an [*arbitrary*]{} $2
\times 2$ matrix $A$ we have $${\ensuremath{\langle1|}} U^{\dagger} A U {\ensuremath{|1\rangle}} = {\ensuremath{\langle0|}} U^{\dagger} \sigma_y A^\mathrm{T}
\sigma_y U {\ensuremath{|0\rangle}},$$
Maximally entangled shadow $$\mathcal{P}^{\mathrm{ent}}_{A \oplus B} = \mathcal{P}_{\frac12 (A + \sigma_y B^\mathrm{T} \sigma_y)},$$ where $A\oplus B$ denotes block matrix.
We write $$\begin{split}
{\ensuremath{\langle\psi|}} (A \oplus B) {\ensuremath{|\psi\rangle}} &=
{\ensuremath{\langle\psi_+|}} ({{\rm 1\hspace{-0.9mm}l}}\otimes U^{\dagger}) (A \oplus B) ({{\rm 1\hspace{-0.9mm}l}}\otimes U)
{\ensuremath{|\psi_+\rangle}} \\
&={\ensuremath{\langle\psi_+|}} (U^{\dagger} \oplus U^{\dagger}) (A \oplus B) (U \oplus U)
{\ensuremath{|\psi_+\rangle}} \\
&= \frac12 \left({\ensuremath{\langle0|}}U^{\dagger}AU {\ensuremath{|0\rangle}} + {\ensuremath{\langle1|}}U^{\dagger}BU
{\ensuremath{|1\rangle}}\right).
\end{split}$$ Now we use lemma and write $$\begin{split}
{\ensuremath{\langle\psi|}} (A \oplus B) {\ensuremath{|\psi\rangle}} &= \frac12 \left({\ensuremath{\langle0|}}U^{\dagger}AU
{\ensuremath{|0\rangle}} + {\ensuremath{\langle1|}}U^{\dagger}BU {\ensuremath{|1\rangle}}\right)\\
&= \frac12 \left({\ensuremath{\langle0|}}U^{\dagger}AU {\ensuremath{|0\rangle}} + {\ensuremath{\langle0|}}U^{\dagger}\sigma_y
B^\mathrm{T}
\sigma_yU
{\ensuremath{|0\rangle}}\right)\\
&= \frac12 \left({\ensuremath{\langle0|}}U^{\dagger}(A+\sigma_y B^\mathrm{T}
\sigma_y)U {\ensuremath{|0\rangle}}\right).
\end{split}$$
This theorem is valid for complex and real entangled shadow. Also we made no assumptions on $A$ and $B$, hence it is valid for non-normal matrices.
Real maximally entangled shadow of $4 \times 4$ matrices
--------------------------------------------------------
Real maximally entangled numerical shadow $\mathcal{P}_A^{{\mathrm{ent} | \mathbb{R}}}$ of a matrix $A$ of size $N=N_1\times
N_2$ is defined similarly to the maximally entangled shadow, but with restriction to the real maximally entangled states.
The following theorem gives a full characterization of the real maximally entangled numerical shadow of $4 \times 4$ matrices.
\[th:real-ent\] Let $A$ be any $4 \times 4$ matrix then we have $$\mathcal{P}_A^{\mathrm{ent} | \mathbb{R}}=
\frac{1}{2} \mathcal{P}_{Z_1^\mathrm{T} A Z_1}^{\mu_\mathbb{R}}+
\frac{1}{2} \mathcal{P}_{Z_2^\mathrm{T} A Z_2}^{\mu_\mathbb{R}},$$ where, $Z_1$ and $Z_2$ are $$Z_1 = \frac{1}{\sqrt 2}
\left(
\begin{smallmatrix}
1 & 0 \\
0 & 1 \\
0 & -1 \\
1 & 0
\end{smallmatrix}
\right) ,
\ \
Z_2 = \frac{1}{\sqrt 2}
\left(
\begin{smallmatrix}
1 & 0 \\
0 & 1 \\
0 & 1 \\
-1 & 0
\end{smallmatrix}
\right).
\label{z1z2}$$
Any real maximally entangled pure state ${\ensuremath{|\psi\rangle}}$ of size four may be written as a vector obtained from the elements of an orthogonal matrix of order two, $${\ensuremath{|\psi\rangle}} =\frac{1}{\sqrt 2} \mathrm{vec}(O(\theta)).$$ First we consider an orthogonal matrix $O(\theta)$ satisfying $\det O(\theta)
= 1$. We have $${\ensuremath{|\psi\rangle}} =
\frac{1}{\sqrt 2}
\left(
\begin{smallmatrix}
\cos \theta \\
\sin \theta \\
-\sin \theta \\
\cos \theta
\end{smallmatrix}
\right)
=
\frac{1}{\sqrt 2}
\left(
\begin{smallmatrix}
1 & 0 \\
0 & 1 \\
0 & -1 \\
1 & 0
\end{smallmatrix}
\right)
\left(
\begin{smallmatrix}
\cos \theta \\
\sin \theta
\end{smallmatrix}
\right).$$ Hence, $${\ensuremath{\langle\psi|}}A{\ensuremath{|\psi\rangle}} = {\ensuremath{\langler|}}Z_1^\mathrm{T} A Z_1 {\ensuremath{|r\rangle}}.
\label{eq:max-ent-p1}$$
Now we consider an orthogonal matrix $O(\theta)$ satisfying $\det O(\theta)
= -1$. We have $${\ensuremath{|\psi\rangle}} =
\frac{1}{\sqrt 2}
\left(
\begin{smallmatrix}
\cos \theta \\
\sin \theta \\
\sin \theta \\
-\cos \theta
\end{smallmatrix}
\right)
=
\frac{1}{\sqrt 2}
\left(
\begin{smallmatrix}
1 & 0 \\
0 & 1 \\
0 & 1 \\
-1 & 0
\end{smallmatrix}
\right)
\left(
\begin{smallmatrix}
\cos \theta \\
\sin \theta
\end{smallmatrix}
\right).$$ Hence, $${\ensuremath{\langle\psi|}}A{\ensuremath{|\psi\rangle}} = {\ensuremath{\langler|}}Z_2^\mathrm{T} A Z_2 {\ensuremath{|r\rangle}}.
\label{eq:max-ent-p2}$$ Combining equations and we get the theorem.
Examples are shown in Figures \[fig:t2-ex-1\] and \[fig:t2-ex-2\]. The theorem is valid for non-normal matrices.
[m[0.5]{}m[0.5]{}]{} &\
& $
A=
\left(
\begin{smallmatrix}
1 & 0 & 0 & 1 \\
0 & {\mathrm{i}}& 0 & 1 \\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -{\mathrm{i}}\end{smallmatrix}
\right)
$
[m[0.5]{}m[0.5]{}]{} &\
& $B=\left(
\begin{smallmatrix}
0.3+0.5{\mathrm{i}}& -0.8-0.2{\mathrm{i}}& 0.4-0.5{\mathrm{i}}& 1\\
0.6-0.8{\mathrm{i}}& -0.8-0.4{\mathrm{i}}& -0.6+0.8{\mathrm{i}}& -0.8+0.8{\mathrm{i}}\\
0.7-0.8{\mathrm{i}}& -0.5-0.4{\mathrm{i}}& -0.8 & 0.7-0.3{\mathrm{i}}\\
0.4+0.6{\mathrm{i}}& -1.-0.8{\mathrm{i}}& -0.4-0.4{\mathrm{i}}& -0.7
\end{smallmatrix}
\right)$
Complex maximally entangled shadow of $4 \times 4$ matrices
-----------------------------------------------------------
\[th:complex-ent\] Given an [*arbitrary*]{} matrix $A$ of order four its complex maximally entangled shadow is equal to the real shadow of matrix $W^\dagger A W$ $$\mathcal{P}^{\mu_\mathrm{ent} }_A = \mathcal{P}^{\mu_\mathbb{R
}}_{W^\dagger A W},$$ where $W$ is the matrix representing the ’magic basis’, $$W = \frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 0 & 1 & {\mathrm{i}}\\
-1 & {\mathrm{i}}& 0 & 0 \\
1 & {\mathrm{i}}& 0 & 0 \\
0 & 0 & 1 & -{\mathrm{i}}\end{pmatrix}.$$
The above theorem is related to the well known fact in the group theory, that $$SO(4) = (SU(2) \times SU(2)) / \mathbb{Z}_2.$$
Any maximally entangled two-qubit state ${\ensuremath{|\psi\rangle}}$ can be written as $${\ensuremath{|\psi\rangle}} = \mathrm{vec}(V), \ {\rm where} \ V \in SU(2).$$ Using a parameterization of $SU(2)$ we can write $$V =
\begin{pmatrix}
{\mathrm{e}}^{{\mathrm{i}}\xi_2} \cos\eta & {\mathrm{e}}^{{\mathrm{i}}\xi_1} \sin \eta \\
-{\mathrm{e}}^{-{\mathrm{i}}\xi_1} \sin \eta & {\mathrm{e}}^{-{\mathrm{i}}\xi_2} \cos\eta
\end{pmatrix}.$$ Reshaping this matrix into a vector of length four we obtain the state $${\ensuremath{|\psi\rangle}} = \frac{1}{\sqrt{2}}
\begin{pmatrix}
\cos \xi_2 \sin \eta + {\mathrm{i}}\sin \xi_2 \cos \eta \\
-\cos \xi_1 \sin \eta + {\mathrm{i}}\sin \xi_1 \sin \eta \\
\cos \xi_1 \sin \eta + {\mathrm{i}}\sin\xi_1 \sin \eta \\
\cos \xi_2 \cos \eta - {\mathrm{i}}\sin \xi_2 \cos \eta
\end{pmatrix}.\label{eq:vectorized}$$
On the other hand, consider the Hopf parameterization of the 3-sphere $S^3$ embedded in ${\mathbb{C}}^2$. A point on this sphere can be expressed as $$\begin{cases}
z_1 = {\mathrm{e}}^{{\mathrm{i}}\xi_1} \sin \eta \\
z_2 = {\mathrm{e}}^{{\mathrm{i}}\xi_2} \cos \eta.
\end{cases}$$
A point on the 3-sphere may be written in real coordinates $(r_1, r_2, r_3,
r_4)$ as $$\begin{pmatrix}
r_1 \\
r_2 \\
r_3 \\
r_4 \\
\end{pmatrix}
=
\begin{pmatrix}
\Re z_1 \\
\Im z_1 \\
\Re z_2 \\
\Im z_2
\end{pmatrix}
=
\begin{pmatrix}
\cos \xi_1 \sin \eta \\
\sin \xi_1 \sin \eta \\
\cos \xi_2 \cos \eta \\
\sin \xi_2 \cos \eta
\end{pmatrix}.
\label{eq:real-vec}$$ Now, using Equation , we can rewrite Equation as $${\ensuremath{|\psi\rangle}} =
\frac{1}{\sqrt{2}}
\begin{pmatrix}
r_3 +{\mathrm{i}}r_4 \\
-r_1 + {\mathrm{i}}r_2 \\
r_1 + {\mathrm{i}}r_2 \\
r_3 - {\mathrm{i}}r_4
\end{pmatrix}
=
\frac{1}{\sqrt{2}}
\begin{pmatrix}
0 & 0 & 1 & {\mathrm{i}}\\
-1 & {\mathrm{i}}& 0 & 0 \\
1 & {\mathrm{i}}& 0 & 0 \\
0 & 0 & 1 & -{\mathrm{i}}\end{pmatrix}
\begin{pmatrix}
r_1\\
r_2\\
r_3\\
r_4
\end{pmatrix}.$$ Hence, we can write $${\ensuremath{\langle\psi|}} A {\ensuremath{|\psi\rangle}} = {\ensuremath{\langler|}} W^\dagger A W {\ensuremath{|r\rangle}},$$ where ${\ensuremath{|r\rangle}}$ is a real vector defined in equation .
Concluding remarks {#sec:remarks}
==================
In this work we analyzed probability distributions on the complex plane induced by projecting the set of quantum states, (i.e. Hermitian, positive and normalized matrices of a given size $N$), endowed with a certain probability measure. In the case of the unique, unitarily invariant Haar measure on the set of complex pure states, this distribution coincides with the standard numerical shadow [@Shd1; @GS2012] of a certain matrix $A$ of size $N$. The case of a normal matrix corresponds to the projection of the unit simplex covered uniformly onto a plane [@shadow1]. If the matrix $A$ is Hermitian, its (complex) numerical shadow is supported on an interval on the real axis, and is equivalent to the $B$–spline with knots at the eigenvalues of $A$.
The real shadow of a matrix corresponds to the Haar measure restricted to the set of real pure states [@shadow3]. For a real symmetric $A$ its real shadow is shown to be equivalent to a to the projection of the unit simplex covered by the Dirichlet measure. The main result of this work consists in Theorem \[th:main-theorem\], which establishes an explicit exact formula for the real shadow of any real symmetric $A$ with prescribed spectrum $(a_1,\dots, a_N)$.
As the real shadow of a matrix corresponds to the Dirichlet distribution with its parameters equal to $k_1= k_2=\dots=k_N=1/2$, it is natural to generalize it by considering also other values of this parameter. For instance, the case of complex shadow $\mathcal{P}_A=\mathcal{P}^{\mathbb{C}}_A$ corresponds to the case $k_i=1$. This fact implies that the real shadow of an extended matrix is equivalent to the complex shadow, $$\mathcal{P}^{\mathbb{R}}_{A\otimes {\mathbb I_2}}(x) =
\mathcal{P}_A(x) .
\label{prodshad}$$ This result allows us to consider the real shadow of a real symmetric matrix $C$ of an even size $2N$ as a generalization of the $B$ spline, which is recovered, if each eigenvalue is doubly degenerated. In general, each of $N$ knot points of the standard $B$–spline can be splitted into two halves, and each eigenvalue $\lambda_i$ of $C$ can be considered as a ’half of the knot point’, as $2N$ points $\{ \lambda_i\}_{i=1}^{2N}$ determine the generalized $B$-spline equal to the real shadow of $C$.
Analyzing the generalized Dirichlet distribution one needs not to restrict the attention to parameters $k_i$ equal to $1/2$ or $1$. For instance, one can consider the shadow of a matrix of an even order with respect to quaternion states which corresponds to the Dirichlet distribution with all parameters equal, $k_i=2$, see \[sec:quaternion-shadow\].
As another example of the restricted shadow we analyzed entangled shadow of a matrix of an order $N$ equal to a composite number. As before we distinguish the shadow with respect to complex (or real) maximally entangled states. Note that these probability distributions in general are supported on non–convex sets. In the simplest case of $N=4$ we found explicit formulae for the complex and real entangled shadows by relating it to the real shadows of suitably transformed matrices. As such shadows visualize projection of the set of complex/real maximally entangled states onto a plane [@shadow3] it is likely to expect that such tools will be useful in studying the structure of the set of maximally entangled states.
Acknowledgements: It is a pleasure to thank John Holbrook for several discussions on real numerical shadow and for providing us a copy of his paper prior to publication. This research was supported by the the Polish National Science Centre (NCN): P. Gawron under the grant number N N516 481840, Z.Pucha[ł]{}a under the grant number DEC-2012/04/S/ST6/00400 while K. [Ż]{}yczkowski and [Ł]{}. Pawela acknowledge support by the grant number N202 090239.
Quaternion shadow {#sec:quaternion-shadow}
=================
In the case of the real shadow one considers random normalized real vectors with distribution invariant to orthogonal transformations. This distribution is induced by a Haar measure on the orthogonal group. In a similar fashion, one can introduce the quaternion shadow, $\mathcal{P}^{\mathbb{H}}_A$, defined as a probability distribution of expectation values taken among random normalized quaternion vectors, with distribution invariant with respect to symplectic operations.
An $N$-vector with quaternion $\mathbb{H}$ entries is replaced by a $2N\times2$ complex matrix, where $a_{0}+a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}$ is mapped to$$\nu\left(a_{0}+a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}\right)
=\left[
\begin{array}
[c]{cc}a_{0}+a_{1}\mathrm{i} & a_{2}+a_{3}\mathrm{i}\\
-a_{2}+a_{3}\mathrm{i} & a_{0}-a_{1}\mathrm{i}\end{array}
\right] .$$ We use $\nu$ to also indicate the map $N$-vectors to $2N\times2$ matrices. Suppose $A$ is a $2N\times2N$ complex Hermitian matrix. Consider the numerical range-type map from the unit sphere in $\mathbb{H}^{N}$ to $\mathbb{R}:$$$\xi\mapsto\frac{1}{2}{\mathrm{Tr}}\left( \nu\left( \xi\right) ^{\dagger}A\nu\left(
\xi\right) \right) .$$ Note $\nu\left( \xi\right) ^{\dagger}A\nu\left( \xi\right) $ is a $2\times2$ complex Hermitian matrix. By direct computation we find that the same values are obtained if $A$ is transformed as follows:$$q\left(\left[
\begin{array}
[c]{cc}a_{11} & a_{12}\\
a_{21} & a_{22}\end{array}
\right]\right) =
\frac{1}{2}
\left[
\begin{array}
[c]{cc}\left( a_{11}+\overline{a_{22}}\right) & \left(
a_{12}-\overline{a_{21}}\right) \\
\left( a_{21}-\overline{a_{12}}\right) & \left(
\overline{a_{11}}+a_{22}\right)
\end{array}
\right] ,$$ and this map is applied to each of the $N^{2}$ $2\times2$ blocks of $A$. Note that these blocks correspond to quaternions of the form$$\left[
\begin{array}
[c]{cc}\alpha & \beta\\
-\overline{\beta} & \overline{\alpha}\end{array}
\right] .$$ Then the numerical range and shadow can be interpreted as those of an $N\times
N$ quaternionic Hermitian matrix. The probability density is Dirichlet, parameter 2, using $N$ real eigenvalues. The transformed matrix $q\left(
A\right) $ has (duplicates) pairs of eigenvalues.
Here is a trivial example:$$A=\left[
\begin{array}
[c]{cccc}0 & 1 & 0 & 0\\
1 & 0 & 1 & 0\\
0 & 1 & 0 & 1\\
0 & 0 & 1 & 0
\end{array}
\right] ,q\left( A\right) =\left[
\begin{array}
[c]{cccc}0 & 0 & 0 & -\frac{1}{2}\\
0 & 0 & \frac{1}{2} & 0\\
0 & \frac{1}{2} & 0 & 0\\
-\frac{1}{2} & 0 & 0 & 0
\end{array}
\right] .$$ The eigenvalues of $A$ are $\pm\frac{1}{2}\pm\frac{1}{2}\sqrt{5}$ and the eigenvalues of $q\left( A\right) $ are $-\frac{1}{2},-\frac{1}{2},\frac
{1}{2},\frac{1}{2}$.
Consider $M_{2}\left( \mathbb{C}\right) $ as an eight-dimensional vector space over $\mathbb{R}$ with the basis:$$\begin{split}
\mathbf{1} & =\left[
\begin{array}
[c]{cc}1 & 0\\
0 & 1
\end{array}
\right] ,\mathbf{i}=\left[
\begin{array}
[c]{cc}\mathrm{i} & 0\\
0 & -\mathrm{i}\end{array}
\right] ,\mathbf{j}=\left[
\begin{array}
[c]{cc}0 & 1\\
-1 & 0
\end{array}
\right] ,\mathbf{k}=\left[
\begin{array}
[c]{cc}0 & \mathrm{i}\\
\mathrm{i} & 0
\end{array}
\right] ,\\
\mathbf{\zeta} & =\left[
\begin{array}
[c]{cc}\mathrm{i} & 0\\
0 & \mathrm{i}\end{array}
\right] ,\mathbf{\zeta i}=\left[
\begin{array}
[c]{cc}-1 & 0\\
0 & 1
\end{array}
\right] ,\mathbf{\zeta j}=\left[
\begin{array}
[c]{cc}0 & \mathrm{i}\\
-\mathrm{i} & 0
\end{array}
\right] ,\mathbf{\zeta k}=\left[
\begin{array}
[c]{cc}0 & -1\\
-1 & 0
\end{array}
\right] ,
\end{split}$$ (Pauli matrices). The basis is orthonormal with the inner product $$\left\langle \alpha,\beta\right\rangle =\frac{1}{2}\operatorname{Re}{\mathrm{Tr}}\left(
\alpha\beta^{\dagger}\right) ,$$ and $\left\langle \alpha,\beta\right\rangle =\left\langle \alpha^{\dagger},\beta^{\dagger}\right\rangle $. Then $q\left( \alpha\right) =\left\langle
\alpha,\mathbf{1}\right\rangle \mathbf{1+}\left\langle \alpha,\mathbf{i}\right\rangle \mathbf{i+}\left\langle \alpha,\mathbf{j}\right\rangle
\mathbf{j+}\left\langle \alpha,\mathbf{k}\right\rangle \mathbf{k}$ and $q\left( \alpha^{\dagger}\right) =q\left( \alpha\right) ^{\dagger}$. Set $\mathbb{H}=\mathrm{span}_{\mathbb{R}}\left\{ \mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}\right\} $ (a $\ast$-subalgebra). Now suppose $\left(
\alpha_{ij}\right) _{i,j=1}^{N}$ is a Hermitian matrix with entries in $M_{2}\left( \mathbb{C}\right) $ (that is, $\alpha_{ji}=\alpha_{ij}^{\dagger}$), and $\left( \beta_{i}\right) _{i=1}^{N}$ is a vector with entries in $\mathbb{H}$.
The following equality holds $${\mathrm{Tr}}\left( \sum_{i,j=1}^{N}\beta_{i}^{\dagger}\alpha_{ij}\beta_{j}\right)
={\mathrm{Tr}}\left( \sum_{i,j=1}^{N}\beta_{i}^{\dagger}q\left( \alpha_{ij}\right)
\beta_{j}\right) .$$
Break up the sum into $i=j$ and $i<j$ parts. Then ${\mathrm{Tr}}\left( \beta_{i}^{\dagger
}\alpha_{ii}\beta_{i}\right) \in\mathbb{R}$ and$$\begin{split}
{\mathrm{Tr}}\left( \beta_{i}^{\dagger}\alpha_{ii}\beta_{i}\right) & =
{\mathrm{Tr}}\left(
\alpha_{ii}\beta_{i}\beta_{i}^{\dagger}\right) =2\left\langle \alpha_{ii},\beta_{i}\beta_{i}^{\dagger}\right\rangle \\
& =2\left\langle q\left( \alpha_{ii}\right) ,\beta_{i}\beta_{i}^{\dagger
}\right\rangle ={\mathrm{Tr}}\left( \beta_{i}^{\dagger}q\left( \alpha_{ii}\right)
\beta_{i}\right) .
\end{split}$$ For $i<j$ consider the typical term$$\begin{split}
{\mathrm{Tr}}\left( \beta_{i}^{\dagger}\alpha_{ij}\beta_{j}\right) +{\mathrm{Tr}}\left( \beta
_{j}^{\dagger}\alpha_{ji}\beta_{i}\right) & ={\mathrm{Tr}}\left( \beta_{i}^{\dagger}\alpha_{ij}\beta_{j}\right) +{\mathrm{Tr}}\left( \beta_{j}^{\dagger}\alpha_{ij}^{\dagger
}\beta_{i}\right) \\
& =2\operatorname{Re}{\mathrm{Tr}}\left( \beta_{i}^{\dagger}\alpha_{ij}\beta_{j}\right)
=2\operatorname{Re}{\mathrm{Tr}}\left( \alpha_{ij}\beta_{j}\beta_{i}^{\dagger}\right) \\
& =4\left\langle \alpha_{ij},\beta_{i}\beta_{j}^{\dagger}\right\rangle
=4\left\langle q\left( \alpha_{ij}\right) ,\beta_{i}\beta_{j}^{\dagger
}\right\rangle \\
& ={\mathrm{Tr}}\left( \beta_{i}^{\dagger}q\left( \alpha_{ij}\right) \beta_{j}\right)
+{\mathrm{Tr}}\left( \beta_{j}^{\dagger}q\left( \alpha_{ji}\right) \beta_{i}\right) ,
\end{split}$$ because $q\left( \alpha_{ji}\right) =q\left( \alpha_{ij}^{\dagger}\right)
=q\left( \alpha_{ij}\right) ^{\dagger}$. This proves the claim.
Every quaternion Hermitian matrix can be diagonalized with symplectic operations, thus when studying quaternion numerical shadow of Hermitian matrices, without loss of generality, we can consider only diagonal matrices with real elements on the diagonal. We note, that for such quaternion matrices, the representation on a block complex matrices gives us $\nu(A) = A\otimes
{{\rm 1\hspace{-0.9mm}l}}_2$. Combining this with relation , we may write the following chain of equalities $$\mathcal{P}_{A\otimes {{\rm 1\hspace{-0.9mm}l}}_4}^{\mathbb{R}} = \mathcal{P}_{A\otimes {{\rm 1\hspace{-0.9mm}l}}_2} = \mathcal{P}_{A}^{\mathbb{H}}.$$
Proofs {#sec:proofs}
======
Indeed (set $t_{N}:=1-\sum_{i=1}^{N-1}t_{i}$)$$\begin{aligned}
\mathcal{E}\left[ \left( 1-rX\right) ^{-\widetilde{k}}\right] &
=\sum_{n=0}^{\infty}\frac{\left( \widetilde{k}\right) _{n}}{n!}\int_{\mathbb{T}_{N-1}}\left( \sum_{i=1}^{N}a_{i}t_{i}\right) ^{n}d\mu_{\mathbf{k}}\nonumber\\
& =\sum_{n=0}^{\infty}\frac{\left( \widetilde{k}\right) _{n}}{n!}r^{n}\sum_{\alpha\in\mathbb{N}_{0}^{N},\left\vert \alpha\right\vert =n}\binom
{n}{\alpha}\frac{1}{\left( \widetilde{k}\right) _{\left\vert \alpha
\right\vert }}{\textstyle\prod_{i=1}^{N}}
\left( k_{i}\right) _{\alpha_{i}}\nonumber\\
& =\sum_{\alpha\in\mathbb{N}_{0}^{N}}r^{\left\vert \alpha\right\vert }{\textstyle\prod_{i=1}^{N}}
\frac{\left( k_{i}\right) _{\alpha_{i}}}{\alpha_{i}!}={\textstyle\prod_{i=1}^{N}}
\left( 1-ra_{i}\right) ^{-k_{i}}.\end{aligned}$$ We used the negative binomial theorem and the multinomial theorem with the multinomial coefficient $\binom{n}{\alpha}=\frac{n!}{\alpha!}$.
The required value is the integral of $d\mu_{\mathbf{k}}$ over the simplex with vertices $\left( 0,0,\ldots,0\right) ,\xi_{i}\left( x\right) $ for $1\leq i\leq N-1$. Set $\xi_{i}^{\prime}\left( x\right) =\frac{a_{N}-x}{a_{N}-a_{i}}$. Change variables to $t_{i}=\xi_{i}^{\prime}\left(
x\right) s_{i}$, then$$\begin{aligned}
1-F\left( x\right) & =\frac{\Gamma\left( \widetilde{k}\right) }{\Gamma\left( k_{N}\right) }\prod_{i=1}^{N-1}\frac{\xi_{i}^{\prime}\left(
x\right) ^{k_{i}}}{\Gamma\left( k_{i}\right) }\int_{\mathbb{T}_{N-1}}{\textstyle\prod_{i=1}^{N-1}}
s_{i}^{k_{i}-1}\left( 1-\sum_{i=1}^{N-1}\xi_{i}^{\prime}\left( x\right)
s_{i}\right) ^{k_{N}-1}ds_{1}\ldots ds_{N-1}\\
& =\frac{\Gamma\left( \widetilde{k}\right) }{\Gamma\left( k_{N}\right)
\Gamma\left( \widetilde{k}-k_{N}\right) }\prod_{i=1}^{N-1}\xi_{i}^{\prime
}\left( x\right) ^{k_{i}}\sum_{\alpha\in\mathbb{N}_{0}^{N-1}}\frac{\left(
1-k_{N}\right) _{\left\vert \alpha\right\vert }}{\left( \widetilde{k}-k_{N}\right) _{\left\vert \alpha\right\vert +1}}\prod_{i=1}^{N-1}\frac{\left( k_{i}\right) _{\alpha_{i}}}{\alpha_{i}!}\xi_{i}^{\prime}\left(
x\right) ^{\alpha_{i}}. \nonumber\end{aligned}$$ The negative binomial series converges when $0\leq\xi_{i}^{\prime}\left(
x\right) <1$ for all $i$, that is, $a_{N-1}<x\leq a_{N}$. Now replace $\xi_{i}^{\prime}\left( x\right) $ by $\frac{a_{N}-x}{a_{N}-a_{i}}$ to obtain the stated formula.
The proof follows the method described in Henrici [@H vol. 1, p. 555]. Use the notation from equation (\[parfrac1\]). Set $f_{i}\left( r\right) ={\displaystyle\prod\nolimits_{j\neq i}}
\left( 1-ra_{j}\right) ^{-k}$, then$${\displaystyle\prod\nolimits_{j=1}^{N}}
\left( 1-ra_{j}\right) ^{-k}=\sum_{j=1}^{k}\frac{\beta_{ij}}{\left(
1-ra_{i}\right) ^{j}}+q_{i}\left( r\right) f_{i}\left( r\right) ,$$ where $q_{i}\left( r\right) $ is a polynomial. Multiply the equation by $\left( 1-ra_{i}\right) ^{k}$ to obtain$$f_{i}\left( r\right) =\sum_{j=1}^{k}\beta_{ij}\left( 1-ra_{i}\right)
^{k-j}+\left( 1-ra_{i}\right) ^{k}q_{i}\left( r\right) f_{i}\left(
r\right) .$$ Apply $\left( \frac{d}{dr}\right) ^{m}$ to both sides and set $r=\dfrac
{1}{a_{i}}$. This cancels out every term on the right side except for $j=k-m$; this term becomes $\beta_{i,k-m}\left( -a_{i}\right) ^{m}m!$. By the generalized product rule$$\left( \frac{d}{dr}\right) ^{m}f_{i}\left( r\right) =\sum_{\alpha
\in\mathbb{N}_{0}^{N},\left\vert \alpha\right\vert =m,\alpha_{i}=0}\binom
{m}{\alpha}\prod_{j=1,j\neq i}^{N}\left( k\right) _{\alpha_{i}}a_{j}^{\alpha_{i}}\left( 1-ra_{j}\right) ^{-k-\alpha_{j}}.$$ Set $r=\frac{1}{a_{i}}$ to get the stated values of $\beta_{i,k-m}$ (note $\left( 1-\frac{a_{j}}{a_{i}}\right) ^{-k-\alpha_{j}}=a_{i}^{k+\alpha_{i}}\left( a_{i}-a_{j}\right) ^{-k-\alpha_{j}}$).
|
---
abstract: 'We classify purely inseparable morphisms of degree $p$ between rational double points (RDPs) in characteristic $p$. Using such morphisms, we show that any RDP admit a finite smooth covering.'
author:
- Yuya Matsumoto
date: '2020/03/24'
title: Inseparable coverings of rational double points in positive characteristic
---
Introduction
============
We work over an algebraically closed field $k$ of characteristic $p > 0$.
We consider $2$-dimensional complete local $k$-algebras that are either rational double points (RDPs for short) or smooth, and we call them *at most RDPs*. In this paper we will classify (Theorem \[thm:main\]) finite purely inseparable morphisms $\Spec B \to \Spec B'$ of degree $p$ between at most RDPs $B$ and $B'$.
We have an application on finite smooth coverings of RDPs. While the universal covering (to be precise, the normalization of the universal covering of the complement of the closed point) of an RDP in characteristic $0$ is always smooth, in characteristic $p > 0$ this may be a simply-connected RDP. Using purely inseparable morphisms between at most RDPs given in Theorem \[thm:main\], we obtain (Theorem \[thm:smooth covering\]) a finite covering $\map{\pi}{\Spec \tilde{B}}{\Spec B}$ of an RDP $B$, by a smooth local ring $\tilde{B}$, that is “unramified” in a certain sense (Definition \[def:unramified\]).
Let us state the classification theorem. Let $\map{\pi}{\Spec B}{\Spec B'}$ be a finite purely inseparable morphism of degree $p$. Then it is given as the quotient by a $p$-closed derivation $D$ on $B$ (unique up to $\Frac(B)^*$-multiple).
The relation between $B$ and $B'$ is reciprocal in the sense that knowing $\map{\pi}{\Spec B}{\Spec B'}$ is equivalent to knowing $\map{\pi'}{\Spec B'}{}{\Spec \pthpower{B}}$. $\pi'$ is also purely inseparable of degree $p$ and hence given as the quotient by a $p$-closed derivation $D'$ on $B'$. We say that the morphisms $\pi$ and $\pi'$ are dual to each other.
We may assume that the fixed locus $\Fix(D)$ of $D$ (Definition \[def:fixed locus\]), which is a closed subscheme of $\Spec B$, satisfies exactly one of the following three conditions:
(a) \[case:empty\] $\Fix(D) = \emptyset$.
(b) \[case:point\] $\Fix(D) = \set{\fm}$.
(c) \[case:non-principal\] $\Fix(D)$ has non-principal divisorial part.
Similarly we may assume that $\Fix(D')$ satisfies exactly one of (\[case:empty\]$'$), (\[case:point\]$'$), (\[case:non-principal\]$'$). Thus there are a priori $3 \times 3$ possibilities.
\[thm:main\] Suppose $B$ and $B'$ are complete $2$-dimensional local rings over $k$, both either smooth or RDP. Suppose $\Spec B \to \Spec B'$ is purely inseparable of degree $p$. We have the following.
- If $B'$ is smooth, then (\[case:empty\]) holds.
- If $B$ is smooth, then (\[case:empty\]$'$) holds.
- (\[case:non-principal\]) holds if and only if (\[case:non-principal\]$'$) holds.
By the last assertion, there remain $5$ possibilities among $3 \times 3$, and in each case we have the following classification.
1. If (\[case:empty\]) and (\[case:empty\]$'$) hold, then both $B$ and $B'$ are smooth. In this case, it is known (Rudakov–Shafarevich [@Rudakov--Shafarevich:inseparable]\*[Theorem 1 and Corollary]{} and Ganong [@Ganong:frobenius]\*[Theorem]{}) that $B = k[[x,y]]$ and $B' = k[[x,y^p]]$ for some $x,y \in B$.
2. \[case:non-fixed\] If (\[case:empty\]) and (\[case:point\]$'$) hold, then $B \cong k[[x,y,z]]/(F) \supset B' \cong k[[x,y,z^p]]/(F)$ and $D(x) = D(y) = 0$, where $F$ is one of Table \[table:derivations on RDPs:non-fixed\].
3. \[case:non-fixed2\] Dually, if (\[case:point\]) and (\[case:empty\]$'$) hold, then either $B \cong k[[x,y]]$ or $B \cong k[[x,y,z]]/(F)$ with $F$ and $D$ as in one of Table \[table:derivations on RDPs:non-fixed2\], up to replacing $D$ with a unit multiple. $B'$ is generated by $x^p,y^p$ and one more element if $B$ is smooth and by $x^p,y^p,z$ if $B$ is an RDP, subject to an equation similar to the one in the dual case in Table \[table:derivations on RDPs:non-fixed\].
4. \[case:both fixed:principal\] Suppose (\[case:point\]) and (\[case:point\]$'$) hold. Then $(\Sing(B), \Sing(B'))$ is one of Table \[table:derivations on RDPs:fixed\], and there are elements $x,y,z,w \in B$ such that $$\begin{aligned}
B &= k[[x,y,z,w]] / (x^p - P(y^p,z,w), w - Q(z,y,x)) \quad \text{and} \\
B' &= k[[w,z,y^p,x^p]] / (w^p - Q(z,y,x)^p, x^p - P(y^p,z,w)),
\end{aligned}$$ with $P$ and $Q$ as in the table up to terms of high degree. Up to a unit multiple, $D$ satisfies $(D(x),D(y),D(z)) = (-Q_y, Q_x, 0)$.
5. \[case:both fixed:non-principal\] Suppose (\[case:non-principal\]) and (\[case:non-principal\]$'$) hold. Let $l^{(\prime)}$ be the order of $\divisorialfix{D^{(\prime)}}$ in $\Pic(B^{(\prime)})$. Then $l = l'$ and $l \divides (p-1)$, and we have $B \cong k[[x,y,z]]/(F)$ with $D(z) = 0$ and $B' \cong k[[w,z,y^p]]/(F')$, where $(l, \Sing(B), \Sing(B'), F, w)$ is one of Table \[table:derivations on RDPs:non-principal\]. Moreover, $\pi$ canonically induces a commutative diagram $$\begin{tikzcd}
\Spec \bar{B} \arrow[r,"\bar{\pi}"] \arrow[d] &
\Spec \bar{B}' \arrow[d] \\
\Spec B \arrow[r,"\pi"] &
\Spec B'
\end{tikzcd}$$ where $\map{\bar{\pi}}{\Spec \bar{B}}{\Spec \bar{B}'}$ is a morphism satisfying the assumptions of this theorem and satisfying (\[case:empty\] or \[case:point\]) and (\[case:empty\]$'$ or \[case:point\]$'$), and moreover equivariant with respect to the $\bZ/l\bZ$-actions on $\Spec \bar{B}$ and $\Spec \bar{B}'$ whose quotients are the vertical maps.
In Table \[table:derivations on RDPs:fixed\], $\positive{q} := \max\set{0, q}$ denotes the positive part of a real $q$.
We use Artin’s notation $D_n^r$, $E_n^r$ [@Artin:RDP] of non-taut RDPs with the following exception: We say that $k[[x,y,z]] / (z^2 + x^2 y + z y^n + z x y^{n-s})$ ($n \geq 2$, $0 \leq s \leq n-1$) in characteristic $2$ to be of type $D_{2n+1}^{s + 1/2}$, instead of Artin’s notation $D_{2n+1}^s$. Consequently, the range of $r$ for $D_{2n+1}^r$ is $\set{\frac{1}{2}, \frac{3}{2}, \dots, \frac{2n-1}{2}}$. Under this notation, the RDP defined by $z^2 + x^2 y + z x y^m + y^n = 0$ ($m \geq 1$, $n \geq 2$) is of type $D_{n+2m}^{n/2}$ regardless of the parity of $n$, and this makes it easier to write Table \[table:derivations on RDPs:fixed\] and the proofs of Theorem \[thm:smooth covering\] and Proposition \[prop:smooth covering etale last\].
We use the convention that $D_3 := A_3$ and that $A_0$ is a smooth point.
\[conv:RDP\] Let $B$ be an at most RDP. For simplicity, we write $\Pic(B) := \Pic((\Spec B)^{\sm})$ and call it the Picard group of $B$. Since we assume that $B$ is complete, it follows from [@Lipman:rationalsingularities]\*[Proposition 17.1]{} that this group is determined from the Dynkin diagram as in Table \[table:Picard group of RDP\] and is independent of the characteristic and the coindex.
We say that a finite morphism $\map{f}{\Spec B'}{\Spec B}$ between at most RDPs is unramified if the restriction of $f$ outside $\fm_B$ is étale. Similarly we say of the universal coverings of RDPs. We note that then, since $\Omega_{B'/k}^1$ is isomorphic to $\Omega_{B/k}^1 \otimes_B B'$ outside the closed point, a derivation $D$ on $B$ induces a derivation on $\Spec(B') \setminus \set{\fm_{B'}}$, hence a derivation on $B'$, which we will denote by $f^*(D)$.
smooth $A_n$ $D_{2m}$ $D_{2m+1}$ $E_6$ $E_7$ $E_8$
-------- ---------------- ---------------- ------------ ------------ ------------ -------
$0$ $\bZ/(n+1)\bZ$ $(\bZ/2\bZ)^2$ $\bZ/4\bZ$ $\bZ/3\bZ$ $\bZ/2\bZ$ $0$
: Local Picard groups of Henselian RDPs (in any characteristic)[]{data-label="table:Picard group of RDP"}
The proof of Theorem \[thm:main\] is given in Section \[sec:proof\] after some preparations on derivations in Section \[sec:derivation\].
Preliminaries on $p$-closed derivations {#sec:derivation}
=======================================
The next formula is well-known.
Let $B$ be an $\bF_p$-algebra, $a$ an element of $B$, and $D$ a derivation on $B$. Then $$(aD)^p = a^p D^p + (aD)^{p-1}(a) D.$$
\[cor:h unit\] Suppose a derivation $D$ on $B$ is $p$-closed in the sense that there exists $h \in B$ with $D^p = hD$.
- For any $a \in B$, $aD$ is also $p$-closed.
- If $B$ is a local domain, $D \neq 0$, $\Image D \subset \fm_B$, $a \in \fm_B \setminus \set{0}$, and $(aD)^p = h_1 aD$, then $h_1 \not\in B^*$.
\[lem:Dh=0\] Let $B$ be an $\bF_p$-algebra, $h$ an element of $B$, and $D$ a $p$-closed derivation on $B$ with $D^p = h D$. Assume either $B$ is reduced, or $\Image D$ contains a regular element of $B$. Then $D(h) = 0$.
We have $h D D = D^p D = D D^p = D(h D) = D(h) D + h D D$, hence $D(h) D = 0$. If $B$ is reduced, then $D(h)^2 = 0$ and $D(h) = 0$. If $D(a) \in \Image D$ is a regular element, then $D(h) D(a) = 0$ and $D(h) = 0$.
$D(h) = 0$ does not hold in general without the assumptions. For example, let $B = k[t]/(t^p)$ and $D = t \partialdd{t}$. Then $D^p = h D$ with $h = 1 + t^{p-1}$, but $D(h) = (p-1)t^{p-1} \neq 0$. (But we also have $D^p = h' D$ with $h' = 1$, and then $D(h') = 0$.)
\[def:fixed locus\] Suppose $D$ is a derivation on a scheme $X$. The *fixed locus* $\Fix(D)$ is the closed subscheme of $X$ corresponding to the sheaf $(\Image (D))$ of ideals generated by $\Image(D) = \set{D(a) \mid a \in \cO_X}$. A *fixed point* of $D$ is a point of $\Fix(D)$.
Assume $X$ is a smooth irreducible variety and $D \neq 0$. Then $\Fix(D)$ consists of its divisorial part $\divisorialfix{D}$ and non-divisorial part $\isolatedfix{D}$. If we write $D = f \sum_i g_i \partialdd{x_i}$ for some local coordinate $x_i$ with $g_i$ having no common factor, then $\divisorialfix{D}$ and $\isolatedfix{D}$ corresponds to the ideal $(f)$ and $(g_i)$ respectively.
Assume $X$ is a smooth irreducible variety and suppose $D \neq 0$ is now a *rational* derivation, locally of the form $f^{-1} D'$ for some regular function $f$ and (regular) derivation $D'$. Then we define $\divisorialfix{D} = \divisorialfix{D'} - \divisor(f)$ and $\isolatedfix{D} = \isolatedfix{D'}$.
If $X$ is only normal, then we can still define $\divisorialfix{D}$ as a Weil divisor.
Rudakov–Shafarevich [@Rudakov--Shafarevich:inseparable] uses the term *singularity* for the fixed locus. We do not use this, as we want to distinguish them from the singularities of the varieties.
Suppose $X_0 \namedto{\pi_0} X_1 \namedto{\pi_1} \dots \namedto{\pi_{n-1}} X_n = \pthpower{X_0}$ is a sequence of purely inseparable morphisms of degree $p$ between $n$-dimensional integral normal varieties, with each $\pi_i$ given by a $p$-closed rational derivation $D_i$ on $X_i$. Then $K_{X_0} = -\sum_{i = 0}^{n-1} (\pi_{i-1} \circ \dots \circ \pi_0)^*( \divisorialfix{D_i})$.
\[cor:dual derivation\] If $\map{\pi}{X = \Spec B}{X' = \Spec B'}$ and $D$, $D'$ are as in Theorem \[thm:main\], then the order of $\divisorialfix{D}$ in $\Pic(B)$ is equal to that of $\divisorialfix{D'}$ in $\Pic(B')$.
By applying the previous proposition to the sequence $X \to X' \to \pthpower{X}$ and using $K_X = 0$, we obtain $\divisorialfix{D} + \pi^*(\divisorialfix{D'}) = 0$ in $\Pic(B)$. Dually we have $\pi'^*(\divisorialfix{\pthpower{D}}) + D' = 0$ in $\Pic(B')$.
\[prop:2-forms\] Suppose $\map{\pi}{X}{X'}$ is a purely inseparable morphism of degree $p$ between smooth varieties of dimension $m$, induced by a $p$-closed rational derivation $D$ such that $\Delta := \Fix(D)$ is divisorial. Then we have isomorphisms $\Omega^m_{X/k}(\Delta) \cong (\pi^* \Omega^m_{X'/k})(p\Delta)$ and $(\pi_* \Omega^m_{X/k}(\Delta))^D \cong \Omega^m_{X'/k}(\pi_* \Delta)$ sending $f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge D(g)^{-1} dg
\mapsto f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge D(g)^{-p} d(g^p)$ if $f_i,g \in \cO_X$, $D(f_i) = 0$ ($1 \leq i \leq m-1$), and $D(g)^{-1} \in \cO_X(\Delta)$, and for the second morphism if moreover $D(f_0) = 0$. In particular, we obtain the Rudakov–Shafarevich formula $K_X = \pi^* K_{X'} + (p-1) \Delta$.
See [@Matsumoto:k3alphap]\*[Section 2.1]{} for the definition of the action of $D$ on $m$-forms. This isomorphism (although it depends on the choice of $D$) may be considered as an analogue of the natural pullback isomorphism $\Omega^m_{X/k} \cong \pi^* \Omega^m_{X'/k}(R)$ for a finite morphism $\map{\pi}{X}{X'}$, with $R = 0$ if and only if $\pi$ is unramified (étale) in codimension $1$. Thus we make the following definition.
\[def:unramified\] We say that a finite purely inseparable morphism $\pi$ of degree $p$ between normal varieties of dimension $m$, equipped with a $p$-closed derivation $D$ giving $\pi$, is *unramified* if $\codim \Fix(D) \geq 2$. If $\pi$ is as in Theorem \[thm:main\], it is unramified if and only if it satisfies (\[case:empty\]) or (\[case:point\]).
We say that a finite morphism $\map{\pi}{X}{Y}$ between normal varieties (of dimension $m$) is unramified if it can be decomposed as $X \namedto{\pi_{\mathrm{i}}} Z \namedto{\pi_{\mathrm{s}}} Y$ with $\pi_{\mathrm{i}}$ a composite of finite purely inseparable morphisms of degree $p$ that are unramified (in the above sense) and $\pi_{\mathrm{s}}$ finite and étale outside a closed subscheme of codimension $\geq 2$.
\[rem:unramified\] An obvious shortcoming of this definition is that it depends on the choice (and the existence) of the derivation. This yields the following difficulties.
- Suppose both $\pi = \pi_1 \circ \pi_2$ and $\pi_2$ are unramified. $\pi_1$ may not be unramified.
- Suppose $\map{\pi}{X}{Y}$ is unramified and moreover $G$-equivariant with respect to actions of a finite group $G$ on $X$ and $Y$. The induced morphism $X/G \to Y/G$ may not be unramified.
For example, assume $p > 2$ and let $\map{\phi}{A_1}{A_2}$ be a purely inseparable isogeny of degree $p$ between abelian surfaces, $\map{f_i}{A_i}{X_i = A_i / \set{\pm 1}}$ the quotient morphisms, and $\map{\phi'}{X_1}{X_2}$ the purely inseparable morphism of degree $p$ induced by $\phi$. Clearly $f_1$ and $f_2$ are unramified. $\phi$ is induced by a derivation $D$ (unique up to scalar) on $A_1$, which is fixed-point-free, and hence $\phi$ is unramified. However there is no regular derivation inducing $\phi'$ (since $D$ is not $[-1]$-invariant) and hence $\phi'$ is not unramified in the sense of Definition \[def:unramified\].
Proof of classification {#sec:proof}
=======================
Reductions
----------
We first note that the conditions (\[case:empty\]), (\[case:point\]), (\[case:non-principal\]) are pairwise exclusive, and that we can replace $D$ so that one of these holds. Indeed, replacing $D$ changes $\Fix(D)$ precisely by a principal divisor.
It follows from [@Matsumoto:k3alphap]\*[Theorem 3.3(2)]{} that if $B'$ is smooth (resp. $B$ is smooth) then (\[case:empty\]) (resp. (\[case:empty\]$'$)) holds.
The equivalence (\[case:non-principal\]) $\iff$ (\[case:non-principal\]$'$) and the equality $l = l'$ follow from Corollary \[cor:dual derivation\]. This case will be considered in Section \[sec:non-principal\].
As mentioned earlier, the case where $B$ and $B'$ are both smooth is a result of Rudakov–Shafarevich and Ganong.
Suppose (\[case:empty\]) holds and $B$ is an RDP. Then by [@Matsumoto:k3mun]\*[Proposition 4.8]{}, there is $F \in k[[x,y,z^p]]$ such that $B = k[[x,y,z]]/(F)$, $D = \partialdd{z}$, and $B' = k[[x,y,z^p]] / (F)$, and moreover all possible $F$ are classified. The result is as in Table \[table:derivations on RDPs:non-fixed\]. Also the case (\[case:empty\]$'$) follows dually, and the result is as in Table \[table:derivations on RDPs:non-fixed2\]. It also follows that if (\[case:empty\]) and (\[case:empty\]$'$) hold then both $B$ and $B'$ are smooth.
Suppose (\[case:point\]) and (\[case:point\]$'$). If moreover $D$ is of multiplicative type (i.e. $h = 1$) then such derivations are classified in [@Matsumoto:k3mun]\*[Proposition 4.10]{}. More generally, if $h \in B^*$, then we may replace $D$ with $h^{-1/(p-1)} D$, which is of multiplicative type (since $D(h) = 0$ by Lemma \[lem:Dh=0\]). Hereafter we assume $h \in \fm$ and $h' \in \fm'$.
By [@Matsumoto:k3alphap]\*[Corollary 3.5]{}, there are partial resolutions $\map{f}{\tilde{X}}{X}$ and $\map{f'}{\tilde{X'}}{X'}$ with a morphism $\map{\tilde{\pi}}{\tilde{X}}{\tilde{X}'}$ as in the diagram $$\begin{tikzcd}
\tilde{X} \arrow[r,"\tilde{\pi}"] \arrow[d,"f"] & \tilde{X'} \arrow[d,"f'"] \\
X \arrow[r,"\pi"] & X' ,
\end{tikzcd}$$ and a derivation $\tilde{D}$ on $\tilde{X}$ with $\tilde{D} = D$ outside the exceptional loci, satisfying the following properties: $\tilde{X'} = \tilde{X}^{\tilde{D}}$, $\divisorialfix{\tilde{D}} = 0$, $\Sing(\tilde{X}) \cap \tilde{\pi}^{-1}(\Sing(\tilde{X}')) = \emptyset$ (in particular, $\Sing(\tilde{X}) \cap \Fix(\tilde{D}) = \emptyset$), and $\Fix(\tilde{D}) \neq \emptyset$. We will refer to the points of $\Fix(\tilde{D})$ as *upper* fixed points of $D$.
Now suppose $p \geq 3$ (we will consider the case of $p = 2$ separately). If $h \in \cO_{\tilde{X}, w}^*$ at some upper fixed (smooth) point $w \in \Fix(\tilde{D})$, then $h \in B \cap \cO_{\tilde{X}, w}^* = B^*$, contradicting our assumption that $h \in \fm$. Thus we may assume $h \in \fm_w$ for any $w \in \Fix(\tilde{D})$. By the classification [@Matsumoto:k3alphap]\*[Lemma 3.6(2)]{} of such quotients $B'_w := \cO_{\tilde{X'},\pi(w)}$, either $p = 3$ and $B'_w$ is either $E_6^0$ or $E_8^0$, or $p = 5$ and $B'_w$ is $E_8^0$. In other words, $B'$ admits one of these RDPs as a partial resolution. Since there is no Dynkin diagram strictly containing $E_8$, the only possibility is that $p = 3$ and $\Sing(B') \in \set{E_7^r, E_8^r}$. $E_7^1$ and $E_8^2$ are impossible since their partial resolution is $E_6^1$, not $E_6^0$, by [@Matsumoto:k3rdpht]\*[Lemma 4.7]{}. Also $E_7^0$ is also impossible, since by replacing $B$ and $B'$ with their universal coverings we obtain a derivation with the same properties and with quotient $E_6^0$, which is impossible by above. Hence $\Sing(B') \in \set{E_8^0, E_8^1}$. Applying the same argument to $D'$, we obtain the same conclusion for $\Sing(B)$.
Summarizing the reductions so far, it remains to consider the following cases, which will be done in subsequent sections.
- $p = 3$, both $B$ and $B'$ are either $E_8^0$ or $E_8^1$, and both $h$ and $h'$ are non-unit at every upper fixed point of $D$ and $D'$. (Section \[sec:3E8\])
- $p = 2$, and both $h$ and $h'$ are non-unit of $B$ and $B'$. (Section \[sec:2\])
- The divisorial part $\divisorialfix{D}$ of $\Fix(D)$ is not principal. (Section \[sec:non-principal\])
\[lem:delta\] Suppose $\map{\pi}{\Spec B}{\Spec B'}$ is as in Theorem \[thm:main\] and satisfies (\[case:point\]). Let $\delta = \dim_k \Image (\fm'/\fm'^{2} \to \fm/\fm^{2})$. Then $\delta \in \set{0,1}$.
We have $\delta < 3$ since $\fm'$ cannot generate $\fm$. Assume $\delta = 2$. We may assume $\fm = (x, y, z)$ and $\fm' = (x, y, z^p)$. If the ideal $(F) = \Ker (k[[x,y,z]] \to B)$ does not have a generator that belongs to $k[[x,y,z^p]]$, then $B' = k[[x,y,z^p]] / (F^p)$, but this cannot be an RDP. Hence $(F)$ is generated by an element of $k[[x,y,z^p]]$, which we may assume to be $F$ itself, and $B' = k[[x,y,z^p]] / (F)$. Then $D$ is proportional to $\partialdd{z}$, contradicting the assumption.
Case of $E_8$ in $p = 3$ {#sec:3E8}
------------------------
Suppose $p = 3$, both $B$ and $B'$ are either $E_8^0$ or $E_8^1$, and both $h$ and $h'$ are non-unit at every upper fixed point of $D$. There should be exactly one upper fixed point of $D$, and its quotient should be $E_6^0$. Let $X_1 \to X$ be the blow-up at the closed point. Then $\Sing(X_1) = E_7^0$. Since there is no derivation on $E_7^0$ fixing precisely the closed point with non-unit $h$ and RDP quotient, this point is not fixed, and the upper fixed point should be on $X_1^{\sm}$.
Suppose $B$ is $E_8^1$. We may assume that $B = k[[x,y,z]] / (z^2 + x^3 + y^5 + x^2 y^3)$. Then the space of derivations on $B$ are generated by the following elements $D_1, D_2, D_3$. Then $D$ has at least two fixed points $(x = 1 + y_1^2 = z_1 = 0)$ on $\Spec B[y/x,z/x] = \Spec k[[x]][y_1,z_1] / (z_1^2 + x + x^3 y_1^3 (y_1^2 + 1)) \subset X_1$, contradiction.
$D_1$ $D_2$ $D_3$
------------- -------------- ---------------------- --------------
$x$ $0$ $z$ $y$
$y$ $z$ $0$ $-x$
$z$ $-y^4$ $-xy^3$ $0$
$x$ $0$ $x z_1$ $x y_1$
$y_1 = y/x$ $z_1$ $- y_1 z_1$ $-1 - y_1^2$
$z_1 = z/x$ $-x^3 y_1^4$ $-x^3 y_1^3 - z_1^2$ $- y_1 z_1$
Dually, $B'$ is not $E_8^1$.
Now suppose $B$ and $B'$ are $E_8^0$. Let $\delta = \dim_k \Image (\fm'/\fm'^{2} \to \fm/\fm^{2})$ as in Lemma \[lem:delta\]. We first show that $\delta = 1$ and that $D \restrictedto{\fm/\fm^2}$ is nilpotent of index $3$. We write $B = \Spec k[[x,y,z]] / (z^2 + x^3 + y^5)$ during the proof of this claim. Then the space of derivations on $B$ are generated by the following elements $D_1, D_2$.
$D_1$ $D_2$
------------- ----------------- -----------
$x$ $0$ $1$
$y$ $z$ $0$
$z$ $-y^4$ $0$
$x$ $0$ $1$
$y_1 = y/x$ $z_1$ $- y_1/x$
$z_1 = z/x$ $- x^3 y_1^4$ $- z_1/x$
$x_2 = x/y$ $- x_2 z_2$ $1/y$
$y$ $y z_2$ $0$
$z_2 = z/y$ $- y^3 - z_2^2$ $0$
Write $D = f_1 D_1 + f_2 D_2$ with $f_i \in B$, and suppose $f_i \equiv f_{i0} + f_{i1} x + f_{i2} y + f_{i3} z \pmod{\fm^2}$ with $f_{ij} \in k$. Since $D$ extends to $X_1$, we have $f_2 \in \fm$ ($f_{20} = 0$). Since $D$ does not fix the origin of $\Spec B[x/y, z/y] = \Spec k[[y]][x_2,z_2] / (z_2^2 + x_2^3 y + y^3) \subset X_1 = \Bl_{\fm} X$, we have $f_{22} \neq 0$. Since $h \not\in B^*$, we have $f_{21} = 0$. Let $\fm_1 = (y_1, z_1)$ be the maximal ideal at the origin of $\Spec B[y/x, z/x] = \Spec k[[x]][y_1,z_1] / (z_1^2 + x + x^3 y_1^5) \subset X_1$, which is a fixed point of $D$. Since the image of this point is $E_6^0$, we have $D(\fm_1) \not\subset \fm_1^2$, hence $f_{10} \neq 0$. Then $v := D(D(x)) - hx$ satisfies $D(v) = (D^3 - h D)(x) = 0$ (by Lemma \[lem:Dh=0\]) and $v \equiv f_{10} f_{22} z \not\equiv 0 \pmod{\fm^2}$. Hence $D \restrictedto{\fm/\fm^2}$ is nilpotent of index $3$ and $\delta \geq 1$.
We note that if $(B = k[[x,y,z]]/(F),\fm)$ is an RDP of type $D_n$ or $E_n$ in characteristic $\geq 3$ and $D$ is a derivation on $B$ with $D(\fm) \subset \fm$ and $D \restrictedto{\fm/\fm^2}$ is nilpotent of index $3$, then the degree $2$ part of $F$ is of the form $l^2$ with $l \in \Ker(D \restrictedto{\fm/\fm^2})$ (otherwise $D(F)$ cannot be zero).
We will show that $B$ admit elements $x,y,z,w$ as in the statement of Theorem \[thm:main\](\[case:both fixed:principal\]). We may assume $\fm = (x,y,z)$, $z \in B'$, and $\fm' = (Y, z, w)$, where $Y := y^3$. We may moreover assume $x^3 \in \fm'^2$ and $w \in \fm^2$. Write $w = Q(z,y,x) \in (z,y,x)^2 \subset k[[z,y,x]]$. Since $x^3 \in \fm'^2$, there exists $u \in k[[x,y,z]]$ and $P \in (Y,z,w)^2 \subset k[[Y,z,w]]$ such that $u F = - x^3 + P(Y, z, Q(z,y,x))$. If $u \not\in k[[x,y,z]]^*$, then the degree $2$ part of $F$ cannot be the square of a linear term contained in $\Ker(D \restrictedto{\fm/\fm^2}) = k z$, contradicting the observation above. Hence $u \in k[[x,y,z]]^*$, and we may assume $u = 1$: $F = - x^3 + P(Y, z, Q(z,y,x))$, and also $F' = -w^3 + \tilde{Q}(z^3, Y, P(Y,z,w))$, where $\tilde{Q}(z^3,Y,P) := Q(z,y,x)^3$. Since $B$ and $B'$ are $E_8$, we obtain the following: $P$ has $z^2$; $Q$ has $y^2$; we may assume $P$ does not have $zw$ nor $w^2$; we may assume $Q$ does not have $yx$ nor $x^2$; $P$ has $Yw$; $Q$ has $zx$. By replacing $x,y,z,w,F,F'$ with scalar multiples, we may assume $F = -x^3 + z^2 + y^3 w + (\dots)$, $F' = (-w + y^2 + z x + (\dots))^3$. Since $0 = D(w) = D(x) Q_x + D(y) Q_y$ and since $Q_x \equiv z \pmod{\fm^2}$ and $Q_y \equiv 2 y \pmod {(z) + \fm^2}$ have no common factor, we may assume $(D(x), D(y)) = (- Q_y, Q_x)$.
Case of characteristic $2$ {#sec:2}
--------------------------
Suppose $p = 2$ and both $h$ and $h'$ are non-unit.
Let $\delta = \dim_k \Image (\fm'/\fm'^{2} \to \fm/\fm^{2})$ as in Lemma \[lem:delta\] and similarly $\delta' = \dim_k \Image (\secondpower{\fm}/(\secondpower{\fm})^{2} \to \fm'/\fm'^{2})$. We have $\delta, \delta' \in \set{0,1}$. We shall show $\delta = \delta' = 1$.
Assume $\delta = 0$, that is, $\fm' \subset \fm^{2}$. Since $x^2,y^2,z^2 \in \secondpower{\Frac B} \subsetneq \Frac B'$, the elements $x^2,y^2,z^2 \subset \secondpower{\fm} \subset \fm'$ cannot generate $\fm'$. In other words, there exists a nonzero linear combination $f$ of $x^2,y^2,z^2$ that belongs to $\fm'^2$. Since $f \in \fm'^2 \subset \fm^4$, (a unit multiple of) $F$ is of the form $f + G$ with $f \in \secondpower{\fm}$ and $G \in \fm^4$. But such a polynomial cannot define an RDP. Hence $\delta = 1$. Dually $\delta' = 1$.
We may assume that $\fm = (x, y, z)$ and $\fm' = (w, y^2, z)$, and that $x^2 \in \fm'^2$ and $w \in (\secondpower{\fm})^2$.
Since $x^2 \in \fm'^2$, we may assume $F = x^2 + P_0(y^2, z) + P_1(y^2, z) w$ with $P_0 \in \fn'^2$ and $P_1 \in \fn'$, where $\fn'$ is the maximal ideal $(y^2, z)$ of $k[[y^2, z]]$. Similarly we may assume $w = Q_0(z, y) + Q_1(z, y) x$ with $Q_0 \in \fn^2$ and $Q_1 \in \fn$, where $\fn$ is the maximal ideal $(z, y)$ of $k[[z, y]]$. Hence we have $$\begin{aligned}
F &= x^2 + P_0(y^2, z) + P_1(y^2, z) (Q_0(z, y) + Q_1(z, y) x), \\
F' &= w^2 + \tilde{Q}_0(z^2, Y) + \tilde{Q}_1(z^2, Y) (P_0(Y, z) + P_1(Y,z) w),
\end{aligned}$$ where $Y := y^2$, and $\tilde{Q}_i(z^2, Y) := Q_i(z, y)^2$.
We may assume $P_0$ does not have $z^2$, and $\tilde{Q}_0$ does not have $Y^2$.
Suppose $(Q_1)_y \in B^*$. Then $D$ defined by $D(x,y,z) = ((Q_0)_y + (Q_1)_y x, Q_1, 0)$ satisfies $B^D = B'$ and $D^2 = (Q_1)_y D$. Moreover $\divisorialfix{D} = 0$ by Corollary \[cor:h unit\]. This contradicts the assumption that $h \notin B^*$. Hence we have $(Q_1)_y \in \fm$, that is, the coefficient of $y$ in $Q_1$ is $0$. Similarly, the coefficient of $z$ in $P_1$ is $0$.
For $F$ to define an RDP, we need either
- $P_0$ has $y^2 z$, $P_1$ has $z^k$, and either $Q_0$ has $z^m y$ or $Q_1$ has $z^l$, or
- $P_0$ does not have $y^2 z$ but has $z^3$, $P_1$ has $y^2$, and $Q_0$ has $zy$ or $y^3$.
Similarly, for $F'$ to define an RDP, we need either
- $\tilde{Q}_0$ has $z^2 Y$, $\tilde{Q}_1$ has $Y^{k'}$, and either $P_0$ has $Y^{m'} z$ or $P_1$ has $Y^{l'}$, or
- $\tilde{Q}_0$ does not have $z^2 Y$ but has $Y^3$, $\tilde{Q}_1$ has $z^2$, and $P_0$ has $Yz$ or $z^3$.
Combining these conditions, one of the following holds (after possibly replacing $(B,D)$ with $(B',D')$).
- $P_0$ has $y^2 z$, $P_1$ has $z^{m'}$, $Q_0$ has $zy$, and $Q_1$ has $y^{m}$. In this case $B$ is $D_{2m'+2}^{\positive{2-m}}$ and $B'$ is $D_{2m+2}^{\positive{2-m'}}$.
- $P_0$ does not have $y^2 z$ but has $z^3$, $P_1$ has $y^2$, $Q_0$ has $zy$, and $Q_1$ has $y^m$. In this case $B$ is $E_7^{\positive{3-m}}$ and $B'$ is $D_{2m+3}^0$.
- $P_0$ does not have $y^2 z$ but has $z^3$, $P_1$ has $y^2$, $Q_0$ does not have $zy$ but has $y^3$, and $Q_1$ has $z$. In this case $B$ and $B'$ are $E_8^2$.
Non-principal case {#sec:non-principal}
------------------
Suppose $\divisorialfix{D}$ is not principal. Let $l$ be the order of $[\divisorialfix{D}]$ in $\Pic(B)$. We have $l \divides (p-1)$ by the Rudakov–Shafarevich formula (Proposition \[prop:2-forms\]). Let $\map{f}{\bar{X} = \Spec \bar{B}}{X = \Spec B}$ be the unramified covering trivializing the class, and let $\bar{\fm} \subset \bar{B}$ be the maximal ideal. Let $\bar{D}$ be a derivation proportional to $f^* D$ with $\divisorialfix{\bar{D}} = 0$ (which exist since $[\divisorialfix{f^* D}] = 0$). Let $\bar{X}' = \Spec \bar{B}' = \bar{X}^{\bar{D}}$.
Let $G = \Gal(\bar{X}/X) = \Gal(\bar{X'}/X') \cong \bZ/l\bZ$. For $g \in G$, the derivation $g^* \bar{D}$ (see Convention \[conv:RDP\]) can be written as $g^* \bar{D} := g^{\circ} \bar{D} (g^{\circ})^{-1}$, where $\map{g^{\circ}}{\bar{B}}{\bar{B}}$ corresponds (contravariantly) to $\map{g}{\bar{X}}{\bar{X}}$. Since $\bar{B}^{g^* \bar{D}} = \bar{B}^{\bar{D}}$ and $\divisorialfix{g^* \bar{D}} = 0 = \divisorialfix{\bar{D}}$, there exists $\beta_g \in \bar{B}^*$ such that $g^* \bar{D} = \beta_g \cdot \bar{D}$. We have $\beta_{hg} = g^{\circ}(\beta_h) \beta_g$. This shows that ($\beta$ is a $1$-cocycle and), since $G \actson \bar{B}/\bar{\fm} = k$ is trivial, the map $\rho \colon G \namedto{\beta} \bar{B}^* \to (\bar{B}/\bar{\fm})^* = k^*$ is a homomorphism.
We will show that this $\map{\rho}{G}{k^*}$ is injective. Let $G_1 := \Ker \rho$. Let $\bar{D}_1 := \sum_{h \in G_1} h^* \bar{D} = (\sum_{h \in G_1} \beta_h) \cdot \bar{D}$. Then $\bar{D}_1$ is $G_1$-invariant, hence descends to a derivation on $\bar{B}^{G_1}$. Since $\sum_{h \in G_1} \beta_h \in \bar{B}^*$ (since $\sum \beta_h \equiv \sum 1 = \abs{G_1} \not\equiv 0 \pmod{\fm}$), $\bar{D}_1$ has no divisorial fixed locus. In other words, the pullback of $\divisorialfix{D}$ to $X/G_1$ is trivial. By the definition of $\bar{X}$, we obtain $G_1 = \set{1}$.
Summarizing the arguments so far, the at most RDP $\bar{X} = \Spec \bar{B}$, the action of the cyclic group $G$, and the derivation $\bar{D}$ satisfy the following properties.
- $\bar{X} \to \bar{X}/G = X$ is induced by a non-principal divisor class of order dividing $p-1$ on an at most RDP $X$.
- $\bar{D}$ is $p$-closed with $\divisorialfix{\bar{D}} = 0$, and $\bar{X}^{\bar{D}}$ is at most RDP.
- There exist units $\beta_g \in \bar{B}$ satisfying $g^* \bar{D} = \beta_g \cdot \bar{D}$, and the homomorphism $\map[\mapsto]{\rho}{g}{(\beta_g \bmod \fm)}$ is injective.
The first two conditions imply that $(\Sing(B), \Sing(\bar{B}), \Sing(\bar{B}^{\bar{D}}))$ is one in Table \[table:covering\] (with $n \geq 1$), that $(\bar{B}, G) \cong (k[[x,y,z]] / \bar{F}, \spanned{g})$ as in the table, and that $(\bar{B}, \bar{D}) \cong (k[[x,y,z]] / (\bar{F} + \varepsilon), u \bar{D})$ as in the table with $u \in \bar{B}^*$ (not necessarily under the same isomorphism).
First suppose that $\bar{B}$ is of type $A_{n-1}$ ($n \geq 1$) and that $\bar{D}$ fixes the closed point. Let $V = \bar{\fm} / \bar{\fm}^2$. Then $g \in G$ and $\bar{D}$ induce $k$-linear endomorphisms on $V$, denoted by the same symbols, satisfying $g \bar{D} g^{-1} = \rho(g) \cdot \bar{D}$. By the classification, the set of eigenvalues of $\bar{D}$ is of the form $\set{i, -i, 0}$ if $n \geq 2$ and $\set{i, -i}$ if $n = 1$ for some $i \in k^*$. Since this set is invariant under multiplication by $\rho(g)$, we obtain $l = 2$, and the nontrivial element of $G$ interchanges the eigenspaces $V_{\pm i}$. By the classification, $n$ is even and the quotient $B = \bar{B}^G$ is of type $D_{n/2+2}$. This is realized by the example $\bar{B} = k[[x,y,z]] / (xy - z^n)$, $g(x,y,z) = (y,x,-z)$, $\bar{D}(x,y,z) = (x,-y,0)$, and $X \to X'$ is $D_{m+2} \to D_{mp+2}$ ($m = n/2 \geq 1$).
Next suppose that $\bar{B}$ is of type $A_{np-1}$ ($n \geq 1$) and that $\bar{D}$ is fixed-point-free. Let $V = \bar{\fm} / \bar{\fm}^2$ and $V' = \Image(\bar{\fm}' \to V)$: these spaces are stable under $g$. We may assume that $\bar{B} = k[[x,y,z]] / (xy - z^{np})$ and $\bar{D} = \partialdd{z}$, and then $V/V' = k z$. We have $\rho(g) = g \restrictedto{V/V'}$ and, by the classification, $(\rho(g), l)$ for a generator $g$ of $G$ is equal to $(1, l)$, $(-1, 2)$, $(-1, 4)$ respectively if $\bar{B}^G$ is of type $A_{lnp-1}$, $D_{np/2+2}$, $D_{np+2}$. Since $\rho(g)$ should be a primitive $l$-th root of $1$, only the second case (where $n$ is even) is appropriate. This case is indeed realized by the example $\bar{B} = k[[x,y,z]] / (xy - z^{np})$, $g(x,y,z) = (y,x,-z)$, $\bar{D}(x,y,z) = (0,0,1)$, and $X \to X'$ is $D_{mp+2} \to D_{m+2}$ ($m = n/2 \geq 1$).
Next suppose that $\bar{B}$ is smooth and that $\bar{D}$ is fixed-point-free. Again, let $V = \bar{\fm} / \bar{\fm}^2$ and $V' = \Image(\bar{\fm}' \to V)$. Since $G$ preserves these spaces, a generator $g$ of $G$ acts on $V'$ by a primitive $l$-th root $\zeta$ of $1$ and on a complement by $\zeta^{-1}$. By taking an appropriate coordinate we may assume $\bar{B} = k[[x,y]]$, $g(x,y) = (\zeta x, \zeta^{-1} y)$, $\bar{D}(x) = 0$, $\bar{D}(y) \neq 0$. Since $\zeta^{-p} = \zeta^{-1}$ (since $p \equiv 1 \pmod{l}$), the action $g(x,y^p) = (\zeta x, \zeta^{-p} y^p)$ on $\bar{B}' = k[[x,y^p]]$ is also symplectic. Then $X \to X'$ is $A_{l-1} \to A_{l-1}$.
Finally suppose $\bar{B}$ is $E_6^1$ in characteristic $3$. This is realized by the example $\bar{B} = k[[x,y,z]] / (z^2 + x^3 + y^3 + x^2 y^2)$, $g(x,y,z) = (y, x, -z)$, $\bar{D}(x,y,z) = (x, -y, 0)$. Then $X \to X'$ is $E_7^1 \to E_7^1$.
In each case, after a coordinate change we obtain an isomorphism $(\bar{B}, \bar{D}, g) \cong (k[[x,y,z]]/(\bar{F}), \bar{D}, g)$ with $\bar{F}, \bar{D}, g$ as in Table \[table:covering\].
$p$ $l$ $B$ $\bar{B}$ $\bar{B}^{\bar{D}}$ $\bar{F}$ $\bar{D}$ $g$
----- ----- ------------------------- ------------ --------------------- ----------------------------- ------------ ----------------------------
any $l$ $A_{l-1}$ smooth smooth $0, 1$ $\zeta x, \zeta^{-1} y$
any $l$ $A_{ln-1}$ $A_{n-1}$ $A_{np-1}$ $xy - z^n$ $x, -y, 0$ $\zeta x, \zeta^{-1} y, z$
any $2$ $D_{n/2+2}$ ($n$ even) $A_{n-1}$ $A_{np-1}$ $xy - z^{n}$ $x, -y, 0$ $y, x, -z$
any $4$ $D_{n+2}$ ($n$ odd) $A_{n-1}$ $A_{np-1}$ $xy - z^{n}$ $x, -y, 0$ $y, -x, -z$
any $l$ $A_{lnp-1}$ $A_{np-1}$ $A_{n-1}$ $xy - z^{np}$ $0, 0, 1$ $\zeta x, \zeta^{-1} y, z$
any $2$ $D_{np/2+2}$ ($n$ even) $A_{np-1}$ $A_{n-1}$ $xy - z^{np}$ $0, 0, 1$ $y, x, -z$
any $4$ $D_{np+2}$ ($n$ odd) $A_{np-1}$ $A_{n-1}$ $xy - z^{np}$ $0, 0, 1$ $y, -x, -z$
$3$ $2$ $E_7^1$ $E_6^1$ $E_6^1$ $z^2 + x^3 + y^3 + x^2 y^2$ $x, -y, 0$ $y, x, -z$
: Etale coverings $\bar{B}$ induced by non-principal divisors on $B$ of order $l$ dividing $p-1$ and derivations on $\bar{B}$ with RDP quotients[]{data-label="table:covering"}
Smooth coverings of RDPs {#sec:smooth coverings}
========================
\[thm:smooth covering\] Suppose $B$ is a complete local RDP in characteristic $p > 0$. Then there exists a finite extension $B \subset C$ of complete local rings that is unramified (in the sense of Definition \[def:unramified\]) with $C$ smooth. More precisely, there exists a sequence $B \subset C_0 \subset C_1 \subset \dots \subset C_{n}$ ($n \geq 0$) of finite extensions of complete local rings such that
- $C_{n}$ is smooth, and all other $C_i$ are normal,
- $\Spec C_0 \to \Spec B$ is the universal covering, and
- $\Spec C_{i+1} \to \Spec C_{i}$ are purely inseparable of degree $p$ and unramified.
We can take all $C_i$ ($i < n$) to be RDPs if and only if $(p, \Sing(B)) \neq (2, E_8^1)$.
In some (not all) cases the order of the étale and purely inseparable coverings can be reversed, although in this case we cannot always take unramified coverings.
\[prop:smooth covering etale last\] Suppose $B$ is as in Theorem \[thm:smooth covering\]. Then there exists a sequence $B = B_0 \subset B_1 \subset \dots \subset B_n \subset C$ ($n \geq 0$) of finite extensions of complete local rings such that
- $C$ is smooth, and all other $B_i$ are RDPs,
- $\Spec C \to \Spec B_n$ is the universal covering, and
- $\Spec B_{i+1} \to \Spec B_{i}$ are purely inseparable of degree $p$,
if and only if $(p, \Sing(B)) \neq (2, E_8^1), (2, D_N^r)$ ($4r > N$).
See Table \[table:pi1\] for the fundamental groups and the universal coverings of RDPs.
First suppose $B$ is $E_8^1$ in characteristic $2$. Then $C_0 = B$ is not smooth and, by Theorem \[thm:main\], it does not admit any purely inseparable covering of degree $p$ that is at most RDP.
We will give a sequence $C_0 \subset C_1 \subset C_2$ of purely inseparable unramified coverings of degree $p$ such that ($C_1$ is a non-RDP and) $C_2$ is an RDP of another type. Let $$\begin{aligned}
C_2 &= k[[x,y,z]] / (x^2 + y^2 z + z^3 (y^3 + z^2 x)), \\
C_1 &= k[[w,z,Y]] / (w^2 + Y^3 + z^4 (z Y + z^3 w)), \\
C_1' &= k[[x',y',z']] / (x'^2 + z'^3 + y'^2 z' (y'^3 + z'x')), \\
B = C_0 &= k[[w',z',Y']] / (w'^2 + Y'^3 + z'^2 (z'^3 + z' Y' w')).\end{aligned}$$ Define $D_2 \in \Der(C_2)$ by $D_2(x,y,z) = (y^2, z^2, 0)$, then $C_2^{D_2} = C_1$ by $Y = y^2$, $w = y^3 + z^2 x$. Define $D_1 \in \Der(C_1')$ by $D_1(x',y',z') = (y'^2, z', 0)$, then $C_1'^{D_1} = B$ by $Y' = y'^2$, $w' = y'^3 + z' x'$. Then $\Fix(D_1)$ and $\Fix(D_2)$ consists of the closed point, hence the coverings are unramified. We have an isomorphism $\map[\isomto]{\phi}{C_1'}{C_1}$ by $\phi(x',y',z') = (\frac{w + Y^2 z^2}{1 + Y z^4}, z, \frac{Y + z^2 w}{1 + Y z^4})$. The composite $\Spec C_2 \to \Spec C_1 = \Spec C_1' \to \Spec B$ is unramified, and $C_2$ is an RDP of type $D_{11}^{1/2}$.
Now suppose $B$ is not $E_8^1$ in characteristic $2$. It is known that the universal covering of an at most RDP is again an at most RDP. Hence we may assume $B$ is simply-connected. By Theorem \[thm:main\] we have the following sequences.
$A_{p^e-1} \from A_{p^{e-1}-1} \from \dots \from {A_0}$.
For $p = 5$: $E_8^0 \from {A_0}$.
For $p = 3$: $E_8^0 \from {A_0}$, $E_8^1 \from E_6^0 \from {A_0}$.
For $p = 2$: $E_8^3 \from E_7^2 \from D_5^{1/2} \from A_1 \from {A_0}$, $E_8^0 \from {A_0}$, $E_7^1 \from D_7^{1/2} \from {A_0}$, $E_7^0 \from {A_0}$, $D_{2m}^0 \from {A_0}$, $D_{2m+1}^{1/2} \from A_1 \from {A_0}$, $D_{4k-l}^{k-l/2} \from D_{4k-2l}^{\positive{k-l}} \from \dots \from
D_{4k-2^n l}^{0} \from {A_0}$ if $l > 0$ and $k - l/2 > 1/2$, where $n$ is the minimum integer with $4k - 2^{n-1} l \leq 0$.
Note that the correspondence $(r, N) = (k-l/2, 4k-l)$ gives a bijection between the sets $\set{(r, N) \in \frac{1}{2} \bZ \times \bZ \mid N \geq 4, 2r - N \in \bZ, 0 \leq r \leq (N/2)-1}$ and $\set{(k, l) \in \bZ \times \bZ \mid k \geq 1, k \geq l/2, (k,l) \neq (1,1),(1,2)}$, and $4r < N$ if and only if $l > 0$.
Consider $C_1$ and $C_1'$ in the proof of the case of $E_8^1$ in characteristic $2$ in Theorem \[thm:smooth covering\] over a field of characteristic $0$. Then they are isomorphic and it is an exceptional unimodal singularity usually denoted by the symbol $E_{13}$, where the index $13$ stands for the Milnor number (i.e. $\dim_k k[[x,y,z]]/(F_x, F_y, F_z)$ for $k[[x,y,z]] / (F)$) in characteristic $0$ (although in characteristic $2$ it is not the Milnor number nor the Tjurina number).
Again this is impossible for $E_8^1$ in characteristic $2$. If $p = 2$ and $B$ belongs to the set $\set{D_N^r \mid 4r > N}$, then by Theorem \[thm:main\] so does any RDP that is a purely inseparable covering of $B$ of degree $p$, and no member of this family has smooth universal covering.
Suppose otherwise. We shall describe $\Spec B_0 \from \Spec B_1 \from \dots \from \Spec B_{n}$ with $B_n$ having smooth universal covering. By Theorem \[thm:smooth covering\], it suffices to consider the cases with $\pi_1(B) \neq 0$.
$A_{n p^e-1} \from A_{n p^{e-1}-1} \from \dots \from A_{n-1}$ ($p \notdivides n$).
$D_{n p^e + 2} \from D_{n p^{e-1} + 2} \from \dots \from D_{n+2}$ ($p \neq 2$, $p \notdivides n$). (Note that this requires a derivation satisfying (\[case:non-principal\]).)
For $p = 3$: $E_7^0 \from A_1$.
For $p = 2$: $E_6^0 \from A_2$.
Suppose $B_1$ and $B_2$ are at most RDPs (over the same field $k$) in characteristic $p > 0$ having isomorphic fundamental groups $\pi_1(B_1) \cong \pi_1(B_2)$. If $(p, \pi_1) \neq (2, \Dih_n)$ for any $n \geq 1$, then $B_1$ and $B_2$ are connected by a finite purely inseparable morphism.
Here $\Dih_n$ is the dihedral group (of order $2n$).
If $\pi_1 = 0$, then both $B_1$ and $B_2$ are connected to $k[[x,y]]$ by Theorem \[thm:smooth covering\].
Suppose $\pi_1 \neq 0$. Theorem \[thm:main\] shows that $B_1$ and $B_2$ are connected by a purely inseparable morphism of degree $p$ if $(B_1,B_2)$ is one of the following: $(A_{n p^e - 1}, A_{n p^{e'} - 1})$, $(D_{n p^e + 2}, D_{n p^{e'} + 2})$ ($p \neq 2$), $(A_1, E_7^0)$ ($p = 3$), $(A_2, E_6^0)$ ($p = 2$). According to Table \[table:pi1\], general cases (with $(p, \pi_1) \neq (2, \Dih_n)$) follow from this.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I thank Hiroyuki Ito, Kentaro Mitsui, and Hisanori Ohashi for helpful comments and discussions.
|
---
abstract: 'The use of discrete material representation in numerical models is advantageous due to the straightforward way it takes into account material heterogeneity and randomness, and the discrete and orientated nature of cracks. Unfortunately, it also restricts the macroscopic Poisson’s ratio and therefore narrows its applicability. The paper studies the Poisson’s ratio of a discrete model analytically. It derives theoretical limits for cases where the geometry of the model is completely arbitrary, but isotropic in the statistical sense. It is shown that the widest limits are obtained for models where normal directions of contacts between discrete units are parallel with the vectors connecting these units. Any deviation from parallelism causes the limits to shrink. A comparison of the derived equations to the results of the actual numerical model is presented. It shows relatively large deviations from the theory because the fundamental assumptions behind the theoretical derivations are largely violated in systems with complex geometry. The real shrinking of the Poisson’s ratio limit is less severe compared to that which is theoretically derived.'
address: 'Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, Brno, 60200, Czech Republic'
author:
- Jan Eliáš
bibliography:
- 'bibliography.bib'
title: ' Elastic properties of isotropic discrete systems: connections between geometric structure and Poisson’s ratio'
---
lattice model, geometry, elasticity, Poisson’s ratio, mesoscale, macroscopic characteristics
Introduction
============
Discrete modeling is a well established technique in mechanics. It allows the explanation or prediction of the complex behavior of heterogeneous, cohesive or granular materials. The main advantages it offers are the straightforward representation of material random heterogeneity, the simple formulation of constitutive equations in vectorial form and also the direct consideration of discrete and oriented cracks. In contrast, the elastic behavior of these models still poses open challenges. Besides the minor issue of the inevitable boundary layer with different elastic properties [@Eli17], the most serious problem lies in the inability of such models to exhibit Poisson’s ratios greater than 1/3 for plane stress simplification and 1/4 for plane strain and three dimensional models ([@BatRot88; @LiaCha-97; @Eli17] or see Eq. \[eq:lim\_gamma\_0\]).
Recently, four remedies providing the full range of Poisson’s ratio in discrete systems were presented. The first one [@AsaIto-15; @AsaAoy-17] introduces artificial auxiliary stresses within an iterative loop to achieve an elastically homogeneous system with an arbitrary Poisson’s ratio. The other three methods are similar as all of them are based on estimation of tensorial stresses or strains. The tensorial quantity is always computed nonlocally, in some neighborhood of the contact or body. These methods take into account the lateral stresses (confinement effect). The second remedy [@CusRez-17] proposes constitutive model as a function of the volumetric and deviatoric strain split. The third remedy [@CelLat-17] adds into the standard vectorial constitutive model terms accounting for the lateral stress. The fourth remedy [@RojZub-18] modifies distance between particles by integration tensorial strain over the body and evaluating body deformation. The stress oscillations caused by the heterogeneity of the material are unfortunately full or partially smeared out. Therefore, such models do not seem to be convenient for studying elastic behavior of highly heterogeneous structures at the mesoscale.
This paper is motivated by the author’s long belief that the Poisson’s ratio of discrete systems can be increased by changing the model geometry. Most of the papers published in this field use contact faces between discrete model units perpendicular to contact vectors [@GraAnt19; @BolSai98]. Examples of models with skewed normals are mostly from the field of granular materials when non-spherical particle shapes are used [@RotBat91; @GarLat-09; @KilDon-19] but also static homogeneous models can be found [@YaoJia-16]. The assumption of perpendicularity is abandoned here allowing model of completely arbitrary geometry to be constructed. Poisson’s ratio is then analyzed using strong assumptions about rotations and translations (Voigt’s hypothesis) in the model adopted according to [@KuhDadd-00]. It is proven here that abandoning the perpendicularity leads only to shrinking of the interval of achievable Poisson’s ratios.
The studied discrete system fills space continuously (without gaps or overlapping) with rigid bodies that possess translational ($\bm{u}$) and rotational ($\bm{\varphi}$) degrees of freedom. It is assumed that the system is isotropic – arbitrary rotation of the domain does not change its geometrical properties in the statistical sense. The rigid bodies interact via contacts defined at their boundaries. The normal and tangential displacement discontinuities $\bm{\Delta}$ at these boundaries are dictated by rigid body kinematics and give rise to normal and tangential forces linearly dependent on the corresponding component of $\bm{\Delta}$. A critical parameter governing the macroscopic Poisson’s ratio is the ratio between tangential and normal contact stiffness, hereinafter denoted by $\alpha$. The parameter $\alpha$ must be non-negative, otherwise the system would exhibit negative stiffness and instability. One can find several examples of these models in literature [@RezZho-17; @FasOsk19; @KanKim-14; @AmaQia-18].
The analytical derivation utilizes the equivalence of virtual work arising in the discrete system and continuum when they are subjected to equal straining. The Boltzmann continuum is used, and therefore the stress tensor must be symmetric (Boltzmann axiom). However, the discrete system yields a non-symmetric stress tensor, as it is a discrete instance of polar (Cosserat) continua instead [@RezCus16]. The virtual work equivalence is therefore accomplished with the help of the symmetrization of the tensor of elastic constants from the discrete model.
Normal and contact vector, volume
=================================
![a) Two dimensional rigid bodies in contact. The shaded area represents a single mechanical element with normal vector ${{\ensuremath{\bm{n}}}}$, contact vector ${{\ensuremath{\bm{t}}}}$, area $A$, centroid $\bm{c}$, length $l$ and volume $V$. b) Normal and contact vector in two and three dimensions.[]{data-label="fig:angles"}](normals2D3D){width="12cm"}
The domain is divided into rigid bodies, each of which has degrees of freedom associated with the translations and rotations of its governing node, $\bm{x}_a$. The contact between two nodes $\bm{x}_a$ and $\bm{x}_b$ is provided by a mechanical element with contact area $A$, length $l=\| \bm{x}_b-\bm{x}_a \|$, unit normal vector ${{\ensuremath{\bm{n}}}}$ and contact vector ${{\ensuremath{\bm{t}}}}=(\bm{x}_b-\bm{x}_a)/l$. The situation is depicted in Fig. \[fig:angles\]a in two dimensions.
We assume that the system has no directional bias, and that therefore all normal directions share the same probability of occurrence. The vector $\bm{n}$ is here defined in the Cartesian coordinate system by two angles, $\xi$ and $\zeta$ $$\begin{aligned}
{{\ensuremath{\bm{n}}}}= \begin{cases} \left(\begin{array}{cc}\cos\xi & \sin\xi \end{array}\right) & \mathrm{in\ 2D} \\
\left(\begin{array}{ccc}\cos\xi \sin\zeta & \sin\xi \sin\zeta & \cos\zeta\end{array}\right) & \mathrm{in\ 3D}\end{cases}\end{aligned}$$ In two dimensions (2D), $\xi$ represents the angle between the $x$ axis and the normal, and uniformly covers the solid angle. In three dimensions (3D), $\zeta$ is the angle between the $z$ axis and the normal, and $\xi$ is the rotation of ${{\ensuremath{\bm{n}}}}$ around the $z$ axis - see Fig. \[fig:angles\]b. For 3D isotropic systems, $\xi$ must also be uniform over the interval from 0 to $2\pi$ and $\zeta$ has the following probability density function $$\begin{aligned}
f_{\xi}(\xi) = \begin{cases} \displaystyle\frac{1}{2\pi} & \mathrm{for}\ \xi\in[0, 2\pi] \\ 0 & \mathrm{otherwise}\end{cases} \quad\quad
f_{\zeta}(\zeta) = \begin{cases} \displaystyle\frac{\sin\zeta}{2} & \mathrm{for}\ \zeta\in[0,\pi] \\ 0 & \mathrm{otherwise}\end{cases}\end{aligned}$$
The second fundamental vector governing the behavior of the contact is the contact vector ${{\ensuremath{\bm{t}}}}$. It is defined relative to the normal vector ${{\ensuremath{\bm{n}}}}$ by angles $\chi$ and $\theta$ - see Fig. \[fig:angles\]b. The requirement of isotropicity dictates that (i) in 2D $\chi$ must have probability density function symmetric around zero and (ii) in 3D $\theta$ must be uniformly distributed over the 0–2$\pi$ interval. $$\begin{aligned}
f_{\theta}(\theta) &= \begin{cases} \displaystyle\frac{1}{2\pi} & \mathrm{for}\ \theta\in[0, 2\pi] \\ 0 & \mathrm{otherwise}\end{cases} \end{aligned}$$ The probability distribution $f_{\chi}$ can be arbitrary (but symmetric around zero in 2D). For the sake of simplicity, it will now be assumed that the maximum angle between ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$ is $\gamma\in\left[ 0,\, \pi \right]$ and that all directions within this range are equally probable. $$\begin{aligned}
\mathrm{in\ 2D:\ } f_{\chi}(\chi) &= \begin{cases} \displaystyle\frac{1}{2\gamma} & \mathrm{for}\ \theta\in[-\gamma, \gamma] \\ 0 & \mathrm{otherwise}\end{cases} &
\mathrm{in\ 3D:\ }f_{\chi}(\chi) &= \begin{cases} \displaystyle\frac{\sin\chi}{1-\cos\gamma} & \mathrm{for}\ \theta\in[0, \gamma] \\ 0 & \mathrm{otherwise}\end{cases} \label{eq:chi_restriction}\end{aligned}$$ This assumption will later be removed in Sec. \[sec:arbitrary\_chi\].
Let us define a second order tensor (rotation matrix) that provides the following relation between ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$ $$\begin{aligned}
{{\ensuremath{\bm{t}}}}= {{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{n}}}}\label{eq:tRn} \end{aligned}$$ In two dimensions, this tensor is the standard rotation matrix $$\begin{aligned}
{{\ensuremath{\bm{\rho}}}}(\chi) = \left[\begin{array}{cc} \cos\chi & -\sin\chi\\ \sin\chi & \cos\chi \end{array}\right]\end{aligned}$$ A more complex situation is in 3D. One can imagine the construction of ${{\ensuremath{\bm{n}}}}$ by taking the vector $\left(\begin{array}{ccc}0&0&1\end{array}\right)$, rotating it along the $y$ axis by angle $\zeta$ and then along the $z$ axis by angle $\xi$ (Fig. \[fig:angles\]b). In the same way, the construction of ${{\ensuremath{\bm{t}}}}$ is achieved via four successive rotations along axes $y$, $z$, $y$ and $z$ by angles $\chi$, $\theta$, $\zeta$ and $\xi$, respectively. $$\begin{aligned}
{{\ensuremath{\bm{n}}}}&= {{\ensuremath{\bm{\rho}}}}_z(\xi)\cdot{{\ensuremath{\bm{\rho}}}}_y(\zeta)\cdot\left(\begin{array}{ccc}0&0&1\end{array}\right) & {{\ensuremath{\bm{t}}}}&= {{\ensuremath{\bm{\rho}}}}_z(\xi)\cdot{{\ensuremath{\bm{\rho}}}}_y(\zeta)\cdot{{\ensuremath{\bm{\rho}}}}_z(\theta)\cdot{{\ensuremath{\bm{\rho}}}}_y(\chi)\cdot\left(\begin{array}{ccc}0&0&1\end{array}\right)\end{aligned}$$ The rotation matrix from Eq. is therefore $$\begin{aligned}
{{\ensuremath{\bm{\rho}}}}(\xi,\zeta,\chi,\theta) = {{\ensuremath{\bm{\rho}}}}_z(\xi)\cdot{{\ensuremath{\bm{\rho}}}}_y(\zeta)\cdot{{\ensuremath{\bm{\rho}}}}_z(\theta)\cdot{{\ensuremath{\bm{\rho}}}}_y(\chi)\cdot{{\ensuremath{\bm{\rho}}}}_y^T(\zeta)\cdot{{\ensuremath{\bm{\rho}}}}_z^T(\xi)\end{aligned}$$
The cosine of angle $\chi$ between ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$ can be calculated using Eq. $$\begin{aligned}
\cos\chi = {{\ensuremath{\bm{n}}}}\cdot{{\ensuremath{\bm{t}}}}= {{\ensuremath{\bm{n}}}}\cdot{{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{n}}}}= {{\ensuremath{\bm{\rho}}}}:\left({{\ensuremath{\bm{n}}}}\otimes{{\ensuremath{\bm{n}}}}\right) = {{\ensuremath{\bm{\rho}}}}:{{\ensuremath{\bm{\nu}}}}\label{eq:cospsi}\end{aligned}$$ where the second order tensor ${{\ensuremath{\bm{\nu}}}}$ is defined according to @KuhDadd-00 as ${{\ensuremath{\bm{\nu}}}}={{\ensuremath{\bm{n}}}}\otimes {{\ensuremath{\bm{n}}}}$. Based on the assumption of no gaps or overlapping between the rigid bodies of the model, the volume of the domain is a summation over the volumes of individual mechanical elements $$\begin{aligned}
V = \displaystyle\sum\limits_e V_e = \displaystyle\sum\limits_e \cos\chi_e\dfrac{A_e l_e}{{{\ensuremath{N_{\mathrm{dim}}}}}} = \displaystyle\sum\limits_e {{\ensuremath{\bm{\rho}}}}_e:{{\ensuremath{\bm{\nu}}}}_e\dfrac{A_e l_e}{{{\ensuremath{N_{\mathrm{dim}}}}}}
\label{eq:volume}\end{aligned}$$ where the number of dimensions ${{\ensuremath{N_{\mathrm{dim}}}}}$ is 2 or 3. Note that the volume of an individual element is negative if $|\chi|>\pi/2$.
Equivalence of virtual work
===========================
The fundamental assumption about system degrees of freedom is taken from [@KuhDadd-00]. It is known as Voigt’s hypothesis and widely used as a homogenization technique (e.g. [@MikJir17]). It is assumed that when a discrete system is subjected to constant strain $\bm{\varepsilon}$, all the rotations are zero and differences in translations are dictated by differences in position $$\begin{aligned}
\varphi&=0 &\bm{u}_b-\bm{u}_a &= \bm{\varepsilon} \cdot \left(\bm{x}_b-\bm{x}_a\right) \label{eq:homdef}\end{aligned}$$ The displacement jump at the contact between cells $a$ and $b$ is, based on the previous assumption, given by the rigid body kinematics of bodies without rotation $$\begin{aligned}
\bm{\Delta} &= \bm{u}_b-\bm{u}_a = l \bm{\varepsilon} \cdot {{\ensuremath{\bm{t}}}}\label{eq:delta}\end{aligned}$$ where $l$ and $\bm{t}$ are respectively the length and contact vector belonging to the element connecting bodies $a$ and $b$. The normal and shear strain directly follow $$\begin{aligned}
e_N &=\frac{{{\ensuremath{\bm{n}}}}\cdot\bm{\Delta}}{l} = {{\ensuremath{\bm{n}}}}\cdot \bm{\varepsilon} \cdot {{\ensuremath{\bm{t}}}}& \bm{e}_T &= \frac{\bm{\Delta} }{l}- e_N{{\ensuremath{\bm{n}}}}= \bm{\varepsilon} \cdot {{\ensuremath{\bm{t}}}}- \left({{\ensuremath{\bm{n}}}}\cdot \bm{\varepsilon} \cdot {{\ensuremath{\bm{t}}}}\right) {{\ensuremath{\bm{n}}}}\label{eq:strainTheoretical}\end{aligned}$$ The stresses read $$\begin{aligned}
s_N &= E_0 e_N & \bm{s}_T &= E_0\alpha\bm{e}_T \label{eq:facetStress}\end{aligned}$$ where $E_0$ is the normal stiffness coefficient and $\alpha$ is the tangential/normal stiffness ratio. Both of the material parameters are considered constant throughout the whole domain.
The virtual work done by a single element is obtained via integration of the product of the stress and the displacement jump (both constant) over the contact face $$\begin{aligned}
\delta W &= \int\limits_A (s_N{{\ensuremath{\bm{n}}}}+ \bm{s}_T)\cdot\delta\bm{\Delta} {\mbox{\,d}}{A} = Al \left(s_N \delta e_N + \bm{s}_T \cdot \delta \bm{e}_T\right) \label{eq:workSingleE1}\end{aligned}$$ The integration is simple because the assumed zero rotations imply constant stresses and the displacement jump over the whole contact face. The total virtual work in the discrete system is the summation of these individual contributions.
To simplify the notation, we introduce the transpose operation $T_{ij}$ on an arbitrary tensor $\bm{A}$ of sufficient order by swapping indices $i$ and $j$. $$\begin{aligned}
\bm{A}_{\dots i \dots j\dots}^{T_{ij}} = \bm{A}_{\dots j \dots i\dots}\end{aligned}$$
Let us now define two additional tensors according to @KuhDadd-00: the fourth order tensor ${{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}$ and the third order tensor ${{\ensuremath{\bm{T}}}}$. $$\begin{aligned}
{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}&=\frac{\bm{1}\otimes\bm{1}}{3}\\
{{\ensuremath{\bm{T}}}}&= 3{{\ensuremath{\bm{n}}}}\cdot \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{13}} - {{\ensuremath{\bm{n}}}}\otimes {{\ensuremath{\bm{n}}}}\otimes {{\ensuremath{\bm{n}}}}\end{aligned}$$ where $\bm{1}$ is the identity matrix of size $N_{\mathrm{dim}}$. Note that ${{\ensuremath{\bm{T}}}}$ is different from the definition in [@KuhDadd-00; @Eli17] because the symmetry implied by equality ${{\ensuremath{\bm{t}}}}={{\ensuremath{\bm{n}}}}$ is no longer present. The transposition $T_{13}$ means that dimensions 1 and 3 are swapped.
With the previously defined tensors ${{\ensuremath{\bm{\nu}}}}$ and ${{\ensuremath{\bm{T}}}}$, Eq. can be rewritten as $$\begin{aligned}
e_N &= \left({{\ensuremath{\bm{\nu}}}}\cdot {{\ensuremath{\bm{\rho}}}}^T\right) : \bm{\varepsilon} & \bm{e}_T &= \left({{\ensuremath{\bm{T}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T\right) : \bm{\varepsilon} \label{eq:facetStrain}\end{aligned}$$ using the transposition $T$ of the second order tensor swapping its two dimensions $T=T_{12}$. The virtual work of a single element (Eq. ) can be rewritten as well. $$\begin{aligned}
\delta W &= \nonumber A l \left(s_N \delta e_N + \bm{s}_T \cdot \delta \bm{e}_T\right)\\
&= A l E_0 \left(\left[\left({{\ensuremath{\bm{\nu}}}}\cdot {{\ensuremath{\bm{\rho}}}}^T\right):\bm{\varepsilon}\right]\left[ \left({{\ensuremath{\bm{\nu}}}}\cdot {{\ensuremath{\bm{\rho}}}}^T\right) : \delta\bm{\varepsilon} \right] + \alpha\left[ \left({{\ensuremath{\bm{T}}}}\cdot {{\ensuremath{\bm{\rho}}}}^T\right):\bm{\varepsilon}\right]\cdot\left[\left({{\ensuremath{\bm{T}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T\right) : \delta\bm{\varepsilon}\right]\right)
\nonumber\\&= A l E_0 \left[\bm{\varepsilon}:\left({{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{\nu}}}}\otimes{{\ensuremath{\bm{\nu}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T\right)^{T_{12}}:\delta\bm{\varepsilon} + \alpha \bm{\varepsilon}:\left({{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{T}}}}^{T_{13}}\cdot{{\ensuremath{\bm{T}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T\right):\delta\bm{\varepsilon} \right] \nonumber\\ &= A l E_0 \bm{\varepsilon}: \left(\left({{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{\nu}}}}\otimes{{\ensuremath{\bm{\nu}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T\right)^{T_{12}} + \alpha {{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{T}}}}^{T_{13}}\cdot{{\ensuremath{\bm{T}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T \right):\delta\bm{\varepsilon}\\
&= \nonumber A l E_0 \bm{\varepsilon}: \left({{\ensuremath{\pmb{\mathscr{N}}}}}+ \alpha{{\ensuremath{\pmb{\mathscr{T}}}}}\right):\delta\bm{\varepsilon}\end{aligned}$$ where $$\begin{aligned}
{{\ensuremath{\pmb{\mathscr{N}}}}}&=\left({{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{\nu}}}}\otimes{{\ensuremath{\bm{\nu}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T\right)^{T_{12}} &
{{\ensuremath{\pmb{\mathscr{T}}}}}&={{\ensuremath{\bm{\rho}}}}\cdot{{\ensuremath{\bm{T}}}}^{T_{13}}\cdot{{\ensuremath{\bm{T}}}}\cdot{{\ensuremath{\bm{\rho}}}}^T\end{aligned}$$
The total virtual work of the discrete assembly is $$\begin{aligned}
\delta W^{\mathrm{dis}} = \sum\limits_e \delta W_e = \sum\limits_e A_e l_e E_0 \bm{\varepsilon}: \left({{\ensuremath{\pmb{\mathscr{N}}}}}_e + \alpha {{\ensuremath{\pmb{\mathscr{T}}}}}_e \right):\delta\bm{\varepsilon} \label{eq:virtW_discrete} \end{aligned}$$
The virtual work of an equally strained elastic isotropic Boltzmann continuum occupying the same volume $V$ is $$\begin{aligned}
\delta W^{\mathrm{con}} = V\bm{\sigma}:\delta\bm{\varepsilon} = V\bm{\varepsilon}:{{\ensuremath{\bm{\mathcal{D}}}}}:\delta\bm{\varepsilon}
\label{eq:virtW_homog} \end{aligned}$$ with the constitutive equation $\bm{\sigma}={{\ensuremath{\bm{\mathcal{D}}}}}:\bm{\varepsilon}$ where ${{\ensuremath{\bm{\mathcal{D}}}}}$ is fourth order tensor of elastic constants.
The equivalence of the discrete and continuous system implies the equality of virtual works (Hill-Mandel condition [@Hil63]) $$\begin{aligned}
\delta W^{\mathrm{dis}} = \delta W^{\mathrm{con}} \label{eq:VWequivalence}\end{aligned}$$ Substituting Eqs. and into Eq. , the following expression for the tensor of elastic constants is derived $$\begin{aligned}
{{\ensuremath{\bm{\mathcal{D}}}}}= \left\langle\frac{1}{V}\sum\limits_e A_e l_e E_0 \left({{\ensuremath{\pmb{\mathscr{N}}}}}_e + \alpha {{\ensuremath{\pmb{\mathscr{T}}}}}_e \right)\right\rangle^{\mathrm{SYM}} \label{eq:D_initial}\end{aligned}$$ The symmetrization is needed because the tensors ${{\ensuremath{\pmb{\mathscr{N}}}}}$ and ${{\ensuremath{\pmb{\mathscr{T}}}}}$ do not possess the symmetries required for the Boltzmann continuum, which are *major* symmetry (derived from the equivalence of mixed derivatives of elastic potential) and *minor* symmetry (derived from symmetry of stress and strain tensors $\sigma_{ij} = \sigma_{ji}$, $\varepsilon_{ij} = \varepsilon_{ji}$). $$\begin{aligned}
\mathrm{major\ symmetry:\quad}{{\ensuremath{{\mathcal{D}}}}}_{ijkl} &= {{\ensuremath{{\mathcal{D}}}}}_{klij} & \mathrm{minor\ symmetry:\quad}{{\ensuremath{{\mathcal{D}}}}}_{ijkl} &= {{\ensuremath{{\mathcal{D}}}}}_{jikl} = {{\ensuremath{{\mathcal{D}}}}}_{ijlk} = {{\ensuremath{{\mathcal{D}}}}}_{jilk}\end{aligned}$$ Because of the non-symmetric stress tensor in the discrete system, the minor symmetry is violated. The symmetric part can be easily obtained using transposition $T_{34}$. $$\begin{aligned}
\left\langle\bullet\right\rangle^{\mathrm{SYM}} = \frac{\bullet+\bullet^{T_{34}}}{2} \label{eq:symmetrization}\end{aligned}$$
Thanks to the assumed statistical independence between the normal and contact vector and the elemental area and length, the summation in Eq. can be broken into the following expression $$\begin{aligned}
{{\ensuremath{\bm{\mathcal{D}}}}}= \frac{E_0}{V}\left\langle\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right] + \alpha \mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right]\right\rangle^{\mathrm{SYM}}\sum\limits_e A_e l_e \label{eq:D}\end{aligned}$$ where $\mathrm{E}\left[\bullet(\bm{x})\right]$ is the mean value of function $\bullet$, which is dependent on vector $\bm{x}$ with the distribution function $f_{\bm{X}}(\bm{x})$ $$\begin{aligned}
\mathrm{E}\left[\bullet(\bm{x})\right] = \int\limits_{-\infty}^{\infty} \dots \int\limits_{-\infty}^{\infty} \bullet(\bm{x}) f_{\bm{X}}(\bm{x}) {\mbox{\,d}}{\bm{x}}\end{aligned}$$
Substituting $V$ from Eq. and utilizing the statistical independence again, one obtains $$\begin{aligned}
{{\ensuremath{\bm{\mathcal{D}}}}}= \frac{{{\ensuremath{N_{\mathrm{dim}}}}}E_0}{\mathrm{E}[{{\ensuremath{\bm{\rho}}}}:{{\ensuremath{\bm{\nu}}}}]}\left\langle\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right] + \alpha \mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right]\right\rangle^{\mathrm{SYM}} \label{eq:D0}\end{aligned}$$
Calculation of expectations
===========================
Two dimensional case
--------------------
In two dimensions, the rotation matrix depends only on angle $\chi$, while the normal ${{\ensuremath{\bm{n}}}}$ depends only on angle $\xi$. The calculation of the mean value can be separated into two steps. Let us first calculate all the quantities dependent solely on ${{\ensuremath{\bm{n}}}}$. $$\begin{aligned}
\mathrm{E}\left[{{\ensuremath{\bm{\nu}}}}\right] &= \int\limits_0^{2\pi} {{\ensuremath{\bm{n}}}}\otimes{{\ensuremath{\bm{n}}}}\frac{1}{2\pi} {\mbox{\,d}}{\xi} = \frac{1}{2}\bm{1}
\\
\mathrm{E}\left[{{\ensuremath{\bm{\nu}}}}\otimes{{\ensuremath{\bm{\nu}}}}\right] &= \int\limits_{0}^{2\pi} {{\ensuremath{\bm{\nu}}}}\otimes{{\ensuremath{\bm{\nu}}}}\frac{1}{2\pi}{\mbox{\,d}}{\xi}= \frac{1}{4}{{\ensuremath{\pmb{\mathscr{I}}}}}+ \frac{3}{8}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\\
\mathrm{E}\left[{{\ensuremath{\bm{T}}}}^{T_{13}}\cdot{{\ensuremath{\bm{T}}}}\right] &= \int\limits_{0}^{2\pi} {{\ensuremath{\bm{T}}}}^{T_{13}}\cdot{{\ensuremath{\bm{T}}}}\frac{1}{2\pi}{\mbox{\,d}}{\xi}= \frac{3}{4}{{\ensuremath{\pmb{\mathscr{I}}}}}- \frac{3}{8}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}- \frac{3}{2}\left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}} \end{aligned}$$ where the fourth order tensor ${{\ensuremath{\pmb{\mathscr{I}}}}}=\mathscr{I}_{ijkl}=(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})/2$ delta is employed; $\delta_{ij}\equiv\bm{1}$ is the the Kronecker delta.
In the second step, theses quantities are used in the calculation of the mean values of terms involving both ${{\ensuremath{\bm{\rho}}}}$ and ${{\ensuremath{\bm{n}}}}$. $$\begin{aligned}
\mathrm{E}\left[{{\ensuremath{\bm{\rho}}}}:{{\ensuremath{\bm{\nu}}}}\right] &= \frac{1}{2}\int\limits_{-\gamma}^{\gamma} {{\ensuremath{\bm{\rho}}}}:\bm{1} \frac{1}{2\gamma}{\mbox{\,d}}{\chi} = \frac{1}{4\gamma}\int\limits_{-\gamma}^{\gamma} 2\cos\chi {\mbox{\,d}}{\chi} = \frac{\sin\gamma}{\gamma} \label{eq:vol2D}
\\
\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right] =& \nonumber \int\limits_{-\gamma}^{\gamma}\int\limits_{0}^{2\pi} {{\ensuremath{\pmb{\mathscr{N}}}}}\frac{1}{2\gamma} \frac{1}{2\pi} {\mbox{\,d}}{\xi}{\mbox{\,d}}{\chi} = \int\limits_{-\gamma}^{\gamma} \left[{{\ensuremath{\bm{\rho}}}}\cdot\mathrm{E}\left[{{\ensuremath{\bm{\nu}}}}\otimes{{\ensuremath{\bm{\nu}}}}\right]\cdot{{\ensuremath{\bm{\rho}}}}^T\right]^{T_{12}} \frac{1}{2\gamma} {\mbox{\,d}}{\chi}\\
=& \int\limits_{-\gamma}^{\gamma}\left[{{\ensuremath{\bm{\rho}}}}\cdot\left(\frac{1}{4}{{\ensuremath{\pmb{\mathscr{I}}}}}+ \frac{3}{8}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)\cdot{{\ensuremath{\bm{\rho}}}}^T\right]^{T_{12}} \frac{1}{2\gamma} {\mbox{\,d}}{\chi} \\
=& \nonumber \frac{3}{4} \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}} + \frac{3\sin2\gamma}{16\gamma}\left(
{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}- \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}} + \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{24}}\right)
\\
\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right] =& \nonumber \int\limits_{-\gamma}^{\gamma}\int\limits_{0}^{2\pi} {{\ensuremath{\pmb{\mathscr{T}}}}}\frac{1}{2\gamma} \frac{1}{2\pi} {\mbox{\,d}}{\xi}{\mbox{\,d}}{\chi} = \int\limits_{-\gamma}^{\gamma} {{\ensuremath{\bm{\rho}}}}\cdot\mathrm{E}\left[{{\ensuremath{\bm{T}}}}^{T_{13}}\cdot{{\ensuremath{\bm{T}}}}\right]\cdot{{\ensuremath{\bm{\rho}}}}^T \frac{1}{2\gamma} {\mbox{\,d}}{\chi}\\
=& \int\limits_{-\gamma}^{\gamma}{{\ensuremath{\bm{\rho}}}}\cdot\left(\frac{3}{4}{{\ensuremath{\pmb{\mathscr{I}}}}}- \frac{3}{8}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}- \frac{3}{2}\left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}}\right)\cdot{{\ensuremath{\bm{\rho}}}}^T \frac{1}{2\gamma} {\mbox{\,d}}{\chi} \\
=& \nonumber \frac{3}{4} \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{24}} - \frac{3\sin2\gamma}{16\gamma}\left(
{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}+ \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}} - \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{24}}\right)\end{aligned}$$
Only the symmetric parts of these expectations are needed. The following identities, which are valid for both the 2D and the 3D model, are derived from Eq. $$\begin{aligned}
\left\langle\left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}}\right\rangle^{\mathrm{SYM}} = \left\langle\left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{24}}\right\rangle^{\mathrm{SYM}} = \frac{{{\ensuremath{\pmb{\mathscr{I}}}}}}{3} \label{eq:symI}\end{aligned}$$ and used to obtain the final symmetric expectations of tensors. $$\begin{aligned}
\left\langle\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right]\right\rangle^{\mathrm{SYM}} &=\frac{\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right] + \mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right]^{T_{34}}}{2} = \frac{1}{4}{{\ensuremath{\pmb{\mathscr{I}}}}}+ \frac{3\sin2\gamma}{16\gamma} {{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\label{eq:bigN2D} \\
\left\langle\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right]\right\rangle^{\mathrm{SYM}} &=\frac{\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right] + \mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right]^{T_{34}}}{2} = \frac{1}{4}{{\ensuremath{\pmb{\mathscr{I}}}}}- \frac{3\sin2\gamma}{16\gamma} {{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\label{eq:bigT2D}\end{aligned}$$
Three dimensional case
----------------------
In three dimensions, integration is substantially more complex. It is performed over four independent angles and cannot be separated since rotation matrix ${{\ensuremath{\bm{\rho}}}}$ depends on all four angles. Calculation by hand is extremely tedious; it was performed by computer instead. The following three integrations were delivered with the help of the Python library for symbolic mathematics, SymPy [@SymPy]. $$\begin{aligned}
\mathrm{E}\left[{{\ensuremath{\bm{\rho}}}}:{{\ensuremath{\bm{\nu}}}}\right] &= \int\limits_{-\gamma}^{\gamma}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi} {{\ensuremath{\bm{\rho}}}}:{{\ensuremath{\bm{\nu}}}}\frac{1}{2\pi} \frac{\sin\zeta}{2} \frac{1}{2\pi} \frac{\sin\chi}{1-\cos\gamma} {\mbox{\,d}}{\xi}{\mbox{\,d}}{\zeta}{\mbox{\,d}}{\theta}{\mbox{\,d}}{\chi} = \cos^2\left(\frac{g}{2}\right) \label{eq:vol3D}
\\
\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right] =& \nonumber \int\limits_{-\gamma}^{\gamma}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi} {{\ensuremath{\pmb{\mathscr{N}}}}}\frac{1}{2\pi} \frac{\sin\zeta}{2} \frac{1}{2\pi} \frac{\sin\chi}{1-\cos\gamma} {\mbox{\,d}}{\xi}{\mbox{\,d}}{\zeta}{\mbox{\,d}}{\theta}{\mbox{\,d}}{\chi} = \\
=& \frac{1}{3} \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}} + \frac{2\cos\gamma + \cos 2\gamma + 1}{20}\left(
{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}+ \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{24}} - \frac{2}{3} \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}}\right)
\\
\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right] =& \nonumber \int\limits_{-\gamma}^{\gamma}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi} {{\ensuremath{\pmb{\mathscr{T}}}}}\frac{1}{2\pi} \frac{\sin\zeta}{2} \frac{1}{2\pi} \frac{\sin\chi}{1-\cos\gamma} {\mbox{\,d}}{\xi}{\mbox{\,d}}{\zeta}{\mbox{\,d}}{\theta}{\mbox{\,d}}{\chi} = \\
=& \frac{2}{3} \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{24}} - \frac{2\cos\gamma + \cos 2\gamma + 1}{20}\left(
{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}+ \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{23}} - \frac{2}{3} \left({{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right)^{T_{24}}\right)\end{aligned}$$
Using identity , the symmetric part yields $$\begin{aligned}
\left\langle\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right]\right\rangle^{\mathrm{SYM}} &=\frac{\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right] + \mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right]^{T_{34}}}{2} =
\frac{2\cos\gamma + \cos 2\gamma + 21}{180} {{\ensuremath{\pmb{\mathscr{I}}}}}+ \frac{2\cos\gamma + \cos 2\gamma + 1}{20} {{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\label{eq:bigN3D}
\\
\left\langle\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right]\right\rangle^{\mathrm{SYM}} &=\frac{\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right] + \mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right]^{T_{34}}}{2} = \frac{39 - 2\cos\gamma - \cos 2\gamma}{180} {{\ensuremath{\pmb{\mathscr{I}}}}}- \frac{2\cos\gamma + \cos 2\gamma + 1}{20} {{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\label{eq:bigT3D}\end{aligned}$$
Relation between the elastic parameters of discrete system and continuum
========================================================================
The mechanical behavior of a linearly elastic isotropic solid is determined by two constants. Here we choose the elastic modulus ($E$) and Poisson’s ratio ($\nu$). The tensor of the elastic constants is expressed using these variables $$\begin{aligned}
{{\ensuremath{\bm{\mathcal{D}}}}}= \begin{cases}\displaystyle\frac{E}{1+\nu}{{\ensuremath{\pmb{\mathscr{I}}}}}+\frac{3E\nu}{1-\nu^2}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}& \mathrm{2D,\,plane\ stress} \\[7pt]
\displaystyle\frac{E}{1+\nu}{{\ensuremath{\pmb{\mathscr{I}}}}}+\frac{3E\nu}{(1+\nu)(1-2\nu)}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}& \mathrm{2D,\, plane\ strain} \\[7pt]
\displaystyle\frac{E}{1+\nu}{{\ensuremath{\pmb{\mathscr{I}}}}}+\frac{3E\nu}{(1+\nu)(1-2\nu)}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}&\mathrm{3D}
\end{cases} \label{eq:Dmacro}\end{aligned}$$ Equation , along with symmetrized expectations , and in two dimensions as well as , and in three dimensions, provides $$\begin{aligned}
{{\ensuremath{\bm{\mathcal{D}}}}}= \begin{cases}
\displaystyle E_0\left[(1+\alpha)\frac{\gamma}{2\sin\gamma}{{\ensuremath{\pmb{\mathscr{I}}}}}+(1-\alpha)\frac{3\cos\gamma}{4}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right] & \mathrm{2D} \\[7pt]
\displaystyle{E_0}\left[
\dfrac{(1-\alpha)(2\cos\gamma+\cos(2\gamma)-39)+60}{30(\cos\gamma+1)}
{{\ensuremath{\pmb{\mathscr{I}}}}}+ \frac{3(1-\alpha)}{5}\cos\gamma {{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}\right] &\mathrm{3D}
\end{cases} \label{eq:Dmeso}\end{aligned}$$
![Macroscopic elastic properties of two and three dimensional discrete systems with all directions of ${{\ensuremath{\bm{t}}}}$ equally probable (limited by $|\chi|<\gamma$) dependent on the tangential/normal stiffness ratio $\alpha$ according to Eqs. and .[]{data-label="fig:elastic_constants_gamma"}](elastic_constants_gamma){width="\textwidth"}
Equality between meso and macroscopic elastic tensors and requires equality between the respective scalar multipliers of tensors ${{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}$ and ${{\ensuremath{\pmb{\mathscr{I}}}}}$. One can modify these algebraic equations into relations between macroscopic parameters $E$ and $\nu$ and mesoscopic parameters $E_0$, $\alpha$ and $\gamma$. $$\begin{aligned}
\nu&=\begin{cases}\dfrac{\left(1-\alpha\right)\sin2\gamma}{4(1+\alpha)\gamma +(1-\alpha)\sin2\gamma} & \mathrm{2D,\,plane\ stress}\\[8pt]
\dfrac{\left(1-\alpha\right)\sin2\gamma}{4(1+\alpha)\gamma +2(1-\alpha)\sin2\gamma} &\mathrm{2D,\,plane\ strain} \\[8pt]
\dfrac{3(1-\alpha)(\cos\gamma+\cos^2(\gamma))}{(1-\alpha)(7\cos\gamma+7\cos^2\gamma-20)+30} & \mathrm{3D}
\end{cases}
\label{eq:nu_gamma}\\
E&=\begin{cases}E_0\dfrac{2(1+\alpha)^2 \gamma^2 + (1-\alpha^2)\gamma\sin 2\gamma}{\sin\gamma(4(1+\alpha)\gamma + (1-\alpha)\sin 2\gamma)} & \mathrm{2D,\,plane\ stress}\\[8pt]
E_0\dfrac{4(1+\alpha)^2 \gamma^2 + 3(1-\alpha^2)\gamma\sin 2\gamma}{\sin\gamma(8(1+\alpha)\gamma + 4(1-\alpha)\sin 2\gamma} &\mathrm{2D,\,plane\ strain} \\[8pt]
E_0\dfrac{2\left[(1-\alpha)(\cos\gamma+\cos^2\gamma-20)+30\right]\left[(1-\alpha)(\cos\gamma+\cos^2\gamma-2)+3\right]}{(1-\alpha)(7\cos\gamma+7\cos^2\gamma-20)+30} & \mathrm{3D}
\end{cases}
\label{eq:E_gamma}\end{aligned}$$ These equations are plotted in Fig. \[fig:elastic\_constants\_gamma\] for the range $\alpha\in[0,\,3]$.
Decreasing $\gamma$ towards zero must yield relations for a discrete system with ${{\ensuremath{\bm{n}}}}={{\ensuremath{\bm{t}}}}$. $$\begin{aligned}
\lim\limits_{\gamma\rightarrow 0}\nu&=\begin{cases} \dfrac{1-\alpha}{3+\alpha} & \mathrm{2D,\,plane\ stress} \\[7pt] \dfrac{1-\alpha}{4} & \mathrm{2D,\,plane\ strain} \\[7pt] \dfrac{1-\alpha}{4+\alpha} & \mathrm{3D} \end{cases}
&
\lim\limits_{\gamma\rightarrow 0}E&=\begin{cases} E_0\dfrac{2+2\alpha}{3+\alpha} & \mathrm{2D,\,plane\ stress} \\[7pt] E_0\dfrac{(1+\alpha)(5-\alpha)}{8} & \mathrm{2D,\,plane\ strain} \\[7pt] E_0\dfrac{2+3\alpha}{4+\alpha} & \mathrm{3D} \end{cases} \label{eq:lim_gamma_0}\end{aligned}$$ Indeed, the calculation of limits provides correct expressions (derived for example in [@BatRot88; @LiaCha-97; @Eli17] under the assumption of the perpendicularity of the contact vector and contact face). They are also identical to those from microplane theory [@CarBaz97].
What are the maximum and minimum Poisson’s ratios that can be achieved? One can differentiate the expression with respect to $\gamma$ and search for a stationary point (leaving out the degenerative case $\alpha=1$). In 2D, such an analysis reveals a local extreme at points $\gamma=0$ and $\gamma\approx2.24670$ (solution of $2\gamma=\tan2\gamma$). In 3D, the stationary points are $\gamma=0$, $\pi$ and $\approx2.09440$ (exactly $\arccos (-0.5)$). Plotting the Poisson’s ratio with respect to the limit angle $\gamma$ (Fig. \[fig:elastic\_constants\_gamma\_gamma\]) shows that the maximum range of $\nu$ is obtained for $\gamma=0$, i.e. when the contact vector equals the normal vector. This is the classic solution stated in Eq. . Increasing $\gamma$ up to $\pi/2$ causes the interval of achievable Poisson’s ratios to shrink to zero. Then, the interval opens again with opposite signs; its width maximizes at $\gamma=2.24670$ (2D) or $\gamma=2.09440$ (3D). The interval of possible values of $\nu$ for these $\gamma$ reads: $\left[-0.122,\,0.098\right]$ for 2D plane stress, $\left[-0.139,\,0.089\right]$ for 2D plane strain and $\left[-0.091,\,0.034\right]$ for 3D, respectively. These maximum and minimum value of Poisson’s ratio occur always for $\alpha=0$ or $\alpha\rightarrow\infty$.
We have proven that under assumption , one cannot increase the Poisson’s ratio limits beyond that which is provided by the model with ${{\ensuremath{\bm{n}}}}={{\ensuremath{\bm{t}}}}$ in Eq. .
![The macroscopic Poisson’s ratio of two and three dimensional discrete systems with all direction of ${{\ensuremath{\bm{t}}}}$ equally probable (limited by $|\chi|<\gamma$) dependent on the limit $\gamma$ according to Eqs. and .[]{data-label="fig:elastic_constants_gamma_gamma"}](elastic_constants_gamma_gamma){width="\textwidth"}
Arbitrary distribution $f_{\chi}(\chi)$ \[sec:arbitrary\_chi\]
==============================================================
This section proves that the same conclusion unfortunately holds for the arbitrary distribution of angle $\chi$. The only restriction on the probability density of $\chi$ applied here comes from the isotropy requirement in 2D which demands that $f_{\chi}(\chi)$ be symmetric around zero. In 3D, $f_{\chi}(\chi)$ can be arbitrary.
Let us denote the following auxiliary integrals $$\begin{aligned}
\int\limits_{\Omega_{\chi}} \cos \chi \,f_{\chi}(\chi) {\mbox{\,d}}{\chi} &= I_1 & \int\limits_{\Omega_{\chi}} \cos(2\chi) f_{\chi}(\chi) {\mbox{\,d}}{\chi} &= I_2
\label{eq:aux_integrals}\end{aligned}$$ $\Omega_{\chi}$ is the domain of the probability distribution function $f_{\chi}(\chi)$ which is the interval $[-\pi,\,\pi ]$ in 2D and $[ 0,\,\pi ]$ in 3D. With the help of the Python symbolic mathematics library SymPy [@SymPy], the expectations needed in Eq. were computed. Using identity $\int_{\Omega_{\chi}} f_{\chi}(\chi) {\mbox{\,d}}{\chi}=1$ resulting from the definition of probability density; and in 2D $\int_{\Omega_{\chi}} \sin(2\chi) f_{\chi}(\chi) {\mbox{\,d}}{\chi}=0$ derived from the symmetry of $f_{\chi}$, the expectations are $$\begin{aligned}
\left\langle\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{N}}}}}\right] + \alpha\mathrm{E}\left[{{\ensuremath{\pmb{\mathscr{T}}}}}\right]\right\rangle^{\mathrm{SYM}} & = \begin{cases}\dfrac{1+\alpha}{4}{{\ensuremath{\pmb{\mathscr{I}}}}}+ \dfrac{3(1-\alpha)I_2}{8}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}& \mathrm{2D}\\
\dfrac{(1-\alpha)(I_2-13)+20}{60}{{\ensuremath{\pmb{\mathscr{I}}}}}+ (1-\alpha)\dfrac{3I_2+1}{20}{{\ensuremath{\pmb{\mathscr{I}}^{\mathrm{vol}}}}}& \mathrm{3D}
\end{cases} \label{eq:arbit_expectations1}\\
\mathrm{E}\left[{{\ensuremath{\bm{\rho}}}}:{{\ensuremath{\bm{\nu}}}}\right] &= I_1 \label{eq:arbit_expectations2}\end{aligned}$$ Substituting these expressions into Eq. , requiring equality with Eq. and solving for unknown $E$ and $\nu$, provides $$\begin{aligned}
\nu &= \begin{cases}\dfrac{(1-\alpha)I_2}{2(1+\alpha)+(1-\alpha)I_2} & \mathrm{2D,\,plane\ stress}\\[8pt]
\dfrac{(1-\alpha)I_2}{2(1+\alpha)+2(1-\alpha)I_2} & \mathrm{2D,\,plane\ strain}\\[8pt]
\dfrac{(1-\alpha)(3I_2+1)}{(1-\alpha)(7I_2-11)+20} & \mathrm{3D}\end{cases}
\label{eq:arbit_nu}
\\
E &= \begin{cases}E_0\dfrac{(1+\alpha)^2 + (1-\alpha^2)I_2}{(2(1+\alpha)+(1-\alpha)I_2)I_1}& \mathrm{2D,\,plane\ stress}\\[8pt]
E_0\dfrac{2(1+\alpha)^2 + 3(1-\alpha^2)I_2}{4(1+\alpha+(1-\alpha)I_2)I_1}& \mathrm{2D,\,plane\ strain}
\\[8pt]E_0\dfrac{((1-\alpha)(I_2-13)+20)(I_2(1-\alpha) + 1+\alpha)}{2((1-\alpha)(7I_2-11)+20)I_1}& \mathrm{3D}\end{cases}
\label{eq:arbit_E}\end{aligned}$$ The Poisson’s ratio predicted by Eq. is plotted in Fig. \[fig:elastic\_constants\_I\].
![Poisson’s ratio of two and three dimensional discrete systems with arbitrary distribution of angle $\chi$ between the normal and contact vector according to Eq. . The variable $I_2$ represents integral $\int_{\Omega_{\chi}} \cos 2\chi f_{\chi}(\chi){\mbox{\,d}}{\chi}$.[]{data-label="fig:elastic_constants_I"}](elastic_constants_I){width="\textwidth"}
Let us focus again on the theoretical limits of Poisson’s ratio provided by Eq. . The meaningful values of $\alpha$ range from 0 to $\infty$. The integral $I_2$ is limited to an interval from -1 to 1 because these are the maximum and minimum values of the continuous function $\cos(2\chi)$, which in the integral is “weighted” by an arbitrary non-negative function with the unit integral over its domain. The only way $I_2=1$ can be obtained is when $f_{\chi}(\chi)$ is a discrete distribution with zero probability everywhere except points $\chi=\pi$, 0 or $-\pi$ (the last of which is possible only in 2D). All of these angles imply that ${{\ensuremath{\bm{t}}}}$ is parallel to ${{\ensuremath{\bm{n}}}}$. The second limit case of $I_2=-1$ is only possible when $f_{\chi}(\chi)$ is zero everywhere except points $\chi=\pm\pi/2$, i.e. when ${{\ensuremath{\bm{t}}}}$ is perpendicular to ${{\ensuremath{\bm{n}}}}$.
Differentiating Eq. with respect to $I_2$ reveals that, except in the degenerative case of $\alpha=1$, there is no stationary point within the investigated domain for both 2D and 3D models. The extreme values therefore lie on the boundaries. One can see in Figure \[fig:elastic\_constants\_I\] that the maximum value of $\nu$ is reached by minimizing $\alpha$ and maximizing $I_2$ or by maximizing $\alpha$ and minimizing $I_2$; the opposite is true for the opposite goal of reaching the minimum Poisson’s ratio.
In 2D, the same Poisson’s ratio limits are obtained for both $I_2=1$ or $I_2=-1$, and these limits are the same as those of the model where ${{\ensuremath{\bm{n}}}}={{\ensuremath{\bm{t}}}}$. Indeed, the equality ${{\ensuremath{\bm{n}}}}={{\ensuremath{\bm{t}}}}$ implies $I_2=1$, and substituting that into Eq. yields Eq. . In 3D, the case $I_2=1$ obtained for parallel ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$ again yields Eq. and the widest range of Poisson’s ratio. The case with perpendicular ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$ (when $I_2=-1$), leads in 3D to a narrower interval as well as any other distribution of $\chi$.
We have proven that an isotropic discrete structure with an arbitrary relation between ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$ cannot increase the Poisson’s ratio limits beyond the limits obtained by the model with ${{\ensuremath{\bm{n}}}}={{\ensuremath{\bm{t}}}}$ in Eq. .
Macroscopic elastic properties of actual discrete systems
=========================================================
Let us now observe the behavior of actual discrete systems and compare it to our analytical formulas. We divide a domain into (generally non-convex) polygons or polyhedrons; this tessellation defines the shapes of rigid bodies. Contact forces ${{\ensuremath{\bm{f}}}}=A(s_N{{\ensuremath{\bm{n}}}}+\bm{s}_T)$ arise at the centroids of the edges (or faces) of polygons (or polyhedrons, respectively). We search for displacements and rotations fulfilling linear and angular momentum balance equations. These equations, for a single discrete body without external load, read $$\begin{aligned}
\sum \limits_e {{\ensuremath{\bm{f}}}}_e = E_0\sum \limits_e A_e \left(e_N^e \bm{n}_e + \alpha \bm{e}_T^e \right) &= \bm{0} \label{eq:linear_momentum}\\
\sum \limits_e {{\ensuremath{\bm{f}}}}_e\cdot{{\ensuremath{\bm{\mathcal{E}}}}}\cdot\bm{r}_e = E_0 \sum \limits_e A_e \left(e_N^e \bm{n}_e + \alpha \bm{e}_T^e \right)\cdot{{\ensuremath{\bm{\mathcal{E}}}}}\cdot\bm{r}_e &= \bm{0} \mathrm{\ in\ 3D\ or\ } 0 \mathrm{\ in\ 2D} \label{eq:angular_momentum}\end{aligned}$$ where $e$ runs over all contacts with body neighbors, $\bm{r}$ is a position vector of the contact force ${{\ensuremath{\bm{f}}}}$ and ${{\ensuremath{\bm{\mathcal{E}}}}}$ is the Levi-Civita permutation symbol. In 3D, the contraction of ${{\ensuremath{\bm{\mathcal{E}}}}}$ from both sides gives the vector product $\bm{b}\cdot{{\ensuremath{\bm{\mathcal{E}}}}}\cdot\bm{a} = \bm{a}\times\bm{b}$.
We limit this section to 2D models, as it is expected that 3D models would yield similar results. Four different 2D tessellation types are considered, namely the *Voronoi*, *randomized Voronoi*, *random* and *centered random* tessellation.
Voronoi tessellation
--------------------
The first tessellation type, referred to as *Voronoi* hereinafter, has parallel normal and contact vectors. It can be obtained via Voronoi tessellation, which is widely used in discrete modeling, or via Power tessellation, which is capable of taking into account the size of the inclusions (mineral aggregates in concrete) associated with rigid bodies [@Eli16].
The *Voronoi* model is created here by placing points randomly into a domain in sequence and accepting only those with a minimum distance from previously placed points that is greater than the length parameter ${\ensuremath{l_{\min}}}$. The sequential placement process is terminated after no point is accepted for a sufficiently large number of trials. The random points serve as nuclei for clipped Voronoi tessellation and their translations and rotations constitute model degrees of freedom.
The macroscopic elastic behavior of the *Voronoi* model is described by Eq. . Numerical verification is performed in [@Eli16] for both the 2D and 3D models, revealing increasing deviation from Eq. with the increasing distance of parameter $\alpha$ from 1. The deviation is caused by the violation of assumption , which is exactly met only for $\alpha=1$. Indeed, assuming $\alpha=1$, applying Eqs. and that were derived from assumption , and using ${{\ensuremath{\bm{t}}}}={{\ensuremath{\bm{n}}}}$, which is valid for *Voronoi* tessellation, the equilibrium equations become $$\begin{aligned}
E_0\sum \limits_e A_e \frac{\bm{\Delta}_e}{l_e} = E_0\bm{\varepsilon}\cdot\sum \limits_e A_e \bm{n}_e =E\bm{\varepsilon}\cdot\int \limits_\Gamma \bm{n} {\mbox{\,d}}{\Gamma} &= 0 \label{eq:lin_mom_Voronoi}\\
E_0\sum \limits_e A_e \frac{\bm{\Delta}_e}{l_e}\cdot{{\ensuremath{\bm{\mathcal{E}}}}}\cdot\bm{r}_e= E_0{{\ensuremath{\bm{\mathcal{E}}}}}:\left[\left(\sum \limits_e A_e \bm{r}_e \otimes \bm{n}_e\right) \cdot \bm{\varepsilon}\right] = E_0{{\ensuremath{\bm{\mathcal{E}}}}}:\left[\int \limits_\Gamma \bm{r} \otimes \bm{n} {\mbox{\,d}}{\Gamma} \cdot \bm{\varepsilon}\right] &= \bm{0} \mathrm{\ or\ } 0 \label{eq:ang_mom_Voronoi}\end{aligned}$$ The sum over the contacts is transformed into integration over the enclosed surface $\Gamma$ of the rigid body. The first integral is the zero vector, while the second integral is the identity matrix multiplied by the rigid body volume. This is derived via component-wise integration with the help of the divergence theorem and unit standard Cartesian basis vector $\bm{j}$. $$\begin{aligned}
\int \limits_\Gamma n_j {\mbox{\,d}}{\Gamma} = \int \limits_\Gamma \bm{j}\cdot {{\ensuremath{\bm{n}}}}{\mbox{\,d}}{\Gamma} = \int \limits_V \nabla\cdot\bm{j}{\mbox{\,d}}{V} &= 0 \\
\int \limits_\Gamma r_i n_j {\mbox{\,d}}{\Gamma} = \int \limits_\Gamma r_i\,\bm{j}\cdot{{\ensuremath{\bm{n}}}}{\mbox{\,d}}{\Gamma} = \int \limits_V \nabla\cdot(r_i\,\bm{j}){\mbox{\,d}}{V} = \int \limits_V \frac{\partial r_i}{\partial x_j}{\mbox{\,d}}{V} &= \begin{cases} V & \mathrm{for\ } i=j \\ 0
& \mathrm{for\ } i\neq j\end{cases} \label{eq:angular_momentum_mod} \end{aligned}$$ Since the first integration is the zero vector, the right-hand side of Eq. is zero and Eq. is exactly satisfied. Substituting the result of the second integration, the right-hand side of Eq. becomes $VE_0{{\ensuremath{\bm{\mathcal{E}}}}}:\bm{\varepsilon}$, which is always zero thanks to the symmetry of the strain tensor and the antisymmetry of the Levi-Civita symbol, and Eq. is exactly satisfied as well.
We have shown that for *Voronoi* tessellation with $\alpha=1$, assumption holds. However, for different $\alpha$ or non-parallel ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$, assumption is incorrect. The actual system is more compliant and has a higher Poisson’s ratio than that predicted by assumption due to the removed constraint on rotations and translations.
![Four types of tessellation created with the same set of nuclei (hollow circles). The centroids of the bodies of *random tessellation* are plotted by solid circles.[]{data-label="fig:tessellations"}](Voronoi){width="13cm"}
![Probability density function $f_{\chi}(\chi)$ for *randomized Voronoi*, *random* and *centered random* tessellation. \[fig:chist\_chi\]](hist_chi){width="8cm"}
Randomized Voronoi tessellation \[sec:rand\_vor\]
-------------------------------------------------
*Randomized Voronoi* tessellation is generated by modifying the *Voronoi* model. We simply move each Voronoi vertex in a random direction by a random distance. The normal vectors, $\bm{n}$, are randomly rotated by such random movements, while the contact vectors, $\bm{t}$, remain intact. The random distance is generated from the uniform distribution in the interval $(0,\,k)$, where $k$ is half of the minimum distance to the closest vertex. This upper limit on $k$ is introduced to prevent overlapping of the rigid bodies. The resulting tessellation has generally non-convex body shapes and non-parallel $\bm{n}$ and $\bm{t}$.
By generating 50 discrete structures of size $150{\ensuremath{l_{\min}}}\times150{\ensuremath{l_{\min}}}$, the distribution function of angle $\chi$ is obtained. Numerical integration of Eq. then provides constants $I_1\approx0.97724$ and $I_2\approx0.91372$. An example of *randomized Voronoi* tessellation as well as the probability distribution function of $\chi$ is shown in Figs. \[fig:tessellations\] and \[fig:chist\_chi\].
Random tessellation
-------------------
The randomization of the Voronoi structure defined in section \[sec:rand\_vor\] is still rather limited and strongly resembles the original *Voronoi* model. When searching for a tessellation with more deviation from the parallelism of ${{\ensuremath{\bm{t}}}}$ and ${{\ensuremath{\bm{n}}}}$, the following process was found to be quite effective. The *random* model is based on two sets of nodes: basic nodes, which are generated by the same sequential process as described above with the minimum distance ${\ensuremath{l_{\min}}}$, and auxiliary nodes (vertices), which are created independently in the same way with the minimum distance ${\ensuremath{l_{\min}}}/2$. Delaunay triangulation is performed on the vertices, and triangles containing a basic node are directly assigned to it. Then, an iterative loop is performed over all unassigned triangles. Whenever a neighboring triangle already belonging to a basic node is found, the unassigned triangle is assigned to the same basic node. The rigid body associated with a given basic node is then a union of all the triangles assigned to that node. The obtained shapes are highly non-convex and typically several contact faces are created between two neighboring bodies.
Again, the distribution function of angle $\chi$ is estimated from 50 discrete structures of the size $150{\ensuremath{l_{\min}}}\times150{\ensuremath{l_{\min}}}$. This is shown along with an example of *random* tessellation in Figs. \[fig:tessellations\] and \[fig:chist\_chi\]. Numerical integration of Eq. provides $I_1\approx0.73516$ and $I_2\approx0.28884$.
Centered random tessellation
----------------------------
The last tessellation type has bodies which are identical to those in the *random* model. The difference is that for *centered random* tessellation, a final step is performed in which the governing nodes bearing the rigid body degrees of freedom are moved into the centroids of the generated bodies. The situation is depicted in Fig. \[fig:tessellations\], where the hollow circles (initial tessellation nuclei) are replaced by solid circles (centroids). Though the shape of the bodies is unchanged, the geometrical characteristics of the tessellation changes because the vectors ${{\ensuremath{\bm{t}}}}$ and lengths $l$ are updated. The statistical evaluation of angle $\chi$ on 50 discrete structures with a size of $150{\ensuremath{l_{\min}}}\times150{\ensuremath{l_{\min}}}$ provides the probability distribution function shown in Fig. \[fig:chist\_chi\] that, when numerically integrated in Eq. , provides $I_1\approx0.78830$ and $I_2\approx0.38688$.
One would assume that since the rigid body shapes are equal for the *random* and *centered random* model, these models would behave equally. We will now prove the opposite by contradiction.
Let us assume two structures with equal shapes and connectivity of the rigid bodies but two different sets of governing nodes (with coordinates $\bm{x}_i$ and $\hat{\bm{x}}_i$, respectively) which bear their degrees of freedom. The normals ${{\ensuremath{\bm{n}}}}$ and contact areas $A$ remain the same because these are dictated solely by rigid body shape. However, the contact lengths and contact vectors are different, $l \neq \hat{l}$ and ${{\ensuremath{\bm{t}}}}\neq\hat{{{\ensuremath{\bm{t}}}}}$. The starting point of the proof by contradiction is that under equal load the movements of the bodies are equal as well, and therefore the displacement jumps at the boundaries are identical, $\bm{\Delta} = \hat{\bm{\Delta}}$. According to the first parts of Eqs. combined with Eq. , the stresses differ by factor $l/\hat{l}$ $$\begin{aligned}
\hat{s}_N &= E_0 \frac{{{\ensuremath{\bm{n}}}}\cdot \bm{\Delta}}{\hat{l}} = s_N\frac{l}{\hat{l}} &
\hat{\bm{s}}_T &= E_0\alpha\left(\frac{\bm{\Delta}}{\hat{l}} - \hat{e}_N{{\ensuremath{\bm{n}}}}\right) = E_0\alpha\left(\frac{\bm{\Delta}}{\hat{l}} - \frac{{{\ensuremath{\bm{n}}}}\cdot \bm{\Delta}}{\hat{l}}{{\ensuremath{\bm{n}}}}\right) = \bm{s}_T\frac{l}{\hat{l}}\end{aligned}$$ The same factor also holds for contact forces as these are only stresses multiplied by areas: $\hat{{{\ensuremath{\bm{f}}}}} = l/\hat{l}{{\ensuremath{\bm{f}}}}$
The first set of forces (${{\ensuremath{\bm{f}}}}$) satisfies equilibrium equations and because we assume it originates from the true solution with the first set of nuclei. We will show now that the second set ($\hat{{{\ensuremath{\bm{f}}}}}$) violates equilibrium equations because the factor $l/\hat{l}$ differs for each element $$\begin{aligned}
\sum \limits_e \hat{{{\ensuremath{\bm{f}}}}}_e =\sum \limits_e {{\ensuremath{\bm{f}}}}_e \frac{l_e}{\hat{l}_e} &\neq \sum \limits_e {{\ensuremath{\bm{f}}}}_e = \bm{0}\\
\sum \limits_e \hat{{{\ensuremath{\bm{f}}}}}_e\cdot{{\ensuremath{\bm{\mathcal{E}}}}}\cdot\bm{r}_e = \sum \limits_e \frac{l_e}{\hat{l}_e} {{\ensuremath{\bm{f}}}}_e\cdot{{\ensuremath{\bm{\mathcal{E}}}}}\cdot\bm{r}_e &\neq \sum \limits_e {{\ensuremath{\bm{f}}}}_e\cdot{{\ensuremath{\bm{\mathcal{E}}}}}\cdot\bm{r}_e = \bm{0} \mathrm{\ or\ } 0\end{aligned}$$ The assumption of the equality of rigid body movements for different governing nodes leads to a contradiction. Therefore, the movements are generally different and the macroscopic elastic properties of the *random* and *centered random* model differ as well.
Comparison of analytical formulas to actual characteristics
-----------------------------------------------------------
A two dimensional model with a size of $150{\ensuremath{l_{\min}}}\times150{\ensuremath{l_{\min}}}$ was generated for all four tessellation types. The structure was loaded by the prescribed translations and rotations along the whole boundary, $\Gamma$: $u^{\Gamma}_1=p x_1$, $u^{\Gamma}_2=q x_2$ and $\varphi^{\Gamma}=0$. The resulting macroscopic strain components are $\varepsilon_{11}=p$, $\varepsilon_{22}=q$ and $\varepsilon_{12}=\varepsilon_{21}=0$, respectively. Alternatively, the strain tensor can be obtained via the linear regression of location-translation dependence, see [@Eli16]. The stress tensor was estimated from all inner contacts with a distance from the boundary greater than $3{\ensuremath{l_{\min}}}$ [@Bag96] $$\begin{aligned}
\bm{\sigma} = \left\langle\sum\limits_e {{\ensuremath{\bm{f}}}}_e \otimes \bm{c}_e\right\rangle^{\mathrm{SYM}}\end{aligned}$$ where $\bm{c}$ is the centroid of the contact face and the symmetrization of the second order tensor reads $\langle \bullet \rangle^{\mathrm{SYM}} = (\bullet + \bullet^{T})/2$.
Since the diagonal strain term $\varepsilon_{12}$ is zero, the macroscopic elastic properties are easily evaluated $$\begin{aligned}
\nu &= \begin{cases} \dfrac{\sigma_{22}\varepsilon_{11}-\sigma_{11}\varepsilon_{22}}{\sigma_{11}\varepsilon_{11}-\sigma_{22}\varepsilon_{22}} & \mathrm{plane\ stress}\\[3mm]
\dfrac{\sigma_{22}\varepsilon_{11}-\sigma_{11}\varepsilon_{22}}{(\sigma_{11}+\sigma_{22})(\varepsilon_{11}-\varepsilon_{22})} & \mathrm{plane\ strain}
\end{cases} \label{eq:poisson_numeric}
\\
E &= \begin{cases}\dfrac{\sigma_{11}^2-\sigma_{22}^2}{\sigma_{11}\varepsilon_{11}-\sigma_{22}\varepsilon_{22}} & \mathrm{plane\ stress}\\[3mm] \dfrac{(\sigma_{11}-\sigma_{22})(\varepsilon_{11}(\sigma_{11}+2\sigma_{22})-\varepsilon_{22}(2\sigma_{11}+\sigma_{22}))}{(\varepsilon_{11}-\varepsilon_{22})^2(\sigma_{11}+\sigma_{22})} & \mathrm{plane\ strain}
\end{cases} \label{eq:E_numeric}\end{aligned}$$
The model was assembled and solved for each tessellation type and various $\alpha$ ratios. The macroscopic characteristics obtained by Eqs. and are plotted in Fig. \[fig:elastic\_constants\_NUM\] along with the derived analytical predictions and . There is reasonable correspondence for *Voronoi* and *random Voronoi* tessellation when $\alpha>0.3$. In all the other cases the predictions severely departed from numerical solution due to the unfulfilled fundamental assumption . However, the elastic tensor evaluated using Eq. on the generated structure correspond in all the cases with one from Eq. when analytical expectations and are used.
![The macroscopic elastic characteristics (Poisson’s ratio and elastic modulus) of a two dimensional discrete system derived analytically and computed numerically for four different tessellation types.[]{data-label="fig:elastic_constants_NUM"}](elastic_constants_NUM){width="15cm"}
Though the correspondence is rather weak, there is a clear trend seen in both the analytical and the numerical results. The Poisson’s ratio shrinks into a narrower interval as the integral $I_2$ grows due to the loss of parallelism between ${{\ensuremath{\bm{n}}}}$ and ${{\ensuremath{\bm{t}}}}$; simultaneously, the elastic modulus increases. We consider this result to be a verification of our conclusion that Poisson’s ratio limits are maximized for Voronoi (or Power) tessellation. Any departure from perpendicularity between ${{\ensuremath{\bm{t}}}}$ and the face plane causes the narrowing of these limits.
Conclusions
===========
Analytical estimations are derived for the macroscopic elastic behavior of (i) discrete isotropic assemblies with (ii) normal and shear force linearly dependent on normal and shear displacement discontinuity and (iii) a continuously filled domain. The derivation takes advantage of a strong assumption about the rotations (assumed zero everywhere) and translations (assumed uniformly distributed over domain) of discrete bodies. Comparison with numerical results reveals that the assumption is only reasonable for a limited range of model types, though the overall trend emerging from the derived equations is confirmed.
- It is proven that the widest limits of Poisson’s ratio are obtained when contact and normal vectors are parallel, such as when Voronoi or Power tessellation is used to generate rigid body shapes.
- Any other model geometry (where normal and contact vectors are not strictly parallel) shrinks the interval of achievable Poisson’s ratio. It is not possible to extend the limits via geometrical manipulations. These limits are $\nu\in\left[-1,\,\sfrac{1}{3}\right]$ for 2D plane stress, $\nu\in\left(-\infty,\,\sfrac{1}{4}\right]$ for 2D plane strain and $\nu\in\left[-1,\,\sfrac{1}{4}\right]$ for 3D models, respectively.
- The mechanical parameters $E_0$ and $\alpha$ are considered constant throughout the whole volume. Also, no overlapping or gaps between rigid bodies are allowed. The abandonment of these conditions seems to be one of the possible ways to proceed further in searching for other methods of extending the Poisson’s ratio interval of discrete models.
- It is shown that the position of the governing node affects model results, even though the actual rigid bodies are identical. It is questionable whether the body centroids, Voronoi sites (generators of the Voronoi diagram) or other points should be used as the governing nodes. It would be advantageous to find a model modification that would remove this dependence.
The discrete models are often used for analyzing inelastic phenomena (mostly fracture). The connection between the angle formed by the normal and contact vectors and the local and global inelastic behavior of the model is an interesting open topic deserving further investigation.
Acknowledgement {#acknowledgement .unnumbered}
===============
Financial support provided by the Czech Science Foundation under project No. GA19-12197S is gratefully acknowledged.
|
---
abstract: 'The C/O ratio is a defining feature of both gas giant atmospheric and protoplanetary disk chemistry. In disks, the C/O ratio is regulated by the presence of snowlines of major volatiles at different distances from the central star. We explore the effect of radial drift of solids and viscous gas accretion onto the central star on the snowline locations of the main C and O carriers in a protoplanetary disk, H$_2$O, CO$_2$ and CO, and their consequences for the C/O ratio in gas and dust throughout the disk. We determine the snowline locations for a range of fixed initial particle sizes and disk types. For our fiducial disk model, we find that grains with sizes $\sim$$0.5$ cm $\lesssim s \lesssim$ 7 m for an irradiated disk, and $\sim$$0.001$ cm $\lesssim s \lesssim$ 7 m for an evolving and viscous disk, desorb at a size-dependent location in the disk, which is independent of the particle’s initial position. The snowline radius decreases for larger particles, up to sizes of $\sim$7 m. Compared to a static disk, we find that radial drift and gas accretion in a viscous disk move the H$_2$O snowline inwards by up to 40 %, the CO$_2$ snowline by up to 60 %, and the CO snowline by up to 50 %. We thus determine an inner limit on the snowline locations when radial drift and gas accretion are accounted for.'
author:
- 'Ana-Maria A. Piso, Karin I. Öberg, Tilman Birnstiel, Ruth A. Murray-Clay'
bibliography:
- 'refs.bib'
title: 'C/O and Snowline Locations in Protoplanetary Disks: The Effect of Radial Drift and Viscous Gas Accretion'
---
Introduction {#sec:intro}
============
The chemical composition of protoplanetary disks affects planet formation efficiencies and the composition of nascent planets. Gas giants accrete their envelopes from the nebular gas. As such, planet compositions are tightly linked to the structure and evolution of the protoplanetary disk in which they form. It is thus essential to understand the disk chemistry and dynamics well enough to (1) predict the types of planet compositions that result from planet formation in different parts of the disk, and (2) backtrack the planet formation location based on planet compositions.
The structures of protoplanetary disks are complex, and affected by a multitude of chemical and dynamical processes (see review by @henning13). From the chemistry perspective, volatile compounds are particularly important. Their snowline locations determine their relative abundance in gaseous and solid form in the disk,. Based on protostellar and comet abundances, some of the most important volatile molecules are H$_2$O, CO$_2$, CO, N$_2$. Recent observations of protoplanetary disks have provided valuable information about the abundances and snowline locations of some of these compounds. For example, the CO snowline has been detected in the disk around TW Hya [@qi13], as well as in the disk around HD 163296 [@mathews13] using line emissions from DCO$^+$. Observations of TW Hya have also revealed a H$_2$O snowline [@zhang13], and more such snowline detections are expected in future ALMA cycles. These observations are currently lacking an interpretive framework that takes into account all important dynamical and chemical processes. Furthermore, such a framework is crucial to connect observed snowline locations to planet formation.
An important consequence of snowline formations in disks is that disks are expected to present different carbon-to-oxygen (C/O) ratios in the gas and in icy dust mantles at different disk radii. This effect was quantified by @oberg11, who considered the fact that the main carries of carbon and oxygen, i.e. H$_2$O, CO$_2$ and CO, have different condensation temperatures. This changes the relative abundance of C and O in gaseous and solid form as a function of the snowline location of the volatiles mentioned above. @oberg11 calculated analytically the C/O ratio in gas in dust as a function of semimajor axis for passive protoplanetary disks and found a gas C/O ratio of order unity between the CO$_2$ and CO snowlines, where oxygen gas is highly depleted. This effect was used to explain claims of detections of superstellar C/O ratios in exoplanet atmospheres (e.g., WASP-12b, @madhu11), which however have been unambiguously refuted (@stevenson14, @kreidberg15). @oberg11 assumed a static disk with no chemical evolution. In reality, dynamical and chemical processes affect the snowline locations and the resulting C/O ratio. Several works have addressed some of these effects. @madhu14 use a steady-state active disk model that includes planetary migration and use the C/O ratio to constrain migration mechanisms. @alidib14 calculate the C/O ratio throughout the disk by incorporating the evolution of solids, i.e. radial drift, sublimation and grain coagulation, as well as the diffusion of volatile vapors. @alidib14 use the 1+1D $\alpha$-disk model of @hughes10, in which the gas drifts outwards in the disk midplane, and thus small particles that are well-coupled to the gas will also advect outward. Their model assumes a cyclical conversion between H$_2$O or CO dust and vapor: large enough particles that are decoupled from the gas drift inwards and start desorbing. Once their sublimation is complete, back-diffusion moves the H$_2$O or CO vapor ouwards to their respective snowlines, where they instantly condense into mm-sized particles that diffuse outwards with the gas while coagulating into larger particles. Once the grains become large enough to decouple from the gas and drift inwards, the cycle restarts. This “‘conveyor belt’” model is based on the pioneering work by @cuzzi04, and @ciesla06 for the evolution of H$_2$O in a viscous disk. This approach leads @alidib14 to find that the gaseous C/O ratio increases with time inside the H$_2$O snowline, approaching unity at 2 AU after $\sim$$10^4-10^5$ years. @thiabaud15 consider additional carbon carrier volatile species in their chemical network, such as CH$_4$, and find that the gas C/O ratio may be enriched by up to four times the Solar value in the outer parts of the disk where CH$_4$ and CO are the only gaseous carriers of C and O. They also include nitrogen carriers such as N$_2$ or NH$_3$, and perform similar calculations for nitrogen.
Each of these studies have considered a specific combination of dynamical and chemical effects. One scenario that has not yet been considered is the combination of radial drift and viscous gas accretion in isolation. Studying these two dynamical processes makes it possible to quantify their separate effect on snowline locations and the C/O ratio at various disk radii.
In this paper, we perform a systematic study to understand the detailed qualitative and quantitative effects of radial drift and gas accretion on the H$_2$O, CO$_2$ and CO snowline locations, and the resulting C/O ratio in gas and dust throughout the protoplanetary disk. More importantly, we obtain a limit on how close to the star the snowline locations can be pushed by radial drift and gas accretion.
This paper is organized as follows. In Section \[sec:model\], we present our disk, radial drift and desorption models, as well as the timescales relevant to the coupled drift-desorption process. We calculate the H$_2$O, CO$_2$ and CO snowline locations as a function of particle size for an irradiated and an evolving disk in Section \[sec:snowlines\], and the resulting C/O ratio throughout the disk in Section \[sec:COratio\]. In Section \[sec:discussion\], we discuss the generality of our results, as well as additional effects on the snowline locations. Finally, we summarize our findings in Section \[sec:summary\].
Model Framework {#sec:model}
===============
We present our protoplanetary disk model for a static, an irradiated, an evolving, and a viscous disk in section \[sec:disk\]. In section \[sec:drift\], we describe our analytic model for the radial drift of solids. We summarize our ice desorption model in section \[sec:desorption\]. Finally, we discuss the relevant timescales for dynamical effects that affect snowline locations in section \[sec:timescales\].
Disk Model {#sec:disk}
----------
To understand the separate effects of radial drift, radial movement of gas throughout the disk due to gas accretion, and accretion heating, we use four separate disk models: *static disk*, which is solely irradiated by the host star and does not take into account gas accretion onto the star or radial drift; *irradiated disk*, which has the same temperature profile as the static disk and does not experience gas accretion or accretional heating, but it takes into account radial drift of solids; *evolving disk*, in which the gas is accreting onto the central star causing the gas surface density to decrease with time, but which does not experience accretion heating; and *viscous disk*, for which the mass flux $\dot{M}$ is constant in time and independent of semimajor axis, and the temperature profile is calculated using both accretional heating and stellar irradiation.
**Static and Irradiated disk.** We adopt a minimum mass solar nebula (MMSN) disk model for a static and and an irradiated disk similar to the prescription of @chiang10. The gas surface density and midplane temperature are
\[eq:disk\] &=&2000 (r/)\^[-1]{} \^[-2]{}\
T &=& 120 (r/)\^[-3/7]{} ,
where $r$ is the semimajor axis. Our surface density profile is flatter than the $\Sigma \propto r^{-3/2}$ used by @chiang10. Our choice is inspired by observations of protoplanetary disks at radii larger than $\sim$20 AU (e.g., @andrews10), which suggest that typical disks may have surface density profiles with $\Sigma \propto r^{-1}$. A slope flatter than $\Sigma \propto r^{-3/2}$ is also more consistent with the temperature profile for a steady-state gas disk (see the Viscous disk heading below and [Appendix \[app:steadystate\]]{}). We use the static disk model to compare our results with those of @oberg11.
**Evolving disk.** We model the evolving disk as a thin disk with an $\alpha$-viscosity prescription [@shakura73]: $$\label{eq:nu}
\nu=\alpha c H.$$ Here $\nu$ is the kinematic viscosity, $\alpha < 1$ is a dimensionless coefficient and we choose $\alpha=0.01$, and $c$, $H$ are the isothermal sound speed and disk scale height, respectively:
\[eq:cdHd\] c &=&\
H&=& ,
where $k_{\rm B}$ is the Boltzmann constant, $\mu$ is the mean molecular weight of the gas, $m_{\rm p}$ is the proton mass, and $\Omega_{\rm k} \equiv \sqrt{G M_*/r^3}$ is the Keplerian angular velocity, with $G$ the gravitational constant and $M_*$ the stellar mass. We choose $M_*=M_{\odot}$ and $\mu=2.35$, corresponding to the Solar composition of hydrogen and helium. The temperature profile for the evolving disk is assumed to be the same as for the irradiated disk and given by Equation (\[eq:diskT\]). From Equations (\[eq:nu\]) and (\[eq:cdHd\]), the viscosity can thus be expressed as a power-law in radius, $\nu \propto r^{\gamma}$, with $\gamma=15/14 \approx 1$ for our choice of parameters. Following @hartmann98, we define $R \equiv r/r_{\rm c}$ and $\nu_{\rm c} \equiv \nu(r_{\rm c})$, where $r_{\rm c}$ is a characteristic disk radius. We choose $r_{\rm c}=100$ AU. The gas surface density is given by the self-similar solution
$$\label{eq:Sigmaact}
\Sigma(R, \tilde t) = \frac{M (2 - \gamma)}{2 \pi r_{\rm c}^2 R^{\gamma}} \tilde t^{-(5/2-\gamma)/(2-\gamma)} \exp{\Big[-\frac{R^{(2-\gamma)}}{\tilde t}}\Big],$$
where $M$ is the total disk mass and
\[eq:T\] t & & + 1\
t\_[c]{} & & ,
where $t$ is time. We choose $M=0.1 M_{\odot}$ (e.g., @birnstiel12), but we note that our results are insensitive to this choice (see Section \[sec:discussion\]). The irradiated and evolving disk surface densities match at $t \approx 5 \times 10^5$ years in the inner disk, but they diverge at distances larger than a few AU due the exponential cutoff in radius of the surface density of the evolving disk (Equation \[eq:Sigmaact\]).
**Viscous disk.** Calculating the midplane temperature self-consistently for an evolving disk that is also actively heated, and thus whose thermal evolution is dominated both by accretion heating and stellar irradiation, is non-trivial. We therefore use instead the Shakura-Sunyaev thin disk steady-state solution to derive the midplane temperature profile, $T_{\rm act}$. The equations governing the evolution of the steady-state disk are listed in [Appendix \[app:steadystate\]]{}. We assume an interstellar opacity for the dust grains given by @bell94, but reduced by a factor of 100. This reduction is due to the fact that disk opacities are lower than the interstellar one. While this scaling is consistent with more detailed models of grain opacities in disks (e.g., @mordasini14), realistic disk opacities are much less sensitive to changes in temperature than the interstellar opacity if substantial grain growth has occurred. However, the disk temperature does not vary significantly across the small region of the disk where accretion heating is important ($r \lesssim 1$ AU). Moreover, using an analytic opacity formula is more convenient since it results in a constant gas surface density in the inner disk region (see below). Our opacity law is thus
$$\label{eq:opacity}
\kappa=\kappa_0 T_{\rm act}^2,$$
where $\kappa_0=2 \times 10^{-6}$. By solving the Equation set (\[eq:diskeq\]) we find $$\label{eq:Tdact}
T_{\rm act}=\frac{1}{4 r} \Big(\frac{3 G \kappa_0\dot{M}^2 M_* \mu m_{\rm p} \Omega_{\rm k}}{\pi^2 \alpha k_{\rm B} \sigma}\Big)^{1/3}.$$ Since both accretion heating and stellar irradiation contribute to the thermal evolution of the disk, we compute the midplane temperature for our viscous disk as $$\label{eq:activeT}
T^4 = T_{\rm act}^4 + T_{\rm irr}^4,$$ where to avoid notation confusion $T_{\rm irr}=T$ from Equation (\[eq:diskT\]), the temperature profile for an irradiated disk. We can then easily determine $c$ and $H$ from Equation (\[eq:cdHd\]), as well as the viscosity $\nu$ from Equation (\[eq:nu\]) for a given $\alpha$. For consistency, we choose $\alpha=0.01$ as in the previous case. Finally, we determine $\Sigma$ from Equation (\[eq:Mdot\]), where we choose $\dot{M}=10^{-8} M_{\odot}$ yr$^{-1}$ based on disk observations (e.g., @andrews10). In the inner portion of our disk ($r \lesssim 1$ AU for our fiducial model with $\dot{M}=10^{-8} M_{\odot}$ yr$^{-1}$), our choice of opacity (Equation \[eq:opacity\]) implies that the disk has a constant surface density with radius (see Equations \[eq:diskeq\]).
Before we proceed forward, we note that our disk models assume a constant stellar luminosity $L_*$, as well as a constant mass accretion rate $\dot{M}$ for the viscous disk. In reality, the stellar luminosity decreases as the host star contracts, which will reduce the disk temperature and push the snowlines inward, as we explain in Section \[sec:neglected\]. For a Solar type star, as our fiducial model assumes, $L_*$ remains relatively constant during the star’s pre-main sequence evolution of $\sim$ 10 Myr [@kennedy06], which is larger than the giant planet formation timescale. Thus the midplane temperature will not change significantly for our model due to variations in stellar luminosity, but it may decrease substantially for smaller stars, pushing the snowline inward (see Section \[sec:neglected\] for details). Realistic mass accretion rates, $\dot{M}$, may vary between $\sim$$10^{-7}$ and $\sim$$10^{-9}$ $M_{\odot}$ yr$^{-1}$ as the disk evolves (e.g., @chambers09, @sicilia10). For $\dot{M}\lesssim 10^{-9} M_{\odot}$ yr$^{-1}$, the disk becomes optically thin and hence depleted of gas, which means giant planets must have formed before $\dot{M}$ becomes too low. @garaud07 find that the snowline locations scale as $r_{\rm snow} \propto \dot{M}^{1/3}$. A factor of 100 reduction in the mass accretion rate will thus move the H$_2$O snowline inwards by a factor of $\sim$$4$ — since accretion heating is dominant only in the inner disk, the CO$_2$ and CO snowline locations are unlikely to be affected by changes in mass accretion rate. The inward movement of the H$_2$O snowline due to the decrease in $\dot{M}$ may be even larger, by up to one order of magnitude [@chambers09]. We thus conclude that changes in $L_*$ throughout time may only modestly affect our results, while changes in $\dot{M}$ may significantly affect our results for the H$_2$O snowline, as its location may be determined by the decline in mass accretion rate rather than radial drift. The time variability of $L_*$ and $\dot{M}$ should be taken into account when drawing more robust conclusions, as well as for different host star and disk properties.
Radial Drift {#sec:drift}
------------
Solid particles orbit their host star at the Keplerian velocity $v_{\rm k} \equiv \Omega_{\rm k} r$. The gas, however, experiences an additional pressure gradient, which causes it to rotate at sub-Keplerian velocity [@weidenschilling77]. Dust grains that are large enough thus experience a headwind, which removes angular momentum, causing the solids to spiral inwards and fall onto the host star. Small particles are well-coupled to the gas, while large planetesimals are decoupled from the gas. From the review by @chiang10, the extent of coupling is quantified by the dimensionless stopping time, $\tau_{\rm s} \equiv \Omega_{\rm k} t_{\rm s}$, where $t_{\rm s}$ is $$\label{eq:ts}
t_{\rm s}= \left\{
\begin{array}{l l}
\rho_{\rm s} s / (\rho c), & \quad s < 9 \lambda/4 \,\,\,\ \text{Epstein drag} \\
4 \rho_{\rm s} s^2 / (9 \rho c \lambda), & \quad s < 9 \lambda/4, \,\text{Re} \lesssim 1 \,\,\,\ \text{Stokes drag.}
\end{array}
\right.$$ Here $\rho$ is the gas midplane density, $\rho_{\rm s}=2$ g cm$^{-3}$ is the density of a solid particle, $s$ is the particle size, $\lambda$ is the mean free path, and Re is the Reynolds number.
For an irradiated disk, the radial drift velocity can be approximated as $$\label{eq:rdotpas}
\dot{r} \approx -2 \eta \Omega_{\rm k} r \Big(\frac{\tau_{\rm s}}{1+\tau_{\rm s}^2}\Big),$$ where $$\label{eq:eta}
\eta \equiv - \frac{\partial P/\partial \ln r}{2 \rho v_{\rm k}^2} \approx \frac{c^2}{2 v_k^2}$$ and $P = \rho c^2$ is the disk midplane pressure.
For an evolving disk, the radial drift velocity has an additional term due to the radial movement of the gas [@birnstiel12], i.e. $$\label{eq:rdotact}
\dot{r} \approx -2 \eta \Omega_{\rm k} r \Big(\frac{\tau_{\rm s}}{1+\tau_{\rm s}^2}\Big) + \frac{\dot{r}_{\rm gas}}{1+\tau_{\rm s}^2},$$ where $\dot{r}_{\rm gas}$ is the radial gas accretion velocity and can be expressed as (e.g., @fkr02) $$\label{eq:vgas}
\dot{r}_{\rm gas} = - \frac{3}{\Sigma \sqrt{r}} \frac{\partial}{\partial r}(\nu \Sigma \sqrt{r})$$ with $\Sigma$ from Equation (\[eq:Sigmaact\]). For the viscous disk (see Section \[sec:disk\]), $\dot{r}_{\rm gas}$ can be expressed more simply using the definition of the mass flux, $\dot{M}=-2 \pi r \dot{r}_{\rm gas} \Sigma$, with $\dot{M}$ fixed and $\Sigma$ obtained from Equation (\[eq:Mdot\]). For our choice of parameters for both the evolving and the viscous disks, we have found that the radial flow of gas(calculated from Equation \[eq:vgas\] for the evolving disk and from $\dot{r}_{\rm gas}=-\dot{M}/(2 \pi r \Sigma)$ for the viscous disk) is always directed inward for our parameter space of interest, in contrast with the model of @alidib14 which assumes that the gas drifts outwards (see Section \[sec:intro\]). For our evolving disk, the gas starts drifting outwards at a radius $r_{\rm switch} \approx 200 AU$, which is however well outside the CO snowline in our model. We thus note that variations in our fiducial disk model parameters (e.g., $T$, $\Sigma$, $\dot{M}$) may cause the gas to flow outwards in the outer parts of the disk, specifically at the CO$_2$ and CO snowline locations. Since drifting particles larger than a few cm are only modestly affected by gas accretion, an outward gas flow would move the CO$_2$ and CO locations further away from the star only for the smallest particles in our model, which are well-coupled to the gas.
Volatile Desorption {#sec:desorption}
-------------------
In order for a volatile species to thermally desorb, it has to overcome the binding energy that keeps it on the grain surface. Following @hollenbach09, the desorption rate per molecule for a species $x$ can be expressed as $$\label{eq:Rdes}
R_{\rm{des}, x} = \nu_x \exp{(-E_x/T_{\rm grain})},$$ where $E_x$ is the adsorption binding energy in units of Kelvin, $T_{\rm grain}$ is the grain temperature, and $\nu_x=1.6 \times 10^{11} \sqrt{(E_x/\mu_x)}$ s$^{-1}$ is the molecule’s vibrational frequency in the surface potential well, with $\mu_x$ the dimensionless mean molecular weight. We assume that the dust and gas have the same temperature in the disk midplane, hence $T_{\rm grain}=T$. For H$_2$O, CO$_2$ and CO, the binding energies $E_x$ are assumed to be 5800 K, 2000 K and 850 K, respectively (@collings04, @fraser01, @aikawa96). We use the desorption rate, $R_{\rm des}$, to estimate the desorption timescale for particles of different sizes as described in section \[sec:timescales\].
Relevant Timescales {#sec:timescales}
-------------------
We can estimate the extent to which radial drift and gas accretion affect desorption by comparing the timescales for desorption, drift and accretion, for solids of different sizes and compositions.
*Desorption timescale.* We assume that the solid bodies are perfect spheres and are entirely composed of only one volatile species, i.e. either H$_2$O, CO$_2$ or CO [^1]. The timescale to desorb a single layer of molecules can then be estimated as $$\label{eq:tdes}
t_{\rm des}=\frac{\rho_{\rm s}}{3 \mu_x m_{\rm p}} \frac{s}{N_x R_{\rm des, x}},$$ where $N_x \approx 10^{15}$ sites cm$^{-2}$ is the number of adsorption sites of volatile $x$ per cm$^2$, assuming that the particle has a smooth surface [@hollenbach09].
*Radial drift timescale.* To order of magnitude, the radial drift timescale can be estimated as $$\label{eq:tdrift}
t_{\rm drift} \sim \Big|\frac{r}{\dot{r}}\Big|,$$ where $\dot{r}$ is the radial drift velocity given by Equation (\[eq:rdotpas\]) for an irradiated disk and by Equation (\[eq:rdotact\]) for an evolving disk.
*Gas accretion timescale.* The timescale for gas accretion onto the central star for an evolving disk is (e.g., @armitage10) $$\label{eq:tgas}
t_{\rm gas, acc} \sim \frac{r^2}{\nu} \sim \frac{1}{2 \alpha \eta \Omega_{\rm k}},$$ with the latter expression derived from Equations (\[eq:nu\]) and (\[eq:eta\]).
![Relevant timescales for dynamical effects in the desorption process: $t_{\rm drift}$ (solid lines), $t_{\rm des}$ (dashed lines) and $t_{\rm gas, acc}$ (dotted lines). The timescales are calculated at three representative locations, i.e. the H$_2$O, CO$_2$ and CO snowlines in the static disk. For our choice of parameters, the snowlines are located at $\sim$0.7 AU (blue lines), $\sim$8.6 AU (green lines) and $\sim$59 AU (red lines), respectively. The horizontal dot-dashed line represents a typical disk lifetime of 3 Myr. The particle size ordering at the minimum $t_{\rm drift}$ is not monotonic in snowline distance due to different drag regimes for those particle sizes at the snowline locations (Epstein drag at the H$_2$O and CO$_2$ snowlines, and Stokes drag at the CO snowline). Similarly, the ordering of $t_{\rm des}$ is not monotonic in snowline distance due to the non-monotony in mean molecular weight between H$_2$O, CO$_2$ and CO (18 $m_{\rm p}$, 44 $m_{\rm p}$ and 28 $m_{\rm p}$, respectively). Radial drift and gas accretion affect desorption in the regions where their respective timescales, i.e. $t_{\rm drift}$ and $t_{\rm gas, acc}$, are comparable to the desorption timescale $t_{\rm des}$.[]{data-label="fig:timescales"}](drift_timescales_betaS1_gas_acc_new2.pdf){width="50.00000%"}
For simplicity, we calculate the radial drift timescale, $t_{\rm drift}$, for an irradiated disk in this section, but most of our conclusions hold true for an evolving disk as well. Figure \[fig:timescales\] shows $t_{\rm des}$, $t_{\rm drift}$ and $t_{\rm gas, acc}$ as a function of particle size at three different locations in the disk, corresponding to the H$_2$O, CO$_2$ and CO snowlines in the static disk. As expected, micron-sized particles desorb on very short timescales of $\sim 1-1000$ years in the close vicinity of their respective snowlines, since the desorption rate depends exponentially on temperature and hence on disk location (see Equation \[eq:Rdes\]). On the other hand, their radial drift timescale exceeds the typical disk lifetime of a few Myr by several orders of magnitude due to their strong coupling with the gas. Thus for small particles in an irradiated disk, the snowline locations and the C/O ratio are the same as for a static disk (see Figure 1 from @oberg11). This is not true for an evolving disk, however, where gas accretion causes even micron-sized particles to drift significantly before desorbing, as we show in section \[sec:snowlines\]. At the other extreme, kilometer-sized particles are unaffected by gas drag and have long desorption timescales ($\gg$1 Myr ), and the snowline locations and C/O ratio remain unchanged in this case as well. This is true for both irradiated and evolving disks, since large planetesimals are decoupled from the gas and hence unaffected by gas accretion onto the host star.
In the particle size regime for which (1) $t_{\rm drift} \lesssim t_{\rm des} \lesssim t_{\rm d}$ ($t_{\rm d}=3$ Myr is the disk lifetime), i.e. for $\sim$$0.5$ cm $\lesssim s \lesssim$ $1000$ cm, or (2) $t_{\rm gas, acc} \lesssim t_{\rm des} \lesssim t_{\rm d}$, i.e. for $\sim$$0.1$ cm $\lesssim s \lesssim$ $10$ cm, radial drift or gas accretion (or both) are faster than thermal desorption, which is of particular interest for our purposes. We note that $t_{\rm gas, acc}<t_{\rm d}$ always holds true. Particles of sizes that satisfy the requirements above will drift significantly due to radial drift or gas accretion before desorbing, thus moving the H$_2$O, CO$_2$ and CO snowlines closer towards the central star and changing the C/O ratio throughout the disk. We quantify these effects in sections \[sec:snowlines\] and \[sec:COratio\].
Snowline Locations {#sec:snowlines}
==================
{width="70.00000%"}
{width="70.00000%"}
In this section we use the model described in section \[sec:model\] to quantify the effects of radial drift (irradiated disk) or radial drift and gas accretion (evolving disk) on the snowline location, for dust particles of different sizes composed of either H$_2$O, CO$_2$ or CO. Specifically, we determine a particle’s final location (i.e., where the particle either fully desorbs or remains at its initial size due to having a desorption timescale longer than the time at which we stop the simulation) as a function of its initial position in the disk, after the gas disk has dissipated. The disk lifetime, $t_{\rm d}$, is particularly relevant since the timescale for giant planet formation must be less than or equal to $t_{\rm d}$. The snowline locations at $t=t_{\rm d}$ throughout the protoplanetary disk determine the disk C/O ratio in gas at this time, and thus the C/O ratio in giant planet atmospheres that have formed *in situ*, before planetesimal accretion or core dredging.
For each species $x$, we determine the final location in the disk of a particle of initial size $s_0$ by solving the following system of coupled differential equations:
\[eq:ddt\] &= & - N\_x R\_[des, x]{}\
&=& ,
where the desorption rate $R_{\rm des, x}$ for each particle type (i.e., composed of H$_2$O, CO$_2$ or CO) is evaluated at $T=T(r)$, and the radial drift velocity $\dot{r}$ is given by Equation (\[eq:rdotpas\]) for an irradiated disk and Equation (\[eq:rdotact\]) for an evolving disk. Equations (\[eq:dsdt\]) and (\[eq:drdt\]) describe the coupled desorption and radial drift, and can be derived straightforwardly from Equation (\[eq:tdes\]). Our initial conditions are $s(t_0)=s_0$ and $r(t_0)=r_0$, where $t_0$ is the initial time at which we start the integration and $r_0$ is the initial location of the particle. We choose $t_0=1$ year, but our result is independent on the initial integration time as long as $t_0 \ll t_{\rm d}$. The desorption timescale $t_{\rm des}$ will then satisfy $s(t_{\rm des})=0$, from which we can determine the desorption distance $r_{\rm des}=r(t_{\rm des})$.
We define the final position of a grain as the disk location it has reached after $t_{\rm d}=3$ Myr, or the radius at which it completely desorbs if that happens after a time shorter than 3 Myr. Figure \[fig:snowlines\] shows our results for H$_2$O, CO$_2$ and CO particles, for both an irradiated and an evolving disk. We do not show the results for the viscous disk as they would complicate the plot without adding any qualitative insight — the results for the viscous disk are quantitatively similar with those of the evolving disk for the CO$_2$ and CO particles, but they are different for the H$_2$O grains, since accretion heating will push the H$_2$O snowline outwards (see Section \[sec:COratio\]). We also show the static snowlines for comparison, which are calculated by balancing adsorption and desorption [@hollenbach09]. Kilometer-sized bodies do not drift or desorb during the disk lifetime neither for an irradiated nor for an evolving disk. Similarly, micron- to mm-sized particles in the irradiated disk do not drift and only desorb if they are located inside the static snowlines. In an evolving disk, however, micron-to mm-sized grains do drift significantly since they move at the same velocity as the accreting gas. For $0.5$ cm $\lesssim s_0 \lesssim$ 700 cm in an irradiated disk and $0.001$ cm $\lesssim s_0 \lesssim$ 700 cm in an evolving disk, we notice that particles of initial size $s_0$ desorb at a particle size dependent radius $r_{\rm des}$ regardless of their original location in the disk. In fact, the only grains that will both drift and evaporate are those that reach their fixed final location (represented by the horizontal curves in Figure \[fig:snowlines\]) within the disk lifetime. We show in section \[sec:COratio\] that this result is essential in determining the C/O ratio throughout the disk for different particle sizes.
Another interesting feature of Figure \[fig:snowlines\] is that particles above a certain size ($\sim$7 m for our choice of parameters) all desorb at the same distance. This is due to the fact that once the large bodies pass the static snowline, they first lose mass, thus eventually following the same evolutionary track as the meter-sized bodies and evaporating at the same location.
{width="80.00000%"}
Intuitively, this fixed $r_{\rm des}$ should be the location in the disk for which $t_{\rm drift} \sim t_{\rm des}$, given an initial particle size. We can calculate this location analytically by equating Equations (\[eq:tdes\]) and (\[eq:tdrift\]) and solving for $r=r_{\rm des}(s)$ for a given particle size $s$. Figure \[fig:an\_vs\_actual\] shows $r_{\rm des}$ calculated analytically using the prescription above as a function of the actual desorption distance calculated numerically. We display this result for the range of particle sizes that desorb at a fixed distance in an irradiated and an evolving disk (see Figure \[fig:snowlines\]). We notice that the analytic approximation accurately reproduces the numerical result for most cases of interest, but it deviates for particles larger than $s \gtrsim10$ cm. For small particles with $\tau_{\rm s} \ll 1$, $t_{\rm drift}$ is a power-law in $r$ (for our parameters, $t_{\rm drift} \propto r^{-1/14}$ for the irradiated disk in the Epstein drag regime), and the Equation set (\[eq:ddt\]) has an explicit analytic solution (see [Appendix \[app:tdriftan\]]{}). Once particles are large enough so that $\tau_{\rm s} \sim 1$, $t_{\rm drift}$ has a more complicated dependence on $r$ (see Equation \[eq:ts\]), and the coupled drift-desorption differential equations have to be integrated numerically to obtain an accurate result.
Given $r_{\rm des}$, we need to only calculate the distance over which particles desorb to determine the location of a snowline. Figure \[fig:s\_t\_a\], left panels, shows the size evolution with time for H$_2$O particles of various initial sizes, starting at three different initial locations in an irradiated disk. Once solid H$_2$O particles begin to evaporate, they do so almost instantly for all explored particle sizes and initial locations. The right panels of Figure \[fig:s\_t\_a\] show that the drifting grains lose most of their mass in a very narrow distance range; moreover, this distance is the same for a given initial particle size, no matter where the particle started drifting at the time $t_0$ when the simulation is started. Figure \[fig:s\_t\_a\] thus demonstrates that solid particles that drift and fully desorb during the lifetime of the protoplanetary disk do so (1) instantaneously, and (2) at a fixed stellocentric distance, regardless of their initial location in the disk. It follows that the H$_2$O, CO$_2$ and CO snowlines are fixed for a given initial particle size and disk model. Both of these conclusions remain valid for an evolving and a viscous disk, as well as for particles composed of CO$_2$ or CO, but the snowline locations will vary between the three disks for a given initial particle size (see Section \[sec:COratio\]). If we do not take into account the time dependence of the mass accretion rate and stellar luminosity (see Section \[sec:disk\]), the C/O ratio will then only depend on disk properties, grain size, and the abundance of H$_2$O, CO$_2$ and CO relative to the H$_2$ abundance in the disk midplane, and [*not*]{} directly on the disk age when only considering drift, accretion and desorption.
C/O Ratio Estimates {#sec:COratio}
===================
![Estimated C/O ratio in gas (solid lines) and in dust (dashed lines) for an irradiated disk (top panel), an evolving disk (middle panel) and a viscous disk (bottom panel). The particle size increases from $\sim$0.05 cm to $\sim$700 cm as indicated by the color bar. The horizontal dotted line represents the stellar value of 0.54. The black lines represent the C/O ratio in gas (solid black line) and dust (dashed black line) for a static disk, with the temperature profile given by Equation (\[eq:diskT\]) for the top two panels and by Equation (\[eq:activeT\]) for the bottom panel. For both the evolving and the viscous disk, the movement of desorbed CO$_2$ gas inside the CO$_2$ snowline, and of desorbed CO$_2$ and H$_2$O gas inside the H$_2$O snowline due to gas accretion will increase the amount of oxygen gas inside the respective snowlines and thus reduce the gas C/O ratio, as shown by the arrows.[]{data-label="fig:CO_ratio"}](C_O_ratio_passive_active_disk_many_colorbar_complete_new2.pdf){width="55.00000%"}
Given our results in Section \[sec:snowlines\], a disk’s C/O ratio in mainly affected by the snowline location for the particle size housing the most mass in ice. Realistic grain size distributions in disks are dominated by large grains (e.g., @dalessio01, @birnstiel12). In Figure \[fig:CO\_ratio\], we display the H$_2$O, CO$_2$, and CO snowline locations as a function of particle size for disks with static chemistry that experience radial drift of solids and gas accretion onto the central star. The minimum snowline distance for a disk is given by the curve corresponding to the maximum particle size it hosts. For grains that have grown to radii larger than $\sim$7 m and are able to drift and desorb, the $\sim$7 m snowline applies (see Section 3).
Drift and gas accretion affect the C/O ratio in a disk both because they move the snowline locations of the main C and O carriers and because they cause solids and gas—which contain different proportions of C and O—to move inward at different rates. As shown in Section 3, the snowline locations depend on disk age only indirectly, through changes in disk properties and grain size. The C/O ratio is a function of the locations of the snowlines and the abundances of H$_2$O, CO$_2$, and CO relative to the H$_2$ abundance in the disk midplane. These abundances evolve over time as solids and gas move inward at different rates.
Figure \[fig:CO\_ratio\] shows the estimated C/O ratio in gas and dust as a function of semimajor axis for an irradiated disk, an evolving disk, and a viscous disk, under the simplifying assumption that the abundance relative to hydrogen for each volatile is fixed, so that drift and accretion affect only the locations of the snowlines. We use the relative number densities of C and O in their different molecular forms (H$_2$O, CO$_2$ and CO) from Table 1 of @oberg11. Snowline locations correspond to $r_{\rm des}$ in Figure \[fig:snowlines\], representing the location at which particles desorb in the absence of readsorption. The C/O ratio for a static disk, where desorption and readsorption balance [@hollenbach09], is shown as a guideline. We note that the true snowline for particles with $r_{\rm des}$ outside the static snowline is the static snowline itself—thus only particles with initial sizes larger than $\sim$0.05 cm are plotted in the three panels, as particles that form snowlines at larger distances (cf. Figure \[fig:snowlines\]) are not true snowlines.
Before discussing the quantitative aspects of this plot, it is essential to acknowledge that our estimates for the C/O ratios in the evolving and viscous disks ignore the movement of the desorbed ices with the accreting gas—the relative fluxes of the volatiles in gaseous and solid form will affect the relative abundance of C and O in gas and dust throughout the disk. As demonstrated in Figure \[fig:s\_t\_a\], this will not affect the snowline locations for particles of a given size, but will change the shape of the C/O curves in between the various snowlines. For example, for the disk parameters and particle sizes displayed in Figure \[fig:CO\_ratio\], water molecules in solid particles drift up to $\sim$1000 times faster across the H$_2$O snowline than do molecules of CO and CO$_2$ vapor that are entrained in the accreting gas. This differential inward motion will result in an increased oxygen gas abundance inside the H$_2$O snowline, and thus a (in some cases much) lower gaseous C/O ratio in this region. Conversely, oxygen gas inside the water snowline will be depleted compared to the static disk if H$_2$O particles grow to planetesimal size and stall their migration between the H$_2$O and CO$_2$ snowlines, leaving only gaseous CO and CO$_2$ to accrete inward. Growth of large planetesimals can therefore increase the C/O ratio in the inner disk.
Figure \[fig:CO\_ratio\] plots snowline curves for particle sizes $\sim0.5$ cm $\lesssim $s$ \lesssim$ 7m. In the outermost disk, H$_2$O, CO$_2$, and CO all solidify. Hence, relative drift across the CO snowline can alter only the abundances of volatiles between the CO$_2$ and CO snowlines, but not the C/O ratio in this region. Interior to the CO$_2$ snowline, however, relative drift is important. We have found that the largest drifting particles in our model ($\sim$7 m) drift faster than the gas at both the H$_2$O and CO$_2$ snowlines. We thus conclude that the C/O ratio interior to the H$_2$O and CO$_2$ snowlines in our evolving and viscous disks will be lower than in the static disk, due to the additional oxygen added to the gas by desorbing H$_2$O and CO$_2$. For these particle sizes, our calculated C/O ratio is an upper limit, as indicated by the arrows in Figure \[fig:CO\_ratio\].
Fundamentally, the elevated C/O ratios interior to the static H$_2$O and CO$_2$ snowlines are simply caused by the inward movement of the snowlines due to radial drift and gas accretion. Qualitatively, this scenario should be robust to changes in total abundances throughout the disk — for example, at the dynamic (non-static) CO$_2$ snowlines, the rapid return of CO$_2$ into gas-phase during CO$_2$ desorption will reduce the C/O ratio interior to the CO$_2$ dynamic snowline, while no major change in gas-phase composition, and therefore C/O ratio, is expected between the static and dynamic snowlines.
As noted earlier, we assume that the total (ice and gas) abundance of each volatile is the same at every radius after ices have migrated. This is a good approximation for the irradiated disk, given that this model, by definition, presents a constant influx of particles at any given radius while the gas is static, and thus the ice and gas surface density remain constant. For the evolving disk, the gas-phase C/O ratio may decrease everywhere interior to the H$_2$O and CO$_2$ snowlines due to the decline in the surface density of solids with time at any given radius. For the viscous disk, the solid abundances at a fixed radius are constant, given that this model is not time-dependent, but the gas-to-solid ratio is not constant, which can result in a substantially lower C/O ratio interior to the H$_2$O and CO$_2$ snowlines compared to the static case (as indicated by the arrows in Figure \[fig:CO\_ratio\]). The main goal of Figure \[fig:CO\_ratio\], however, is to show the different snowline radii in static and non-static disks, and therefore the locations in the disk where the gas-phase C/O ratio is reduced or increased, rather than provide a quantitative estimate of the magnitude of the C/O increase or decrease.
The snowline locations in these disks exhibit several interesting features. For the irradiated disk, only grains larger than $\sim$0.5 cm drift, desorb and thus move the snowline compared to the static disk. In contrast, even $\sim$micron-sized grains drift and desorb for the evolving disk, since they flow towards the host star together with the accreting gas. For the same particle size, the snowline locations are slightly closer to the central star in the evolving disk, due to the fact that the accreting gas adds an additional component to the drift velocity of the solids (cf. Equation \[eq:rdotact\]). The addition of accretional heating in the viscous disk moves the H$_2$O snowline outwards. This is due to the fact that accretional heating dominates in the inner disk, where high temperatures cause the grains to evaporate further away from the star. Once $r\gtrsim1-2$ AU, stellar irradiation dominates the thermal evolution of the disk, and therefore the CO$_2$ and CO snowlines locations are the same as in the evolving and viscous disks.
Perhaps the most interesting feature is the fact that the snowlines are pushed inwards as the grain size increases. While the plot only shows the snowlines and C/O ratio for particle sizes up to $\sim$7 m, we have found that bodies larger than $\sim$7 m evaporate at the same location as the meter-sized planetesimals (see Section \[sec:snowlines\]). However, the contribution of kilometer-sized bodies to the snowline location is modest, since they only drift if they are located very close to the snowline. Thus the innermost snowlines (depicted in red in Figure \[fig:CO\_ratio\]) set the limit on how close in the H$_2$O, CO$_2$ and CO snowlines can be pushed due to radial drift and gas accretion on to the host star. For a grain size distribution with a maximum particle size different than our model, one can pick out the appropriate minimum snowline locations from this plot. For our choice of parameters, the minimum snowline radii are: $r_{\rm{H_2O}} \approx 0.3$ AU for the irradiated disk, $r_{\rm{H_2O}} \approx 0.26$ AU for the evolving disk and $r_{\rm{H_2O}} \approx 0.63$ AU for the viscous disk; $r_{\rm{CO_2}} \approx 3.7$ AU for the irradiated disk, $r_{\rm{CO_2}} \approx 3.4$ AU for both the evolving and the viscous disks; $r_{\rm{CO}} \approx 30$ AU for the irradiated and both the evolving and the viscous disks. For comparison, $r_{\rm{H_2O}} \approx 0.67$ AU[^2], $r_{\rm{CO_2}} \approx 8.6$ AU and $r_{\rm{CO}} \approx 59$ AU for the static disk. For the viscous disk model, which is the most realistic, radial drift and gas accretion push the snowline locations inwards by up to $\sim$$40$ % for H$_2$O, by up to $\sim$$60$ % for CO$_2$, and by up to $\sim$$50$ % for CO. We note that the H$_2$O snowline in all disks is significantly closer to the host star compared with Solar system models, which place the H$_2$O snowline between $\sim$$2.7$ to $\sim$$3.1$ AU (@hayashi81, @podolak04, @martin12). This is partially because we choose a colder disk model, as well as the fact that gas accretion rates decrease over time, moving the snowline location inwards (see also @garaud07 and Section \[sec:neglected\]). @min11 find that the location of the H$_2$O snowline is highly sensitive to the gas mass accretion rate $\dot{M}$ (equal to $10^{-8} M_{\odot}$ yr$^{-1}$ in our model) and the dust opacity $\kappa$ (Equation \[eq:opacity\]). Higher values of $\dot{M}$ and $\kappa$ would increase the accretional component of the disk temperature (cf. Equation \[eq:Tdact\]), which would push the H$_2$O snowline in the viscous disk outwards to match the Solar system snowline. At the same time, the snowline location in Solar type stars may be as close as $\sim$1 AU [@mulders15], further in than the H$_2$O snowline in our Solar system.
Observations of the CO snowline in TW Hya [@qi13] have found its location at a disk midplane temperature of 17 K (at 30 AU for the TW Hya specific temperature profile). The inferred desorption temperature corresponds to the CO desorption temperature in a static disk, or to desorption from very small grains in an evolving disk, i.e. from grains that are too small to drift substantially. This suggests that the outer TW Hya disk is dominated by small grains, since larger particles would push the snowline location inwards, and therefore to higher desorption temperatures. This may seem contradictory to observations of grain growth in disks in general and in TW Hya in particular [@wilner00]. However, recent observations have revealed that grain growth is concentrated to the inner disk [@perez12] and outer disk snowlines may therefore be close to the ones expected in a static disk.
Discussion {#sec:discussion}
==========
{width="\textwidth"}
![Desorption or final distance as a function of initial position in the disk for H$_2$O particles of initial size of 1 m, for total disk masses $M=0.1 M_{\odot}$ (solid lines) and $M=0.01 M_{\odot}$ (dashed lines). The timescales at which we stop the simulations are $10^4$ yr (cyan curve), $10^5$ yr (red curve), 1 Myr (green curve) and 3 Myr (blue curve). A lower disk mass does not change the snowline location.[]{data-label="fig:varMd"}](desorption_distance_varying_Md.pdf){width="50.00000%"}
![Desorption or final distance as a function of initial position in the disk for H$_2$O particles of initial size of 1 m, for our fiducial disk (solid lines) and for a transition disk with an inner cavity at $r_0=4$ AU (dashed lines). The timescales of the simulations and their color code are the same as in Figure \[fig:varMd\]. Particles that start inside the cavity drift towards the original snowline, while particles that start outside the gap stop shortly after crossing the gap edge, due to being trapped in a pressure maximum.[]{data-label="fig:cavity"}](desorption_distance_transition_disk_1000.pdf){width="50.00000%"}
Generality of Results: Dependence on Disk Parameters {#sec:incond}
----------------------------------------------------
In this section we investigate how variations in our fiducial parameters, the total disk mass, disk age, and disk structure, affect the calculated snowline locations and the C/O ratio. All previous results assumed a disk lifetime $t_{\rm d}=3$ Myr, the typical disk lifetime and the expected time scale for giant planets to accrete their gaseous atmopsheres (e.g., @pollack96, @piso14). Some gas accretion may occur at earlier times, however,before the core is fully formed (e.g., @rafikov06). Recent models such as aerodynamic pebble accretion [@lambrechts12] suggest that rapid core growth on timescales of $10^5$ years is possible. The composition of giant planet atmospheres, and specifically their C/O ratio, can thus depend on the abundance of H$_2$O, CO$_2$ and CO in gas at earlier times than $t_{\rm d}$ in the disk evolution.
Figure \[fig:timeplots\] shows the particle desorption or final distance as a function of a particle’s initial location in the disk, for ice particles of initial sizes of 10 cm and 1 m, composed of either H$_2$O, CO$_2$ or CO. These sizes are important since radial drift timescales are shortest for particles within this size range (see Figure \[fig:timescales\]) — these are the particles whose drift and desorption evolution should be most strongly affected by variations in disk conditions. We choose the evolving disk as a disk model, and we stop the simulations after $10^4$ yr, $10^5$ yr, 1 Myr and $t_{\rm d}=3$ Myr, respectively. The most important result of these plots is that particles of a given size always desorb at the same disk radii, the 3 Myr snowline, regardless of simulation stopping time. Particles that start at large stellocentric distances do not desorb within the shorter timeframes, e.g. $10^4$ or $10^5$ years, but they do evaporate at a fixed radius if their initial location is closer to the host star. While the amount of material that moves through the disk changes with time, the radius at which particles desorb and the snowline locations are thus independent of the time elapsed, and our results for the snowline locations are valid throughout the time evolution of the protoplanetary disk.
We choose as a fiducial model a total disk mass $M=0.1 M_{\odot}$. Observationally, disk masses span at least an order of magnitude around Solar type stars [@andrews13]. We thus explore the effect of disk mass on the location of snowlines. Figure \[fig:varMd\] shows the desorption or final distance as a function on the initial location of a H$_2$O particle with initial size of 1 m, for two total disk masses: $M=0.1 M_{\odot}$, our fiducial model, and $M=0.01 M_{\odot}$. Similarly to Figure \[fig:timeplots\], we perform our calculations for an evolving disk. The simulations are stopped after the same timeframes as those in Figure \[fig:timeplots\]. The location of the H$_2$O snowline is the same for both disks (the same holds true for the CO$_2$ and CO snowlines). The C/O ratio is thus insensitive to the choice of $M$. We note that our conclusions regarding the disk age and total mass are only valid if the snowline itself does not move with time or disk mass (see also Sections \[sec:disk\] and \[sec:neglected\]).
We also apply our evolving disk model to a transition disk, i.e. a protoplanetary disk with an inner cavity significantly depleted of gas. We choose a disk with an inner gap of radius $r_0=4$ AU, consistent with observations of TW Hya [@zhang13], and with the gas surface density in the gap reduced by a factor of 1000. Figure \[fig:cavity\] shows the desorption or final distance for a H$_2$O particle of initial size of 1 m, with the simulation stopped at the same timescales as in Figures \[fig:timeplots\] and \[fig:varMd\]. Particles that start at an initial distance interior to the gap drift towards the original snowline, while grains located exterior to the gap stop shortly after crossing the gap edge, due to the decrease in gas pressure inside the cavity, thus forming a snowline at $\sim$$3.8$ AU. This is qualitatively consistent with the observations of @zhang13, which show that the H$_2$O snowline is pushed outwards in a transition disk compared to a full disk. Our model framework is thus generally valid for more complicated disk structures as well.
Model Extensions {#sec:neglected}
----------------
Our goals in this paper were (1) to gain a qualitative and quantitative understanding of the effect of radial drift and gas accretion onto the central star on snowline locations and the C/O ratio in disks, and (2) to obtain a limit on how close in the snowlines can be pushed due to drift and gas accretion. We have thus used a simplified model and out of necessity neglected potentially significant dynamical and chemical processes. In what follows, we discuss these limitations and their effects. We note that our future work will address some of these issues.
We summarize in Table 1 the potential physical and chemical processes occurring in disks and their effect on snowline locations compared to a static disk. For the sake of completeness, Table 1 also includes the processes addressed in this paper, i.e. radial drift and gas accretion. The neglected effects are discussed in more detail below.
1. **Particle growth.** While our model assumes a range of particle sizes, each size is considered initially fixed for a given grain before it drifts and desorbs, since we do not take into account particle coagulation. However, grain growth has been observed in protoplanetary disks (e.g., @ricci10, @perez12), as well as theoretically constrained (e.g., @birnstiel10, @birnstiel12). In Section \[sec:COratio\] we have shown that larger grains move the snowline locations closer in, but those locations remain fixed above a certain particle size. Particle growth will thus initially push the snowlines inwards. This is consistent with particle growth models, which predict a maximum particle size often around or below the particle sizes that drift the fastest [@birnstiel12]. As the largest grains contain most of the solid mass, grain growth models should produce snowlines corresponding to our snowline location estimates for the largest grains in the particle size distribution. However, once the solids grow larger than km-sized and form planetesimals, they are no longer affected by drift or desorption, and the snowline reduces to that of a static disk.
2. **Turbulent diffusion.** The radial drift model presented in Section \[sec:drift\] only considers a laminar flow and thus ignores turbulence. However, the disk gas also experiences turbulent diffusion (e.g., @birnstiel12, @alidib14). Turbulence causes eddies and vertical mixing, which are likely to reduce the radial drift velocity of the solids (e.g., @youdin07). Additionally, the flow of H$_2$O, CO$_2$ and CO vapor will diffuse radially. Back-diffusion across the snowline will change the shape of the snowline, as well as the C/O ratio in gas and dust both inside and outside of the snowline, due to the reduction of gas-phase volatile abundance interior to the snowline.
**Process** **Effect**
------------------------------ ----------------------------
Radial drift $\leftarrow$ [^3]
Gas accretion $\leftarrow$ [^4]
Particle growth $\leftarrow$ [^5]
Turbulent diffusion $\rightarrow$ $\leftarrow$
Particle fragmentation $\rightarrow$ $\leftarrow$
Grain morphology $\rightarrow$
Particle composition $\rightarrow$ $\leftarrow$
Disk gaps and holes $\rightarrow$
Accretion rate evolution $\rightarrow$ $\leftarrow$
Stellar luminosity evolution $\leftarrow$
Non-static chemistry $\rightarrow$ $\leftarrow$
: The effects of dynamical and chemical processes on snowline shapes and locations
3. **Particle fragmentation.** Frequent particle collisions in disks cause them to fragment (e.g., @birnstiel12). The fragmentation of meter- to km-sized particles will move the snowlines outwards, as smaller particles desorb faster and further out from the host star (cf. Figures \[fig:snowlines\] and \[fig:CO\_ratio\]). Large boulders, which neither drift nor desorb, may become e.g. meter-sized due to collisions and subsequent fragmentation, which will cause them to drift significantly before desorbing, pushing the snowlines inwards. Thus fragmentation can move the snowline locations in either radial direction — specifically, fragmentation leads to a certain grain size distribution, and the largest particles in this size distribution are the ones that determine the position of the snowline.
4. **Grain morphology.** Our model assumes that the ice particles are perfect, homogeneous spheres. However, this is not a very good approximation, since grain growth can be fractal rather than compact (@zsom10, @okuzumi12, @krijt15). The inhomogeneity due to cracks in the grain structure will cause the particles to desorb faster. They will therefore drift less before evaporating and will move the snowlines less far inward.
5. **Particle composition.** The ice particles in our model are assumed to be fully made of either H$_2$O, CO$_2$ or CO. More realistically, grains may have a layered structure, such as an interior composed of non-volatile materials (e.g., sillicates) covered by an icy layer. The ice thus only constitutes a fraction of the total particle mass, which accelerates its desorption and pushes the snowlines outwards. The grains may also be composed of a mixture of H$_2$O, CO$_2$ and CO ices, which will increase the binding energies of the more volatiles species, moving the snowlines inwards.
6. **Disk gaps and holes.** The snowline locations will be different for transition disks, which have inner cavities significantly depleted of gas (e.g., @espaillat12, @vandermarel15), or pre-transitional disks, which have a gap between an inner and outer full disk (e.g., @kraus11). The decrease in gas pressure in these gaps or holes will reduce the particles’ drift velocity close to the gap edge, thus slowing them down and pushing the snowline outwards.
7. **Accretion rate evolution.** Our viscous disk model assumes a constant mass accretion rate $\dot{M}$. However, $\dot{M}$ decreases over time, which lowers the accretional component of the disk temperature (Equation \[eq:Tdact\]), thus pushing the snowline location inwards if the disk is optically thick (@garaud07). Once $\dot{M}$ reaches low enough values for the snowline to become optically thin, the snowline location moves outwards [@garaud07]. During the giant planet formation stage of a few Myr, however, $\dot{M}$ steadily decreases with time [@chambers09], which may result in the inward movement of the H$_2$O snowline by up to one order of magnitude, significantly larger than the inward movement caused by radial drift (cf. Section \[sec:COratio\]). We thus acknowledge that the location of the H$_2$O snowline may be set by the mass accretion rate evolution rather than the drift of solids.
8. **Stellar luminosity evolution.** As the host star contracts during its pre-main sequence phase, its luminosity decreases, which reduces the disk temperature and pushes the snowline locations inwards. @kennedy06 found that the snowline is unlikely to move significantly during the pre-main sequence phase for Solar type stars, but it may move inward by a factor of $\sim$$15-20$ for $M_* \sim 0.25 M_{\odot}$ due to the stellar contraction.
9. **Time dependent chemistry.** As the goal of this paper was to explore only the dynamical effects on snowline locations and the C/O ratio in disks, we have assumed a simple, static chemical model. In reality, the chemistry in most of the disk is expected to be time-dependent.In the inner disk, chemistry approaches equilibrium due to intense sources of ionizing radiation (e.g., @ilgner04), while in the outer disk high energy radiation and cosmic rays are the key drivers of chemistry, which is no longer in equilibrium (e.g., @vandishoeck06). A multitude of chemical evolution models have been developed (see references in @henning13), many of which contain tens or hundreds of chemical reactions. Due to the complexity of these chemical models, most of them are decoupled from disk dynamics. The effect of disk chemistry on snowline locations, shape, time evolution, or the C/O ratio is therefore difficult to estimate.
Summary {#sec:summary}
=======
We study the effect of radial drift of solids and viscous gas accretion onto the central star on the H$_2$O, CO$_2$ and CO snowline locations and the C/O ratio in a protoplanetary disk, assuming static chemistry. We develop a simplified model to describe the coupled drift-desorption process and determine the time evolution of particles of different sizes throughout the disk. We assume that the solid particles are perfect, homogeneous spheres, fully composed of either H$_2$O, CO$_2$ or CO. We apply our model to an irradiated disk, an evolving disk, and a viscous disk that also takes into account stellar irradiation. We determine the desorption or final location of drifting particles after a time equal to the disk lifetime, and use this result to set an inner limit for the location of the H$_2$O, CO$_2$ and CO snowlines. Our results can be summarized as follows:
1. Radial drift and gas accretion affect desorption and move the snowline locations inward compared to a static disk for particles with sizes $\sim$$0.5$ cm $\lesssim s \lesssim$ 7 m for an irradiated disk and $\sim$$0.001$ cm $\lesssim s \lesssim$ 7 m for an evolving disk.
2. For our simplified model that does not account for the effects outlined in Section \[sec:neglected\], particles with sizes in the above range desorb almost instantaneously once desorption has begun, and at a fixed location in the disk that only depends on the particle size and the gas accretion rate. Thus for each particle size there is a fixed and uniquely determined H$_2$O, CO$_2$ or CO snowline.
3. The results of our numerical simulation are in agreement with the analytic solution of the drift-desorption system of differential equations if the stopping time $\tau_{\rm s} \ll 1$. We present an explicit analytic solution for the desorption distance in this regime.
4. Since realistic grain size distributions are dominated in mass by the largest particles, the H$_2$O, CO$_2$ and CO snowlines are those created by the largest drifting particles in our model. This corresponds to the innermost snowlines that we determine. Our model thus sets a limit on how close to the central star the snowlines can be pushed by radial drift and gas accretion.
5. The snowline locations move inwards as the particle size increases; the innermost snowline is set by particles with initial size $s \sim 7$ m in our model — bigger particles drift too slowly to make it further in before desorbing (see Section \[sec:snowlines\]). Gas accretion causes even micron-sized particles to drift, desorb and move the snowline location compared to a static disk. A viscous disk that includes accretion heating moves the H$_2$O snowline outwards compared to an evolving disk, but has no effect on the CO$_2$ and CO snowline locations, for our particular choice of mass accretion rate $\dot{M}$ and midplane opacity $\kappa$.
6. For our fiducial model, which considers particles with sizes between $10^{-3}$ and $10^8$ cm, the innermost H$_2$O, CO$_2$ and CO snowlines are located at 0.3 AU, 3.7 AU and 30 AU for an irradiated disk, 0.26 AU, 3.4 AU and 30 AU for an evolving disk, and 0.63 AU, 3.4 AU and 30 AU for a viscous disk with accretion heating. Compared to a static disk, radial drift and gas accretion move the snowlines by up to 60 % for H$_2$O and CO$_2$, and by up to 50 % for CO. For the viscous disk, however, which is the most realistic of the three models since it takes into account accretion heating, the H$_2$O snowline location moves inwards by up to 40 %.
7. Our C/O estimates confirm the conclusions of @oberg11 that the C/O ratio in gas may be enhanced compared to the stellar value throughout most of the disk, with the C/O ratio reaching its maximum value between the CO$_2$ and CO snowlines. We note, however, that our results for the C/O ratio do not take into account the radial movement of the desorbed ices with the accreting gas in the evolving and viscous disks, which may significantly decrease the C/O ratio in gas inside the H$_2$O and CO$_2$ snowlines. We plan to address this issue in a future paper.
8. For a constant gas mass accretion rate $\dot{M}$ and stellar luminosity $L_*$, the snowline locations are independent of the time at which we stop our simulation and of the total disk mass, as long as the disk midplane remains optically thick.
Our model does not address additional effects, such as gas diffusion, grain composition and morphology, or complex time-dependent chemical processes. Future work will address some of these dynamical and chemical processes, with the goal of obtaining more realistic results for the snowline locations, shapes and time evolution, and the resulting effect on the C/O ratio.
Steady-state active disk solution {#app:steadystate}
=================================
Following @shakura73 and @armitage10, the steady-state solution for a geometrically thin, optically thick actively accreting disk with an $\alpha$-prescription for viscosity is governed by the following set of equations:
\[eq:diskeq\] & = & c H\
c\^2 & = &\
& = &\
H & = &\
T\_[act]{}\^4 & = & T\_[surf]{}\^4\
& = &\
& = &\
T\_[surf]{}\^4 & = & \_[k]{}\^2\
& = & \_0 T\_[act]{}\^2 ,
where $T_{\rm surf}$ is the surface temperature of the disk and the other quantities are defined in the main text. This is a system of nine equations with nine unknowns ($\nu$, $c$, $H$, $T_{\rm act}$, $\rho$, $\Sigma$, $\tau$, $T_{\rm surf}$, $\kappa$) that can be solved numerically once $\alpha$ and $\kappa_0$ are specified.
Desorption distance analytic solution {#app:tdriftan}
=====================================
For a particle of size $s$ that desorbs and satisfies $\tau_{\rm s} \ll 1$ ($\tau_{\rm s}$ is the dimensionless stopping time, defined in Section \[sec:drift\]), we can derive an explicit analytic solution for the particle’s desorption distance in an irradiated disk. For $\tau_{\rm s} \ll 1$, a particle is in the Epstein drag regime (see Equation \[eq:ts\]) and its drift velocity $\dot{r}$ (Equation \[eq:rdotpas\]) can be approximated as $$\label{eq:rdotapp}
\dot{r} \approx -2 \eta \Omega_{\rm k} r \tau_{\rm s}.$$ By using Equations (\[eq:tdes\]) and (\[eq:tdrift\]) and setting $t_{\rm drift}=t_{\rm des}$, we can express a particle’s desorption distance as
$$\label{eq:tan}
r_{\rm des}=\Bigg(\frac{d}{q \,C} \,\, \mathcal{W}\Big[\frac{(B/A)^{-q/d} q\,C}{d}\Big]\Bigg)^{\frac{1}{q}},$$
where $\mathcal{W}$ is the Lambert-W function, $q=3/7$ is the power-law coefficient in Equation (\[eq:diskT\]), $d=-\frac{1}{2}+p-q$ with $p=1$ the power-law coefficient in Equation (\[eq:disksigma\]), and
A & = & r\_0\^d\
B & = &\
C & = & r\_0\^[-q]{},\
where $r_0=1$ AU, $\rho_0=\rho(r_0)$ and $c_0=c(r_0)$.
[n]{}n
[n]{}y
[^1]: We discuss the validity of these simplifications in section \[sec:discussion\].
[^2]: For the viscous disk, we calculated the static snowline location using the same temperature profile as that of the viscous disk, for consistency purposes. Thus $r_{\rm{H_2O}} \approx 0.98$ AU for the static disk in this scenario.
[^3]: The arrows signify how a process affects the snowline: $\leftarrow$ means that the snowline is pushed closer to the host star compared to the static snowline, $\rightarrow$ means that the snowline is pushed further from the host star compared to the static snowline. The presence of both arrows means that the process may have both effects on the snowline location.
[^4]: Gas accretion pushes the snowlines inwards compared to the snowline locations in a static disk. However, accretional heating may push the snowline outwards compared to an evolving disk without accretion heating.
[^5]: As stated in the main text, if particles grow to km-sizes and above, the snowline remains the same as that of a static disk.
|
---
abstract: |
To analyze whole-genome genetic data inherited in families, the likelihood is typically obtained from a Hidden Markov Model (HMM) having a state space of $2^n$ hidden states where $n$ is the number of meioses or edges in the pedigree. There have been several attempts to speed up this calculation by reducing the state-space of the HMM. One of these methods has been automated in a calculation that is more efficient than the naïve HMM calculation; however, that method treats a special case and the efficiency gain is available for only those rare pedigrees containing long chains of single-child lineages. The other existing state-space reduction method treats the general case, but the existing algorithm has super-exponential running time.
We present three formulations of the state-space reduction problem, two dealing with groups and one with partitions. One of these problems, the maximum isometry group problem was discussed in detail by Browning and Browning [@Browning2002]. We show that for pedigrees, all three of these problems have identical solutions. Furthermore, we are able to prove the uniqueness of the solution using the algorithm that we introduce. This algorithm leverages the insight provided by the equivalence between the partition and group formulations of the problem to quickly find the optimal state-space reduction for general pedigrees.
We propose a new likelihood calculation which is a two-stage process: find the optimal state-space, then run the HMM forward-backward algorithm on the optimal state-space. In comparison with the one-stage HMM calculation, this new method more quickly calculates the exact pedigree likelihood.
author:
- 'B. Kirkpatrick, K. Kirkpatrick'
bibliography:
- 'pedigree.bib'
title: 'Optimal State-Space Reduction for Pedigree Hidden Markov Models'
---
Introduction
============
#### Motivation.
Statistical calculations on pedigrees are the principal method behind the most accurate disease-association approaches [@Risch1996; @Thornton2007]. In those approaches, the aim is to find the regions of the genome that are associated with the presence or absence of a disease among related individuals. Furthermore, pedigree likelihoods are used to estimate fine-scale recombination rates in humans [@Coop2008], where there are few other approaches for making these estimates. Many implementations of the likelihood estimates for pedigrees exist [@Fishelson2005; @Abecasis2002; @Sobel1996]. Estimates of probabilities on pedigrees are of great interest to computer scientists because they give an important example of graphical models which model probability distributions by using a graph whose edges are conditional probability events and whose nodes are random variables [@Lauritzen2003]. Methods for reducing the state-space of a pedigree graphical model could generalize to other graphical models, as noted also by Geiger et al [@Geiger2009].
#### The Problem Summary.
Hidden Markov Models (HMMs) analyzing the genotypes of related individuals have running time $O(m2^n)$ where $m$ is the number of sites and $n$ is the number of meioses in the pedigree. Therefore, it is desirable to find more efficient algorithms. Any partitioning of the state space into $k$ ensemble states (i.e., states with identical emission probabilities and Markovian transition probabilities) will improve the running time of an HMM to $O(mk)$, even if the ensembles are not optimal. Since the HMMs have an exponential state space and a running time polynomial in the size of the state space, even an exponential algorithm for finding ensemble states can improve the running time of the HMM calculations.
#### Literature Review.
Donnelly [@Donnelly1983] introduced the idea of finding ensemble states for the IBD Markov model, and used a manual method for finding the symmetries for several examples of two-person pedigrees. Browning and Browning [@Browning2002] formalized the requirements for symmetries that describe ensemble states in a new HMM. They gave the first algorithm for finding the maximal set of isometries that preserves the Markov property and the IBD information. Their algorithm which is based on enumerating permutations appears to have worst-case running time of at least of $O(n!2^{2n})$, where $n$ is the number of meioses in the pedigree. However, the running time of their algorithm is difficult to analyze. They also left open the question of whether groups other than isometry groups could give useful state-space reductions [@Browning2002].
McPeek [@McPeek2002inference] presented a detailed formulation of the condensed identity states and an algorithm. Most recently Geiger et al [@Geiger2009] discussed a similar problem using the language of partitions. In this paper, we will prove that the partition problem and the isometry problem are identical. Geiger, et al. gave a special-case state-space reduction involving only partitions that collapse simple lineages (multiple generations with a single child per generation and with the non-lineage parents being founders). Several people have introduced algorithms for finding symmetries for systems applications [@Lorentson2001; @Junttila2004].
#### Our Contribution.
Inspired by the work of Browning and Browning [@Browning2002], we look for maximal ensembles of the hidden states that can be used to create a new HMM with a much more efficient running-time. We introduce an improved algorithm for finding the maximal ensemble states, sets of hidden states, that preserve both the Markov property and the identity by descent (IBD) information of the individuals of interest.
Related work includes the algorithms of Geiger, et al [@Geiger2009], and Browning and Browning [@Browning2002]. Geiger, et al. found isometries of a limited type in $O(n^2)$ time and demonstrated their practical value. Browning and Browning introduced a super-exponential algorithm that finds a maximal group of isometries, i.e. the largest number of elements. However, they did not draw any conclusions about whether their method finds the group with the maximal orbit sizes.
We introduce an $O(n2^{2n})$ maximal-ensemble algorithm for finding a permutation group on the $2^n$ vertices of the hypercube, and for producing the most efficient ensemble states (i.e. the smallest partition of the state-space that respects the IBD and Markov properties and has the maximal partition sets and minimal number of sets in the partition). We prove that the optimal partition is a solution to the maximal isometry group problem that Browning and Browning introduced. Therefore, both their algorithm and ours finds the optimal partition of the state space which can be described using a group of isometries having a maximal number of elements. However, our algorithm is much faster, having a coefficient $n$ instead of $n!$.
We also introduce a bootstrap version of the maximal-ensemble algorithm which takes advantage of the isometries introduced by Geiger, et al. [@Geiger2009] and the well-known founder isometry. By enumerating one representative from each set of the partition induced by the known isometries, we can create a bootstrap maximal-ensemble algorithm that runs in $O(nk2^{n})$ time where $n$ is the number of meioses in the pedigree, and $k$ is the number of partitions from the known isometries.
Problem Description
===================
Consider a pedigree graph, $P$, having individuals $V$ as nodes and having $n$ meioses with each meiosis being a directed edge from parent to child. Let $I$ being the set of individuals of interest, perhaps because we have data for those individuals. While it might be algorithmically convenient to assume that $I = V$, it is impractical. Many of the ancestral individuals in the pedigree are likely deceased, and genetic samples are unavailable.
An *inheritance state or vector* is a binary vector $x$ with $n$ bits where each bit indicates which grand-parental allele, paternal or maternal, was copied for that meiosis. The equivalent *inheritance graph*, $R_x$, has two nodes per individual (one for each allele) and edges from inherited parental alleles to their corresponding child alleles. Individuals of interest are called *identical by descent (IBD)* if a particular founder allele was copied to each of the individuals. In general, the inheritance graph is a collection of trees, since each allele is copied from a single parent.
The set of all inheritance states (binary $n$-vectors) is the $n$-dimensional hypercube $\mathcal{H}_{n}$, with $2^n$ vertices. The inheritance process is modeled as a symmetric random walk on $\mathcal{H}_{n}$, with the time dimension of the walk being the distance along the genome. At equilibrium, the walk has uniform probability of being at any of the hypercube vertices. From vertex $x$ in $\mathcal{H}_n$, a step is taken to a neighboring vertex after an exponential waiting time with parameter $\lambda=n$. For each individual zygote, with one meiosis, this is a Poisson process with parameter $\lambda=1$ and genome length roughly $30$.
There is a discrete version of this random walk, which is often used for hidden Markov models (HMMs) that compute the probability of observing the given data by taking an expectation over the possible random walks on the hypercube. Let $X$ be a Markov process, $\{X_t:
t=1,2,...,m\}$ for $m$ loci with a state space $\mathcal{H}_{n}$ consisting of all the inheritance states of the pedigree. The recombination rate, $\theta_t$, is the probability of recombination per meiosis, between a neighboring pair of loci, $t$ and $t+1$. If $t$ and $t+1$ are separated by distance $d$, then the Poisson process tells us that the probability of an odd number of recombinations is $\theta_t = 1/2(1-e^{-2\lambda d})$. The natural distance on $\mathcal{H}_{n}$ is the Hamming distance, $|x \oplus y|$, for two states $x$ and $y$, where $\oplus$ is the XOR operation and $|\cdot|$ is the $L^1$-norm in $\mathbb{R}^n$. Then the probability of transitioning from $x$ to $y$ is $$Pr[X_{t+1}=y|X_t=x] = \theta_t^{|x \oplus y|}(1-\theta_t)^{n-|x \oplus y|}.$$ Figure \[fig:halfsibs\] shows an example HMM with three genomic sites. The states of the HMM are shown in circles on the right.
![[**Two Half-Siblings.**]{} (Left Panel) A pedigree with two non-founders of which two are half-siblings together with their common parent. Circles and boxes represent female and male individuals, while the two black dots for each person represent their two chromosomes or alleles. Edges are implicitly directed downward from parent to child. The alleles of each individual are ordered, so that the left allele, or paternal allele, is inherited from the person’s father, while the right, maternal allele is inherited from the mother. The two siblings are the only labeled individuals. Their genomes are shown in color so that the same color indicates inheritance from the same ancestor. For convenience, the genotype of each person is homozygous. (Right Panel) The HMM for the genotypes from the left panel. At each site in the genome, the possibles states are the vectors in ${\cal
H}_n$. In each circle an inheritance state is drawn as an inheritance graph and the inheritance states for a sigle site are arranged in a column. The allowed transitions between neighboring sites are a complete bipartite graph (due to space, only a fraction of the edges are drawn). The nodes with a slash through them are inheritance states that are not allowed by the data. The red nodes and edges are the path for the actual inheritance states indicated by the yellow and blue in the left panel. However, this is only one of several paths of inheritance states that are consistent with the data. ](genome.eps "fig:") ![[**Two Half-Siblings.**]{} (Left Panel) A pedigree with two non-founders of which two are half-siblings together with their common parent. Circles and boxes represent female and male individuals, while the two black dots for each person represent their two chromosomes or alleles. Edges are implicitly directed downward from parent to child. The alleles of each individual are ordered, so that the left allele, or paternal allele, is inherited from the person’s father, while the right, maternal allele is inherited from the mother. The two siblings are the only labeled individuals. Their genomes are shown in color so that the same color indicates inheritance from the same ancestor. For convenience, the genotype of each person is homozygous. (Right Panel) The HMM for the genotypes from the left panel. At each site in the genome, the possibles states are the vectors in ${\cal
H}_n$. In each circle an inheritance state is drawn as an inheritance graph and the inheritance states for a sigle site are arranged in a column. The allowed transitions between neighboring sites are a complete bipartite graph (due to space, only a fraction of the edges are drawn). The nodes with a slash through them are inheritance states that are not allowed by the data. The red nodes and edges are the path for the actual inheritance states indicated by the yellow and blue in the left panel. However, this is only one of several paths of inheritance states that are consistent with the data. ](genome2.eps "fig:")
\[fig:halfsibs\]
We define potential ensembles of states as being the orbits of a group of symmetries. Let $G$ be a group that acts on the state space $\mathcal{H}_{n}$ of $X$. A symmetry is a bijection $\psi \in G$ where $\psi$ is a permutation on $2^n$ elements, the vertices of $\mathcal{H}_{n}$. An *orbit* of $G$ acting on $\mathcal{H}_{n}$ is the set $$\omega(y) = \{x | x = \psi(y) ~\textrm{and}~ \psi \in G\},$$ and we write the set of all orbits of $G$ as ${\Omega}(G) = \{\omega(y): y \in \mathcal{H}_{n}\}$.
Conventional algorithms for computing likelihoods of data have an exponential running time, because the state space of the HMM is exponential in the number of meioses in the pedigree. We propose new ways to collapse hypercube vertices into ensemble states for a new HMM that has a more efficient running time. In particular we are interested in optimal ensemble states that preserve certain relationship structures: the Markovianness of the random walk and the emission probabilities. We will first discuss the Markov property and then discuss the constraints on ensemble states that the emission probabilities provide.
Markov Property
---------------
Let $\{X_t\}$ be a stationary, reversible Markov chain with state space $\Omega$, such as the chain corresponding to the hidden states of the pedigree HMM.
Let $Y$ be a new processes, $\{Y_t: t=1,2,...,m\}$ having states $\Omega(G) = \{\omega_1,...,\omega_k\}$ which are the orbits of some group $G$. This new Markov chain is coupled to the original such that when $X_t=x \in
\omega \in \Omega(G)$, $Y_t = \omega$, and $Y_t$ is a projection of $X_t$ into a smaller state space. Define the transition probabilities for process $Y_t$ as $$\begin{aligned}
\label{eqn:transitions}
Pr[X_{t+1} \in \omega_j | X_t= x] = \sum_{y \in \omega_j} Pr[X_{t+1}=y|X_t=x] ~~\mathrm{for}~ x \in \omega_i,~\textrm{for}~ \omega_i,\omega_j \in \Omega(G).\end{aligned}$$ We will call $Y_t$ the expectation chain since $$\begin{aligned}
Pr[Y_{t+1}=\omega_j | Y_t=\omega_i] &=& {\mathbb{E}}[E_j | X_t=x],\end{aligned}$$ where $E_j$ is the event that $X_{t+1} \in \omega_j$.
Since $X_t$ is stationary and reversible, the necessary and sufficient condition [@Burke1958] for $Y_t$ to also be Markov is that $$\label{eqn:markov}
\sum_{y \in \omega_j} Pr[X_{t+1}=y|X_t=x_1] = \sum_{y \in \omega_j} Pr[X_{t+1}=y|X_t=x_2]$$ for all $x_1,x_2 \in \omega_i$ for all $i$, and for all $\omega_j$. Therefore any group whose orbits satisfy this set of equations can be used to create a new Markov chain $Y_t$.
From Equations (\[eqn:transitions\]) and (\[eqn:markov\]) we see that the stationary distribution of Markov chain $Y_t$ is $Pr[Y_t =
\omega_i] = \sum_{y \in \omega_i} \pi_y$ where $\pi_y$ is the stationary distribution of $X_t$. For pedigree HMMs, the stationary distribution of $X_t$ is uniform, $\pi_y = 1/2^n$, therefore the expectation chain for some group that satisfies Equation (\[eqn:markov\]) will have a stationary distribution $Pr[Y_t = \omega_i] = |\omega_i| / 2^n$.
For pedigree Markov chains, Equation (\[eqn:markov\]) becomes, for $s = \theta/(1-\theta)$ and $0 < \theta < 0.5$, $$\label{eqn:markovped}
\sum_{y \in \omega_j} s^{|y \oplus x_1|} = \sum_{y \in \omega_j} s^{|y \oplus x_2|} ~~~\forall x_1,x_2 \in \omega_i.$$ If the expectation chain $Y_t$ corresponding to pedigree Markov chain $X_t$ satisfies this equation, we say that it satisfies the *Markov property*. Notice that these polynomials are identical if and only if the coefficients of like powers are equal.
Browning and Browning [@Browning2002] gave an algorithm that searches for a maximal group of isometries where the group was maximal in the number of group elements. A group, $G$, of isometries has orbits $\Omega(G) = \{\omega_1,...,\omega_k\}$ such that $|T(x) \oplus T(y)| = |x \oplus y|$ for all $T \in G$, $y \in \omega_j$ and $x \in \omega_i$ for all $i,j$. This means that the transition probabilities are related by $$Pr[X_{t+1}=y | X_t = x] = Pr[X_{t+1}=T(y) | X_t=T(x)].$$ Browning and Browning left open the question of whether any symmetry groups satisfying Equation (\[eqn:markovped\]) must be equivalent to a group of isometries (meaning that it has the same orbits). We answer this question. Theorem \[thm:isometry\] proves that for any permutation satisfying Equation \[eqn:markovped\], there is always a group of isometries having the same orbits as the permutation sets.
\[thm:isometry\] Let $S$ be a group of permutations of $\mathcal{H}_{n}$ whose orbits ${\Omega}(S)$ satisfy Equation (\[eqn:markovped\]). Then there exists a group of isometries $G$ having the same orbits as $S$: that is, for every $T \in G$ and all $x,y \in \mathcal{H}_{n}$, $|y \oplus x| = |T(y) \oplus T(x)|$, and the set of orbits ${\Omega}(G)$ is equal to ${\Omega}(S)$.
We prove this by constructing a generating set $A$ for $G$. First, let the identity permutation $\pi_e$ be in $A$. Then for each orbit $\omega$ of $S$, and each pair of points $x_1$ and $x_2$ in $\omega$, we will construct a permutation $\pi_{x_1,x_2}$ to add to the generating set $A$. If $x_1=x_2$, then $\pi_{x_1,x_2} =
\pi_e$ which is already in $A$. If $x_1 \ne x_2$ then $\pi_{x_1,x_2}$ will be a composition of disjoint two-cycles, in particular including the cycle $(x_1~ x_2)$. Our generating set $A$ will then be the union of all these permutations, so by construction it will generate a group $G = \langle A \rangle$ having the same orbits as $S$.
For fixed $x_1,x_2\in \omega$, the two-cycles comprising $\pi_{x_1,x_2}$ are constructed as follows: For each $k = 1, \dots, n$, define $a_k := \# \{y \in \omega: |y \oplus x_1|=k \}$ and $b_k := \# \{z \in \omega: |z \oplus x_2|=k \}$, which implies by Equation (\[eqn:markovped\]) that $a_k s^k = b_k s^k$ for each $k$, and hence $a_k = b_k$, since $s > 0$ and polynomials in $s$ are uniquely determined by their coefficients and powers. Then, for each given $y_1 \in \omega$ that is distinct from both $x_1$ and $x_2$, there exists $z_1$ such that $|y_1 \oplus x_1| = |z_1 \oplus x_2| =k$, because $a_k \ge 1$, a consequence of the fact that $y_1 \in A_k := \{y \in \omega: |y \oplus x_1|=k \}$. In other words, $z_1:= y_1 \oplus (x_1 \oplus x_2)$, and the cycle is $c_1:=(y_1~z_1)$.
Proceed similarly for $y_2 \in {\cal H}_n \setminus \{x_1, x_2, y_1, z_1\}$, defining $z_2:= y_2 \oplus (x_1 \oplus x_2)$, and the cycle $c_2:=(y_2~z_2)$, and so on for each $y_i \in {\cal H}_n \setminus \{x_1, x_2, y_1, z_1, \dots, y_{i-1}, z_{i-1}\}$, with $z_i:= y_i \oplus (x_1 \oplus x_2)$ and $c_i:=(y_i~z_i)$. Then we define the permutation $\pi_{x_1,x_2} := c_1 \circ c_2 \circ ... \circ c_{2^{n}}$. In particular it has the cycle $(x_1~ x_2)$ in its composition, since when $y=x_1$, we have $z=x_2$. Notice also that the definitions of $z_i$ imply that $$\begin{aligned}
y_i \oplus y_j = y_i \oplus y_1 \oplus y_j \oplus y_1 =z_i \oplus z_1 \oplus z_j \oplus z_1 = z_i \oplus z_j; \\
y_i \oplus z_j = y_i \oplus x_1 \oplus z_j \oplus x_1 =z_i \oplus x_2 \oplus y_j \oplus x_2 = z_i \oplus y_j.\end{aligned}$$ Hence by taking $L^1$ norms, the permutation $\pi_{x_1,x_2}$ is an isometry with respect to Hamming distance.
Furthermore, the group $G = \langle A \rangle$ will have the same orbits as $S$, since for each orbit $\omega$ and each pair $x_1,x_2\in \omega$, the cycle $(x_1~ x_2)$ will appear in some permutation, and no pair of points from different orbits will appear as a cycle in any permutation.
This proof complements the result from Browning and Browning regarding the fact that isometry groups always satisfy Equation \[eqn:markovped\]. Indeed, we will state the complete result as a corollary.
A group $S$ has orbits $\Omega(S)$ satisfying Equation \[eqn:markovped\] if and only if there is an isometry group $G$ whose orbits $\Omega(G)$ are identical to $\Omega(S)$.
Browning and Browning [@Browning2002] showed that all isometry groups $G$ satisfy Equation \[eqn:markovped\]. Theorem \[thm:isometry\] completes the proof.
It is a well-known fact in algebra that any partition can be the orbits of some symmetry group, and that the orbits of any symmetry group are a partition [@Durbin2000]. We will recapitulate this simple result next.
\[cor:partition\] A partition satisfies Equation \[eqn:markovped\] if and only if it is equivalent to the orbits of some isometry group.
Assume we are given a partition $\{W_1,...,W_k\}$ of set ${\cal H}_n$ where $W_i \cup W_j = \emptyset$, $\cup_{i} W_i = {\cal H}_n$ and the partition satisfies Equation \[eqn:markovped\]. We will create a symmetry group $S$ whose orbits $\Omega(S) = \{W_1,...,W_k\}$. This is easily done. For each set in the partition $W_i$, create a permutation with a single cycle $\pi_i = (y_1~y_2~...~y_l)$ where all $y_j \in W_i$. Make a generating set $A = \{\pi_i : 1 \le i \le k\}
\cup \pi_e$ where $\pi_e$ is the identity permutation. Then group $S
= \langle A \rangle$ clearly has orbits $\Omega(s) = \{W_1,...,W_k\}$. By Theorem \[thm:isometry\] there is an isometry group with the same orbits.
Assume we are given an isometry group $G$. Clearly, by Browning and Browning’s proof [@Browning2002], the orbits define a partition $\Omega(G)$ that satisfies Equation \[eqn:markovped\].
Browning and Browning [@Browning2002] also showed that any isometry $T:\mathcal{H}_{n} \to \mathcal{H}_{n}$ can be uniquely written as $T=\pi \circ \phi_a$ where $\pi$ is a permutation on $n$ elements, the bits of the hypercube vertex, and $\phi_a$ is a switch function where $\phi_a(x) = a \oplus x$ where $\oplus$ is the bit-wise XOR operation.
An isometry describes some aspect of the pedigree graph. For example, an isometry consisting of a switch and the identity permutation can be used to enumerate one element from each orbit by simply fixing the 1-bit’s value and then enumerating all possible values for the other switch bits. On the other hand, an isometry consisting of the identity switch (all zero) and a permutation of one cycle can be used to enumerate one element for each orbit by listing the 1-prefixes of the permuted bits, (i.e. for three bits, the representatives are $000$, $100$, $110$, and $111$).
Emission Property
-----------------
The Markov property is not enough to ensure that the HMM based on Markov chain $Y_t$ has the same likelihood as the HMM for $X_t$. In order to ensure this, we introduce a property on the emission probabilities, namely that all the elements in one orbit must have identical emission probabilities. We call these orbits the emisison partition, since they are induced by the emission probability. In order to define this object, we need to introduce some more notation.
Recall that $R_x$ is the inheritance graph for inheritance path $x$. The relationship structures we wish to preserve are the IBD relationships on the individuals of interest $I$. Let $I_m$ be the maternal alleles of all the individuals of interest and $I_p$ be the paternal alleles of all the individuals of interest. The inheritance graph $R_x$ is a forest of trees; let $CC(R_x)$ refer to the connected components of $R_x$ which are labeled with $I_m \cup I_p$. The same-labeled connected components induce a partition $$D_x = \{y \in \mathcal{H}_{n} | CC(R_y) = CC(R_x)\}.$$ We call the parition $\{D_x | \forall x\}$ the identity states, since it indicates a particular identity-by-descent (IBD) relationship among the labeled individuals. These have been well studied [@Jacquard1972; @Thompson1974; @Karigl1982].
Looking at a small example, containing two siblings who are the individuals of interest and their two parents, we see that the identity states are:
$$D_{0000}=\{0000, 0101,1010,1111\},$$
$$D_{1000}=\{1000,0010,1101,0111\},$$
$$D_{0100}=\{0100,1110,0001,1011\},$$
$$D_{1100}=\{1100,0110,1001,0011\}.$$ But if we think carefully about this example, there is symmetry in the pedigree, namely swapping the two parents, that does not appear in this partition. Due to this reason, we need to consider the following object.
Let $Pr[D|X_t]$ be the probability that the state $X_t$ of the HMM emits the observed data $D$ in state $t$. Then the partition induced on the state space by the emission probablity is the *emission partition* containing all distinct sets $E_x$: $$E_x = \{y \in \mathcal{H}_n ~| ~Pr[D=d|X_t=x] = Pr[D=d|X_t=y] ~\forall d \}$$ and $$Pr[D=d| X_t = x] = \sum_{\tilde{d}~consistent~with~R_y} \frac{1}{2^{h(d)}} \prod_{c \in CC(R_x)} Pr[c(\tilde{d})]$$ where $d$ is a vector of sets, $\tilde{d}$ is a vector of tuples that is an ordered version of $d$, meaning that $d_i \equiv \tilde{d}_i$ while removing the order information from $\tilde{d}_i$, and $c(\tilde{d})$ gives the allele of $\tilde{d}$ that is assigned to that connected component, and $h(d)$ is the number of heterzygous sites in $d$. Note that each connected component is a tree, and has exactly one founder. Also, the identity states are consistent with these probabilities, but the identity states are a sub-partition of the emission partition. Specifically, from our previous example, $0100 \notin D_{1000}$, but $0100 \in E_{1000}$. Indeed, the emission partition for the example is $D_{0000}, \{D_{1000},D_{0100}\}, D_{1100}$.
We say that the expectation Markov chain $Y_t$ satisfies the *emission property* if and only if it preserves the emission partition in order for the corresponding HMM to have the correct likelihood. To preserve the emission partition, all the group elements $T \in G$ must satisfy $T(y) \in E_x$ for all $y \in E_x$ and for all $x$.
Now, it is necessary to compute the $E_x$ quickly. The naïve algorithm would be slow, since we would have to consider all pairs $x,y$ and all possible data $d$. Neither can we use the methods in the literature dealing with *condensed identity states* [@Jacquard1972; @Thompson1974; @Karigl1982], because the literature computes pedigree-free condensed identity states. That calculation takes the sets from the identity states and applies permutations of the form $\pi_{i} = (i_m~i_f)$ to swap the alleles of an individual of interest $i \in I$. However these permutations can violate the inheritance rules specified by a fixed pedigree. For the example above, take vector $1010 \in D_{0000}$ and swap the alleles of the second child $\pi_2(1010) = 1001 \in D_{1100}$. This clearly produces a partition that is not the emission partition, and so it would violate the property that we wish to enforce. Several works on optimal state space reduction for pedigree HMMs have discussed the condensed identity states [@Browning2002; @McPeek2002inference] for state-space reduction. It would appear that they did not formulate the emission partition that was mentioned by Geiger, et al. [@Geiger2009] and that is used here.
The main difference between $D$ and $E$ partitions is that the probability $Pr[D=d| X_t = x]$ has a product over indistinguishable connected components, whereas the identity states distinguishes each connected component. This partition must answer the question of which connected components are exchangeable. Let $I'$ be the individuals of interest having parents who are not individuals of interest. So, we can rewrite $E_x$ as follows: $$E_x = \{y \in \mathcal{H}_n ~|~ \exists \phi ~\textrm{a proper isomorphism s.t.}~ CC(R_x) = CC(\phi(R_y)) \}$$ where an isomorphism $\phi$ is *proper* if and only if $\phi$ is an isomorphism from $R_y$ to $R_x$ where for all $i\in I' \cup V \setminus I$, either $\phi(i_f) = i_f$ and $\phi(i_m) = i_m$ or $\phi(i_f) = i_m$ and $\phi(i_m) = i_f$. This definition of $E_x$ is easier to compute, because now we can do an $O(n)$ check to see if the forest of trees in $x$ and $y$ are isomorphic, which leads to an $O(n2^{2n})$ calculation. However, we can do better.
From the above definition, we see that in order for two inheritance paths to be isomorphic, the pedigree graph itself (as opposed to the inheritance graph) must have a automorphism. If we can identify all the relevant automorphisms for the pedigree graph, then we can make a set $A$ of permutations (one for each automorphism), and use a group theoretic algorithm for obtaining the orbits of $\langle A \rangle$ acting on the partition $\{D_x~|~ \forall x \in \mathcal{H}_n\}$ to obtain the desired emission partition.
First to obtain the automorphisms of the graph, we will employ a naïve strategy. Let $i \in I' \cup V \setminus I$ be an individual of interest. Recall that any proper isomorphism must map one branch of $i$’s ancestral lineage to the other branch. In order to be consistent, for the set $J = \{i\} \cup \{j ~|~ j \textrm{ full sib of } $i$\}$, the automorphism must $\phi(j_m) = j_f$ for $j \in J$. Considering $i$’s parents and proceeding backward in time, the sub-pedigree connected to the ancestors forms a directed acyclic graph (dag) with in-degree two. Without loss of generality, we can assume that this sub-pedigree has no individuals in $I \setminus \{i\}$, because if there were there would be no proper automorphism, and there is a descendant of the ancestors not in $I$ it can be trivially removed from the pedigree [@McPeek2002inference]. Therefore, we may consider only the tree of direct ancestors branching backward in time. At each branch point, $b$, in this tree, we assign an indicator $\gamma_b = 1$ if the father is to the left and the mother to the right. There are $O(2^n)$ assignments of these variables $\{\gamma_b~|~\forall b\}$. For each possible assignment, perform an $O(n)$ graph-traversal operation to check whether this assignment is an automorphism. We take the first automorphism $\phi$ that we find, because any other $\phi'$ from the same lineage will satisfy $CC(\phi(R_x)) = CC(\phi'(R_x))$ for all inheritance paths $x$.
Now that we have the automorphisms, we can write them as isometries and put them in set $A$ and consider the orbits of the group $\langle A \rangle$ acting on the identity states. These orbits are the emission partition. To obtain these orbits, we will use the well-known orbit algorithm from computational group theory. Notice, that we with to apply this algorithm to the existing partition $M := \{D_x| \forall x \}$. Take one set $D_x \in M$ and initialize its orbit as $O_x := \{D_x\}$. At the end of the following procedure $O_x$ will contain all the elements in $x$’s orbit. For every element $D_x \in O$ and every automorphism permutation $a \in A$, compute $y := a(z)~~\forall z \in D_x$. If $y \notin O_x$, then this $y$ and all the elements in its set $D_y$ are added to $O_x$ and $D_y$ is removed from $M$. This procedure is repeated until $M$ is empty. Notice that $CC(y) = CC(a(z))$ is compared to $CC(x)$ to determine if $y$ is is in $O_x$.
Since the comparison $CC(y) = CC(x)$ can be computed in linear time. The running time to obtain the automorphisms is $O(n2^n)$ and the orbit algorithm runs in $O(n2^n)$ time. This means that obtaining autormorphisms of the pedigree is preferable to checking pairs of inheritance vectors for isomorphism.
Examples {#sec:example}
--------
We will consider two examples, here. The first is a three generation pedigree while the second is a result that applies to all two generation pedigrees.
### Three-Generation Pedigree
For example, given 4 meioses for two half-cousins, $A$ and $B$, with one shared grandparent, their common grandparent and their respective parents who are half-siblings, we have 16 hypercube vertices (see Figure \[fig:halfcousins\]). Our individuals of interest are $I =
\{A,B\}$. The emission partition is, in this case, identical to the identity states and contains the sets $E_1=\{A_p\}\{A_m,B_m\}\{B_p\}$ and $E_2=\{A_p\}\{A_m\}\{B_m\}\{B_p\}$, since these are the only partitions of alleles of individuals $I$ that have non-empty sets in the emission partition. The emission partition induced on the hypercube vertices is: $E_{x_1}=\{1001,1111\}$ and $E_{x_2}=\mathcal{H}_n \setminus E_{x_1}$.
Notice that in this instance we cannot use the emission partition $\{E_x| ~\forall x\}$ as the state space of a new Markov chain. For example, if we were to let $Z_t$ be a Markov chain on the partition given by the emission partition, then the Markov criteria would fail to hold. Specifically, consider state $x_1=0001$ and $x_2=0011$. Then by checking Equation (\[eqn:markov\]), we have $\sum_{y \in E_{x_1}} Pr[X_t = y | X_t = 0001] = \theta(1-\theta)^3 + \theta^3(1-\theta)$ but $\sum_{y \in E_{x_1}} Pr[X_t = y | X_t = 0011] = 2 \cdot \theta^2(1-\theta)^2$.
The largest partition of ${\mathcal H}_n$ that satisfies the Markov criteria is $P_J=\{1001,1111\},$ $P_R=\{0010,0100\},$ $P_G=\{1011,1101\},$ $P_B=\{0000,0110\},$ $P_K=\{0011,0101,1010,1100\},$ and $P_L=\{0001,0111,1000,1110\}$. Let $H$ be the matrix of pair-wise Hamming distances between all the vertices of the hypercube. Then the transition probabilities take the form: For example, $Pr[Y_{t+1}=P_L | Y_t=P_K] = 2 \theta(1-\theta)^3 + 2 \theta^3(1-\theta)$.
![[**Two Half-Cousins.**]{} (Left Panel) A pedigree with four non-founders of which two are half-cousins together with their common grandparent. As before, the two black dots for each person represent their two alleles, and the alleles of each individual are ordered, so that the left allele, or paternal allele, is inherited from the person’s father, while the right, maternal allele is inherited from the mother. The two cousins are labeled $A$ and $B$. It is easy to see that the only possible IBD is between alleles $A_m$ and $B_m$, the maternal alleles of individuals $A$ and $B$, respectively. (Right Panel) This makes the four male founders irrelevant to the question of IBD. The four meioses are labeled in the order of their bits, left-to-right,and the inheritance states are represented in binary as $x_1 x_2 x_3 x_4$. Let $x_i=0$ if that allele was inherited from the parent’s paternal allele, and $x_i=1$ if from the maternal allele. For instance, $A$ and $B$ are IBD only for inheritance states $1001$ and $1111$. ](half-cousins.eps)
\[fig:halfcousins\]
Notice that this partition can be expressed as the orbits of a group of isometries, because $G = \langle (1~4), (2~3), \phi_{0110} \rangle$ does not violate the IBD class.
### Two-Generation Pedigrees
\[lem:twogen\] For any two-generation pedigree, the partition defined by the emission partition, $C = \{E_x | ~\forall x\}$, satisfies the Markov Property.
We can establish this by finding a group of isometries whose orbits are the emission partition. This group has the generating set $A$ where $A = \{\phi_f : \forall f\} \cap \{\pi_m: \forall m\}$ and $\phi_f$ and $\pi_m$ are defined as follows. For founder $f$, $\phi_f$ is a switch having bits set as follows. Let $i_1,..,i_c$ be the meioses from founder $f$ to each of the founders $c$ children. Then $\phi_{fi} = 1$ if $i = i_j$ for some $j$ and $\phi_{fi} = 0$ otherwise. Let $m = (f_1,f_2)$ which are untyped monogamous married founding pairs. Then $\pi_m = c_1 \circ c_2 \circ ... \circ c_k$ is a permutation composed of $k$ disjoint cycles, one for each child. For child $i$ with meiosis bits $i_0,i_1$, $c_i = (i_0~ i_1)$. The group of isometries $G = \langle A \rangle$.
Now, we simply need to establish that the emission partition $C$ are the orbits of this group $G$. There is no element $T \in G$ that maps $x \in E_{x_1}$ to $y \in E_{x_2}$, since every $\phi_f$ and $\pi_m$ map the bits of $x$ in ways that maintain $CC(R_x)$. Now, we simply need to show that for any $x_1,x_2 \in E_x$, there is always some element $T \in G$ such that $y = T(x)$. Consider each connected component in $CC(R_x)$ where $x$ and $y$ differ. The alleles connected in this connected component must all share inheritance through one of the founder bits of the common parents. If there is only one common parent, the switch for that founder must map between $x$ and $y$ in the bits for that connected component. If there are two common parents, then there must exist a composition of two founder switches and the founder permutation that maps between $x$ and $y$ for the bits in that connected component. The complete map $T$ is simply the composition of the isometries for each connected component.
In the next section we will introduce the Maximal Ensemble Problem, and we will soon see that this lemma provides a fast method to obtain the optimal partition for two-generation pedigrees.
The State-Space Reduction Problem
---------------------------------
There have been three state-space reduction problems posed, we restate these here. Given the original pedigree state space $\mathcal{H}_n$, there are three ways to reduce the state space.
Maximum Ensemble Problem [@Geiger2009]
: Find the partition, $\{W_1,...,W_k\}$ of $\mathcal{H}_n$ that satisfies both the Markov property and the emission property and that maximizes the sizes of the sets: $\max
\sum_{i=1}^k |W_i|$.
Maximum Isometry Group Problem [@Browning2002]
: Find the isometry group $G$ of maximal size whose orbits $\Omega(G)$ satisfy the emission property.
Maximum Symmetry Group Problem
: Find the symmetry group $G$ of maximal size whose orbits $\Omega(G)$ satisfy both the Markov property and the emission property.
We have already proven that all symmetry groups that satisfy the Markov property have an isometry group with equivalent orbits. This means that the later two problems are identical. Indeed since these last two problems are equivalent, we will refer to them collectively as the [**Maximum Group Problem.**]{} The remaining question is the relationship between the maximum ensemble problem and the maximum isometry group problem. We will first introduce a Maximum Ensemble Algorithm and use it to prove that the solution to the Maximum Ensemble Problem is unique. Using the uniqueness result, we will be able to prove the equivalence of the Maximum Ensemble and Maximum Isometry Group Problems.
Maximum Ensemble Algorithm {#algorithm}
==========================
We will introduce an algorithm that solves the Maximum Ensemble Problem and that helps us establish the uniqueness of the solution. We will then use uniqueness to prove that this algorithm also solves the Maximum Group Problem.
Consider the emission partition containing, $E_x$ for all $x$ of interest. Of course the sets in the emission partition are disjoint. Consider the $(2^n)!$ permutations on the vertices of the hypercube. Naively, these are all candidate permutations for our group, if we wish to find the maximal group. However in this section, we focus on finding the partition that yields the maximum ensemble solution. Given the state space, the partition can be found in linear time.
We do this by iteratively sub-partitioning the partition according to the coefficients and powers appearing in Equation \[eqn:markovped\]. See Algorithm \[alg:bipartition\]: Bipartition which takes as input a subpartition of the emission partition. This recursion is possible since the Markov property must produce a partition that is a sub-partition of the emission partition (i.e. in order to respect the emission partition). Indeed, as shown in Lemma \[lem:partition\], any pair of vectors $x_1,x_2$ that violate the Markov property must appear in separate sets of the partition. This recursive approach will at worst produce a partition with each element in its own set.
Algorithm \[alg:bipartition\]: Bipartition only needs to compute the $2^n \times 2^n$ matrix of distances between IBD vectors, as well as do some bookkeeping. So, the total running time is $O(2^{2n})$. Since the iterative sub-partitioning at minimum splits sets in two and does not introduce new inequalities, the number of iterations of the partition algorithm is $O(log(2^n))= O(n)$. One iteration of Algorithm \[alg:bipartition\]: Biartition requires $O(2^{2n})$ time for each iteration, since we have to check the $2^n \times 2^n$ matrix of distances between partition elements. So, the total running time is $O(n 2^{2n})$.
Now, we need to establish the correctness and uniqueness of the partition.
\
$P$: current subpartition of the emission partition\
\
$P'$: violates fewer equations of the Markov property\
\
$P' = \emptyset$ $C_{i0} = W_i$ $C_{i1} = \emptyset$ $a_k = 0$ for all $0 \le k \le n$ $s_{x'} = 0$ for all $x' \in C_{i0}$ Let $x_1 \in C_{i0}$ be a fixed element of $C_{i0}$. $b_k = 0$ for all $0 \le k \le n$ Let $k = |y \oplus x|$ $a_k++$ $b_k++$ $s_{x} = 1$ {Bipartition $W_i$} $C_{i0} \gets C_{i0} \setminus \{x\}$ $C_{s_x} \gets C_{s_x} \cup \{x\}$ $P' \gets P' \cup \{C_{i0},C_{i1}\}$ RETURN $P'$
\[lem:partition\] Let $W_i, W_j$ be two sets of the partition such that $x_1,x_2 \in W_i$ and $x_1,x_2$ violate the Markov property in Equation \[eqn:markovped\], i.e. such that $$\sum_{y\in W_j} s^{|y \oplus x_1|} \ne \sum_{y \in W_j} s^{|y \oplus x_2|}.$$ Then even if $W_j$ is subdivided, $x_1,x_2$ continue to violate Equation \[eqn:markovped\].
This is proven by a simple property of polynomials. Since $$\sum_{y\in W_j} s^{|y \oplus x_1|} \ne \sum_{y \in W_j} s^{|y \oplus x_2|},$$ there must be at least one power for which the polynomial coefficients disagree. Let $a_k$ and $b_k$ be the coefficients from the left- and right-had sides respectively. Let $A(k) = \{y : |y \oplus x_1|=k \}$, so that $a_k = |A(k)|$, and let $B(k) = \{y : |y \oplus x_2|=k \}$, so that $b_k
= |B(k)|$. Let $C,D$ be any bipartition of $W_j$. Therefore $C$ and $D$ induce a partition of $A(k)$ and $B(k)$. Specifically $A(k)$ is partitioned into sets $A(k) \cap C$ and $A(k) \cap D$, while $B(k)$ is partitioned into $B(k) \cap C$ and $B(k) \cap D$. Since $|A(k)| \ne |B(k)|$, then at least one of $$|A(k) \cap C| \ne |B(k) \cap C|$$ or $$|A(k) \cap D| \ne |B(k) \cap D|.$$ Therefore at least one of $$\sum_{y\in C} s^{|y \oplus x_1|} \ne \sum_{y \in C} s^{|y \oplus x_2|},$$ or $$\sum_{y\in D} s^{|y \oplus x_1|} \ne \sum_{y \in D} s^{|y \oplus x_2|}.$$
(Loop Invariant.) \[lem:invariant\] Once $C_{i0}$ is added to $P'$, it is never subdivided again in any iteration. This is equivalent to stating the invariant that for any $i$, $$\sum_{y \in W_j} s^{|y \oplus x_1|} = \sum_{y \in W_j} s^{|y \oplus x_2|} ~~\forall~ x_1,x_2 \in C_{i0} ~~\forall~ W_j \in P'$$
Notice that the above invariant is a consequence of both the loop “foreach $W_j \in P$” and of the Bipartition algorithm. For the base case $C_{i0} = \emptyset ~~\forall i$, and the invariant holds trivially.
Now we need to inductively prove that the invariant holds. Assume that for some $i$, the invariant holds. Now, consider the loop for a fixed $W_j \in P$. $W_j$ may be partitioned into some $C_{j0}$ and $C_{j1}$. Our task is to prove that for the new partition of $W_j$ the invariant holds, i.e. that $$\sum_{y \in C_{j0}} s^{|y \oplus x_1|} = \sum_{y \in C_{j0}} s^{|y \oplus x_2|} ~~\forall~ x_1,x_2 \in C_{i0}.$$
From the invariant, we have $\sum_{y \in W_j} s^{|y \oplus x_1|} = \sum_{y
\in W_j} s^{|y \oplus x_2|} ~~\forall x_1,x_2 \in C_{i0}$. Fix $k$ and define the set $$A(k,x_1):= \{y \in W_j : |y \oplus x_1| = k\} ~~\forall~ x_1 \in C_{i0},$$ then the coefficient of the $k$th power in the equation is $|A(k,x_1)|$. Futhermore, we have $|A(k,x_1)|=|A(k,x_2)|$ for all $x_1,x_2 \in
C_{i0}$.
Notice that $C_{j0}$ was created with the property that $\sum_{x_1 \in
C_{i0}} s^{|x_1 \oplus y_1|} = \sum_{x_1 \in C_{i0}} s^{|x_1 \oplus y_2|}$ for all $y_1,y_2 \in C_{j0}$. Define the set $$B(k,x_1) := \{y_1 \in C_{j0}: |x_1 \oplus y_1| = k\}~~\forall~ x_1 \in C_{i0},$$ and its mirror set $$D(k,y_1) := \{x_1 \in C_{i0} : |x_1 \oplus y_1|=k\}~~\forall~ y_1 \in C_{j0}.$$ Notice that $A(k, x_1) \cap C_{j0} = B(k,x_1)$ for all $x_1\in C_{i0}$.
Now we will use the property $|D(k,y_1)| = |D(k,y_2)|$ for all $y_1,y_2 \in C_{j0}$ to prove that $|B(k,x_1)| = |B(k,x_2)|$ for all $x_1,x_2 \in C_{i0}$. Let $\phi:C_{j0} \to C_{j0}$ be a bijective map on $C_{j0}$ such that $\phi(x_1)=x_2$. Pick a bijective map $\pi:C_{i0} \to
C_{i0}$ that maps elements of $D(k,y_1)$ to elements of $D(k,\phi(y_1))$. Now, we will show that $y_1 \in B(k,x_1)$ if and only if $\phi(y_1) \in B(k,\pi(x_1))$. Now $y_1 \in B(k,x_1) = A(k,x_1) \cap C_{j0}$, so this is equivalent to $x_1 \in D(k,y_1)$, which in turn is true if and only if $\pi(x_1) \in D(k,\phi(y_1))$, or if and only if $\phi(y_1) \in A(k,\pi(x_1))$. Then since $\phi(y_1) \in C_{j0}$, we have that $\phi(y_1)
\in B(k, \pi(x_1))$.
This proves that $|B(k,x_1)| = |B(k,x_2)|$ for all $x_1,x_2 \in C_{i0}$. Therefore we have $$\sum_{y \in C_{j0}} s^{|y \oplus x_1|} = \sum_k |B(k,x_1)| s^{k} ~~\forall~ x_1 \in C_{i0}.$$ Therefore, we have the invariant that $$\sum_{y \in C_{j0}} s^{|y \oplus x_1|} = \sum_{y \in C_{j0}} s^{|y \oplus x_2|} ~~\forall~ x_1,x_2 \in C_{i0}$$
(Uniqueness of Solution.) \[thm:uniqueness\] The Maximum Ensemble Algorithm finds the unique solution to the Maximum Ensemble Problem.
The partitioning algorithm produces a partition that respects the emission partition, since it begins with the partition given by the emission partition and sub-partitions it. The algorithm also produces partitions that respect the Markov property, since it iteratively sub-partitions the emission partition until the Markov property is satisfied. Notice that the algorithm is guaranteed to find such a partition since the trivial partition, i.e. the original state space, satisfies the Markov property. Since partition sets are only divided if they violate the Markov property, the algorithm necessarily finds an optimal partition. Only the proof of uniqueness remains.
By Lemma \[lem:partition\] the solution is invariant to the order in which the bipartitions are made, since any $x_1,x_2$ which violate the Markov property must be put into separate sets of the partition at some point. Indeed, by Lemma \[lem:invariant\] we know that once $C_{i0}$ is created, it is never partitioned again. Since we begin with a unique partition, the emission partition, the sequence of $C_{i0}$, created by different calls to Algorithm \[alg:bipartition\], will be the final sets in the partition, up to reordering. Therefore the Maximum Ensemble Algorithm finds the unique partition which is the solution to the Maximum Ensemble Problem.
Equivalence
===========
Now, using the uniqueness of a partition as the solution to the Maximum Ensemble Problem, we can prove equivalence of the Maximum Ensemble Problem and the Maximum Isometry Group Problem.
(Equivalence of Maximum Ensemble Problem and Maximum Isometry Group Problem) \[thm:equivalence\] A partition $\{W_1,W_2,...,W_k\}$ is a solution to the Maximum Ensemble Problem if and only if there is an isometry group $G$ that is a solution to the Maximum Group Problem having orbits $\Omega(G)$ equivalent to the partition: for all $\omega$, we have $\omega
\in \Omega(G)$ if and only if there exists a set in the partition $W_j$ such that $W_j = \omega$.
First, due to Corollary \[cor:partition\], we know that only isometry groups satisfy the Markov property. Any partition which is a solution for the Maximum Ensemble Problem is also, in particular, the orbits of a group of isometries, $G$. Assume that $G$ is not the maximal isometry group. Because, if not, then there must be some isometry which can be added. And, if it were added, it would join two orbits into one. Therefore joining two sets of the partition into one, which contradicts the assumption that the partition was maximal. Furthermore, since $G$ satisfies the emission property, it’s orbits must be a subpartition of the emission partition. There is no other group $G'$ with larger size, since the solution to the Maximum Ensemble Problem is unique (Theorem \[thm:uniqueness\]). The Maximum Ensemble Problem is a solution to the Maximum Group Problem.
For the converse we argue by contrapositive. Assume that partition $\{W_1,...,W_k\}$ is not a solution to the Maximum Ensemble Problem but that it satisfies Equation \[eqn:markovped\]. Then there must exist some partition $\{V_1,...,V_l\}$ such that $\sum_{i=1}^k |W_i| <
\sum_{j=1}^l |V_j|$. This inequality is strict by the uniquness Theorem \[thm:uniqueness\]. There must exist some $i$, $i',$ and $j$, such that $W_i \subset V_j$ and $W_{i'} \subset V_j$.
By Corollary \[cor:partition\], there are groups $G^W$ and $G^V$ with orbits $\{W_1,...,W_k\}$ and $\{V_1,...,V_l\}$, respectively. Choose $x_1 \in W_i \cap V_j$ and $x_2 \in W_{i'} \cap V_j$. Then $\pi_{x_1,x_2}$ from Theorem \[thm:isometry\] will be in $G^V$ and not in $G^W$. Therefore, $G^W$ is not a solution to the Maximal Isometry Group Problem; proving the claim.
Bootstrapping with Known Isometries
===================================
As noted by Geiger et al. [@Geiger2009], there are two types of isometries that can be detected easily. There are the founder isometries and the chain isometries where there is an outbred lineage consisting of multiple ungenotyped generations.
The founder isometries apply only to ungenotyped founders and are switches on the bits for the edges adjacent to the founder. Specifically, if ${i_1},...,{i_c}$ are the meiosis bits between the ungenotyped founder and each of the $c$ children of the founder, then the switch is given by the bit vector $X_{i} = 1$ if $i = i_j$ for some $j$ and $X_{i} = 0$ otherwise. Since the founder alleles are indistinguishable (due to the missing genotype), we can fix one bit adjacent to the founder and enumerate the other bits adjacent that founder. These founder isometries can be found in $O(n)$ time.
The chain isometries apply to a lineage of $l$ individuals, from oldest to youngest $i_1,i_2,...,i_l$ where each individual has exactly one parent from the lineage, one founder parent, one child, and no siblings, except $i_l$ which may have any number of siblings. All individuals except the most recent must be ungenotyped. The isometry is then the permutation on every bit, except the oldest, i.e. $\pi =
(1_1~i_2~i_3~...~i_l)$ Please see Geiger, et al. [@Geiger2009] and Browning and Browning [@Browning2002] for examples. These chain isometries can be found in $O(n^2)$ time.
It would seem that there are other classes of isometries which can be found quickly, such as the permutations shown in the example in Section \[sec:example\]. The exact algorithms for finding other classes of isometries remain an open problem. Furthermore, it is unknown whether all the isometries in the maximal group can be found efficiently.
Representatives
---------------
Let $A$ be a generating set of isometries that generate group $G =
\langle A \rangle$, such as the founder and chain isometries. In order to compute the bootstrap maximum ensemble states, We need to obtain the orbits of $G$ acting on ${\cal H}_n$. We can obtain them in $O(k|A|o)$ time where $k$ is the number of orbits and $o = \max_{x\in {\cal H}_n} |\omega(x)|$, provided that orbit membership can be checked in constant time.
Let $M = {\cal H}_n$ initially. We take any vector $x$ out of $M$ and find its orbit $O$. Initially let $O = \{x\}$. Now, for every $x \in O$ and every $a \in A$, compute $y = a(x)$. If $y \notin O$, add $y$ to $O$ and remove $y$ from $M$. Repeat until $M$ is empty.
Following this proceedure, we have all of the orbits of $G$ acting on ${\cal H}_n$. For each orbit, we will fix a representative to use in the bootstrap maximal ensemble algorithm.
Bootstrap Maximal Ensemble
--------------------------
Now that we have $k$ representatives, one from each orbit of group $G
= \langle A \rangle$, we can introduce a bootstrap version of the Maximal Ensemble algorithm. In this case, we can compute Equation (\[eqn:markovped\]) once per representative.
First, we need to partition our representatives according to the set of the emission partition that they belong to. Consider the emission partition, $\{E_x | ~\forall x \}$, and partition the representatives into these sets. Also partition ${\cal H}_n$ according to the emission partition. These two equivalent partitions define our initial partitions.
Now, we can recursively sub-divide the representatives whenever Equation (\[eqn:markovped\]) is violated. Notice that we can compute this equation with $x$ being the representative and $\omega_j$ is some set of the current partition of ${\cal H}_n$. Each time we subdivide the partition of the representatives, we need to also subdivide the partition of ${\cal H}_n$ in the equivalent fashion. Suppose that we have representative $x$ that we have put into a new set in the representative partition. We obtain the equivalent partition of ${\cal H}_n$ by creating a new set containing $x$ and all the vectors $y \in \omega(x)$ the orbit of $x$ under the action of $G$. The recursive subdivision continues until the Markov property is satisfied.
Since the recursive sub-partitioning at minimum splits sets in two, the number of iterations required is $O(n)$. Checking the Markov properties for each iteration requires $O(k2^{n})$ time where $k$ is the number of representatives, since we have to check the $k
\times 2^n$ matrix of distances, or sums of distances, between partition elements. So, the total running time is $O(n k 2^{n})$.
Running Times
=============
Notice that the naive calculation of Equation (\[eqn:transitions\]) requires $O(k2^n)$ time where $k \le 2^n$ is the number of sets in the partition and $n$ is the number of meioses in the pedigree. The calculation is as follows, for each set $W_i$ in the partition, choose a representative $x \in W_i$. For each of the sets in the partitions $W_j$, compute the transition probability $Pr[X_{t+1} \in W_j | X_t =
x]$. This last step seems to require enumeration of the inheritance paths.
The running time of the state-space reduction is the running time of the ensemble algorithm and the running time of the transition calculation. It is interesting to note that calculating the transition probabilities in Equation \[eqn:transitions\] is faster than the HMM forward-backward algorithm having running time $O(m2^{2n})$. This means there is potential to improve the state-space reduction running time, if there is a more efficient maximal ensemble algorithm.
Regardless of whether the over all running time of the state-space reduction is determined by calculating the transition function or the ensemble states, all the algorithms here produces savings when the forward-backward algorithm is run. This is because a $k$-set partition of the states results in the forward-backward algorithm having $O(mk)$ running time where $m$ is the number of sites. Furthermore, since the original state space has an $O(m2^{2n})$ forward-backward algorithm and the ensemble algorithm is $O(n2^{2n})$, the ensemble algorithm is more efficient when $n < m$ which is typically the case. The bootstrap algorithm is even more efficient having a running time of $O(nk2^n)$.
Simulation Results
==================
We simulated pedigrees under a Wright-Fisher model with monogamy where each pair of monogamous individuals has Poisson distributed number of offspring. There are $n$ individuals per generation and $\lambda$ is the mean number of offspring per monogamous pair. The individuals of interest, $I$, are the extant individuals, i.e. those in the most recent generation or, equivalently, the nodes with out-degree zero. These pedigrees have no inter-generational mating due to how the Wright-Fisher model is defined. To get a half-sibling pedigree, each edge of the pedigree had 50% chance of have a new parent drawn at random. Since monogamy was not preserved during this random process, the resulting pedigree had half-siblings.
Running the simulation process and the maximal ensemble algorithm 100 times produced Figure \[fig:simulation\]. The maximal ensemble algorithm produces exponential reductions in the size of the state-space. Whether the relationships have half-siblings seems not to influence the practical applicability of the maximal ensemble algorithm (data not shown).
In practice, the maximal ensemble algorithm seems limited to pedigrees of roughly 14 meioses while the bootstrap maximal ensemble algorithm seems limited to about 18 meioses. Of course, both methods yield the same reduced state space. Given the practical success of the bootstrap maximal ensemble algorithm, we recommend that the bootstrap maximal ensemble algorithm be employed for state-space reduction.
\[fig:simulation\]
Discussion
==========
Even though past efforts at state-space reduction have focused on finding groups of isometries on the edges of the pedigree graph, it is clear that this is an equivalent problem to finding the optimal partition of the state space that respects the Markov property. Although the paper mostly discusses the pedigree state-space, the maximum ensemble algorithm is general to any HMM.
Even if some isometries can be obtained efficiently, for example the founder and chain isometries, computation of the transition probabilities according to Equation \[eqn:transitions\] seems to require enumeration of the inheritance vectors. The naive algorithm requires $O(k2^n)$ where $k$ is the number of orbits and $n$ is the number of meioses in the pedigree. Due to this fact, and the fact that the forward-backward algorithm for pedigree HMMs has running time $O(m2^{2n})$, it is an advantage to use exponential algorithms to find the maximal state-space reduction. Indeed, the maximal ensemble algorithm we introduce here has running time $O(n2^{2n})$ which yields more efficient HMM algorithms when $n < m$ where $n$ is the number of meioses in the pedigree and $m$ is the number of sites.
In addition to introducing the maximal ensemble algorithm, we introduced a bootstrap maximal ensemble algorithm which runs in $O(nk2^{n})$ where $k$ is the number of orbits of the bootstrap isometry group. This allows our algorithm to take advantage of know isometries such as the founder and chain isometries.
It would appear that there might be an $O(2^{2n})$ algorithm for the maximum ensemble problem. This can be seen by the looking at the for loop of Algorithm \[alg:bipartition\]: Bipartition that says “foreach $x \in A_0$ do”. This could easily be changed to “foreach $A_{\delta}$ and foreach $x \in A_{\delta}$ do”. However, this algorithm appears to require sorting the sets in the emission partition in increasing order by size. We do not consider the details of this improved algorithm due to space considerations.
In practice, the maximal ensemble algorithm obtains exponential reductions in the state-space required for an HMM likelihood calculation. The algorithm operates on up to about 14 meioses.
There are several open problems of interest. First, the computational complexity of the maximum ensemble problem is open. Second, an open problem is the computational complexity of finding the transition rates after having determined the partition of the state space. Although naive algorithms are exponential, it is unclear whether there are approximation algorithms or polynomial-time algorithms for special cases.
Another very interesting direction is approximation algorithms where instead of guaranteeing equality in Equation (\[eqn:markovped\]), we could allow for bounded inequalities. Let $Y_t$ be the approximate Markov chain and $X_t$ be the original Markov chain. The idea is that a bound on the inequality for the transition probabilities of $Y_t$ would allow for a larger reduction in the state-space. In addition, we would hope that the bound on the inequality would guarantee that the deviation of $Y_t$’s stationary distribution is bounded relative to the stationary distribution of $X_t$.
#### Acknowledgments.
Many thanks go to Yun Song for suggesting the problem and to Eran Halperin for the random pedigree simulation.
|
---
abstract: |
The purpose of this paper is to exhibit fine structure for polyhedral products $Z(K; (\underline{X},\underline{A}))$, and polyhedral smash products $\widehat{Z}(K; (\underline{X},\underline{A}))$. Moment-angle complexes are special cases for which $(X,A) = (D^2,S^1)$ There are three main parts to this paper.
1. One part gives a natural filtration of the polyhedral product together with properties of the resulting spectral sequence in Theorem \[thm:spectral sequence\]. Applications of this spectral sequence are given.
2. The second part uses the first to give a homological decomposition of $\widehat{Z}(K; (\underline{X},\underline{A}))$ CW pairs $(\underline{X},\underline{A})$.
3. Applications to the ring structure of $Z(K; (\underline{X},\underline{A}))$ are given for CW-pairs $(X,A)$ satisfying suitable freeness conditions.
address:
- 'Department of Mathematics, Rider University, Lawrenceville, NJ 08648, U.S.A.'
- 'Department of Mathematics CUNY, East 695 Park Avenue New York, NY 10065, U.S.A.'
- 'Department of Mathematics, University of Rochester, Rochester, NY 14625, U.S.A.'
author:
- 'A. Bahri'
- 'M. Bendersky'
- 'F. R. Cohen'
- 'S. Gitler'
title: A spectral sequence for polyhedral products
---
\[section\] \[thm\][Corollary]{} \[thm\][Lemma]{} \[thm\][Proposition]{}
\[thm\][Definition]{} \[thm\][Example]{} \[thm\][Remark]{}
[**This paper is dedicated to Samuel Gitler Hammer who brought us much joy and interest in Mathematics** ]{}
**[Introduction]{}** {#Introduction}
====================
The subject of this paper is the homology of polyhedral products $Z(K; (\underline{X},\underline{A}))$, and polyhedral smash products $\widehat{Z}(K; (\underline{X},\underline{A}))$ [@davis.jan; @buchstaber.panov; @bbcg; @bbcg2; @bbcg3]. Definitions are listed in section \[definitions\] of this paper.
One of the purposes of this article is to give the Hilbert-Poincaré series for the polyhedral product $Z(K; (\underline{X},\underline{A}))$ in terms of
1. the kernel, image, and cokernel of the induced maps $$H^*(X_i) \to H^*(A_i)$$ for all $i$, and
2. the full sub-complexes of $K$.
This computation was also worked out in [@cartan] using more geometric methods.
This is achieved by analysis of a spectral sequence abutting to the cohomology of the polyhedral product $Z(K; (\underline{X},\underline{A}))$ by filtering this space with the left-lexicographical ordering of simplices. The method applies to a generalized multiplicative cohomology theory, $h^*$ as well. The spectral sequence is then used to describe some features of the ring structure of $h^*(\zk)$.
Qibing Zheng [@zheng] gives an alternative description of the cohomology of a polyhedral product. Our methods are distinct from his and the presentation of the computational results assumes a different form. Unlike the spectral sequence developed here, his collapses at the $E_2$ term.
**[Definitions, and main results]{}** {#definitions}
=====================================
The basic constructions addressed in this article are defined in this section. First recall the definition of an abstract simplicial complex.
\[defin: simplicial complex\]
1. Let $K$ denote an abstract simplicial complex with $m$ vertices labeled by the set $[m]=\{1,2,\ldots, m\}$. Thus, $K$ is a subset of the power set of $[m]$ such that an element given by a $(k-1)$-simplex $\sigma$ of $K$ is given by an ordered sequence $\sigma = (i_1,\ldots, i_k)$ with $1 \leq i_1 <\cdots < i_k \leq m$ such that if $\tau \subset \sigma$, then $\tau$ is a simplex of $K$. In particular the empty set $\emptyset$ is a subset of $\sigma$ and so it is in $K$. The vertex set of $\sigma$, $\{i_1,\ldots, i_k \}$ will be denoted $|\sigma|$.
2. Given a sequence $I = (i_1, \ldots, i_k)$ with $1 \leq i_1 <\ldots <
i_k \leq m $, define $K_I \subseteq K$ to be the [*full sub-complex* ]{} of $K$ consisting of all simplices of $K$ which have all of their vertices in $I$, that is $K_I = \{\sigma \cap I \ | \
\sigma \in K\}.$
3. In case $I = (i_1, \ldots, i_k)$, define $X^I = X_{i_1} \times X_{i_2} \times \ldots \times X_{i_k}.$
4. Let $\Delta[m-1]$ denote the abstract simplicial complex given by the power set of $[m]=\{1,2,\ldots, m\}$.
Let $h^*$ be a generalized, multiplicative cohomology theory and $(\underline{X},\underline{A})$ denote a collection of based CW pairs $\{(X_i, A_i,x_i)\}^m_{i=1}$. We will also assume $h^*(X_i)$ and $h^*(A_i)$ are finite type, i.e. $h^*(X_i)$ and $h^*(A_i)$ are generated as $h^*$ modules by classes, $\{x_{\ell}\}$ and $\{a_{\ell}\}$ respectively with finitely many generators in each degree.
For a generalized cohomology theory, $h^*$, we now describe a strong freeness condition on $\xa$ that will be imposed in Section \[sec:spectral sequence\].
The strong freeness condition assumes that the long exact sequence
$$\overset{\delta}{\to} \widetilde{h}^*(X_i/A_i) \overset{\ell}{\to} h^*(X_i) \overset{\iota}{\to} h^*(A_i) \overset{\delta}{\to} \widetilde{h}^{*+1}(X_i/A_i) \to$$
can be written in terms of explicit, free $h^*$ modules $E_i, B_i, C_i$ and $W_i$.
\[def:bas\]
The pair $\xa$ is said to satisfy a strong $h^*$ freeness condition if there are free $h^*$-modules $E_i, B_i, C_i$ and $W_i$ satisfying
1. $ h^*(A_i)= E_i\oplus B_i $ $ (B_i \ni 1 \subset h^0(A_i)).$
2. $h^*(X_i)= B_i \oplus C_i$ where $B_i \underset{\simeq}{\overset{\iota}{\to}} B_i, \quad C_i \overset{\iota}{\mapsto} 0$
3. $ \widetilde{h}^*(X_i/A_i)=C_i \oplus W_i$. where $C_i \underset{\simeq}{\overset{\ell}{\to}} C_i, \quad E_i \underset{\simeq}{\overset{\delta}{\to}} W_i \overset{\ell}{\mapsto} 0$
The goal of the spectral sequence is to compute the cohomology of the polyhedral product defined below. Our answer will be given in terms of the strong $h^*$ free decomposition described in Definition \[def:bas\]. In particular the description of the cohomology is only natural with respect to mappings of $h^*(X_i)$ and $h^*(A_i)$ which preserve the chosen strong $h^*$ decomposition. This point is further developed at the end of section \[revisited\].
In the following definition $\mathcal{K}$ denotes the category of simplicial complexes and $\mathcal{CW}_* $ is the category of based CW pairs.
\[defin:gmac\]
1. The [*polyhedral product* ]{} determined by $(\underline{X},\underline{A})$ and $K$ denoted $$Z(K;(\underline{X},\underline{A}))$$ is defined using the functor $$D: \mathcal{K} \to
\mathcal{CW}_{\ast}$$ as follows: For every $\sigma$ in $K$, let $$D(\sigma) =\prod^m_{i=1}Y_i,\quad {\rm where}\quad
Y_i=\left\{\begin{array}{lcl}
X_i &{\rm if} & i\in \sigma\\
A_i &{\rm if} & i\in [m]-\sigma
\end{array}\right.$$ with $D(\emptyset) = A_1 \times \ldots \times A_k$.
2. The polyhedral product is $$Z(K;(\underline{X},\underline{A}))=\bigcup_{\sigma \in K}
D(\sigma)= \mbox{colim} D(\sigma)$$ where the colimit is defined by the inclusions, $d_{\sigma,\tau}$ with $\sigma \subset \tau$ and $D(\sigma)$ is topologized as a subspace of the product $X_1 \times
\ldots \times X_k$. The [*polyhedral product*]{} is the underlying space $Z(K;(\underline{X},\underline{A}))$ with base-point $\underline{*}
= (x_1, \ldots, x_k) \in Z(K;(\underline{X},\underline{A}))$.
3. In the special case where $X_i = X$ and $A_i = A$ for all $1 \leq i \leq
m$, it is convenient to denote the polyhedral product by $Z(K;(X,A))$ to coincide with the notation in [@denham.suciu].
A direct variation of the structure of the polyhedral product follows next. Spaces analogous to polyhedral products are given next where products of spaces are replaced by smash products, a setting in which non-degenerate base-points are required. We will always assume the pairs $(X,A)$ are based CW pairs, in which case the base point condition is always satisfied.
The (reduced) suspension of a (pointed) space $(X,*)$ $$\Sigma(X)$$ is the smash product $$S^1 \wedge X.$$
\[defin:smash.product.moment.angle.complex\] Given a polyhedral product $Z(K;(\underline{X},\underline{A}))$ obtained from $(\underline{X},\underline{A}, \underline{*})$, the [*polyhedral smash product*]{} $$\widehat{Z}(K;(\underline{X},\underline{A}))$$ is defined to be the image of $Z(K;(\underline{X},\underline{A}))$ in the smash product $X_1
\wedge X_2 \wedge \ldots \wedge X_k$.
The image of $D(\sigma)$ in $\widehat{Z}(K;
(\underline{X},\underline{A}))$ is denoted by $\widehat{D}(\sigma)$ and is $$Y_1 \wedge Y_2 \wedge \ldots \wedge Y_k$$ where $$Y_i=\left\{\begin{array}{lcl}
X_i &{\rm if} & i\in \sigma\\
A_i &{\rm if} & i\in [m]-\sigma.
\end{array}\right.$$
In case it is important to distinguish the pair $(\underline{X},\underline{A})$, the notations $D(\sigma;(\underline{X},\underline{A}, \underline{*}))$, and $\widehat{D}(\sigma;(\underline{X},\underline{A}))$ will be used.
As in the case of $Z(K;(\underline{X},\underline{A}))$, note that $\widehat{Z}(K;(\underline{X},\underline{A}))$ is the colimit obtained from the spaces $ \widehat{D}(\sigma) .$
\[defin:smash.products\] Consider an ordered sequence $I = (i_1, \ldots, i_k)$ with $1 \leq
i_1 <\ldots < i_k \leq m$ together with pointed spaces $Y_1, \ldots,
Y_m$. Then
1. the length of $I$ is $|I|= k$,
2. the notation $I \subseteq [m]$ means $I$ is any increasing subsequence of $(1,\ldots, m)$,
3. $Y^{[m]}=Y_1 \times \ldots \times Y_m,$
4. $Y^{I} = Y_{i_1} \times Y_{i_2} \times \ldots \times
Y_{i_k},$
5. $\widehat{Y}^{I} = Y_{i_1} \wedge \ldots \wedge Y_{i_k},$
Given a sequence $I = (i_1, \ldots, i_k)$ with $1 \leq i_1 <\ldots < i_k \leq m $, define $K_I \subseteq K$ to be the [*full sub-complex* ]{} of $K$ consisting of all simplices of $K$ which have all of their vertices in $I$, that is $K_I = \{\sigma \cap I \ | \
\sigma \in K\}.$ This notation is used for the first decomposition proven in [@bbcg; @bbcg2] stated next.
\[thm:decompositions.for.general.moment.angle.complexes\] Let $K$ be an abstract simplicial complex with $m$ vertices. Given $(\underline{X},\underline{A}) =\{(X_i, A_i, x_i)\}^m_{i=1}$ where $(X_i,A_i,x_i)$ are pointed triples of CW-complexes there is a natural pointed homotopy equivalence $$H: \Sigma(Z(K;(\underline{X},\underline{A})))\to \Sigma(\bigvee_{I \subseteq [m]}
\widehat{Z}(K_I;(\underline{X_I},\underline{A_I}))).$$
A second result in [@bbcg; @bbcg2] is stated next where $|lk_{\sigma}(K)|$ denotes the geometric realization of the link of $\sigma$ in $K$.
\[thm:null.A\] Let $K$ be an abstract simplicial complex with $m$ vertices and $\overline{K}$ its associated poset. Let $(\underline{X},\underline{A})$ have the property that the inclusion $A_i\subset X_i$ is null-homotopic for all $i$. Then there is a homotopy equivalence $$\widehat{Z}(K;(\underline{X},\underline{A}))\to\bigvee\limits_{\sigma\in
K} |\Delta(\overline{K}_{<\sigma})|*\widehat{D}(\sigma)$$ where $$|\Delta(\overline{K}_{<\sigma})| = |lk_{\sigma}(K)|$$ the link of $\sigma$ in $K$.
In particular if $X_i$ is contractible for all $i$ there is a homotopy equivalence
$$\widehat{Z}(K;(\underline{X},\underline{A})) \to |K| \ast \widehat{A}^K.$$ Furthermore, there is a homotopy equivalence $$\Sigma (Z(K;(\underline{X},\underline{A})))
\to \Sigma(\bigvee\limits_{I\in [m]}(\bigvee\limits_{\sigma\in
K_I}|\Delta((\overline{K}_I)_{<\sigma})|*\widehat{D}(\sigma))).$$
Theorem \[thm:null.A\] for the case $X_i$ contractible for all $i$ is called the [*wedge lemma*]{}.
A filtration on $\zk$ is next described. The purpose of introducing this filtration is that there is an associated spectral sequence which is the subject of the article. The spectral sequence converges to the cohomology of $\zk$.
\[defin: lexicographical ordering\]
The $(m-1)$-simplex $\Delta[m-1]$ is totally ordered by the left-lexicographical ordering of all faces defined as follows: $$\sigma = (i_1,i_2,..., i_s) < \tau = (j_1,j_2,..., j_t)$$ if and only if either
1. $ 1 \leq s < t \leq m$ or
2. $t = s$, and there exists an integer $n$ such that $(i_1,i_2,..., i_n) = (j_1,j_2,..., j_n)$ but\
$i_{n+1} < j_{n+1}$.
There are $$(1+ \binom{m}{1} + \binom{m}{ 2}+ \binom{m}{ 3} + \cdots + \binom {m} { m-1} +\binom {m}{m}) = 2^m$$ faces in $\Delta[m-1]$ (including the empty set) which are totally ordered by the integers $q$ such that $0 \leq q \leq 2^m-1$.
Furthermore, let $\sigma_0$ denote the emptyset $\emptyset$; thus $\sigma_0 \leq \sigma $ for all $\sigma$ in $\Delta[m-1]$.
The weight of a face $\sigma$ is that integer $q,$ denoted by $wt(\sigma)$ where $q$ is the position of $\sigma$ in this total left lexicographical ordering of the simplices.
The $(m-1)$-simplex $\Delta[m-1]$ is filtered by requiring $$F_t\Delta[m-1] = \cup_{wt(\sigma) \leq t}\sigma.$$
This filtration of $\Delta[m-1]$ induces a filtration of $K$ as given next.
\[defin: filtration of polyhedral products\] The $(m-1)$-simplex $\Delta[m-1]$ is filtered by the left lexicographical ordering of all faces as in Definition \[defin: lexicographical ordering\]. Let $K$ be a simplicial complex with $m$ vertices.
Filter $K$ by $$F_tK = K \cap F_t\Delta[m-1].$$
Filter the polyhedral product $Z(K;X,A))$ and polyhedral smash product $\widehat{Z}(K;(X,A))$ by
1. $$F_tZ(K;(X,A)) = \cup_{wt(\sigma) \leq t} D(\sigma;(X,A)),$$ and
2. $$F_t\widehat{Z}(K;(X,A)) = \cup_{wt(\sigma) \leq t} \widehat{D}(\sigma;(X,A)).$$
Record this information stated as the next lemma.
\[lem: filtration of a simplicial complex\] There is a total ordering of all of the faces of a simplicial complex $K$ given by
1. the left-lexicographical ordering of all of the faces of $\Delta[m-1]$, and
2. the induced ordering via the natural inclusion $$K \subset \Delta[m-1].$$
Furthermore, inclusions $$L \subset K$$ induced by an embedding of simplicial complexes with $m$ vertices is order preserving, and filtration preserving where $F_tK = K \cap F_t\Delta[m-1]$ as listed in Definition \[defin: filtration of polyhedral products\]. Namely, the inclusion $L \subset K$ induced by an embedding of simplicial complexes with $m$ vertices is a morphism of filtered complexes $$F_*L \subset F_*K.$$
This filtration of $K$ induces a filtration of the polyhedral product $Z(K;X,A))$ and polyhedral smash product $\widehat{Z}(K;(X,A))$ given by
1. $$F_tZ(K;(X,A)) = \cup_{wt(\sigma) \leq t} D(\sigma;(X,A)),$$ and
2. $$F_t\widehat{Z}(K;(X,A)) = \cup_{wt(\sigma) \leq t} \widehat{D}(\sigma;(X,A)).$$
Furthermore, the natural quotient map $Z(K;X,A)) \to \widehat{Z}(K;(X,A))$ is filtration preserving.
The filtration constructed in Lemma \[lem: filtration of a simplicial complex\] is exploited in [@cartan].
\[defin: notation for spectral sequence\]
1. If $\sigma \in K$, write $$(\underline{X}/\underline{A})^{\sigma}$$ for the smash product $(X_{i_1}/A_{i_i})\wedge \cdots \wedge (X_{i_q}/A_{i_q})$ where $I=(i_1, \cdots , i_q)$ is as in Definition \[defin:smash.products\] and $\sigma$ has vertex set $I$.
2. Write $$\underline{A}^{\sigma^c}$$ for the product $ A_{j_1} \times \cdots \times A_{j_{k-q}} $ where $\sigma \cup \{j_{1}, \cdots, j_{k-q}\}=[m]$, and $\sigma^c $ denotes the complement of $\sigma$. In particular for $\sigma =\emptyset, A^{\sigma^c}=A_1 \times \cdots \times A_k$.
Half-smash products are basic in this setting with their definition as follows.
\[defin: half smash products\] Let $$(X, x_0) \ \mbox{and} \ (Y,y_0)$$ denote pointed spaces. Define $$X\rtimes Y = (X \times Y)/(x_0 \times Y),$$ and $$X \ltimes Y = (X \times Y)/ (X \times y_0).$$
An example is given next.
\[exm:filtrations\] Let $K$ denote the simplicial complex with two vertices $\{1,2\}$ and with one edge $(1,2)$. Then $Z(K; (\underline{X},\underline{A})) = X_1 \times X_2.$ The filtration of $X_1 \times X_2$ given in Definition \[defin: lexicographical ordering\] is stated next.
1. $F_0Z(K; (\underline{X},\underline{A})) = A_1 \times A_2$,
2. $F_1Z(K; (\underline{X},\underline{A})) = X_1 \times A_2$,
3. $F_2Z(K; (\underline{X},\underline{A})) = (X_1 \times A_2) \cup (A_1\times X_2)$, and
4. $F_3Z(K; (\underline{X},\underline{A})) = X_1 \times X_2$.
Let $$F_i = F_iZ(K; (\underline{X},\underline{A}))$$ in this example. If $(X_i,A_i)$ are pairs of finite CW-complexes, there are homeomorphisms
1. $F_1/F_0 \to (X_1/A_1 \times A_2)/ (* \times A_2) \to X_1/A_1 \rtimes A_2$,
2. $F_2/F_1 = (X_1 \times A_2) \cup (A_1\times X_2)/ (X_1 \times A_2 ) \to
A_1 \ltimes( X_2/A_2),$ and
3. $F_3/F_2 = X_1 \times X_2/(X_1\times A_2 \cup A_1\times X_2 \to (X_1/A_1)\wedge (X_2/A_2)$.
Letting $[x]$ denote image of the projection of $x \in X_1$ to $X_1/A_1$ then the homeomorphism in (2) is given by $$(X_1 \times A_2) \cup (A_1 \times X_2)/(X_1 \times A_2) \simeq (A_1 \times X_2)/(A_1 \times A_2) \overset{p}{\to} A_1 \ltimes( X_2/A_2)$$ with $p(a \times x)=a \times [x]$
The homeomorphism in (3) is a special case of Lemma \[lem: cofibrations\]
The filtrations and their associated graded for the smash polyhedral products are exhibited next.
1. $F_0\widehat{Z}(K; (\underline{X},\underline{A})) = A_1 \wedge A_2$,
2. $F_1\widehat{Z}(K; (\underline{X},\underline{A})) = X_1 \wedge A_2$,
3. $F_2\widehat{Z}(K; (\underline{X},\underline{A})) = (X_1 \wedge A_2) \cup (A_1\wedge X_2)$, and
4. $F_3\widehat{Z}(K; (\underline{X},\underline{A})) = X_1 \wedge X_2$.
Let $$F_i = F_i\widehat{Z}(K; (\underline{X},\underline{A}))$$ in this example where $(X_i,A_i)$ are assumed to be pairs of finite CW-complexes. There are homeomorphisms
1. $F_1/F_0 \to (X_1/A_1) \wedge A_2,$
2. $F_2/F_1 = (X_1 \wedge A_2) \cup (A_1\wedge X_2)/ X_1 \wedge A_2 \to A_1 \wedge (X_2/A_2),$ and
3. $F_3/F_2 = X_1 \times X_2/(X_1\times A_2 \cup A_1\times X_2 \to (X_1/A_1)\wedge (X_2/A_2)$.
Given a filtered space, there is a natural spectral sequence associated to that filtration. The next theorem records the properties of the resulting spectral sequence of a filtered space in the context of polyhedral products with the left-lexicographical ordering obtained from Definition \[defin: lexicographical ordering\].
\[thm:spectral sequence\] The left-lexicographical ordering of simplices induces spectral sequences of a filtered spaces
1. $E_r(K; \xa) \Rightarrow \widetilde{h}^*(\zk) $ with $$E_1(K; \xa) = \underset{\sigma \in K}{\bigoplus} \widetilde{h}^*( (\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c} ),$$
and a spectral sequence
2. $E_r(\widehat{K}; \xa) \Rightarrow \widetilde{h}^*(\zkh) $
$$E_1^{s,t}(\widehat{Z}(K; (\underline{X}, \underline{A}))) = \underset{\sigma \in K}{\bigoplus} \widetilde{h}^*( (\underline{X}/\underline{A})^{\sigma}) \wedge \widehat{\underline{A}}^{\sigma^c} ).$$
The notation defined in \[defin: notation for spectral sequence\]. The grading, $s$ is the index of the simplex, $\sigma$ in the left-lexicographical ordering. $t$ is the cohomological degree.
The differentials satisfy $$d_r: E_r^{s,t} \to E_r^{s+r,t+1}.$$
Furthermore, the spectral sequence is natural for embeddings of simplicial maps, $L \subset K$ with the same number of vertices and with respect to maps of pointed pairs $\xa \to (\underline{Y},\underline{B})$.
The natural quotient map $$Z(K; (\underline{X}, \underline{A})) \to \widehat{Z}(K; (\underline{X}, \underline{A}))$$ induces a morphism of spectral sequences, and the stable decomposition of Theorem \[thm:decompositions.for.general.moment.angle.complexes\] induces a morphism of spectral sequences.
We remark that $\zk$ and the spectral sequence commutes with colimits in $\xa$.
We also note that we only use the fact that the inclusions $F_t \subset F_{t+1}$ are cofibrations. This follows from the hypothesis that that $(X_i,A_i)$ are finite CW pairs. The argument generalizes to NDR pairs.
Some consequences of this spectral sequence are worked out below. An explicit description of the cohomology of $Z(K; (\underline{X},\underline{A}))$ with field coefficients $\mathbb F$ will be given next followed by a section on examples. The answers for cohomology are given in terms of kernels and cokernels of $$H^i(X_j) \to H^i(A_j).$$
\[defin:SR.ideals.again\]
Assume that $K$ is a simplicial complex and the pointed pairs $(\underline{X},\underline{A},\underline{*})$ are of finite type. Assume that the maps $A_i \to X_i$ induce split surjections in cohomology with field coefficients $\mathbb F$. Consider the kernel of $H^i(X_j) \to H^i(A_j)$ together with the elements $x_j \in \mbox{kernel}(H^i(X_j) \to H^i(A_j))$ together with the two-sided ideal generated by all such $x_{i_1}\otimes x_{i_2} \otimes \cdots \otimes x_{i_t}$ with $(i_1, i_2, \cdots, i_t)$ not a simplex in $K$, denoted $$SR(K; (\underline{X},\underline{A})).$$
A result which is analogous to Theorem $2.35$ of [@bbcg] follows next.
\[thm:Cartan\_and\_split.monomorphisms\] Let $K$ be an abstract simplicial complex with $m$ vertices. Assume that $$(\underline{X},\underline{A},\underline{*})$$ are pointed triples of connected CW-complexes of finite type for all $i$ for which cohomology is taken with field coefficients $\mathbb F$. If the maps $A_i \to X_i$ induce split surjections in cohomology, then the induced map $$H^*(X^{[m]}) \to H^*(Z(K; (\underline{X},\underline{A})))$$ is an epimorphism of algebras which is additively split. Furthermore, there is an induced isomorphism of algebras $$H^*(X^{[m]})/SR(K; (\underline{X},\underline{A})) \to H^*(Z(K; (\underline{X},\underline{A}))).$$
Recall that a map $A_i \to X_i$ induces a split monomorphism in integer homology if and only if it induces a split monomorphism with field coefficients for every prime field $\mathbb F_p$ and the rational numbers. A corollary of Theorem \[thm:Cartan\_and\_split.monomorphisms\] which follows immediately is stated next.
\[thm:Cartan\_and\_split.monomorphisms.INTEGRALLY\] Let $K$ be an abstract simplicial complex with $m$ vertices. Assume that $$(\underline{X},\underline{A}, \underline{*})$$ are pointed triples of connected CW-complexes of finite type for all $i$ for which cohomology is taken with coefficients $\mathbb Z$. Assume that the maps $A_i \to X_i$ induce split monomorphisms in homology over $\mathbb Z$, then the induced map $$H^*(X^{[m]};\mathbb Z) \to H^*(Z(K; (\underline{X},\underline{A})); \mathbb Z)$$ is an epimorphism of algebras which is additively split.
The proof of Theorem \[thm:Cartan\_and\_split.monomorphisms\] works just as well for any multiplicative cohomology $h^*$ and CW pairs with $h^*(X_i)$ and $h^*(A_i)$ finitely generated free $h^*$ modules.
**A spectral sequence for the cohomology of $Z(K,\xa)$ and $\zkh$** {#sec:spectral sequence one}
===================================================================
The object of this section is to construct the spectral sequences of Theorem \[thm:spectral sequence\]. In subsequent sections these spectral sequences will be used to compute the cohomology of $Z(K; \xa)$ when $\xa$ satisfies suitable flatness conditions. The spectral sequences of Theorem \[thm:spectral sequence\] are precisely those obtained by filtering the spaces $Z(K,\xa)$ and $\widehat{Z}(K; (\underline{X}, \underline{A}))$ by finite filtrations induced by the left-lexicographical ordering. Since these spectral sequences arise by finite filtrations, the spectral sequences converge in the strong sense. It remains to identify the associated graded $E_0$ as well as $E_1$, and the first differential.
The next three lemmas give the identification of $E_0$.
Suppose that $(X, x_0) \ \mbox{and} \ (Y,y_0)$ are both pointed CW complex, then the right and left half-smash products were defined in Definition \[defin: half smash products\] by $$X\rtimes Y = (X \times Y)/(x_0 \times Y), \ \mbox{and} \
X \ltimes Y = (X \times Y)/ (X \times y_0).$$ A useful lemma follows in which $X_{+}$ denotes $X$ with a disjoint base-point added.
\[lem: cofibrations.two\]
Let $$(X, x_0), \ \mbox{and} \ (Y,y_0)$$ be pointed, finite CW pairs. Then there are homotopy equivalences
1. $\Sigma(X \rtimes Y ) \to \Sigma(X \wedge Y) \vee \Sigma(X)$,
2. $\Sigma(X \ltimes Y ) \to \Sigma(X \wedge Y) \vee \Sigma(Y)$, and
3. $X \ltimes Y = (X \times Y)/ (X \times y_0) \to X_{+} \wedge Y.$
\[lem: cofibrations\]
Let $$(X_i, A_i)$$ be finite CW pairs. Let $\Lambda$ denote the subspace of $$X_1 \times X_2 \times \cdots \times X_n$$ given by $$\Lambda = \cup_{1 \leq i \leq m} X_1 \times X_2 \times \cdots \times X_{i-1} \times A_i \times X_{i+1} \times \cdots \times X_m.$$ There is a natural homeomorphism $$\theta: X_1 \times X_2 \times \cdots X_m/\Lambda \to (X_1/A_1) \wedge (X_2/A_2) \wedge \cdots \wedge (X_m/A_m).$$
Let $[x_i]$ denote the image of $x_i \in X_i$ of the projection $X_i \to X_i/A_i.$ The natural map $$\phi: X_1 \times X_2 \times \cdots X_m\to (X_1/A_1) \wedge (X_2/A_2) \wedge \cdots \wedge (X_m/A_m)$$ which send $(x_1,\cdots,x_m) $ to $([x_1], \cdots, [x_m])$ is a continuous surjection. The class $(x_1,\cdots,x_m) $ maps to the base point in $(X_1/A_1) \wedge (X_2/A_2) \wedge \cdots \wedge (X_m/A_m)$ if and only if at least one of the factors, $x_i$ maps to the base point in $X_i/A_i$. Equivalently at least one of the factors $ x_i \in A_i$. In particular $\phi$ factors through a map, $\theta$. By construction $\theta$ is a bijection. The lemmas follow for compact CW complexes with $A_i$ closed in $X_i$, since the target space is Hausdorff while the domain is compact. Thus the natural map is a homeomorphism.
The lemmas extends to locally finite CW complexes by taking a limit over finite skeleta.
Another useful lemma follows.
\[lem: half.smashcofibrations\]
Let $$(Y_i, A_i)$$ be finite, pointed CW pairs. Then there is a homeomorphism $$(Y_1 \times Y_2 )/(A_1 \times Y_2) \to (Y_1/A_1) \rtimes (Y_2).$$
Thus there are homeomorphisms
$$\begin{gathered}
(Y_1 \times Y_2 \times \cdots \times Y_n )/(A_1 \times Y_2 \times \cdots \times Y_n ) \to \\
(Y_1/A_1) \rtimes (Y_2 \times \cdots \times Y_n ) \to (\cdots (Y_1/A_1) \rtimes (Y_2)) \rtimes Y_3) \cdots \rtimes Y_n ).\end{gathered}$$
The natural quotient map is a continuous bijection by inspection. Since all spaces are finite complexes with $A_i$ closed in $Y_i$, the target space is Hausdorff while the domain is compact. Thus the natural map is a homeomorphism.
In the next lemma, abbreviate $F_sZ(K;(X,A))$ by $F_sZ$, and $F_s\widehat{Z}(K;(X,A))$ by $F_s\widehat{Z}$. Let $(X_i,A_i), i= 1,\cdots, m$ be finite, pointed CW pairs. The filtrations of Definition \[defin: filtration of polyhedral products\] are given by
1. $$F_tZ(K;(X,A)) = F_tZ = \cup_{wt(\sigma) \leq t} D(\sigma;(X,A)),$$ and
2. $$F_t\widehat{Z}(K;(X,A)) = F_t\widehat{Z} = \cup_{wt(\sigma) \leq t} \widehat{D}(\sigma;(X,A))$$
have the property that the natural quotient map $$Z(K;X,A)) \to \widehat{Z}(K;(X,A))$$ is filtration preserving, and are natural for morphisms of simplicial complexes $$L \to K$$ which induce an isomorphism of sets on the vertices.
The next step is to identify the filtration quotients $F_sZ/ F_{s-1}Z$ as well as $F_s\widehat{Z}/F_{s-1}\widehat{Z}.$
\[lem:associated graded\]
Let $(X_i,A_i), i= 1,\cdots, m$ be finite, pointed CW pairs. The filtrations of Definition \[defin: filtration of polyhedral products\] have the following property. If $\sigma$ is the maximal simplex occurring in $$F_tZ(K;(X,A)) = F_tZ = \cup_{wt(\sigma) \leq t} D(\sigma;(X,A)),$$ then the natural quotient maps $$F_tZ/ F_{t-1}Z \to
(\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c}\; \; \text{and}\;\;\;
F_t\widehat{Z}/F_{t-1}\widehat{Z} \to
(\underline{X}/\underline{A})^{\sigma} \wedge \widehat{\underline{A}^{\sigma^c}}$$ are homeomorphisms.
Suppose $F_sK$ is obtained from $F_{s-1}K$ by attaching an $n$-simplex $\Delta$. For simplicity we may, after relabelling, assume $\Delta$ has vertices $\{1,2 \cdots, n+1\}$.
$\Delta$ and its boundary, $\partial \Delta$ can be viewed as simplicial complexes on the vertex set $[m]$. The vertices $\{n+2, \cdots, m\}$ are not zero simplices. In the literature they are referred to as ghost vertices.
It follows from [@GT] that there is a commutative diagram of cofibrations
$$\label{diag:pushout}
\begin{array}{ccccc}
Z(\partial \Delta; \xa)&\overset{j}{ \to} & Z(\Delta; \xa) & \to &C\\
\downarrow&&\downarrow&& \Vert \\
F_{s-1}Z & \to & F_sZ & \to &C
\end{array}$$
Now observe that
1. $Z(\Delta;\xa) = X_1 \times \cdots \times X_{n+1} \times A_{n+2} \times \cdots \times A_m$.
2. $Z(\partial \Delta; \xa ) = Z(\partial \overline{\Delta} ; \xa) \times A_{n+2} \times \cdots \times A_m$ where $\partial \overline{\Delta}$ is a simplicial complex on $[n+1]$. While $\partial \Delta$ is a simplicial complex on $[m]$ with the set of $0-$simplices $=\{1,\cdots,n+1\}$ and ghost vertices $=\{n+2, \cdots m\}$.
3. If $\partial \overline{\Delta} \subset [n+1]$ then
$$Z(\partial \overline{\Delta}; \xa) = \underset{q}{\bigcup} X_1 \times \cdots \times {{\mathcal A}}_q \times \cdots X_{n+1}.$$
4. By lemma \[lem: cofibrations\], $$X_1 \times \cdots \times X_{n+1}/Z(\partial \overline{\Delta}; \xa) = (X_1/A_1) \wedge \cdots \wedge (X_{n+1}/A_{n+1}).$$
5. It follows from Lemma \[lem: half.smashcofibrations\] that for $A \subset X$ a CW pair the cofiber of $A \times Y \hookrightarrow X \times Y$ is the right half smash $(X/A) \rtimes Y$.
The following description of $C$ follows from the top row of diagram \[diag:pushout\] and these observations
$$C = F_sZ/F_{s-1}Z= (X_1/A_1) \wedge \cdots \wedge (X_{n+1}/A_{n+1}) \rtimes (A_{n+2} \times \cdots \times A_m).$$
A similar argument using part (2) of Lemma \[lem: cofibrations\] shows that
\[lemma:relhatspace\] $$\widehat{F}_sZ/\widehat{F}_{s-1}Z = (X_1/A_1) \wedge \cdots \wedge (X_{n+1}/A_{n+1}) \wedge (A_{n+2} \wedge \cdots \wedge A_m) = \widehat{D}(\sigma; (\underline{X/A},\underline{A}).$$
This completes the description of the $E_1$ page and the proof of \[thm:spectral sequence\].
\[cor: filtration of a simplicial complex\] Let $(X_i,A_i), i= 1,\cdots, m$ be finite CW pairs. The filtrations of Definition \[defin: filtration of polyhedral products\] given by
1. $$F_tZ(K;(X,A)) = F_tZ = \cup_{wt(\sigma) \leq t} D(\sigma;(X,A)),$$ and
2. $$F_t\widehat{Z}(K;(X,A)) = F_t\widehat{Z} = \cup_{wt(\sigma) \leq t} \widehat{D}(\sigma;(X,A))$$
have the property that the natural quotient map $$Z(K;X,A)) \to \widehat{Z}(K;(X,A))$$ is filtration preserving, and are natural for morphisms of simplicial complexes $$L \to K$$ which induce an isomorphism of sets when restricted to the vertices.
Then there is a spectral sequence abutting to $h^*(Z(K;(X,A)))$ obtained from these filtrations for which the $E^{s,t}_0$-term is specified by
The $E^{s,t}_1$-term is specified by
1. $E^{s,t}_1Z(K;(X,A)) = h^{t}(F_sZ,F_{s-1}Z)= h^{t}(\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c})$, and
2. $E^{s,t}_1\widehat{Z}(K;(X,A)) = h^{t}(F_s\widehat{Z},F_{s-1}\widehat{Z})= h^{t}( (\underline{X}/\underline{A})^{\sigma} \wedge \widehat{\underline{A}^{\sigma^c}})$.
This spectral sequence has the following properties.
1. The spectral sequence is natural for embeddings of simplicial complexes $$L \subset K$$ with the same number of vertices.
2. The spectral sequence is natural for morphisms of simplicial complexes $$L \to K$$ which are order preserving (of the left lexicographical ordering).
3. There is a finite filtration of $h^*(\zk)$ such that $E_{\infty}$ is the associated graded group of this filtration.
An immediate application of the spectral sequence is a computation of $h^*(K; \xa)$ as a ring when $h^*(X_i) \to h^*(A_i) $ is surjective for all $i$ and a freeness condition is satisfied.
\[exm:splitting\]
Suppose $(X_i,A_i)$ is a CW pair, $h^*(X_i) \to h^*(A_i) $ is surjective for all $i$ and $h^*(A_i) $ and $h^*(X_i/A_i)$ are free $h^*$ modules. Then $$\widetilde{h}^*(\zk) =
\underset{\sigma \in K }{\bigoplus} \widetilde{h}^*( (\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c} ).$$
The hypothesis implies $\widetilde{h}^*(X_i/A_i)\to h^*(X_i)$ is injective, and by the freeness assumption, $h^*(C) \to h^*( Z(\Delta; \xa) ) $ in the following diagram is injective.
$$\begin{array}{ccccccccccc}
0\leftarrow &h^*(Z(\partial \Delta; \xa)) & \leftarrow &h^*( Z(\Delta; \xa) ) & \leftarrow & h^*(C) & \leftarrow0 \\
&\uparrow&&\uparrow&& \Vert \\
\leftarrow & h^*(Z(K_{q-1} , \xa)) & \leftarrow &h^*( Z(K_q,\xa) )&\overset{j}{ \leftarrow} & h^*(C) &\overset{\delta}{ \leftarrow}
\end{array}$$ which implies $j$ is injective. Hence $\delta=0$. This implies the differentials in the spectral sequence are zero.
This is particularly interesting in the cases where the surjectivity or the freeness conditions are satisfied for $h^*$, but not for ordinary cohomology. For example $h^*=K^*$ and $(X_i,A_i) = \newline (SO(2n+1), SO(2n))$.
Specializing $h^*$ to ordinary cohomology with coefficients in a field we can now prove theorem \[thm:Cartan\_and\_split.monomorphisms\]. The surjectivity condition of Proposition \[exm:splitting\] implies $H^*(X_i/A_i)$ is a subring of $H^*(X_i).$ Therefore $I$ in the following corollary is an ideal in $H^*(X_1) \otimes \cdots \otimes H^*(X_m).$
Assume $H^*$ is cohomology with coefficients in a field and that $(X_i,A_i)$ is a CW pair, such that $H^*(X_i) \to H^*(A_i) $ is surjective for all $i$. Then there is a ring isomorphism
$$H^*(\zk) = H^*(X_1) \otimes \cdots \otimes H^*(X_m)/I$$ where $I$ is the ideal generated by $\widetilde{H}^*(X_{j_1}/A_{j_1}) \otimes \cdots \otimes \widetilde{H}^*(X_{j_t}/A_{j_t}), $ with $(j_1,\cdots,j_t)$ not spanning a simplex in $K$.
The hypothesis implies there is a split short exact sequence $$0 \to \widetilde{H}^*(X_i/A_i) \to H^*(X_i) \to H^*(A_i) \to 0$$for all $*>0$.
After choosing a splitting we may write $H^*(X_i) = H^*(A_i) \oplus \widetilde{H}^*(X_i/A_i)$.
The tensor product $H^*(X_1) \otimes \cdots \otimes H^*(X_m)$ may now be written as a sum of terms of the form $H^*(Y_1) \otimes \cdots \otimes H^*(Y_m)$ where $Y_i$ is $A_i$ or $X_i/A_i$. In particular Proposition \[exm:splitting\] implies the natural map of rings $ H^*(X_1) \otimes \cdots \otimes H^*(X_m) \to H^*(\zk)$ is surjective with kernel given by the ideal $I$.
The Stanley-Reisner ring is the cohomology ring in the special case, $\xa = (CP^{\infty},\ast)$
It follows from [@Coxeter] and [@buchstaber.panov Example 6.40] that $Z(K; (D^1,S^0))$ is an orientable surface if $K$ is the boundary of an $n$-gon. The next application of the spectral sequence is to determine the genus of this surface. The genus is determined by the Euler characteristic of $H^*(Z(K; (D^1,S^0))$ which is the Euler characteristic of the $E^1$ page.
The boundary of an $n-$gon has $n$ $0$-simplicies and $n$ $1$-simplicies. To compute the Euler characteristic, the ranks of $\widetilde{H}^*( (\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c} )$ are computed.
1. For $\sigma =\emptyset$, $H^*( (\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c}) = H^*(\overbrace{S^0 \times \cdots \times S^0}^n) $ where $\overbrace{S^0 \times \cdots \times S^0}^n$ is $2^n$ distinct points. So the unreduced homology has $2^n$ $0$-dimensional classes.
2. For $\sigma $ a 0-simplex
[100pt]{} $$\begin{gathered}
(\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c} =\\
S^1 \rtimes \overbrace{S^0 \times \cdots \times S^0}^{n-1}=\\
S^1 \wedge ( 2^{n-1} \mbox{ points })_+ = S^1 \wedge (\underset{2^{n-1}}{\vee} S^0) $$\end{gathered}$$
So $\widetilde{H}^*((\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c)}$ has $ 2^{n-1}$ $1-$ dimensional classes. There are $n$ $0$-simplices so there are a total of $n(2^{n-1})$ $1$-dimensional classes.
3. If $\sigma $ is a 1-simplex, the computation is similar.
[100pt]{} $$\begin{gathered}
(\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c} =\\
S^2 \rtimes \overbrace{S^0 \times \cdots \times S^0}^{n-2}=\\
S^2 \wedge ( 2^{n-2} \mbox{ points })_+ = S^2 \wedge (\underset{2^{n-2}}{\vee} S^0)$$\end{gathered}$$
which contributes $2^{n-2}$ $2-$ dimensional classes. There are $n$ $1$-simplices. So there are a total of $n(2^{n-2})$ 2-cells.
So the Euler charateristic of $E_1$ and hence of $Z(K; (D^1,S^0))$ is $(4-n)2^{n-2}$. This proves a theorem of Coxeter, [@Coxeter], i.e. if $K$ is the boundary of an $n$-gon $Z(K; (D^1,S^0))$ is a surface of genus $1 + (n -4)2^{n-3}$.
M. Davis first computed the Euler characteristic of $Z(K;(D^1,S^0))$ [@Davis], but the analogous spectral sequence argument as given in the above example also gives the following result.
\[prop:genus\] if $K$ is a simplicial complex with $m$ vertices which has $t_n$ n-simplices then
$$\chi(Z(K;(D^1,S^0)) = \Sigma (-1)^{n+1} t_n 2^{m-n-1}.$$ (The empty simplex is considered to be a $(-1)$-simplex, $t_{-1}=1$).
**Computing the differentials.** {#sec:spectral sequence}
================================
For CW pairs satisfying a freeness condition, the differentials in the spectral sequence are shown to be determined by the coboundaries of the long exact sequences of the pairs $(X_i,A_i)$. This result is used to compute the generalized cohomology of $\zkh$.
We recall the description of the $E_1$ page of the spectral sequences constructed in section \[sec:spectral sequence one\].
$$\label{thm:h*spectral sequence}$$
1. $E_r(\zk) \Rightarrow \widetilde{h}^*(\zk) $ with
$$E_1(\zk) = \underset{\sigma \in K}{\bigoplus} \widetilde{h}^*( ( \underline{X}/\underline{A})^{\sigma}
\rtimes ( A^{\sigma^c})).$$
2. $E_r(\zkh) \Rightarrow \widetilde{h}^*(\zkh) $ with
$$E_1(\zkh) = \underset{\sigma \in K}{\bigoplus} \widetilde{h}^*( ( \underline{X}/\underline{A})^{\sigma}
\wedge ( \widehat{A}^{\sigma^c}).$$
We next compute the differentials in the spectral sequence. In order to do so the strong freeness assumption, Definition \[def:bas\], on the pairs $(X_i,A_i)$ is imposed.
Using the Künneth theorem the $E_1$ page of the spectral spectral sequence converging to $\widetilde{h}^*(\zkh)$ is isomorphic to a direct sum of $h^*-$ modules
$$\widetilde{h}^*( ( \underline{X} /\underline{A})) ^{\sigma}
\bigotimes \underset{ j_i \notin \sigma}\otimes (\widetilde{h}^*(A_{j_i}))$$
Each summand is a tensor product of the cohomology of $X_i/A_i$ (which is a sum of $C_i$ and $W_i$) and the cohomology of $A_i$ (which is the sum of $E_i$ and $B_i$). After expanding the tensor product the $E_1$ page becomes a sum of tensor products of $E_i, C_i, B_i$ and $W_i$. Hence any class in $ E^{s,t}_1$ is a sum of classes $$y_1 \otimes \cdots \otimes y_m$$ with $y_j \in E_i, C_i, B_i$ or $W_i$.
There are coboundary maps $$\delta_i: H^*(A_i) \supset E_i \to W_i \subset H^{*+1}( X_i/A_i).$$
\[coboundary\] There is a coboundary map, $\delta$ defined on the $h^*-$generators $y_1 \otimes \cdots \otimes y_m$ of $$\widetilde{h}^*( ( \underline{X} /\underline{A})) ^{\sigma}
\bigotimes \underset{ j_i \notin \sigma}\otimes (\widetilde{h}^*(A_{j_i}))$$ by the coboundary maps $\delta_i$ and the graded Leibniz rule.
A monomial, $$y_1 \otimes \cdots \otimes y_m \in \widetilde{h}^*( ( \underline{X} /\underline{A})) ^{\sigma}
\bigotimes \underset{ j_i \notin \sigma}\otimes (\widetilde{h}^*(A_{j_i}))$$ defines a simplex, $\sigma(y_1 \otimes \cdots \otimes y_m)$ as follows. There are two indexing sets determined by $ y_1 \otimes \cdots \otimes y_m$.
$$I_1 = \{i \in [m] |\quad y_i \in C_i\}$$ $$I_2 =\{ i \in [m] |\quad y_i \in W_i\}$$
The $h^*$-modules $C_i$ and $W_i$ are summands of $\widetilde{h}^*(X_i/A_i)$. The $E_1$ page of the spectral sequence is a sum of terms with factors of $\widetilde{h}^*(X_i/A_i)$ indexed by the simplices of $K$. It follows that $\sigma(y_1 \otimes \cdots \otimes y_m) = I_1 \cup I_2$ is a simplex of $K$.
The weight of $\sigma(y_1 \otimes \cdots \otimes y_m)$ in the left-lexicographical ordering of the simplices of $K$ is the filtration of $y_1 \otimes \cdots \otimes y_m$ in the spectral sequence. The simplex $\sigma(y_1 \otimes \cdots \otimes y_m)$ is called the support of $y_1 \otimes \cdots \otimes y_m$.
\[diff\]
Asssume $(\underline{X},\underline{A})$ satisfies the strong freeness condition in definition \[def:bas\], then
$$y = \underset{\ell}{\Sigma} \quad y_{1_{\ell}} \cdots y_{m_{\ell}}$$ survives to $E^{s,*}_r$ if
1. $\sigma( y_{1_{\ell}} \cdots y_{m_{\ell}})$ has weight $s$ for all $\ell$
and, with $\delta$ as in Definition \[coboundary\]
2. $$\delta(y) = \underset{t}{\Sigma} \quad
\overline{y}_{1_{t}} \cdots \overline{y}_{m_{t}}$$ where $$\mbox{ weight } \sigma(\overline{y}_{1_{t}} \cdots \overline{y}_{m_{t}})\geq s+r$$ for all $t$.
Then $$d_r(y) = \underset{wt(\sigma(\overline{y}_{1_{t}} \cdots \overline{y}_{m_{t}}))=s+r}{\Sigma} \overline{y}_{1_{t}} \cdots \overline{y}_{m_{t}}$$
\[ssd2\] We illustrate Theorem \[diff\] for $h^*=H^*$, $K$ a simplicial complex on $3$ points, i.e. $K=$ $3$ distinct points, $K=$ an edge and a disjoint point, $K=$ two edges meeting at a common vertex, $K=$ the boundary of the $2$ simplex and finally $K=$ the $2$ simplex. $\xa =(D^1,S^0)$. There are the generators, $e_0 \in \widetilde{H}^0(S^0) $ and $w_1 \in \widetilde{H}^1(S^1)$.
We first build up the spectral sequence for three distinct points then add one edge at a time followed by the two simplex.
- Three distinct points.
$$\begin{array}{ccccc}
\mbox {filtration}& 0 &1&2&3\\
& e_0 \otimes e_0 \otimes e_0& w_1 \otimes e_0 \otimes e_0 & e_0 \otimes w_1 \otimes e_0 & e_0 \otimes e_0\otimes w_1
\end{array}$$
\[diff\] implies there is a differential from filtration $0$ to filtration $1$.
- Add one edge.
$$\begin{array}{ccccccccc}
\mbox {filtration}& 0& &1&&2&3&4\\
& 0&& 0 && e_0 \otimes w_1 \otimes e_0 & e_0 \otimes e_0\otimes w_1&w_1 \otimes w_1 \otimes e_0
\end{array}$$ \[diff\] implies a differential from filtration 2 to filtration 4.
- Add another edge
$$\begin{array}{cccccccc}
\mbox {filtration}& 0 &1&2&3&4&5\\
& 0& 0 & 0& e_0 \otimes e_0\otimes w_1&0& w_1 \otimes e_0 \otimes w_1
\end{array}$$ \[diff\] implies a differential from filtration 3 to 5.
- Add another edge to form $\partial \Delta^2$
$$\begin{array}{cccccccc}
\mbox {filtration}& 0 &1&2&3&4&5&6 \\
& 0& 0 & 0& 0 &0& 0 &e_0 \otimes w_1 \otimes w_1
\end{array}$$
- Add $\Delta^2$
$$\begin{array}{ccccccccc}
\mbox {filtration}& 0 &1&2&3&4&5&6 &7\\
& 0& 0 & 0& 0 &0& 0 &e_0 \otimes w_1 \otimes w_1 & w_1 \otimes w_1 \otimes w_1
\end{array}$$ There is a differential from filtration 6 to filtration 7.
The filtration in the spectral sequence is induced by the\
left-lexicographical order of the simplices which are added one at a time. The weight of a simplex is its position in this order. In particular the empty simplex has index $0$. The proof is an induction on the index of the simplex being added. In the proof, $\widetilde{h}^*(F_s \widehat{K})$ is written $F_s$. Inductively assume $F_{s-1}$ is given by Theorem \[diff\]. The induction starts with $F_0 = \widetilde{h}^*(A_1) \otimes \cdots \otimes \widetilde{h}^*(A_m)$.
Recall the definition of the differentials in the spectral sequence. The spectral sequence is induced by the exact couple
(1.1,2)node\[above \][$F_t$]{} –( 0,-2)node\[below \] [$\widetilde{h}^*(\widehat{D}(\sigma_t; (\underline{X}/\underline{A},\underline{A}))$]{} node\[pos=0.4 ,right\][$\widetilde{h}^*(j)$]{} node\[pos=1.05\][$\bullet$]{} node\[pos=-0.01\][$\bullet$]{}; (-2.9,2.0)node\[above\][$F_{t-1}$]{} – (.8,2) node\[pos=0.5, left, above\][$\widetilde{h}^*(i)$]{} node\[pos=-.025\][$\bullet$]{}; (-3,2)–(-.1,-2)node\[pos=0.4, left\][$\delta$]{};
Where the maps are induced by the inclusions, $i: F_{t-1} \widehat{K} \to F_t\widehat{K}$, and the projections $j:F_t \widehat{K} \to F_t \widehat{K}/F_{t-1}\widehat{K} = \widehat{D}(\sigma_t; (\underline{X}/\underline{A},\underline{A}))$ where $\widehat{D}(\sigma_t; (\underline{X}/\underline{A},\underline{A}))$ is defined in Lemma \[lemma:relhatspace\]. $\delta $ denotes the coboundary map.
The differential on a class $\alpha \in \widetilde{h}(\widehat{D}(\sigma_t; (\underline{X}/\underline{A},\underline{A})))$ which has survived to $E_{s-t}$ is defined by pulling $\widetilde{h}^*(j)(\alpha)$ back to $\beta \in F_{s-1}$ followed by the coboundary. i.e. $ d_{s-t}(\alpha)= \gamma = \delta(\beta)$
(1,2)node\[above \][$F_t$]{} –( 0,-2)node\[below left\] [$\alpha \in \widetilde{h}^*(\widehat{D}(\sigma_t; (\underline{X}/\underline{A},\underline{A})))$]{} node\[pos=0.4 ,left\][$\widetilde{h}^*(j)$]{} node\[pos=1.05\][$\bullet$]{} node\[pos=-0.01\][$\bullet$]{};
(1.2,2)– (5.9,2)node\[pos=0.4 , left, above \][$\widetilde{h}^*(i)$]{};
(6,1.9)node\[above \][$\beta \in F_{s-1}$]{}– ( 5,-2)node\[below \][$\gamma \in h^{*+1}(\widehat{D}(\sigma_{s}; (\underline{X}/\underline{A},\underline{A})))$]{} node\[pos=0.4, left\][$\delta$]{} node\[pos=1.05\][$\bullet$]{} node\[pos=-0.024\][$\bullet$]{};
The diagram defining the differential fits into a larger diagram.
(1,2)node\[above \][$F_t$]{} –( 0,-2)node\[below left\] [$\alpha \in \widetilde{h}^*(\widehat{D}(\sigma_t; (\underline{X}/\underline{A},\underline{A})))$]{} node\[pos=0.4 ,left\][$\widetilde{h}^*(j)$]{} node\[pos=1.05\][$\bullet$]{} node\[pos=-0.01\][$\bullet$]{}; (-3,2)node\[above\][$F_{t-1}$]{} – (.8,2) node\[pos=0.5, left, above\][$\widetilde{h}^*(i)$]{} node\[pos=-.025\][$\bullet$]{}; (1.2,2)– (6,2)node\[pos=0.4 , left, above \][$\widetilde{h}^*(i)$]{}; (6.2,2) – (10, 2)node\[above \][$F_{s}$]{} node\[pos=1.025\][$\bullet$]{}; (6,1.9)node\[above \][$\beta \in F_{s-1}$]{}– ( 5,-2)node\[below \][$h^{*+1}(\widehat{D}(\sigma_{s}; (\underline{X}/\underline{A},\underline{A})))$]{} node\[pos=0.4, left\][$\delta$]{} node\[pos=1.05\][$\bullet$]{} node\[pos=-0.024\][$\bullet$]{}; (6, 2.7) – (6, 5)node\[pos=0.4, left\][$\theta$]{} node\[above\][$\widetilde{h}^*(\widehat{Z}(\partial \Delta; \xa))$]{} node\[pos=1.024\][$\bullet$]{};
(6.2, 5)– (9.9, 5)node\[above\][$\widetilde{h}^*(\widehat{Z}(\Delta; \xa))$]{} node\[pos=1.025\][$\bullet$]{}; (10,4.8)–(10,2.5);
(5.8,5) .. controls +(left:2cm) .. (4.8,-2.03) node\[pos=.9, left \][$\widetilde{\delta}$]{};
Where the right square is induced by the pushout diagram in Lemma \[lem:associated graded\] and $\widetilde{\delta}$ is the coboundary map associated to the cofibration $$\widehat{Z}(\partial \Delta;(\underline{X},\underline{A})) \to \widehat{Z}( \Delta;(\underline{X},\underline{A})) \to \widehat{D}(\sigma_{s}; (\underline{X}/\underline{A},\underline{A})).$$
By induction $\beta$ is a sum of terms of the form $y_1 \otimes \cdots \otimes y_m$. We write $(y_1 \otimes \cdots \otimes y_m)_{\ell}$ for the terms that appear in $\beta$
$$\beta = \underset{\ell}{\Sigma} (y_1 \otimes \cdots \otimes y_m)_{\ell}.$$ The differential of $\alpha$ is given by $$d(\alpha) = \widetilde{\delta}(\theta(\underset{\ell}{\Sigma} (y_1 \otimes \cdots \otimes y_m)_{\ell}))$$ where $\widetilde{\delta}$ is given by the formula in part (3) of Theorem \[diff\].
The following Lemma \[lem: strongly isomorphic homology and homology of smash products\], motivates the definition of [*strongly isomorphic $h^*-$homology*]{}, which was also defined in [@cartan].
\[defin:maps between infinite symmetric products\]
The pairs $(\underline{Y},\underline{B})$ and $(\underline{X},\underline{A})$ are said to have [**strongly isomorphic $h^*-$cohomology**]{} provided
1. there are $h^*-$isomorphisms $$\beta_j: h^*(Y_j) \to h^*(X_j),$$ $$\alpha_j: h^*(B_j) \to h^*(A_j),$$ and $$\gamma_j:h^*(Y_j/B_j)\to h^*(X_j/A_j)$$
2. there is an commutative diagram of exact sequences
$$\begin{CD}
@>{}>> \bar{H}_i(B_j ) @>{{\lambda_j}_*}>> \bar{H}_i(Y_j ) @>{}>> H_*(Y_j/B_j)@>{\delta}>> \\
&& @VV{\alpha_j}V @VV{\beta_j}V @VV{\gamma_j}V \\
@>{}>> \bar{H}_i(A_j) @>{{\iota_j}*}>> \bar{H}_i(X_j) @>{}>> H_*(X_j/A_j)) @>{\delta}>>
\end{CD}$$.
where $\lambda_j: B_j \subset Y_j$, and $\iota_j: A_j \subset X_j$ are the natural inclusions.
3. The maps of triples $$(\alpha_j,\beta_j,\gamma_j): (H_*(Y_j), H_*(B_j), H_*(Y_j/B_j) ) \to (H_*(X_j), H_*(A_j), H_*(X_j/A_j)$$ which satisfy conditions $1-2$ are said to [**induce a strong homology isomorphism**]{}.
We will sometimes say that pairs of $h^*$ modules with maps satisfying conditions $1-3$ are strongly $h^*$ isomorphic without reference to any spaces. The next corollary is an immediate consequence of \[diff\].
\[cor:specSeqstrong isomorphism\] Assume $(X_i,A_i)$ are CW pairs for all $i$ which satisfy the freeness conditions of \[diff\]. Then the spectral sequence $E_r(\zk) \Rightarrow \widetilde{h}^*(\zk) $ depends only on the strong $h^*$ cohomology isomorphism type of the pairs, $(X_i,A_i)$.
Specifically the filtration, differentials and extensions depend only on $K$ and the $h^*$ cohomology isomorphism type of the pairs, $(X_i,A_i)$.
A straightforward, recursive application of the splitting for the right smash, Lemma \[lem: cofibrations.two\], shows that there is a homotopy equivalence
$$\label{splittingofSS}\Sigma ( \underset{I \subset [m]}{\bigvee} [ \underset{\sigma \in K_I}{\bigvee} ( \underline{X}/\underline{A})^{\sigma}
\wedge (\widehat{A}^{I-|\sigma|})] )\to
\Sigma( \underset{\sigma \in K}{\bigvee} ( \underline{X}/\underline{A})^{\sigma}
\rtimes ( A^{\sigma^c})).$$
This implies a splitting of $E_1(\zk)$ into a sum over $I$ of $E_1(\widehat{Z}(K_I; \xa))$. Hence Theorem \[thm:decompositions.for.general.moment.angle.complexes\] appears at the level of the $E_1$ pages of the spectral sequences in equation . The differentials respect the splitting. The next corollary records this.
\[cor:SSsplitting\] Assume $(X_i,A_i)$ are CW pairs which satisfy the freeness conditions of \[diff\]. Then $$E_r(Z(K; \xa)) = \underset{I \subset [m]}{\bigoplus}E_r(\widehat{Z}(K_I; \xa_I)).$$
The following example illustrates the main ideas in the proof. The notation is as in (\[def:bas\]).
$$\label{ideaProof}$$
- The splitting at $E_1$ is a consequence of the presence of the classes $1\in B_i$ defined in \[def:bas\] which appears in $E_r(Z(K; \xa)) $ but not in $E_r(\widehat{Z}(K_I; \xa_I))$.
- The differentials only involve $E_i$ and $W_i$, hence the spectral sequences are modules over $B_i$ and $C_i$.
Let $h^*=$ ordinary cohomology, $K$ the simplicial complex of two disjoint points, $\{1,2\}$, and $X_i = D^2 \bigvee S^3$, $A_i = S^1 \subset D^2 \subset X_i$ for $i=1,2$. Since $i$ is determined by the coordinate in the subsequent tensor products, we omit it from the notation.
The modules of (\[def:bas\]) are as follows: $$E ={{\mathbb Z}}\{e_1\}\in H^1(S^1), \quad B = {{\mathbb Z}}\{1\} \in H^0(D^2 \bigvee S^3),$$$$C={{\mathbb Z}}\{c_3\} \in H^3(D^2 \bigvee S^3), \quad W={{\mathbb Z}}\{w_2\} \in H^2((D^2/S^1) \bigvee S^3).$$
For this case, the decomposition given by equation has the form
$$E_1(Z(K; \xa)) =
\begin{array}{cccccccccc}
H^*(A \times A) & \bigoplus & H^*(X/A \rtimes A) & \bigoplus & H^*(A \ltimes X/A)\\
\Vert && \Vert && \Vert\\
(e_1 \oplus 1)\otimes ( 1 \oplus e_1) & \bigoplus &(c_3 \oplus w_2) \otimes (e_1 \oplus 1) & \bigoplus & (e_1 \oplus 1) \otimes (c_3 \oplus w_2)
\end{array}$$ which equals
$$\begin{array}{llll}
(1 \otimes 1) \oplus
(e_1 \otimes 1) \oplus (1 \otimes e_1) \oplus(e_1 \oplus e_1) \\
\hspace{26 \jot} \bigoplus \\
(c_3 \otimes e_1) \oplus (c_3 \otimes 1) \oplus (w_2 \otimes e_1) \oplus (w_2 \otimes 1) \\
\hspace {26 \jot} \bigoplus\\
(e_1 \otimes c_3) \oplus (1 \otimes c_3) \oplus (e_1 \otimes w_2) \oplus (1 \otimes w_2)\\
\end{array}$$
Now arrange the summands according to the location of the unit
$$\begin{array}{lllll}
(e_1 \otimes e_1) \bigoplus (c_3\otimes e_1) \oplus (w_2 \otimes e_1) \bigoplus (e_1 \otimes c_3) \oplus (e_1 \otimes w_2)\\
(e_1 \otimes 1) \bigoplus (c_3 \otimes 1) \oplus (w_2 \otimes 1)\\
(1 \otimes e_1) \bigoplus (1 \otimes c_3) \oplus (1 \otimes w_2)\\
(1 \otimes 1)
\end{array}$$
The first line is $E_1(\widehat{Z}(K_{\{1,2\}}; \xa))$, the second line is $E_1(\widehat{Z}(K_{\{1\}}; \xa))$ and the third line is $E_1(\widehat{Z}(K_{\{2\}}; \xa))$. The last line is the unit in $H^0(Z(K; \xa) $ which appears as $E_1(\widehat{Z}(K_{\emptyset}; \xa))$. This decomposition is an example of the first observation of (\[ideaProof\]).
There is a differential in the first line from $e_1 \otimes e_1$ to $w_2 \otimes e_1$, a differential in the second line from $e_1 \otimes 1$ to $w_2 \otimes 1$ and a similar differential from $1 \otimes e_1$ to $ 1 \otimes w_2$. The differentials are zero on $1$ and $c_3$. This illustrates the second point of \[ideaProof\].
Using the units we have splitting of the spectral sequence.
$$\label{algebraicSplitting}
E_r(Z(K; \xa)) = \underset{I\subset [m]}{\bigoplus} E_r(\widehat{Z}(K_I; \xa_I)) \bigotimes 1^{[m]-I}$$
where $1^{[m]-I}$ denotes a factor of $1$ in each coordinate in the complement of $I$.
\[Proof of Corollary \[cor:SSsplitting\].\] First some equalities that were used in \[ideaProof\] and are relevant to proving \[cor:SSsplitting\] that follow from the Künneth formula are listed. We assume $X$ and $Y$ have free $h^*$ cohomology.
1. $$\widetilde{h}^*(X \times Y) =(\widetilde{h}^*(X)\otimes \widetilde{h}^*(Y)) \oplus( \widetilde{h}^*(X) \otimes 1)\oplus (1 \otimes \widetilde{h}^*(Y)).$$
This is the algebraic version of the homotopy splitting $$\Sigma (X \times Y) = \Sigma (X \wedge Y) \vee \Sigma (X) \vee \Sigma (Y).$$
2. $$\widetilde{h}^*(X \rtimes Y) =[ \widetilde{h}^*(X)\otimes \widetilde{h}^*(Y)] \oplus[ \widetilde{h}^*(X) \otimes 1] .$$
This is the algebraic version of Lemma \[lem: cofibrations.two\].
3. More generally, with $\sigma = \{1,2,\cdots, n\}$.
$h^*( (\underline{X}/\underline{A})^{\sigma} \rtimes \underline{A}^{\sigma^c})$ $$= \widetilde{h}^*( (X_1/A_1) \wedge \cdots \wedge (X_n/A_n) \bigwedge (A_{n+1} \times \cdots \times A_m)) \bigoplus \widetilde{h}^*((X_1/A_1) \wedge \cdots \wedge (X_n/A_n)) \otimes 1 \cdots \otimes 1$$ $$=\underset{ \{i_1,\cdots i_p \} \subset \{1, \cdots, n\}}{\bigoplus} \widetilde{h}^*( (X_1/A_1)) \otimes \cdots \otimes \widetilde{h}^*(X_n/A_n) \otimes \widetilde{h}^*(A_{i_1}) \otimes \cdots \otimes \widetilde{h}^*(A_{i_p})) \otimes 1 \otimes \cdots \otimes 1$$ where the units appear in the factors with coordinates in $[m]$ not in the set $\{i_1,\cdots, i_p \}$.
Summing over all simplices gives the isomorphism in Corollary \[cor:SSsplitting\].
**The Cohomology of the polyhedral product.** {#revisited}
==============================================
The freeness conditions of Section \[sec:spectral sequence\] are assumed for $\xa$ throughout this section the goal of which is to compute $h^*(\zkh)$ in terms of the strong $h^*$-cohomology isomorphism type of the pairs $(X_i,A_i)$ and the cohomology of sub-complexes of $K$. A consequence is a formula for the reduced Poincare series for $\widetilde{H}^*(\zk)$.
$K$ denotes a simplicial complex with $m$ vertices. $E_i, B_i$, $C_i$ and $W_i$ are as in Definition \[def:bas\].
Write $E_1$ for $E_1(\widehat{Z}(K; \xa).$
Recall the following from Section \[sec:spectral sequence\].
1. $E_1$ is a sum of $h^*-$modules $$\widetilde{h}^*( ( \underline{X}/\underline{A})^{\sigma}
\wedge ( \widehat{A}^{\sigma^c}))$$ which is a sum of tensor products of $E_i, B_i, C_i, W_i$.
In particular a typical summand in $E_1$ may be written in the form
$$\label{JST}
E^J \otimes W^L \otimes C^S \otimes B^T,$$
with $ J \cup L \cup S \cup T = [m]$ and $J, L, S, T$ mutually disjoint.
2. The indexing set $L \cup S$ is a simplex in $K$ (see Definition \[coboundary\]).
3. By Theorem \[diff\] the differentials in the spectral sequence are induced by the coboundary maps, $\delta_i: \widetilde{h}^*(A_i) \to \widetilde{h}^*(X_i/A_i)$. with $\delta_i$ mapping $E_i$ to $W_i$.
We next show that $E_r$ is a sum of simpler spectral sequences. To this end fix $S$ and $T$. The next definition, which is simply a reparametrization of $C^S \otimes B^T$ is for convenient bookkeeping.
Let $I \subset [m], \quad I=(i_1,\cdots , i_p)$.
Let $\sigma \subset I$ be a simplex in $K$. Define
$Y^{I,\sigma} = Y_1 \otimes \cdots \otimes Y_p \mbox{where} $
$ Y_t= \begin{cases}
C_{i_t} & \mbox{ if } i_t \in \sigma \\
B_{i_t} & \mbox{ if } i_t \notin \sigma
\end{cases}$
In the notation of (\[JST\]) $\sigma=S$ and $I = S \cup T$.
By definition, $Y^{I,\sigma}$ is fixed because $S$ and $T$ are fixed . Now consider the sum $$\Big[ \underset{\substack{J\cup L \cup I=[m] \\ L \cup \sigma \in K}}{\bigoplus}E^J\otimes W^L \Big]\otimes Y^{I,\sigma}$$ which is a sub-sum of the $E_1$ page of the spectral sequence. Since $J, L$ and $I$ are mutually disjoint, and $L \cup \sigma \in K$ it follows that $L$ is a simplex of $K$ belonging to the link of $\sigma$ in the complement of $I$, which is now defined.
\[def:N(Isigma)\] If $I \subset [m]$ and $\sigma \in K$, $\sigma \subset I$ then the link of $\sigma$ in the complement of $I$, $N(I,\sigma),$ is the simplicial complex, $N(I,\sigma)$ on vertex set $[m]-I$ such that
$$\overline{\sigma} \in N(I,\sigma) \Leftrightarrow \overline{\sigma} \cup \sigma \in K.$$
Note that
1. $N(I,\sigma)$ is indeed a simplicial complex since $\sigma^{\prime} \subset \overline{\sigma} $ implies $\sigma^{\prime} \cup \sigma \in K$ which implies $\sigma^{\prime} \in N(I,\sigma)$.
2. If $N(I,\sigma) =\emptyset$ then $\sigma \cup \{v\} \notin K$ for all $v \in [m]-I$. This implies $N(I,\sigma) =\emptyset \Leftrightarrow \sigma$ is a maximal simplex in $K$.
3. $N(I, \emptyset)= K_{[m]-I}$.
4. $N(|\sigma|,\sigma) = lk_{\sigma}(K)$, the link of $\sigma$ in $K$.
Since all differentials take place between the terms $E^J$ and $W^L$ in $E_r$ it follows that for fixed $I,\sigma$ $$\Big[ \underset{\substack{J\cup L \cup I=[m] \\ L \cup \sigma \in K}}{\bigoplus}E^J\otimes W^L \Big]\otimes Y^{I,\sigma}$$ is a sub spectral sequence of $E_r$ with all differentials taking place within the brackets. In particular
$$\Big[ \underset{\substack{J\cup L \cup I=[m] \\ L \cup \sigma \in K}}{\bigoplus}E^J\otimes W^L \Big]$$is the $E_1-$page of a spectral sequence.
This spectral sequence is next identified as a spectral sequence $$\Big[ E_r(\widehat{Z}(N(I,\sigma); (\underline{CV},\underline{V})))\Big]$$ for some collection of CW complexes, $\{V_i\}$.
Recall that $h^*(A_i) = E_i \oplus B_i$ for free $h^*-$modules $E_i, B_i$. Let $$\{e^{\ell}_i\}$$ be a set of generators for $E_i$.
Set $$V_i = \bigvee S^{|e^{\ell}_i|}$$
where $|e^{\ell}_i|$ is the dimensions of generator $e^{\ell}_i$.
The $V_i$ are constructed so that $(0, \widetilde{h}^*(V_i))$ is strongly $h^*$ cohomology isomorphic to $(0,E_i)$. [*Strongly $h^*$ cohomology isomorphic*]{} is defined in definition \[defin:maps between infinite symmetric products\] and the paragraph before corollary \[cor:specSeqstrong isomorphism\]. By construction the spectral sequence $$E_r(\widehat{Z}((N(I,\sigma); (\underline{CV},\underline{V}))$$is isomorphic to the spectral sequence $$\Big[ \underset{\substack{J\cup L \cup I=[m] \\ L \cup \sigma \in K}}{\bigoplus}E^J\otimes W^L\Big].$$ The spectral sequence converges to $h^*(\widehat{Z}(N(I,\sigma));(\underline{CV},\underline{V}))$. By the wedge lemma, \[thm:null.A\] $$h^*(\widehat{Z}(N(I,\sigma); (\underline{CV},\underline{V}) = h^*(\Sigma |N(I,\sigma|) \otimes E^{[m]-I}.$$
Combining these results gives the following calculation of the cohomology of the polyhedral smash product functor, which by the algebraic splitting theorem, \[cor:SSsplitting\], gives the cohomology of the polyhedral product functor for CW pairs satisfying the freeness condition. The difference between the spectral sequence converging to $h^*(Z(K; \xa)$ and the one converging to $\widetilde{h}^*(\widehat{Z}(K; \xa)$ is describe in the proof of Corollary \[cor:SSsplitting\].
The extensions in the spectral sequence converging to $$h^*(\zkh)$$ appear as extension is the spectral sequence converging to $h^*(\Sigma |N(I,\sigma|) \otimes E^{[m]-I}$. So we may write $ E^{[m]-I}\otimes \widetilde{h}^*(\Sigma |N(I,\sigma|)$ below rather than the associated graded group.
\[Ps\]
1. $$\widetilde{h}^*(\zkh)) = \underset{I\subset [m], \sigma \in K}{\bigoplus} E^{[m]-I} \otimes \widetilde{h}^*(\Sigma|N(I,\sigma)|) \otimes Y^{I,\sigma}.$$as $h^*-$modules with $ \widetilde{h}^*(\Sigma \emptyset) =1$. The factors, $B_i$ of $Y^{I,\sigma}$ are subsets of the **reduced cohomology** of $A_i$. i.e. $1\in B_i$ as defined in Definition \[def:bas\] does not appear.
2. Similarly $$h^*(\zk)) = \underset{I \subset [m], \sigma \in K}{\bigoplus} E^{[m]-I} \otimes \widetilde{h}^*(\Sigma|N(I,\sigma)|) \otimes Y^{I,\sigma}.$$as $h^*-$modules with $ \widetilde{h}^*(\Sigma \emptyset) =1$, but now the factors, $B_i$ of $Y^{I,\sigma}$ are subsets of the **un-reduced cohomology** of $A_i$. i.e. $ 1$ as defined in Definition \[def:bas\] may appear in a coordinate of $Y^{I,\sigma}$.
A convenient reformulation of Theorem \[Ps\] is given in the next corollary.
\[srstar\] $$h^*=\underset{}{\underset{ \underset{I \cup J =[m] } {I \cap J =\emptyset} }{\bigoplus} }\widetilde{h}^*(\Sigma | N_J|) \otimes E^J \otimes h^*(X^I)/(R)$$ where $I=\{i_1, \cdots, i_t\}$, $X^I=X_{i_1} \times \cdots \times X_{i_t}$, and $(R) $ is the ideal generated by $ C^{S} \subset H^*(X^I)$ where $S$ is not simplex of $K$.
The reduced Poincare series for $H^*(\zkh) $ is recorded as the following corollary.
\[spectralsequencePS\]
$$\overline{P}(H^*(\zkh)) = \underset{I, \sigma}{\Sigma} \quad t\overline{P}(Y^{I,\sigma})\times \overline{P}(H^*(|N(I,\sigma)|) \times \overline{P}(E^{[m]-I}).$$
where $\overline{P}(H^*(\emptyset))= 1/t$.
The Poincare series was first computed in [@cartan].
\[SSexample\] We illustrate Corollary \[spectralsequencePS\] with the example of $K $ a simplicial complex with 3 vertices and edges $\{1,3\}, \{1,2\}$. $H^*(X) ={{\mathbb Z}}\{ b_4, c_6\}, \quad H^*(A)={{\mathbb Z}}\{e_2, b_4\} $. The cases in the example are indexed by the $I$ in Corollary \[spectralsequencePS\] starting with the empty set and building up to $I=\{1,2,3\}$. For each $I$ there are the sub cases indexed by the simplices $\sigma \subset I$.
- For $I=\emptyset$. the only possible simplex, $\sigma$, is the empty set and $N(I,\emptyset$) is contractible. So there is no contribution to the Poincare series.
- The next case is $I=\{1\}$. There are two possible simplices, namely $\sigma=\emptyset$ and $\sigma=\{1\}$.
1. $\sigma = \emptyset$.
In this case $Y^{I,\emptyset}= b_4.$ and $|N(I,\emptyset)| = |\{ \{2\},\{3\}\}|=S^0$ which contributes
$$t\overline{P}(Y^{I,\sigma}) \overline{P}(H^*(|N(I,\sigma)|) \overline{P}(E^{[m]-I}) =t (t^4)(t^2)^2=t^9$$to the Poincare series.
2. $\sigma=\{1\}$.
In this case $Y^{I,\sigma} = c_6.$ and $|N(I,\sigma)|= |\{2\},\{3\}| =S^0$. Thus the term
$$t\overline{P}(Y^{I,\sigma}) \overline{P}(H^*(|N(I,\sigma)|) \overline{P}(E^{[m]-I})=t(t^6)(t^2)^2 = t^{11}$$ is contributed to the Poincare series.
- Similarly for $I=\{2\}$ there are the cases
1. $\sigma=\emptyset$ with $N(I,\sigma)=$ the edge $ \{1,3\}$ which is contractible.
2. $\sigma = \{2\}$ with $N(I,\sigma) = \{1\}$ which is also contractible.
- $I= \{3\}$. This is similar to $I=\{2\}$.
1. $\sigma=\emptyset$, $N(I,\sigma)=$ the edge $ \{1,2\}$ which is contractible.
2. $\sigma = \{3\}$, $N(I,\sigma) = \{1\}$ which is contractible.
- For $I= \{1,2\}$ there are $4$ possible simplices
1. $\sigma= \emptyset$ with $N(I,\emptyset)= \{3\}$ which is contractible.
2. $\sigma = \{1\}$ with $N(I,\sigma) = \{3\}$ which is contractible.
3. $\sigma = \{2\}$, with $Y^{I,\sigma} = b_4 \otimes c_6$ and $N(I,\sigma)=\emptyset$ which contributes $$t\overline{P}(Y^{I,\sigma}) \overline{P}(H^*(|N(I,\sigma)|) \overline{P}(E^{[m]-I})=(t^6t^4)(t^2)=t^{12}.$$to the Poincare series.
4. $\sigma=\{1,2\}$
$Y^{I,\sigma}= c_6\otimes c_6$, $N(I, \{1,2\}) = \emptyset$. So this case contributes $t^{14}$ to the Poincare series.
- $ I= \{1,3\}$ is identical to $I=\{1,2\}$ so we get a contribution of $t^{12}$ and $t^{14}$ to the Poincare series.
- $I=\{2,3\}$
1. The cases there are $3$ possible simplices: $\sigma = \emptyset, \quad \{2\}$, and $\sigma= \{3\}$. For all $3$ simplices $N(I,\sigma)=\{1\}$ which is contractible.
- Finally $I=\{1,2,3\} =K$. For all $\sigma$, $N(I,\sigma)=\emptyset$.
The sub simplices of $K$ contribute to the Poincare series as follows:
1. 1. $\sigma = \{1\}, Y^{I,\sigma}= c_6 \otimes b_4 \otimes b_4$
2. $\sigma = \{2\}, Y^{I,\sigma}= b_4 \otimes c_6 \otimes b_4$
3. $\sigma = \{3\}, Y^{I,\sigma}= b_4 \otimes b_4 \otimes c_6$
each contributes $ t^{14}$ to $\overline{P} .$
2. 1. $\sigma = \{1,2\}, Y^{I,\sigma}= c_6 \otimes c_6 \otimes b_4$
2. $\sigma = \{1,3\} , Y^{I,\sigma}= c_6 \otimes b_4 \otimes c_6$
each contributes $t^{16} $to $\overline{P}.$
3. $\sigma = \emptyset,Y^{I,\sigma}=b_4 \otimes b_4 \otimes b_4$, contributes $t^{12} $ to $\overline{P}$
Adding all the terms we get have the Poincare series for the cohomology of\
$\zkh$.
$$\overline{P}(H^*(\zkh)) = t^9+t^{11}+3t^{12}+5t^{14}+2t^{16}.$$
The next two corollaries describe summands in $h^*(\zkh)$. The summand in Corollary \[cor:summand\] depends on the $E_i$. The summand in corollary \[cor:5.8\] is natural since $C_i$ is the kernel of the map $H^*(X_i) \to H^*(A_i)$ which is a functor of the pairs $(X_i,A_i)$.
\[cor:summand\] $\widetilde{h}^{*}(\Sigma|K|) \otimes E_1 \otimes \cdots \otimes E_m$ is a summand $\widetilde{h}^*(\zkh)$.
This corresponds to the summands with $I=\emptyset$ in Theorem \[Ps\]. Specifically if $I=\emptyset$ then $\sigma=\emptyset$ and $N(\emptyset, \emptyset)=K$.
Corollary \[cor:summand\] generalizes [@bbcg3 Theorem 1.12] which describes the cohomology of $\widehat{Z}(K; (\underline{CX},\underline{X}))$. In this case $B_i=C_i=0$ and $E_i=\widetilde{H}^*(X_i).$ The only summand is $I=\emptyset$.
However Theorem 1.12 of [@bbcg3] is not a consequence of Corollary \[cor:summand\] since the complicated bookkeeping involved to evaluate all the differentials in the spectral sequence were subsumed by [@bbcg3 Theorem 1.12] which was used to prove Theorems \[Ps\] and therefore Corollary \[cor:summand\].
\[cor:5.8\]
Let $I \subset h^*(X_1 \times \cdots \times X_m) $ be the ideal generated by $C_{i_1}\otimes \cdots \otimes C_{i_q} $ where $(i_1,\cdots, i_q)$ is not a simplex in $K$. Then $h^*(X_1 \times \cdots \times X_m)/(I) $ is a sub-ring of $\widetilde{h}^*(\zk)$.
There are the maximal summands, $Y^{[m],\sigma}$. $$\underset{\sigma \in K}{\bigoplus} Y^{[m],\sigma} \simeq h^*(X_1 \times \cdots \times X_m)/(I)$$ as $h^*$ modules. The inclusion, $$\iota: \zk \to X_1 \times \cdots \times X_m$$ induces a surjection in cohomology onto $\underset{\sigma \in K}{\bigoplus} Y^{[m],\sigma} $. To prove corollary \[cor:5.8\] it suffices to show that $I $ is isomorphic to the kernel of $h^*(\iota)$.
We may write $h^*(X_i) = B_i \oplus C_i$. The tensor product $h^*(X_1) \otimes \cdots \otimes h^*(X_m)$ may now be written as a sum of terms of the form $S_1 \otimes \cdots \otimes S_m$ where $S_i$ is $B_i$ or $C_i$. The map of rings, $ h^*(X_1) \otimes \cdots \otimes h^*(X_m) \to h^*(\zk)$ is surjective with kernel given by the ideal $I$.
\[rem:naturality\] The spectral sequence is natural with respect to maps of pairs $$(X_i,A_i) \to (Y_i,D_i)$$ (In fact it is a functor of strong $h^*-$cohomology maps of pairs as described in Corollary \[cor:specSeqstrong isomorphism\]). However the description of the cohomology of $\zk$ given in Theorem \[Ps\] is not natural. It depends on the choice of splitting of $h^*(X_i)$ as $B_i \oplus C_i$ and splitting of $h^*(A_i)$ as $E_i \oplus B_i$. We now describe how the decomposition in Theorem \[Ps\] and the naturality of the spectral sequence interact.
To this end suppose there are maps of long exact sequences
$$\label{eq:naturalityofdecomposition}
\begin{array}{ccccccccccc}
\cdots \overset{\delta}{\leftarrow}&h^*(A_i) &\leftarrow&h^*(X_i)&\leftarrow & \widetilde{h}^*(X_i/A_i) & \leftarrow \cdots\\
&\uparrow{g_i} && \uparrow{f_i} && \uparrow\\
\cdots \overset{\delta}{\leftarrow} &h^*(D_i)&\leftarrow&h^*(Y_i) &\leftarrow & \widetilde{h}^*(Y_i/D_i) &\leftarrow \cdots
\end{array}$$
Diagram \[eq:naturalityofdecomposition\] induces a map of spectral sequences
$$E_r(Z(K;(\underline{Y},\underline{D}))) \overset{(f,g)^*}{\to} E_r(\zk)$$and a map
$$h^*(Z(K;(\underline{Y},\underline{D}))) \overset{\ell}{\to} h^*(\zk).$$ The map $\ell$ is now described in terms of the decompositions given by Definition \[def:bas\].
For each $i$ there is a decomposition of the $h^*-$modules in the the top row
$$h^*(A_i) = E_i \oplus B_i, \quad h^*(X_i) = B_i \oplus C_i.$$
The bottom row has a corresponding decomposition
$$h^*(D_i) =E_i^{\prime} \oplus B_i^{\prime}, \quad h^*(Y_i) = B_i^{\prime} \oplus C_i^{\prime}.$$
Suppose $\alpha \in h^*(Z(K; (\underline{Y}, \underline{D})))$ is a class appearing in a summand $$(E^{\prime})^{[m]-I} \otimes \widetilde{h}^*(\Sigma|N(I,\sigma)|) \otimes (Y^{\prime})^{I,\sigma}$$ of the decomposition of $h^*(Z(K;(\underline{Y},\underline{D}))) $ given by Theorem \[Ps\].
Specifically $$\label{alphadecomposition}
\alpha=\underset{J}{\bigotimes} (e_j^{\prime}) \otimes n \otimes \underset{|\sigma|}{\bigotimes} ( c_s^{\prime})\otimes \underset{L}{\bigotimes} (b^{\prime}_{\ell})$$ where $I=L \cup |\sigma|$, $J=[m]-I$ and $n \in \widetilde{h}^*(\Sigma|N(I,\sigma)|)$
In order to compute $\ell(\alpha) \in h^*(\zk)$ we note that the decompositions of $h^*(A_i)$ and $h^*(X_i) $ into $E_i, B_i$ and $C_i$ imply unique representations $$g_i(e_i^{\prime})= e_i+\overline{b}_i, \hspace {.2 in} f_i(b_i^{\prime})=b_i+\overline{c}_i \mbox{ and } f_i(c_i^{\prime}) = c_i$$ where $e_i \in E_i,\quad b_i$ and $ \overline{b}_i \in B_i, \quad$ $\overline{c}_i$ and $c_i \in C_i$.
Formally substitute $e_i+\overline{b}_i$ for $e^{\prime}_i$, $b_i+\overline{c}_i$ for $b_i^{\prime}$ and $c_i$ for $c^{\prime}_i$ in (\[alphadecomposition\]). The resulting expression is a sum of terms with factors $e_i,$ $ b_i,$ $ \overline{b}_i,$ $\overline{c}_i,$ $c_i$ and $n \in \widetilde{h}^*(\Sigma|N(I,\sigma)|)$. Each summand determines a summand in $\ell(\alpha)$. There are a number of cases.
The easiest case is the summand without any over-lined factors. For this term the map of spectral sequences $(f,g)^*$ respects the decomposition of Definition \[def:bas\] at the $E_1$ page and contributes the summand
$$\underset{J}{\bigotimes} (e_j) \otimes n \otimes \underset{|\sigma|}{\bigotimes} ( c_s) \underset{L}{\bigotimes} (b_{\ell})$$ to $\ell(\alpha)$.
Now suppose there are terms of the formal sum with non-zero $\overline{b}_i$ factors i.e. there is an indexing set $Q \subset J$ where $\overline{b}_q$ is not zero for $q \in Q$. In this situation there are formal summands in
$$E^{J\setminus Q} \otimes n \otimes C^{|\sigma| } \otimes B^{L \cup Q}.$$
In terms of the decomposition of Theorem \[Ps\] these terms contribute classes in
$$E^{J \setminus Q} \otimes \widetilde{h}^*(\Sigma |N(I \cup Q,\sigma|)) \otimes C^{|\sigma|} \otimes B^{ L \cup Q}$$ to $\ell(\alpha)$ (recall that $I = L \cup |\sigma|$).
The simplicial complex $N(I \cup Q,\sigma)$ is a sub-simplicial complex of $N(I ,\sigma)$. To prove this suppose $\tau \in N(I \cup Q,\sigma)$ then $\tau \cup \sigma \in K$ and $|\tau| \subset J \setminus Q \subset J$ so $\tau \in N(I,\sigma)$.
The formal summand $$E^{J\setminus Q} \otimes n \otimes C^{|\sigma| } \otimes B^{L \cup Q}.$$
contributes
$$E^{J\setminus Q} \otimes \iota^*(n) \otimes C^{|\sigma| } \otimes B^{L \cup Q}.$$
to $\ell(\alpha)$ where $\iota^*$ is induced by the inclusion
$$\iota^*: \widetilde{h}^*(\Sigma |N(I,\sigma)|) \to \widetilde{h}^*(\Sigma |N(I \cup Q,\sigma)|).$$
Indeed $\iota^*$ at the cochain level is the map which sends the dual of a simplex, $\tau$ to zero if $\tau$ is not a simplex in $N(I \cup Q,\sigma)$ and to the dual of $\tau$ if $\tau$ is a simplex in $N(I \cup Q,\sigma)$. This agrees with the map $(f,g)^*$.
Finally suppose there are terms of the formal sum with non-zero $\overline{c}_i$ factors. i.e there are indexing sets $P \subset L$ where $\overline{c}_{p}\neq 0$ for $p \in P$. These formal summands are classes in
$$E^J \otimes n \otimes C^{|\sigma| \cup P} \otimes B^{ L\setminus P}.$$
In terms of the decomposition of Theorem \[Ps\] these terms will contribute classes in
$$E^J \otimes \widetilde{h}^*(\Sigma |N(I,\sigma \cup P|)) \otimes C^{|\sigma| \cup P} \otimes B^{ L\setminus P}.$$
We shown that $N(I,\sigma \cup P)$ is a sub complex of $N(I,\sigma)$ if $\sigma\cup P$ is a simplex in $K$ (otherwise thie summand lies in the zero group). Suppose $\sigma\cup P$ is a simplex in $K$, say $\tau$. Let $ \rho \in N(I,\tau)$. Then $\rho$ has vertices in $J$ and $\rho \cup \sigma \cup P$ is a simplex in $K$ which implies $\rho \cup \sigma$ is also a simplex in $K$ and $\rho \in N(I,\sigma)$. The summands of $\ell(\alpha)$ with $\overline{c}_p$ factors are represented by classes which jump filtration in the spectral sequence with $n$ replaced by $\iota^*(n)$ ($\iota: N(I,\tau) \to N(I,\sigma)$ the inclusion).
This description of $\ell$ will be applied in (\[productmap\]).
**Products** {#sec: products}
============
The purpose of this section is to describe the ring structure in $H^*(\zk ,\mathbb{ R})$ with $\mathbb{R}$ a commutative ring. (there are similar results for $h^*(\zk)$). As in the previous sections $(X_i,A_i)$ are assumed to be based CW pairs which satisfy the freeness condition of Definition \[def:bas\]. We write $H^*(\zk)$ for $H^*(\zk ,\mathbb{ R})$ in the sequel.
Recall the computation of $H^*(\zk)$ as an $h^*-$module given by \[Ps\].
$$h^*(\zk)) = \underset{I \subset [m], \sigma \in K}{\bigoplus} E^{[m]-I} \otimes \widetilde{h}^*(\Sigma|N(I,\sigma)|) \otimes Y^{I,\sigma}.$$
In particular $h^*(\zk)$ is generated, as an $h^*-$module by monomials $$n \otimes a_1 \otimes \cdots \otimes a_m$$ with $a_i \in E_i, B_i$ or $C_i$ and
$$n \in \begin{cases}
\widetilde{H}^*(\Sigma |N(I,\sigma)|) & \mbox{ if }
\sigma = \{i| a_i \in C_i\}\subset [m] \mbox{ is a simplex in } K \\
0 & \mbox{ otherwise. }
\end{cases}$$where $I=\{i| a_i \in C_i \mbox{ or } B_i\} \subset [m] $ and $N(I,\sigma)$ is defined in Definition \[def:N(Isigma)\].
To describe $$H^*(\zk)$$ as a ring it suffices to define a paring on the summands of Theorem \[Ps\]
$$\Big[ E^{[m]-I_1} \otimes \widetilde{H}^*(\Sigma|N(I_1,\sigma_1)|) \otimes Y^{I_1,\sigma_1}\Big] \otimes \Big[E^{[m]-I_2} \otimes \widetilde{H}^*(\Sigma|N(I_2,\sigma_2)|) \otimes Y^{I_2,\sigma_2}\Big] \overset{\cup}{\to} H^*(\zk)$$
Specifically suppose that $$\alpha = n_{\alpha} \otimes a_1 \otimes \cdots \otimes a_m \in E^{[m]-I_1} \otimes \widetilde{H}^*(\Sigma|N(I_1,\sigma_1)|) \otimes Y^{I_1,\sigma_1}$$
where $n_{\alpha}\in \widetilde{H}^*(\Sigma|N(I_1,\sigma_1)|)$ and $a_i \in E_i, B_i$ or $C_i$.
$$\gamma = n_{\gamma} \otimes g_1 \otimes \cdots \otimes g_m \in E^{[m]-I_2} \otimes \widetilde{H}^*(\Sigma|N(I_2,\sigma_2)|) \otimes Y^{I_2,\sigma_2}$$
where $n_{\gamma} \in \widetilde{H}^*(\Sigma|N(I_2,\sigma_2)|) $ and $g_i \in E_i, B_i $ or $C_i$.
We will describe $\alpha \cup \gamma$ in terms of a coordinate wise multiplication of $a_i$ and $g_i$ and a paring
$$\label{pairing}H^*(\Sigma | N(I_1, \sigma_1)|) \otimes H^*(\Sigma |N(I_2,\sigma_2)|) \to H^*(\Sigma |N(I_3, \sigma_3)|)$$
where $(I_3,\sigma_3)$ will be defined in terms of $(I_1,\sigma_1)$ and $(I_2, \sigma_2)$
The pairing, (\[pairing\]) will be defined in terms of the $\ast-$product defined in [@bbcg3] which we now recall.
Writing $ Z(K)$ for $\zk$ and $\widehat{Z}(K_I)$ for $\widehat{Z}(K_I; (\underline{X}, \underline{A})_I)$, [*partial diagonals*]{}
$$\Delta^{J,L}_I:\widehat{Z}(K_I)\to
\widehat{Z}(K_J)\wedge\widehat{Z}(K_L)$$ ($J\cup L=I$) are defined which fit into a diagram
$$\begin{CD}
Z(K) &@>\widehat{\Delta}>>
&Z(K)\wedge Z(K) \\
@V \Pi VV&&@VV\Pi \wedge \Pi V\\
\widehat{Z}(K_I) &@> \Delta^{J,L}_I >>&
\widehat{Z}(K_J)\wedge\widehat{Z}(K_L).
\end{CD}$$ where $\widehat{\Delta}$ is the reduced diagonal and $\Pi$ is the projection.
The definition of the partial diagonals and projections are as follows (the notation is as in Definitions \[defin:gmac\] and \[defin:smash.product.moment.angle.complex\]).
1. For $I \subset [m]$ and $\sigma \in K$ there is the projection followed by the collapsing map $$\pi: D(\sigma) \to D(\sigma \cap I) \to \widehat{D}(\sigma \cap I).$$ These composites are compatible with the maps in the colimit and induce the vertical maps $$\Pi : Z(K) \to \widehat{Z}(K_I).$$
2. Let $W^{J,L}_I$ be defined by $$W^{J,L}_I =
\begin{cases}
Y_i & \mbox{ if } i \in I - J \cap L \\
Y_i \wedge Y_i & \mbox{ if } i \in J \cap L
\end{cases}$$ where $Y_i$ is either $X_i$ or $A_i$ as in Definition \[defin:smash.product.moment.angle.complex\].
There is a homoemorphism
$$Sh: W^{J,L}_I \to \widehat{Y}^J \wedge \widehat{Y}^L$$ given by the evident shuffle which is compatible with the maps in the colimit.
3. Define $$Y^I \to \widehat{Y}^J \wedge \widehat{Y}^L$$ by first mapping into $W^{J,L}_I$ by
1. the identity of $Y_i$ if $i \in I - J \cap L$ or
2. the diagonal of $Y_i$ if $ i \in J \cap L$
followed by $Sh$. The maps induce a map of colimits which define the partial diagonal, $\Delta^{J,L}_I$.
Given cohomology classes $u\in H^p(\widehat{Z}(K_J)), v\in H^q(\widehat{Z}(K_L))$, the [*$\ast-$product* ]{} is defined by $$\label{starproductI}u*v=(\Delta^{J,L}_I)^*(u\otimes v) \in H^{p+q}(\widehat{Z}(K_I)).$$ In [@bbcg3] the ring structure of $H^*(Z(K,(\underline{X},\underline{A}))$ is shown to be induced by the $\ast$-product.
The special case of $(\underline{X},\underline{A})=(D^1,S^0)$ is particularly important.
The splitting of Theorem, \[thm:decompositions.for.general.moment.angle.complexes\], and Theorem \[thm:null.A\] imply there are homotopy equivalences
$$\Sigma Z(K;(D^1,S^0)) \to \underset{I \subset [m]}{\bigvee} \Sigma \widehat{Z}(K_I; (D^1,S^0))$$ and $$\widehat{Z}(K_I;(D^1,S^0))\stackrel{\simeq}{\to}|K_I|*(\widehat{S}^0)^I\simeq \Sigma |K_I|.$$
With $I = J \cup L$ a pairing
$$\label{equa:starD1} H^*(\Sigma |K_J|) \otimes H^*(\Sigma |K_L|) \to H^*(\Sigma|K_I|)$$
is induced by the partial diagonals map $\Delta^{J,L}_I.$ $$\Sigma |K_I| \simeq \widehat{Z}(K_I;(D^1,S^0)) \overset{\Delta_I^{J,L}}{\to} \widehat{Z}(K_J;(D^1,S^0)) \wedge \widehat{Z}(K_L ;(D^1,S^0)) \simeq \Sigma|K_J| \wedge \Sigma |K_L|.$$
\[ringstar\] The product
$$\Big[n_{\alpha} \otimes a_1 \otimes \cdots \otimes a_m \Big]
\cup \Big[n_{\gamma} \otimes g_1 \otimes \cdots \otimes g_m \Big] \in \hz$$is given by the $\ast-$product of $n_{\alpha}$ and $n_{\beta}$ composed with an inclusion map described in Lemma \[lem:diagfactor\] and a coordinate wise product defined as follows:
1. If $a_i, g_i \in H^*(X_i)$ the product in the $i$-th coordinate is the product in $H^*(X_i).$
2. If $a_i, g_i \in H^*(A_i) $ the product is induced by the product in $H^*(A_i)$.
3. if $a_i \in E_i, g_i \in C_i$ or $g_i \in E_i, a_i \in C_i$ the product is zero.
The rest of this section is devoted to proving Theorem \[ringstar\].
A description of the diagonal map $$\zk \to \zk \times \zk$$and the partial diagonal maps $$\widehat{Z}(K_I; \xa_I) \to \widehat{Z}(K_J;\xa_J)\wedge \widehat{Z}(K_L; \xa_L)$$ are now given. The description uses the following lemma where the notation
$$[ \xa^{J.L}_I ]_i =\begin{cases}
(X_i,A_i) & \mbox{ if } i \in I -J \cap L\\
(X_i \wedge X_i, A_i \wedge A_i) & \mbox{ if } i \in J \cap L
\end{cases}$$ is used in part (c).
\[Zdiagonal\]
1. Suppose $A_m\subset X_m, B_m \subset Y_m$ then there is a natural map $$Sh: Z(K; (\underline{X \times Y}, \underline{ A \times B})) \to Z(K; (\underline{X},\underline{A})) \times Z(K; (\underline{Y},\underline{B}))$$ where $$(\underline{X \times Y}, \underline{ A \times B})) = \{(X_m \times Y_m, A_m \times B_m)\}.$$
2. The diagonal maps of pairs, $ (X_m,A_m) \to (X_m \times X_m, A_m \times A_m)$ defines a map $$\widetilde{\Delta}: (\underline{X},\underline{A}) \to ( \underline{X \times X}, \underline{A \times A})$$ which induces a map of spectral sequences.
$$E_r(Z(K; (\underline{X \times X}, \underline{A \times A}))) \to E_r(Z(K; \xa))$$
The diagonal map is the composite $$\Delta: \zk \overset{\widetilde{\Delta}}{\to} Z(K; (\underline{X \times X}, \underline{A \times A}) ) \overset{Sh}{\to} \zk \times \zk.$$
3. The partial diagonal map of pairs $\widehat{\Delta}^{J,L}_I: \xa \to \xa^{J.L}_I $
$$\widehat{\Delta}^{J,L}_I = \begin{cases}
\mbox{ the identity } & \mbox{ if } i \in I - J \cap L\\
\mbox{ the reduced diagonal } & \mbox{ if } i \in J \cap L.
\end{cases}$$ induces a map of spectral sequences. $$E_r(\widehat{Z}(K_I; \xa^{J,L}_I)) \to E_r(\widehat{Z}(K_I; \xa_I))$$
$\Delta^{J,L}_I$ is the composite $$\widehat{Z}(K_I; \xa_I) \overset{ \widehat{\Delta}^{J,L}_I }{\to} \widehat{Z}(K_I; \xa^{J,L}_I) \overset{Sh}{\to} \widehat{Z}(K_J; \xa_J) \wedge \widehat{Z}(K_L; \xa_L).$$
(a): From the definition of the polyhedral product functor, Definition \[defin:gmac\], $Z(K; (\underline{X \times Y}, \underline{ A \times B}))$ is a colimit of spaces, $D(\sigma,(\underline{X \times Y}, \underline{ A \times B}) )$. By shuffling the factors, there is a map $D(\sigma,(\underline{X \times Y}, \underline{ A \times B}) )\to D(\sigma, (\underline{X}, \underline{A})) \times D(\sigma, (\underline{Y}, \underline{B}))$. The maps are compatible with the maps into the spaces of the colimit defining $Z(K; (\underline{X},\underline{A})) \times Z(K; (\underline{Y},\underline{B}))$, proving (a).
(b): The diagonal $$\Delta: \zk \to \zk \times \zk$$ is induced at the level of $D(\sigma)$ by $\widetilde{\Delta}$ followed by a shuffle. It follows from (a) that $\Delta$ factors as indicated.
(c): Is similar.
Since the product in $H^*(\zk)$ is determined by the $\ast-$product, as in (\[starproductI\]) it suffices to compute the map induced by $\Delta^{J,L}_I$. It follows from Lemma \[Zdiagonal\] that $\Delta^{J,L}_I$ decomposes as the composition of the map induced by the shuffle followed by the map induced by $\widehat{\Delta}^{J,L}_I $. The shuffle map may be computed using the fact that the spectral sequence, and hence $H^*(\widehat{Z}(K; \xa)$ is a functor of the strong $H^*-$cohomology type (Corollary \[cor:specSeqstrong isomorphism\]) and the decomposition in Definition \[def:bas\]. The map induced by $\widehat{\Delta}^{J,L}_I$ is computed using the naturally discussed in remark \[rem:naturality\].
We first describe $(\widehat{\Delta}^{J,L}_I)^*$. To this end the decomposition of Definition \[def:bas\] is described for the pair $(\underline{X \wedge X}, \underline{A \wedge A})$.
There is the long exact sequence $$\overset{\delta}{\leftarrow} \widetilde{H}^*(A_i \wedge A_i) \overset{i}{\leftarrow} \widetilde{H}^*(X_i
\wedge X_i) \leftarrow \widetilde{H}^*(X \wedge X/A \wedge A) \leftarrow$$ The image, kernel and cokernel of Definition \[def:bas\] for the above exact sequence will be denoted $\widehat{B}_i, \widehat{C}_i$ and $\widehat{E}_i$ respectively. In terms of $B_i,C_i,E_i$ associated to the pair $(X_i,A_i)$
$$\label{hats}
\begin{array}{lcl}
\widehat{B}_i & =& B_i \otimes B_i\\
\widehat{C}_i & =& \Big(C_i \otimes C_i\Big) \oplus \Big( B_i \otimes C_i\Big) \oplus \Big(C_i \otimes B_i\Big)\\
\widehat{E}_i &=& \Big(E_i \otimes E_i\Big) \oplus \Big(E_i \otimes B_i \Big)\oplus \Big(B_i \otimes E_i\Big)
\end{array}$$
The diagonal induces the product in $H^*(X_i)$ and $H^*(A_i)$
$$\label{productmap}
\begin{array}{lcl}
\widehat{B}_i & \to &H^*(X_i) = B_i \oplus C_i\\
\widehat{C}_i & \to & C_i \\
\widehat{E}_i &\to & H^*(A_i) = E_i \oplus B_i
\end{array}$$
The map of long exact sequences induced by the diagonal
$$\begin{array}{ccccccccccc}
\cdots \overset{\delta}{\leftarrow}&h^*(A_i) &\leftarrow&h^*(X_i)&\leftarrow & \widetilde{h}^*(X_i/A_i) & \leftarrow \cdots\\
&\uparrow{\Delta_i^*} && \uparrow{\Delta^*_i} && \uparrow\\
\cdots \overset{\delta}{\leftarrow} &h^*(A_i \wedge A_i)&\leftarrow&h^*(X_i\wedge X_i) &\leftarrow & \widetilde{h}^*(X_i\wedge X_i/A_i \wedge A_i) &\leftarrow \cdots
\end{array}$$
is an example of the more general situation described at the end of Section \[revisited\].
The map
$$\widehat{\Delta}^{J,L}_I: \widehat{Z}(K_I; \xa_I) \to \widehat{Z}(K_I; \xa^{J,L}_I)$$is induced by a map of pairs $\xa_I \to \xa^{J,L}_I$ which at $E_{\infty}$ is the product described in (\[productmap\]).
The details follow. We first have to adjust the indexing sets in Theorem \[Ps\]. The vertex set of the simplicial complex $K_I$ is $I$ not $[m]$. Since $I$ now denotes the vertex set we cannot use it in the notation for the factor $Y^{I,\sigma}$. We replace $I$ in Theorem \[Ps\] with $F$. With these modifications
$$H^*(\widehat{Z}(K_I; \xa^{J,L}_I))$$is a sum of groups $$\underset{F \subset I, \sigma \in K_I}{\bigoplus} \overline{E}^{I-F} \otimes \widetilde{H}^*(\Sigma |N(F,\sigma)|) \otimes \overline{Y}^{F,\sigma}$$
where a factor $\overline{E}_i$ of $\overline{E}^{I-F} $ is $\widehat{E}_i$ if $i \in J \cap L$ and $E_i$ otherwise. The factors of $ \overline{Y}^{F,\sigma}$ are defined analogously. Specifically a factor $\overline{C}_i$ of $\overline{Y}^{F,\sigma}$ is $\widehat{C}_i$ if $i \in J \cap L$ and $C_i$ otherwise. A factor $\overline{B}_i$ of $\overline{Y}^{F,\sigma}$ is $\widehat{B}_i$ if $i \in J \cap L$ and is $B_i$ otherwise.
Similarly $$H^*(\widehat{Z}(K_I; \xa_I ))$$ is a sum
$$\underset{F^{\prime} \subset I , \tau \in K_I}{\bigoplus} E^{I-F^{\prime}} \otimes \widetilde{H}^*(\Sigma|N( F^{\prime} ,\tau)|) \otimes Y^{F^{\prime},\tau}.$$
The map $$\widehat{\Delta}^{J,L}_I: \widehat{Z}(K_I; \xa_I) \to \widehat{Z}(K_I; \xa^{J,L}_I)$$
induces a map in cohomology which restricted to each summand is a map
$$\overline{E}^{I-F} \otimes \widetilde{H}^*(\Sigma |N(F,\sigma)|) \otimes \overline{Y}^{F,\sigma} \to \underset{F^{\prime} \subset I, \tau \in K_I}{\bigoplus} E^{I-F^{\prime}} \otimes \widetilde{H}^*(\Sigma|N(F^{\prime},\tau)|) \otimes Y^{F^{\prime},\tau}.$$
This map is computed using the naturality discussion at the end of Section \[revisited\] with maps on $\widehat{E}_i, \widehat{B}_i,$ and $\widehat{C}_i$ given by (\[productmap\]). Specifically the map induced by the product on $\widehat{E}_i$ may have summands in $B_i$ thus enlarging $F$ to a larger indexing set, $F^{\prime}$. Also some of the terms $\widehat{B}_i$ which appear in $\overline{Y}^{F,\sigma}$ map via the product to summands with a factor of $C_i$ enlarging the simplex $\sigma$ to $\tau$. Thus we have proven lemma \[lem:diagfactor\] below. The sum is over all $F^{\prime} \subset I$ and $\tau$.
\[lem:diagfactor\] $$(\widehat{\Delta}^{J,L}_I)^*: \overline{E}^{I-F} \otimes \widetilde{H}^*(\Sigma |N(F,\sigma)|) \otimes \overline{Y}^{F,\sigma} \to \underset{F^{\prime},\tau}{\bigoplus} E^{I-F^{\prime} } \otimes \widetilde{H}^*(\Sigma|N(F^{\prime},\tau)|) \otimes Y^{F^{\prime},\tau},$$ where $ F^{\prime} \supset F, \tau \supset \sigma$, the product, (\[productmap\]), induces the maps on the factors in $\overline{E}$ and $\overline{Y}$ and $ \widetilde{H}^*(\Sigma |N(F,\sigma)|) \to \widetilde{H}^*(\Sigma |N(F^{\prime},\tau)|)$ is induced by the inclusions $$N(F^{\prime},\tau) \to N(F,\sigma).$$ $ \square$
Intuitively the diagonal induces the product on the coordinates. Because of the mixing of $E's, B's$ and $C's$ the cohomology of the links map to the cohomology of the resulting sub-links .
Note that (\[hats\]) implies neither $E_i \otimes C_i$ nor $C_i \otimes E_i$ appear in $H^*(\widehat{Z}(K_I; \xa^{J,L}_I))$. This implies the product of classes in $\zk $ involving $C_i$ and $E_i$ must be zero.
An important special case is that of wedge decomposable spaces.
A collection of spaces, $\xa$ is wedge decomposable if for all $i$ $$X_i = B_i \vee C_i$$ and $$A_i = B_i \vee E_i$$ where $$E_i \to B_i \vee C_i$$ is null homotopic.
In this case there is none of the mixing of the products complicating the map in Lemma \[lem:diagfactor\].
If $\xa$ is wedge decomposable then the product of
$$\Big[n_{\alpha} \otimes a_1 \otimes \cdots \otimes a_m \Big]
\cup \Big[n_{\gamma} \otimes g_1 \otimes \cdots \otimes g_m \Big] \in \hz$$is given by the $\ast-$product of $n_{\alpha}$ and $n_{\beta}$ and a coordinate wise product defined as follows:
1. If $a_i, g_i \in H^*(B_i)$ the product in the $i$-th coordinate is the product in $H^*(B_i).$
2. If $a_i, g_i \in H^*(C_i) $ the product is induced by the product in $H^*(C_i)$.
3. If $a_i, g_i \in H^*(E_i) $ the product is induced by the product in $H^*(E_i)$.
4. The product is zero otherwise.
\[exm:productOnD1S0\] We illustrate Lemma \[lem:diagfactor\] for $H^*(\widehat{Z}(K; (D^1,S^0)))$.
$E_i$ is generated by a zero dimensional class, $t_0$ for all $i$. The product $E_i \otimes E_i \to E_i$ is an isomorphism ( $t_0 \otimes t_0 \mapsto t_0$). In particular $F=F^{\prime}$.
In this case the pair $(D^1 \wedge D^1, S^0 \wedge S^0) $ is homotopy equivalent to $(D^1,S^0)$ and $\widehat{\Delta}^{J,L}_I$ is a homotopy equivalence.
As a consequece the $\ast-$product, (\[equa:starD1\])
$$H^*(\Sigma |K_J|) \otimes H^*(\Sigma |K_L|) \to H^*(\Sigma|K_I|)$$
is completely determined by the map induce by the shuffle
$$Sh^*: \widetilde{H}^*(\widehat{Z}(K_J; (D^1,S^0)_J)) \otimes \widetilde{H}^*( \widehat{Z}(K_L; (D^1,S^0)_L)) \to \widetilde{H}^*( \widehat{Z}(K_I; (D^1,S^0)^{J,L}_I) ).$$
In order to describe the suffle map for general $\xa$ satisfying the freeness condition it is convenient to give yet another description of the spectral sequence. Motivated by the proof of Theorem \[Ps\] we introduce variables, $t_i$ and $s_i$ into the $E_1$ term of the spectral sequence. The degree of $t_i = 0$ and the degree of $s_i=1$.
For $K$ a simplicial complex with vertex set $[m]$ recall the $E_1$ term is a sum of groups
$$\widetilde{H}^*(X_i/A_i)^{\tau} \otimes \widetilde{H}^*(\widehat{A}_i)^{\tau^c}$$ ($\tau$ a simplex of $K$). Which in turn is a sum of groups
$$\label{sumofGroups}C^{\sigma}\otimes W^{\sigma^{\prime}} \otimes E^P \otimes B^Q$$
Where $\sigma$ and $\sigma^{\prime }$ are simplices of $K$ whose union is is a simplex $\tau \in K$ and $P \cup Q = \tau^c$. In the notation of Theorem \[Ps\] $, Q \cup |\sigma| = I$, $P \cup |\sigma^{\prime}| = [m]-I$. The filtration of this summand is the weight of the simplex $\tau$ in the left lexicographical ordering of the simplices.
Replace $E_i$ with $t_i E_i$ and $W_i$ with $s_i E_i$ in (\[sumofGroups\]) and arrange the sum as follows
$$\label{sumGroupsII}
E_1=\underset{I\cup J =[m], I \cap J =\emptyset, \sigma \in K_I}{\underset{I,J, \sigma}{\bigoplus} }\Big( \underset{P \cup |\sigma^{\prime}|=J, \sigma \cup \sigma^{\prime} \in K}{\underset{P, \sigma^{\prime}}{\bigoplus}} E^J \otimes Y^{I,\sigma} t^P s^{\sigma^{\prime} }\Big)$$
Define a differential on $E_1$ by $\delta t_i = s_i$, $\delta s_i=0$. The proof of Theorem \[Ps\] implies the following proposition.
\[prop:e1withts\]
1. The spectral sequence $$E_r(\widehat{Z}(K;\xa) \Rightarrow H^*(\widehat{Z}(K; \xa))$$ is isomorphic to the spectral sequence with $E_1$ term $$\underset{I\cup J =[m], I \cap J =\emptyset, \sigma \in K_I}{\underset{I,J, \sigma}{\bigoplus} }\Big( \underset{P \cup |\sigma^{\prime}|=J, \sigma \cup \sigma^{\prime} \in K}{\underset{P, \sigma^{\prime}}{\bigoplus}} E^J \otimes Y^{I,\sigma} t^P s^{\sigma^{\prime} }\Big)$$ and differential $\delta t_i = s_i$.
2. Let $N(I,\sigma)$ be the simplicial complex defined in Definition \[def:N(Isigma)\]. Then $$\underset{P \cup |\sigma^{\prime}|=J, \sigma \cup \sigma^{\prime} \in K}{\underset{P, \sigma^{\prime}}{\bigoplus}} E^J \otimes Y^{I,\sigma} t^P s^{\sigma^{\prime} }$$ is isomorphic to $$\Big(E^J \otimes Y^{I,\sigma}\Big) \otimes E_1(\widehat{Z}(N(I,\sigma) ; (D^1, S^0))$$as differential groups.
It will be convenient to write $N_J$ for $ |N(I,\sigma|)$. So the summands of $H^*(\zkh)$ have the form $$E^J \otimes H^*(\Sigma |N_J|) \otimes Y^{I,\sigma}.$$
The shuffle map is induced by $$\widehat{D}(\sigma_I) \to \widehat{D}(\sigma_J) \wedge \widehat{D}(\sigma_L)$$ where $\sigma_J=\sigma_I \cap J, \sigma_L = \sigma_I \cap L$. This map induces the shuffle on the $E, C$ and $B$ terms and induces the shuffle on the $t_i,p_i$ terms. Specifically the shuffle map induces a map
$$Sh^*: \big(E^{J_1} \otimes Y^{I_1,\sigma_1} t^{P_1} s^{\sigma_1^{\prime} }\big) \otimes
\big(E^{J_2} \otimes Y^{I_2,\sigma_2} t^{P_2} s^{\sigma_2^{\prime} } \big)\to \bigoplus \overline{E}^J \otimes \overline{Y}^{I,\sigma} t^P s^{\sigma^{\prime} }$$ with $P_1 \cup |\sigma_1^{\prime}| = J_1, P_2 \cup |\sigma_2^{\prime}| = J_2.$ $\overline{E}$ and $\overline{Y}$ are define in the paragraph before Lemma \[lem:diagfactor\] and $P \cup |\sigma^{\prime}|=J$.
From (\[hats\]) it follows that $Sh^*$ is zero if there is a coordinate, $i$ with $E_i$ a factor in $E^{J_1} \otimes Y^{I_1,\sigma_1} t^{P_1} s^{\sigma_1^{\prime} }$ and a factor $C_i$ in $E^{J_2} \otimes Y^{I_2,\sigma_2} t^{P_2} s^{\sigma_2^{\prime} } $ (or visa-verse). Hence there is no loss of generality to assume
$$\label{condition}
J_1 \cap |\sigma_2| = \emptyset = J_2 \cap |\sigma_1|$$
Next notice that $Sh^*$ takes any coordinate with a factor of $E_i$ to $\overline{E}_i$, any coordinate involving $C_i$ to $\overline{C}_i$ and coordinates with both factors in $B_i$ to $\overline{B}_i$.
Hence the image of $Sh^*$ takes values in the summand
$$\overline{E}^{J_1 \cup J_2} \otimes \overline{Y}^{I_1\cap I_2, \sigma_1 \cup \sigma_2}t^P s^{\sigma^{\prime}}$$ where $P \cup |\sigma^{\prime}|=J_1 \cup J_2$ and $\sigma^{\prime} \cup \sigma_1 \cup \sigma_2 $ is a simplex in $K$. In particular $Sh^*$ is zero on factors were $\sigma_1 \cup \sigma_2$ is not a simplex in $K$. It is a consequence of (\[condition\]) that $|\sigma_1 \cup \sigma_2| \subset I_1 \cap I_2$.
Next observe that the shuffle map on $t_i$ and $s_i$ is exactly the map induced by the shuffle map for $$H^*(\Sigma |N_{J_1}|) \otimes H^*(\Sigma |N_{J_2}|) \to H^*(\Sigma |N_{J=J_1 \cup J_2}|)$$ Which by example \[exm:productOnD1S0\], is the $\ast-$product. Using work of Cai [@LC] we will give a chain level formula for the $\ast-$product in section \[section7\].
Thus $Sh^*$ is the shuffle map on the $E, B$ and $C$ factors and the $\ast$-product on the cohomology of the links. Lemma \[lem:diagfactor\] and the computation of $Sh^*$ prove Theorem \[ringstar\].
We have shown that the map induced by the shuffle only depends on the ring structure of the cohomology of the subcomplexes of $K$ and the strong homology type of $H^*(X_i)$ and $H^*(A_i)$. The contributions of the ring structure of $H^*(X_i)$ and $H^*(A_i)$ to $H^*(\zkh)$ appear in the diagonal map, $\widehat{\Delta}^{J,L}_I$. So decomposing the partial diagonal into the diagonal composed with the shuffle separates the combinatorial contribution to the ring structure of $\zk$ from the cup product structure of the cohomology of $\xa$.
1. Q. Zheng, [@zheng], also describes a product in $\hz$.
2. Proposition \[prop:e1withts\] generalizes to a multiplicative cohomology theory, $h^*$, satisfying the flatness condition.
[**The Cohomology of $Z(K;(\underline{D^1}, \underline{S^0}))$**]{} {#section7}
===================================================================
Assuming suitable freeness conditions the results of Section \[sec: products\] determine the cohomology ring $H^*(\zk) $ in terms of the rings $H^*(X_i)$, $H^*(A_i)$ and the star product on the cohomology of the links. The goal of this section is to complete this description by describing the star product.
Recall that the product
$$H^*(\zk) \otimes H^*(\zk) \to H^*(\zk)$$
is the composite of the map induced by the shuffle
$$Sh^*:H^*(\zk) \otimes H^*(\zk) \to H^*(Z(K;(\underline {X \times X}, \underline{A\times A})))$$
and the map induced by the diagonal $$\Delta^*: H^*(Z(K;(\underline {X \times X}, \underline{A\times A}))) \to H^*(\zk).$$
On the summands of Theorem \[Ps\] it was shown in section \[sec: products\] that the shuffle has the form
$$\begin{gathered}
Sh^*: \big(E^{J_1} \otimes H^*(\Sigma|N_{J_1}|)) \otimes Y^{I_1,\sigma_1}\big) \otimes
\big(E^{J_2} \otimes H^*(\Sigma |N_{J_2}|) \otimes Y^{I_2,\sigma_2} \big) \\ \to
\overline{E}^{J_1 \cup J_2} \otimes H^*(\Sigma |N_{J_1\cup J_2}|) \otimes \overline{Y}^{I_1 \cap I_2,\sigma_1 \cup \sigma_2}
\end{gathered}$$
where the pairing
$$\label{shuffelLinks} H^*(\Sigma|N_{J_1}|)) \otimes H^*(\Sigma |N_{J_2}|) \to H^*(\Sigma |N_{J_1\cup J_2}|)$$
is induced by the $\ast-$product.
In section \[sec: products\] it was also shown that the diagonal has the form
$$\overline{E}^{J_1 \cup J_2} \otimes H^*(\Sigma |N_{J_1\cup J_2}|) \otimes \overline{Y}^{I_1 \cap I_2,\sigma_1 \cup \sigma_2} \to E^{J^{\prime}} \otimes H^*(\Sigma |N_{J^{\prime}}|) \otimes Y^{F^{\prime}, \tau}$$
with the map on the link $$\label{diagLink}H^*(\Sigma |N_{J_1\cup J_2}|) \to H^*(\Sigma |N_{J^{\prime}}|)$$ induced by an inclusion $\iota: N_{J^{\prime}} \to N_{J_1 \cup J_2}$.
We describe the $\ast-$product and inclusion map on the links by constructing a filtered chain complex for $H^*(Z(K; (D^1,S^0)))$. The spectral sequence associated to this filtered complex is the spectral sequence $E_r(Z(K; (D^1,S^0)))$.
For $I \subset [m]$ recall that there are classes, $s_i$ of degree $1$ and $t_i$ of degree $0$ such that the $E_1$ page of the spectral sequence for $\widehat{Z}(K_I;(D^1,S^0))$ is generated by $$y_{\sigma} =y_1\otimes \cdots \otimes y_m$$ where $\sigma$ is a simplex in $K_I$ and $$y_i=
\begin{cases}
s_i & \mbox{ if } i \in \sigma \\
t_i & \mbox{ if } i \notin \sigma \mbox{ and } i \in I\\
1 & \mbox{ if } i \notin I.
\end{cases}$$ Where $1$ is the multiplicative unit.
Define a cochain complex, $C_{K_I}$ freely generated over $\mathbb{Z}$ by $\{y_{\sigma}| \sigma \in K_I\} $ with differential $$d(y_{\sigma} )= \underset{\tau}{\Sigma} \mbox{ } (-1)^{n(\tau)} y_{\tau}$$ where, for $\sigma$ an $n-$simplex, the sum is over all $n+1$ simplices $\tau$ such that $\sigma \subset \tau \in K_I$. The integer $n(\tau)$ is defined by the usual graded sign convention for a derivation. i.e. there is a differential $\delta$ acting on each factor by $\delta(t_i) = s_i$, $\delta(s_i)=\delta(1) =0$. $d$ is defined on $y_{\sigma}$ by extending $\delta $ to $y_{\sigma}$ by the graded Leibniz rule.
Define $C_K$ by $$\label{def:chaincomplex} C_K = \underset{I\subset [m]}{\bigoplus} C_{K_I}.$$ The following is proven in [@LC].
\[prop:chaincomplex\] There is an isomorphism of groups
$$H^*(C_K) = H^*(Z(K; (D^1,S^0)))$$
$C_K$ is the dual of the standard chain complex for $\underset{I}{\bigvee} \Sigma |K_I|$. The proposition now follows from Theorem \[thm:null.A\].
The left lexicographical ordering of the simplices of $K$ induce a filtration on $C_K= \underset{I}{\bigoplus} \mathbb{Z}\{y_{\sigma}| \sigma \in K_I\} .$ The spectral sequence associated to this filtration is easily seen to be the spectral sequence $E_r(Z(K;(D^1,S^0)))$.
In [@LC] Cai gives a non-commutative product on the chain complex (\[def:chaincomplex\]) which induces the cup product in $H^*(Z(K; (D^1,S^0)))$. This product specializes to the $\ast-$product of Theorem \[ringstar\].
Following [@LC] we define a non commutative product on $C_K$ by extending the following product on the classes $t_i$ and $s_i$: $$\label{caiproduct}
t_i t_i = t_i,\quad
t_i s_i = 0,\quad
s_i t_i = s_i,\quad
s_i s_i = 0$$
to $C_K$ by the graded shuffle.
The product (\[caiproduct\]) induces the $\ast-$product, (\[shuffelLinks\]), on the cochain complexes for $N_{J_1} $ and $N_{J_2}$ as follows. We suppose there are simplices $\alpha \in N_{J_1}$ and $\beta \in N_{J_2}$. There are the generators $y_{\alpha}$ and $y_{\beta}$ of the cochains of $N_{J_1}$ and $N_{J_2}$ respectively. The graded shuffle product of $y_{\alpha}$ and $y_{\beta}$ followed by the coordinate wise product defined in (\[caiproduct\]) determine a signed monomial, $y_{\gamma}$ in the $t_i$’s and $s_i$’s. Here $\gamma = \{i|s_i \mbox{ appears in the } i-th \mbox{ coordinate of the monomial} \}$. The $\ast-$product, (\[shuffelLinks\]) on cochains is given by
$$y_{\alpha} \otimes y_{\beta} \mapsto \pm y_{\gamma} \mapsto \begin{cases}
\pm y_{\gamma} & \mbox{ if } \gamma \cup \sigma_1 \cup \sigma_2 \in K \\
0 & \mbox{ otherwise }
\end{cases}$$
The map (\[diagLink\]) is induced by the map of cochains dual to the inclusion. Namely if $y_{\gamma}$ is a generator of the cochains of $N_{J_1 \cup J_2}$ then $\iota^*(y_{\gamma})$ is given by
$$y_{\gamma} \mapsto
\begin{cases}
y_{\gamma} & \mbox{ if } \gamma \in N_{J^{\prime}} \\
0 & \mbox{ otherwise }
\end{cases}$$
[**Acknowledgments**]{}. The first author was supported in part by a Rider University Summer Research Fellowship and grant number 210386 from the Simons Foundation; the third author was supported partially by DARPA grant number 2006-06918-01.
[99]{}
A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, [*The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces*]{}, Advances in Math., 225 (2010), 1634-1668. arXiv:0711.4689.
A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, [*Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces*]{}, PNAS (2009) 106:12241–12244.
A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, [*Cup-products in generalized moment-angle complexes*]{}, Math. Proc. Camb. Phil. Soc., 153 (2012), 457-469.
A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, [*A Cartan formula for the homology of polyhedral products*]{}, In preparation. V. Buchstaber and T. Panov, [*Torus actions and their applications in topology and combinatorics*]{}, AMS University Lecture Series, (2002), [**24**]{}, .
L. Cai, [*On products in a real moment-angle manifold*]{}, arXiv:1410.5543v2\[math.AT\] 25 Oct 2014. H. S. M. Coxeter, [*Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues* ]{}, Proc. London Math. Soc. (1938) s2-43 (1), P.33–62.
M. Davis,[*The Euler characteristic of a polyhedral product*]{} arXiv:1103.3034v1 \[math.GT\].
M. Davis and T. Januszkiewicz, [*Convex polytopes, Coxeter orbifolds and torus actions*]{}, Duke Math. J., **62** (1991), no. 2, 417-451.
G. Denham and A. Suciu, [*Moment-angle complexes, monomial ideals and Massey products*]{}, Pure Appl. Math. Q., **3** (2007), no. 1, 25–60.
J. Grbić and S. Theriault, [*The homotopy type of the polyhedral product functor for shifted complexes*]{}, arXiv:1109.2728v2 \[math.AT\] 20 Oct 2011
Q. Zheng, [*The homology coalgebra and cohomology algebra of generalized moment-angle complexes*]{}, arXiv: arXiv:1201.4917v1 \[math.AT\].
|
---
abstract: 'We study the possibility that the transition from hadron matter to quark matter at vanishing temperatures proceeds via crossover, similar to the crossover behavior found with lattice QCD studies at high temperatures. The purpose is to examine astrophysical consequences of this postulate by constructing hybrid star sequences fulfilling current experimental data.'
address:
- |
Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Dubna, Russia\
Instituto de Fisica, Universidad Autonoma de San Luis Potosi, S.L.P., Mexico
- 'Physics Department, Faculty of Science, University of Zagreb, Zagreb, Croatia'
- |
Institut Fizyki Teoretycznej, Uniwersytet Wroc[ł]{}awski, Wroc[ł]{}aw, Poland\
Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Dubna, Russia
- 'Institut Fizyki Teoretycznej, Uniwersytet Wrocławski, Wrocław, Poland'
author:
- |
David Edwin Alvarez Castillo\
Sanjin Benić\
David Blaschke\
Rafa[ł]{} [Ł]{}astowiecki
title: Crossover transition to quark matter in heavy hybrid stars
---
Introduction
============
Recent simulations of quantum chromodynamics (QCD) on the lattice [@Aoki:2006we; @Bazavov:2011nk] show that the hadron-to-quark matter transition in the region of small quark chemical potential ($\mu\simeq 0$) in the QCD phase diagram is a crossover. To what extent this result persists in the regime for cold ($T=0$), dense ($\mu_B=3\mu > m_N$) matter is an open question eventually to be answered by heavy-ion collision experiments of the third generation such as NICA and FAIR.
An alternative is to measure masses and radii of compact stars, where the Bayesian analysis is used to invert the Tolman-Oppenheimer-Volkoff (TOV) equations [@Steiner:2012xt] and obtain the most probable equation of state (EoS) corresponding to a chosen set of observational constraints. The challenge within this approach is a reliable measurement of neutron star radii. We quote results from millisecond pulsar timing analyses [@Bogdanov:2012md] rather than burst sources used in [@Steiner:2012xt].
Prompted by recent findings of a second $2M_\odot$ neutron star [@Demorest:2010bx; @Antoniadis:2013pzd] we reexamine the EoS for dense matter by constructing non-rotating sequences and studying the possibility of a hybrid star with $2M_\odot$.
The hybrid EoS is constructed from a non-local Nambu–Jona-Lasinio model (nl-NJL) [@Contrera:2010kz; @Hell:2011ic; @Benic:2013eqa] with appreciable vector interaction strength [@Contrera:2012wj], while for the nuclear matter we use the DD2 EoS [@Typel:1999yq; @Typel:2009sy].
In this work we abandon the standard Maxwell construction for a first order phase transition, anticipating instead a crossover transition described by an interpolation of the pressure $p(\mu_B)$ as a thermodynamic potential for the above EoS. This procedure is equivalent to the one described in [@Masuda:2012ed] using energy density $\varepsilon$ versus baryon density $n_B$, which corrects an earlier suggested inappropriate construction in the $p(n_B)$ plane [@Masuda:2012kf]. The construction leads to a characteristic stiffening on the hadron-dominated side followed by a softening and smooth joining to the quark-dominated side of the EoS. While the former maybe due to quark substructure effects (Pauli blocking) initiating the hadron dissociation (Mott effect), the appearance of finite size structures (pasta phases) at the quark-hadron interface [@Yasutake:2012dw] and strong hadronic fluctuations [@Herbst:2010rf] might be responsible for the latter.
Equation of state
=================
The thermodynamic potential for quark matter is provided by the nl-NJL model $$\label{Omega}
\Omega = \Omega_\mathrm{cond} + \Omega_\mathrm{kin}^\mathrm{reg}
+\Omega_\mathrm{reg}^\mathrm{free}~,$$ $$\Omega_\mathrm{cond} =
\frac{\sigma_1^2+\kappa_\mathbf{p}^2\sigma_2^2
+\kappa_{p_4}^2\sigma_3^2}{2G_S}
-\frac{\omega^2}{2G_V}~,
\label{eq:omcond}$$ $$\Omega_\mathrm{kin}^\mathrm{reg} =
-2N_f N_c\int\frac{d^4 p}{(2\pi)^4}\log\left[\frac{\mathbf{p}^2A^2(\tilde{p}^2)
+ \tilde{p}_4^2C^2(\tilde{p}^2)
+ B^2(\tilde{p}^2)}{\tilde{p}^2
+ m^2}\right]~,$$ $$\Omega_\mathrm{reg}^\mathrm{free} =
- \frac{N_c}{24\pi^2}\left[2\tilde{\mu}^3 \tilde{p}_F
- 5m^2 \tilde{\mu} \tilde{p}_F
+ 3m^4 \log\left(\frac{\tilde{p}_F+\tilde{\mu}}{m}\right)\right]~.$$ Here $A(p^2) = 1+\sigma_2 f(p^2)$, $B(p^2)=1+\sigma_1 g(p^2)$ and $C(p^2) = 1+\sigma_3 f(p^2)$, where $f(p^2)$ and $g(p^2)$ are appropriately chosen formfactors [@Contrera:2010kz]. We denote $\tilde{p} = (\mathbf{p},\tilde{p}_4)$, $\tilde{p}_4 = p_4-i\tilde{\mu}$. The vector channel is introduced as a background field, similar to the way the Polyakov loop is introduced in NJL models; via renormalization of the quark chemical potential $\tilde{\mu} = \mu-\omega$ and completed by a classical term in the thermodynamic potential (\[eq:omcond\]).
The pressure corresponds to the thermodynamic potential in equilibrium by $p_Q(\mu) = -\Omega$, where the latter is found from Eq. \[Omega\] for a given chemical potential as a minimum with respect to variations of the mean fields $$\frac{\partial \Omega}{\partial (\sigma_1, \sigma_2, \sigma_3)} = 0~.$$ The value of the vector meanfield $\omega$ is found from the constraint of a given baryon density $n_B=\partial p_Q/\partial \mu_B$, namely $\omega = G_V \ n_B(\tilde{\mu}_B)~.$
For describing dense nuclear matter we choose the DD2 EoS [@Typel:1999yq; @Typel:2009sy]. The transition region is constructed by a Gaussian interpolation $$p(\mu_B) =
\left\{ \begin{array}{ll}
p_H(\mu_B) \ , & \mu_B < \bar{\mu} \\
\left[p_H(\mu_B)-p_Q(\mu_B)\right]e^{-(\mu_B-\bar{\mu})^2/\Gamma^2}+p_Q(\mu_B) \ , & \mu_B > \bar{\mu} \end{array}
\right.
\label{eq:eoscross}$$ where $\bar{\mu}$ and $\Gamma$ are parameters controlling the onset and the width of the transition, respectively. This approach is equivalent to the crossover construction in the $\epsilon-n_B$ plane [@Masuda:2012ed] where a $\tanh$ function was used, but corrects the inappropriate construction suggested in [@Masuda:2012kf]. Note that in Refs. [@Blaschke:2013rma; @Blaschke:2013ana] the crossover construction was utilized for interpolating between quark matter EoS for two different values of the vector coupling thus mimicking its medium dependence. There the transition from the hadronic to the quark phase is seen as a sharp first order, while here we assume a smooth crossover.
![Left panel shows the crossover construction Eq. (\[eq:eoscross\]) in the $p-\mu_B$ plane. On the right panel we give the EoS used in this work.[]{data-label="fig:eos"}](DD2vsnlPNJL.eps "fig:"){width="45.00000%"} ![Left panel shows the crossover construction Eq. (\[eq:eoscross\]) in the $p-\mu_B$ plane. On the right panel we give the EoS used in this work.[]{data-label="fig:eos"}](eos_pepsilon.eps "fig:"){width="45.00000%"}
There is an obvious benefit from our approach. With the Maxwell construction $p$ as a function of energy density $\epsilon$ in the transition region is flat, making the EoS soft, while the crossover construction leads to a stiffening in the transition region.
We set $\bar{\mu}=\mu_c$, where $\mu_c$ is the onset of quark matter in the nl-NJL model and use the minimal possible $\Gamma$ consistent with causality. This leaves $\eta_V = G_V/G_S$ as a free parameter. The resulting EoS are shown in the right panel of Fig. \[fig:eos\].
Astrophysical implications
==========================
![The left panel shows sequences in the $M-R$ plane, while the right panel gives mass as a function of central density for these sequences. The orange diamonds represent the onset of the crossover region, while orange triangles denote its end corresponding to the onset of pure quark matter in the core of the neutron star. The orange plusses are the maximum mass configurations for the given EoS. []{data-label="fig:tov"}](massradii.eps "fig:"){width="45.00000%"} ![The left panel shows sequences in the $M-R$ plane, while the right panel gives mass as a function of central density for these sequences. The orange diamonds represent the onset of the crossover region, while orange triangles denote its end corresponding to the onset of pure quark matter in the core of the neutron star. The orange plusses are the maximum mass configurations for the given EoS. []{data-label="fig:tov"}](massdensity_v2.eps "fig:"){width="45.00000%"}
It is known that within the Maxwell construction scheme hybrid stars with small or almost zero vector coupling do not reach $2M_\odot$ before turning unstable [@Klahn:2013kga]. The softness of quark matter is then regulated by a strong vector coupling channel. However, the delay the quark matter onset caused by large vector couplings can be compensated by a strong diquark coupling. In total, if quark matter appears through a first order transition, model calculations indicate that stable stars may require large couplings in both vector and diquark channels.
In this work we limit ourselves to the region of small vector coupling ($\eta_V<0.17$) and offer an alternative mechanism that compensates such relative softness of the quark EoS. We solve the TOV equations by using the EoS with the crossover construction (\[eq:eoscross\]). The results shown in Fig. \[fig:tov\] are able to predict stable stars reaching and exceeding $2M_\odot$ already with a small $\eta_V$.
Our calculations show that quark matter appears for sequences with mass heavier than $M\sim M_\odot$, and central densities higher than $n_c \sim 2n_0$ where $n_0 = 0.16$ fm$^{-3}$. For vector couplings $\eta_V >0.05$ sequences with masses above $2M_\odot$ do not have pure quark matter in their cores. If we vaguely consider the crossover region as a mixed phase of hadrons and quarks, we might describe these stars to have a hadronic mantle and a mixed phase core. For mild vector coupling $\eta_V = 0.05$ the sequence with pure quark matter lies on the verge of stability, as shown by the triangle on the dashed line in the right panel of Fig. \[fig:tov\]. In addition, it is interesting to note that this sequence lies within the $1~\sigma$ band of both the $2M_\odot$ constraint for PSR J0348-0432 [@Antoniadis:2013pzd] and PSR J1614-2230 [@Demorest:2010bx].
The possibility of having only a mixed phase in the $M>2M_\odot$ stars is due to the tension between the vector coupling and the width of the crossover. With lower values of the vector coupling hadronic matter and quark matter EoS lie closer to each other in the $p-\mu_B$ plane so that a smaller crossover region is needed to achieve a causal EoS. By increasing the vector coupling the onset density of quark matter increases and the two EoS separate, requiring a larger crossover region.
Conclusions
===========
Requiring the transition from hadron to quark matter being unique, any viable hybrid EoS model should be able to fulfill the constraint of the recent observational lower limit of $2M_\odot$ for the maximum mass of the corresponding hybrid star sequence. For a wide class of NJL models this is not possible with the standard Maxwell construction unless quark matter is in a color superconducting state and has a strongly repulsive vector meanfield.
We have offered an alternative based on the requirement that the transition between quark and hadron matter is a crossover. We have found that hybrid star configurations reaching or even exceeding the $2 M_\odot$ mass constraint have cores comprised of a mixed phase of quarks and hadrons. This conclusion is similar to the one drawn when promoting the local charge neutrality condition to a global one via Gibbs construction [@Orsaria:2012je].
Our work might be considered as a first step towards a microscopically based construction of the transition from hadron to quark matter either via pasta phases or beyond mean-field studies taking into account the quark substructure of hadrons and their dissociation in the dense medium [@Wergieluk:2012gd; @Blaschke:2013zaa].
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are grateful for exciting discussions on the subject of crossover EoS constructions to T. Fischer, H. Grigorian, P. Haensel, T. Hatsuda, M. Hempel, T. Klähn, J. Lattimer, T. Maruyama, K. Masuda, I. Mishustin, J. Schaffner-Bielich, S. Typel, D. N. Voskresensky and N. Yasutake; and on neutron star radius measurements to M. C. Miller and J. Trümper. This work was supported in part by the COST Action MP1304 “NewCompStar” and by NCN within the “Maestro” programme under contract No. DEC-2011/02/A/ST2/00306. D.A-C. received funding by the Bogoliubov-Infeld programme for his collaboration with the University of Wroclaw. S.B. was supported by the Ministry of Science, Education and Sports of Croatia through contract No. 119-0982930-1016. D.B. acknowledges support by the Russian Fund for Basic Research under grant No. 11-02-01538-a and by EMMI for his participation in the Rapid Reaction Task Force Meeting “Quark Matter in Compact Stars” at FIAS Frankfurt.
[10]{}
Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature [**443**]{} (2006) 675. A. Bazavov [*et al.*]{}, Phys. Rev. D [**85**]{} (2012) 054503. A. W. Steiner, J. M. Lattimer and E. F. Brown, Astrophys. J. [**765**]{} (2013) L5. S. Bogdanov, Astrophys. J. [**762**]{} (2013) 96. \[arXiv:1211.6113 \[astro-ph.HE\]\]. P. Demorest [*et al.*]{}, Nature [**467**]{} (2010) 1081. J. Antoniadis [*et al.*]{}, Science [**340**]{} (2013) 6131. G.A. Contrera, M. Orsaria and N.N. Scoccola, Phys. Rev. D [**82**]{} (2010) 054026. T. Hell, K. Kashiwa and W. Weise, Phys. Rev. D [**83**]{} (2011) 114008. S. Benic, D. Blaschke, G.A. Contrera and D. Horvatic, arXiv:1306.0588. G.A. Contrera, A.G. Grunfeld and D. Blaschke, arXiv:1207.4890 \[hep-ph\]. S. Typel and H. H. Wolter, Nucl. Phys. A [**656**]{} (1999) 331. S. Typel, G. Ropke, T. Klahn, D. Blaschke and H. H. Wolter, Phys. Rev. C [**81**]{}, 015803 (2010) K. Masuda, T. Hatsuda and T. Takatsuka, PTEP [**2013**]{} (2013) 7, 073D01. K. Masuda, T. Hatsuda and T. Takatsuka, Astrophys. J. [**764**]{} (2013) 12. N. Yasutake [*et al.*]{}, arXiv:1208.0427 \[astro-ph.HE\]. T.K. Herbst, J.M. Pawlowski and B.-J. Schaefer, Phys. Lett. B [**696**]{} (2011) 58. D. Blaschke [*et al.*]{}, PoS ConfinementX [****]{} (2012) 249; \[arXiv:1302.6275 \[hep-ph\]\]. D. Blaschke, D. E. Alvarez-Castillo and S. Benic, PoS CPOD2013 (2013) 063. T. Klähn, D. Blaschke and R. [Ł]{}astowiecki, Phys. Rev. D [**88**]{} (2013) 085001. M. Orsaria, H. Rodrigues, F. Weber and G. A. Contrera, Phys. Rev. D [**87**]{} (2013) 023001. A. Wergieluk, D. Blaschke, Yu.L. Kalinovsky and A. Friesen, Phys. Part. Nucl. Lett. [**7**]{} (2013) in press; \[arXiv:1212.5245 \[nucl-th\]\].
D. Blaschke, D. Zablocki, M. Buballa and G. Röpke, arXiv:1305.3907 \[hep-ph\].
|
---
abstract: 'Large scale monitoring systems, enabled by the emergence of networked embedded sensing devices, offer the opportunity of fine grained online spatio-temporal collection, communication and analysis of physical parameters. Various applications have been proposed and validated so far for environmental monitoring, security and industrial control systems. One particular application domain has been shown suitable for the requirements of precision agriculture where such systems can improve yields, increase efficiency and reduce input usage. We present a data analysis and processing approach for distributed monitoring of crops and soil where hierarchical aggregation and modelling primitives contribute to the robustness of the network by alleviating communication bottlenecks and reducing the energy required for redundant data transmissions. The focus is on leveraging the fog computing paradigm to exploit local node computing resources and generate events towards upper decision systems. Key metrics are reported which highlight the improvements achieved. A case study is carried out on real field data for crop and soil monitoring with outlook on operational and implementation constraints.'
author:
- 'Grigore Stamatescu, Cristian Drăgana, Iulia Stamatescu, Loretta Ichim and Dan Popescu[^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'med19\_refs.bib'
title: '**IoT-Enabled Distributed Data Processing for Precision Agriculture** '
---
INTRODUCTION
============
Internet of Things (IoT) systems are based on distributed sensing, computing and communication devices that collaborate in order to monitor and control physical processes. These enable the collection of real world data at an unprecedented scale and resolution which can then be used to improve the models that define the understanding and help the forecasting of the processes, be it technical, social or environmental. New data processing infrastructure are thus needed to store and retrieve the information collected in an online manner while providing mechanisms to run the analysis and control algorithms based on this data. Beyond conventional environmental monitoring as initial key driver of IoT design, current domains include (smart) cities, industry and agriculture. Finally the outcomes of the analysis are either handled in closed loops for control actions or they are supplied to hierarchical entities for decision support.
Among the applications areas mentioned above, precision agriculture represents one of the salient areas where IoT-enabled systems can improve the quality, productivity and increase automation [@8355152]. Main challenges in this field relate to reducing input use: water, fertiliser, work, and obtaining better crop yields which is demanded by the market to keep food costs low under the strains of increasing global population. By having access to reliable, on-line information, relayed over distributed networks, domain specialists can oversee tangible improvements [@8372905].
The conceptual and practical challenges that we approach in the design of such systems is related to efficient data reduction and management which impacts directly the congestion and energy metrics of the deployed network. This is performed by proposing a hierarchical data processing architecture in accordance to fog computing design principles. Fog computing as a concept has initially emerged as a computing organisation alternative to leverage intelligent network edge devices which make up modern IoT systems [@anawar2018fog]. The limited computing resources available on these embedded devices are thus exploited to reduce the large quantities of collected data and transmit only higher level information pieces upstream. Given the large heterogeneity the processing primitive can run of the edge nodes range from basic threshold detection and averaging up to more advanced outlier detection and embedded learning algorithms. Wireless sensor networks (WSN) are an enabling technology to deploy fog computing systems [@8394851; @8679064] where hundreds to thousands of sensing nodes self organise intro and communicate over low power radio channels. As with the case with agriculture, large areas can thus be covered with multi-hop communication networks as the networking protocols rely on cluster heads, gateways and hubs serving as intermediary data concentrators. One alternative definition presents fog systems in opposition or as complementary to conventional centralised and large scale cloud infrastructures. The complex functionality of the cloud platform is broken down at the field level over functional or spatially distributed entities which collaborate to achieve a common monitoring, event-detection and control case. In the precision agriculture use case this can help implement an optimised distributed irrigation or fertiliser dosage schemes accounting for local properties and variance of soil, micro-climate and crop particularities. The need to integrate fog computing with cloud computing in this particular scenario lays with the fact that joint observations can be derived when federating high-level information across multiple farms.
The main novelty of the paper is justified by the application of fog computing data aggregation and modelling primitives in the context of IoT-enabled smart agriculture, a highly active area of research currently. The subsequent contributions of the paper can be argued:
- system architecture for hierarchical data processing and analysis based on field level IoT devices;
- data aggregation methodology based on the fog computing paradigm under precision agriculture constraints.
RELATED WORK
============
In [@guardo2018fog] a fog computing framework for precision agriculture is introduced. The two tiered system is able to reduce significantly the data transmitted in the network. Reducing the computational loads, and most important, the cloud computing costs associated with centralised processing is highlighted as an essential benefit of the fog approach. The authors of [@8521668] propose a hybrid IoT for smart farming in rural areas. The communication network uses 6LoWPAN local radio for the field interfaces while long range connections are implemented over WiFi. A 6LoWPAN border router and dedicated gateway are used to assure cross-domain integration of the networks from field level, intermediate long range relays and cloud. Network requirements for smart agriculture applications are also discussed in terms of throughput, latency and mobility support. These offer a good reference to quantify the data aggregation potential in conjunction with the sensing and control requirements. A distributed computing architecture is presented in [@ferrandez2018precision] which the agricultural system basic components such as: crop, soil, climate, water and nutrients, energy. The messaging system is standardised around the Message Queuing Telemetry Transport (MQTT) to interlink sensors, actuators, communication nodes, devices and subsystems [@7845442]. A decision tree is designed for irrigation control and integrated on the edge devices for in situ decision making. At the top level cloud services supply data through an end-user dashboard for high level decision support.
[@kamienski2019smart] introduce an intelligent irrigation system based on distributed sensor using the LoRA long range, low rate, nodes and gateways. The FIWARE infrastructure is leveraged as data management middleware platform which provides the support services. Several operation scenarios are discussed based on the scalability requirements, in terms of tens of thousands of nodes. Reference computational resource assessment for cpu, memory and network is also reported. Large scale IoT monitoring is discussed in [@popescu2018collaborative]. The focus is on the ground level clustering mechanisms that support the timely collection of data and generating of the field level monitoring events. Aerial robotic platform support is provided through suitable high level control of trajectories for data collection and backhaul. Data reduction is achieved by thresholding over locally computing moving averages in conjunction with expert knowledge adapted to the monitored processes. Several radio access technologies are available to achieve reliable transmissions [@7090210].
SYSTEM ARCHITECTURE AND METHODOLOGY
===================================
SYSTEM ARCHITECTURE FOR DATA COLLECTION AND PROCESSING
------------------------------------------------------
The proposed system architecture that we have designed for the purpose of efficient data collection and processing in precision agriculture is illustrated in Figure \[fig:perf1\]. It consists of the following information and physical layers: field layer, fog computing layer, cloud computing layer, data presentation layer, which are linked by cross-layer upstream and downstream data and control information flows. The layer functionality is detailed next:
- Field layer: includes the actual sensors deployed in the precision agriculture application to measure the physical parameters of interest; these include air temperature, air humidity, solar radiation, soil temperature at various depths, windspeed and rainfall; the field layer can also be expanded to accommodate intelligent actuators e.g. for irrigation or fine grained nutrient dosage, to execute commands incoming from higher level systems;
- Fog Computing layer: the fog nodes collect data from the sensors and run the data processing primitives for intelligent aggregation in order to reduce network traffic and energy expenditure; the main idea is to locally derive basic model characteristics of the particular process which are sent to the cloud in compact form; correlations between the sensed variables can also be exploited at this level for local decisions thus avoiding completely the increased cost and latency of the upper layers;
- Cloud computing layer: data is streamed towards a common cloud platform; regarding the particular implementation we use the ThingSpeak [@thingspeak] platform in conjunction with Matlab algorithm development for higher level processing routines; at the cloud layer the model parameters allow the reconstruction of the time series characteristics if needed, while accounting for the inherent modelling errors;
- Data presentation layer: is concerned with the front-end software systems that present the outcomes of the data analysis to end-users or decision makers with the ability to provide mobile access and timely alerts in the case of event detection; parametrisation of the process by domain experts is also achieved at this layer.
{width="0.7\paperwidth"}
A more detailed algorithm flowchart is provided in Figure \[fig101\]. It includes the steps for algorithm description which runs on the fog computing node.
![Fog Computing algorithm[]{data-label="fig101"}](img/algorithm.pdf){width="\columnwidth"}
In-field measurements are uploaded to the IoT application in two ways depending on the type of information: events and measurements. Note that, a primary batching procedure is usually available for most of the monitoring systems, basically consisting of performing minimum, maximum and mean value during a specific period of time. We consider this as the starting point for further local data processing.
*Primary batch aggregation* Note that, a primary batching procedure is usually available for most of the monitoring systems, basically consisting of performing minimum, maximum and mean value during a specific period of time. We consider this as the starting point for further local data processing.
For instance, batches are defined within 30 minutes. Once a new batch is available, $min, max$ and $mean$ values are computed (step A).
*Check for outliers procedure* For each batch of measurements, an outliers’ check procedure is performed, considering an acceptance bandwidth of data variance for the measured value around the mean (step B). The procedure outputs an event if the minimum or maximum values exceeds the thresholds. The event $E$ is defined as:
$$E=\left \{ e(x_{i})\in Q, T_{min}<x_{i}<T_{max} \right \}$$
where:
- $x_{i}$ is the measured value at iteration $i$
- $ T_{min}$ and $ T_{max}$ are thresholds computed as: $$T_{min}=mean(1-w)$$ $$T_{max}=mean(1+w)$$ where $w$ is a weight for acceptance bandwidth size define.
*Relevant data extraction* Aggregated data sets are achieved based on different methods. All seek for relevant data point, aiming to a reduced size set providing at the same time a satisfying reconstruction of the initial data.
One effective method, in terms of data volume, is based on using the $min$ and $max$ values extraction, computed for 24 hours. It is obvious that this method is suitable only for measurements that follow a regular shape during time, with insignificant variations during a day. A measurement for which this method is suitable is the soil temperature.
Instead, change detection is a common method applicable for irregular shaped data sets. This method follows extraction of data points where trend changes occur.
Given a set of data point $(x_{i}, y_{i}), i = 1, ..., n$, trend $t_{i}$ is followed for each pair $x_{i}, x_{i+1}$, such that for
$$\begin{aligned}
x_{i+1}- x_{i}>\delta \implies t(i)=1\\
x_{i+1}- x_{i}<\delta \implies t(i)=-1\\
x_{i+1} = x_{i} \implies t(i)=0
\end{aligned}$$
Then, if $t(i)\neq t(i+1)$ means that a trend change is detected. The coresponding data point $x_(i+1)$ is added to the relevant data set.
Relevant data extraction (step C) is performed when a set of primary aggregated batches is available.
DATA AGGREGATION
----------------
One reference method of extracting high level information from sensor data is Symbolic Aggregate Approximation (SAX) [@Keogh:2005:HSE:1106326.1106352]. It operates by assigning label symbols to segments of the time series thus porting it in a unified lower dimension representation. It belongs to the family of time series data mining techniques leading to non-parametric modelling. Ranges are identified through the data histogram or in a uniform manner. The method provides linear complexity and opens up the use and application of multiple statistical learning tools. Parametrisation of SAX is highly important by defining the number of segments and the alphabet size which can influence the quality and robustness of the result.
The background on which SAX has been defined is established by PAA [@Chakrabarti:2002:LAD:568518.568520] where symbols are attributed to the aggregated numerical values listed by PAA. Several discrete event models can incorporate the resulting aggregated segments e.g. Markov models in order to compute the probability of the observed patterns for future observations. According to the PAA method description, starting with a time series $X$ of length $n$, this is approximated into a vector $\bar{X}=(\bar{x}_1,...,\bar{x}_M)$ of any length $M\leq n$, with $n$ divisible by $M$. Each element of the vector $\bar{x}_i$ is calculated by:
$$\bar{x}_i=\frac{M}{n}\sum^{(n/M)i}_{j=n/M(i-1)+1}x_j$$
The dimensionality of the time series is thus reduced from $n$ to $M$ samples by initially dividing the original data into $M$ equally sized frame and then compute the mean values for each frame. A new sequence is achieved by putting the mean values together which is considered to be the PAA transform (approximation) of the original data. With regard to computational considerations, the PAA transform complexity can be reduced from $O(NM)$ to $O(Mm)$ with $m$ being the number of frames as tuning parameter of the method. The distance measure between two time series vector approximations $\bar X$ and $\bar Y$ is defined as:
$$D_{PAA}(\bar X,\bar Y)=\sqrt{\frac{n}{M}}\sqrt{\sum^M_{i=1}(\bar x_i - \bar y_i)}$$
It has been shown by the proposers of the method that PAA satisfies the lower bounding condition and guarantees no false dismissals such that:
$$D_{PAA}(\bar{X},\bar{Y}) \leq D(X,Y)$$
INTERPOLANT METHODS
-------------------
The Cloud-based application rebuilds data sets by estimates based on interpolation mechanisms. For performance evaluation we showcase three methods: the common linear interpolant (also referred as piecewise linear interpolant ) and two closely related interpolants, cubic *spline* and shape preserving *Piecewise Cubic Hermite Interpolating Polynomial (pchip)*.
Given a set of data points $\left ( x_{i}, y_{i} \right ), \left ( x_{i+1}, y_{i+1} \right ), ..., $ $\left ( x_{n}, y_{n} \right )$, the linear interpolation is defined as the concatenation of linear interpolants between each pair of data points, thus a set of straight lines between each data points. Any pair of data points with $x_{i}\neq x_{i+1}$ determines a unique polynomial $p$ of degree less than two whose graph passes through the two points with the property:
$$p(x_{i})=y_{i}$$
with the form:
$$p(x)=a_{1}x+a_{0}$$
a 1-D linear interpolation.
In general, given $n$ points $\left ( x_{i}, y_{i} \right ), i = 1, ..., n$, with disting $x_{i} $, a polynomial of degree less than $n$ whose graph passes through the $n$ points denoted $P_{n}(x)$, is expressed in the *Lagrange* form as:
$$P_{n}(x)=\sum_{i=1}^{n}\Bigg(\prod_{\begin{matrix}
{\scriptstyle j=1}\\
{\scriptstyle j\neq i}\\
\end{matrix}}^{n}\frac{x-x_{j}}{x_{i}-x_{j}}\Bigg)y_{i}
\label{poly_lag}$$
The Lagrange form in can be written out in power form of an interpolating polynomial as,
$$P_{n}(x)=a_{1}x^{n-1}+a_{2}x^{n-2}+ ... +a_{n-1}x+a_{n}
\label{poly_pow}$$
where the coefficients $a_{k}$ are computed through a system of linear equations: $$\begin{bmatrix}
x^{n-1}_{1} & x^{n-2}_{1} & ... & x_1 & 1\\
x^{n-1}_{2} & x^{n-2}_{2} & ... & x_2 & 1\\
\vdots & \vdots & \vdots & \vdots & \vdots \\
x^{n-1}_{n} & x^{n-2}_{n} & ... & x_n & 1\\
\end{bmatrix}
\begin{bmatrix}
a_{1}\\
a_{2}\\
\vdots\\
a_{n}
\end{bmatrix}=
\begin{bmatrix}
y_{1}\\
y_{2}\\
\vdots\\
y_{n}
\end{bmatrix}$$
Considering this, a piecewise linear interpolant is produced by first computing the divided difference:
$$\delta_{i}:=\frac{y_{i+1}-y{i}}{x_{i+1}-x{i}}
\label{div_dif}$$
Then the interpolant is constructed as:
$$P(x)=y_{i}+\delta_{i}(x-x_{i})$$
Further, for piecewise cubic polynomials, considering an interval $x_{i}\leq x\leq x_{i+1}$ let $h_{i} := x_{i+1}-x{i}$ be the length of an $i^{th}$ interval and $d_{k}:=P'(x_{i})$. Therefore, using this derivative it is possible to adjust the interpolant in order to enforce smoothness, by forcing the pair of derivatives from consecutive piecewice cubics to agree.
All piecewise cubic hermite interpolating polynomials are continuous and have a continuous first derivative. In particular, *spline* is oddly smooth, meaning that it’s second derative also varies continously.
Instead, *pchip* is not as smooth as *spline*, it is actually designed so that it never overshoots the data. The slopes are chosen so that $P(x)$ preserves the shape of data and also respects monotonicity.
EXPERIMENTAL RESULTS
====================
We collect experimental data from a network of field devices installed on site at an experimental research farm. Form the long term monitoring dataset we select a sample for analysis that covers one month of data. The data is preprocessed for missing values, noise removal and averaged over 30 minute intervals.
We first illustrate the application of the SAX method on the measured values for soil temperature and solar radiation in Figure \[fig11\] and Figure \[fig13\]. Segment levels codify the evolution of the respective time series and provide a compact representation with considerable impact on the data storage and transmission requirements at the fog node. Finer grained patterns can be observed by zooming in at the daily level as is illustrated in Figure \[fig13\]. Based on the selected segment labels, if the expected value deviates significantly by entering a different label segment, an event detection primitive can trigger a communication message from the node upstream.
![Symbolic Aggregation Approximation - Soil Temperature[]{data-label="fig11"}](img/soiltemp_sax.pdf){width="\columnwidth"}
![Symbolic Aggregation Approximation - Solar Radiation[]{data-label="fig12"}](img/solarrad_sax.pdf){width="\columnwidth"}
![Solar Radiation - Day level aggregation[]{data-label="fig13"}](img/solarrad_day4.pdf){width="\columnwidth"}
In order to evaluate reconstructed data consistency, achieved through different estimating algorithms, more precisely the proposed interpolants, some well known goodness-on-fit statistics are performed:
- Sum of squares of errors (SSE) - measures the total deviation of the response values from the fit to the response values and is defined as: $$SSE=\sum_{i=1}^{n}w_{i}(x_{i}-P(x_{i}))^{2}$$
where $w_{i}$ is the weight for the $i^{th}$ error between estimated $i^{th}$ value and the empiric data
- R-square - measures how successful the fit is in explaining the variation of the data and is expressed as: $$R-square=1-\frac{SSE}{SST}$$
where
$$SST=\sum_{i=1}^{n}w_{i}(x_{i}-\bar{x_{i}})^{2}$$
where $\bar{x_{i}}$ is the mean value of $x_{i}$ dataset.
- Root mean square error (RMSE) - is an estimate of the standard deviation of the random component in the data and is expressed as:
$$RMSE=\sqrt{\frac{SSE}{n}}$$
Results are summarised in Table I.
[|\*[4]{}[p[1.8cm]{}|]{}]{} & pchip RMSE & sline RMSE & interp1q RMSE\
soil temperature & 0.0852 & 0.1104 & 0.0795\
solar radiation & 0.2627 & 0.3691 & 0.2551\
![Soil Temperature[]{data-label="fig01"}](img/soil_temp.pdf){width="\columnwidth"}
![Solar Radiation[]{data-label="fig02"}](img/solar_rad.pdf){width="\columnwidth"}
![Solar Radiation - Histogram[]{data-label="fig013"}](img/solar_rad_hist.pdf){width="0.63\columnwidth"}
![Soil Temperature - Histogram[]{data-label="fig014"}](img/soil_temp_hist.pdf){width="0.63\columnwidth"}
Figure \[fig01\] and Figure \[fig02\] graphically depict the results of applying the alternative methods of interpolation on the two time series. In Figures \[fig013\] and \[fig014\] the histograms quantify the associated data reduction between the raw input data and the interpolant methods presented.
For this case, the monotonicity property of *pchip* is more desirable than the smoothness property of *spline*, which in some places overshoots the data, thus one may prefer the good behavior of the shape preserving *pchip* method. Note that, as with the linear interpolation, when there are two consecutive points with the same value, the interpolant is constant over that interval. This behaviour was expected and it is appropriate in this context.
Even if the metrics indicate better fitting for linear interpolation through the studied cases, one can choose the *pchip* method, given that the results are quite close and it does a much more visual pleasing representation, in particular better modelling the peeks and following the expected behaviour around the baseline.
CONCLUSIONS
===========
The paper presented a system architecture and distributed data processing application based on IoT in precision agriculture. By exploiting the dense spatial and temporal distributions of the sensing nodes, intelligent data reduction through aggregation and model reconstruction is illustrated for significants benefits for network congestion and energy efficiency. As the results achieved show promise, future work is focused on extensive evaluation for online decision making by domain experts in order to improve the reconstructed data quality.
[^1]: \*This work has been funded by the Romanian Space Agency (ROSA), through the project “Integrated Multi-Agent Aerial Robotic System for Exploring Terrestrial Regions of Interest” (MAARS), contract no. 185/2017.
[^2]: The authors are with the Department of Automatic Control and Industrial Informatics, University Politehnica of Bucharest, 060042 Bucharest, Romania. Grigore Stamatescu is also with the Institute of Technical Informatics, Technical University of Graz, 8010 Graz, Austria [gstamatescu@tugraz.at]{}
|
---
abstract: 'This paper introduces an Internet of Things (IoT)-based data acquisition and monitoring scheme for insulin pumps. The proposed work employs embedded system hardware (Keil LPC1768-board) for data acquisition and monitoring. The hardware is used as an abstract layer between the insulin pump and the cloud. Diabetes data are secured before they are sent to the cloud for storage. Each patient’s record is digitally signed using a secure hash algorithm mechanism. The proposed work will protect the patient’s records from being breached from unauthorized entities, and authenticates them from improper modifications. The design is tested and verified using $\mu$Vision studio, the Keil board mentioned above, and an ALARIS 8100 infusion pump. Moreover, a test case for a real cloud example is presented with the help of the Center of Computationally Assisted System and Technology. This center provided the infrastructure service to test our work.'
author:
-
bibliography:
- 'bibo.bib'
title: '**IoT-Based Secure Embedded Scheme for Insulin Pump Data Acquisition and Monitoring**\'
---
IoT; Security; Embedded system.
Introduction
============
The physical devices that are linked as Internet of Things (IoT) are continuously increasing [@al2015internet]. These devices are allowed to mimic human being’s senses. For example, the use of a smart home as an IoT-based application can turn on the air conditioning system when sensing residents are close [@atzori2010internet][@kazi2015iot]. The industrial world has moved toward the use of IoT by adding Internet connection to small electronic boards [@ungurean2014iot]. Moreover, the connections between different primitives are made possible through mobile communications [@hinge2013mobile].
Recent improvements in IoT design helps with the support of some health care systems. An example is tracking patient’s records and biomedical devices using the Internet [@catarinucci2015iot]. Medical devices for diabetic care have also joined the world of IoT through supporting versatile design options. However, security issues need to be addressed to ensure device security and the patient’s privacy. A system with an authentic security mechanism is required to guarantee the security of patient’s records. One of the existing methods that can be easily implemented in hardware is the Secure Hash Algorithm (SHA) [@harsha2014design]. The SHA is an official hash algorithm standard that was standardized by the National Institute of Standards and Technology (NIST). SHA is compatible with hardware-level implementation, and this makes it one of the more desirable methods for hardware designers to use to secure and authenticate their designs [@pub2012secure].
The implementation of IoT technology in hardware has become crucial for high performance. The benefit of using hardware to manipulate the increased size of patient’s records is that the hardware allows for high-speed computation to manipulate and retrieve records. Therefore, hardware designers have moved toward the use of IoT hardware units in their designs, these units support high-speed computation power for IoT related functions [@boppudi2014data]. This paper introduces an IoT-based embedded system for insulin pump data acquisition and monitoring. Data related to a patient’s diabetes disease along with other health records are stored on the cloud. All these data need to be secured and authenticated when they are retrieved from the cloud. We use the SHA mechanism in our design to support security and authentication.
The rest of the paper is organized as follows. Section \[sec:section2\] describes related work. The proposed methodology is presented in Section \[sec:section3\]. Results are detailed in Section \[sec:section4\]. Section \[sec:section5\] concludes the paper.
Related Work {#sec:section2}
============
There is a lot of recent work in the employment of IoT applications for health monitoring and control [@catarinucci2015iot][@harsha2014design][@rahmani2015smart; @hsueh2016next; @liu2015secure]. A novel IoT-aware smart architecture for automatic monitoring and tracking of the patient, personnel, and biomedical devices, was presented in [@catarinucci2015iot]. The proposed work built a smart hospital system relying on Radio Frequency Identification (RFID), Wireless Sensor Network (WSN), and smart mobile. The three hardware components were incorporated together through a local network, to collect surrounding environment and patient’s physiology related parameters. The collected data is sent to a control center in real-time, which makes all the data available for monitoring and management by the specialist through the Internet.
A smart E-health care system for ubiquitous health monitoring is proposed in [@rahmani2015smart]. The proposed work exploits ubiquitous health care gateways to provide a higher-level of services. The Gateways are the bridging point between IoT and applications (software or hardware). This work studied significant ever-growing demands that have an important influence on health care systems. The proposed work suggests an enhanced health care environment where control center burdens are transferred to the gateways. The security of this scheme is taken into consideration as the system deals with substantial health care data.
A personalized health care scheme for the next generation wellness technology has been proposed in [@hsueh2016next]. The security of patient’s records was addressed in case of data storage and retrieval over the cloud. The proposed work established a patient-based infrastructure allowing multiple service providers including the patient, service providers, specialists, and researchers to access the stored data. Their work was implemented on a cloud-based platform for testing and verification. Liu *et al*. [@liu2015secure] have presented a scheme for secure sharing of personal health records in the cloud. The health records are ciphered before they are stored in the cloud. The proposed work uses Cipher-Text Attribute-Based Signcryption Scheme (CP-ABSC) as an access control mechanism. Using this scheme, they are able to get fine-grained data access over the cloud.
The use of embedded micro-controllers for data monitoring and acquisition has also been previously explored. The Keil LPC1768 micro-controller has been used in two different schemes for medical device control [@harsha2014design][@boppudi2014data]. In [@harsha2014design], an online design of patient’s data monitoring system was presented. The proposed work employed an Advanced RISC Machine (ARM) Cortex M3 microprocessor embedded on the Keil LPC1768 board. Pulse, temperature, and gas sensors were used to collect the patient’s medical parameters. The LPC1768 board was used as the hardware layer between the Internet and the medical sensors. A data acquisition and control system using the ARM Cortex M3 microprocessor was also presented in [@boppudi2014data]. Monitored sensor data are sent to the Internet using an Ethernet-controlled interface. The interface was built using an Keil LPC1768 board. The proposed work employed two sensing devices (temperature and accelerator-meter) to collect data from the surrounding environment. The collected readings are sent to the Internet through the Ethernet interface. According to the uploaded readings, a specialist can change the behavior of the device through an Internet browser.
In the next section, we will present our proposed IoT-based secure data acquisition and monitoring scheme. The integration between the embedded architecture and the cloud-computing based storage will be discussed in detail.
Proposed Methodology {#sec:section3}
====================
We provide some background information before getting into the details of the proposed methodology. In the subsequent text, we provide a brief description of the secure hash standard.
Background: Secure Hash Algorithm
---------------------------------
SHA takes a message with an arbitrary size and produces a message hash through some calculations. The process is defined in equation (\[eq:hash\]).
$$h = H(M)
\label{eq:hash}$$
where, $M$ is the input message and $h$ is the digest generated using the hash algorithm $H$.
In our scheme, the SHA2/256 standard is employed for securing the patient’s records. Data are signed with the SHA2/256 before they are stored in the cloud. The stored records are made available for research centers and medical institutions. Figure \[fig:sha\] depicts the general procedure that is used to compute the hash for any given message. The input message of size less than $2^{64}$ is padded first to make its size a multiple of $512$. The full message is divided into equal size blocks of 512-bits. The blocks are then processed sequentially using compression function $F$, and Initial Hash Value ($IHV_0$) that are defined in [@pub2012secure].
![SHA general architecture: Padding, dividing, compression, and computation of SHA-256.[]{data-label="fig:sha"}](sha.pdf){width="\columnwidth"}
At the end of the process, the hash value that is generated from the last block produces the final $256$ bits hash. A detailed description of the secure hash algorithm can be found in [@pub2012secure].
General Architecture of the Proposed Scheme
-------------------------------------------
The proposed design for the secure IoT-based embedded system is depicted in Figure \[fig:general\]. A patient using the Alaris 8100 infusion pump will take preset insulin doses regularly. The Alaris infusion pump is controlled and monitored by the Keil Cortex M3 board through a serial connection. All dosage related records that are given to the patient are sent to the cloud through the Keil board using an Ethernet connection. To secure and authenticate the recorded data, they are digitally signed using the SHA compression function. The signature and patient’s records are then stored in the cloud.
In the cloud, a Secure Socket Shell (SSH) is provided to entities authorized to access the data. For instance, a physician can follow up with a patient’s case using a mobile application or a web browser. Furthermore, research institutions are given the authorization to access health records upon agreements made between patient, medical centers, and research institutions.
![General architecture of the proposed scheme.[]{data-label="fig:general"}](general2.pdf){width="\columnwidth"}
The security and authenticity of the health care records are verified using the SHA signature. The SHA value is computed after the health records or prescription commands are generated. Then the generated SHA is appended to the corresponding data (health record or preset control command). The health record and its signature are remained correlated in all places (cloud, hospital, and patient’s side). For instance, the physician in the hospital confirms that the record is received without altering using SHA signature. When the health record is received at the hospital, SHA computation will be carried out. The resultant SHA will be compared with the appended SHA value. Once both SHA values are equal, the record will be affirmed to its corresponding patient. Otherwise, the health record will be discarded as is does not belong to the patient.
In the case of the preset control command, this command is generated from the hospital and appended with its corresponding hash value. The preset control command and the SHA signature are sent through the cloud to the infusion pump. At the patient’s end, the hardware takes the responsibility to check the genuineness of the received control command by SHA computation and comparison. The Keil micro-controller computes the SHA value for the received preset control command and then compares the result with the appended SHA value. Once authorized, the preset control command is passed to the infusion pump for a new schedule. Figure \[fig:Circuit\] shows the connection between the Keil LPC1768 board and the Alaris 8100 infusion pump.
![Connection of Alaris Infusion Pump 8100 with Keil 1768 PCB board[]{data-label="fig:Circuit"}](Circuit.jpg){width="\columnwidth"}
In the case of a fault exception, all Cortex-M processors (including Keil LPC1768) have a fault exception mechanism embedded inside the processor. If any fault is detected, the corresponding exception handler will be executed [@alkim2016newhope]. The hardware setup was done at the NDSU-Electrical and Computer Engineering laboratories. The infusion pump was first disassembled, interfaced with the Keil micro-controller, and then programmed using a serial cable and the Keil $\mu$Vision studio 5.
Results and Discussion {#sec:section4}
======================
The proposed design has been tested and verified using data from [@BibEntry2018Oct]. The sample data includes glucose levels in the patient’s body during a 24 hour period, a patient’s profile information, and the patient’s medical information. A snipped portion of the sample data is shown in Figures \[fig:sample\_orig\]. Figure \[fig:sample\_mod\] shows the modified sample data. The data are stored in the cloud regularly, where each copy has its designated SHA value. To ensure the integrity of data, SHA-256 is applied to both sides (cloud and patient) after any query from either side.
![Snipped health record from the original sample[]{data-label="fig:sample_orig"}](sample_orig)
![Snipped health record from the modified sample[]{data-label="fig:sample_mod"}](sample_mod)
The record is valid if its generated hash value on the patient’s side is the same as the hash value on the cloud side. Table \[tab:sample\] shows a valid record hash. However, two different hash values are depicted for the same record in Table \[tab:sample\_modified\], because the received record on the patient’s side has been altered. Accordingly, the corresponding hash value has also been altered. The micro-controller will detect the alteration and discard the received record.
[ll]{} **Cloud side**:&
----------------------------------------
`14b93acf-ccdcbe40-ea3795be-c1073498-`
`51a96c90-6cedfc9c-49d8e2cf-a141befb`
----------------------------------------
: SHA-256 HASH VALUES OF THE SAMPLE DATA ON BOTH SIDES.
\
**Patient side**:&
----------------------------------------
`14b93acf-ccdcbe40-ea3795be-c1073498-`
`51a96c90-6cedfc9c-49d8e2cf-a141befb`
----------------------------------------
: SHA-256 HASH VALUES OF THE SAMPLE DATA ON BOTH SIDES.
\
\[tab:sample\]
[ll]{} **Cloud side**:&
----------------------------------------
`14b93acf-ccdcbe40-ea3795be-c1073498-`
`51a96c90-6cedfc9c-49d8e2cf-a141befb`
----------------------------------------
: SHA-256 HASH VALUES OF THE ORIGINAL AND MODIFIED SAMPLE DATA ON BOTH SIDES.
\
**Patient side**:&
----------------------------------------
`358c4f29-f0e2bb60-8efa35d4-a88a6b3b-`
`58939ffd-deebf824-8065c195-b834b8cd`
----------------------------------------
: SHA-256 HASH VALUES OF THE ORIGINAL AND MODIFIED SAMPLE DATA ON BOTH SIDES.
\
\[tab:sample\_modified\]
The patient’s intervention for the proposed design is limited to turn on and off the infusion pump. However, the future work of our design will upgrade the patient’s privileges, e.g., change the infusion pump schedule according to predefined levels.
Conclusion and Future work {#sec:section5}
==========================
We have presented a secure IoT-based embedded data acquisition and control scheme. The work employed three modules: Keil micro-controller, LPC1768 board, and Alaris 8100 infusion pump. Secure Hash Algorithm standard SHA-256 is used to ensure the authenticity of the system. The authenticity of the proposed work was verified with a cloud storage utility using a real sample record. The results show that any altering in the health record is going to be identified immediately, thus the patient remains safe from false prescriptions. In future, we plan to apply the proposed scheme to hand-held glucose devices.
Acknowledgment {#acknowledgment .unnumbered}
==============
This publication was funded by a grant from the United States Government and the generous support of the American people through the United States Department of State and the United States Agency for International Development (USAID) under the Pakistan - U.S. Science & Technology Cooperation Program. The contents do not necessarily reflect the views of the United States Government.
|
---
abstract: 'Thin accretion discs around massive compact objects can support slow pressure modes of oscillations in the linear regime that have azimuthal wavenumber $m=1$. We consider finite, flat discs composed of barotropic fluid for various surface density profiles and demonstrate–through WKB analysis and numerical solution of the eigenvalue problem–that these modes are stable and have spatial scales comparable to the size of the disc. We show that the eigenvalue equation can be mapped to a Schrödinger-like equation. Analysis of this equation shows that all eigenmodes have discrete spectra. We find that all the models we have considered support negative frequency eigenmodes; however, the positive eigenfrequency modes are only present in power law discs, albeit for physically uninteresting values of the power law index $\beta$ and barotropic index $\gamma$.'
author:
- |
Tarun Deep Saini$^{1,3}$, Mamta Gulati$^{1,2,4}$, S. Sridhar$^{2,5}$\
$^{1}$ Indian Institute of Science, Bangalore, Karnataka, India, 560 012\
$^{2}$ Raman Research Institute, Sadashivanagar, Bangalore, Karnataka, India, 560 080\
$^{3}$ tarun@physics.iisc.ernet.in\
$^{4}$ mgulati@rri.res.in\
$^{5}$ ssridhar@rri.res.in
title: Slow pressure modes in thin accretion discs
---
accretion discs; hydrodynamics; waves; methods: analytical
Introduction
============
Low-mass discs orbiting massive compact bodies are a feature of many astronomical systems. When the dynamics of a disc is dominated by the Newtonian gravitational force of the central body, the disc may be considered nearly Keplerian. In a purely Keplerian potential eccentric orbits do not precess because the orbital frequency is equal to the epicyclic frequency. In a nearly Keplerian disc there is a small difference between the orbital and epicyclic frequencies. This could be due to the self-gravity of the disc, thermal pressure in a gas disc, and random motions in a collisionless disc. This difference in frequencies manifests as a precession of eccentric orbits at rates that are small compared to the orbital and epicyclic frequencies. Then the disc may be able to support large-scale, slow, lopsided modes [@kat83; @sst99; @lg99; @st99]. In the linear regime, these modes have azimuthal frequency, $m=1$, whose first systematic investigation is due to @tre01. He studied slow modes in various types of discs (fluid, collisionless and softened gravity), with the focus largely on the effect of the self-gravity of the disc. In particular, a WKB analysis was used to show that the fluid disc can support large-scale slow modes when the Mach number, ${\mathscr M}$, is much larger than the Toomre $Q$ parameter (both parameters are defined in § 2). The assumption behind this analysis is that the self-gravity of the disc dominates fluid pressure. However, such is not the case for thin accretion discs around white dwarfs and neutron stars. Indeed, for a disc around a white dwarf [@fkr02], we can estimate ${\mathscr M}\sim 50$ and $Q\sim 10^{10}$; hence the analysis of [@tre01] is not directly applicable. The goal of this paper is to study large-scale $m=1$ slow modes in thin accretion discs, where $Q\gg {\mathscr M}\gg 1$.
In § 2 we use the WKB approximation to establish that pressure, in the absence of self-gravity, can enable slow, $m=1$ modes in thin accretion discs. The linear eigenvalue problem for slow pressure modes (or “p-modes”) is formulated in § 3 for a flat barotropic disc, which is axisymmetric in its unperturbed state. When appropriate boundary conditions are chosen, the eigenvalue equation reduces to a Sturm-Liouville problem. Since the differential operator is self-adjoint, the eigenfrequencies are real: therefore all p-modes are stable, and the eigenfunctions form a complete set of orthogonal functions. We also map the eigenvalue equation into a Schrödinger-like equation, which is useful in the interpretation of our numerical results. In § 4 we present numerical results for a variety of discs, namely (i) an approximation to the Shakura-Sunyaev thin disc, (ii) the Kuzmin disc which is more centrally concentrated, (iii) power-law discs. Of particular interest is the nature of the eigenfrequency; whether it is positive or negative. This has bearing on the excitation of these modes, because they are stable and will not grow spontaneously through a non-viscous instability. Comparison with earlier work, summary and conclusions follow in § 5.
Slow pressure modes
===================
Consider a flat thin disc of fluid orbiting a central mass $M$. The fluid is assumed to be barotropic and the disc is described by a surface density profile ${\Sigma}$. In our analysis we ignore viscous forces, assuming that they adjust to maintain a quasi-stationary flow with a small radial velocity, and have little effect on the perturbed flow. Thus, we start with the continuity and Euler equations in cylindrical polar coordinates $$\begin{aligned}
&&{\frac{\partial{\Sigma}}{\partial{t}}}+ \frac{1}{R}{\frac{\partial{}}{\partial{R}}}(R\Sigma v_R) + \frac{1}{R}{\frac{\partial{}}{\partial{\phi}}}(\Sigma v_{\phi}) = 0\,,
\nonumber\\
&&{\frac{\partial{v_R}}{\partial{t}}}+v_R{\frac{\partial{v_R}}{\partial{R}}}+\frac{v_{\phi}}{R}{\frac{\partial{v_R}}{\partial{\phi}}}-\frac{{v_{\phi}}^2}{R}=
- \frac{GM}{R^2} -{\frac{\partial{}}{\partial{R}}}
(\Phi+h)\,,\nonumber\\
&&{\frac{\partial{v_{\phi}}}{\partial{t}}}+v_R{\frac{\partial{v_{\phi}}}{\partial{R}}}+\frac{v_{\phi}}{R}{\frac{\partial{v_{\phi}}}{\partial{\phi}}}+\frac{{v_R}{v_{\phi}}}{R}
= -\frac{1}{R} {\frac{\partial{}}{\partial{\phi}}}(\Phi+h)\,,
\label{eq:euler}\end{aligned}$$ where $v_R$ and $v_{\phi}$ are the radial and azimuthal components of the fluid velocity, $h$ is the enthalpy per unit mass, and $\Phi$ is the gravitational potential due to the disc. For a barotropic fluid with an equation of state given by $p=D{\Sigma}^{\gamma}$ (where $D>0$ is a constant), the isentropic sound speed and enthalpy are given by $$\begin{aligned}
&&{c_s}^2 = \gamma\,D\,\Sigma^{\gamma-1}\,,\\
\label{eq:enthalpy}
&&h=\frac{\gamma{D}}{\gamma-1}{\Sigma}^{\gamma-1} =\frac{c_s^2}{\gamma-1}\,,\end{aligned}$$
Precession rate in the unperturbed disc
---------------------------------------
We assume that the radial velocity of the unperturbed flow is much smaller than the azimuthal velocity and set it identically equal to zero; this assumption is justified below at the end of § 2.2. The unperturbed disc is assumed to be axisymmetric, therefore all $\phi$ derivatives are set to zero. Gas flows along circular orbits, with centrifugal balance maintained largely by the gravitational attraction of the central mass (with small but non trivial contributions from gas pressure and disc self-gravity). The azimuthal and radial frequencies, $\Omega >0$ and $\kappa >0$ respectively, associated with nearly circular orbits are given by $$\begin{aligned}
\Omega^2 &=& \frac{GM}{R^3} + \frac{1}{R}{\frac{d}{dR}}(\Phi_0 + h_0)\,,\nonumber\\
\kappa^2 &=& \frac{GM}{R^3} + \frac{d^2}{dR^2}(\Phi_0 + h_0) + \frac{3}{R}{\frac{d}{dR}}(\Phi_0 + h_0)\,,
\label{eq:freqs}\end{aligned}$$ where the subscript ‘$0$’ indicates unperturbed quantities. The Mach number of the flow ${\mathscr M}(R)= R\Omega(R)/c_s(R) \gg 1$. The dominant contribution to both $\Omega^2$ and $\kappa^2$ is due to the central mass, with small corrections coming from the disc self-gravity $(\Phi_0)$ and enthalpy $(h_0)$. Let us define the small parameter $\epsilon\ll 1$ as the larger of $(\Sigma_0 R^2/M)$ and $(h_0R/GM)$. The apsides of the nearly circular orbit of a fluid element precesses at a rate given by, $$\begin{aligned}
{\dot{\varpi}}&=& \Omega - \kappa\nonumber\\
&=& -\frac{1}{2\,\Omega} \left( \frac{d^2}{dR^2} + \frac{2}{R}{\frac{d{}}{d{R}}}\right)
\left(\Phi_0 + h_0\right) + O(\epsilon^2)\,.
\label{eq:pomegadot}\end{aligned}$$ Note that, in contrast to @tre01, we have retained the contribution from gas pressure (i.e. enthalpy). In fact, in thin accretion discs around compact stars, disc self-gravity is negligible and the contribution to ${\dot{\varpi}}$ is almost entirely from gas pressure. The goal of this section is to establish that these discs have large-scale slow modes driven only by pressure. In the WKB analysis of linear modes given below we follow the presentation due to @tre01.
The WKB approximation
---------------------
We consider linear perturbations (of the velocity, surface density etc) of the form $$A(R)\exp{\left[i\left(\int^R k(R) dR + m\phi - \omega t\right)\right]}\,,$$ where $k(R)$ and $m$ are the radial and azimuthal wavenumbers, respectively, and $\omega$ is the angular frequency of the mode. In the tight-winding limit where $\vert k(R)R\vert\gg 1$, a dispersion relation between $\omega$ and $k(R)$ can be derived [@saf60; @bt08]: $$\left(\omega - m\Omega\right)^2 = \kappa^2 - 2\pi G\Sigma_0\vert k\vert +
c_s^2k^2\,.
\label{eq:drfull}$$ The disc is stable to axisymmetric $(m=0)$ perturbations if and only if $$Q\equiv \frac{c_s\kappa}{\pi G\Sigma_0} > 1\,.
\label{eq:q}$$ This is readily satisfied in thin accretion discs around compact stars. @tre01 showed that the dispersion relation for $m=1$ modes can be written as $$\omega = {\dot{\varpi}}+ \frac{\pi G\Sigma_0\vert k\vert}{\Omega} -
\frac{c_s^2k^2}{2\Omega} + \frac{1}{\Omega}O({\dot{\varpi}}^2, \omega^2)\,.
\label{eq:dr}$$
In @tre01 it is argued that, when the pressure is negligible compared to disc self-gravity (i.e. $c_s\approx 0$), the WKB dispersion relation of equation (\[eq:dr\]) admits large-scale ‘$\vert k(R)R\vert \sim 1$’ modes with frequencies $\omega\sim{\dot{\varpi}}\sim
(\Sigma_0 R^2/M)\Omega$. It may be verified that the condition of negligible pressure implies that the Mach number ${\mathscr M}\gg Q$. However, as we have argued in the introduction, this inequality is violated for thin accretion discs around compact stars where the opposite is true, i.e. $Q\gg {\mathscr M}\gg 1$. Hence we need to consider a situation that is complementary to the analysis of @tre01. Disc self-gravity being negligible in accretion discs, the precession rate is determined entirely by the gas pressure. Then equation (\[eq:pomegadot\]) can be written as $${\dot{\varpi}}= -\frac{1}{2\,\Omega} \left[\frac{d^2h_0}{dR^2} +
\frac{2}{R}{\frac{d{h_0}}{d{R}}}\right] + O(1/{\mathscr M}^4) \sim \frac{\Omega}{{\mathscr M}^2}\,,
\label{eq:pomh}$$ (see @kat83) and we can approximate equation (\[eq:dr\]) as $$\omega = {\dot{\varpi}}- \frac{c_s^2k^2}{2\Omega} + \frac{1}{\Omega}O({\dot{\varpi}}^2, \omega^2)\,.
\label{eq:drslow}$$ For a disc with a non zero inner radius, eigenmodes must satisfy the Bohr-Sommerfeld quantization condition, given by $$\oint k(R)\,dR = \left(n+\frac{3}{4}\right)2\pi \,,\qquad n=0,1,2,\ldots
\label{eq:quant}$$ Thus, there exists a [*prima facie*]{} case for modes with frequencies $\omega\sim{\dot{\varpi}}$, with radial wavenumbers, $$k(R)\sim \left\vert\frac{\Omega{\dot{\varpi}}}{c_s^2}\right\vert^{1/2} \sim \frac{1}{R}\,,
\label{eq:k}$$ comparable to the radial scale of the disc. However, this tentative conclusion is based on a WKB analysis which may not be valid for modes with $k(R)R\sim 1$. This is the motivation for our studies of the eigenvalue problem for slow modes given below. Henceforth we ignore disc self-gravity altogether and consider only the effect of gas pressure.
We now justify the assumption made at the beginning of § 2.1, that the radial velocity in the unperturbed disc is small, and may be ignored when studying slow modes. The time scale of radial spreading of the disc is, $t_{vis}\sim R^2/\nu \sim
\mathscr{M}^{2}/(\alpha\Omega)$, where $\alpha \ll 1$ is the Shakura–Sunyaev viscosity parameter. The frequency of a slow mode is $\omega\sim {\dot{\varpi}}\sim \Omega\mathscr{M}^{-2}$. Therefore $\omega
t_{vis}\sim \alpha^{-1}\gg 1$ implies that the radial spreading occurs over many slow mode periods.
Formulation of the eigenvalue problem
=====================================
Eigenvalue equation
-------------------
The linearized Euler, continuity and enthalpy equations that govern the perturbed flow are $$\begin{aligned}
&&{\frac{d{v_{R1}}}{d{t}}} - 2\Omega(R){v_{\phi 1}}=-{\frac{\partial{h_1}}{\partial{R}}}\,,
\label{eq:pert1}\\
&&{\frac{d{v_{\phi 1}}}{d{t}}} -2B(R)v_{R1} = -\frac{1}{R}{\frac{\partial{h_1}}{\partial{\phi}}}\,, \label{eq:pert2}\\
&&{\frac{d{{\Sigma}_1}}{d{t}}} + \frac{{\Sigma}_0}{R}{\frac{\partial{v_{\phi 1}}}{\partial{\phi}}} + \frac{1}{R}{\frac{\partial}{\partial{R}}}(R{\Sigma}_{0}v_{R1}) = 0,\label{eq:pert:cont}\\
&&h_1 = c^2_{s0}\frac{{\Sigma}_1}{{\Sigma}_0}, \label{eq:pert:eth}\end{aligned}$$ where the subscript ‘0’ stands for the unperturbed quantities and ‘1’ for the first order perturbed quantities. The Oort’s parameter $B(R)$ is related to the epicyclic frequency through $\kappa^2(R) =
-4\Omega(R) B(R)$, and $d/dt = (\partial/\partial t +
\Omega\partial/\partial\phi)$ is the convective derivative with respect to the unperturbed flow.
We consider non-axisymmetric perturbations with azimuthal wave number $m=1$, of the form $T_1=T_{a}(R) \exp\left[ i(\phi-\omega t)\right]$, where $T_1$ stands for any perturbed quantity. Substituting this form in equations (\[eq:pert1\]), (\[eq:pert2\]), (\[eq:pert:cont\]) and (\[eq:pert:eth\]) yields $$\begin{aligned}
&&i(\Omega-\omega){v_{Ra}}-2\Omega{v_{\phi a}}+{\frac{d{h_a}}{d{R}}}=0\,\label{eq:pert:11}\,,\\
&&i(\Omega-\omega){v_{\phi a}}-2B{v_{R a}}+\frac{i h_a}{R} =0\,\label{eq:pert:21}\,,\\
&& i (\Omega - \omega) {\Sigma}_a + \frac{i{\Sigma}_0}{R} v_{\phi a} +\frac{1}{R} {\frac{d}{dR}}(R{\Sigma}_0 v_{Ra})=0\,,\label{eq:pert:cont1}\\
&& h_a = c^2_{s0}\frac{{\Sigma}_a}{{\Sigma}_0}\,.\label{eq:pert:eth1}\end{aligned}$$ Solving equations (\[eq:pert:11\]) and (\[eq:pert:21\]) for the velocity amplitudes we obtain $$\begin{aligned}
v_{R a} &=& -\frac{i}{\Delta}\left[(\Omega-\omega){\frac{d{h_a}}{d{R}}}+\frac{2\Omega}{R}h_a\right]\,,\label{eq:pert:12}\\
v_{\phi a} &=& \frac{1}{\Delta}\left[-2B{\frac{d{h_a}}{d{R}}}+\frac{\Omega-\omega}{R}h_a\right]\,,\label{eq:pert:22}\end{aligned}$$ where $ \Delta = \kappa^2-(\Omega-\omega)^2$. These equations, along with equation (\[eq:pert:eth1\]), when substituted in (\[eq:pert:cont1\]) yields $$\begin{aligned}
&&\Bigg[ {\frac{d^2}{dR^2}}+ \left\lbrace {\frac{d}{dR}}\ln\left(\frac{R{\Sigma}_0}{\Delta}\right)\right \rbrace {\frac{d}{dR}}+\nonumber\\
&&
\frac{2\Omega}{R(\Omega - \omega)}\left\lbrace {\frac{d}{dR}}\ln\left(\frac{{\Sigma}_0 \Omega}{\Delta}\right)\right \rbrace -
\frac{1}{R^2}\Bigg]h_a = \frac{h_a\Delta}{c^{2}_{s0}}\,.\label{eq:gold}\end{aligned}$$ which is the eigenvalue problem for [*undriven modes*]{}, with eigenvalue $\omega$ and eigenfunction $h_a$. This equation is a special case of equation (13) of @gold79 [@tlai09], where $m = 1$, and the external and self gravity perturbations are set equal to zero. It can be noted that this equation becomes singular at $\Omega = \omega$ and $\Delta = 0$. The former corresponds to the corotation resonance and the latter corresponds to the Lindblad Resonances (LR). Below we discuss the validity of equation (\[eq:gold\]) at LR; a similar analysis holds for the singularity at corotation radius but we do not discuss it in this paper since, as is argued later, for the slow modes the corotation radius has to lie outside the disc.
The system of equations (\[eq:pert:11\])—(\[eq:pert:eth1\]) describe an undriven, autonomous system. At the Lindblad resonance the algebraic equations (\[eq:pert:11\]) and (\[eq:pert:21\]), become indeterminate if no conditions are imposed on the enthalpy perturbations $h_a$. It is easily seen that these equations become consistent if $$-\frac{i(\Omega - \omega)}{2B} = \frac{2i\Omega}{(\Omega - \omega)} = \frac{dh_a/dR}{ih_a/R}\,.$$ The first equality follows from $\Delta = 0$. Rearranging the second equality yields $$\left[ {\frac{d}{dR}}(R^2 h_a) - \frac{\omega}{\Omega}\left(R^2 {\frac{d{h_a}}{d{R}}}\right)\right]\Bigg|_{\rm LR} = 0.\label{eq:ILR:1}$$ This condition must be satisfied at the Lindblad resonances for all undriven modes. However, equation (\[eq:ILR:1\]) may not be satisfied if the disc is driven by external forcing and may lead to curious dynamics around the LR and transport of angular momentum away from the LR due to external torquing [@gold79]. In this work we confine our investigations to free modes of an undriven disc, therefore, as the above discussion shows, nothing special happens at the LR.
[**Slow Mode Approximation**]{}: We now make the ansatz that the perturbed flow supports frequencies that are small in comparison to the circular frequency, i.e., $\vert\omega\vert\ll\Omega$. Therefore, when $\omega\neq 0$, the disc must be finite, with outer radius such that the orbital frequency at the outer edge is much greater than $\vert\omega\vert$. Applying the slow mode approximation to equation (\[eq:gold\]) we obtain $$\frac{c^2_{s0}R^{3/2}}{{\Sigma}_0 \Omega}{\frac{d}{dR}}\left(\frac{{\Sigma}_0 \Omega}{ R^{3/2}
\,\Delta}{\frac{d}{dR}}(R^2 h_a)\right) = R^2 h_a\,.\label{eq:apgold}$$ Similar to equation (\[eq:gold\]), equation (\[eq:apgold\]) too is singular at the Lindblad resonances, however the singularity at the corotation radius has gone away since this equation has been derived under the slow mode condition, $\vert\omega\vert\ll\Omega$. The condition $\Delta=0$ implies that at some radius, either $\omega=\Omega -\kappa$ or $\omega=\Omega +\kappa$. Since $\kappa
\simeq \Omega + O(\epsilon^2)$, we see that the second equality cannot be satisfied under the slow mode approximation. It is straightforward to see that the radius where this would be satisfied would be larger than the corotation radius due to the fact that the Keplerian circular frequency falls off monotonically with radius. Therefore, there are no outer Lindblad resonance singularities for slow modes. However, the Inner Lindblad Radius (ILR), where $\omega={\dot{\varpi}}(R)$, could very well lie inside the disc. Due to the fact that the disc surface density is completely arbitrary, there could in general be more than one ILRs. To make the problem well posed under the slow mode approximation, at the ILRs, the condition (\[eq:ILR:1\]) reduces to $${\frac{d}{dR}}{(R^2 h_a)} \bigg |_{\rm ILR} = 0\,.\label{eq:ILR}$$ We shall see later that the numerical solutions of equation (\[eq:eigen\]) satisfy this condition and the velocity amplitude at the ILR remains finite; thus the linear approximation remains valid and nothing special happens at the ILR.
From equation (\[eq:pomh\]), ${\dot{\varpi}}/\Omega \simeq O({\mathscr
M}^{-2})$. This allows us to approximate $\kappa^2 \simeq \Omega^2$, leading to $B\simeq -\Omega/4$. Using these in equation (\[eq:pert:22\]) we obtain $$\begin{aligned}
\label{eq:vphi}v_{\phi a} &=& \frac{\Omega}{2\Delta}\left[{\frac{d{h_a}}{d{R}}}+\frac{2h_a}{R}\right]\,,\nonumber\\
&=& \frac{\Omega}{2\Delta} \frac{1}{R^{2}} {\frac{d{}}{d{R}}} \left (R^{2}h_a \right)\,.\end{aligned}$$ Differentiating equation (\[eq:apgold\]) and using equation (\[eq:vphi\]) we obtain, $$\label{eq:eigen} {\frac{d}{dR}}\left[\left(\frac{c^2_{s0} R^{3/2}}{{\Sigma}_0 \Omega}\right) {\frac{d{\Theta}}{d{R}}}\right] + \frac{2R^{3/2}}{{\Sigma}_0} ({\dot{\varpi}}-\omega)\Theta = 0\,,$$ where we have used the variable $\Theta=R^{1/2}{\Sigma}_0\,v_{\phi a}$ and $\Delta \simeq 2\Omega(\omega-\dot{\varpi})$, which is valid under the slow mode approximation.
Slow modes as a Sturm-Liouville problem
---------------------------------------
Before we proceed to specific examples, we have to choose the boundary conditions that we impose to solve equation (\[eq:eigen\]). We first cast the equation in a dimensionless form by choosing a radius ${R_{\star}}$, at which we evaluate various quantities, ${\Sigma}_{\star}, c_{s\star},
{\dot{\varpi}}_{\star}$ and $\Omega_{\star}$. We introduce the parameter $x=R/{R_{\star}}$, and similarly for a quantity $H$, we use $H'=H/H_{\star}$, leading to the Sturm-Liouville form of the eigen equation $$\label{eq:sl}{\frac{d{}}{d{x}}}\left({\rm P}(x){\frac{d{\Theta}}{d{x}}}\right) +\left({\rm Q}(x) + \lambda {\rm W}(x) \right)\Theta=0\,,$$ where $$\begin{aligned}
{\rm P}(x) & =\frac{{c'_{so}}^2{x^{3/2}}}{\Sigma'_0{\Omega'}}, \quad {\rm W}(x) = \frac{2x^{3/2}}{\Sigma'_0}\,,\nonumber\\
{\rm Q}(x) & = \frac{2x^{3/2}\dot{\varpi'}}{\Sigma'_0}, \quad \Theta =x^{1/2}{v_{\phi_a}}{\Sigma'_0}\,.
\nonumber
\label{eq:pqw}\end{aligned}$$ In equation (\[eq:sl\]), $\lambda = -\omega
\mathscr{M_{\star}}^2/\Omega_{\star} $ is defined with a negative sign to make an explicit correspondence with the Schrödinger’s equation, to be introduced in § 3.3. Henceforth we reserve the term “eigenvalue” for $\lambda$, and use either “frequency” or “eigenfrequency” for $\omega$.
We consider discs with an inner edge at $R_{\rm inner}$, and an outer edge at $R_{\rm outer}$, and we choose ${R_{\star}}=R_{\rm inner}$. We now argue that the parameters of the disc and the central mass for astrophysically interesting discs are such that the slow mode condition can be easily satisfied everywhere inside the disc. In a Keplerian disc the slow mode condition $|\omega|\ll\Omega$ is satisfied everywhere in the disc if it is satisfied at the disc outer radius, this leads to $$\left(\frac{R_{\rm outer}}{R_{\star}}\right)^{3/2}\ll\frac{{\mathscr M_{\star}}^2}{|\lambda|}\,,$$ where we have used $\lambda =
-\omega\mathscr{M_{\star}}^2/\Omega_{\star} $, and $\Omega(R_{\rm
outer}) = \Omega_{\star}(R_{\star}/ R_{\rm outer})^{3/2}$. Typical expected values for $\mathscr{M_{\star}}$ are in the range $10^{4}$–$10^{6}$ [@fkr02]. Most examples we consider have surface densities that decline by $R_{\rm outer}/R_{\star}\simeq
30$–$50$, therefore we see that for an eigenmode to be slow through out the disc, $|\lambda|$ has to be much smaller than $\sim
10^5$–$10^9$. We shall see in the examples that this condition is comfortably satisfied.
We integrate the eigen equation in the range $1<x<x_{\rm outer}$. We assume that the perturbations obey the boundary conditions, $$\Theta(1)=\Theta(x_{\rm outer})=0\,.$$ We note that these boundary conditions make the differential operator in equation (\[eq:sl\]) self-adjoint: therefore the eigenvalues $\lambda$ are real, and all slow p-modes are stable. The complete set of eigenfunctions also form a complete basis; however, we note that not all eigenvalues are slow, and thus we do not expect this set to describe the evolution of arbitrary perturbations, but only the ones that obey the slow mode condition, $\Omega\gg\vert\omega\vert$.
![The effective potentials for our barotropic approximations to the SS disc. The solid line corresponds to ${\rm V}_1(x)$ and the dashed line to ${\rm V}_2(x)$, described in § 4. The positive values of $\lambda$ can provide both Type I and Type II eigenvalues as described in the text. The negative values, although seemingly allowing eigenstates lead to no such solution. Note that positive values of $\lambda$ correspond to the negative frequency modes.[]{data-label="fig1"}](fig_1){width="50.00000%"}
Effective potential and WKB approximation
-----------------------------------------
In the usual WKB approximation we substitute the trial solution $$\Theta(x) = A(x)\exp\left[\frac{i}{\mu}\int^x \tilde{k}dx\right]\,,$$ in the following equation: $$\mu^2 {\frac{d{}}{d{x}}}\left({\rm P}(x){\frac{d{\Theta}}{d{x}}}\right) +\left({\rm Q}(x) + \lambda {\rm W}(x) \right)\Theta=0\,.$$ Here $\mu$ is an ordering parameter which is finally set equal to unity. $A(x)$ and $\tilde{k}(x)$ are the amplitude and the wavevector respectively. Collecting terms of zeroth order in $\mu$ leads to the dispersion relation $$\label{eq:disp} \tilde{k}^2 = \frac{{\rm Q}(x) + \lambda {\rm W}(x)}{{\rm P}(x)}
= \frac{2\Omega}{c_s^2}\left({\dot{\varpi}}- \omega\right)\,,$$ which is identical to equation (\[eq:drslow\]). However, we find that this dispersion relation, together with the Bohr-Sommerfeld quantization condition of equation (\[eq:quant\]) predicts eigenvalues that compare poorly with those obtained from numerical integration of the Sturm-Liouville equation. Hence we have reformulated equation (\[eq:sl\]), using new variables $\eta(x) =
\sqrt{{\rm P}(x)}$ and $\Psi = \sqrt{{\rm P}(x)}\Theta$. Then equation (\[eq:sl\]) takes the Schrödinger-like form $$\Psi'' + K^2(x)\Psi = 0\,,$$ where $$K^{2}(x)=\frac{1}{\eta^{2}(x)}\left[{\rm Q}(x) +\lambda{{\rm W}(x)}-\eta(x)\eta''(x)\right]\,,$$ which on defining ${\rm V}(x) = ( -{\rm Q}(x) +\eta(x)\eta''(x))/{\rm W}(x)$ can be written as $$\label{eq:pot} K^{2}(x)=\frac{{\rm W}(x)}{\eta^{2}(x)}\left[\lambda-{\rm V}(x)\right]\,.$$ This dispersion relation differs from equation (\[eq:drslow\]) and seems to better describe the numerical solutions, giving a match with the numerically obtained eigenvalues up to a few per cent, as can be seen in Table 1.
Note that $K^2(x)$ in equation (\[eq:pot\]) differs from the standard form for the Schrödinger equation by the factor ${\rm
W}(x)/\eta^{2}(x)$. However, it is very useful for discussions of the turning points, where $K^2(x)=0$, separating classically accessible regions from the forbidden ones. The solution is oscillatory where $K^2>0$, implying $\lambda > {\rm V}(x)$, and non-oscillatory otherwise. Since the disc is finite the eigen spectrum is always discrete and there are two distinct types of spectra:
[**Type I**]{}: This occurs when there is at least one turning point within the disc. In the case of a single turning point we can have oscillatory solution on either side of the turning point, depending on the form of $K^2(x)$. If there are more than one turning point then we could either have oscillatory behaviour confined between the turning points or outside, such as the case of the SS disc discussed below.
[**Type II**]{}: This occurs when there are no turning points within the disc and the discreteness of the spectra depends entirely on the size of the disc.
To obtain real eigenvalues we need to consider the possibility of satisfying $K^2>0$ in a bounded region, which could either be bounded by one or both of the disc boundaries or by turning points. Thus a useful first step is to plot this potential for the problem at hand. If the potential allows regions that can support bound states, we search for solutions numerically, and to verify our results we use the WKB approximation.
Let us consider the case of the SS disc shown in Fig \[fig1\]. On the negative side the potential blows up at the inner edge at $x=1$, where the perturbations are assumed to vanish. If $\lambda < 0$ is to be a valid eigenvalue, then it must satisfy the quantization condition $$\label{eq:WKBQ} \int_1^a K(x)\,dx = \left(n+\frac{3}{4}\right)\pi\,,\qquad n=0,1,2,\ldots$$ where, $x=a > 1$ is a turning point which separates a classically accessible region (I) on the left from a forbidden region (II) on the right. The case of positive eigenvalues, $\lambda > 0$, is more interesting. If the eigenvalue is positive and smaller than the maximum value of ${\rm V}(x)$, then there are two turning points (say, $x=a, b$ with $b > a$) separating three distinct regions. The classically forbidden region lies between $x=a$ and $x=b$, separating the two classically accessible regions $(1, a)$ and $(b, x_{\rm
outer})$. If $\lambda$ is greater than the maximum of ${\rm V}(x)$, then whole disc is classically allowed. Below we present numerical results on eigenvalues and eigenfunctions, and use WKB approximation to understand them. It turns out that WKB approximation is very useful even for the case of small quantum numbers.
![The effective potential for the Kuzmin disc. Positive values of $\lambda$ lead to a discrete spectrum of both Type I and Type II eigenvalues. There are no negative eigenvalues.[]{data-label="fig2"}](fig_2){width="50.00000%"}
Numerical results
=================
As discussed in the last section, we consider finite disc and we expect the spectra of equation (\[eq:sl\]) to be discrete. These modes would have observational consequences since they would rotate at a definite frequency around the disc. The perturbations in the enthalpy would lead to azimuthal variations in the temperature and density across the disc, which might be observable depending on the amplitude of perturbations.
Since very little is known about the surface density profiles of the discs we carry out a simplistic calculation based on certain standard forms of the disc as test cases. We also consider the generic power law profile. Some of the profiles considered below are formally infinite in size, however as noted above, we need to keep the disc finite. This would imply that the surface density would abruptly fall to zero at the outer disc radius. This is unphysical and we expect that there would be a thin transition region that would deviate from the density profile being considered near $x_{\rm outer}$. If the eigenvalues and eigenmodes are not very sensitive to $x_{\rm outer}$, then this is not an issue and in our numerical investigation we indeed find this to be true. In all the examples below $x_{\rm outer} = 50$. In Table 1 we give the first a few eigenvalues for $x_{\rm outer} = 50$ and $x_{\rm outer} = 200$, as $\lambda_{50}$ and $\lambda_{200}$ respectively. It can be noticed that eigenvalues do not change substantially.
![The allowed region (hatched) in the $\beta$-$\gamma$ space to obtain Type I negative eigenvalues (corresponding to the positive frequency modes) for the power law discs (case (ii)). Most of the region is either unphysical or uninteresting.[]{data-label="fig3"}](fig_3){width="50.00000%"}
[**Shakura-Sunyaev (SS) discs**]{}: We first consider the standard model of an accretion disc proposed by Shakura and Sunyaev [@ss73]. The surface density and temperature of this disc are given by (see @fkr02): $$\begin{aligned}
\Sigma_{\rm SS} &=& 5.2\,\alpha^{-4/5}{\dot{M}}_{16}^{7/10}{m_1}^{1/4}{R_{10}}^{-3/4}f^{14/5}
{\text{~g~cm}}^{-2}\,,
\label{eq:sigmass}\\[1em]
T_{\rm SS} &=& 1.4\times 10^4\alpha^{-1/5}{\dot{M}}_{16}^{3/10}{m_1}^{1/4}{R_{10}}^{-3/4}f^{6/5}\,{\rm K}\,,
\label{eq:tempss}\end{aligned}$$ where $$f =\left[1-\left(\frac{R_*}{R_{10}}\right)^{1/2}\right]^{1/4}\,.$$ ${\dot{M}}_{16}$ is the mass accretion rate in the units of $10^{16}$g s$^{-1}$, $m_1$ is the mass of disc in solar mass units, $R_{10}$ is radius in the units of $10^{10}$cm, $R_*$ is the radius of the central object in the units of $10^{10}$cm, and $\alpha$ is the Shakura-Sunyaev viscosity parameter.
Although the SS disc is not based on a barotropic model, we find that a barotropic disc with index $\gamma=2$ serves as a reasonable approximation. We have considered two cases:
1. Choosing $\Sigma_0 = \Sigma_{\rm SS}$ of equation (\[eq:sigmass\]), and deriving the temperature profile for $\gamma=2$, we find that the effective potential is $${\rm V_1}(x)= \frac{344 - 590\sqrt{x}+225x}{800\,x^{9/4}\,(1-\sqrt{1/x})^{13/10}}\,.$$
2. Choosing $T_0 = T_{\rm SS}$ of equation (\[eq:tempss\]), and deriving the surface density profile for $\gamma=2$, we find that the effective potential is $${\rm V_2}(x)= \frac{12 - 22\sqrt{x}+ 9x}{32\,x^{9/4}\,(1-\sqrt{1/x})^{3/2}}\,,$$
where we have used the natural length scale given by the inner disc radius, which we have adopted for conversion of our eigen equation into a dimensionless from. As Fig (\[fig1\]) shows, ${\rm{V}_1}(x)$ and ${\rm{V}}_2(x)$ are quite similar to each other. The effective potential blows up at the inner edge of the disc and steadily climbs up above zero and then decreases asymptotically. In the immediate vicinity of the inner edge a negative $\lambda$ gives an oscillatory solution, and to the right of it is a classical turning point. Beyond the turning point the solution is exponentially decaying. This suggests that discrete, Type I, negative eigenvalues might exist, however, both the numerical search and the WKB quantitation condition (\[eq:WKBQ\]), fail to find such discrete eigenvalues. For $0<\lambda\lesssim 0.01$, we find discrete, Type I eigenvalues for which the oscillatory behaviour is outside the region bounded by two turning points. For $\lambda\gtrsim 0.01$, we find discrete eigenvalues of Type II, where the separation between neighbouring eigenvalues decreases with increasing values of the outer radius of the disc.
[**The Kuzmin Disc**]{}: In contrast to the SS discs, the Kuzmin disc has a centrally concentrated surface density, and hence offers a distinct case in which to study slow modes. The surface density profile in this case is given by $$\Sigma(R) = \frac{aM_D}{2\pi\,(R^2+a^2)^{3/2}}\,,$$ where $M_D$ is the mass in the disc and $a$ is the core radius. The surface density extends all the way to $R=0$. If we take $a$ as the size of the inner radius of the disc then we can rescale our equation by choosing $R_{\star}=a$, leading to the effective potential $${\rm V}(x)=\frac{3(x^4(5-4\gamma+3\gamma^2)+x^2(6-8\gamma)+1)}{8x^{1/2}(1+x^2)^{(3\gamma+1)/2}}.
\label{eq:potkuz}$$ In Fig (\[fig2\]) we have plotted this potential for $\gamma=4/3$. As may be seen, this case qualitatively resembles Fig (\[fig1\]), and the discussion in § 2.4 carries through. For negative values of $\lambda$, we can infer from Fig (\[fig2\]) that $K^2(x)<0$, so wave like solutions are not possible. For positive values of $\lambda$ in the range $0<\lambda\lesssim 0.3$, we can have Type I eigenvalues, but this region of $\lambda$ is further divided into two parts: for $0<
\lambda \le {\rm V}(1)$, there is only one turning point and oscillatory behaviour is possible for radius greater than the turning point, and for ${\rm V}(1)<\lambda\lesssim 0.3$, there are two turning points and oscillatory behaviour is possible outside the region bounded by the two turning points. For $\lambda\gtrsim 0.3$ the eigenvalues are of Type II. This behaviour is confirmed by numerical integration of the eigenvalue equation. We can also admit values of $\gamma$ other than $4/3$. However, the effective potential in equation (\[eq:potkuz\]) retains the general shape of Fig (\[fig2\]), and the conclusions stated above remain valid.
![The allowed region (cross-hatched) in the $\beta$-$\gamma$ space to obtain discrete positive eigenvalues (corresponding to the negative frequency modes), corresponding to Type I eigenvalues for case (iii), for the power law discs.[]{data-label="fig4"}](fig_4){width="50.00000%"}
![Two eigenfunctions (Type-I eigenvalue) obtained for the power law case (case iii) with $\beta=\gamma=2$. The higher quantum number leads to a more radially extended eigenmode. Star marks the position of the turning point.[]{data-label="fig5"}](fig_5){width="50.00000%"}
![The allowed region (hatched) in the $\beta$-$\gamma$ space for the positive Type I eigenvalues (corresponding to negative frequency eigenmodes), for case (iv) for the power law discs.[]{data-label="fig6"}](fig_6){width="50.00000%"}
![Two eigenfunctions (Type-I eigenvalue) obtained for the power law case (case iv) with $\beta= -2, \gamma=2$. Star marks the position of the turning point.[]{data-label="fig7"}](fig_7){width="50.00000%"}
[**Power Law Discs**]{}: Certain physical models (e.g. @ny94) naturally lead to scale invariant discs that follow a power law profile. Although in these models there is no associated length scale, we choose to truncate the disc at the inner disc radius thus leading to the form $$\Sigma_0=\Sigma_{\star}\,x^{\beta}\,,$$ where, $\Sigma_{\star} = \Sigma(R_{\star})$ and $\beta$ is the power law index. The potential is given by $${\rm V}(x)= C(\beta,\gamma) x^{\nu}\,,$$ where $$\begin{aligned}
C(\beta,\gamma) &=& \frac{1}{8}\left(3 + \beta\left(-4 + \gamma(4 + \gamma\beta)\right)\right)\,,\\
\nu(\beta,\gamma)&=& \gamma\beta - \beta-1/2\,. \end{aligned}$$
There are four distinct possibilities:
1. $C(\beta , \gamma)< 0$, $\nu(\beta , \gamma)<0$: The region in $\beta$-$\gamma$ space satisfying these conditions is plotted in Fig (\[fig3\]). The regions where these conditions are satisfied are those with (1) $\beta$ positive and $\gamma$ negative or (2) very small negative $\beta$ and very large $\gamma$. Both these cases are unphysical. Note that there is a region with large, positive $\beta$, and small, positive $\gamma$, and these are also physically uninteresting.
2. $C(\beta , \gamma)< 0$, $\nu(\beta , \gamma)>0$: It can be verified that for $\nu(\beta , \gamma)>0$ $C(\beta , \gamma)$ is always positive and hence it is impossible to satisfy these conditions.
3. $C(\beta , \gamma)> 0$, $\nu(\beta , \gamma)>0$: These two constraints give us a region in the $\beta$-$\gamma$ space, displayed in Fig (\[fig4\]), that admits physically reasonable values. For ${\rm V}(x_{\rm inner})<\lambda< {\rm V}(x_{\rm
outer})$, we have one turning point admitting Type I eigenvalues. In Fig (\[fig5\]) we plot two examples of eigenmodes for the case $\beta=\gamma=2$. The eigenfunctions for small quantum numbers are found to be centrally concentrated while they extend to larger radii for larger quantum numbers. It should be noted that our solutions are regular at the turning points[^1]. The first few eigenvalues for this case are tabulated in Table 1, where we find an excellent match with the WKB eigenvalues. Outside this range of positive $\lambda$ there are no turning points and hence only Type II eigenvalues are possible.
4. $C(\beta , \gamma)> 0$, $\nu(\beta , \gamma)<0$: The region in $\beta$-$\gamma$ space satisfying these constraints is plotted in Fig (\[fig6\]). For $0<\lambda< {\rm V}(1)$, we have one turning point admitting Type I eigenvalues. Here the classically accessible region is bounded by the turning point on left and $x_{outer}$ on right. Numerical solution for $\beta = -2$ and $\gamma = 2$ are plotted in Fig (\[fig7\]). Comparison with Fig (\[fig5\]) shows that the solutions are more radially extended, with well separated peaks. For $\lambda > {\rm V}(1)$ there is no turning point and the eigenvalues are of Type II.
Discussion and Conclusions
==========================
We have presented a theory of slow $m=1$ linear pressure modes (or “p-modes”) in thin accretion discs around massive compact objects, such as white dwarfs and neutron star. These modes are enabled by the small deviation from a purely Keplerian flow, due to fluid pressure rather than disc self-gravity. For simplicity we have taken the fluid to be barotropic. Our formulation largely follows that of @tre01, although there is a key difference: using the WKB approximation, @tre01 argued that fluid discs for which disc self-gravity dominates fluid pressure can support slow modes, if the Mach number ${\mathscr M}$ is much larger than the Toomre $Q$ parameter. This condition may be satisfied in relatively cool discs, but not for thin accretion discs around white dwarfs or neutron stars. In these discs, $Q\gg {\mathscr M}\gg 1$, and the analysis in @tre01 does not apply, because disc self-gravity is negligible when compared with fluid pressure in thin accretion discs. This implies that the precession rate of the apsides $({\dot{\varpi}})$ of a fluid element in a nearly circular orbit is determined by the fluid pressure; to order of magnitude, ${\dot{\varpi}}\sim (\Omega/{\mathscr M}^2)$.
\[tab:comp\]
$n$ WKB $\lambda$ Numerical $\lambda_{50}$ Numerical $\lambda_{200}$
----- --------------- -------------------------- ---------------------------
0 13.42 13.86 13.86
1 30.15 30.73 30.73
2 52.31 53.06 53.06
3 79.99 80.92 80.91
4 113.22 114.32 114.31
5 151.99 153.29 153.26
6 196.31 197.83 197.75
7 246.18 247.96 247.80
8 301.60 303.72 303.40
9 362.57 365.16 364.54
10 429.09 432.31 431.24
: The eigenvalues for the power law model with are tabulated here for comparison between those obtained numerically and those obtained using the WKB approximation. The columns $\lambda_{50}$ and $\lambda_{200}$ are the eigenvalues corresponding to $x_{\rm{outer}} = 50$ and $200$ respectively. The match between numerical and WKB eigenvalues is within a few per cent, and remains good even for small quantum numbers. The eigenvalues are not very sensitive to precise value of $x_{\rm{outer}}$
.
A WKB analysis was used first to argue that thin accretion discs can support large-scale $(k(R) R \sim 1)$, $m=1$ p-modes with small angular frequencies, $\omega\sim{\dot{\varpi}}\sim (\Omega/{\mathscr M}^2)$. As noted by @tre01, these long wavelength modes may dominate the appearance of the disc, and are not expected to be damped by viscosity. We derived an eigen equation for slow linear modes and showed that it is identical to a Sturm-Liouville problem. The differential operator being self-adjoint implies that the eigenvalues are all real, so that all slow p-modes are stable. This corresponds to the result in @tre01 that all slow modes of the softened gravity disc are stable. We solved the Sturm-Liouville problem numerically for a variety of unperturbed discs, and summarise our results below.
1. The first corresponds to two different kinds of barotropic approximations to the Shakura-Sunyaev thin disc, which have modes with negative eigenfrequencies.
2. The second is the Kuzmin disc, which is more centrally concentrated. As earlier, this too supports only negative eigenfrequencies.
3. Power-law discs can support modes with negative eigenfrequencies for reasonable values of $\beta$ and $\gamma$. For certain combinations of these parameters power law discs can support positive eigenfrequencies as well; however the range of parameters turn out to be physically uninteresting.
If slow modes are stable, it is necessary to consider how they could be excited. Since they have azimuthal wavenumber $m=1$, we need look for excitation mechanisms which possess the same symmetry, at least in the linear limit. One possibility is from the stream of matter from the secondary star that feeds the accretion disc. When viewed in the rest frame of the primary, the region where the stream meets the outer edge of the disc rotates in a prograde sense with angular frequency equal to the orbital frequency of the binary system. Since the orbital frequency of the binary can be much smaller than the orbital frequency of the gas in the accretion disc, there is the possibility of the resonant excitation of a slow mode if it has positive frequency. In the cases we have considered, we find only negative frequencies belonging to a discrete spectrum, allowing only for non–resonant driving.
There is, however, another alternative that does not rely on external sources of excitation. @zl05 have studied linear waves in thin accretion discs and applied their theory to slow $m=1$ modes around black holes. They used the pseudo-Newtonian gravitational potential of @pw80 to model the general relativistic effects due to a Schwarzschild black hole. In this case ${\dot{\varpi}}$ is due to the deviation of the pseudo-Newtonian potential from a Kepler potential, so their slow modes are not driven by pressure (as is true in all the cases we have considered); hence more detailed comparison with our work is not possible. What is interesting is that they find that their slow modes to have negative energy and angular momentum, and suggest that slow modes may be excited spontaneously through the action of viscous forces. This possibility should be examined in the context of the p-modes we have studied. However, this would require reformulating the eigenvalue problem taking into account viscous effects.
Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition, Princeton University Press. Frank, J., King, A., & Raine, D. J. 2002, Accretion Power in Astrophysics, Cambridge University Press. Goldreich, P., & Tremaine, S. 1979, , 233, 857 Kato, S. 1983, , 35, 249 Lee, E., & Goodman, J. 1999, , 308, 984 Narayan, R., & Yi, I. 1994, , 428, L13 Paczynsky, B., & Wiita, P. J. 1980, , 88, 23 Safronov, V. S. 1960, Annales d’Astrophysique, 23, 979 Shakura, N. I., & Sunyaev, R. A. 1973, , 24, 337 Sridhar, S., Syer, D., & Touma, J. 1999, Astrophysical Discs - an EC Summer School, 160, 307 Sridhar, S., & Touma, J. 1999, , 303, 483 Tremaine, S. 2001, , 121, 1776 Tsang, D., & Lai, D. 2009, , 396, 589 Zhang, L., & Lovelace, R. V. E. 2005, , 300, 395
[^1]: The positions of the ILRs, obtained from, $ \lambda = -{\rm{Q}(x)}/{\rm{W}(x)}$ are slightly different from the turning points obtained from the equation $ \lambda = -(\rm{Q}(x) - \eta(x)\eta''(x))/{\rm{W}(x)}$.
|
---
address:
- |
Institute of Theoretical Physics, Warsaw University, ul. Hoza 69, PL-00-681 Warsaw, Poland\
E-mails: bohdang@fuw.edu.pl,pliszka@fuw.edu.pl
- |
Davis Institute for High Energy Physics, UC Davis, CA, USA\
E-mail: jfgucd@physics.ucdavis.edu
author:
- 'B. Grzadkowski and J. Pliszka'
- 'J. F. Gunion'
title: 'Measuring the relative CP-even and CP-odd Yukawa couplings of a Higgs boson at a muon-collider Higgs factory'
---
It is believed that the scalar sector is an inherent component of the theory of elementary interactions and, one or more physical Higgs, or Higgs-like bosons will, sooner or later, be discovered. Since the CP nature of Higgs bosons is a model dependent feature[@hhg] its determination would not only provide information concerning the mechanism of CP violation but would also restrict possible extensions of the Standard Model (SM) of electroweak interactions and therefore reveal the structure of fundamental interactions beyond the SM. A muon collider with transversely polarized beams is the only place where CP properties of a second generation fermion Yukawa coupling could be probed. This is the subject of our more complete analysis[@hcpmupmum] which is summarized in this talk.[^1] We follow the line of our previous works[@gghgp] where we have tried to unveil the CP-nature of Higgs bosons in a model-independent way.
The attractive possibility of s-channel Higgs boson production at a muon collider has been discussed before[@bbgh] together with the possibility of the measurement of CP violation in the muon Yukawa couplings[@bbgh; @soni; @hcpmupmum]. The latter is based on the fact that in any muon collider design[@pstareport; @euroreport] there is a natural beam polarization on the order of 20%[@rajat] that allows for a rare possibility of direct Higgs boson production with known polarization of initial state particles.
The cross section for the Higgs boson resonance production:$\mu^+\mu^-\!\rightarrow\!R $ depends on transverse $P_T^\pm$ and longitudinal $P_L^\pm$ beam polarizations and $\anti\mu r \exp( i \delta\gamma_5)\mu$ muon Yukawa coupling in the following way: $$\sigma_S(\zeta)=
\sigma_S^0 \left[1+P_L^+P_L^-+P_T^+P_T^-\cos(2\delta+\zeta)\right]
\label{sigform}$$ where $\zeta$ is the angle in the transverse plane between beam polarizations and $\sigma_S^0$ is the unpolarized cross section. We stress that only the transverse polarization term is sensitive to $\delta$ of the muon Yukawa coupling. Since it is proportional to the product of the transverse polarizations, it is essential to have large $P_T^+$ and $P_T^-$, as obtained by applying stronger cuts while selecting muons from decaying pions (which, however, causes a reduction of luminosity). To compensate, more intensive proton source or ability to repack muon bunches will be needed. Another speculative option, would be high, up to 50%, polarization obtained by a phase-rotation technique[@kaplan] which would lead to less luminosity reduction.
While varying $\zeta$ one can observe a maximum at $\zeta=-2\delta$ and a minimum at $\zeta=\pi-2\delta$. Thus, studying $\zeta$ dependence is essential for resolving the $\delta$ value. A muon collider offers the unique possibility of a setup which in a natural way provides a scan over different $\zeta$ values. We will not discuss this option here. Our results will correspond to a configuration with four fixed $\zeta$ values: 0,90,180 and 270. Even though this cannot be accomplished experimentally, due to the spin precession in the accelerator ring, it can be well approximated by a simple but realistic setup[@hcpmupmum] that yields the same results as the fixed $\zeta$ analysis at the expense of 50% luminosity increase.
In order to illustrate the ability to reject different Higgs boson CP scenarios we can assume that the measured data is mimicked by the SM Higgs boson.
-------- ---------------- ----------- ----------- ----------- -----------
P\[%\] L\[pb$^{-1}$\] 1$\sigma$ 3$\sigma$ 1$\sigma$ 3$\sigma$
20 150 0.94 – – –
39 75 0.30 1.14 0.41 –
48 75 0.20 0.64 0.27 0.93
45 150 0.15 0.50 0.21 0.69
-------- ---------------- ----------- ----------- ----------- -----------
: 1 and 3 $\sigma$ exclusion limits on $\delta$ for $b\bar{b}$ final state for various luminosity and polarization configurations. Beam energy spread 0.003% and $b\bar{b}$ tagging efficiency $54\%$ have been assumed.
\[tab\_results\]
For given luminosity $L$ and polarization $P$ we can place 1 and 3 $\sigma$ limits on the $\delta$ value for the observed resonance, assuming the $\delta=0$ SM is input. The limits for Higgs boson masses of 110 and 130 GeV are presented in table \[tab\_results\]. For the expected yearly luminosity of $L=100~{\rm pb}^{-1}$, even several years of running at the natural 20% polarization would be insufficient for useful limits. However, 1$\sigma$ limits for the $P=39\%$ option (with reduced $L$) do give a rough indication of the CP nature of the resonance. 3$\sigma$ limits in 110-130 GeV mass range require either $>40\%$ polarization or $<50\%$ luminosity loss. (The requirements are less stringent for a 110 GeV Higgs boson.) We stress that there is no other way the measurement of the muon Yukawa $\delta$ can be done and that operation in the transverse polarization mode should not interfere with most of the other studies.
\[12pt\]\[7pt\][$(\delta)$]{} $\zeta=0$ $\zeta=\pi/2$ $\zeta=\pi$ $\zeta=3\pi/2$
---------------------------------- ----------- --------------- ------------- ----------------
\[10pt\]\[5pt\][$(0)$]{} $1+P^2$ 1 $1-P^2$ 1
\[5pt\]\[5pt\][$(\pi/4)$]{} 1 $1-P^2$ 1 $1+P^2$
\[5pt\]\[5pt\][$(\pi/2)$]{} $1-P^2$ 1 $1+P^2$ 1
\[5pt\]\[5pt\][$(0)+(\pi/2)$ ]{} 1 1 1 1
: Event number pattern for different Higgs models as a function of $\zeta$, assuming $P_L^\pm=0$ and $P_T^\pm=P$; see Eq. (\[sigform\]).
\[patterns\]
For a heavier resonance, operation of a muon collider as an s-channel Higgs boson factory is justified only if the branching ratio $BR(R\rightarrow \mu^+\mu^-)$ is enhanced. Then, the analysis sketched above applies as well. Such enhancement arises in the Minimal Supersymmetric Standard Model (MSSM) at large $\tan\beta$. If the pseudoscalar mass is large ($m_A >300\textrm{GeV}$), the $H$ and $A$ masses will be similar. For $\tan\beta>8$ degeneracy may be such that we will not be able to see two separate peaks but only a single peak with both CP-even and CP-odd components. It would be crucial to distinguish such a case from a single CP-violating Higgs boson which may appear e.g. in MSSM[@cp-phases]. Table \[patterns\] illustrates the very distinct event number pattern as a function of $\zeta$ that would yield the needed discrimination. The event rate for any single, CP conserving or CP violating Higgs boson, has a minimum and maximum as a function of $\zeta$. In contrast, overlapping CP-even and CP-odd resonances result in a pattern independent of $\zeta$. We have found that for simple MSSM test cases with $m_A= 300-400$ GeV and $\tan\beta> 8$ (for which we cannot see separate resonance peaks) even natural 20% polarization will allow us to distinguish two overlapping resonances from any single one at more than the 3$\sigma$ level. Higher polarization will allow for a precise measurement of the relative contribution from the CP-even and the CP-odd component.
To summarize, we have presented results of a realistic study of measuring the CP properties of the muon Yukawa couplings in Higgs boson production at a muon collider with transversely polarized beams. We have found that transverse polarization is essential for determining the CP nature of the muon Yukawa couplings. In particular, a collider with $P\!\approx\!40\%$ and at least 50% of the original luminosity retained will ensure that the CP nature of the produced scalar resonance will be revealed.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank S. Geer, R. Raja and R. Rossmanith for helpful conversations on experimental issues. This work was supported in part by the U.S. Department of Energy, the U.C. Davis Institute for High Energy Physics, the State Committee for Scientific Research (Poland) grant No. 2 P03B 014 14 and by Maria Sklodowska-Curie Joint Fund II (Poland-USA) grant No. MEN/NSF-96-252.
[99]{} J.F. Gunion, H.E. Haber, G. Kane and S. Dawson in [*The Higgs Hunters Guide*]{}, (Addison-Wesley Publishing Company, Redwood City, CA, 1990).
B. Grzadkowski, J.F. Gunion and J. Pliszka, [hep-ph/0003091](http://arXiv.org/abs/hep-ph/0003091.html)
J.F. Gunion, B. Grzadkowski, X.-G. He, , [hep-ph/9605326](http://arXiv.org/abs/hep-ph/9605326.html); J.F. Gunion and J. Pliszka, , [hep-ph/9809306](http://arXiv.org/abs/hep-ph/9809306.html); B. Grzadkowski and J. Pliszka, , [hep-ph/9907206](http://arXiv.org/abs/hep-ph/9907206.html). V. Barger, M.S. Berger, J.F. Gunion and T. Han, , [hep-ph/9602415](http://arXiv.org/abs/hep-ph/9602415.html). , [hep-ph/9504330](http://arXiv.org/abs/hep-ph/9504330.html).
D. Atwood and A. Soni, [hep-ph/9505233](http://arXiv.org/abs/hep-ph/9505233.html); S.Y. Choi and J.S. Lee, [hep-ph/9909315](http://arXiv.org/abs/hep-ph/9909315.html).
C.M. Ankenbrandt [*et al*]{}, [, 081001 (1999)](http://prst-ab.aps.org/ejnls/przfetch/abstract/PRZ/V2/E081001/index.html). “Prospective Study of Muon Storage Rings at CERN”, edited by B. Autin, A. Blondel and J. Ellis ([CERN 99-02](http://preprints.cern.ch/index.html), ECFA 99-197).
R. Raja and A. Tollestrup, [](http://arXiv.org/abs/hep-ex/9801004.html) and private communication.
D.M. Kaplan, [physics/0001037](http://arXiv.org/abs/physics/0001037.html). We thank S. Geer for providing details regarding this option.
A. Pilaftsis, [](http://arXiv.org/abs/hep-ph/9803297.html), [](http://arXiv.org/abs/hep-ph/9805373.html); D.A. Demir,[](http://arXiv.org/abs/hep-ph/9901389.html); A. Pilaftsis and C. E. Wagner, [](http://arXiv.org/abs/hep-ph/9902371.html)
[^1]: Presented by J. Pliszka.
|
---
abstract: 'Reduction of frustration was the driving force in an approach to social balance as it was recently considered by Antal *et al.* \[ T. Antal, P. L. Krapivsky, and S. Redner , Phys. Rev. E [**72**]{} , 036121 (2005). \]. We generalize their triad dynamics to $k$-cycle dynamics for arbitrary integer $k$. We derive the phase structure, determine the stationary solutions and calculate the time it takes to reach a frozen state. The main difference in the phase structure as a function of $k$ is related to $k$ being even or odd. As a second generalization we dilute the all-to-all coupling as considered by Antal *et al.* to a random network with connection probability $w<1$. Interestingly, this model can be mapped onto a $k$-XOR-SAT problem that is studied in connection with optimization problems in computer science. What is the phase of social balance in our original interpretation is the phase of satisfaction of all clauses without frustration in the satisfiability problem of computer science. Nevertheless, although the ideal solution without frustration always exists in the cases we study, it does not mean that it is ever reached, neither in the society nor in the optimization problem, because the local dynamical updating rules may be such that the ideal state is reached in a time that grows exponentially with the system size. We generalize the random local algorithm usually applied for solving the $k$-XOR-SAT problem to a $p$-random local algorithm, including a parameter $p$, that corresponds to the propensity parameter in the social balance problem. The qualitative effect is a bias towards the optimal solution and a reduction of the needed simulation time. We establish the mapping between the $k$-cycle dynamics for social balance on diluted networks and the $k$-XOR-SAT problem solved by a $p$-random local algorithm.'
author:
- Filippo Radicchi
- Daniele Vilone
- Sooeyon Yoon
- 'Hildegard Meyer-Ortmanns'
title: 'Reducing Frustration in Spin Systems: Social Balance as an XOR-SAT problem'
---
Introduction {#intro_sec}
============
Recently Antal *et al.* [@antal] proposed a triad dynamics to model the approach of social balance. An essential ingredient in the algorithm is the reduction of frustration in the following sense. To an edge (or link) in the all-to-all topology is assigned a value of $+1$ or $-1$ if it connects two individuals who are friends or enemies respectively. The sign $\pm 1$ of a link we call also its spin. If the product of links along the boundary of a triad is negative, the triad is called frustrated (or imbalanced), otherwise it is called balanced (or unfrustrated). The state of the network is called balanced if all triads are balanced. If the balanced state is achieved by all links being positive the state is called “paradise”. The algorithm depends on a parameter $p \in [0,1]$ called propensity which determines the tendency of the system to reduce frustration via flipping a negative link to a positive one with probability $p$ or via flipping a positive link to a negative with probability $1-p$. For an all-to-all topology Antal *et al.* predict a transition from imbalanced stationary states for $p<1/2$ to balanced stationary states for $p\geq 1/2$. Here the dynamics is motivated by social applications so that the notion of frustration from physics goes along with frustration in the psychological sense.\
Beyond frustration in social systems, within physics, the notion is familiar from spin glasses. It is the degree of frustration in spin glasses which determines the qualitative features of the energy landscape. A high \[low\] degree of frustration corresponds to many \[few\] local minima in the energy landscape. In terms of energy landscape it was speculated by Sasai and Wolynes [@wolynes] that is the low degree of frustration in a genetic network which is responsible for the few stable cell states in the high-dimensional space of states.\
Calculational tools from spin-glass theory like the replica-method [@parisi] turned out to be quite useful in connection with generic optimization problems (as they occur, for example, in computer science) whenever there is a map between the spin-glass Hamiltonian and a cost function. The goal in finding the ground state-energy of the Hamiltonian translates to the minimization of the costs. A particular class of the optimization problems refers to the satisfiability problems. More specifically one has a system of $B$ Boolean variables and $Q$ logical constraints (clauses) between them. In this case, minimizing the costs means minimizing the number of violated constraints. In case of the existence of a non-violating configuration the problem is said to be satisfiable, it has a zero-ground state energy in the Hamiltonian language. Here it is obvious that computer algorithms designed to find the optimal solution have to reduce the frustration down to a minimal value. So the reduction of frustration is in common to very different dynamical processes.\
The algorithms we have to deal with belong to the so-called incomplete algorithms [@garey; @weigt; @semerjian] characterized by some kind of Monte-Carlo dynamics that tries to find the solution via stochastic local moves in configuration space, starting from a random initial configuration. It either finds the solution “fast” or never (this will be made more precise below). Among the satisfiability problems there are the $k$-SAT ($k$S) problems [@cook; @mezard; @mezard2], for which actually no frustration-free solution exists above a certain threshold in the density of clauses imposed on the system. In this case the unsatisfiability is not a feature of the algorithm but intrinsic to the problem. However, there is a special case of $k$S problems, so-called $k$-XOR-SAT ($k$XS) problems [@weigt; @semerjian; @mezard2; @cocco] which are always solvable by some global algorithm, but poses a challenge for finding the solution by some kind of Monte-Carlo dynamics, very similar to the one used for solving the $k$S problem, where actually no solution may exist. Now it is these $k$XS problems and their solutions that are related to the social balance dynamics.\
In particular it can be easily shown [@mezard; @mezard2; @cocco] that the satisfiability problem $3$S (and also the subclass $3$XS) can first be mapped onto a $3$-spin model that is a spin-glass, and as we shall show below, the $3$-spin glass model can next be mapped onto the triad dynamics of Antal *et al.* [@antal]. The $k$XS problem is usually studied for diluted connections because the interesting changes in the phase structure of the $k$XS problem appear at certain threshold parameters in the dilution, while the all-to-all case is not of particular interest there.\
Dilution of the all-to-all topology is not only needed for the mapping to the $3$XS problem in its usual form. It is also a natural generalization of the triad dynamics considered in [@antal] for social balance. A diluted network is more realistic than an all-to-all topology by two reasons: either two individuals may not know each other at all (this is very likely in case of a large population size) or they neither like or dislike each other, but are indifferent as emphasized in [@cartwright] as an argument for the postulated absence of links. For introducing dilution into the all-to-all network considered by Antal *et al.* it is quite natural to study random Erdös-Rényi networks [@erdos] for which two nodes are connected by a link with probability $w$. On the other hand, dilution in the $k$XS problem is parameterized by the ratio $\alpha$ of number of clauses over number of variables (variables in the corresponding spin model or number of links in the triad dynamics). We will determine the map between both parameterizations.\
\
In the first part of this paper (section \[model\]) we generalize the triad dynamics to $k$-cycle dynamics, driven by the reduction of frustration, with arbitrary integer $k$. In the context of *social balance* theory, Cartwright and Harary [@cartwright] introduced the notion of balance describing a balanced state with all $k$-cycles being balanced and $k$ not restricted to three. We first study such model on fully connected networks (section \[complete\]). For given fixed and integer $k\geq 3$ in the updating rules, we draw the differential equations of the time evolution due to the local dynamics (section \[evolution\]) and we predict the stationary densities of $k$-cycles, $k$ arbitrary integer, containing $j \leq k$ negative links (section \[stationary\]). As long as $k$ is odd (section \[odd\]) in the updating dynamics, the results are only quantitatively different from the case of $k=3$ considered in [@antal]. An odd cycle of length three is, however, not an allowed loop in a bipartite graph, for which links may only exist between different type of vertices so that the length of a loop of minimal size in a bipartite graph is four. In addition, a $4$-cycle with four negative links (that is four individuals each of which dislikes two others) is balanced and not frustrated, although it may be called the “hell”, so it does not need to be updated in order to reduce its frustration. (To call the hell with four negative links balanced is not specific for the notion of frustration in physics; also in social balance theory it is the product over links in the loop which counts and decides about balance or frustration [@roberts].) This difference is essential as compared to the triad dynamics, in which a triad of three unfriendly links is always updated. It has important implications on the phase structure as we will show. For even values of $k$ and larger than four, again there are only quantitative differences in the phase structure as compared to $k=4$ (section \[even\]).\
As in [@antal] , for odd values of $k$, we shall distinguish between stationary states in the infinite volume limit that can be either balanced (for $p \geq 1/2$) or frustrated (for $p< 1/2$) since it is not possible to reach the paradise in a finite time. They are predicted as solutions of mean field equations. In numerical simulations, fluctuations about their stationary values do not die out in the phase for $p< 1/2$ so that some frustration remains, while for $p \geq 1/2$ frozen states are always reached in the form of the paradise although other balanced states with a finite amount of negative links are in principle available, but are quite unlikely to be realized during the finite simulation time. They are exponentially suppressed due to their small weight in configuration space. We calculate the time it takes to reach a frozen state at and above the phase transition (section \[time\_odd\]). For even values of $k$ we have only two types of stationary frozen states, “paradise” and “hell” with all links being positive and negative, respectively. In this case the time to reach the frozen states at the transition can be calculated in two ways. The first possibility applies for both even and odd values of $k$ and is based on calculating the time it takes until a fluctuation is of the same order in size as the average density of unfriendly links. The second one, applicable to the case of even values of $k$, can be obtained by mapping the social system to a Markov process known as the Wright-Fisher model for diploid organisms [@wright], for which the decay time to one of the final configurations (all “positive” or all “negative” genes) increases quadratically in the size $N$ of the system (section \[sec:time\_even\]).\
In the second part we generalize the $k$-cycle dynamics to diluted systems (section \[sec:diluted\]). The dilution, originally given in terms of the probability for connecting two links in a random Erdös-Rényi network [@erdos], is then parameterized in terms of the dilution parameter $\alpha$, and the results for stationary and frozen states and the time needed to reach them will be given as a function of $\alpha$ (section \[sec:ratio\]). The original triad dynamics of Antal *et al.* with propensity parameter $p$ on a diluted network contains, as special case, the usual Random-Walk SAT (RWS) algorithm for finding the solution of the $3$XS problem corresponding to the choice of $p=1/3$ in the triad dynamics. Therefore it is natural to generalize the RWS algorithm for generic $p\in [0,1]$ and to study the modifications in the performance of the algorithm as a function of $p$ (section \[sec:RWS\]). For the $k$S problem, and similarly for the $k$XS problem, there are three thresholds in $\alpha$, $\alpha_d$, $\alpha_s$, and $\alpha_c$ with $\alpha_d<\alpha_s<\alpha_c$. Roughly speaking, the threshold $\alpha_d$ corresponds to a dynamical transition between a phase in which the RWS algorithm finds a solution in a time linearly increasing with the size of the system for $\alpha<\alpha_d$, and exponentially increasing with the system size for $\alpha>\alpha_d$. The value $\alpha_s$ characterizes a transition in the structure of the solution space, from one cluster of exponentially many solutions ($\alpha<\alpha_s$) to exponentially many clusters of solutions ($\alpha>\alpha_s$). Finally, $\alpha_c$ refers to the transition between satisfiable and unsatisfiable $k$S problems, this means that for these models not all constraints can be satisfied simultaneously in the UNSAT-phase for $\alpha>\alpha_c$ so that a finite amount of frustration remains. Above this last threshold lies a value of $\alpha,\; \alpha=\alpha_m$, such that for $\alpha>\alpha_m$ the mean field approximation is justified that was used for the maximum value of $\alpha$ in the all-to-all topology of the triad dynamics of [@antal]. We shall study the influence of the parameter $p$ on the value of $\alpha_d$ (section \[sec:alphad\]) and on the Hamming distance for $\alpha$ smaller $\alpha_s$ or larger $\alpha_s$ (section \[sec:alphas\]). Moreover we will show how the choice of $p$ changes the possibility to find a solution for the $k$XS problem (section \[sec:pc\]) and we will determine the validity range of the mean-field approximation (section \[sec:alpham\]). As it turns out, the parameter $p$ introduces some bias in the RWS, accelerating the convergence to “paradise” and reducing the explored part of configuration space. On the other hand, an inappropriate choice of $p$ or too much dilution may prevent an approach to paradise. Fluctuations in the wrong direction, increasing the amount of frustration, go along with improved convergence to the balanced state.
The Model for Social Balance {#model}
============================
We represent individuals as vertices (or nodes) of a graph and a relationship between two individuals as a link (or edge) that connects the corresponding vertices. Moreover we assign to a link $(i,j)$ between two nodes $i$ and $j$ a binary spin variable $s_{i,j}=\pm 1$, with $s_{i,j}=1$ if the individuals $i$ and $j$ are friends , and $s_{i,j}=-1$ if $i$ and $j$ are enemies. We consider the standard notion of *social balance* extended to cycles of order $k$ [@cartwright; @heider]. In particular a cycle of order $k$ (or a $k$-cycle) is defined as a closed path between $k$ distinct nodes $i_1$, $i_2$, …, $i_k$ of the network, where the path is performed along the links of the network $(i_1,i_2)$ , $(i_2,i_3)$ , …, $(i_{k-1},i_k)$, $(i_k,i_1)$. Given a value of $k$ we have $k+1$ different types $T_0$, $T_1$, …, $T_j$, …, $T_k$ of cycles of order $k$ containing $0$, $1$, …, $j$, …, $k$ negative links, respectively. A cycle of order $k$ in the network is considered as balanced if the product of the signs of links along the cycle equals $1$, otherwise the cycle is considered as imbalanced or frustrated. Accordingly, the network is considered as balanced if each $k$-cycle of the network is balanced.\
We consider our social network as a dynamical system. We perform a local unconstrained dynamics obtained by a natural generalization of the local triads dynamics, recently proposed by Antal *et al.* [@antal]. We first fix a value of $k$. Next, at each update we choose at random a $k$-cycle $T_j$. If this $k$-cycle $T_j$ is balanced ($j$ is even) nothing happens. If $T_j$ is imbalanced ($j$ is odd) we change one of its link as follows: if $j<k$, then $T_j \to \; T_{j-1}$ occurs with probability $p$, while $T_j \to \; T_{j+1}$ occurs with probability $1-p$ ; if $j=k$, then $T_{j} \to \; T_{j-1}$ happens with probability $1$. During one update, the positive \[negative\] link which we flip to take a negative \[positive\] sign is chosen at random between all the possible positive \[negative\] links belonging to the $k$-cycle $T_j$. One unit of time is defined as a number of updates equal to $L$, where $L$ total number of links of the network. In Figure \[fig:example\] we show a simple scheme that illustrates the dynamical rules in the case $k=4$ (A) and $k=5$ (B). It is evident from the figure that for even values of $k$ the system remains the same if we simultaneously flip all the spins $s_{i,j}
\to -s_{i,j}$ $\forall \; (i,j)$ and make the transformation $p
\to 1-p$. The same is not true for odd values of $k$. The reason is that a $k$-cycle with only “unfriendly” links is balanced for even values of $k$, while it is imbalanced for odd values of $k$. The presence or absence of this symmetry property for even values of $k$ or odd, respectively, is responsible for very different features in the phase structure. This will be studied in detail in the following sections.
![Dynamical rules in case of $k=4$ (A) and $k=5$ (B). The cycles containing an odd number of “unfriendly” links are considered as imbalanced and evolve into balanced ones. Full and dashed lines represent “friendly” and “unfriendly” links respectively.[]{data-label="fig:example"}](example){width="47.00000%"}
Complete graphs {#complete}
===============
We first consider the case of fully connected networks. Later we extend the main results to the case of diluted networks in section \[sec:diluted\]. In a complete graph every individual has a relationship with everyone else. Let $N$ be the number of nodes of this complete graph. The total number of links of the network is then given by $L={N \choose 2}$, while the total number of $k$-cycles is given by $M={N \choose k}$. ${x \choose y}$ is the standard notation of the binomial coefficient. It counts the total number of different ways of choosing $y$ elements out of $x$ elements in total, while it is $0 \leq y\leq x$ , with $x,y \in \mathbb{N}$. Moreover we define $M_j$ as the number of $k$-cycles containing $j$ negative links, and $m_j= \; M_j/\;M$ the respective density of $k$-cycles of type $T_j$. The total number of positive links $L^+$ is then related to the number of $k$-cycles by the relation $$L^+ = \frac{\sum_{i=0}^k\left(k-i\right) \; M_i}{\left(N-2\right)!
\;/\; \left(N-k\right)!}\;\;\; .\label{eq:link_positive}$$ A similar relation holds for the total number of negative links $L^-$ $$L^- = \frac{\sum_{i=0}^k i \; M_i}{\left(N-2\right)! \;/\; \left(N-k\right)!}\;\;\;.
\label{eq:link_negative}$$ In particular, in Eq.s (\[eq:link\_positive\]) and (\[eq:link\_negative\]) the numerators give us the total number of positive and negative links in all the $k$-cycles, respectively, while the same denominator comes out from the fact that one link belongs to $(N-2)(N-3)\cdots(N-k+1)=\left(N-2\right)!/ \left(N-k\right)!$ different $k$-cycles. Furthermore the density of positive links is $\rho=L^+/L=1-\sum_{i=0}^k\; i \; m_i$, while the density of negative links is $1-\rho$.
Evolution Equations {#evolution}
-------------------
In view of deriving the mean field equations for the unconstrained dynamics, introduced in the former section \[model\], we need to define the quantity $M^+_j$ as the average number of $k$-cycles of type $T_j$ which are attached to a positive link. This number is given by $$%\begin{equation}
M^+_j = \frac{\left(k-j\right) \;\; M_j }{L^+}\;\;\; ,
%\label{eq:average_positive}$$while similarly $$%\begin{equation}
M^-_j = \frac{j \;\; M_j }{L^-}
%\label{eq:average_negative}$$counts the average number of $k$-cycles of type $T_j$ attached to a negative link. In term of densities we can easily write $$m^+_j = \frac{\left(k-j\right) \;\; m_j }{\sum_{i=0}^k\; \left(k-i\right) \; m_i}
\label{eq:density_positive}$$ and $$m^-_j = \frac{j \;\; m_j }{\sum_{i=0}^k\; i \; m_i}\;\;\;.
\label{eq:density_negative}$$ Now let $\pi^+$ be the probability that a link flips its sign from positive to negative in one update event and $\pi^-$ the probability that a negative link changes its sign to $+1$ in one update event. We can write such probabilities as $$\pi^+ = \left(1-p\right) \; \sum_{i=1}^{(k-1)/2} \; m_{2i-1}
\label{eq:prob_positive_odd}$$ and $$\pi^- = p \; \sum_{i=1}^{(k-1)/2} \; m_{2i-1} \; + \; \; m_k \;\;\;,
\label{eq:prob_negative_odd}$$ valid for the case odd values of $k$. For even values of $k$, these probabilities read $$\pi^+ = \left(1-p\right) \; \sum_{i=1}^{k/2} \; m_{2i-1}
\label{eq:prob_positive_even}$$ and $$\pi^- = p \; \sum_{i=1}^{k/2} \; m_{2i-1} \; \; \; .
\label{eq:prob_negative_even}$$ Since each update changes $\left(N-2\right)! / \left(N-k\right)!$ $k$-cycles, and also the number of updates in one time step is equal to $L$ update events, the rate equations in the mean field approximation can be written as $$\left\{
\begin{array}{l}
\frac{d}{dt}\; m_0 \; = \; \pi^- \; m^-_1 \; - \; \pi^+ \; m^+_0
\\
\\
\begin{array}{ll}
\frac{d}{dt}\;m_1 \; = \; & \pi^+ \; m^+_0 \; + \; \pi^- \; m^-_2 \; +
\\
& - \; \pi^- \; m^-_1 \; - \; \pi^+ \; m^+_1
\end{array}
\\
\vdots
\\
\begin{array}{ll}
\frac{d}{dt}\;m_j \; = \; & \pi^+ \; m^+_{j-1} \; + \; \pi^- \; m^-_{j+1} \; +
\\
& - \; \pi^- \; m^-_{j} \; - \; \pi^+ \; m^+_{j}
\end{array}
\\
\vdots
\\
\begin{array}{ll}
\frac{d}{dt}\;m_{k-1} \; = \; & \pi^+ \; m^+_{k-2} \; + \; \pi^- \; m^-_{k} \; +
\\
& - \; \pi^- \; m^-_{k-1} \; - \; \pi^+ \; m^+_{k-1}
\end{array}
\\
\\
\frac{d}{dt}\;m_k \; = \; \pi^- \; m^-_{k-1} \; - \; \pi^- \; m^-_k
\end{array}
\right.
\;\;\;.
\label{eq:mean_field}$$ We remark that the only difference between the cases of odd values of $k$ and even values of $k$ comes from Eq.s (\[eq:prob\_positive\_odd\]) and (\[eq:prob\_negative\_odd\]), and Eq.s (\[eq:prob\_positive\_even\]) and (\[eq:prob\_negative\_even\]), respectively. This difference is the main reason why the two cases odd values of $k$ and even values of $k$ lead to two completely different behavior and why we treat them separately in the following section \[stationary\].
Stationary states {#stationary}
-----------------
Next let us derive the stationary states from the rate equations (\[eq:mean\_field\]) that give a proper description of the unconstrained dynamics of $k$-cycles in a complete graph. Imposing the stationary condition $\frac{d}{dt}\;m_j = 0$ , $\forall \; 0
\leq j \leq k$, we easily obtain $$m^+_{j-1}\; = \; m^-_j \;\;\;, \; \forall \; 1 \leq j \leq k \;\;\; .
\label{eq:stationary1}$$ Then, forming products of the former quantities appearing in Eq.(\[eq:stationary1\]), we have $$%\begin{equation}
m^+_{j-1}\; m^-_{j+1}\; = \; m^+_j\; m^-_j\;\;\; , \; \forall \; 1 \leq j \leq k \;\;
%\label{eq:stationary2}$$and, using the definitions of Eq.s (\[eq:density\_positive\]) and (\[eq:density\_negative\]), we finally obtain $$\left(k-j+1\right)\left(j+1\right)
\; m_{j-1} \; m_{j+1} \; \; = \; \; \left(k-j\right) j \left(\; m_j\; \right)^2\;\;\; ,
\label{eq:stationary3}$$ valid $\forall \; 1 \leq j \leq k$. Moreover the normalization condition $\sum_i \; m_i \; = \; 1$ should be satisfied. Furthermore, in the case of stationary, the density of friendships should be fixed, so that we should impose that $\pi^+ \; = \;
\pi^-$.
### The case of odd values of $k$ {#odd}
In the case of odd values of $k$, the condition for having a fixed density of friendships reads $$m_k \; = \; \left(1 - 2p\right) \; \sum_{i=1}^{(k-1)/2} \; m_{2i-1}\;\;\;,
\label{eq:stationary_odd1}$$ where we used Eq.s (\[eq:prob\_positive\_odd\]) and (\[eq:prob\_negative\_odd\]). In principle the $k$ equations of (\[eq:stationary3\]) plus the normalization condition and the fixed friendship relation (\[eq:stationary\_odd1\]) determine the stationary solution. For $k=3$ Antal *et al.* [@antal] found $$m_j \; = \; {3 \choose j} \; \rho_\infty^{3-j} \; \left(1-\rho_\infty\right)^j\;\;, \; \forall \; 0 \leq j \leq 3 \;\;\; ,
\label{eq:antal1}$$ where $$\rho_\infty \; = \;
\left\{
\begin{array}{ll}
1/\left[ \sqrt{3\left(1-2p\right)} +1 \right] & \textrm{ , if } p \leq 1/2
\\
1 & \textrm{ , if } p \geq 1/2
\end{array}
\right.
\label{eq:antal2}$$ is the stationary density of friendly links. In the same manner also the case $k=5$ can be solved exactly with the solution $$m_j \; = \; {5 \choose j} \; \rho_\infty^{5-j} \; \left(1-\rho_\infty\right)^j \;\;, \; \forall \; 0 \leq j \leq 5 \;\;\; ,
\label{eq:k5a}$$ where $$\rho_\infty \; = \;
\left[ \sqrt{5\left(1-2p\right) \left(1+\sqrt{1+\frac{1}{5(1-2p)}}\right)} +1 \right]^{-1}
\label{eq:k5b}$$ for $p\leq 1/2$, while $\rho_\infty=1$ for $p\geq 1/2$.\
In Figure \[fig:t5\] we plot the densities $m_j$ given by Eq.(\[eq:k5a\]) and the stationary density of friendly links $\rho_\infty$ given by Eq.(\[eq:k5b\]) as function of $p$. Moreover we verified the validity of the solution performing several numerical simulations on a complete graph with $N=64$ nodes (full dots). We compute numerically the average density of positive links after $10^3$ time steps, where the average is done over $10^2$ different realizations of the system. At the beginning of each realization we select at random the values of the signs of the links, where each of them has the same probability to be positive or negative, so that $\rho_0=0.5$. The numerical results perfectly reproduce our analytical predictions.
![(Color online) Exact stationary densities $m_j$ for the cycles of order $k=5$ from Eq.(\[eq:k5a\]) and stationary density of friendly links $\rho_\infty$ from Eq.(\[eq:k5b\]), both as a function of the dynamical parameter $p$. Numerical results are also reported for a system with $N=64$ vertices. Each value (full dot) is obtained by averaging the density of friendly links reached after $10^3$ time steps over $10^2$ different realizations with random initial conditions ($\rho_0 = 0.5$).[]{data-label="fig:t5"}](t5){width="47.00000%"}
\
As one can easily see, both solutions (\[eq:antal1\]) and (\[eq:k5a\]) are just binomial distributions. This means that the densities of a cycle of order $k=3$ or a cycle of order $k=5$ with $j$ negative links are simply given by the probability of finding these densities on a complete graph in which each link is set equal to $1$ with probability $\rho_\infty$ or equal to $-1$ with probability $1-\rho_\infty$. (As already noticed in [@antal], this result may come a bit as a surprise, because the $3$-cycle or here the $5$-cycle dynamics seems to be biased towards the reduction of frustration, on the other hand it is a bias for individual triads without any constraint of the type that the frustration of the whole “society” should get reduced.)\
For odd values of $k>5$, a stationary solution always exists. This solution becomes harder to find as $k$ increases, because the maximal order of the polynomials involved increases with $k$ (for $k=3$ we have polynomials of first order, for $k=5$ polynomials of second order, for $k=7$ of third and so on). So it becomes impossible to find the solution analytically as the maximal order of solvable equations is reached. Nevertheless we can give an approximate solution using a self-consistent approach as we shall outline in the following. We suppose that the general solution for the stationary densities is of the form $$m_j \; = \; {k \choose j} \; \rho_\infty^{k-j} \; \left(1-\rho_\infty\right)^j \;\;, \; \forall \; 0 \leq j \leq k \;\;\; ,
\label{eq:kgen}$$ Eq.(\[eq:kgen\]) is an appropriate ansatz as we can directly see from the definition of the density of friendly links $\rho_\infty
= 1-\sum_{i=0}^k i\; m_i= 1- (1-\rho_\infty)$, where the last term comes out as mean value of the binomial distribution. ( Actually such self-consistency condition is satisfied by any distribution of the $m_j$s with mean value equal to $1-\rho_\infty$. ) Moreover the ansatz for the stationary solution in the form of Eq.(\[eq:kgen\]) has the following features: first it is valid for the special cases $k=3$ and $k=5$, and second, it is numerically supported. In Figure \[fig:test\] we show some results obtained by numerical simulations. We plot the densities $m_j$ for different values of $k$ \[ $k=7$ (A) , $k=9$ (B), $k=11$ (C) and $k=21$ (D) \] and different values of $p$ \[ $p=0$ (black circles) , $p=0.3$ (red squares) , $p=0.44$ (green diamonds) and $p=0.49$ (blue crosses) \]. We performed $50$ different realizations of a system of $N=64$ vertices, where the densities are extrapolated from $10^6$ samples ($k$-cycles) at each realization and after $5\cdot 10^2$ time steps of the simulations (so that we have reached the stationary state). The initial values of the signs are chosen to be friendly or unfriendly with the same probability ($\rho_0=0.5$). The full lines are given by Eq.(\[eq:kgen\]) for which the right value of $\rho_\infty$ is given by the average stationary density of friendly links and the average is performed over all simulations. Furthermore, we numerically check whether Eq.(\[eq:kgen\]) holds, with the same $\rho_\infty$ if we measure the densities of cycles also of order $k' \neq k$ and moreover, whether it holds during the time while using the time dependent density of friendly links $\rho(t)$ instead of the stationary one $\rho_\infty$. Since all these checks are positive, we may say that if at some time the distribution of friendly links (and consequently of unfriendly links) is uncorrelated, it will stay so forever.
![(Color online) Stationary densities $m_j$ for the $k$-cycles with $j$ negative links and different values of $k$ \[ $k=7$ (A) , $k=9$ (B), $k=11$ (C) and $k=21$ (D) \], and for different values of $p$ \[ $p=0$ (black circles) , $p=0.3$ (red squares) , $p=0.44$ (green diamonds) and $p=0.49$ (blue crosses) \]. The numerical results (symbols) represent the histograms extrapolated from $10^6$ samples and over $50$ different realizations of the network. In particular the initial values of the spins are equally likely at each realization (so that $\rho_0=0.5$), the distributions are sampled after $5\cdot 10^2$ time steps and the system size is always $N=64$. The prediction of Eq.(\[eq:kgen\]) is plotted as a full line and the value of $\rho_\infty$ used is taken from the simulations as the average value of the stationary density of positive links.[]{data-label="fig:test"}](test){width="47.00000%"}
\
Let us assume that the ansatz (\[eq:kgen\]) is valid, we then evaluate the unknown value of $\rho_\infty$ self-consistently by imposing the condition that the density of friendly links is fixed at the stationary state $$\pi^+ \; = \; \pi^- \;\;\; \Leftrightarrow \;\; \left(1-2p\right)
\sum_{i=1}^{(k-1)/2}\; m_{2i-1}\; = \; m_k \;\;\;.$$ In particular we can write $$\sum_{i=1}^{(k-1)/2}\; m_{2i-1} \; + \; m_k \; = \;
\sum_{i=1}^{(k+1)/2}\; m_{2i-1} \; = \xi\;, \label{eq:kgen_a}$$ and so $$m_k\; = \; \left(1-2p\right)\left(\xi-\; m_k\right)\;$$ from which $$\rho_\infty\; = \; 1-\left[ \frac{\xi \left(1-2p\right)}{2\left(1-p\right)}\right]^{1/k}\;\;\;,
\label{eq:kgen_fin}$$ for $p \leq 1/2$, while $\rho_\infty=1$ for $p \geq 1/2$. In particular we notice that Eq.(\[eq:kgen\_fin\]) goes to zero as $k \to \infty$ for $p < 1/2$, because $0 \leq \xi \leq 1$.. This means that in the limit of large $k$ the stationary density of friendly links takes the typical shape of a step function centered at $p=1/2$, with $\rho_\infty=0$ for $p<1/2$ and $\rho_\infty=1$ for $p>1/2$. This is exactly the result we find for the case even values of $k$ (see the next section \[even\]), and it is easily explained since in the limit of large $k$ the distinction between the cases odd values of $k$ and $k$ even should become irrelevant.\
Furthermore it should be noticed that $\xi$ defined in Eq.(\[eq:kgen\_a\]) is nothing more than a sum of all odd terms of a binomial distribution. For large values of $k$ we should expect that the sum of the odd terms is equal to the sum of the even terms of the distribution, so that $$\xi \; = \; \sum_{i=1}^{(k+1)/2}\;
m_{2j-1} \; \simeq \frac{1}{2} \; \simeq \; \sum_{i=0}^{(k-1)/2}\; m_{2j} \;\;\;,$$ because of the normalization. In Figure \[fig:theory\] we plot the quantity $(1-\rho_\infty)^k$ obtained by numerical simulations for different values of $k$ \[ $k=3$ (black circles) , $k=5$ (red squares) , $k=7$ (blue diamonds) , $k=9$ (violet triangles), $k=11$ (orange crosses) \] as a function of $p$. Each point represents the average value of the density of positive links (after $10^3$ time steps) over $10^2$ different realizations. The system size in our simulations is $N=64$, while, at the beginning of each realization, the links have the same probability to have positive or negative spin ($\rho_0 = 0.5$). From Eq.(\[eq:kgen\_fin\]) we expect that the numerical results collapse on the same curve $\xi(1-2p)/(2-2p)$, depending on the parameter $\xi$. Imposing $\xi=1/2$ \[dashed line\] we obtain an excellent fit for all values of $p$. Only for small values of $p$ the fit is less good than for intermediate and large values of $p$, which is explained by the plot in the inset of Figure \[fig:theory\]. There Eq.(\[eq:kgen\_a\]) is shown as function of $p$ for $k=3$ (black dotted line) and for $k=5$ (red full line). The values of $m_j$ are taken directly from the binomial distribution of Eq.(\[eq:kgen\]) with values of $\rho_\infty$ known exactly from Eq.s (\[eq:antal2\]) and (\[eq:k5b\]) for $k=3$ and $k=5$, respectively. We can see how well the approximation $\xi=1/2$ works already for $k=3$ and how it improves for $k=5$, with the only exception for small values of $p$ where $\xi > 1/2$. Furthermore we see that $\xi < 1/2$ for $p
\simeq 1/2$, but in this range the dependence on $\xi$ of Eq.(\[eq:kgen\_fin\]) becomes weaker since the factor $\xi(1-2p)$ tends to zero anyway.
![(Color online) Numerical results (symbols) and approximate solution (dashed line) for the function $\left(1-\rho_\infty\right)^k$, depending on the stationary density of positive links $\rho_\infty$ and the parameter $k$ \[ $k=3$ (black circles) , $k=5$ (red squares) , $k=7$ (blue diamonds) , $k=9$ (violet triangles) , $k=11$ (orange crosses) \], as a function of the dynamical parameter $p$. The theoretical result, plotted here as a dashed line, is given by Eq.(\[eq:kgen\_fin\]) for $\xi=1/2$. This prediction is in good agreement with the numerical results obtained by averaging the density of friendly links after $10^3$ time steps over $10^2$ different realizations. The system size is $N=64$. Each simulation starts with random initial conditions ($\rho_0=0.5$). Moreover, as we can see from the inset, the value of $\xi$ calculated for $k=3$ (red full line) and for $k=5$ (black dotted line) is very close to $1/2$ for an extended range of $p$.[]{data-label="fig:theory"}](theory){width="47.00000%"}
### The case of even values of $k$ {#even}
The stability of a $k$-cycle with all negative links in the case of even $k$ (see Figure \[fig:example\]) has deep implications on the global behavior of the model. Actually the elementary dynamics is now symmetric. Only the value of $p$ gives a preferential direction (towards a completely friendly or unfriendly cycle) to the basic processes. With odd $k$, for $p < 1/2$ the tendency of the dynamics to reach the state with a minor number of positive links in the elementary processes (involving no totally unfriendly cycles) is overbalanced by the process $T_{k}\rightarrow T_{k-1}$ which happens with probability one, so that in the thermodynamical limit the system ends up in an active steady state with a finite average density of negative links due to the competition between the basic processes. Instead, for even $k$, nothing prevents the system from reaching the “hell”, that is a state of only negative links, because here a completely negative cycle is stable. Only for $p=1/2$ we expect to find a non-frozen fluctuating final state, since in this case the elementary dynamical processes are fully symmetric. Imposing the stationary conditions on the system we do not get detailed information about the final state. As we can see from Eq.s (\[eq:prob\_positive\_even\]) and (\[eq:prob\_negative\_even\]), for $p\neq1/2$ the only possibility to have $\pi^+=\pi^-$ is the trivial solution for which both probabilities are equal to zero, so that the system must reach a frozen configuration, while for $p=1/2$, $\pi^+$ and $\pi^-$ are always equal, in this case we expect the system to reach immediately an active steady state. In order to describe more precisely the final configuration of this active steady state, it is instructive to consider the mean-field equation for the density of positive links. For generic even value of $k$, it is easy to see that the number of positive links increases in updates of type $T_{2j-1}\rightarrow
T_{2(j-1)}$ with probability $p$, whereas it decreases in updates of type $T_{2j-1}\rightarrow T_{2j}$ with probability $1-p$, so that the mean field equation that governs the behavior of the density of friendly links is given by $$\frac{d\rho}{dt}=(2p-1)\rho(1-\rho)\cdot\sum_{i=1}^{k/2}
{k \choose 2i-1}\cdot\rho^{k-2i}(1-\rho)^{2(i-1)}\;\;\;.
\label{mfr}$$ For $p\neq1/2$ we have only two stationary states, $\rho_{\infty}=0$ and $\rho_{\infty}=1$ (the other roots of the steady state equation are complex). It is easily understood that for $p<1/2$ the stable configuration is $\rho_{\infty}=0$, while for $p>1/2$ it is $\rho_{\infty}=1$. In contrast, for $p=1/2$ we have $\rho(t)=$const at any time, so that $\rho_{\infty}=\rho(t=0)=\rho_0$. These results are confirmed by numerical simulations. Moreover, the convergence to the thermodynamical limit is quite fast, as it can be seen in Figure \[thl\], where we plot the density of friendly links $\rho_\infty$ as a function of $p$ for the system sizes $N$ \[ $N=8$ (dotted line), $N=16$ (dashed line) and $N=32$ (full line) \] and for $k=4$. Each curve is obtained from averages over $10^3$ different realizations of the dynamical system. In all simulations the links get initially assigned the values $\pm 1$ with equal probability, so that $\rho_0=0.5$.
![Behavior of the stationary density of friendly links $\rho_{\infty}$ as a function of $p$ for three (small) values of $N$ \[ $N=8$ (dotted line), $16$ (dashed line) and $32$ (full line) \] and for $k=4$. The values of the initial configuration are randomly chosen to be $\pm 1$ with density of friendly links $\rho_0=0.5$. The curves are obtained from averages over $10^3$ different realizations.[]{data-label="thl"}](therm_lim.eps){width="47.00000%"}
Frozen configurations {#frozen}
---------------------
When all $k$-cycles of the network are balanced we say that the network itself is balanced. In particular, in the case of our unconstrained dynamics we can say that if the network is balanced it has reached a frozen configuration. The configuration is frozen in the sense that no dynamics is left since the system cannot escape a balanced configuration. Furthermore it was proven [@cartwright] that if a graph (not only a complete graph) is balanced it is balanced independently of the choice of $k$ and that the only possible balanced configurations are given by bipartitions of the network in two subgroups (or “cliques”), where all the individuals belonging to the same subgroup are friends while every couple of individuals belonging to different subgroups are enemies (this result is also known as *Structure Theorem* [@roberts]). In the case of even values of $k$ the latter result is still valid if all the individuals of one subgroup are enemies, while two individuals belonging to different subgroups are friends. It should be noticed that one of the two cliques may be empty and therefore the configuration of the paradise (where all the individuals are friends) is also included in this result, as well as, for the case even values of $k$, the hell with all individuals being enemies . In the following we will combine our former results about the stationary states (section \[stationary\]) with the notion of frozen configurations in order to predict the probability of finding a particular balanced configuration and the time needed for freezing our unconstrained dynamical process. For clarity we analyze the cases of odd values of $k$ and even values of $k$ separately, again.
### Freezing time for odd values of $k$ {#time_odd}
Let $0 \leq N_1 \leq N$ be the size of one of the two cliques. Therefore the other clique will be of size $N-N_1$. In such a frozen configuration the total number of positive and negative links are related to $N_1$ and $N$ by $$L^+ = \frac{N_1\left(N_1-1\right)}{2}+\frac{\left(N-N_1\right)\left(N-N_1-1\right)}{2}
\label{eq:positive_frozen}$$ and $$L^- = N_1\left(N-N_1\right)
\label{eq:negative_frozen}$$ respectively. As we have seen in the former section \[odd\], for odd values of $k$ and $p<1/2$, all the $k$-cycles are uncorrelated during the unconstrained dynamical evolution, if we start from an initially uncorrelated configuration. In such cases, we can consider our system as a purely random process in which the values of the spins are chosen at random with a certain probability. In particular, the probability of a link to be positive is given by $\rho$ , the density of positive links ( $1-\rho$ is the probability for a link to be negative). The probability of reaching a frozen configuration, characterized by two cliques of $N_1$ nodes and $N-N_1$ nodes, is then given by $$P(\rho,N_1) = {N \choose N_1}
\rho^{\frac{N(N-1)}{2}-N_1(N-N_1)} \left( 1 -\rho \right)^{N_1(N-N_1)} \;\;\; .
\label{eq:frozen_prob}$$ The binomial coefficient ${N \choose N_1}$ in Eq.(\[eq:frozen\_prob\]) counts the total number of possible bi-partitions into cliques with $N_1$ and $N-N_1$ nodes ( i.e. the total number of different ways for choosing $N_1$ nodes out of $N$), and each of these bi-partitions is considered as equally likely because of the randomness of the process. We should also remark that in Eq.(\[eq:frozen\_prob\]) we omit the time dependence of $\rho$, while the density of positive links $\rho$ follows the following master equation $$\frac{d\rho}{dt}=
(1-\rho)^k+(2p-1) \sum_{i=1}^{(k-1)/2} {k \choose 2i-1} \rho^{2i-1} (1-\rho)^{k-2i+1}
\;\;\;.$$ Eq.(\[eq:frozen\_prob\]) shows that the probability of having a frozen configuration with cliques of $N_1$ and $N-N_1$ nodes is extremely small, because the number of the other equiprobable configurations with the same number of negative and positive links is equal to ${L \choose L^-} \gg {N \choose N_1}$, where $L^-$ should satisfy Eq.(\[eq:negative\_frozen\]). This allows us to ignore the transient time to reach the stationary state (we expect that the system goes to the stationary state exponentially fast for any $k$, as shown in [@antal] for $k=3$) and consider the probability for obtaining a frozen configurations as $$P\left(\rho_\infty \right) = \sum_{N_1=0}^N P\left(\rho_\infty,N_1\right) \;\;\; .
\label{eq:frozen_prob_odd}$$ This probability provides a good estimate for the order of magnitude in time $\tau$ that is needed to reach a frozen configuration, because $\tau \sim 1/P\left(\rho_\infty
\right)$. Unfortunately this estimate reveals that the time needed for freezing the system becomes very large already for small sizes $N$ (i.e. $\tau$ increases almost exponentially as a function of $L\sim N^2$). This means that it is practically impossible to verify this estimate in numerical simulations.\
\
At the transition, for the dynamical parameter $p=1/2$ we can follow the same procedure as used by Antal *et al.* [@antal]. The procedure is based on calculating the time it takes until a fluctuation in the number of negative links reaches the same order of magnitude as the average number of negative links. In this case the systems happens to reach the frozen configuration of the paradise due to a fluctuation. The number of unfriendly links $L^- \equiv A(t)$ can be written in the canonical form [@kampen] $$A(t)=La(t)+\sqrt{L}\eta(t)\;\;\;,
\label{qwe}$$ where $a(t)$ is the deterministic part and $\eta(t)$ is a stochastic variable such that $\langle\eta\rangle=0$. Let us consider the elementary processes $$A\longrightarrow\left\{\begin{array}{ll}
A-1 & \textrm{ , rate } \quad M_k \\
A-1 & \textrm{ , rate } \quad p\sum_{i=1}^{(k-1)/2}M_{2i-1} \\
A+1 & \textrm{ , rate } \quad (1-p)\sum_{i=1}^{(k-1)/2}M_{2i-1}
\end{array}
\right.
\label{elpr1}$$ and therefore $$A^2\longrightarrow\left\{\begin{array}{ll}
A^2-2A+1 & \textrm{ , rate} \quad M_k \\
A^2-2A+1 & \textrm{ , rate} \quad p\sum_{i=1}^{(k-1)/2}M_{2i-1} \\
A^2+2A+1 & \textrm{ , rate} \quad (1-p)\sum_{i=1}^{(k-1)/2}M_{2i-1}
\end{array}
\right.
\;.
\label{elpr2}$$ We can then write the following equations for the mean values of $A$ and $A^2$ $$%\begin{equation}
\frac{d\langle A\rangle}{dt}=-\langle M_k\rangle+(1-2p)\sum_{i=1}^{(k-1)/2}\langle M_{2i-1}\rangle
%\label{fluc1}$$and $$%\begin{equation}
\begin{array}{ll}
\frac{d\langle A^2\rangle}{dt}= & \langle(1-2A)M_k\rangle+
\\
& +p\left\langle(1-2A)\sum_{i=1}^{(k-1)/2}M_{2i-1}\right\rangle+
\\
& +(1-2p)\left\langle(1+2A)\sum_{i=1}^{(k+1)/2}M_{2i-1}\right\rangle
\end{array}
\;\;\;.
%\label{fluc2}$$For $p=1/2$ we obtain $$\frac{d\langle A\rangle}{dt}=-\langle M_k\rangle
\label{fluc3}$$ and $$%\begin{equation}
\frac{d\langle A^2\rangle}{dt}=\langle M_k\rangle+\sum_{i=1}^{(k-1)/2}
\langle M_{2i-1}\rangle-2\langle AM_k\rangle \;\;\;.
%\label{fluc4}$$Since it is $\langle A\rangle\sim a$ and $\langle M_k\rangle\sim
a^k$, we get from Eq.(\[fluc3\]) $$\frac{da}{dt}=-a^k \;\;\; ,
\label{fluc4B}$$ from which $$a(t)\sim t^{-\frac{1}{k-1}}\;\;\;.
\label{fluc5}$$ On the other hand, considering that $d\langle A\rangle^2/dt=2\langle
A\rangle\cdot d\langle A\rangle/dt$ and by definition $\sigma=\langle A^2\rangle-\langle A\rangle^2=\langle\eta^2\rangle$, we have $$\frac{d\sigma}{dt}=\langle M_k\rangle+\sum_{i=1}^{(k-1)/2}
\langle M_{2i-1}\rangle-2(\langle AM_k\rangle-\langle A\rangle\langle M_k\rangle)\;\;\;.
\label{fluc6}$$ Moreover we can write $$%\begin{equation}
\begin{array}{rl}
\langle AM_k\rangle-\langle A\rangle\langle M_k\rangle=
&\langle(La+\sqrt{L}\eta)M_k\rangle-La\langle M_k\rangle=
\\
= & \sqrt{L}\langle\eta M_k\rangle
\end{array}\;\;\;.
%\label{fluc7}$$It is easy to see that $\langle\eta M_k\rangle\sim\langle
\eta A^k\rangle=\langle\eta(La+\sqrt{L}\eta)^k\rangle$, so that $$\begin{array}{ll}
\langle\eta M_k\rangle\sim & \langle\eta\cdot(L^ka^k+kL^{k-1/2}a^{k-1}
\eta+\dots+L^{k/2}\eta^k)\rangle=
\\
& = kL^{k-1/2}a^{k-1}\langle\eta^2\rangle+O(\langle\eta^3\rangle)\;\;\;.
\end{array}
\label{fluc8}$$ Dividing Eq.(\[fluc6\]) by Eq.(\[fluc4B\]) and using Eq.(\[fluc8\]) we get $$\frac{d\sigma}{da}=-\left[2ka^{k-1}\sigma-\sum_{i=1}^{(k+1)/2}
{k\choose 2i-1}a^{2i-1}(1-a)^{k-2i+1}\right]. \label{fluc9}$$ Here we have taken into account that $$\langle M_j\rangle\sim{k\choose j}a^j(1-a)^{k-j}\;\;\;.
\label{eq:a}$$ It is straightforward to find the solution of Eq.(\[fluc9\]) as $$\sigma(a)=Ca^{2k}+\frac{\gamma_k}{a}+\dots+\frac{\gamma_0}{a^{k-2}}\;\;\;,$$ with $C$ and $\gamma_j$ suitable constants. From Eq.(\[fluc5\]), for $t\to \infty$ we have $$\sigma\sim a^{-(k-2)}\sim t^{\frac{k-2}{k-1}}\;\;\; .$$ For $\eta\sim\sqrt{\sigma}$, we finally obtain $$%\begin{equation}
\eta\sim t^{\frac{k-2}{2(k-1)}}\;\;\;.
%\label{fluc10}$$In general, the system will reach the frozen state of the paradise when the fluctuations of the number of negative links become of the same order as its mean value. (Note that in this case the mean-field approach is no longer valid.) Then, in order of finding the freezing time $\tau$ we have just to set equal the two terms on the right hand side of Eq.(\[qwe\]). $$La(\tau)\sim\sqrt{L}\eta(\tau)\;\;\;.
\label{eq:condition}$$ Since $L\sim N^2$, we get a power-law behavior $$\tau\sim N^\beta
\label{eq:freezing_time_odd}$$ with exponent $\beta$ as a function of $k$ according to $$\beta=2\frac{k-1}{k}\;\;\;.
\label{fluc11}$$ It is worth noticing that in the limit $k \to \infty$ we obtain $\beta=2$, which is the same result as in the case of even values of $k$ as we shall see soon. The analytical results of this subsection are well confirmed by simulations, cf. Figure \[fig:time\_odd\]. There we study numerically the freezing time $\tau$ as a function of the system size $N$ for different odd values of $k$ \[ $k=3$ (black circles) , $k=5$ (red squares) , $k=7$ (blue diamonds) , $k=9$ (violet triangles) and $k=15$ (orange crosses) \]. The freezing time is measured until all links have positive sign and paradise is reached. Other frozen configurations are too unlikely to be realized. Each point stands for the average value over a different number of realizations of the dynamical system \[ $100$ realizations for sizes $N \leq 64$ , $50$ realizations for $64<N\leq 256$ and $10$ realizations for $N>256$ \], where the initial configuration is always chosen as an antagonistic society (all the links being negative so that $\rho_0=0$) to reduce the statistical error. The standard deviations around the averages have sizes comparable with the symbol sizes. The full lines stands for power laws with exponents given by Eq.(\[fluc11\]). They perfectly fit with the numerical measurements.
![(Color online) Numerical results (full dots) for the freezing time $\tau$ as a function of the system size $N$ and for various $k$ \[ $k=3$ (black circles) , $k=5$ (red squares) , $k=7$ (blue diamonds) , $k=9$ (violet triangles) and $k=15$ (orange crosses) \]. Each point is given by the average value over several realizations \[ $100$ realizations for sizes $N \leq 64$ , $50$ realizations for $64<N\leq 256$ and $10$ realizations for $N>256$ \]. Moreover as initial configuration of each realization the links are chosen all negative ($\rho_0=0$, antagonistic society) in order to reduce the statistical error (the standard deviation is comparable with the symbol size) caused by the small number of realizations at larger sizes of the system. The full lines have slope $2(k-1)/k$ as expected from Eq.(\[fluc11\]). The inset shows the numerical results for the freezing time $\tau$ , for different values of $k$ (the same as in the main plot), as a function of the system size $N$ and for $p=3/4$. Each point of the inset is given by the average over $10^3$ different realizations with initial antagonistic society.[]{data-label="fig:time_odd"}](time_odd){width="47.00000%"}
For $p>1/2$ the freezing time $\tau$ scales as $$\tau \sim \ln{N}\;\;\;.
\label{eq:time_pbigger}$$ The derivation would be the same as in the paper of Antal *et al.* [@antal]. It should be noticed that for $p>1/2$ the paradise is reached as soon as $k$ increases. For simplicity let $p=1$ and imagine that the system is at the closest configuration to the paradise, for which only one link in the system has negative spin. This link belongs to $R = (N-2)!/(N-k)!$ different $k$-cycles. At each update event we select one $k$-cycle at random out of $M = {N \choose k}$ total $k$-cycles. This way we have to wait a number of update events $E \sim M/R$ until the paradise is reached, which leads to a freezing time $\tau \sim E/L$, with $L$ the total number of links independent on $k$, so that $$\tau \sim \frac{1}{k!}\;\;\;.
\label{eq:time_biggerp}$$ For values of $1/2<p<1$ the $k$-dependence of $\tau$ should be weaker than the one in Eq.(\[eq:time\_biggerp\]), but anyway $\tau$ should be a decreasing function of $k$. The inset of Figure \[fig:time\_odd\] shows the numerical results obtained for $p=3/4$ as a function of the size of the system $N$. The freezing time $\tau$ is measured for different values of $k$. We plot the average value over $10^3$ different realizations with initial condition $\rho_0=0$.
### Freezing time for even values of $k$ {#sec:time_even}
In the case of even values of $k$ and $p=1/2$ the master equation for the density of positive links \[ Eq.(\[mfr\]) \] reads as $d\rho/dt=0$. Therefore, the density of friendly links, $\rho$, should be constant during time for an infinite large system. In finite size systems the dynamics is subjected to non-negligible fluctuations. This allows to understand the scaling features of the freezing time $\tau$ with the system size. The order of the fluctuations is $\sqrt{L}$ because the process is completely random as we have seen for the case odd values of $k$ and $p<1/2$. Differently from the latter case, for even values of $k$ and $p=1/2$ the system has no tendency to go to a fixed point determined by $p$ because $d\rho/dt=0$. We can view the dynamical system as a Markov chain, with discrete steps in time and state space, for which the transition probability for passing from a state with $L^-(t-1)$ negative at time $t-1$ to a state with $L^-(t)$ negative links at time $t$ is given by $$\begin{array}{l}
P\left[ \; L^-(t)\; | \; L^-(t-1) \; \right] =
\\
= {L \choose L^-(t)} \left(\frac{L-L^-(t-1)}{L} \right)^{L-L^-(t)} \left(\frac{L^-(t-1)}{L} \right)^{L^-(t)}\;\;\;.
\end{array}
\label{eq:markov}$$ So that the probability of having $L^-(t)$ negative links at time $t$ is just a binomial distribution where the probability of having one negative link is given by $\frac{L^-(t-1)}{L}$, the density of negative links at time $t-1$. This includes both the randomness of the displacement of negative links and the absence of a particular fixed point dependent on $p$. The Markov process, with transition probability given by Eq.(\[eq:markov\]), is known under the name of the Wright-Fisher model [@wright] from the context of biology. The Wright-Fisher model is a simple stochastic model for the reproduction of diploid organisms (diploid means that each organism has two genes, here named as “$-$” and “$+$”), it was proposed independently by R.A. Fisher and S. Wright at the beginning of the thirties [@wright]. The population size of genes in an organism is fixed and equal to $L/2$ so that the total number of genes is $L$. Each organism lives only for one generation and dies after the offsprings are made. Each offspring receives two genes, each one selected with probability $1/2$ out of the two genes of a parent of which two are randomly selected from the population of the former generation. Now let us assume that there is a random initial configuration of positive and negative genes with a slight surplus of negative genes. The offspring generation selects its genes randomly from this pool and provides the pool for the next offspring generation. Since the pools get never refreshed by a new random configuration, the initial surplus of negative links gets amplified in each offspring generation until the whole population of genes is “negative”. Actually the solution of the Wright-Fisher model is quite simple. The process always converges to a final state with $L^-=0$ \[$L^+=L$\] or $L^-=L$ \[$L^+=0$\], corresponding to our heaven and \[hell\] solutions for even values of $k$. The average value over several realizations of the same process depends on the initial density of friendly links $\rho_0$ according to $$\langle L^- \rangle = \rho_0 \delta(0) + (1-\rho_0) \delta(L)\;\;\; ,$$ where $\delta(x)=1$ for $x=0$ and $\delta(x)=0$ otherwise. Furthermore, on average, the number of negative links decays exponentially fast to one of the two extremal values $$\langle L^- (t) \rangle \simeq L
\left\{
\begin{array}{l}
e^{-t/L}
\\
1 - e^{-t/L}
\end{array}
\right. \;\;\;.$$ with typical decay time $$\tau \sim L \sim N^2\;\;\;.
\label{eq:freezing_time_even}$$ This result is perfectly reproduced by the numerical data plotted in Figure \[fig:time\_even\]. The main plot shows the average time needed to reach a balanced configuration as a function of the size of the system $N$ and for different values of $k$ \[ $k=4$ (black circles) , $k=6$ (red squares) , $k=8$ (blue diamonds) and $k=12$ (violet crosses) \]. The averages are performed over different numbers of realizations depending on the size $N$ \[ $1000$ realizations for sizes $N \leq 128$ , $500$ realizations for $128<N\leq 384$ and $50$ realizations for $N=384$ and $N=512$, and $10$ realizations for $N=1024$ \]. The dashed line in Figure \[fig:time\_even\] has, in the log-log plane, a slope equal to $2$, all numerical data fit very well with this line. Furthermore it should be noticed that there is no $k$-dependence of the freezing time $\tau$, as it is described by Eq.(\[eq:markov\]). This is reflected by the fact that $\tau$ is the same for all the values of $k$ considered in the numerical measurements.
![(Color online) Numerical results for the freezing time $\tau$ as a function of the system size $N$ and for various even values of $k$ \[ $k=4$ (black circles) , $k=6$ (red squares) , $k=8$ (blue diamonds) and $k=12$ (violet crosses) \] and for $p=1/2$. Each point is given by the average value over several realizations \[ $100$ realizations for sizes $N \leq 64$ , $50$ realizations for $64<N\leq 256$ and $10$ realizations for $N>256$ \]. Moreover, at the beginning of each realization the links are chosen to be positive or negative with the same probability ($\rho_0=0.5$). The dashed line has, in the log-log plane, slope $2$ as expected in Eq.(\[eq:freezing\_time\_even\]). The inset A) shows the numerical results for the freezing time $\tau$ , for different values of $k$ (the same as in the main plot), as a function of the system size $N$ and for $p=3/4$. Each point of the inset is given by the average over $10^3$ different realizations with random initial conditions. The full lines are all proportional to $\ln{N}$ as expected. The inset B) shows the not-normalized probability $P(N_1)$ as a function of the ratio $N_1/N$ and for different values of the system size $N$ \[ $N=6$ (full line), $N=8$ (dashed line) and $N=10$ (dotted line) \]. As one can see, $P(N_1)$ is extremely small for values of $0 < N1 <
N$ already for $N=10$.[]{data-label="fig:time_even"}](time_even){width="47.00000%"}
Nevertheless there is a difference between our model and the Wright-Fisher model that should be noticed. During the evolution of our model there is the possibility that the system freezes in a configuration different from the paradise ( $L^-=0$ ) or the hell ($L^-=L$ ). The probability of this event is still given by Eq.(\[eq:frozen\_prob\]), with $r=L^+(N_1)/L$ as the stationary condition \[ $L^+(N_1)$ is given by Eq.(\[eq:positive\_frozen\]) \]. In this way Eq.(\[eq:frozen\_prob\]) gives us $P(N_1)$, the not-normalized probability for the system to freeze in a balanced configuration with two cliques of $N_1$ and $N-N_1$ nodes, respectively. It is straightforward to see that $P(N_1)=1$ for $N_1=0$ or for $N_1=N$, so that the paradise has a non-vanishing probability to be a frozen configuration. Differently for any other value of $0 < N_1 < N$, $P(N_1)$ decreases to zero faster than $1/N$. This means that for values of $N$ large enough it is appropriate to forget about the intermediate frozen configurations and to consider the features of our model as being very well approximated by those of the Wright-Fisher model. In the inset B) of Figure \[fig:time\_even\] the function $P(N_1)$ is plotted for different values of $N$ \[ $N=6$ (full line), $N=8$ (dashed line) and $N=10$ (dotted line) \] with $N_1$ a continuous variable for clarity of the figure (we approximate the factorial with the Stirling’s formula). Obviously $P(N_1)$ disappears for $0 < N_1 < N$ as $N$ increases, already for reasonably small values of $N$.\
The dependence $\tau \sim N^2$ can also be obtained using the same procedure as the one in section \[time\_odd\] for the case odd values of $k$ and $p=1/2$. In particular for even values of $k$ we can rewrite Eq.(\[elpr1\]) according to $$A\longrightarrow\left\{\begin{array}{ll}
A-1 & \textrm{ , rate}\quad p\sum_{i=1}^{k/2}M_{2i-1} \\
A+1 & \textrm{ , rate}\quad (1-p)\sum_{i=1}^{k/2}M_{2i-1}
\end{array}
\right .
\label{elpr1_even}$$ and therefore Eq.(\[elpr2\]) according to $$A^2\longrightarrow\left\{\begin{array}{ll}
A^2-2A+1 & \textrm{ , rate}\quad p\sum_{i=1}^{k/2}M_{2i-1} \\
A^2+2A+1 & \textrm{ , rate}\quad (1-p)\sum_{i=1}^{k/2}M_{2i-1}
\end{array}
\right. \;\;\;.
\label{elpr2_even}$$ For $p=1/2$ we have $$\frac{d\langle A\rangle}{dt}=0
\label{fluq1_even}$$ and $$%\begin{equation}
\frac{d\langle A^2\rangle}{dt}=\sum_{i=1}^{k/2} \langle M_{2i-1} \rangle\;\;\;.
%\label{fluq2_even}$$Eq.(\[fluq1\_even\]) tells us that $a\sim \langle A\rangle=$const, so that we have $$\eta\sim \sqrt{t}\;\;\;,$$ remebering Eq.(\[eq:a\]). As in the previous case, for determining the freezing time we impose the condition that the average value is of the same order as the fluctuations \[Eq.(\[eq:condition\])\], and, for $L\sim
N^2$, we obtain again Eq.(\[eq:freezing\_time\_even\]).\
\
For even values of $k$ and for $p \neq 1/2$ the time $\tau$ needed for reaching a frozen configuration scales as $\tau \sim \ln{N}$. In the inset of Figure \[fig:time\_even\] numerical estimates of $\tau$ for $p=3/4$ and different values of $k$ demonstrate this dependence on the size $N$ of the system. Each point is obtained from averaging over $10^3$ different simulations with the same initial conditions $\rho_0=0.5$. Again, as in the case of $k$ odd and $p>1/2$, $\tau$ is a decreasing function of $k$ and the same argument used for obtaining Eq.(\[eq:time\_biggerp\]) can be applied here.
Diluted Networks {#sec:diluted}
================
In this section we extend the former results, valid in the case of fully connected networks, to diluted networks. Real networks, apart from very small ones, cannot be represented by complete graphs. The situation in which all individuals know each other is in practice very unlikely. As mentioned in the introduction, links may be also missing, because individuals neither like nor dislike each other but are just indifferent. In the following we analyze the features of dynamical systems, still following the unconstrained $k$-cycle dynamics, but living on topologies given by diluted networks.\
For diluted networks there is an interesting connection to another set of problems that leads to a new interpretation of the social balance problem in terms of a certain $k$-SAT ($k$S) problems (SAT stands for satisfiability) [@cook; @mezard; @mezard2]. In such a problem a formula $F$ consists of $Q$ logical clauses $\left\{C_q\right\}_{q=1,\ldots,Q}$ which are defined over a set of $B$ Boolean variables $\left\{x_i=0,1\right\}_{i=1,\ldots,B}$ which can take two possible values $0=$*FALSE* or $1=$*TRUE*. Every clause contains $k$ randomly chosen Boolean variables that are connected by logical $OR$ operations ($\bigvee$). They appear negated with a certain probability. In the formula $F$, all clauses are connected by logical $AND$ operations ($\bigwedge$) $$F=\bigwedge_{q=1}^Q C_q\;\;\;,$$ so that all clauses $C_q$ should be simultaneously satisfied in order to satisfy the formula $F$. A particular formulation of the $k$S problem is the $k$-XOR-SAT ($k$XS) problem [@weigt; @semerjian; @mezard2; @cocco], in which each clause $C_q$ is a parity check of the kind $$C_q= x_{i_1}^q+x_{i_2}^q+ \ldots + x_{i_k}^q \;\;\; \textrm{mod }2\;\;\;,
\label{eq:xor}$$ so that $C_q$ is *TRUE* if the total number of true variables which define the clause is odd, while otherwise the clause $C_q$ is *FALSE*. It is straightforward to map the $k$XS problem to our former model for the case odd values of $k$. Actually, each clause $C_q$ corresponds to a $k$-cycle \[$Q \equiv M$\] and each variable $x_v$ to a link $(i,j)$. Furthermore \[$B \equiv L$\] with the correspondence $s_{i,j}=1$ for $x_v=1$, while $s_{i,j}=-1$ for $x_v=0$. For the case of even values of $k$, one can use the same mapping but consider as clause $C_q$ in Eq.(\[eq:xor\]) its negation $\overline{C_q}$. In this way, when the number of satisfied variables $x_i^q$ is odd the clause $\overline{C_q}$ is unsatisfied for odd values of $k$, while $\overline{C_q}$ is satisfied for even values of $k$.\
Moreover a typical algorithm for finding a solution of the $k$S problem is the so-called Random-Walk SAT (RWS). The procedure is the following [@weigt; @semerjian]: select one unsatisfied clause $C_q$ randomly, next invert one randomly chosen variable of its $k$ variables $x_{i^*}^q$; repeat this procedure until no unsatisfied clauses are left in the problem. Each update is counted as $1/B$ units of time. As one can easily see, this algorithm is very similar to our unconstrained dynamics apart from two aspects. First, in our unconstrained dynamics we use the dynamical propensity parameter $p$, while it is absent in the RWS. Second, in our unconstrained dynamics we count also the choice of a balanced $k$-cycle as update event, although it does not change the system at all. Because of this reason, the literal application of the original algorithm of unconstrained dynamics has very high computational costs if it is applied to diluted networks. Apart from the parameter $p$, we can therefore use the same RWS algorithm for our unconstrained dynamics of $k$-cycles. This algorithm is more reasonable because it selects at each update event only imbalanced $k$-cycles which are actually the only ones that should be updated. In case of an all-to-all topology there are so many triads that a preordering according to the property of being balanced or not is too time consuming so that in this case our former version is more appropriate. In order to count the time as in our original framework of the unconstrained dynamics, we should impose that, at the $n$-th update event, the time increases as $$t_n\; \; = \;\; t_{n-1}\;\;+ \;\;\frac{1}{L}\; \cdot \; \frac{\alpha}{\alpha^{(n-1)}_{u}}\;\;\;.
\label{eq:time_scaling}$$ Here $\alpha=M/L$ stands for the ratio of the total number of $k$-cycles of the system (i.e. total number of clauses) and the total number of links (i.e. total number of variables). The parameter $\alpha$ is called the “dilution” parameter, it can take all possible values in the interval $\left[0 , {L \choose
k}/L\right]$. $\alpha^{(n-1)}_{u}=\sum_{i=1}^{(k+1)/2}M_{2i-1}/L$ is the ratio of the total number of imbalanced (or “unsatisfied”) $k$-cycles over the total number of links, in particular $\alpha^{(n-1)}_{u}$ is computed before an instant of time at which the $n$-th update event is implemented. Therefore the ratio $\alpha/\alpha^{(n-1)}_{u}$ gives us the inverse of the probability for finding an imbalanced $k$-cycle, out of all, balanced or imbalanced, $k$-cycles, at the $n$-th update event. This is a good approximation to the time defined in the original unconstrained dynamics. It should be noticed that this algorithm works faster in units of this computational time, but the simulation time should be counted in the same units as defined for the unconstrained dynamics introduced in section \[model\].\
The usual performance of the RWS is fully determined by the dilution parameter $\alpha$. For $\alpha \leq \alpha_d$ the RWS always finds a solution of the $k$S problem within a time that scales linearly with the number of variables $L$. In particular for the $k$XS problem $\alpha_d=1/k$. For $\alpha_d < \alpha <
\alpha_c$ the RWS is still able to find a solution for the $k$S problem, but the time needed to find the solution grows exponentially with the number of variables $L$. For the case of the $3$XS problem $\alpha_c \simeq 0.918$. $\alpha_d$ is the value of the dilution parameter for which we have the “dynamical” transition, depending on the dynamics of the algorithm while $\alpha_c$ represents the transition between the SAT and the UNSAT regions: for values of $\alpha \geq \alpha_c$ the RWS is no longer able to find any solution for the $k$S problem, and in fact no such solution with zero frustration exists for the $k$S problem. Furthermore there is a third critical threshold $\alpha_s$, with $\alpha_d < \alpha_s <\alpha_c$. For values of $\alpha < \alpha_s$ all solutions of the $k$S problem found by the RWS are located into a large cluster of solutions and the averaged and normalized Hamming distance inside this cluster is $\langle d \rangle \simeq 1/2$. For $\alpha > \alpha_s$ the solutions space splits into a number of small clusters (that grows exponentially with the number of variables $L$) , for which the averaged and normalized Hamming distance inside each cluster is $\langle d \rangle \simeq 0.14$, while the averaged and normalized Hamming distance between two solutions lying in different clusters is still $\langle d \rangle \simeq 1/2$ [@cocco]. For the special case of the $3$XS problem $\alpha_s$ was found as $\alpha_s\simeq 0.818$.\
In order to connect the problems of social balance on diluted networks and the $k$XS problem on a diluted system we shall first translate the parameters into each other. First of all we need to calculate the ratio $\alpha=M/L$ between the total number of $k$-cycles of the network and the total number of links $L$ (section \[sec:ratio\]). Next we consider the standard RWS applied to the $k$XS problem taking care of the right way of computing the time as it is given by the rule (\[eq:time\_scaling\]) and the introduction of the dynamical parameter $p$ (section \[sec:RWS\]). In particular we focus on the “dynamical” transition at $\alpha_d$ (section \[sec:alphad\]) and the transition in solution space concerning the clustering properties of the solutions at $\alpha_s$ (section \[sec:alphas\]). The dynamical parameter $p$, formerly called the propensity parameter, leads to a critical value $p_c$ above which it is always possible to find a solution within a time that grows at most linearly with the system size (section \[sec:pc\]). Finally, in section \[sec:alpham\] we decrease the dilution, i.e. increase $\alpha$ to $\alpha_m$ such that for $\alpha \geq \alpha_m$ the system is fully described by the mean field equations of the former sections. We focus on the simplest case $k=3$, but all results presented here for $k=3$ should be qualitatively valid for any value of $k \geq 3$.
Ratio $\alpha$ for random networks {#sec:ratio}
----------------------------------
Let us first consider Erdös-Rényi networks [@erdos] as a diluted version of the all-to-all topology that we studied so far. An Erdös-Rényi network, or a random network, is a network in which each of the ${N \choose 2}$ different pairs of nodes is connected with probability $w$. The average number of links is simply $\langle L \rangle = w {N \choose 2}$. The average number of cycles of order $k$ is given $\langle M \rangle = w^k {N
\choose k}$, so that the average ratio $\langle \alpha \rangle$ can be estimated as $$\langle \alpha \rangle \simeq w^{k-1} \frac{2N^{k-2}}{k!} \;\;\; .
\label{eq:average_ratio}$$ In Figure \[fig:ratio\] we plot the numerical results obtained for the ratio $\alpha$ as a function of the probability $w$, in the particular case of cycles of order $k=3$. The reported results, from bottom to top, have been obtained for values of $N=16, 32, 48, 64, 96, 128, 192$ and $256$. Each point is given by the average over $10^3$ different network realizations. In particular these numerical results fit very well with the expectations (full lines) of Eq.(\[eq:average\_ratio\]), especially for large values of $N$ and/or small values of $w$. Furthermore the critical values $\alpha_d =1/3$ , $\alpha_s
=0.818$ and $\alpha_c =0.918$ (dotted lines) are used for extrapolating the numerical results of $w_d$ (open circles), $w_s$ (open squares) and $w_c$ (gray squares) respectively \[see the inset of Figure \[fig:ratio\]\]. $w_i \; , \; i=d,s,c$ is the value of the probability for which the ratio $\alpha_i \; , \;
i=d,s,c$ is satisfied. As expected, they follow the rule $w_i =
\sqrt{3\alpha_i/N} \; , \; i=d,s,c$ predicted by Eq.(\[eq:average\_ratio\]) for $k=3$.
![Numerical results (full dots) for the ratio $\alpha=M/L$ between the total number of cycles $M$ of order $k=3$ and the total number of links $L$ as a function of the probability $w$ for different sizes of Erdös-Rényi networks. In particular the numerical results refer to different network size $N$: from bottom to top $N=16, 32, 48, 64, 96, 128, 192$ and $256$. Each point is given by the average over $10^3$ network realizations. The full lines are the predicted values given by Eq.(\[eq:average\_ratio\]), while the dotted lines denote the critical values $\alpha_d =1/3$ , $\alpha_s =0.818$ and $\alpha_c
=0.918$ as described in detail in the text. In particular the numerical values of the probability $w$ for which these three critical values of $\alpha$ are realized are denoted by $w_d$ (open circles), $w_s$ (open squares) and $w_c$ (gray squares) respectively, they are plotted in the inset, where the full lines are extrapolated by Eq.(\[eq:average\_ratio\]) as $w_i =
\sqrt{3\alpha_i/N} \; , \; i=d,s,c$. The two upper curves for $w_s$ and $w_c$ almost coincide.[]{data-label="fig:ratio"}](alpha){width="47.00000%"}
\
According to the isomorphism traced between the $k$XS problem and the social balance for $k$-cycles, from now on we will not make any distinction between the words problem and network, variable and link, $k$-clause and $k$-cycle, value and sign (or spin), false and negative (or unfriendly), true and positive (or friendly), satisfied and balanced (or unfrustrated), unsatisfied and imbalanced (or frustrated), etc….
$p$-Random-Walk SAT {#sec:RWS}
--------------------
So far we have established the connection between the $k$XS problem and the social balance for $k$-cycles, proposed in this paper. In particular we have determined how the dilution parameter $\alpha$ is related to diluted random networks parameterized by $w$. In this section we extend the known results for the standard RWS of [@weigt; @semerjian] to the $p$-Random-Walk SAT ($p$RWS) algorithm, that is the RWS algorithm extended by the dynamical parameter $p$ that played the role of a propensity parameter in connection with the social balance problem. The steps of the $p$RWS are as follows:
1. [Select randomly a frustrated clause between all frustrated clauses.]{}
2. Instead of randomly inverting the value of one of its $k$ variables, as for an update in the case of the RWS, apply the following procedure:
- [if the clause contains both true and false variables, select with probability $p$ one of its false variable, randomly chosen between all the false variables belonging to the clause, and flip it to the true value;]{}
- [if the clause contains both true and false variables, select with probability $1-p$ one of its true variable, randomly chosen between all the true variables belonging to the clause, and flip it to the false value;]{}
- [if the clause contains only false values ($k$ should be odd), select with probability $1$ one of its false variables, randomly chosen between all the false variables belonging to the clause, and flip it to the true value.]{}
3. [Go back to point 1 until no unsatisfied clauses are present in the problem.]{}
The update rules of point 2 are the same used in the case of $k$-cycle dynamics and illustrated in Figure \[fig:example\] for the cases $k=4$ (A) and $k=5$ (B). For the special case of $3$XS problem, the standard RWS algorithm and the $p$RWS algorithm coincides for the dynamical parameter $p=1/3$.
### Dynamical transition at $\alpha_d$ {#sec:alphad}
The freezing time $\tau$, that is the time $\tau$ needed for finding a solution of the problem, abruptly changes its behavior at the dynamical critical point $\alpha_d =
1/k$.\
Figure \[fig:time\_diluted\] reports the numerical estimate of the freezing time $\tau$ as a function of the dilution parameter $\alpha$ and for different values of the dynamical parameter $p$ \[ $p=0$ (circles) , $p=1/3$ (squares) , $p=1/2$ (diamonds) and $p=1$ (crosses) \]. As one can easily see, for $p=1/3$ and $p=0$, $\tau$ drastically changes around $\alpha_d$, increasing abruptly for values of $\alpha> \alpha_d$. For $p=1/2$ and for $p=1$ this drastic change is not observed. This is understandable from the fact that both values of $p$ provide a bias towards paradise, while $p=1/3$ corresponds to a random selection of one of the three links of a triad as in the original RWS and $p=0$ would favor the approach to the hell if it were a balanced state. The simulations are performed over a system with $L=10^3$ variables. Moreover each point stands for the average over $10^2$ different networks and $10^2$ different realizations of the dynamics on such topologies. At the beginning of each simulation the variables take the value $1$ or $0$ with the same probability. The inset shows the relation between the time $\tau^*$ calculated using the standard RWS and the time $\tau$ calculated according to Eq.(\[eq:time\_scaling\]). The almost linear relation (the dashed line has a slope equal to one) between $\tau^*$ and $\tau$ means that there is no qualitative change between the two different ways of counting the time.
![Time $\tau$ for reaching a solution for a system of $L=1000$ variables as a function of the ratio $\alpha$ and for different values of the dynamical parameter $p$ \[ $p=0$ (circles), $p=1/3$ (squares), $p=1/2$ (diamonds) and $p=1$ (crosses) \]. The $p$RWS performed for $p=1/3$ shows a critical behavior around $\alpha_s=1/3$: for values of $\alpha \leq \alpha_s$, $\tau$ grows almost linearly with $\alpha$, while it jumps to an exponential growth with $\alpha$ for $\alpha> \alpha_s$. The same is qualitatively true for $p=0$, but the time $\tau$ needed for reaching a solution increases more slowly with respect to the case $p=1/3$ for $\alpha > \alpha_s$. For $p=1/2$ and $p=1$ there seems to be no drastic increment of $\tau$ for $\alpha >
\alpha_s$. Moreover the inset shows the dependence of $\tau^*$, the freezing time as calculated in the standard RWS [@weigt; @semerjian], on the freezing time $\tau$ calculated according to Eq.(\[eq:time\_scaling\]). The almost linear dependence of $\tau^*$ on $\tau$ (the dashed line has slope one) explains that there is no qualitative change if we describe the dynamical features of the system in terms of $\tau$ or $\tau^*$ as time used by the simulations.[]{data-label="fig:time_diluted"}](time2){width="47.00000%"}
Following the same argument as in [@weigt], we can specify for the update event at time $t$ the variation of the number of unsatisfied clauses $M_t^{(u)}$ as $$\Delta M_t^{(u)}=-\left(k \alpha_u(t) +1\right)+k \alpha_s(t) = k \alpha - 2 k \alpha_u(t)-1\;\;\;,$$ because, by flipping one variable of an unsatisfied clause, all the other unsatisfied clauses which share the same variable become satisfied, while all the satisfied clauses containing that variable become unsatisfied. In the thermodynamic limit $L\to \infty$, one can impose $M_t^{(u)}=L\alpha_u(t)$. Moreover, the amount of time of one update event is given by Eq.(\[eq:time\_scaling\]) so that we can write $$\dot{\alpha}_u(t)=\frac{\alpha_u(t)}{\alpha}\left(k\alpha-2k\alpha_u(t)-1\right)\;\;\;.
\label{eq:differential}$$ Eq.(\[eq:differential\]) has as stationary state (or a plateau) at $$\alpha_u=\frac{k\alpha-1}{2k}\;\;\;. \label{eq:plateu}$$ Therefore, when the ratio $\alpha$ (that is the ratio of the number of clauses over the number of variables) exceeds the critical “dynamical” value $$\alpha_d = \frac{1}{k}\;\;\;,
\label{eq:weigt}$$ the possibility of finding a solution for the problem drastically changes. This result was already found by [@weigt; @semerjian]. While for values of $\alpha \leq \alpha_d$ we can always find a solution because the plateau of Eq.(\[eq:plateu\]) is always smaller or equal to zero, for $\alpha>\alpha_d$ the solution is reachable only if the system performs a fluctuation large enough to reach zero from the non-zero plateau of Eq.(\[eq:plateu\]). In Figure \[fig:timebehav\] we report some numerical simulations for $\alpha_u$ as a function of the time for different values of $p$ \[ A) $p=0$ , B) $p=1/3$ , C) $p=1/2$ , D) $p=1$ \] and for different values of the dilution parameter $\alpha$ \[ $\alpha=0.3$ (black, bottom) , $\alpha=0.5$ (red, middle) , $\alpha=0.85$ (blue, top) \]. The numerical values \[full lines\] are compared with the numerical integration of Eq.(\[eq:differential\]) \[dashed lines\]. They fit very well apart from large values of $t$, for $\alpha=0.85$ and for $p=1/2$ or $p=1$. The initial configuration in all cases is that of an antagonistic society ($x_i=0 \;\;\; , \; \forall \; i=1,\ldots ,L$), while the number of variables is $L=10^4$.
![(Color online) Time behavior of the ratio $\alpha_u$ of unsatisfied clauses for different values of $p$ \[ A) $p=0$ , B) $p=1/3$ , C) $p=1/2$ , D) $p=1$ \] and for different values of the dilution parameter $\alpha$ \[ $\alpha=0.3$ (black, bottom) , $\alpha=0.5$ (red, middle) , $\alpha=0.85$ (blue, top) \]. Numerical results of simulations \[full lines\] are compared with the numerical integration of Eq.(\[eq:differential\]) \[dashed lines\] leading to a very good fit in all cases, except for $\alpha=0.85$ and for $p=1/2$ and $p=1$. The initial configuration in all the cases is the one of an antagonistic society ($x_i=0 \;\;\; , \; \forall \; i=1,\ldots ,L$), while the number of variables is $L=10^4$.[]{data-label="fig:timebehav"}](timebeah1){width="47.00000%"}
### Clustering of solutions at $\alpha_s$ {#sec:alphas}
In order to study the transition in the clustering structure of solutions at $\alpha_s$, we numerically determine the Hamming distance between different solutions of the same problem. More precisely, given a problem of $L$ variables and $M$ clauses, we find $T$ solutions $\left\{x_i^r \right\}_{i=1,\ldots ,L}^{r=1,\ldots , T}$ of the given problem. This means that we start $T$ times from a random initial configuration and at each time we perform a $p$RWS until we end up with a solution. We then compute the distance between these $T$ solutions as normalized Hamming distance $$\langle d \rangle = \frac{1}{L\cdot T(T-1)}\sum_{r,s=1}^T
\sum_{i=1}^L \left| x_i^r - x_i^s \right| \;\;\;.
\label{eq:Hamming}$$ The numerical results for $L=20$ are reported in Figure \[fig:dist\]. We average the distance over $T=10^2$ trials and over $10^2$ different problems for each value of $\alpha$. As expected for $p=1/3$ \[squares\] the distance between solutions drops down around $\alpha_s$ (actually it drops down before $\alpha_s$ because of the small number of variables). For different values of $p$ \[ $p=0$ (circles) , $p=1/2$ (diamonds) and $p=1$ (crosses) \], the $p$RWS is less random and $\langle d \rangle$ drops down before $\alpha_s$ (or at least before the point at which the case $p=1/3$ drops down). In particular, if we plot (as in the inset) the distance $\langle d \rangle$ as a function of $p$ and for different values of $\alpha$ \[$\alpha=0.3$ (full line) , $\alpha=0.5$ (dotted line) and $\alpha=0.85$ (dashed line)\] we see a clear peak of the distance $\langle d \rangle$ around $p=1/3$. This suggests that a completely random, unbiased RWS always explores a large region in phase space, it leads to a larger variety of solutions.
![Normalized Hamming distance $\langle d \rangle$ \[ Eq.(\[eq:Hamming\]) \] between solutions as a function of the ratio $\alpha$ and different values of the dynamical parameter $p$ \[ $p=0$ (circles) , $p=1/3$ (squares) , $p=1/2$ (diamonds) and $p=1$ (crosses)\]. For the standard RWS ($p=1/3$) the distance drops down around the critical point $\alpha_s$. Different values of $p$ perform not-really random walks and lead to effective values of $\alpha_s$ smaller than the former one. The inset shows the dependence of $\langle d \rangle$ on the dynamical parameter $p$. As it is shown for different values of $\alpha$ \[$\alpha=0.3$ (full line) , $\alpha=0.5$ (dotted line) and $\alpha=0.85$ (dashed line)\] the peak of the distance between solutions is for a $p$RWS which is really random, that is for $p=1/3$. All the points here , in the main plot as well as in the inset, are obtained for a system of $L=20$ variables. Each point is obtained averaging over $10^2$ different networks and on each of these networks the average distance is calculated over $10^2$ solutions. At the beginning of each simulation the value of one variable is chosen to be $1$ or $0$ with equal probability.[]{data-label="fig:dist"}](dist){width="47.00000%"}
### SAT/UNSAT transition at $\alpha_c$ {#sec:pc}
Differently from the general $k$S problem, the $k$XS problem is known to be always solvable [@semerjian] and the solution corresponds to one of the balanced configurations as described in section \[frozen\] for the all-to-all topology. Nevertheless the challenge is whether the solutions can be found by a local random algorithm like RWS. In the application of the RWS it can happen that the algorithm is not able to find one of these solutions in a “finite” time, so that the problem is called “unsatisfied”. The notion is made more precise in [@cocco]. For practical reasons the way of estimating the critical point $\alpha_c$ that separates the SAT from the UNSAT region is related to the so-called algorithm complexity of the RWS. Here we follow the prescription of [@weigt; @semerjian; @schoning]. Fixed $k=3$ and calling a RWS with initial random assignment of the variables followed by $3L$ update events one trial, one needs a total number of trials $T \gg \left(4/3\right)^L$ for being “numerically” sure to be in the UNSAT region. In fact if after $T$ trials no solution is found, the problem is considered as “unsatisfied” .\
The introduction of the dynamical parameter $p$ can strongly “improve” the performance of RWS. For $p \neq 1/3$ the $p$RWS updates the variables following a well prescribed direction: the tendency is to increases the number of negative variables for $p<1/3$ and to decrease their number for $p > 1/3$. In particular, as we have seen in the former sections, for $p \geq 1/2$ the $p$RWS approaches the configuration of the paradise for the largest value of $\alpha={L \choose k}/L \gg \alpha_c$ and in a time that goes as $\tau \sim L^\beta$, so that there is no UNSAT region at all if we apply the former criterion for the numerical estimate of the UNSAT region. Clearly, if the bias goes in the wrong direction, the performance gets worse.\
In this section we briefly give a qualitative description about the SAT/UNSAT region for the $p$RWS due to the dynamical parameter $p$. Let us define as $^+p_c$ \[$^-p_c$\] the minimum \[maximum\] value of $p$ for which the system can be satisfied. Given a problem with $\alpha L$ clauses we follow the algorithm: 1) Set $p=1$ \[$p=0$\] ; 2) set an initial random configuration and apply the $p$RWS ; 3) if the $p$RWS finds the solution in a number of updates less than $U\cdot L$ , decrease \[increase\] $p$ and go to point 2) ; 4) if not $^+p_c=p$ \[$^-p_c=p$\]. This procedure can be performed up to the desired sensitivity for the numerical estimate of $^+p_c$ \[$^-p_c$\]. The idea of defining an upper $^+p_c$ and lower critical value $^-p_c$ for the dynamical parameter $p$ is related to the fact that for $p=1/3$ the $p$RWS has most trouble to find the solution. Figure \[fig:pcritic\]B and Figure \[fig:pcritic\]C show the numerical results for $^+p_c$ and $^-p_c$ as a function of the dilution parameter $\alpha$. The number of variables is $L=10^3$. We report the results for different values of the waiting time $T=L \cdot U$ \[ $U=1$ (circles) , $U=2$ (squares) , $U=3$ (crosses) , $U=10$ (crosses) \]. Each point is averaged over $10$ different problems and $10$ different $p$RWS applied to each problem. Qualitatively it is seen that for $\alpha \leq \alpha_d$ the problem is always solvable ( $^+p_c=0$ and $^-p_c=1$ ) , while for $\alpha >
\alpha_d$ one needs $p \neq 1/3$ for solving the problem. Of course the numerical values for $^+p_c$ and $^-p_c$ depend on the waiting time until the $p$RWS reaches a solution. Here, for simplicity we do not wait long enough for seeing a similar behavior around $\alpha_c$ instead of $\alpha_d$. Furthermore, in Figure \[fig:pcritic\]A we report the probability $P$, that is the ratio of success over the number of trials, for solving the problem as a function of $p$ for $\alpha=\alpha_c$. The waiting time is $U=1$ (circles) , $U=2$ (squares) , $U=3$ (diamonds) , $U=10$ (crosses) and $U=100$ (triangles), respectively. The probabilities are calculated over $10^2$ trials for each point ($10$ different problems times $10$ $p$RWS for each problem). As the waiting time increases the upper critical value $^+p_c$ for finding for sure the solution decreases ( $^+p_c \simeq 0.8$ for $U=1$ , $^+p_c \simeq 0.7$ for $U=2$ , $^+p_c \simeq 0.6$ for $U=3$ and for $U=10$ , $^+p_c \simeq 0.5$ for $U=100$ ). This means that even for less biased search, solutions can be found, while $^-p_c$ is zero for the waiting time reported here, no value of $p<p_c$ leads to a solution. This is as expected. If the variables are almost all negative it is harder to find a solution of the problem (the paradise is a solution while the hell for $k$ odd is not).
![Numerical estimate of the upper $^+p_c$ (B) and lower $^-p_c$ (C) critical values of $p$ \[see the text for their definition\] as a function of the dilution parameter $\alpha$. Here $L=10^3$ and the different symbols corresponds to different maximum waiting times $T=U\cdot L$ \[ $U=1$ (circles) , $U=2$ (squares) , $U=3$ (diamonds) , $U=10$ (diamonds) \]. Each point is given by the average over $10$ different problems for each value of $\alpha$ and $10$ different $p$RWS for each problem (with random initial condition). Moreover in (A) we show the probability $P$ that the $p$RWS finds a solution at $\alpha
=0.918 \simeq \alpha_c$ as a function of $p$. We used different waiting times \[ $U=1$ (circles) , $U=2$ (squares) , $U=3$ (diamonds) , $U=10$ (crosses) , $U=100$ (triangles) \]. See the text for further comments.[]{data-label="fig:pcritic"}](pcriticnew){width="47.00000%"}
### Mean-field approximation down to $\alpha_m$ {#sec:alpham}
By construction the “topology” of a $k$S problem is completely random (for this reason is sometimes called explicitly as Random $k$-SAT problem). Each of the $L$ variables can appear in one of the $\alpha L$ clauses with probability $v=\frac{1}{L}+\frac{1}{L-1}+\ldots + \frac{1}{L-k}$. In particular for $L \gg k$ one can simply write $v\simeq
\frac{k}{L}$. Then the probability $P_r$ that one variable belongs to $r$ clauses can be described by the Poisson distribution $$P_r = \frac{\left(\alpha k \right)^r}{r!} e^{-\alpha k} \;\;\; ,
\label{eq:probdil}$$ with mean value $\langle r \rangle = \alpha k$ and variance $\sigma_r = \sqrt{\alpha k}$. $P_r$ is plotted in Figure \[fig:probdil\], where the numerical results \[ symbols , $r=0$ (black circles) , $r=1$ (red squares) , $r=2$ (blue diamonds) and $r \geq 3$ (violet crosses) \] are compared to the analytical expectation \[ lines , $r=0$ (black full line) , $r=1$ (red dotted line) , $r=2$ (blue dashed line) and $r \geq 3$ (violet dotted-dashed line) \].
![Probability $p_r$ that one variable belongs to $r$ clauses as function of the dilution parameter $\alpha$. The symbols stand for numerical results obtained over $10^3$ different realizations for $L=128$ variables \[ $r=0$ (black circles) , $r=1$ (red squares) , $r=2$ (blue diamonds) and $r \geq 3$ (violet crosses) \]. The lines stand for analytical predictions of Eq.(\[eq:probdil\]) \[ $r=0$ (black full line) , $r=1$ (red dotted line) , $r=2$ (blue dashed line) and $r \geq 3$ (violet dotted-dashed line) \].[]{data-label="fig:probdil"}](probdil){width="47.00000%"}
If we start from an antagonistic society (all variables false) the minimum value of the dilution $\alpha_m$ needed to reach the paradise (if $p \geq 1/2$) is that all variables belong to at least one clause. This means that $P_0 < 1/L$, from which $$\alpha_m = \frac{\ln{L}}{k} \;\;\; .
\label{eq:meanfield}$$ It is interesting to note that the same criterion applies for any $p$. In Figure \[fig:dendil\] we plot the absolute value of the difference $\left| \; ^{(m)}\rho_\infty - \; ^{(t)}\rho_\infty \; \right| $, between $^{(t)}\rho_\infty$, the theoretical prediction for the stationary density of true variables, \[ Eq.(\[eq:antal2\]) \] and the numerically measured value $^{(m)}\rho_\infty$, as a function of the dilution parameter $\alpha$. $^{(m)}\rho_\infty$ is obtained as the average of the density of friendly links (registered after a waiting time $T=200.0$ , so that is effectively the stationary density) over $50$ different problems and $50$ different $p$RWS for each problem. The results reported here are for $L=128$ (open symbols) and $L=256$ (gray filled symbols) and for different values of $p$ \[ $p=0$ (circles) , $p=1/3$ (squares) , $p=1/2$ (diamonds) , $p=1$ (triangles) \]. The initial conditions are those of an antagonistic society. The dashed lines are proportional to $e^{-3 \alpha}$. Figure \[fig:dendil\] shows that the mean-field approximation of Eq.(\[eq:antal2\]) becomes exponentially fast true as the system dilution decreases. Moreover, as for the cases $p=0$ and $p=1/3$, we can observe that the difference $\left| \; ^{(m)}\rho_\infty - \; ^{(t)}\rho_\infty \; \right| $ is always smaller than for $p=1/2$ and $p=1$. Qualitatively this means that the dilution $\alpha$ of the system needed to reach the theoretical expectation of Eq.(\[eq:antal2\]) is smaller than $\alpha_m$ for $p<1/2$. In general we can say that $\alpha_m$ is a function of $p$: $\alpha_m=\alpha_m(p)$, and $\alpha_m$ is the minimum value of the dilution of the system for which we can effectively describe the diluted system as an all-to-all system for all the values of $p$. Moreover, it should be noticed that for $\alpha > \alpha_m$ almost all variables belong to at least three clauses \[ see Figure \[fig:probdil\] \]. This fact allows the $p$RWS to explore a larger part of configuration space. Let us assume that one variable belongs to less than three clauses: an eventual update event that flips this variable so that the one triad becomes balanced, can never increase the number of unsatisfied clauses by frustrating other clauses it belongs to. This reminds us to the situation in an energy landscape in which an algorithm gets stuck in a local minimum when it never accepts a change in the “wrong” direction, i.e. towards larger energy.
![Difference $\left| \; ^{(m)}\rho_\infty - \;
^{(t)}\rho_\infty \; \right| $ between $^{(t)}\rho_\infty$ the theoretical prediction for the stationary density of friendly variables \[ Eq.(\[eq:antal2\]) \] and the numerically measured value $^{(m)}\rho_\infty$, as a function of the dilution parameter $\alpha$. $^{(m)}\rho_\infty$ is obtained as the average of the density of friendly links (registered after a waiting time $T=200.0$ , so that is effectively stationary ) over $50$ different problems and $50$ different $p$RWS for each problem. The results displayed here are obtained for $L=128$ (open symbols) and $L=256$ (gray filled symbols) and for different values of $p$ \[ $p=0$ (circles) , $p=1/3$ (squares) , $p=1/2$ (diamonds) , $p=1$ (triangles) \]. The initial conditions are those of an antagonistic society. The dashed lines are proportional to $e^{-3 \alpha}$.[]{data-label="fig:dendil"}](dendil){width="47.00000%"}
Summary and conclusions {#summary}
=======================
In the first part of this paper we generalized the triad dynamics of Antal *et al.* to a $k$-cycle dynamics [@antal]. Here we had to distinguish the cases of even values of $k$ and odd values of $k$. For all values of integer $k$ there is again a critical threshold at $p_c=1/2$ in the propensity parameter. For odd $k$ and $p<p_c$ the paradise can never be reached in the thermodynamic limit of infinite system size (as predicted by the mean field equations which we solved exactly for $k=5$ and approximately for $k>5$). In the finite volume, in principle one could reach a balanced state made out of two cliques (a special case of this configuration is the “paradise” when one clique is empty). However, the probability for reaching such type of frozen state decreases exponentially with the system size so that in practice the fluctuations never die out in the numerical simulations. For $p>1/2$ the convergence time to reach the paradise grows logarithmically with the system size. At $p=1/2$ paradise is reached within a time that follows a power law in the size $N$, where we determined the $k$-dependence of the exponent. In particular, the densities of $k$-cycles with $j$ negative links, here evolved according to the rules of the $k$-cycle dynamics, could be equally well obtained from a random dynamics in which each link is set equal to $1$ with probability $\rho_\infty$ or equal to $-1$ with probability $1-\rho_\infty$. This feature was already observed by Antal *et al.* for $k=3$ [@antal]. It means that the individual updating rules which seem to be “socially” motivated in locally reducing the social tensions by changing links to friendly ones, end up with random distributions of friendly links. The reason is a missing constraint of the type that the overall number of frustrated $k$-cycles should not increase in an update event. Such a constrained dynamics was studied by Antal *et al.* in [@antal], but not in this paper.\
For even values of $k$, the only stable solutions are “heaven” (i.e. paradise) and “hell” for $p>1/2$ and $p<1/2$, respectively, and the time to reach these frozen configurations grows logarithmically with $N$. At $p_c=1/2$ other realizations of the frozen configurations are possible, in principle. However, they have negligible probability as compared to heaven and hell. Here the time to reach these configurations increases quadratically in $N$, independently of $k$. This result was obtained in two ways. Either from the criterion to reach the stable state when a large enough fluctuation drops the system into this state (so we had to calculate how long one has to wait for such a big fluctuation). Alternatively, the result could be read off from a mapping to a Markov process for diploid organisms ending up in a genetic pool of either all “$+$”-genes or all “$-$”-genes. The difference in the possible stable states of diploid organisms and ours consists in two-clique stable solutions that are admissible for the even $k$-cycle dynamics, in principle, however such clique states have such a low probability of being realized that the difference is irrelevant.\
The difference in the exponent at $p_c$ and the stable configurations above and below $p_c$ between the even and odd $k$-cycle dynamics was due to the fact that “hell”, a state with all links negative as in an antagonistic society, is a balanced state for even $k$, not only by the frustration criterion of physicists, but also according to the criterion of social scientists [@cartwright].\
\
As a second natural generalization of the social balance dynamics of Antal *et al.* we considered a diluted network. One way of implementing the dilution is via a random Erdös-Rényi network, characterized by the probability $w$ for connecting a randomly chosen pair of nodes. Here we focused our studies to the case $k=3$. The mean-field description and the results about the phase structure remain valid down to a certain degree of dilution, characterized by $w_m$. This threshold for the validity of the mean-field description practically coincides with the criterion whether a single link belongs to at least three triads (for $w>w_m$) or not ($w<w_m$). If it does so, an update event can increase the number of frustrated triads. For $w<w_m$, or more precisely $w<w_d<w_m$ it becomes easier to realize frozen configurations different from the paradise. Isolated links do not get updated at all and isolated triads can freeze to a “$+$”-“$-$”-configuration. The time to reach such a frozen configuration (in general different from the paradise) grows then only linearly in the system size. Also the solution space, characterized by the average Hamming distance between solutions, has different features below and above another threshold, called $w_s$ with $w_d<w_s<w_m$. Therefore one of the main differences between the all-to-all and the sufficiently diluted topology are the frozen configurations. For the all-to-all case we observed the paradise above $p_c$ for odd values of $k$ and even values of $k$ and the hell for even values of $k$ below $p_c$, in the numerical simulations, because the probability to find a two-clique-frozen configuration was calculated to be negligibly small. For larger dilution, also other balanced configurations were numerically found, as mentioned above, and the time passed in the numerical simulations for finding these solutions followed the theoretical predictions.\
In section \[sec:diluted\] we used, however, another parameterization in terms of the dilution parameter $\alpha$, that was the ratio of triads (clauses) over the number of links \[we gave anyway an approximated relation between $\alpha$ and $w$ in Eq.(\[eq:average\_ratio\])\]. The reason for using this parameterization was a mapping of the $k$-cycle social balance of networks to a $k$-XOR-SAT ($k$XS) problem, that is a typical satisfiability problem in optimization tasks. We also traced a mapping between the “social” dynamical rules and the Random-Walk SAT (RWS) algorithm, that is one approach for solving this problem in a random local way. As we have shown, the diluted version of the $3$-cycle social dynamics with propensity parameter $p=1/3$ corresponds to a $3$XS problem solved by the RWS algorithm in its standard form (as used in [@weigt; @semerjian]).\
The $k$XS problem is always solvable like the $k$-cycle social balance, for which a two cliques solution always exists due to the structure theorem of [@cartwright], containing as a special solution the so-called paradise. The common challenge, however, is to find this solution by a local stochastic algorithm. The driving force, shared by both sets of problems, is the reduction of frustration. The meaning of frustration depends on the context: for the $k$-cycle dynamics it is meant in a social sense as a reduction of social tension, for the $k$XS problem it corresponds to violated clauses. The mathematical criterion is the same. The local stochastic algorithm works in a certain parameter range, but outside this range it fails. The paradise is never reached for a propensity parameter $p<1/2$, independently of $k$. Similarly, the solution of the $k$XS problem is never found if the dilution parameter is larger than $\alpha_c$, and the RWS algorithm needs an exponentially long time already for $\alpha>\alpha_d$, with $\alpha_d<\alpha_c$.\
We generalized the RWS algorithm, usually chosen for solving the $k$-SAT ($k$S) problem as well as the $k$XS problem, to include a parameter $p$ that played formerly the role of the propensity parameter in the social dynamics ($p$RWS). The effect of this parameter is a bias towards the solution so that $\alpha_d$, the threshold between a linear and an exponential time for solving the problem, becomes a function of $p$. Problems for which the $p$RWS algorithm needed exponentially long for $p=1/3$, now become solvable within a time that grows less than logarithmically in the system size for $p>1/2$ and less than power-like in the system size for $p=1/2$. Along with the bias goes an exploration of solution space that has on average a smaller Hamming distance between different solutions than in the case of the $\frac{1}{3}$RWS algorithm that was formerly considered [@weigt; @semerjian].\
\
Our paper has illustrated that the reduction of frustration may be the driving force in common to a number of dynamical systems. So far we were concerned about “artificial” systems like social systems and satisfiability problems. It would be interesting to search for natural networks whose evolution was determined by the goal of reducing the frustration, not necessarily to zero degree, but to a low degree at least.
It is a pleasure to thank Martin Weigt for drawing our attention to Random $k$-SAT problems in computer science and for having useful discussions with us while he was visiting the International University Bremen as an ICTS-fellow.
[100]{}
T. Antal, P. L. Krapivsky, and S. Redner , Phys. Rev. E [**72**]{} , 036121 (2005).
M. Sasai, and P.G. Wolynes , Proc. Natl. Acad. Sci. USA [**100**]{} , 2374-2379 (2003).
M. Mézard, G. Parisi, and M.A. Virasoro , *Spin Glass Theory and Beyond* , (World Scientific , Singapore , 1987).
M.R. Garey, and D.S. Johnson , *Computer and Intractability: A Guide to the Theory of NP-Completeness* (Freeman , San Francisco , 1979).
W. Barthel , A.K. Hartmann , and M. Weigt , Phys. Rev. E [**67**]{} , 066104 (2003).
G. Semerjian, and R. Monasson , Phys. Rev. E [**67**]{}, 066103 (2003).
F. Ricci-Tersenghi, M. Weigt, and R. Zecchina , Phys. Rev. E [**63**]{} , 026702 (2001) ; M. Mézard, F. Ricci-Tersenghi, and R. Zecchina , J. Stat. Phys. [**111**]{} , 505-533 (2003).
S. Cook , in *Proceedings of the $3$rd Annual ACM Symposium on Theory of Computing* , p. 151 (Association for Computing Machinery , New York , 1971) ; J.M. Crawford, and L.D. Auton, in *Proc. $11$th Natl. Conf. on Artif. Intell. (AAAI-93)* , p. 21 , (AAAI Press , Menlo Park , California , 1993) ; B. Selman, and S. Kirkpatrick , Science [**264**]{} , 1297 (1994).
R. Monasson, and R. Zecchina , Phys. Rev. Lett. [**76**]{} , 3881 (1996) ; R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and L. Troyansky, Nature (London) [**400**]{} , 133 (1999) ; G. Biroli, R. Monasson, and M. Weigt , Eur. Phys. J. B [**14**]{} , 551-568 (2000) ; M. Mézard, G. Parisi, and R. Zecchina , Science [**297**]{} , 812-815 (2002).
S. Cocco, O. Dubois, J. Mandler, and R. Monasson , Phys. Rev. Lett. [**90**]{} , 047205 (2003).
D. Cartwright , and F. Harary , Psychol. Rev. [**63**]{} , 277-293 (1956) ; F. Harary, R.Z. Norman, and D. Cartwright , *Structural Models: An Introduction to the Theory of Directed Graphs* (John Wiley & Sons , New York , 1965).
P. Erdös, and A. Rényi , Publications of Mathematical Institute of the Hungarian Academy of Sciences [**5**]{} , 17-61 (1960) ; P. Erdös, and A. Rényi , Acta Mathematica Scientia Hungary [**12**]{} , 261-267 (1961).
F. Harary , Mich. Math. J. [**2**]{} , 143-146 , (1953-54) ; F.S. Roberts , Electronic Notes in Discrete Mathematics (ENDM) [**2**]{} , (1999) , <http://www.elsevier.nl/locate/endm> ; N.P. Hummon, and P. Doreian , Social Network [**25**]{} , 17-49 (2003).
S. Wright , Genetics [**16**]{} , 97-159 , (1931) ; R.A. Fisher , *The genetical theory of natural selection* (Clarendon Press, Oxford , 1930).
F. Heider , Psychol. Rev. [**51**]{} , 358-374 (1944) ; F. Heider , J. Psychology [**21**]{} , 107-112 (1946) ; F. Heider , *The Psychology of Interpersonal Relations* (John Wiley & Sons , New York , 1958) ; S. Wasserman, and K. Faust , *Social Network Analysis: Methods and Applications* (Cambridge University Press , New York , 1994).
N. G. Van Kampen , *Stochastic Processes in Physics and Chemistry* (North Holland, Amsterdam, 2005).
U. Scöning , Algorithmica [**32**]{} , 615-623 (2002).
|
---
abstract: |
We study the Brownian motion of particles trapped by optical tweezers inside a colloidal glass (Laponite) during the sol-gel transition. We use two methods based on passive rheology to extract the effective temperature from the fluctuations of the Brownian particles. All of them give a temperature that, within experimental errors, is equal to the heat bath temperature. Several interesting features concerning the statistical properties and the long time correlations of the particles are observed during the transition.
[effective temperature, out of equilibrium systems, colloids, glasses, optical trap, Brownian motion, passive and active rheology]{}
author:
- |
Pierre Jop , Artyom Petrosyan and Sergio Ciliberto$^{\ast}
$[^1]\
\
title: '[*Effective Temperature in a Colloidal Glass*]{}'
---
Introduction
============
The glasses and colloids are interesting examples of out of equilibrium systems where the relaxation towards the equilibrium may last much more than a reasonable observation time. One of the problem which has been widely theoretically studied is the definition of an effective temperature $T_{eff}$ in these systems using the fluctuation dissipation relation (FDR). This relation is an extension of the Fluctuation Dissipation Theorem for an out of equilibrium system and $T_{eff}$ is defined as the ratio between the correlation and the response function[@Cugliandolo]. Using numerical simulations it has been shown that in several models for out of equilibrium systems the effective temperature defined via FDR is higher than the temperature of the thermal bath and it is a good definition of temperature in the thermodynamic sense. However the experimental results are more confused. It has been observed that $T_{eff}$ in polymer, spin glasses and colloids may depend on the experimental conditions. For example in the dielectric measurements in polymer $T_{eff}$ depends on the quenching rate and it may be huge because of the presence of intermittent bursts. The same behavior is observed on mechanical variables. Instead in a colloid (Laponite) during the sol-gel transition the electric observables give a $T_{eff}$ which is quite large whereas measurements done on thermal rheometer indicate that within experimental errors there is no violation of the Fluctuation theorem because the $T_{eff}$ is always equal to the temperature of the thermal bath.(A discussion on the $T_{eff}$ obtained from dielectric measurements and mechanical measurements can be found in ref.[@Bellon02]). The rheological measurement described in ref.[@Bellon02] was a global measurement and one may wonder whether an experiment of microrheology give or not the same results. This experiment can be performed using as a probe a Brownian particle using active and passive microrheology. These kind of experiments are interesting also from another point of view because one may studies whether the properties of the Brownian motion are affected by the fact that the surrounding fluid (the thermal bath) is out of equilibrium. Several experiments of Brownian motion inside a Laponite solution have been done by different groups using various techniques. The results are rather contradictory. Let us resume them. Abou et al. find an increase and then a decrease of $T_{eff}$ as a function of time [@Abou06]. Bartelett et al. find an increase of $T_{eff}$ [@Greinert07]. In contrast Jabbari-Farouji et al. do not observe any change and confirm the results on the thermal rheometer of ref [@Bellon02]. The purpose of this article is to describe the results of the measurement of the Brownian motion of a particle inside a Laponite solution using a combination of different techniques proposed in previous references. All the techniques do not show, within experimental errors, any increase of $T_{eff}$ which remains equal to that of the thermal bath for all the duration of the sol-gel transition. Thus the result of this papers agrees with those of Jabbari-Farouji et al. [@Jabbari-Farouji07] and of Bellon et al. [@Bellon02]. The article is organized as follow. In section 2 we describe the experimental apparatus and the various techniques used to measure response and fluctuations. In section 3 we describe the experimental results of the various experimental techniques. In section 4 we discuss the result and we conclude.
Experimental set up
===================
We measure the fluctuations of the position of one or several silica beads trapped by an optical tweezers during the aging of the Laponite. The laser beam ($\lambda$=980 nm) is focused by a microscope objective ($\times$63) 20 $\mu$m above the cover-slip surface to create a harmonic potential well where a bead of 1 or 2 $\mu$m in diameter ($2r$) is trapped. We can trap several beads if the laser is rapidly swept from a position to another to form a multi trap system. The Laponite mass concentration is varied from 1.2 to 3% wt for different ionic strengths. These conditions allow us to obtain either a gel or a glass according to the phase diagramm found in the literature [@bonn_PRE_2004]. The Laponite is filtered with a 0.45 $\mu$m filter to avoid the formation of aggregates. The aging time $t_w$ is measured since the end of this filtering process. Particular attention has been paid in the cell construction indeed the properties of the Laponite are very sensitive to the experimental conditions. First the Laponite solution is prepared under nitrogen atmosphere and second the cell is completely sealed using Gene Frame adhesive spacers in order to avoid evaporation and the use of vacuum grease, used in other experiments. Indeed this grease, whose PH is much smaller than 10, quickens the evolution of the suspension. In order to measure $T_{eff}$ for the particle motion two techniques have been used. The first one is based on the laser modulation technique as proposed in ref.[@Greinert07]. The second is based on the Kramers-Kronig relations with two laser intensities and it combines the advantages of two methods one proposed in ref.[@Jabbari-Farouji07] and in ref.[@Greinert07].
Laser modulation technique
--------------------------
Following the method used in [@Greinert07], the stiffness of the optical trap is periodically switched between two different values ($k_1=6.34$ pN/$\mu$m and $k_2=14.4$ pN/$\mu$m) every 61 seconds by changing the laser intensity. The position of the bead is recorded by a quadrant photodiode at the rate of 8192 Hz, then we compute the variance of the position over the whole signal, $\left<\delta x^2\right>=\left< x^2-\left<x\right>^2\right>$, where the brackets stand for average over the time. To avoid transients, each record is started 20 seconds after the laser switch to be sure that the system has relaxed toward a quasi stationary state. Assuming the equipartition principle still holds in this out-of-equilibrium system, the $T_{eff}$ is computed as in [@Greinert07] the expression of the effective temperature and of the Laponite elastic stiffness $K_{Lap}$ is the following: $$k_BT_{eff}={
(k_2 - k_1)\left<\delta x_1^2\right>\left<\delta x_2^2\right>
\over \left<\delta x_1^2\right>-\left<\delta x_2^2\right> }
\label{eq:temperature}$$ $$K_{Lap}=
{k_1\ \left<\delta x_1^2\right>- k_2\left<\delta x_2^2\right>
\over
\left<\delta x_1^2\right>-\left<\delta x_2^2\right>}
\label{eq:KLap}$$
This technique, although quite interesting and simple, has the important drawback that, being a global measurement, it has no control of what is going on the different frequencies. To overcome this problem we have used the following method.
Kramers-Kronig and modulation technique
---------------------------------------
This combines the laser modulation technique described in the previous section and the passive rheology technique based on Kramers-Kronig relations. The fluctuation dissipation relations relate the spectrum $S_i(\omega)$ of the fluctuation of the particle position to the imaginary part $\alpha_i''$ of the response of the particle to an external force, specifically :
$$S_i(\omega,t_w)= {4 \ k_B T_{eff} \over \omega}
\alpha_i''(\omega,t_w) \label{eq:FDR}$$
with $i=1$ and $i=2$ for the spectra measured with the trap stiffness $k_1$ and $k_2$ respectively. We recall that for a particle inside a newtonian fluid $G=1/\alpha_i= k_i+ i\nu
\omega$ with $\nu=6\pi\eta r$ and $\eta$ the fluid viscosity. In Eq. \[eq:FDR\] the dependence in $t_w$ takes into account the fact that the properties of the fluid changes after the preparation of the Laponite. If one assumes that $T_{eff}$ is constant as a function of frequency (hypothesis that can be easily checked a posteriori) then the real part $\alpha_i'$ of the response is related to $\alpha_i''$ by the Kramers-Kronig relations [@KK_R] that is: $$\alpha_i'(\omega,t_w)={ 2\over \pi } P\int_0^\infty {\xi
\alpha''(\xi,t_w) \over \xi^2-\omega^2} d\xi ={ 1\over 2 \pi k_B
T } P\int_0^\infty {\xi^2 S_i(\xi,t_w) \over \xi^2-\omega^2} d\xi
\label{eq:KrKr}$$ where $P$ stands for principal part of the integral and assumed that $T_{eff}=T$ to write the second equality using Eq. \[eq:FDR\]. However it can be easily (details will be given in a longer report) shown that using the two measurements at $k_1$ and $k_2$ we get: $$T_{eff}(\omega,t_w)= T_{bath} \left({k_1- k_2 \over
G'_1(\omega,t_w)- G'_2(\omega,t_w) }\right)$$ and $G_i'(\omega,t_w)=k_i+K_{Lap}(\omega,t_w)$. It is clear that if one finds a dependence of $T_{eff}$ on $\omega$ this method cannot be used because $\alpha''(\omega,t_w)$ is not simply proportional to $S_i(\omega,t_w) \omega$ as assumed in Eq. \[eq:KrKr\].
Experimental results
====================
Laser modulation method
-----------------------
Fig. \[fig:spectra\]a) shows the power spectra of the particle fluctuations inside Laponite at concentration measured at four different $t_w$ with the trap stiffness $k_1=6.34$ pN/$\mu$m. We see that as time goes on the low frequency component of the spectrum increases. That is the frequency cut-off $(k_i+K_{Lap})/\nu$ decreases mainly because of the increasing of the viscosity. At very long time this cut-off is well below $0.1Hz$. In Fig. \[fig:spectra\]b) we plot the variance of the particle measured for the same data of Fig. \[fig:spectra\]a) on time windows of length $\tau=61$ s. The variance remains constant for a very long time and they begin to decrease because of the increase o the gel stiffness. Using these data and Eq. \[eq:temperature\] and Eq. \[eq:KLap\] one can compute $T_{eff}$ and $K_{Lap}$. The results for $T_{eff}$ and $K_{Lap}$ are shown in Fig. \[fig:Teff\]a) and \[fig:Teff\]b) respectively.
We find that $T_{eff}$ is constant at the beginning and is very close to $T_{bath}=294$ K, then when the jamming occurs, that is when $K_{Lap}$ increases, it becomes more scattered without any clear increase with $t_w$, contrary to Ref. [@Greinert07]. We now make several remarks. First, we point out that the uncertainty of their results are underestimated. The error bars in Fig. \[fig:Teff\]a) are here evaluated from the standard deviation of the variance using Eq. 2 in Ref [@Greinert07] at the time $t_w$. Although they are small for short time $t_w$, ($\Delta T_{eff}/T_{eff}\leq 10\%$), they increase for large $t_w$. This is a consequence of the increase of variabilities of $\left<\delta x_i^2\right>$ as the colloidal glass forms. This point is not discussed in in Ref. [@Greinert07] and we think that the measurement errors are of the same order or larger than the observed effect. The results depend on the length of the analyzing time window and the use of the principle of energy equipartition becomes questionable for the following reasons.
First, these analyzing windows cannot be made too large because the viscoelastic properties of Laponite evolve as a function of time. Second, the corner frequency of the global trap (optical trap and gel), the ratio of the trap stiffness to viscosity, decreases continuously mainly because of the increase of viscosity. At the end of the experiment, the power spectrum density of the displacement of the bead shows that the corner frequency is lower than $0.1$ Hz. We thus observe long lived fluctuations, which could not be taken into account with short measuring times. This problem is shown on Fig. \[fig:Teff\]b). We split our data into equal time duration $\Delta \tau$, compute the variance and average the results of all samples. The dotted line represents the duration 3.3 s chosen in [@Greinert07]. At the beginning of the experiment, the variance of the displacement is constant for any reasonable durations of measurement. However, we clearly see that this method produces an underestimate of $\left<\delta x^2\right>$ for long aging times, specially when the viscoelasticity of the gel becomes important. Long lived fluctuations are then ignored.
Passive rheology
----------------
This new method, using Kramers-Kronig relation, allows us to test the dependence of the effective temperature on the frequency. The figure \[fig:krkr\]a) shows the real part of $G$, which corresponds to the global elastic modulus of the gel and of the laser, for both trap stiffness. This numerical method is very sensitive to the spectrum. Thus, before computing the elastic modulus, we average the spectrum to obtain smooth curves. The uncertainties would then misplace the curve rather than produce a noisy curve. The increase with time is consistent with the increase of the strength of the gel. The elastic behavior of the Laponite is also more pronounced at high frequency. The last decrease of the curves at very high frequencies is due to the numerical method. Indeed, the frequency cutoff should be set at least a decade below the frequency of the data acquisition: data above 200 Hz is not reliable. We see that the curves of each stiffness are well separated except at the end of the measurement, where the results are not accurate due to the large difference between the optical stiffness and the Laponite stiffness.
From these data, we compute the ratio of the effective temperature to the bath temperature along the ageing process. These results are shown on Fig. \[fig:krkr\]b) for three different frequencies ($f=1$ Hz, 10 Hz and 100 Hz). We first note that the three curves are almost identical. This means that the effective temperature does not depends on the frequency. Second, the temperature ratio is close to 1. The dispersion of the data is rather small at early times and again increases when the stiffness of the gel overcomes those of the optical trap. This comes from the uncertainties on the elastic modulus which become larger than the difference between the two curves. This dispersion may also give, for very long time, negative temperatures, not shown in Fig. \[fig:krkr\]b) which is an expanded view. Even if this method is here less accurate than the previous one, it allows us to verify that the effective temperature is the same for all frequencies.
Conclusion
==========
Finally, a remark has to be made concerning the mean position of the bead during aging. When the stiffness of the gel becomes comparable to the optical one, the bead starts to move away from the centre the optical trap. We observe a drift of the bead position at long time, which could lead to the escape of the bead. Moreover we have performed simultaneous measurements with a multiple trap using a fast camera showing that at very long $t_w$ the mean trajectories of beads separated by 7 $\mu$m are almost identical. This proves that one must pay attention when interpreting such measurements, specially on the duration of measurements. We also have seen that the way the sample is sealed can accelerate the formation of the gel and the drift of the bead by changing the chemical properties in the small sample. We have used different types of cell, Laponite concentrations, bead sizes, stiffness of the optical trap. In each case we do not find any increase of the effective temperature.
In conclusion, our results show no increase of $T_{eff}$ in Laponite and are in agreement with those of Jabbari-Farouji [@Jabbari-Farouji07], who measured fluctuations and responses of the bead displacement in Laponite over a wide range of frequency and found that $T_{eff}$ is equal to the bath temperature.
Acknowledgement
===============
This work has been partially supported by ANR-05-BLAN-0105-01.
[11]{}
L. Cugliandolo, J. Kurchan and L. Peliti, Phys. Rev. E, 55(4), (1997), pp. 3898 - 3914.
, [ Physica D]{}, [168]{} (2002), 325.
, [ Phys. Rev. Lett.]{} [93]{} (16), (2006), 160603. , [Phys. Rev. Lett.]{} [97]{} (2006), 265702.
[Phys. Rev. Lett.]{} [98]{} (2007), 108302. H. Tanaka, J. Meunier, D.Bonn Phys. Rev 69 (2004) 031404.
B. Schnurr, F. Gittes, F.C. MacKintosh, C.F. Schmidt, Macromolecules 30 (1997), 7781 .
[^1]: $^\ast$Corresponding author. Email:sergio.ciliberto@ens-lyon.fr
|
---
abstract: 'In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly $c$-convex potentials arising in optimal transportation belong to $W^{2,1+\kappa}_{\rm loc}$ for some $\kappa>0$. This generalizes some recents results [@DepFi; @DFS; @S] concerning the regularity of strictly convex Alexandrov solutions of the Monge-Ampère equation with right hand side bounded away from zero and infinity.'
address:
- 'Scuola Normale Superiore, p.za dei Cavalieri 7, I-56126 Pisa, Italy'
- 'The University of Texas at Austin, Mathematics Dept. RLM 8.100, 2515 Speedway Stop C1200, Austin, Texas 78712-1202, USA'
author:
- Guido De Philippis
- Alessio Figalli
title: 'Sobolev regularity for Monge-Ampère type equations'
---
Introduction
============
Let $\Omega\subset \R^n$ be a bounded open set. We want to investigate the regularity of solutions to Monge-Ampère type equations of the form $$\label{eq:MA1}
\det\big(D^2u-\A(x,Du)\big)=f \qquad \text{in $\Omega$},$$ where $f \geq 0$ and $\A(x,p)$ is a $n\times n$ symmetric matrix.
This class of equations naturally arises in optimal transportation, and in reflector and refractor shape design problems. In these applications, the matrix $\A$ and the right and side $f$ are given by $$\A(x,D u(x))=-D_{xx} c(x,T_u(x)), \qquad f(x)=\left|\det\bigl(D_{xy}c(x,T_u(x))\bigr)\right| \frac{\rho_0(x)}{\rho_1(T_u(x))},$$ where $c(x,y)$ represents the cost function, $\rho_0$ and $\rho_1$ are probability densities, and $T_u$ is the optimal transport map sending $\rho_0$ onto $\rho_1$. Under a twist assumption on the cost (see [**(C2)**]{} below), the map $T_u$ is uniquely determined through the relation $$-D_xc(x,T_u(x))=Du(x).$$ Moreover, when $\A\equiv 0$ the above equation reduces to the classical Monge-Ampère equation.
The regularity for the above class of equations has received a lot of attention in the last years [@FKM; @FL; @Liu; @LTWC2a; @Loe; @MTW; @TW1; @TW2]. In particular, under some necessary structural conditions on $\A$ (see [**(C1)**]{} below), one can show that if $f$ is smooth then $u$ is smooth as well [@LTWC2a; @MTW; @TW1; @TW2]. In addition, it is proved in [@FKM] that solutions are locally $C^{1,\alpha}$ when $f$ is merely bounded away from zero and infinity (see also [@FL; @Liu]).\
Recently, the authors introduced new techniques to address the Sobolev regularity of $u$ when $\A\equiv 0$: more precisely, under the assumption that $f$ is bounded away from zero and infinity, it is proved in [@DepFi] that $D^2 u \in L\log L_{\rm loc}(\Omega)$, and with a variant of the same techniques this result has been improved in [@DFS] to $u \in W^{2,1+\kappa}_{\rm loc}(\Omega)$ for some $\kappa>0$ (see also [@S]). Let us mention that these results played a crucial role in [@ACDF1; @ACDF2] to show the existence of distributional solutions to the semi-geostrophic system.
The aim of this paper is to extend the $W^{2,1+\kappa}_{\rm loc}$ regularity to the general class of Monge-Ampère equations in . Apart from its own interest, it seems likely that this result could have applications in the study of generalized semi-geostrophic system on Riemannian manifolds [@CDRS], in particular on the sphere [@Loe2] and its perturbations [@DG1; @DG2; @FR; @FRVsn].
In order to describe our result, we need to introduce some more notation and the main assumptions on the cost functions.\
Let $X\subset \R^n$ be an open set, and $u:X \to \R$ be a $c$-convex function, i.e., $u$ can be written as $$\label{eq:cconv}
u(x)=\max_{y \in \overline Y} \{-c(x,y) +\lambda_y\}$$ for some open set $Y\subset \R^n$, and $\lambda_y \in \R$ for all $y \in \overline Y$. We are going to assume that $u$ is an Alexandrov solution of inside some open set $\Omega \subset X$, i.e., $$\left|{\partial}^cu(E) \right|= \int_E f\qquad \text{for all $E\subset \Omega$ Borel},$$ where $${\partial}^cu(E) :=\bigcup_{x \in E}{\partial}^cu(x), \qquad
{\partial}^cu(x):=\{y \in \overline Y \,:\,u(x)=-c(x,y)+\lambda_y\},$$ and $|F|$ denotes the Lebesgue measure of a set $F$. It is well-known that, in order to prove some regularity results, needs to be coupled with some boundary conditions: for instance, when $\A\equiv 0$ and $f\equiv 1$, solutions are smooth whenever they are strictly convex, and to obtain strict convexity some suitable boundary conditions are needed [@CA1; @CA3].
For the general case in , let $u$ be a $c$-convex function associated to an optimal transport problem, and for any $y \in \overline Y$ define the contact set $$\Lambda_y:=\{x \in X \,:\,u(x)=-c(x,y)+\lambda_y\}.$$ Under some structural assumptions on the cost functions (which we shall describe below) and some convexity hypotheses on the supports of the source and target measure, it has been proved in [@FKM] that $u$ is an Alexandrov solution of inside $X$, and it is strictly $c$-convex (i.e., for any $y \in {\partial}^cu(X)$ the contact set $\Lambda_y$ reduces to one point) provided $f$ is bounded away from zero and infinity.
Here, since we want to investigate the interior regularity of $u$, instead of assuming that $u$ comes from an optimal transportation problem where the supports of the source and target measure enjoy some global “$c$-convexity” property, we work assuming directly that $u$ is a strictly $c$-convex Alexandrov solution near some point $\bar x \in X$, and we prove regularity of $u$ in a neighborhood of $\bar x$. This has the advantage of making our result more general and flexible for possible future applications.
Hence, we assume that there exist $(\bar x, \bar y)\in X\times Y$ such that $\Lambda_{\bar y}=\{\bar x\}$, we consider a neighborhood $\Omega$ of $\bar x$ given by $$\label{eq:omega}
\Omega:=\{x \in X\,:\,u(z)<-c(x,\bar y)+\lambda_y+\delta\},$$ where $\delta>0$ is a small constant chosen so that $\Omega \subset \subset X$ and $\partial^c u(\Omega)\subset \subset Y$ (such a constant $\delta$ exists because $\Lambda_{\bar y}:=\{\bar x\}$). Also, we assume that $u$ is an Alexandrov solution of $$\label{eq:MA}
\left\{
\begin{array}{ll}
\det\big(D^2u-\A(x,Du)\big)=f & \text{in $\Omega$},\\
u=-c(\cdot,\bar y)+{\rm const} & \text{on $\partial \Omega$}.
\end{array}
\right.$$
Before stating our result, let us introduce the main conditions on the cost function: let $\Omega$ be as above, and let $\U\subset\subset Y$ be a open neighborhood of $\partial^c u(\Omega)$. We define $$\label{cnorm}
\normm c\normm :=\|c\|_{C^3(\overline \Omega \times \overline \U)}+\|D_{xxyy}c\|_{L^\infty(\overline \Omega \times \overline \U)}+\left\|\log |\det D_{xy}c|\right\|_{L^\infty(\overline \Omega \times \overline \U)},$$ and assume that the following hold:
1. $\normm c\normm <\infty$.
2. For every $x\in \Omega$ and $p:=-D_xc(x,y)$ with $y \in \U$, it holds $$\label{MTWc}
D_{p_kp_\ell}\A_{ij}(x,p)\xi_i\xi_j\eta_k \eta_\ell\ge 0,\qquad \forall \, \xi,\eta \in \R^n,\, \xi \cdot \eta=0,$$ where $\A$ is defined through $c$ by $\A_{ij}(x,p):=-D_{x_ix_j}c(x,y)$, and we use the summation convention over repeated indices.
Let us point out that, up to reduce the size of $\Omega$ and $\U$ (this is possible because $\Omega \to \{\bar x\}$ and $\partial^cu(\Omega) \to \{\bar y\}$ as $\delta\to 0$), as a consequence of [**(C0)**]{} (more precisely, from the fact that $\det D_{xy}c(\bar x,\bar y)\neq 0$ and by the implicit function theorem) we can assume that the following holds:
1. For every $(x,y)\in \Omega\times \U$, the maps $x\in \Omega \mapsto - D_yc(x,y)$ and $y\in \U \mapsto - D_xc(x,y)$ are diffeomorphisms on their respective ranges.
We also notice that, because of the boundary condition $u=-c(\cdot,\bar y)+{\rm const}$ on $\partial \Omega$, if $f$ is bounded away from zero and infinity inside $\Omega$, then any $c$-convex Alexandrox solution of is strictly $c$-convex inside $\Omega$ (this is an immediate consequence of [@FKM Remark 7.2]). Here is our result:
\[w21eps\]Let $u:\Omega \to \R$ be a $c$-convex Alexandrov solution of . Assume that $c$ satisfies conditions [**(C0)**]{}-[**(C2)**]{}, and that $0<\lambda\le f\le 1/\lambda$. Then $u\in W_{\rm loc}^{2,1+\kappa}(\Omega)$ for some $\kappa>0$.
Theorem \[w21eps\] generalizes the corresponding result for the classical Monge-Ampère equation to the wider class of equations considered here. With respect to the arguments in [@DepFi; @DFS], additional complications arise from the fact that, in contrast with the classical Monge-Ampère equation, in general is not affinely invariant.\
*Acknowledgements:* AF is partially supported by NSF Grant DMS-0969962. Both authors acknowledge the support of the ERC ADG Grant GeMeThNES. The first author thanks the hospitality of the Mathematics Department at the University of Texas at Austin, where part of this work has been done.
Notation and preliminary results
================================
Through all the paper, we call *universal* any constant which depends only on the data, i.e., on $n$, $\Omega$, $\U$, $\lambda$, and $\normm
c\normm$. We use $C$ to denote a universal constant larger than $1$ whose value may change from line to line, and we use the notation $a\approx b$ to indicate that the ratio $a/b$ is bounded from above and below by positive universal constants.\
An immediate consequence of the definition of $c$-convexity is that, for any $x_0\in X$, there exists $y_0\in \overline Y$ such that $$u(x)\ge - c(x,y_0)+u(x_0)+c(x_0,y_0) \quad \forall \,x\in X,$$ and in this case $y_0 \in {\partial}^cu(x_0)$. If in addition $u \in C^2$, then it is easily seen that $Du(x_0)=-D_xc(x_0,y_0)$ and $D^2u(x_0)\ge -D_{xx} c(x_0, y_0)=\A(x_0,Du(x_0))$, where $\A$ is defined in [**(C1)**]{} above. In particular equation is degenerate elliptic when restricted to $c$-convex function.
It has been discovered independently in [@FKM] and [@Liu] that, because of [**(C1)**]{}, for any $x_0 \in \Omega$ and $y_0 \in {\partial}^cu(x_0)$, through the change of variables $x\mapsto q(x):=-D_y c (x, y_0)$ the function $$\label{baru}
\bar u(q):=u(x(q))+c(x(q),y_0)-u(x_0)-c(x_0,y_0)$$ has convex level sets inside $\Omega$ (here and in the sequel $x(q)$ denotes the inverse of $q(x)$, which is well defined because of [**(C2)**]{}). Moreover $\bar u$ is $\bar c$-convex, where $$\label{cctilda}
\bar c(q,y):=c(x(q),y)-c(x(q),y_0),$$ see [@FKM Theorem 4.3].
Since $u$ solves one can check by a direct computation that $\bar u$ solves $$\label{MAbaru}
\det\big(D^2\bar u-\B(q,D\bar u)\big)=g,$$ with $$\label{eq:bij}
\B_{ij}(q,p)=-D_{q_iq_j} \bar c(q,T_{\bar u}(q)) \quad \text{and}
\quad g(q)=f(x(q))\left[\det D_{xy}c(x(q),T_{\bar u}(q))\right]^{-2},$$ where $T_{\bar u}$ is the map uniquely identified by the relation $D\bar u(q)=-D_qc(q,T_{\bar u}(q))$. Moreover it holds $$\label{0inunpunto}
\B_{ij}(\cdot ,0)\equiv 0,\qquad D_{p}\B_{ij}(\cdot ,0)\equiv 0,$$ so using Taylor’s formula we can write $$\label{biist}
\B_{ij}(q,D\bar u)=\B_{ij,k\ell}(q, D\bar u)\partial_k \bar u\partial_l \bar u$$ where $$\label{bijst}
\B_{ij,k\ell}(q, D\bar u(q)):=\int_0^1 D_{p_kp_\ell}\B_{ij}(q,\tau D\bar u(q))\, d\tau.$$ In addition, since condition [**(C1)**]{} is tensorial [@MTW; @Loe; @KM] and $\normm c\normm$ involves only mixed fourth derivative, it is easily seen that $\normm \bar c\normm \approx \normm c\normm$ and $\B$ satisfies the same assumptions as $\A$. In particular [**(C1)**]{} and imply that $$\label{MTWB}
\B_{ij,k\ell}\xi_i\xi_j \eta_k\eta_\ell\ge 0\quad \forall \, \xi\cdot \eta=0.$$
Given a $C^1$ $c$-convex function as above, for any $x_0\in \Omega$, $y_0=T_u(x_0)$, and $h\in \R^+$, we define the *section* centered at $x_0$ of height $h$ as $$S^u_h(x_0):=\{x \in \Omega\,:\,\ u(x)\le -c(x,y_0)+u(x_0)+c(x_0,y_0)+h\}.$$ Assuming that $S_h^u(x_0)\subset \Omega$, through the change of variables $x \mapsto q(x):=-D_y c(x,y_0)$ this section is transformed into the *convex* set $$Q^{\bar u}_h(q_0):=-D_y c(S^u_h(x_0),y_0)=\{q:\ {\bar u}(q)\le h\}.$$ When no confusion arises, we will often abbreviate $S_h(x_0)$ and $Q_h(q_0)$ for $S^u_h(x_0)$ and $Q^{\bar u}_h(q_0)$.
We also recall [@john] that, given an open bounded convex set $Q$, there exists an ellipsoid $E$ such that $$\label{eq:john}
E\subset Q\subset n E,$$ where the dilation is done with respect to the center of $E$. We refer to it as the John ellipsoid of $Q$, and we say that $Q$ is normalized if $E=B(0,1)$. An immediate consequence of is that any open bounded convex set $Q$ admits an affine transformation $L$ such that $L(Q)$ is normalized. Hence, given $u$ and $S_h^u(x_0)$ as above, we can consider $\bar u$, his section $Q_h^{\bar u} (q_0)$, and the normalizing affine transformation $L$. Then we define $\bar w : L(Q_h) \to \R $ as $$\label{barbarw}
\bar w ( q'):=(\det L)^{2/n} \bar u(q), \qquad \,q':=Lq.$$ It is easy to check that $\bar w$ solves $$\label{MAbarw}
\det\big(D^2 \bar w(q')-\C( q', D\bar w(q'))\big)=g(L^{-1}q'),$$ where $$\C( q', D\bar w(q')):=(\det L)^{2/n}(L^*)^{-1} \B\left(L^{-1}q',(\det L)^{-2/n} L^*D \bar w(q')\right) L^{-1}$$ Up to an isometry, we can assume that $$E=\biggl\{q :\ \sum_{i=1}^n \frac{q_i^2}{r_i^2} \le 1\biggr\},$$ with $r_1\le \ldots\le r_n$. Then $L^{-1} ={\rm diag}(r_1, \dots,r_n)$, and $$\label{Cijst}
\C_{ij}( q',D\bar w(q'))=\C_{ij,k\ell}( q', D\bar w(q'))\partial_k \bar w\partial_\ell \bar w$$ with $$\label{CB}
\C_{ij,k\ell}(q',D\bar w(q'))=(r_1\dots r_n)^{2/n} \frac{r_ir_j}{r_k r_\ell }\B_{ij,k\ell}(q, D\bar u(q)),$$ see . Moreover, by (or again because of the tensorial nature of condition [**(C1)**]{}) $$\label{MTWC}
\C_{ij,k\ell}\xi_i\xi_j \eta_k\eta_\ell\ge 0\quad \forall \, \xi\cdot \eta=0.$$
Still with the same notation as above, we also define the *normalized size* of a section $S_h(x_0)$ as $$\label{normalizedsize}
\aalpha(S_h(x_0))=\aalpha(Q_h(q_0)):=\frac{|L|^2}{(\det L)^{2/n}}.$$ Notice that, even if $L$ may not be unique, $\aalpha$ is well defined up to universal constants. In case $u$ is $C^2$ in a neighborhood of $x_0$, by a simple Taylor expansion of $\bar u$ around $q_0$ it is easy to see that there exists $h(x_0)>0$ small such that $$\label{hessubar}
\aalpha(S_h(x_0))=\aalpha(Q_h(q_0))\approx |D^2 \bar u (q_0)|\qquad \forall\, h\le h(x_0),$$ where $q_0:=q(x_0)$. Since $u$ and $\bar u$ are related by a diffeomorphism, the following lemma holds:
\[eccen\] Let $\Omega'\subset \Omega$, and $u\in C^2(\Omega')$ be a strictly $c$-convex function such that $\|Du\|_{L^\infty(\Omega')}$ is universally bounded. Then there exists a universal constants $M_1$ such that the following holds: For every $x_0 \in \Omega$ there exists a height $\bar h(x_0)>0$ such that if $|D^2 u(x_0)|\ge M_1$, then $$\label{eq:ecc}
|D^2 u(x_0)|\approx \aalpha (S_h(x_0)) \qquad \forall\, h\le \bar h(x_0).$$
Differentiating twice the relation we obtain $$D_{qq} \bar u=D_{q} x D_{xx} u+D_{qq} x D_xu+D_q xD_{xx} c +D_{qq}x D_x c,$$ which implies that $$\label{eq:ecc2}
\nu|D^2 \bar u(q_0)|-C\bigl(1+|Du(x_0)|\bigr)\le |D^2 u(x_0)|\le \frac{1}{\nu}|D^2 \bar u(q_0)|+C\bigl(1+|Du(x_0)|\bigr)$$ for some universal constants $\nu,C>0$. Since by assumption $Du$ is universally bounded inside $\Omega'$, follows by and , provided $M_1$ is sufficiently large.
We show now some geometric properties of sections and some estimates for solutions of which will play a major role in the sequel. Here, the dilation of a section $S_h(x)$ is intended with respect to $x$.
\[secprop\] Let $u$ be a $c$-convex Aleksandrov solution of with $ 0 < \lambda \leq f \leq 1/\lambda$. Then, for any $\Omega' \subset \subset \Omega''\subset\subset \Omega$, there exists a positive constant $\rho=\rho(\Omega',\Omega'')$ such that the following properties hold:
- $S^u_h(x) \subset \Omega''$ for any $x\in \Omega'$, $0 \leq h\le 4\rho$.
- There exist $0<\alpha_1< \alpha_2$ universal such that for all $\mu \in (0,1)$ $$\mu^{\alpha_2} S^u_h(x) \subset S_{\lambda h}^u(x)\subset\mu^{\alpha_1} S_h^u(x)$$ for any $x\in\Omega'$, $0 \leq2 h \le \rho$.
- There exists a universal constant $\sigma<1$ such that, if $S_h^u(x)\cap S_h^u(y) \ne \emptyset$, then $S_h^u (y)\subset S_{h/\sigma}^u(x)$ for any $x,y\in\Omega'$, $0 \leq h \le \sigma\rho$.
- $\cap_{0 <h \leq \rho} S_h^u(x)=\{x\}$.
Points (i) and (iv) follow from the strict $c$-convexity of $u$ shown in [@FKM section 7], and the fact that the modulus of strict $c$-convexity is universal (this last fact follows by a simple compactness argument in the spirit of [@CA2 Theorem 1’]).
Point (iii) corresponds the engulfing property of sections proved in [@FKM Theorem 9.3].
The second inclusion in point (ii) follows from [@FKM Lemma 9.2][^1]. For the first one, it is enough to show that there exists a universal constant $\bar s \in (0,1)$ such that $$\label{eq:bars}
\bar s Q^{\bar u}_h(\bar q) \subset Q^{\bar u} _{h/2}(\bar q)$$ and then iterate this estimate (here $\bar u$ is defined as in , and $\bar q:=q(x)$). To prove , let $E_{2h}$ be the John ellipsoid associated to $Q^{\bar u}_{2h}(\bar q )$, and assume without loss of generality that that $E_{2h}$ is centered at the origin. By convexity of the sections in this new variables, $$\bar s (Q^{\bar u}_h(\bar q)-\bar q)+\bar q\subset Q^{\bar u}_h(\bar q)\subset Q^{\bar u}_{2h}(\bar q)\subset nE_{2h} \qquad \forall\,\bar s \in (0,1).$$ Observe now that, for any $q\in Q^{\bar u}_h(\bar q)$, we have (recall that $\bar u(\bar q)=0$) $$\label{eq:barubars}
\bar u( \bar s (q-\bar q)+\bar q)=\bar s \int_0^1 D\bar u ((1-t\bar s)\bar q+t \bar s q)\cdot (q-\bar q)\,dt.$$ Since $q,\bar q \in n E_{2h} $ we have $q-\bar q \in 2n E_{2h}$, hence $$\label{freddo}
(q-\bar q)/2n \in E_{2h}\subset Q_{2h}(\bar q).$$ Moreover, by convexity of $Q_{2h}(\bar q)$, $(1-t\bar s)\bar q+t \bar s q\in Q_h(\bar q)\subset \tau_0 Q_{2h}(\bar q) $ for some universal $\tau_0<1$ (see [@FKM Lemma 9.2]). Defining the “dual norm” $\|\cdot\|^*_{\mathcal K}$ associated to a convex set $\mathcal K$ as $$\| a\|^*_{\mathcal K} :=\sup_{\xi \in \mathcal K} a\cdot \xi,$$ it follows from [@FKM Lemma 6.3] that $$\label{gradual}
\|D\bar u(q)\|^*_{Q_{2h}(\bar q)}
=\|-D_q\bar c(q, T_{\bar u}(q))\|^*_{Q^{\bar u}_{2h}(\bar q)}\le C h \quad \forall\, q\in Q^{\bar u}_{h}(\bar q).$$ Thus, thanks to and we get $$\begin{split}
\bar u( \bar s (q-\bar q)+\bar q)&=2n \bar s \int_0^1 D\bar u ((1-t\bar s)\bar q+t \bar s q)\cdot\frac{ (q-\bar q)}{2n}\,dt\\
&\le 2n \bar s \int_0^1\|D\bar u((1-t\bar s)\bar q+t \bar s q))\|^*_{Q_{2h}(\bar q)}dt\le 2n\bar s C h\le h/2,
\end{split}$$ provided $\bar s$ is small enough. This proves the desired inclusion.
As shown for instance in [@DFS], an easy consequence of property (iii) is the following Vitali-type covering theorem.
\[vitali\] Let $u,f,\Omega',\Omega'',\rho,\sigma$ be as in Proposition , let $D$ be a compact subset of $\Omega'$, and let $\{S_{h_x}(x)\}_{x \in D}$ be a family of sections with $h_x \leq \rho$. Then we can find a finite number of these sections $\{S_{h_{x_i}}(x_i)\}_{i=1,\dots,m}$ such that $$D \subset \bigcup_{i=1}^m S_{ h_{x_i}}(x_i), \qquad \mbox{with $\{S_{\sigma h_{x_i}}(x_i) \}_{i=1,\dots,m}$ disjoint.}$$
We now want to show that sections at the same height have a comparable shape. For this, we first recall the following estimate from [@FKM]:
\[prop:Alex\] Let $u,f,\Omega',\Omega'',\rho$ be be as in Proposition , and let $S_h(x)$ be a section of $u$ for some $x \in \Omega'$ and $h\leq \rho$. Then $$\label{alest}
|S_h(x_0)|\approx h^{n/2}.$$
\[rmk:gradientbound\] Estimates and have the following important consequence: consider the function $\bar u$ defined in , fix one of its sections $Q_{h}$ such that $Q_{2h}\subset \Omega''$ with $\Omega''$ as above, normalize $Q_h$ using its corresponding John’s transformation $L$, and define $\bar w$ as in . Since $(\det L)^{-2/n}\approx |E_h|\approx \operatorname*{osc}_{Q_{h}}\bar u\approx \operatorname*{osc}_{Q_{2h}} \bar u $ (by ) and $E_h\subset Q_{2h}$, we deduce the universal gradient bound $$\label{grad}
\begin{split}
\sup_{ L(Q_h)}|D \bar w |&= (\det L)^{2/n}\sup_{Q_h}|(L^*)^{-1}D \bar u |\\
&\le C(\det L)^{2/n} \sup_{Q_h}\|D \bar u\|^*_{E_h}\\
&\le C(\det L)^{2/n} \sup_{Q_h}\|D \bar u\|^*_{Q_{2h}}\\
&\le C(\det L)^{2/n} \operatorname*{osc}_{Q_{2h}} \bar u\le C.
\end{split}$$
\[altezzecomp\] Let $u,f,\Omega',\Omega'',\rho$ be as in Proposition . Then for any $0\leq h\leq \rho$ there exist two radii $r=r(h)$ and $R=R(h)$ such that, for every $x_0\in \Omega'$, if $E$ is the John ellipsoid associated to $S^u _h(x_0)$, then, up to a translation, $$B_r(0)\subset E\subset B_R(0).$$
Let $r_1\le \ldots\le r_n$ be the axes of $E$. Since $r_n\leq {\rm diam} (E)\le C$ and by $$h^{n/2}
\approx |E|\approx r_1\cdot \ldots\cdot r_n \leq {\rm diam}(E)^{n-1}r_1,$$ we obtain the desired lower bound on $r_1$.
Obviously analogous properties holds for the section $Q_h^{\bar u}(q_0)$.
Notice that Proposition \[secprop\](ii) applied to the (convex) sections of $\bar u$ implies the following: given $x \in \Omega''$ and $h \leq \rho$, let $r_1\leq \ldots\leq r_n$ denote the axes of the John ellipsoid associated to $Q_h(x)$. Then $$\label{eq:inclusions}
r_n\le Cr_1^{\alpha_3}$$ for some universal exponent $\alpha_3<1$ and a constant $C(\Omega',\Omega'')$.
To see this just normalize $Q_\rho(x)$ using $L$ and notice that, by [@FKM Theorem 6.11] , ${\rm dist }\bigl(x,\partial \bigl( L(Q_\rho(x)\bigr)\bigr)\ge 1/C$ for some universal constant $C$. Thus, up to enlarge $C$, $$\Bigg(\frac{h}{C\rho}\Bigg)^{\alpha_2}B_1(x)\subset L(Q_{h})\subset \Bigg(\frac{Ch}{\rho}\Bigg)^{\alpha_1}B_1(x).$$ Since, by Lemma \[altezzecomp\], sections of height $\rho$ have bounded eccentricity (i.e., $|L|\approx C(\Omega',\Omega'')$), this implies the claim with $\alpha_3:=\alpha_2/\alpha_1$.
We now observe that $\aalpha(Q_h)\approx r_n^2/(r_1\ldots r_n)^{2/n}$, from which we deduce that $$r_1^2\le C\frac{r_n^2} {\aalpha(Q_h)}.$$ In particular, this and imply $$r_n \leq C r_1^{\a_3} \leq C\frac{r_n^{\a_3}} {\aalpha(Q_h)^{\a_3/2}},$$ that is $$r_n \leq \frac{C} {\aalpha(S_h)^{\beta}},\qquad \text{with} \quad\beta:=\frac{\alpha_3}{2-2\alpha_3}.$$ Hence, since $S_h$ is linked to $Q_h$ by a diffeomorphism with universal $C^1$ norm, and ${\rm diam}(S_h)\le {\rm diam}(nE_{h})=2 n r_n$, we get $$\label{diamsec}
{\rm diam}(S_h)\le \frac{\bar C} {\aalpha(S_h)^{\beta}}, \qquad \beta,\bar C>0 \text{ universal}.$$
$W^{2,1+\kappa}$ estimates
==========================
Applying first a large dilation to $\Omega$ we can assume that $B(0,1) \subset \Omega$, and by a standard covering argument (see for instance [@DepFi Section 3]) it suffices to prove the $W^{2,1+\kappa}$ regularity of $u$ inside $B(0,1/2)$. Also, by an approximation argument[^2], it is enough to prove the result when $u \in C^2$. Hence Theorem \[w21eps\] is a consequence of the following:
\[W21epsloc\]Let $u\in C^2$ be a $c$-convex solution of with $\Omega \supset B(0,1)$. Then there exist universal constants $\kappa$ and $C$ such that $$\label{eqW21epsloc}
\int_{B(0,1/2)}|D^2 u|^{1+\kappa}\le C.$$
We start with the following lemma:
\[lemmaL\^1\]Let $u$ be as above, $x_0\in B(0,3/4)$, and $h>0$ such that $S_{2h}(x_0)\subset B(0,5/6)$. Consider the function $\bar u$ as in , its section $Q_h=Q_h(q_0)$ with $q_0:=q(x_0)$, and (up to a rotation) let $E_h=\left\{\sum x_i^2/r_i^2\le 1\right\}$ be the John ellipsoid associated to $Q_h(q_0)$. Denote by $L$ be the affine transformation that normalizes $Q_h$, and define $\bar w$ and $\C_{ij,k\ell}$ as in and respectively. Then $$\label{L^1}
\int_{L(Q_h)}\big| \partial_{ij} \bar w-\C_{ij,k\ell} \partial_k \bar w \, \partial_\ell \bar w\big| \le C$$ for some universal constant $C$.
Since, by the $c$-convexity of $u$ (which is preserved under change of variables), the matrix $\bigl(\partial_{ij} \bar w
-\C_{ij,k\ell} \partial_k \bar w \, \partial_\ell \bar w\bigr)_{i,j=1,\ldots,n}$ is non-negative definite, it is enough to estimate $$\int_{L(Q_h)}\sum_{i=1}^n\big(\partial_{ii} \bar w-C_{ii,st} \partial_k\bar w \, \partial_l\bar w\big)$$ from above.
Using the bounds $\mathcal H^{n-1}\bigl(\partial\bigl( L(Q_h)\bigr)\bigr)\le C(n)$ (since $L(Q_h)$ is a normalized convex set) and $|D\bar w|\le C$ (see ), we see that first term is controlled from above by $$\label{1}
\int_{L(Q_h)} \Delta \bar w = \int_{\partial\left( L(Q_h)\right)} D\bar w \cdot \nu \le \mathcal H^{n-1}\bigl(\partial\bigl( L(Q_h)\bigr)\bigr) \sup_{ L(Q_h)}|D \bar w |\le C.$$ For the second term, we claim the following: there exists a universal constant $C$ such that $$\label{2}
\inf_{L(Q_h)}\sum_{i=1}^n \C_{ii,k\ell} \partial_k \bar w \, \partial_\ell \bar w \ge -C$$ To see this we write $$\sum_{i=1}^n \C_{ii,k\ell} \partial_k \bar w \, \partial_\ell \bar w
=\sum_{i=1}^n \sum_{k,\ell\ne i} \C_{ii,k\ell} \partial_k \bar w \, \partial_\ell \bar w+2\sum_{i=1}^n\sum_{k\ne i}\C_{ii,ik} \partial_i \bar w \, \partial_k \bar w+\sum_{i=1}^n \C_{ii,ii} \partial_i \bar w \, \partial_i\bar w.$$ We first observe that, since for any $i=1,\ldots,n$ the vector $(\partial_1 \bar w,\dots,\partial_{i-1}\bar w,0,\partial_{i+1} \bar w,\dots,\partial_n\bar w) $ is orthogonal to coordinate vector $e_i$, the first term in the right hand side is non-negative by condition [**(C1)**]{}.
Concerning the second and the third term, taking into account the definition of $\C_{ij,k\ell}$ in we can rewrite them as $$\label{eq:23term}
2\sum_{i=1}^n\sum_{k\ne i}(r_1 \dots r_n)^{2/n}\frac {r_i}{r_k} \B_{ii,ik} \partial_i \bar w \, \partial_k \bar w+\sum_{i=1}^n (r_1 \dots r_n)^{2/n}\B_{ii,ii} \partial_i \bar w \, \partial_i\bar w.$$ Observe that, by , $$\label{eq:ri h}
(r_1 \dots r_n)^{2/n}\approx h.$$ In addition, by the Lipschitz regularity of $\bar u$ (which is simply a consequence of the fact that $u$ is locally Lipschitz inside $\Omega$), $$\label{eq:ri h2}
h/r_k\leq C \qquad \forall\,k=1,\ldots,n.$$ Since $\|D\bar w\|_\infty\leq C$ (see ) and the size of $\B$ is controlled by $\normm \bar c\normm \approx \normm c\normm$, by and we see that the expression in is universally bounded.
This proves , which combined with concludes the proof.
\[lem:hessian\]With the same notation and hypotheses as in Lemma \[L\^1\], let $\sigma$ be as in Proposition \[vitali\]. Then there exists a universal constant $C$ such that $$\label{mis}
\big|\{\tilde q\in L(Q_{\sigma h}): {\rm Id}/C \le\partial_{ij} \bar w-\C_{ij,k\ell} \partial_k \bar w \, \partial_\ell \bar w \le C\,{\rm Id} \}\big|\ge \frac{1}{C}.$$
Since $\sigma$ is universal and $L(Q_h)$ is normalized, by Proposition \[secprop\](ii) we get $$|L(Q_{\sigma h})|\approx |L(Q_h)|\approx 1.$$ So, using Lemma \[lemmaL\^1\] and Chebychev inequality, we deduce the existence of a universal constant $C$ such that $$|\{\tilde q \in L(Q_{\sigma h}): \partial_{ij} \bar w-\C_{ij,k\ell} \partial_k \bar w \, \partial_\ell \bar w\le C\,{\rm Id} \}|\ge \frac{1}{C}.$$ Since by the product of the eigenvalues of the matrix $\bigl(\partial_{ij} \bar w-\C_{ij,k\ell} \partial_k \bar w\bigr)_{i,j=1,\ldots,n}$ is of order one, whenever the eigenvalues are universally bounded from above, they also have to be universally bounded also from below. Hence, up to enlarging the value of $C$, this proves .
\[undoscaling\] Recalling the definition of $\aalpha( Q_h)=\aalpha(S_h)$, we can rewrite both and in terms of $\bar u$ and $Q_h=Q_h(q_0)$, obtaining that $$\int_{Q_h}\big| \partial_{ij} \bar u -\B_{ij,k\ell} \partial_k \bar u \, \partial_l \bar u\big| \le C\aalpha(Q_h)\Big|\big\{x\in Q_{\sigma h}: \aalpha(Q_h)/C \le \big|\partial_{ij} \bar u-\B_{ij,k\ell} \partial_k \bar u \, \partial_l \bar u \big|\le C\aalpha(Q_h) \big\}\Big|$$ (see for instance the proof of [@DFS Lemma 3.2]). In terms of $u$, this estimate becomes $$\label{main}
\int_{S_h}|D^2 u-\A(x,Du)| \le C_0\aalpha(S_h)\Big|\big\{x\in S_{\sigma h}: \aalpha(S_h)/C_0 \le |D^2 u-\A(x,Du)|\le C_0\aalpha(S_h) \big\}\Big|,$$ where $S_h=S_h(x_0)$ with $x_0$ an arbitrary point inside $B(0,3/4)$, and $C_0$ is universal.
Let $M\gg1$ to be fixed later, set $R_{0}:=3/4$, and for all $m\ge 1$ define $$\label{rk}
R_m:=R_{m-1}-\bar CM^{-\beta}.$$ with $\bar C$ and $\beta$ as in . Let us use denote $B(0,R_m)$ by $B_{R_m}$, set $$\label{eq:F}
F(x):=\bigl|D^2 u(x)-\A(x,Du(x))\bigr|,$$ and define $$\label{dk}
D_m:=\left\{x\in B_{R_m}:\ F(x)\ge M^m\right\}.$$ Thanks to Proposition \[secprop\], there exists $\rho>0$ universal such that $S_h(x)\subset B(0,5/6)$ for any $x\in B(0,{3/4})$ and $h \leq 2\rho$, and by Lemma \[altezzecomp\] applied with $h=\rho$ we get $\aalpha(S_\rho(x))\approx 1$. In addition, since [**(C0)**]{} implies that $|\A(x,Du)|\leq C_1$ inside $B(0,1)$ for some $C_1$ universal, we see that $\bigl||D^2u| - F\bigr| \leq C_1$. Hence, using Lemma \[eccen\], we deduce that if $M \gg M_1+C_1$ then there exists a small universal constant $\nu>0$ such that $$\aalpha(S_h(x))\ge \nu M^m \qquad \forall\, x \in D_m,\, \,h \leq \min\{\bar h(x),\rho\},\, \,m \geq 1.$$ So, by choosing $M\ge \max\{ 1/\nu^4, M_1\}$ (so that $\nu M^{m+1} \geq M^{m+1/2}/\nu$), by continuity we obtain that, for every point in $D_{m+1}$, there exists $h_x \in (0,\min\{\bar h(x),\rho\})$ such that $\aalpha(S_{h_x}) \in (\nu M^{m+1/2}, M^{m+1/2}/\nu)$. In particular, by we have ${\rm diam} (S_{h_x})\le \bar C M^{-\beta}$, which implies that (recall ) $$\label{eq:inclusion}
\bigcup_{x\in D_{m+1}} S_{h_x} \subset B\left(0,R_{m+1}+\bar C M^{-\beta}\right)= B_{R_m}.$$ According to Proposition \[vitali\], we can cover $D_{m+1}$ with finitely many sections $\{S_{h_{x_j}}\}_{x_j \in D_{m+1}}$ such that $ S_{\sigma h_{x_j}}$ are disjoint. Then and imply (recall ) $$\begin{split}
\int_{D_{m+1}}F \le \sum_j \int_{ S_{h_{x_j}}}F&\le \sum_j C_0\,\aalpha(S_{h_{x_j}})|\{x\in S_{\sigma h_{x_j}}: \aalpha(S_{h_{x_j}})/C_0 \le F\le C_0\aalpha(S_{h_{x_j}}) \}|\\
&\le \sum_j \frac{C_0}{\nu} \,M^{m+1/2}|\{x\in S_{\sigma h_{x_j}}: \nu M^{m+1/2}/C_0 \le F\le C_0 M^{m+1/2}/\nu \}|.
\end{split}$$ Assuming now that $\sqrt{M}\ge C_0/\nu$ and recalling , we obtain $$\label{stimedk}
\begin{split}
\int_{D_{m+1}}F\le \sum_j \int_{ S_{h_{x_j}}}F&\le \sum_j \frac{C_0}{\nu} \,M^{m+1/2}|\{x\in S_{\sigma h_{x_j}}: \nu M^{m+1/2}/C_0 \le F\le C_0 M^{m+1/2}/\nu \}|\\
&\le \sum_j \frac{C_0}{\nu} \,M^{m+1/2}|\{x\in S_{\sigma h_{x_j}}: M^m \le F\le M^{m+1} \}|\\
&= \sum_j \frac{C_0}{\nu} \,M^{m+1/2}|\{x\in S_{\sigma h_{x_j}}: M^m \le F\le M^{m+1} \}\cap B_{R_m}|\\
&\le \frac{ C_0\sqrt{M}}{\nu} \int_{D_m\setminus D_{m+1}}F.
\end{split}$$ Adding $\frac{C_0\sqrt{M}}{\nu} \int_{D_{m+1}}F$ to both sides of the previous inequality, we obtain $$\Bigg(1+\frac {C_0\sqrt{M}}{\nu}\Bigg) \int_{D_{m+1}}F \le \frac{ C_0\sqrt{M}}{\nu} \int_{D_m}F.$$ which implies $$\int_{D_{m+1}}F\le (1-\tau)\int_{D_m}F$$ for some small constant $\tau=\tau(M)>0$. We finally fix $M$ so that it also satisfies $$\label{k0}
\sum_{m\ge 1} \bar CM^{-m\beta}\le \frac 1 4.$$ In this way $R_m \geq 1/2$ for all $m \geq 1$, so that the above inequalities and the definition of $R_m$ imply $$\int_{\{F\ge M^m\}\cap B(0,{1/2})}F\le \int_{D_m}F\le (1-\tau)^m\int_{D_{0}}F\le C(1-\tau)^m$$ (here we used that $\int_{B(0,{3/4})}F\le C$, which can be easily proved arguing as in the proof of Lemma \[L\^1\]). Thus, choosing $\kappa>0$ such that $1-\tau=M^{-2\kappa}$, we deduce that $$|\{F\ge t\}\cap B(0,{1/2})|\le\frac{1}{t} \int_{\{F\ge t\}\cap B(0,{1/2})}F \le Ct^{-1-2\kappa}$$ for some $C>0$ universal, which implies that $F \in L^{1+\kappa}(B(0,{1/2}))$. Recalling the definition of $F$ (see ) and that $|\A(x,Du)|\leq C$ inside $B(0,{1})$ (by [**(C0)**]{}), this concludes the proof.
[99]{}
*Existence of Eulerian solutions to the semi-geostrophic equations in physical space: the 2-dimensional periodic case*. Comm. Partial Differential Equations, to appear.
*A global existence result for the semi-geostrophic equations in three dimensional convex domains*. Preprint 2012.
Ann. of Math. (2), [**131**]{} (1990), no. 1, 129–134.
Comm. Pure Appl. Math., [**44**]{} (1991), 965–969.
J. Amer. Math. Soc., [**5**]{} (1992), 99–104.
J. Fluid Mech., [**531**]{} (2005), 123–157.
J. Reine Angew. Math., [**646**]{} (2010), 65–115.
Methods Appl. Anal., [**18**]{} (2011), no. 3, 269–302.
Invent. Math., to appear.
Math. Ann., to appear.
Preprint, 2011.
Calc. Var. Partial Differential Equations, [**35**]{} (2009), no. 4, 537–550.
Comm. Pure Appl. Math., [**62**]{} (2009), no. 12, 1670–1706.
Amer. J. Math., [**134**]{} (2012), no. 1, 109–139.
In Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pages 187–204. Interscience, New York, 1948.
J. Eur. Math. Soc. (JEMS), [**12**]{} (2010), no. 4, 1009–1040.
Calc. Var. Partial Differential Equations, [**34**]{} (2009), no. 4, 435–451.
Comm. Partial Differential Equations, [**35**]{} (2010), 165-184.
Acta Math., [**202**]{} (2009), no. 2, 241–283.
Arch. Ration. Mech. Anal., [**199**]{} (2011), no. 1, 269–289.
Arch. Ration. Mech. Anal., [**177**]{} (2005), no. 2, 151–183.
Preprint 2012.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), [**8**]{} (2009) 143-174.
Arch. Ration. Mech. Anal., [**192**]{} (2009), 403-418.
[^1]: To be precise, in [@FKM] the dilation is done with respect to the center of the John ellipsoid, and not with respect to the “center” $x$ of the section. However, it is easy to see that the same statement holds also in this case.
[^2]: To approximate our solution with smooth ones, it suffices to regularize the data and then:\
- either apply [@LTWC2a Remark 4.1] (notice that, by Proposition \[secprop\](iv) and [@FKM Theorem 8.2], $u$ is strictly $c$-convex and of class $C^1$ inside $\Omega$);\
- or approximate our cost $c$ with cost functions satisfying the strong version of [**(C1)**]{} and apply [@LTWC2a Theorem 1.1].
|
---
abstract: 'Recent theoretical results on heavy flavor production and decay in the framework of perturbative QCD are reviewed. This includes calculations for top production at hadron colliders, inclusive charmonium production and the comparison between the singlet and octet mechanisms. Predictions for heavy flavor production in annihilation will be discussed in some detail, covering both the threshold and the high energy region. The first results in NLO for heavy flavor decays will also be reviewed.'
---
\#1\#2\#3[[*Act. Phys. Pol.* ]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Act. Phys. Austr.*]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nucl. Phys.* ]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Lett.* ]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev.* ]{}[**D \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. Lett.* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Reports* ]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Comp. Phys. Commun.* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nucl. Inst. Meth.* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Reports* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Sov. J. Nucl. Phys.* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*JETP Lett.* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*JETP Lett.* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Z. Phys.* ]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Prog. Theor. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nouvo Cim. *]{}[**\#1A**]{} (\#2) \#3]{}
=12.5cm =18.5cm
[**TTP96-53[^1]**]{}\
[**hep-ph/9612372**]{}\
November 1996
[**Heavy Flavor Production and Decay**]{}\
[**J. H. Kühn[^2]**]{}\
[*Institut für Theoretische Teilchenphysik, Universität Karlsruhe,\
D-76128 Karlsruhe, Germany\
*]{}
Introduction
============
Heavy flavor production and decay have developed into benchmark reactions for perturbative QCD. The large energy scale inherent in most of these reactions allows for a separation between hard and soft momentum transfers. The former can be treated perturbatively, the nonperturbative matrix elements which encode the remaining information can either be determined experimentally, or integrated out by considering sufficiently inclusive information such that perturbation theory alone is adequate.
Significant progress has been achieved recently in a number of topics. The predictions for top production at hadron colliders have been scrutinized by several authors. In particular the role of soft gluon resummation has been emphasised and the dependence explored (Section 2). Inclusive charmonium production at hadron and colliders has been studied theoretically and experimentally. A fairly complex picture seems to emerge, with different mechanisms playing a role in various reactions (Section 3). The inclusive cross section for heavy flavor production in annihilation has been studied in a variety of papers. Far above threshold an expansion in $m^2/s$ is adequate and has been successfully applied to $Z$ decays to bottom quarks, or to charm production just below the $b\bar b$ threshold. For a prediction above, but relatively close to threshold a different strategy has been employed, which is based on a combination of analytical and numerical methods. For an adequate treatment of top quark production in the threshold region its large decay rate and the interplay between gluon radiation from the production and the decay process must be taken into account. These topics will be reviewed in section 4. The leading QCD correction to weak decays of heavy flavors have been evaluated quite some time ago. Results are available for the rate, the spectrum and for angular distributions. To match the level of precision claimed by the proponents of the Heavy Quark Effective Theory, next to leading order predictions are required from perturbation theory. First steps into this direction have been made and will be reviewed in section 5.
Top production in hadronic collisions
=====================================
The theoretical framework and the (semi-) analytical results for the top production cross section in NLO have been developed nearly a decade ago [@EQ; @nason1]. The predictions for $\sqrt{s}=1.8$ GeV and $m_t$ = 180 GeV from various authors are listed in Table \[table:history\].
---------------------------------------------------------------------------------------------
$\protect\sigma$ \[pb\]
---------------------------- ----------------------------------------------------------------
Altarelli et al. [@nason1] $3.52$ (DFLM)
$4.10$ (ELHQ)
Laenen et al. [@Laenen] $ \left.\begin{array}{ll} 3.5 &\quad(\mu^2=4m^2)\\
3.8 &\quad(\mu^2=m^2)\\
4.05 &\quad(\mu^2=m^2/4)
\end{array}
\right\}$ MRSD
Resummation
Laenen et al. [@Laenen] $
\left.\begin{array}{c} 3.86\\4.21\\4.78\end{array}
\right\} $ vary $\mu_0$
Berends et al. [@Berends] 4.8 central value
Berger et al. [@Berger] 4.8 “principal value res.”
Catani et al. [@Catani] $4.05^{+0.62}_{-0.52}$
---------------------------------------------------------------------------------------------
: History of predictions for the production cross section for $\protect\sqrt{s} =1.8$ TeV and $m_t = 180$ GeV.[]{data-label="table:history"}
For $m_t$ = 175 GeV the cross section increases by about 0.7 pb. The uncertainty in the factorization and renormalisation scale leads to an uncertainty of roughly 10%. Recently the issue of soft gluon resummation has been raised. The original arguments [@Laenen; @Berger] leading to a large positive shift of roughly 10% have been refuted in [@Catani]. No consensus has yet been reached on the magnitude of these effects. Increasing from the nominal value of around 0.11, which has been frequently used in these calculations, to 0.120 leads to an increase by about 5%. Within the combined uncertainties theory and experiment are in very good agreement (Fig. \[fig:1\]).
All these calculations are based on a perturbative treatment of the threshold region. In principle one should, however, incorporate the leading terms of order $\pi \alpha_s/\beta$. The resulting modifications are small for $t\bar t$ in a color octet which is the dominant configuration at the TEVATRON (see section 2.2.2 in [@top]).
Inclusive Charmonium Production
===============================
High energy hadron-hadron and $e^-p$ colliders are charmonium factories. A variety of production mechanisms have been discussed in the literature. Contributing with different relative strengths in the various reactions they can be disentangled only through a systematic study of different processes. In particular the question of color singlet versus octet production has stimulated a number of detailed investigations.
Inelastic $\jpsi$ production in photon-photon reactions provides a relatively clean testing ground. The dominant subprocess at the parton level +g+ g can produce directly a ($c\bar c$) color singlet state. Incorporating also the one loop perturbative corrections [@Kraemer], satisfactory agreement between theory and experiment is observed for the $\jpsi$ energy distributions and the total production cross section as well (Fig. \[fig:2\]).
This success of the color singlet model (CSM) (where quarkonium (color singlet!) states are required to be produced through a purely perturbative mechanism) is in marked contrast with its failure in purely hadronic collisions. The dominant subprocesses in the CSM are based on the conversion of a virtual gluon into $\jpsi$ or $\chi_J$ plus two or one gluon respectively. The combination of additional powers of with the small phase space gives rise to sizable suppression factors. This perturbative treatment of soft gluon radiation may be inadequate and an alternative approach has been advocated in [@cit1]. The cross section for charmonium production is decomposed into a sum of terms consisting of the cross section for ($c\bar c$) states in a specific angular momentum and color state times the nonperturbative matrix element of an operator characterizing the conversion probability into $\jpsi$: (p|p+ x) = \_n (p|pc|c(n))\^\_n. These matrix elements are effectively free parameters to be determined in different experiments. This approach is closely related in its spirit to the color evaporation model formulated a long time ago; it provides, however, a more firm theoretical formulation. Adjusting the parameters appropriately, a satisfactory description of the data is obtained.
The clean initial state configuration typical for annihilation is ideal to investigate the relative importance of different production mechanisms. Two distinctly different situations have been considered: high energy reactions like $Z$ decays with large event rates available at LEP and alternatively the 10 GeV region that can be explored at present at CESR or in the near future at the $B$-meson factories. Three mechanisms have been identified at which contribute in the high energy region with comparable rates. The reaction [@keu] Zc|c+X requires the production of two $c\bar c$ pairs with a rate proportional to $\alpha_s^2|R(0)|^2$. The second mechanism [@keu2] is the splitting of a virtual gluon in a color octet $c\bar c$: Zq|q (c|c)\_8 with the subsequent nonperturbative conversion of $(c\bar c)_8$ into $\jpsi$. The rate for this mechanism is proportional to $\alpha_s^2\langle{\cal O}^8\rangle$ where the second factor characterizes the nonperturbative matrix element. The third, color singlet, contribution [@hag] Zq|q gg is strongly suppressed by the factor $\alpha_s^4|R(0)|^2$ and, furthermore, by the small phase space. The branching ratios of the three reactions are given by $0.8\cdot 10^{-4}$, $1.9\cdot 10^{-4}$, $0.2\cdot 10^{-4}$, respectively. The total inclusive rate is reasonably consistent with the observations by the OPAL collaboration [@OPAL] of $(1.9\pm 0.7 \pm 0.5 \pm 0.5)\cdot 10^{-4}$. However, a statement about the dominance of any of these processes seems premature. The analysis of $\jpsi$ energy and momentum distributions, however, could help to settle this issue.
Also $B$ meson factories and CESR give rise to a large sample of events with prompt $\jpsi$ production. Two mechanisms have been proposed which might well describe complementary kinematical regions. The leading process in the CSM e\^+ e\^- +gg is proportional to $\alpha_s^2|R(0)|^2$. It leads to a three body final state and hence to a continuous energy distribution (Fig. \[fig:3\]).
Predictions for the rate, the angular and the momentum distribution and the polarization can be found in [@Driesen]. The alternative approach [@bra] is based on “color octet production”, $e^+e^-\to (c\bar
c)_8+g$. The rate is of order and multiplied by a nonperturbative matrix element. The $\jpsi$ energy is essentially fixed at $E_{max}=(s+m^2_\psi)/(2\sqrt{s})$. The angular distribution is proportional to $(1+\cos^2\theta)$. These features are identical to the predictions of the “color evaporation model” [@Fritsch] formulated a long time ago. An excess of $\jpsi$ at this special kinematical point with the predicted angular distribution would be a strong indication for this “octet mechanism”. The angular distribution of the $\jpsi$ in the CSM is of the form $1+\alpha(y)
\cos^2\theta$ where $\alpha(y)$ depends on $y\equiv E_\psi /E_{Beam}$ and approaches roughly $-0.8$ at the endpoint (Fig. \[fig:4\]).
This difference will be crucial in disentangling the two mechanisms.
Heavy flavor production in annihilation
=======================================
$Z\to b\bar b$
--------------
Experimental studies of various partial and of the total $Z$ decay rate have been performed recently with a new level of sophistication. The relative error in $\Gamma_b$ has been lowered to about $0.5 \cdot
10^{-2}$ corresponding to $\delta \Gamma_b \approx 2.5 $ MeV, the uncertainty in the total decay rate which is also influenced by $\Gamma_b$ amounts to about 3 MeV. In comparison with $\Gamma_d$ or $\Gamma_u$ two important differences have to be taken into account for $\Gamma_b$. The first, relatively straightforward aspect is related to the bottom mass. In Born approximation the correction from the phase space suppression of the axial part of the rate is predicted to be $-6m_b^2/M_Z^2$ corresponding to $-4$ MeV. In [@CK] it has been demonstrated that this number is drastically modified by QCD corrections. The bulk of these, the large logarithms, can be absorbed by reexpressing the result in terms of the running mass thus reducing the correction to $-1.6$ MeV. (For a detailed discussion and further references see [@Report].) The second contribution to the $Z\to
b\bar b$ decay has its origin in the double triangle diagrams with two gluon intermediate states. It is present for the axial rate only. The contribution of order $\alpha_s^2$ was calculated quite some time ago for arbitrary $m_t^2/M_Z^2$. Formally it is proportional to $\ln m_t^2/M_Z^2$ and thus seems to diverge in the limit of large $\ln
m_t^2/M_Z^2$. However, additional logarithms of $m_t^2/\mu^2$ are induced by the running of $\alpha_s$ which have to be controlled at the same time. The structure of leading logs was analysed in [@Chet], the constant terms of $\alpha_s^3$ in [@ChetTar]. The combined effect of order $\alpha_s^2$ and $\alpha_s^3$ from these “singlet terms” amounts $\delta \Gamma_b =
-1.8$ MeV. It is clear that the sum of mass and singlet terms must be taken into consideration in any precision analysis.
Intermediate energies
---------------------
The $Z$ decay rate is well described in the massless approximation plus terms of order $m_b^2/M_Z^2$. However, for a prediction at lower energies, an increasing series of terms in the $m^2/s$ expansion is needed. The comparison between the complete calculation and a limited number of terms in the $m^2/s$ expansion indicates that the first three terms are sufficient to describe the cross section from high energies down to $s\approx 8m^2$. With this motivation in mind the quartic terms of order $\alpha_s^2$ have been calculated in [@Quart]. In this way an adequate prediction between roughly 14 GeV and $M_Z$ is available for $b\bar b$ production, and similarly for $c\bar c$ production from roughly 5 to 6 GeV up to the bottom quark threshold [@Rhad] (Fig. \[fig:6\]).
In view of the large statistics available at CESR and at a future $B$-meson factory a detailed theoretical study has been performed in [@Teubner] which demonstrates the potential for this potentially most precise and clean determination of $\alpha_s$.
The NLO calculation for arbitrary $m^2$ and s
---------------------------------------------
A few GeV above charm, bottom, or top threshold measurements can in principle be performed at a $\tau$-charm factory, at a $B-$meson factory and a future linear collider. With a relative momentum of the quarks exceeding for instance 3 GeV perturbative QCD should be applicable also in this region. It is, therefore, desirable to push the theoretical prediction as close as possible towards the threshold. The two-loop calculation has been performed more than 40 years ago [@Kallen]. The imaginary part of those three-loop diagrams which originate from massless fermion loop insertions in the gluon propagator (“double bubble diagrams”) were calculated analytically in [@HKT]. Real and imaginary parts of the purely gluonic correction (and of the double bubble diagrams) were calculated in a semianalytical approach [@ChKS] that will be sketched in this subsection.
The polarization function can be written in the form\
&& + ([\_s]{})\^2 ( C\_F\^2 \_A + C\_FC\_A \_[NA]{} + C\_FTn\_l \_[l]{} + C\_FT \_[F]{} ) where $n_l$ denotes the number of light quark species. Each one of the $\Pi_j$ is analytical in the complex $q^2$ plane with a cut from $4m^2$ to $+\infty$. For small $q^2$ they can be expanded in a Taylor series (q\^2,m\^2) = \_[n>0]{}C\_n ( [q\^24m\^2]{})\^n The renormalization condition $\Pi(q^2=0,m^2)=0$ has already been implemented. The evaluation of the Taylor coefficients amounts to the calculation of three loop tadpole integrals with an increasing number of mass insertions – up to 16 for $C_8$ which is the present limit for the evaluation with the help of algebraic programs.
In the large $q^2$ region a similar expansion can be performed. For this case the expansion has been performed up to terms of order $(m^2/q^2)^0$ and $(m^2/q^2)^1$. Additional information can be obtained about the behavior close to threshold. Leading and subleading terms can be deduced from the influence of the Coulomb potential in the nonrelativistic region, combined with the knowledge about the logarithmic corrections of the perturbative QCD potential. To extend the range of convergence from $q^2<4m^2$ to the full analyticity domain an appropriate variable transformation has to be performed. The data from the three kinematical regions are finally integrated in a Padé approximation which leads to stable results for $\Pi(q^2)$ and $R(s)$ at the same time. The result for the three dominant pieces are shown in Fig. \[fig:71\] where it is compared to the leading terms close to the threshold and to the high energy approximation.
[c]{}\
\
Toponium and top quarks in the threshold region
-----------------------------------------------
Top quarks were treated as stable particles in the previous section. Although adequate away from the threshold, this approximation is inadequate in the “would-be” toponium region. For a mass of the top quark around 175 GeV a decay rate $\Gamma_t\approx 1.5$ GeV is predicted, corresponding to a toponium width of 3 GeV. The resonances are thus completely dissolved [@rep26; @rep43], and the individual peaks are merged into a step function like threshold cross section. Quarkonium physics ceases to exist. The large decay rate introduces, however, a cutoff which eliminates all nonperturbative aspects of the interquark potential. Large momentum tails beyond P\_[cut]{} 24 GeV or, alternatively, distances above r0.01 [fm]{} are irrelevant for the description of the $t\bar t$ system [@FK; @JezKT; @JezT; @Sum]. The impact of the large rate is clearly visible in Fig. \[fig:8\].
=12.cm
The predictions for three different top masses $m_t=150$ GeV, 180 GeV, and 200 GeV corresponding to $\Gamma_t=$ 0.81 GeV, 1.57 GeV, and 2.24 GeV demonstrate the strong influence of $\Gamma_t$ on the shape of the cross section. The shape is furthermore significantly modified by initial state radiation and the spread in the beam energy.
Additional information is encoded in the momentum distribution of top quarks, the “Fermi motion” which can be traced through the decay products $W+b$. This distribution is essentially equivalent to the square of the wave function in momentum space and can, for unstable particles, be evaluated [@JezKT; @JezT; @Sum] with Green’s function techniques (Fig. \[fig:9\]).
=12.cm
Various experimental studies have demonstrated the potential of a linear collider to determine $m_t$ to a precision of perhaps even 200 MeV by measuring the cross section and the momentum distribution simultaneously.
Highly polarized top quarks are required for a variety of precision studies of top decays. In the threshold region this is easily achieved. In fact, even with unpolarized beams top quarks are longitudinally polarized (with a polarization around $-0.4$) as a consequence of the nonvanishing axial part of the neutral current. Longitudinally polarized beams lead to a fully polarized sample of top quarks.
Another step in complication is achieved by considering the interference between the dominant $S$ and the suppressed $P$ wave contributions. The relative size of these effects is of order $\beta\sim 0.1$. It leads to a forward–backward asymmetry [@top122] and furthermore to an angular dependent quark polarization perpendicular to the beam direction [@top123]. A detailed discussion of these effects, in particular of the role of the normal polarization and of rescattering corrections, can be found in [@us]. The small polarization of top quarks normal to the production plane is a particularly sensitive measure of the interquark potential.
Additional complications are introduced through the rescattering [@us; @Sumino] between $b$ quark jets and the spectator, and by relativistic corrections [@JezT] of order $\alpha_s^2$. These effects will be important for the quantitative comparison between theory and experiment and the extraction of a precise value for $m_t$, $\Gamma_t$ and $\alpha_s$ from threshold studies.
Towards NLO in Heavy Flavor Decays
==================================
Semileptonic weak decays of bottom mesons and top quarks are particularly clean probes of the fundamental properties of quarks, their masses and mixing angles. Decay rates are, however, influenced by QCD effects, a large part of which can be calculated in PQCD. Leading order corrections to practically all quantities of interest are available: for the decay rate of charmed and bottom quark from [@CabMai] and for top quarks from [@jk]. Lepton decay spectra have been calculated in [@jkdis], the energy distribution of hadrons in [@had]. Leptons from the decays of polarized quarks exhibit a nontrivial angular distribution [@pol; @cjkk94] and even lepton mass effects have been incorporated in these calculations [@tau; @Motyka]. A compact summary of most of these QCD corrections can be found in [@sum].
Different techniques to determine the degree of $b$ or top polarization have been investigated in [@cjkk94; @ame]. The analysis of moments of the lepton momentum distribution, or the ratio of charged vs. neutral lepton moments appear to be particularly promising.
The corrections are often sizable, in particular those to the decay rate. In order to fix the scale in the running coupling constant and to gain confidence in the numerical result, a calculation of NLO corrections to the rate, if not the spectrum, is necessary. Purely fermionic loops have been considered in [@Smith; @Cz2]. In the limit $m_t^2\gg m_W^2$ the result is particularly simple \_t &=& \_[Born]{} with \_[Born]{} = [G\_F m\_t\^3 8]{} If we adopt the BLM prescription the large coefficient leads to a large shift in the effective scale for $\alpha_s$: $\mu_{BLM} =
0.12m_t$. Similarly large correction factors have been observed [@wise] for the decay of $b$ into $l\nu$ plus a charmed or $u$ quark.
It should be emphasized that the magnitude of NLO corrections $\sim(\alpha_s(m_b^2)/\pi)^2 \approx (0.07)^2$ is well comparable with correction terms obtained in Heavy Quark Effective Theory — typically of order $(\Lambda/m_b)^2 \approx (0.05)^2$. Transitions at zero recoil i.e. for the final state with $p_c = {m_c\over m_b} p_b$, are particularly clean from the theoretical point of view. No uncalculable form factor is present, allowing to determine $V_{cb}$ with remarkable precision. The first calculation of the full NLO QCD corrections has therefore been performed at zero recoil [@czprl]. Two important simplifications are present in this case:
- no real radiation has to be considered,
- only relatively simple two loop integrals arise which can be calculated in a series expansion.
The resulting NLO corrections are smaller than the leading ones by about a factor 4, reducing thus the theoretical error by a significant factor. Evidently these results can be considered a first important step towards a complete NLO calculation of the heavy quark decay rate.
[99]{} W. Beenakker, H. Kuijf, W. L. van Neerven and J. Smith, .\
R. Meng, G. A. Schuler, J. Smith and W. L. van Neerven, .\
W. Beenakker, W. L. van Neerven, R. Meng, G. A. Schuler and J. Smith, [*Nucl. Phys.*]{} [**B351**]{} (1991) 507.
P. Nason, S. Dawson and R. K. Ellis, ; [*ibid.*]{} [**B**]{}327 (1989) 49; Erratum B335 (1990) 260.\
G. Altarelli, M. Diemoz, G. Martinelli, and P. Nason, [*Nucl. Phys.*]{} [ **B308**]{} (1988) 724.
F.A. Berends, J.B. Tausk, and W.T. Giele, [*Phys. Rev.*]{} [**D47**]{} (1993) 2746.
E. Laenen, J. Smith, and W.L. van Neerven, ; .
E.L. Berger and H. Contopanagos, .
S. Catani, preprint hep-ph/9610413.
S. Catani et al., preprint hep-ph/9602208.
J.H. Kühn, [*Theory of Top Quark Production and Decay*]{}, TTP96-18, to be published in the Proceedings of the 1995 SLAC Summer School.
M. Krämer, J. Zunft, P. Zerwas, and J. Steegborn, [*Phys. Lett.*]{} [ **B348**]{} (1995) 657.
G.T. Bodwin, E. Braaten, and G.P. Lepage, .
V. Barger, K. Cheung, and W.Y. Keung, .
K. Cheung, W.Y. Keung, and T.C. Yuan, ;\
P. Cho, .
K. Hagiwara et al., ; Erratum .
OPAL Collaboration, G. Alexander et al., .
V.M. Driesen, J.H. Kühn, and E. Mirkes, and refs. therein.
E. Braaten and Yu-Qi Chen, .
H. Fritzsch and J.H. Kühn, .
J.H. Kühn, A. Djouadi, and P.M. Zerwas, .\
K.G. Chetyrkin and J.H. Kühn, .
K.G. Chetyrkin, J.H. Kühn, and A. Kwiatkowski, [*QCD Corrections to the $e^+e^-$ Cross Section and the $Z$ Boson Decay Rate: Concepts and Results*]{}, TTP96-06, to be published in Physics Reports.
K.G. Chetyrkin and J.H. Kühn, .
K.G. Chetyrkin and O.V. Tarasov, .\
S.A. Larin, T. van Ritbergen, and J.A.M. Vermasseren, .
K.G. Chetyrkin and J.H. Kühn, .
K.G. Chetyrkin and J.H. Kühn, .
K.G. Chetyrkin, J.H. Kühn, and T. Teubner, Karlsruhe preprint TTP 96-35, hep-ph/9609411.
G. Källen and A. Sabry, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. [ **29**]{} (1955) No. 17; J. Schwinger, [*Particles, Sources, and Fields*]{}, Vol. II, (Addison-Wesley, New York, 1973).
A.H. Hoang, J.H. Kühn, and T. Teubner, . K.G. Chetyrkin, J.H. Kühn, and M. Steinhauser, , and preprint hep-ph/9606230, in print in Nucl. Phys. B.
J.H. Kühn, .\
J.H. Kühn, .
I. Bigi, Y.L. Dokshitzer, V.A. Khoze, J. Kühn and P. Zerwas, .
V.S. Fadin and V.A. Khoze, .\
J.M. Strassler and M.E. Peskin, .
M. Jeżabek, J.H. Kühn and T. Teubner, .
M. Jeżabek and T. Teubner, .
Y. Sumino, K. Fujii, K. Hagiwara, H. Murayama, C.-K. Ng, .
H. Murayama and Y. Sumino, .
R. Harlander, M. Jeżabek, J.H. Kühn and T. Teubner, .
R. Harlander, M. Jeżabek, J.H. Kühn, and M. Peter, hep-ph/9604328, Karlsruhe preprint TTP 95-48, in print in Z. Phys. [**C**]{}.
K. Fujii, T. Matsui, and Y. Sumino, .
N. Cabibbo and L. Maiani, .
M. Je[ż]{}abek and J. H. K[ü]{}hn, [*Nucl. Phys.*]{} [**B314**]{} (1989) 1.
M. Je[ż]{}abek and J. H. K[ü]{}hn, [*Nucl. Phys.*]{} [**B320**]{} (1989) 20.
A. Czarnecki, M. Je[ż]{}abek and J. H. K[ü]{}hn, [*Acta Phys. Pol.*]{} [**B20**]{} (1989) 961; E: ibid, [**B23**]{} (1992) 173.
A. Czarnecki, M. Je[ż]{}abek and J. H. K[ü]{}hn, [*Nucl. Phys.*]{} [**B351**]{} (1991) 70.
A. Czarnecki, M. Je[ż]{}abek, J. G. K[ö]{}rner, and J. H. K[ü]{}hn, .
A. Czarnecki, M. Je[ż]{}abek and J. H. K[ü]{}hn, [*Phys. Lett.*]{} [**B346**]{} (1995) 335.
M. Jeżabek and L. Motyka, hep-ph/9609352, [*Acta Phys. Pol.*]{} [ **B27**]{} (1996), in print.
A. Czarnecki and M. Je[ż]{}abek, [*Nucl. Phys.*]{} [**B427**]{} (1994) 3.
B. Mele and G. Altarelli, .
B. H. Smith and M. B. Voloshin, [*Phys. Lett.*]{} [**B340**]{} (1994) 176.
A. Czarnecki, [*Acta Phys. Pol.*]{} [**B26**]{} (1995) 845.
M. Luke, M.J. Savage, and M.B. Wise, .
A. Czarnecki, [*Phys. Rev. Lett.*]{} [**76**]{} (1996) 4124.
[^1]: The complete paper, including figures, is also available via anonymous ftp at ftp://ttpux2.physik.uni-karlsruhe.de/, or via www at\
http://www-ttp.physik.uni-karlsruhe.de/cgi-bin/preprints/
[^2]: Supported by BMBF contract 057KA92P.\
Invited talk presented at the “Cracow International Symposium on Radiative Corrections”, Cracow, Poland, 1996.
|
---
author:
- Taolue Chen
- Matthew Hague
- 'Anthony W. Lin'
- Philipp Rümmer
- Zhilin Wu
bibliography:
- 'references.bib'
title: 'Decision Procedures for Path Feasibility of String-Manipulating Programs with Complex Operations'
---
### Acknowledgments {#acknowledgments .unnumbered}
We are grateful to the anonymous referees for their constructive and detailed comments. T. Chen is supported by the Engineering and Physical Sciences Research Council under Grant No. [EP/P00430X/1]{} and the Australian Research Council under Grant No. [DP160101652, DP180100691]{}. M. Hague is supported by the Engineering and Physical Sciences Research Council under Grant No. [EP/K009907/1]{}. A. Lin is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no 759969). P. Rümmer is supported by the Swedish Research Council (VR) under grant 2014-5484, and by the Swedish Foundation for Strategic Research (SSF) under the project WebSec (Ref. RIT17-0011). Z. Wu is supported by the National Natural Science Foundation of China under Grant No. 61472474, 61572478, and 61872340, the INRIA-CAS joint research project “Verification, Interaction, and Proofs”.
|
---
abstract: 'The Soft X-ray Telescope (SXT) on board Yohkoh revealed that the ejection of X-ray emitting plasmoid is sometimes observed in a solar flare. It was found that the ejected plasmoid is strongly accelerated during a peak in the hard X-ray emission of the flare. In this paper we present an examination of the GOES X 2.3 class flare that occurred at 14:51 UT on 2000 November 24. In the SXT images we found “multiple” plasmoid ejections with velocities in the range of 250-1500 km s$^{-1}$, which showed blob-like or loop-like structures. Furthermore, we also found that each plasmoid ejection is associated with an impulsive burst of hard X-ray emission. Although some correlation between plasmoid ejection and hard X-ray emission has been discussed previously, our observation shows similar behavior for multiple plasmoid ejection such that each plasmoid ejection occurs during the strong energy release of the solar flare. As a result of temperature-emission measure analysis of such plasmoids, it was revealed that the apparent velocities and kinetic energies of the plasmoid ejections show a correlation with the peak intensities in the hard X-ray emissions.'
author:
- 'N. Nishizuka, H. Takasaki, A. Asai and K. Shibata'
title: 'Multiple Plasmoid Ejections and Associated Hard X-ray Bursts in the 2000 November 24 Flare'
---
Introduction
============
The Soft X-ray Telescope [SXT; @tsu91] on board Yohkoh [@oga91] revealed that a soft X-ray emitting plasma ejection, or plasmoid ejection, is sometimes observed in solar flares [e.g. @shi95]. It was also found that the plasmoids show blob-like or loop-like shapes and that the strong acceleration of the plasmoid ejection occurs during the peak time of the hard X-ray emission [@tsu97; @ohy97]. Their ejection velocities are typically several hundred km s$^{-1}$ and the ejected plasma is heated to more than 10 MK before the onset of the ejection [@ohy97; @ohy98]. They often start to rise up gradually a few tens of minutes before the onset of a hard X-ray burst and are then strongly accelerated just before or at the impulsive phase of the flare. A similar kinetic evolution is also seen in the case of coronal mass ejections [CMEs; @zha01; @kim05a; @kim05b].
Similarly, slowly drifting radio structures, observed at the beginning of the eruptive solar flares in the 0.6-1.5 GHz frequency range, have been interpreted as the radio signatures of plasmoid ejection [@kli00; @kha02; @karl02]. Hudson et al. (2001), moreover, identified a rapidly moving hard X-ray source associated with a moving microwave source and an X-ray plasmoid ejection. Kundu et al. (2001) also identified moving soft X-ray ejecta associated with moving decimetric/metric radio sources observed by the Nançay radioheliograph. Sui et al. (2004) also found a plasmoid ejection in hard X-ray images with RHESSI satellite.
In the standard model of solar flares, so-called CSHKP model [@car64; @stu66; @hir74; @kop76], a filament/plasmoid ejection is included. However, it does not necessarily stress the importance of the role of plasmoid ejection explicitly. Shibata et al. (1995) and Shibata & Tanuma (2001) extended the CSHKP model by unifying reconnection and plasmoid ejection and stressed the importance of the plasmoid ejection in a reconnection process. The model is called the “plasmoid-induced-reconnection” model. In that model, the plasmoid inhibits reconnection and stores magnetic energy in a current sheet. Then, once it is ejected, inflow is induced because of the mass conservation, resulting in the enhancement of reconnection rate and the acceleration of the plasmoid due to the faster reconnection outflow. Moreover, reconnection theories predict several plasmoids of various scales are generated. The dynamics of plasmoid formation in the solar flare and their subsequent plasmoid ejection affect the reconnection rate in the nonlinear evolution. Therefore, plasmoid ejections are observational evidence of magnetic reconnection of solar flares. Since plasmoid ejections have been observed in both long duration events and compact flares [@shi95], it is shown that the magnetic reconnection model may be applicable even for the compact flares that do not show the other typical features of the magnetic reconnection.
On the basis of the results of magnetohydrodynamic (MHD) numerical simulations, Kliem et al. (2000) suggested that each individual burst in the slowly pulsating structure is generated by suprathermal electrons, accelerated in the peak of the electric field in the quasi-periodic and bursting regime of the magnetic field reconnection. This is the so-called “impulsive bursty” reconnection [@pri85]. In that regime, several plasmoids can be formed successively as a result of the tearing and coalescence instabilities [@fin77; @kli00]. The repeated formation of magnetic islands can induce magnetic reconnection and their subsequent coalescence [@taj87]. These processes even have a cascading form: secondary tearing, tertiary tearing, and so on, always on smaller and smaller spatial scales [@tan01; @shi01]. Furthermore the formed plasmoids can merge into larger plasmoids. Tanuma et al. (2001) also showed that an increase in the velocity of the plasmoid ejection leads to an increase in the reconnection rate, and Bárta et al. (2008) analyzed the dynamics of plasmoids formed by the current sheet tearing.
The unsteady reconnection mentioned above can release a large amount of energy in a quasi-periodic way. The energy released in the upward direction can be observed as plasmoid ejections or coronal mass ejections, while that in the downward direction as downflows [@mck99; @asa04a] and impulsive bursts at the footpoints of the coronal loops. Bursty energy release in solar flares has been observed as highly time variable hard X-ray bursts and microwave bursts [@fro71; @den85; @kan70; @kip83; @ben92; @asc02]. Benz & Aschwanden (1992) and Aschwanden (2002) argued that these impulsive bursts suggest the existence of highly fragmented particle acceleration regions. This fragmented structure of solar flares indicates that a flare is an ensemble of a vast amount of small-scale energy releases and the fractal/turbulent structure of the current sheet can be expected [see also @nis09]. Recently the kinematics of multiple plasmoids have been studied by full-particle simulations and how the particles interact with their surroundings has been explained [e.g. @dra06; @pri08; @dra09; @dau09]. It is interesting to note that the stochastic acceleration mechanism [e.g. @ben87; @bro85; @mil96; @liu06] may be related to particle acceleration in fractal/turbulent current sheet [see also @str88; @kow09; @sam09].
Karlický et al. (2004) showed a unique series of slowly drifting structures during one flare, from which the authors proposed that it indirectly maps a formation of several plasmoids and their interactions. However in most of the previous studies, only one plasmoid or one drifting structure was reported during the solar flare. In this paper, we present for the first time the direct observations of multiple X-ray emitting plasmoid ejections associated with a single solar flare observed by Yohkoh/SXT (firstly reported by Takasaki 2006). In section 2, we describe the multiple plasmoid ejection events. Then we analyzed the data in section 3 by examining in detail the relationship between the multiple plasmoid ejections and the nonthermal hard X-ray emissions using Yohkoh data. Finally we discuss the dynamic features of magnetic reconnection and the roll of plasmoid ejections in the particle acceleration in a solar flare in section 4.
Observation
===========
Overview
--------
A series of homologous flare-CME events occurred in NOAA Active Region 9236 from 2000 November 24 to November 26. These events have been reported by several researchers. Nitta & Hudson (2001) found that the CME-flare events of the homologous flares show quite similar characteristics in both their coronal/photospheric magnetic structures and their CME properties. Zhang & Wang (2002) compared the homologous flares in detail through the use of multiwavelength observations. Wang et al. (2002) reported that the activities of these flares was driven or triggered by newly emerging magnetic flux, which appeared on the western side of the leading sunspot in this active region. Figure 1a-1c show snapshot images of the preceding sunspot in NOAA 9236 and an associated two-ribbon flare, which occurred on 2000 November 24 observed in white light, ultraviolet and soft X-ray emission. Takasaki et al. (2004) performed a comparison of the physical parameters between the individual flares and from this they could confirm that the plasmoid-induced-reconnection model is reasonable. They then showed that the interaction between the new emerging magnetic flux loops and the pre-existing magnetic field was essential for producing the homologous flares and plasmoid ejections in the active region.
These ejections were followed by a single halo-CME which occurred at 15:30 UT on 2000 November 24. Figure 2 shows a CME image observed with the Large Angle Spectroscopic Coronagraph [LASCO; @bru95] that occurred following the flare [e.g. see @nit01; @zha02]. The core of the CME was observed traveling in the northwest direction. Moon et al. (2003) found a good correlation between CME speed and the GOES X-ray peak flux of the associated flares in this series of homologous flare-CME events.
Impulsive hard X-ray bursts were also observed in this flare with the Hard X-ray Telescope [HXT; @kos91] on board Yohkoh (see Fig. 1d). A pair of hard X-ray sources was located at the footpoints of the coronal arcade. We used the hard X-ray emission data observed with the H-band (52.7-92.8 keV) of HXT, whose temporal resolution was 0.5 s.
Multiple Plasmoid Ejections
---------------------------
We focus on a GOES X2.3 class flare that occurred in NOAA 9236 (N19$^{\circ}$, W06$^{\circ}$) at 14:51 UT on 2000 November 24. This flare was one of the homologous flares which were described previously. In this flare, we observed seven plasmoid ejections in the soft X-ray images of the flare taken with Yohkoh/SXT. We mainly used the partial frame images with half- and quarter-resolution for the analysis. The spatial resolutions are about 5” and 10”, respectively. We used the sandwich (AlMg) filter images which were taken with 20 second cadence. Figure 3 and Supplement Movie 1 show the temporal evolution of the flare, which is made of full-, half- and quarter-resolution images taken with Yohkoh/SXT AlMg filter (negative images). Full-, half- and quarter-resolution images are different in their spatial resolutions, field of views and exposure times. Full-resolution images are of short exposure time and focus on the brightest region of the active region, such as the two ribbon structure. On the other hand, quarter-resolution images are of longer exposure time so that they are applicable for the detection of large-scale and faint phenomena such as plasmoid ejections. The black vertical line in the middle of Figure 3 shows the saturation of the quarter-resolution images. We identified seven major ejections which we named P1-P7. In Figure 4 (Supplement Movie 2), Figure 5 and Figure 6, we marked each plasmoid ejection with a circle. Each plasmoid ejection can be seen more clearly in quarter-resolution images of Figure 5, while half-resolution images of Figure 6 are convenient to see plasmoids just after ejections. The fields of view of quarter- and half-resolution images are also shown in Figure 3.
Figure 7 shows the temporal evolution of soft X-ray emission observed with SXT. A full resolution image is inset on Figure 7a. Figure 7a-7c shows three of the plasmoid ejections denoted as P1, P4 and P7. In Figure 7d-7f we overlaid contour images of the soft X-ray emission, which show the time evolution of the plasmoid ejection. The directions of the ejections are indicated by the arrows in the panels. To make clear the traveling of the plasmoids, we also overlaid the contours of the SXR images taken at different times (e.g. 15:09:19 UT, 15:09:39 UT and 15:09:59 UT for Fig. 7d) on Figures 7d, 7e, and 7f. From these contour images, we can roughly outline the position and the size of the bright cores of the plasmoids. We can also measure the velocity $V$ by taking the time difference of these contour images. Those size and apparent velocity of the plasmoids are listed in Table 1. Here we note that these are the “apparent” velocities, and the motions in the line of sight are ignored. Therefore, the actual velocity of the plasmoid $V/\cos\theta$ will be greater than the plane-of-the-sky value, where $1/\cos\theta$ representing the expected deprojection over a reasonable range of angles $\theta$ to the plane of the sky. The trajectory of the plasmoid is not necessarily in the straight lines as shown with the arrows in Figure 7. In order to take into account those non-straight motions, we measured the apparent velocity of each plasmoid by averaging the velocities derived from each time differences. In the following discussions, we consider the apparent velocity as the actual velocity.
We can see several plasmoids were ejected in the northwest direction, and one plasmoid ejected in the southwest direction in Figure 4 (supplement movie 2) and Figure 5, which is marked as P6 in this paper. Each plasmoid has a unique velocity, brightness and size. The first ejection was that of P1, which was followed by P2-P4 which were successively ejected as a group in the same direction. P1 shows blob-like structure, while P2-P4 seem to be a part of an expanding loop. We are not able to clearly define the structure of P5 and P6 due to the faint emission, although we can surely identify P5 and P6 traveling outward from the active region. P7 is the brightest ejection of blob-like structure. It starts to rise up gradually at 15:12 UT and ejected/accelerated upwards at 15:14 UT.
In the LASCO CME images we can no longer identify the fine structure corresponding to the individual plasmoid ejections, though some complicated structures can be observed (see Fig.2). This is probably because the ejected plasmoids merge into a single CME. The average velocity of the CME listed in the CME event catalog[^1] is about 1245 km s$^{-1}$. This apparent velocity of the CME is faster than those of the plasmoid ejections P1-P7 summarized in Table 1. This observational fact qualitatively suggests that the merged plasmoids are continuously accelerated as they are ejected into interplanetary space, as shown by Cheng et al. (2003), although we can hardly identify the one-to-one relation between them.
Analysis and Results
====================
Time Slice Images of Plasmoid Ejections and Comparison to the Hard X-ray Bursts
-------------------------------------------------------------------------------
Here we focus on the time evolution of the plasmoid ejections. We used a time slice image of the plasmoid ejections as we show in Figure 8. The horizontal axis is the time from 15:06 UT to 15:18 UT, and the vertical axis is the 1D image (negative images) using a slit line placed along the direction of the several plasmoid ejections (P1, P3, P4, P5 and P7). This time slice image is made of half-resolution images. The position of the slit line is shown in Figure 6 and 7a. Those ejections seen in the time slice image are marked with signs $\square$, $\triangle$ $\dots$ in Figure 8b. We can also identify further additional faint ejections in the time slice image. Some ejections travel along the slit lines, while others travel on a path which is slightly different from the slit line. As a result, the visibility of each plasmoid ejection is different.
Initially P1, P2 and P3 are slowly accelerated, then strongly accelerated during the initial impulsive phase of the hard X-ray emission (15:07:40-15:08:40 UT) followed closely by the ejection of P4. A group of plasmoids gradually rise up 15:09:40- 15:12:20 UT followed by the faint ejection P5. P6 is ejected in a different direction (southwest) and does not cross the slit line, so it does not appear in Figures 8a,b. P7 is the brightest ejection and the most clearly visible in Figures 8a,b. The apparent velocities of the plasmoids along the slit can also be derived from the slopes of the fitted lines in Figure 8b. We note that these are the apparent velocities measured from the time slice images of Figure 8b and different from the velocities measured from contour plot images in Figure 7. This is because the former shows the front velocity of thinner density plasma, while the latter shows the velocity of the thick core part of the plasmoids. In Figure 8a and 8b, the plasmoid ejections roughly start to rise at the apparent height of approximately 50” ($\sim$35 Mm) and propagate into the upper atmosphere. This probably means that reconnection occurs at around or just below the height of $\sim$35 Mm. In this paper, we set the start position of plasmoid ejections as the height ($\sim$35 Mm) and define the start times of ejections as the time when each plasmoid crosses the height (which is shown with dotted line in Fig. 8b). It is noted that Figure 6 and Figure 8a,b are drawn with half-resolution images, but Figure 5 is a series of quarter-resolution images. Therefore we see more of the earlier phase of plasmoid ejection in Figure 6 and 8 than in Figure 5, and so Figure 8 is more appropriate to determine the time of plasmoid ejection. We confirmed that the start times are comparable to those defined above.
In Figure 8c, we show the light curve of the hard X-ray emission obtained with the H-band (52.7-92.8 keV) of Yohkoh/HXT and the GOES soft X-ray light curve. We can distinguish the hard X-ray bursts into three separate periods: the first period, A (15:08-15:09 UT), is the brightest phase of the hard X-ray emission, the second period, B (15:09-15:11 UT), show gradual enhancement and the last period, C ($\sim$15:14 UT), is an isolated hard X-ray peak. The plasmoid ejections of P1-P3 seem to be ejected during the peak time of period A. P1 seems to be ejected just before the hard X-ray peaks in Figure 8c. This is consistent with the report by Ohyama & Shibata (1997) who showed that plasmoids are ejected at or just before the hard X-ray peak. During period B, a group of plasmoids gradually rise up and are followed by the faint ejection of P4 and P5, while the brightest plasmoid P7 is ejected during period C. The correspondence of the plasmoid ejections and hard X-ray peaks are shown by the arrows in Figure 8b. Since the bursts in period A are superposed and very complex, it is difficult to identify the exact correspondence between the plasmoid ejections (P1-P3) and the hard X-ray bursts.
We made a correlation plot of the times of the hard X-ray bursts and those of the plasmoid ejections (Figure 9a) to make the relation clearer. Both the horizontal and vertical axes show the times (UT). The horizontal ([*light gray*]{}) and the vertical ([*dark gray*]{}) lines illustrate the times of the plasmoid ejections and those of the hard X-ray bursts, respectively. The thickness of the lines shows an estimation of the error. Figure 9b shows hard X-ray (52.7-92.8 keV) light curves obtained with H-band of Yohkoh/HXT. The thin solid line shows the points where the times of the plasmoid ejections correspond to those of the hard X-ray bursts. These results appear to show that several plasmoid ejections coincide with hard X-ray bursts.
Here we note that, although the intense HXR emissions indicate that strong energy releases occur at those times, it does not necessarily mean that certain amount of plasmoids, that is, notable plasmoids are ejected. We have already noticed that there are many fainter ejections, which also tend to appear at HXR bursts, while some of the hard X-ray intensity fluctuations have no associated ejections in Figures 8 and 9. On the other hand, the HXR burst for P5 and P6 does not show a sharp summit but a gentle hump. This is probably because ejections P5 and P6 are parts of continuous outflows from the active region during this time range. This may also suggest a milder energy release compared with the others, resulting into a gradual enhancement of hard X-ray emission between 15:09:40 UT and 15:11:00 UT.
Temperature Diagnostics of Seven Plasmoid Ejections
---------------------------------------------------
We also studied temperature diagnostics on the soft X-ray emitting plasma using the SXT filter ratio method [@har92]. We used the half-resolution images taken with the Be and thick Al filters of SXT for the analysis of plasmoid ejections. We subtracted the background photon flux of plasmoids. The temperature and emission measure are determined using the two filter data of the Be and thick Al filters. The size of the plasmoid $S$ was measured from the contour plot images in Figure 7b, which shows the lower limit of the observable size (summarized in Table 1). We assume that the X-ray emitting plasma, measured by the filter ratio method, fills the plasmoid with a filling factor of 1. We assumed that the volume of the plasmoid is $S^{3/2}$, such that the line of sight width of a plasmoid is equal to the square-root of the size $S$. Since the observation times of the Be and thick Al filters are not exactly the same, we used two images from the thick Al filter which were taken just before and just after the Be filter observation. The error shown in Table 1 mostly results from combining these two images. This error is much greater than the background photon noise [see @ohy97 in more detail]. As for plasmoid ejections P2 and P3, there are no Be and thick Al filter images, because P2 and P3 are out of field of view of half-resolution images as shown in Figure 6. In these cases we assumed the temperatures of P2 and P3 as $10^7$ K.
Table 1 summarizes the physical parameters, such as temperature, emission measure, density, mass, thermal energy and kinetic energy of each plasmoid, of the seven plasmoid ejections identified in Figure 5. Since the size of the plasmoids that we derived from the images is just the lower limit, the mass, thermal and kinetic energies calculated with the size are also the lower limit. In Table 1, each plasmoid shows a typical temperature of 10$^7$ K, a density of 10$^9$ cm$^{-3}$ and an apparent velocity of 200-1400 km s$^{-1}$, which is similar to results of previous studies [i.e. @ohy97; @ohy98]. The kinetic energy of each plasmoid ejection seems to be comparable to or twice as large as their thermal energy. We also estimate the total flare energy from the full-resolution images (spatial resolution 2.5") of the same filters at the peak time of GOES soft X-ray emission. Since the only part of the total flare energy is converted into plasmoid ejections, the total energy of plasmoid ejections is smaller than the total energy of solar flare.
Summary and Discussion
======================
We analyzed the GOES X2.3 class flare which occurred at 14:51 UT on 2000 November 24. We found multiple plasmoid ejections from a single flare. Furthermore, each plasmoid ejection seems to be associated with a peak in the hard X-ray emission. In Figure 8, the plasmoid ejections seem to occur at the height of approximately 50" ($\sim$ 35 Mm). This tells us that reconnection occurs at the height of approximately 35 Mm. The horizontal light gray lines in the top panel of Figure 9 show that the times of plasmoid ejections, when they reach the height of 35 Mm, seems to be well correlated to the peak time of hard X-ray emission, which is consistent with previous studies [e.g. @ohy97]. Since the peak in hard X-ray emission indicates strong energy release, we have demonstrated that each plasmoid ejection occurs during a period of strong energy release, suggesting a series of impulsive energy releases in a single flare.
We also performed a temperature and emission measure analysis and investigated the physical parameters of the plasmoid ejections shown in Table 1. Figure 8 and Table 1 show that the hard X-ray bursts in period A have large intensities in correlation with the large kinetic energy of plasmoid ejections P1-P4. Conversely, the hard X-ray bursts in period B and C have small intensities in correlation with the kinetic energy of plasmoid ejections P5-P7. Figure 10a shows the relation between the kinetic energy of plasmoid ejection and the intensity of the corresponding hard X-ray peak emission. We can see a rough tendency that the larger kinetic energy of plasmoid ejections is associated with the brighter hard X-ray peak emission, and vice versa. The hard X-ray emission is known to show energy release rate [e.g. Neupert effect; @neu68; @asa04b], which leads us to the following equation: $$I_{HXR} \Delta t \sim \frac{dI_{SXR}}{dt} \Delta t \propto \frac{dE_{th}}{dt} \Delta t \sim \Delta E_{th} \sim \Delta E_{kin}
\sim \frac{1}{2}mV_{pl}^2$$ where we assumed that the released thermal energy of a solar flare in a short period is comparable to the kinetic energy of the plasmoid ejection. This means that a plasmoid ejection with large hard X-ray emission, and therefore with large energy release, can be accelerated strongly. A similar kinetic evolution is also seen in the case of CMEs [@yas09].
Figure 10b shows the relation between the apparent velocity and the intensity of the corresponding hard X-ray peak emissions. There seems to be a correlation between the plasmoid velocities and the hard X-ray emissions, although it is difficult to measure the velocities of the plasmoids precisely due to the faint emission. Similar to the above equations, we can derive the following relation between hard X-ray emission and the kinetic energy of a plasmoid ejection: $$I_{HXR} \sim \frac{dI_{SXR}}{dt} \propto \frac{dE_{th}}{dt} \sim \frac{B^2}{4\pi}v_{in}L^2 \propto V_{CME} \sim V_{pl}$$ Here $B$ is the typical magnetic field in a current sheet, $v_{in}$ is the inflow velocity and $L$ is the characteristic length of the inflow region. The inflow velocity $v_{in}$ is thought to be about 0.01$v_{A}$ from direct observations [@yok01; @nar06]. It has been also known that inflow can be controlled by the plasmoid ejection. As a plasmoid is ejected out of the current sheet, the density in the current sheet decreases, and the inflow is enhanced to conserve the total mass under the condition that incompressibility is approximately satisfied. Here, we assume that $v_{in}$ is proportional to the plasmoid velocity $V_{pl}$. Then, we find the relation $I_{HXR}$ $\propto$ $V_{pl}$. Moreover, if we can further assume the $V_{pl}$ is proportional to CME velocity $V_{CME}$, then $I_{HXR}$ $\propto$ $V_{CME}$. This is consistent with the result of Yashiro & Gopalswamy (2009). A correlation between the energy release rate, which is represented by $I_{HXR}$, and the plasmoid velocity $V_{pl}$ is also successfully reproduced by a magnetohydrodynamic simulation [@nisd09]. In the simulation, they clearly showed that the plasmoid velocity controls the energy release rate (i.e. reconnection rate) in the nonlinear evolution.
Observations of plasmoid ejections have been paid attention to as evidence of magnetic reconnection, though they found only one plasmoid ejection per one solar flare. However, magnetic reconnection theory suggests that the impulsive bursty regime of reconnection or fractal reconnection is associated with a series of plasmoids of various scales. It is known that magnetic reconnection is an effective mechanism for energy release in a solar flare. Once the current sheet becomes thin enough for the tearing instability to occur, repetitive formation of magnetic islands and their subsequent coalescence drives the “impulsive bursty” regime of reconnection [@fin77; @taj87; @pri85]. Furthermore, such reconnection can produce a fractal structure in the current sheet, which is not only theoretically predicted but has also been observed [i.e. @shi01; @nis09]. The plasmoids generated in this cycle control the energy release by inhibiting magnetic reconnection in the current sheet and/or by inflow driven by the ejection. The plasmoid ejection enhances reconnection and promotes further plasmoid ejections from the current sheet. Similar intermittent energy release from a solar flare has been reported as multiple downflows associated with hard X-ray bursts in the impulsive phase [@asa04a]. McKenzie et al. (2009) estimated the total amount of energy of multiple downflows and showed that it is comparable to the total amount of energy released from the magnetic field, which is consistent with the magnetic reconnection model. Tanuma et al. (2001) showed through numerical simulations that plasmoid ejections are closely coupled with the reconnection process and the greatest energy release occurs when the largest plasmoid is ejected. If the strong energy release corresponds to magnetic reconnection, we may conclude that this is evidence of unsteady magnetic reconnection in a solar flare and that plasmoids have a key role in energy release and particle acceleration.
We first acknowledge an anonymous referee for his/her useful comments and suggestions. We also thank A. Hillier for his careful reading and correction of this paper. We also thank J. Kiyohara for her help in making the movies and T. Morimoto for his help in finding the multiple plasmoid ejection events. We made extensive use of YOHKOH/SXT and HXT data. This work was supported in part by the Grant-in-Aid for Creative Scientific Research “The Basic Study of Space Weather Prediction” (Head Investigator: K.Shibata) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan, and in part by the Grand-in-Aid for the Global COE program ”The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan.\
Asai, A., Yokoyama, T., Shimojo, M., & Shibata, K., 2004a, , 605, L77 Asai, A., Yokoyama, T., Shimojo, M., Masuda, S., Kurokawa, H. & Shibata, K., 2004b, , 611, 557 Aschwanden, M. J. 2002, Particle Acceleration and Kinematics in Solar Flares (Dordrecht: Kluwer) Bárta, M., Vršnak, B., & Karlický, M. 2008, A&A, 477, 649 Benz, A. O., & Smith, D. F., 1987, Sol. Phys., 107, 299 Benz, A. O. & Aschwanden, M. J. 1992, in Lect. Notes in Physics, Vol. 399: Eruptive Solar Flares, eds. Z. Svestka, B. V. Jackson, & M. E. Machado (New York: Springer), 106 Brueckner, G. E. et al., 1995, , 162, 357 Brown, J. C., & Loran, J. M., 1985, , 212, 245 Carmichael, H., 1964, in The Physics of Solar Flares, ed. W. N. Hess (NASA SP-50; Washington, DC: NASA), 451 Cheng, C. Z., Ren, Y., Choe, G. S., & Moon, Y. -J., , 2003, 596, 1341 Daughton, W., Roytershteyn, V., Albright, B. J., Karimabadi, H., Yin, L., Bowers, K. J., 2009, , 103, 065004 Dennis, B. R., 1985, Sol. Phys., 100, 465 Drake, J. F., Swisdak, M., Che H., & Shay, M. A., 2006, Nature, 433, 553 Drake, J. F., Cassak, P. A., Shay, M. A., Swisdak, M., & Quataert, E., 2009, , 700, L16 Finn, J. M. & Kaw, P. K. 1977, Phys. Fluids, 20, 72 Frost, K. J. & Dennis, B. R., 1971, , 165, 655 Handy, B. N., et al. 1999, , 187, 229 Hara, H., Tsuneta, S., Lemen, J. R., Acton, L. W., McTiernan, J. M., 1992, PASJ, 44, L135 Hirayama, T., 1974, , 34, 323 Hudson, H. S., Kosugi, T., Nitta, N. V., & Shimojo, M., 2001, , 561, L211 Kane, S. R., & Anderson, K. A., 1970, , 162, 1003 Karlický, M., Fárník, F., & Mészárosová, H. 2002, A&A, 395, 677 Karlický, M., Fárník, F., & Kurucker, S., 2004, A&A, 419, 365 Khan, J. I., Vilmer, N., Saint-Hilaire, P., & Benz, A. O. 2002, A&A, 388, 363 Kim, Y.-H., Moon, Y.-J., Cho, K.-S., Kim, K.-S., & Park, Y. D., 2005a, , 622, 1240 Kim, Y.-H., Moon, Y.-J., Cho, K.-S., Bong, S.-C., & Park, Y. D., 2005b, , 635, 1291 Kiplinger, A., L., Dennis, B. R., Emslie, A. G., Frost, K. J., & Orwig, L. E., 1983, , 265, L99 Kliem, B., Karlický, M. & Benz, A. O. 2000, A&A, 360, 715 Kopp, R. A., & Pneuman, G. W., 1976, , 50, 85 Kosugi, T. et al. 1991, Sol.Phys., 136, 17 Kowal, G., Lazarian, A., Vishniac, E. T., & Otmianowska-Mazur, K., 2009, , 700, 63 Kundu, M. R., Nindos, A., Vilmer, N., Klein, K. -L., Shibata, K., Ohyama, M., 2001, , 559, 443 Liu, S., Petrosian, V., & Mason, G. M., 2006, , 636, 462 McKenzie, D. E. & Hudson, H. S. , 1999, 519, L93 McKenzie, D. E., & Savage, S. L., 2009, , 697, 1569 Miller, J. A., LaRosa, T. N., & Moore, R. L. 1996, , 461, 445 Moon, Y.-J., Choe, G. S., Wang, H., Park, Y. D., & Cheng, C. Z., 2003, Korean Astron. Soc., 36, 61 Narukage, N., & Shibata, K., 2006, , 637, 1122 Neupert, W. M., 1968, , 153, L59 Nitta, N. V. & Hudson, H. S., 2001, Geophys. Res. Lett., 28, 3801 Nishida, K., Shimizu, M., Shiota, D., Takasaki, H., Magara, T., & Shibata, K., 2009, , 690, 748 Nishizuka, N., Asai, A., Takasaki, H., Kurokawa, H., & Shibata, K., 2009, , 694, L74 Ogawara, Y., Takano, T., Kato, T., Kosugi, T., Tsuneta, S., Watanabe, T., Kondo, I., Uchida, Y., 1991, , 136, 1 Ohyama, M. & Shibata, K. 1997, Publ. Astron. Soc. Japan, 49, 249 Ohyama, M. & Shibata, K. 1998, , 499, 934 Priest, E. R., 1985, Rep. Prog. Phys. 48, 955 Pritchett, P. L. 2008, Phys. Plasma, 15, 102105 Samtaney, R., Loureiro, N. F., Uzdensky, D. A., Schekochihin, A. A., & Cowley, S. C., 2009, , 103, 105004 Shibata, K., Masuda, S., Shimojo, M., Hara, H., Yokoyama, T., Tsuneta, S., Kosugi, T., Ogawara, Y., 1995, , 451, L83 Shibata, K. & Tanuma, S., 2001, Earth Planets Space, 53, 473 Strauss, H. R., 1988, , 326, 412 Sturrock, P. A., 1966, Nature, 211, 695 Sui, L., Holman, G. D., & Dennis, B. R., 2004, , 612, 546 Tajima, T., Sakai, J., Nakajima, H., Kosugi, T., Brunel, F., Kundu, M. R., 1987, , 321, 1031 Takasaki, H., Asai, A., Kiyohara, J., Shimojo, M., & Terasawa, T., Takei, Y., Shibata, K., 2004, , 613, 592 Takasaki, H., 2006, PhD thesis, Kyoto University Tanuma, S., Yokoyama, T., Kudoh, T. & Shibata, K. 2001, , 551, 312 Tsuneta, S. et al. 1991, , 136, 37 Tsuneta, S., 1997, , 483, 507 Wang, H., Gallagher, P., Yurchyshyn, V., Yang, G., & Goode, P. R., 2002, , 569, 1026 Yashiro, S. & Gopalswamy, N., 2009, in Proc IAU Symp. 257, Universal Heliophysical Processes, eds. N. Gopalswamy & D. F. Webb (Cambridge: Cambridge Univ. Press), 233 Yokoyama, T., Akita, K., Morimoto, T., Inoue, K., Newmark, J., 2001, , 546, L69 Zhang, J., Dere, K. P., Howard, R. A., Kundu, M. R., & White, S. M., 2001, , 559, 452 Zhang, J. & Wang, J., 2002, , 566, L117
[ccccccccccccrl]{} & (UT) & (K) & (cm$^{-3}$) & (10$^{18}$cm$^2$) & (10$^9$cm$^{-3}$) & (10$^{13}$g) & (km/s) & (10$^{28}$erg) & (10$^{28}$erg) & & (CTS/sec/sc)\
P1 & 15:09:19-15:09:59 & 6.98-7.09 & 45.32-45.33 & $>$3.9-4.7 & 2.76-2.79 & $>$2.63-2.66 & 654-818 & $>$5.63-8.91 & $>$2.51-3.28 & NorthWest & 1000\
P2 & 15:09:59-15:10:59 & (7.00) & (45.34) & $>$4.6-5.8 & (2.65) & $>$(3.69) & 573-654 & ($>$6.05-7.88) & ($>$3.69) & NorthWest & 1300\
P3 & 15:10:19-15:10:59 & (7.00) & (45.32) & $>$4.7-8.3 & (2.59) & $>$(3.60) & 982-1470 & ($>$17.4-38.9) & ($>$3.60) & NorthWest & 1500\
P4 & 15:10:39-15:11:19 & 7.02-7.07 & 45.80-45.80 & $>$3.1-7.1 & 4.85 & $>$4.30 & 654-828 & $>$9.19-14.7 & $>$4.50-5.05 & NorthWest & 950\
P5 & 15:13:33-15:14:27 & 7.02-7.08 & 45.73-45.74 & $>$1.5-3.1 & 5.76-5.83 & $>$1.12-1.13 & 360-844 & $>$0.72-4.03 & $>$1.17-1.36 & NorthWest & 120\
P6 & 15:13:33-15:13:51 & 7.07-7.11 & 45.49-45.50 & $>$1.6-5.2 & 4.37-4.42 & $>$0.85-0.86 & 360-720 & $>$0.55-2.22 & $>$0.99-1.11 & SouthWest & 130\
P7 & 15:16:27-15:18:43 & 6.97-7.03 & 46.05-46.06 & $>$3.2-7.8 & 6.95-7.03 & $>$3.99-4.04 & 254-409 & $>$1.29-3.38 & $>$3.72-4.33 & NorthWest & 150\
Total & - & - & - & - & - & - & - & 43.5-84.7 & 20.2-22.4 &&&&\
Flare & 15:18:33-15:18:53 & 6.73-7.08 & 48.27-48.96 & 9.33 & 137-304 & 391-865 & - & - & 210-1040 &&&&\
[^1]: CME event catalog: http://cdaw.gsfc.nasa.gov/CME\_list/
|
[**A UNIFIED CONFORMAL MODEL\
FOR FUNDAMENTAL INTERACTIONS\
WITHOUT DYNAMICAL HIGGS FIELD**]{}
[**Marek Pawłowski**]{}$^{\ast\dagger}$\
Soltan Institute for Nuclear Studies, Warsaw, POLAND\
[**and** ]{}\
[**Ryszard Raczka**]{}$^\ast\ddagger$\
Soltan Institute for Nuclear Studies, Warsaw, POLAND\
and\
Interdisciplinary Laboratory for Natural and Humanistic Sciences\
International School for Advanced Studies (SISSA), Trieste, ITALY\
\
[July,$\;\;\;$1994]{}
-12cm
**Abstract**
A Higgsless model for strong, electro–weak and gravitational interactions is proposed. This model is based on the local symmetry group $SU(3)\times SU(2)_{L}\times U(1)\times C$ where $C$ is the local conformal symmetry group. The natural minimal conformally invariant form of total lagrangian is postulated. It contains all Standard Model fields and gravitational interaction. Using the unitary gauge and the conformal scale fixing conditions we can eliminate all four real components of the Higgs doublet in this model. However the masses of vector mesons, leptons and quarks are automatically generated and are given by the same formulas as in the conventional Standard Model. In this manner one gets the mass generation without the mechanism of spontaneous symmetry breaking and without the remaining real dynamical Higgs field. The gravitational sector is analyzed and it is shown that the model admits in the classical limit the Einsteinian form of gravitational interactions.
.5cm
------------------------------------------------------------------------
Introduction
============
The recent evidence for top quark production with the top mass estimated as $m_t=174\pm 10^{+13}_{-12} GeV$ [@fnal] implies that the Higgs particle – if exists – may have the mass of the order of 1TeV: in fact the central value of $m_H$ implied by the present data of $m_t$ and $m_W$ was estimated by Hioki and Najima [@hioki] at $m_H\approx 1700 GeV$ with an enormous error however. Since in the lowest order $\lambda={1\over2} ({m_H\over v })^2$ one can afraid that the Higgs self-coupling $\lambda$ would be also very large ($\lambda\approx 25$ for the central value of $m_H$ given by Hioki and Najima). Such strong Higgs self-interaction would mean that the loops with Higgs particles would dominate all other contributions. Therefore the perturbative predictions in Standard Model(SM) for many quantities become unreliable. Hence the predictive power of the SM and its consistency may be questionable
The Higgs particle with such a large mass becomes suspicious. It is natural therefore to search for a modification of SM in which all confirmed by experiment particles would exist but the Higgs particle as the observed object would be absent.
We show in this work that such a modification of SM is possible under the condition that one joints to strong and electro–weak interactions also the gravitational interaction. This extension of the class of SM interactions is in fact very natural. Indeed whenever we have the strong and electro–weak interactions of elementary particles, nuclea, atoms or other objects we have also at the same time the gravitational interactions. It seems natural therefore to consider an unified model for strong, electro–weak and gravitational interactions which would describe simultaneously all four fundamental interactions. It is well known that gravitational interactions give a negligible effect to most of strong or electro–weak elementary particle processes. We show however that they may play the crucial role in a determination of the physical fields and their masses in the unified model and that their presence allows to eliminate all Higgs fields from the final lagrangian.
In turn we recall that in the conventional Standard Model the Higgs mechanism of spontaneous symmetry breaking (SSB) provides a simple and effective instrument for mass generation of weak gauge bosons, quarks and leptons. However, despite of many efforts of several groups of experimentalists [@LEP] the postulated Higgs particle of the SM was not observed. Hence one might expect that the model for strong and electro–weak interactions supplemented by the gravitational interaction in which all dynamical Higgs fields may be eliminated can provide a natural frame–work for a description of elementary particle fundamental interactions.
In order to construct a new form of total lagrangian for the theory of strong and electro–weak interactions extended by the gravitational interactions we observe that the gauge symmetry $SU(3)\times SU(2)_L \times U(1)$ of the fundamental interactions may be naturally extended by the local conformal symmetry. The choice of the unitary gauge condition for $SU(2)_L$ gauge group allows to eliminate the three out of four Higgs fields from the complex Higgs doublet. In turn the choice of the scale fixing condition connected with the local conformal symmetry allows to eliminate the last Higgs field. In that manner all four Higgs fields can be gauged away completely! It is remarkable that in spite of the elimination of all Higgs fields in our model the vector meson, lepton and quark masses are generated and at the tree level they are given by the same analytical formulas as in the conventional SM.
Thus it may be that the dynamical real Higgs field and the associated Higgs particles are in fact absent and it is therefore not surprising that they could not be detected in various experiments [@LEP].
We review in Section 2 the present problems with a very massive Higgs particle. Next in Section 3 we discuss the properties of local conformal symmetry and its representations in field space of arbitrary spin. We present in Section 4 the form of the total lagrangian of our unified theory of electro–weak, strong and gravitational interactions determined by the gauge and the local conformal invariance. The noteworthy feature of the obtained lagrangian is the lack of the Higgs mass term $\mu^2 \Phi^\dagger\Phi$. We show next that using the unitary gauge condition and the conformal scale fixing condition we can eliminate all dynamical Higgs fields from the theory! We show in Section 5 that in spite of the lack of dynamical Higgs fields the masses of vector mesons, leptons and quarks are generated and at the tree level are given by the same analytical expressions in terms of coupling constants as in the conventional SM.
We stress that the renormalizability of our model depends on the value of the new coupling constant $\beta$ which determines the properties of gravitational sector. We discuss in Section 6 the variant of our model with $\beta\not=0$. This leads to the model with massive vector mesons which is nonrenormalizable. In order to get definite perturbative predictions – especially for electro–weak processes – we have to introduce the ultraviolet cutoff $\Lambda$. We show the close connection between the large Higgs mass $m_H$ and $\Lambda$. We illustrate this relation in the case of universal electro–weak parameters $\varepsilon_{N1}$, $\varepsilon_{N2}$ and $\varepsilon_{N3}$ of Altarelli [*et al.*]{} [@altarelli] for which we show that the difference between SM results for $\varepsilon_{Ni}$ and in our model is essentially proportional to $\log {\Lambda^2\over m_H^2}$; thus if one chooses $\Lambda\cong m_H$ one obtains the same analytical formulas for $\varepsilon_{Ni}$ in SM and in our Higgsless model. We show also how using so called General Equivalence Theorems one can calculate the high energy limit for various processes in our model.
We present in Section 7 the analysis of the gravitational sector in the unified model. We show that our unified model after determination of the unitary gauge and scale fixing leads already at classical level to the conventional gravitational theory with Einstein–Hilbert lagrangian implied by the conformal Penrose term contained in the unified lagrangian.
We present in Section 8 the special version of our model (with $\beta=0$) which may lead to perturbatively renormalizable model of fundamental interactions. We discuss shortly some open problems of this formulation of the unified theory.
Finally we discuss in Section 9 three alternatives for a description of fundamental interactions which are given by the conventional SM or its extensions, Higgsless renormalizable SM and nonrenormalizable Higgsless models. We discuss also some open problems connected with derivation of predictions in low and high energy regions from nonrenormalizable Higgsless models.
The present work is the extension of our previous paper [@babbage] and contains the answer to several questions raised by its readers.
Difficulties with Standard Model Higgs particle.
================================================
We shall argue that the recently announced [@fnal] evidence for the top quark with the mass
$$m_t=174\pm 10^{+13}_{-12} GeV \eqno (2.1)$$ may lead to a serious conceptual and calculational problems in the Standard Model. The relatively heavy top quark with the mass (2.1) – heavier than expected on the base of LEP1-CDF-UA1 data [@elis], [@oldaltarelli] – shifts up the expected region of SM Higgs mass and consequently also the area of expected Higgs quartic self-coupling $\lambda$. The analysis of the value of Higgs mass following from the one-loop formula for the $W$–meson mass carried out by Hioki and Najima [@hioki] leads to the central value $$m_H\cong1700GeV. \eqno (2.2)$$
Since the Higgs self–coupling constant $\lambda$ and the Higgs mass are connected at the tree level by the formula
$$\lambda ={1\over 2} ({m_H\over{<\phi >}})^2, \hskip1cm {<\phi
>}=246GeV \eqno (2.3)$$ the value (2.2) implies that
$$\lambda\cong 25. \eqno (2.4)$$
This looks very dangerous; however to be honest we should mention that within the present experimental errors for $m_t$ given by FNAL result and for other experimental quantities being the input for the estimation (2.2) there is a considerable admissible deviation for $m_H$ from the central value $1700GeV$ [@passarino][@pietnascie]. Consequently $m_H$ and therefore also $\lambda$ may be much smaller.
Despite the fact that the present electro-weak data are not very conclusive the result (2.1) compels many authors to consider the possibilities of large Higgs mass and strong Higgs self–coupling more seriously. The super–strong Higgs self–coupling (like (2.4) or even smaller) would evidently break–down the perturbative calculations for many processes for which Higgs loops with $\lambda$-coupling contributes. For instance the two-loop perturbation expansion for the partial width decay $\Gamma(H\rightarrow\bar{f}f)$ of the Higgs particle into the fermion – antifermion pair can be written in the form
$$\Gamma(H\rightarrow\bar{f}f)=\Gamma_0[1+0.11({m_H\over 1TeV})^2 -
0.78({m_H\over 1TeV})^4] \eqno (2.5)$$ where $\Gamma_0$ is the partial width in the Born approximation and the second and third term in the bracket represent the one- and the two-loop contributions respectively [@knie].
We see that with increasing $m_H$ the importance of the two-loop contribution rapidly increases: for $m_H>375GeV$ the two-loop contribution dominates the one-loop and for $m_H>1200GeV$ the width becomes negative! This demonstrates the complete breakdown of perturbation theory for the Higgs mass of the order of 1TeV.
We see therefore that the supposition that the real Higgs field and the corresponding Higgs particle exists in the SM may lead to rather fundamental conceptual and calculational difficulties. Therefore it seems justified at present to look for a modification of SM in which all experimentally confirmed facts would be reproduced but the Higgs particle as the observed object would not exist.
Recently there were proposed several Higgsless models for electro–weak and strong interactions. In particular Shildknecht and collaborators proposed the Higgsless massive vector boson model [@schild] and they have compared some of its predictions with the predictions of the conventional SM. In the work [@dipdip] it was proposed a Higgsless SM with nonrenormalizable current–current and dipol–dipol interactions. Finally in [@sigmaSM] it was proposed a gauged $\sigma$–model for electro–weak interactions.
It seems to us that our Higgsless model based on the extension of electro–weak and strong interactions by gravitational interactions which leads to the extension of gauge symmetry by the local conformal symmetry presents a most natural frame–work for fundamental interactions.
Local conformal symmetry
========================
Let $M^{3,1}$ be the pseudo–Riemannian space time with the metric $g_{{\alpha \beta}}$ with the signature $(+,-,-,-)$. Let $\Omega(x)$ be a smooth strictly positive function on $M^{3,1}$. Then the conformal transformation in $M^{3,1}$ is defined as the transformation which changes the metric by the formula $$g_{{\mu\nu}}(x)\rightarrow
\tilde{g}_{{\mu\nu}}(x)=\Omega^2(x) g_{{\mu\nu}}(x). \eqno (3.1)$$ The set of all conformal transformations forms the multiplicative abelian infinite–dimensional group $C$ with the obvious group multiplication law.
It is evident from (3.1) that $(M^{3,1},g_{\mu\nu})$ and $(M^{3,1},\tilde g_{\mu\nu})$ have identical causal structure and conversely it is easy to show that any two space times which have identical causal structure must be related by a local conformal transformation.
The conformal transformations occur in many problems in general relativity. In particular Canuto et. al. proposed the scale–covariant theory of gravitation, which provides an interesting alternative for the conventional Einstein theory [@canuto].
It should be stressed that a conformal transformation is not a diffeomorphism of space time. The physical meaning of the conformal transformations follows from the transformation law of the length element $$dl(x)= \sqrt{-g_{ij}dx^i dx^j} \hskip1cm \rightarrow \hskip1cm
\tilde{dl(x)}=\Omega(x) dl(x). \eqno (3.2)$$ Hence a local conformal transformation changes locally the length scale. Since in some places of the Earth one utilizes [*the meter*]{} as the length scale, whereas in other places one utilizes [*the feet*]{} or [*the ell*]{} as the length scale one my say that one utilizes the local conformal transformations in everyday live. Similarly one verifies that the conformal transformation changes locally the proper time $$ds(x)=\sqrt{g_{\mu\nu}dx^{\mu}dx^\nu} \hskip1cm \rightarrow \hskip1cm
d\tilde s(x)=\Omega(x)ds(x).$$
Since the physical phenomena should be independent of the unit chosen locally for the length, the proper time, mass etc. the group $C$ of local conformal transformations should be a symmetry group of physical laws.
In order to avoid any confusion we stress that the group $C$ has nothing in common with the 15 parameter conformal group $SO(4,2)$ defined locally in the $M^{3,1}$by the action of Poincare, dilatation and special conformal transformations.
Comparing the physical meaning of local conformal transformations and the local gauge $SU(2)_L$ transformations of SM associated with the concept of the weak isospin it seems that the conformal transformations are not less natural symmetry transformation than the nonabelian gauge transformations in the SM.
We shall give now a construction of the representation of the conformal group $C$ in the field space. Let $\Psi$ be a tensor or spinor field of arbitrary spin. Define the map $$\Omega\rightarrow
U(\Omega)$$ by the formula $$\tilde{\Psi}(x)=U(\Omega)\Psi(x)=\Omega^s(x)\Psi(x), \hskip1cm s\in R
\eqno (3.3)$$
The number $s$ is determined by the condition of conformal invariance of field equation. We say that field equation for $\Psi$ is conformal invariant if there exist $s\in R$ such that $\Psi(x)$ is a solution with the metric $g_{{\mu\nu}}(x)$ if and only if $\tilde\Psi(x)$ given by (3.3) is a solution with the metric $\tilde{g}_{{\mu\nu}}(x)$. The number $s$ is called the conformal weight of $\Psi$ [@birel], [@wald], [@casta]. It is evident that the map $\Omega\rightarrow U(\Omega)$ defines the representation of $C$ in the field space.
Using the above definitions one can calculate the conformal weight for a field of arbitrary spin. Let for instance $F_{\mu\nu}$ be the Maxwell field on $(M^{3,1}, g)$ which satisfies the equation $$g^{\mu\sigma}\nabla_{\sigma}F_{\mu\nu}=0$$ $$\nabla_{[\sigma}F_{\mu\nu]}=0.$$
Using the definition of the covariant derivative $\tilde\nabla_\sigma$ with respect to $\tilde g_{\mu\nu}$ metric and (3.3) one obtains $$\tilde{g}^{\mu\sigma}\tilde\nabla_{\sigma}(\Omega^s
F_{\mu\nu})=(n-4+s)\Omega^{s-3}
g^{\mu\sigma}F_{\mu\nu}\nabla_{\sigma}\Omega$$ $$\tilde\nabla_{[\sigma}(\Omega^s F_{\mu\nu]})=s\Omega^{s-1}
(\nabla_{[\sigma}\Omega)F_{\mu\nu]}.$$
We see that for $n\not=4$ the Maxwell equations are not conformally invariant. For $n=4$ the Maxwell equations are invariant if the conformal weight $s$ equals to zero.
Similarly one can show that the Yang–Mills field strength ${F_{\mu\nu}}^a$ has the conformal weight $s=0$ whereas the massless Dirac field has the conformal weight $s=-{3\over 2}$. It is noteworthy that the scalar massless field $\Phi$ satisfying the Laplace–Beltrami equation $$\triangle\Phi=0$$ is not conformal invariant. In fact it was discovered by Penrose that one has to add to the Lagrangian on $(M^{3,1},g)$ the term $$-{1\over 6}R\Phi^2$$ where $R$ is the Ricci scalar, in order that the corresponding field equation is conformal invariant with the conformal weight $s=-1$ [@penrose].
A unified model for strong, electro–weak and gravitational interactions
=======================================================================
We postulate that the searched unified theory of strong, electro–weak and gravitational interactions will be determined by the condition of invariance with respect to the group $G$ $$G=SU(3)\times
SU(2)_{L}\times U(1)\times C \eqno (4.1)$$ where $C$ is the local conformal group defined by (3.1). Let $\Psi$ be the collection of vector meson, fermion and scalar fields which appear in the conventional minimal SM for electro–weak and strong interactions. Then the minimal natural conformal and $SU(3)\times SU(2)_{L}\times
U(1)$ –gauge invariant total lagrangian $L(\Psi)$ may be postulated in the form:
$$L = [L_{{G}}+L_{{F}}+L_{Y}+
L_{\Phi} + \beta\partial_\mu|\Phi|\partial^\mu|\Phi|
- {1\over 6}(1+\beta)R\Phi^{\dagger}\Phi +
L_{{grav}} ] \sqrt{-g} \eqno (4.2)$$
Here $L_{{G}}$ is the total lagrangian for the gauge fields $A^{a}_{\mu}$, $W^{b}_{\mu}$ and $B_{\mu}$, $a=1,...,8$, $b=1,2,3$ associated with $SU(3)\times SU(2)_{L}\times U(1)$ gauge group $$L_G=-{1\over 4}{F^a}_{\mu\nu}{F^a}^{\mu\nu}-
{1\over 4}{W^b}_{\mu\nu}{W^b}^{\mu\nu}-
{1\over 4}{B}_{\mu\nu}{B}^{\mu\nu}, \eqno (4.3)$$ and ${F^a}_{\mu\nu}$, ${W^b}_{\mu\nu}$ and ${B}_{\mu\nu}$ are the conventional field strengths of gauge fields in which the ordinary derivatives are replaced by the covariant derivatives e.g. $${B}_{\mu\nu}=\nabla_\mu B_\nu - \nabla_\nu B_\mu, \eqno (4.4)$$ etc.; $L_{{F}}$ is the lagrangian for fermion field interacting with the gauge fields; $L_{Y}$ represents the Yukawa interactions of fermion and scalar fields; $L_{\Phi}$ is the lagrangian for the scalar fields $$L_{\Phi}=(D\Phi)^{\dagger}(D\Phi) - \lambda(\Phi^{\dagger}\Phi)^2 \eqno
(4.5)$$ where $D$ denotes the covariant derivative with connections of all symmetry groups. Notice that the condition of conformal invariance does not admit the Higgs mass term $\mu^2\phi^{\dagger}\phi$ which assures the mechanism of spontaneous symmetry breaking and mass generation in the conventional formulation.
The term $$\beta\partial_\mu|\Phi|\partial^\mu|\Phi| \eqno (4.6)$$ is gauge invariant. It may be surprising that (4.6) depends on $|\Phi|$. Observe however that the lagrangian $L_\Phi$ can be written in the form $$(D\Phi)^{\dagger}(D\Phi)=\partial_\mu|\Phi|\partial^\mu|\Phi|
+|\Phi|^2L_\sigma(g(\Phi),W,B) \eqno (4.7)$$ where $L_\sigma(g(\Phi),W,B)$ is a gauged–sigma–model–like lagrangian and $$\Phi = {\phi_u\choose \phi_d}= g(\Phi){0\choose |\Phi|}, \hskip.5cm
g(\Phi)={1\over|\Phi|}\pmatrix{\bar\phi_d & \phi_u \cr
-\bar\phi_u & \phi_d} \eqno (4.8)$$ where $g(\Phi)$is $SU(2)_L$ gauge unitary matrix.
We see therefore that the term like (4.6) is already present in the conventional $L_\Phi$ lagrangian.
The term $$- {1\over 6}(1+\beta)R\Phi^{\dagger}\Phi \eqno (4.9)$$ is the Penrose term. which assures that the lagrangian (4.2) is conformal invariant.
The last term in (4.2) is the Weyl term $$L_{{grav}} = -\rho C^2, \hskip1cm\rho>0, \eqno (4.10)$$ where $C_{\alpha\beta\gamma}^\delta$ is the Weyl tensor which is conformally invariant. Using the Gauss–Bonnet identity we can write $C^2$ in the form $$C^2=2(R^{{\mu\nu}}R_{{\mu\nu}}-{1\over 3}R^2). \eqno (4.11)$$
We see that the condition of conformal invariance does not admit in (4.2) the conventional gravitational Einstein lagrangian $$L=\kappa^{-2} R\sqrt{-g}, \hskip1cm \kappa^2=16\pi G. \eqno (4.12)$$
It was shown however by Stelle [@Stelle] that quantum gravity sector contained in (4.2) is perturbatively renormalizable whereas the quantum gravity defined by the Einstein lagrangian (4.12) coupled with matter is nonrenormalizable [@deser]. Hence, for a time being it is an open question which form of gravitational interaction is more proper on the quantum level. We show in Section 7 that the Einstein lagrangian (4.12) may be reproduced by Penrose term if the physical scale is properly determined. In Section 8 we discuss the role of quantum effects which may reproduce the lagrangian (4.12) and give the classical Einstein theory as the effective induced gravity.
Notice that conformal symmetry implies that all coupling constants in the present model are dimensionless.
The theory given by (4.2) is our conformally invariant proposition alternative to the standard Higgs–like theory with SSB. Its new, most important feature is the local conformal invariance. It means that simultaneous rescaling of all fields (including the field of metric tensor) with a common, arbitrary, space–time dependent factor $\Omega(x)$ taken with a proper power for each field (the conformal weight) will leave the Lagrangian (4.2) unaffected. The symmetry has a clear and obvious physical meaning [@narlikar], [@wald]. It changes in every point of the space–time all dimensional quantities (lengths, masses, energy levels, etc) leaving theirs ratios unchanged. It reflexes the deep truth of the nature that nothing except the numbers has an independent physical meaning.
The freedom of choice of the length scale is nothing but the gauge fixing freedom connected with the conformal symmetry group. In the conventional approach we define the length scale in such a way that elementary particle masses are the same for all times and in all places. This will be the case when we rescale all fields with the $x$–dependent conformal factor $\Omega(x)$ in such a manner that the length of the rescaled scalar field doublet is fixed i.e.
$$\tilde\Phi^{\dagger}\tilde\Phi={v^2 \over 2}=const. \eqno (4.13)$$
(We shall discuss the problem of mass generation in details in Section 5.)
The scale fixing for the conformal group (4.13) is distinguished by nothing but our convenience. Obviously we can choose other gauge fixing condition, e.g. we can use the freedom of conformal factor to set
$$\sqrt{-\tilde{\tilde g}}=1; \eqno (4.14)$$ this will lead to other local scales but it will leave physical predictions unchanged.
Consider, for example the scale fixing condition (4.14). Imposing (4.14) on the conformal invariant theory given by (4.2) we obtain the lagrangian $\tilde{\tilde L}(\tilde{\tilde \Psi},
\tilde{\tilde V_\mu}, \tilde{\tilde \Phi}, \tilde{\tilde g}_{\mu\nu})$ describing dynamics of the fields $\tilde{\tilde \Psi}$, $\tilde{\tilde V_\mu}$, $\tilde{\tilde \Phi}$, $\tilde{\tilde
g}_{\mu\nu})$. The arguments of $\tilde{\tilde L}$ stand for all fermion, vector, scalar and tensor fields of the model and fulfill the condition (4.14). $\tilde{\tilde L}$ is no longer conformal invariant as the scale was fixed by (4.14). We can change variables of $\tilde{\tilde L}$ according to the rule
$$\tilde \Psi=\Biggl({\sqrt{2}|\tilde{\tilde \Phi}|\over v}
\Biggr)^{-3/2}\tilde{
\tilde \Psi} \eqno (4.15a)$$ $$\tilde V_\mu=\tilde{\tilde V_\mu} \eqno (4.15b)$$ $$\tilde g_{\mu\nu}=\Biggl({\sqrt{2}|\tilde{\tilde \Phi}|\over v}
\Biggr)^2
\tilde{\tilde g}_{\mu\nu}\eqno (4.15c)$$ $$\tilde \Phi=\Biggl({\sqrt{2}|\tilde{\tilde \Phi}|\over
v}\Biggr)^{-1}\tilde{\tilde \Phi}\eqno (4.15d)$$ where $\tilde g$ is no longer restricted but $\tilde \Phi$ fulfills (4.13) what follows from (4.15d). Such a change of variable is an example of conformal transformation but, as was said, it is not a symmetry of $\tilde{\tilde L}$. In fact we have
$$\tilde{\tilde L}(\tilde{\tilde
\Psi}(\tilde \Psi, \tilde V_\mu, \tilde \Phi, \tilde
g_{\mu\nu}), \tilde{\tilde V_\mu}(\tilde \Psi, \tilde V_\mu,
\tilde \Phi, \tilde g_{\mu\nu})
, \tilde{\tilde \Phi}(\tilde \Psi, \tilde V_\mu, \tilde \Phi,
\tilde g_{\mu\nu})
, \tilde{\tilde g}_{\mu\nu}(\tilde \Psi, \tilde V_\mu, \tilde
\Phi, \tilde g_{\mu\nu}))=$$ $$=\tilde L(\tilde \Psi, \tilde V_\mu,
\tilde \Phi, \tilde g_{\mu\nu}) \eqno (4.16)$$ where $\tilde L(\tilde \Psi, \tilde V_\mu, \tilde \Phi, \tilde
g_{\mu\nu})$ is the lagrangian which one would obtain by imposing the scale fixing condition (4.13) directly on (4.2). It should be stressed that the functional form of $\tilde{\tilde L}$ in terms of its arguments is different than $\tilde L$ of its arguments (compare (5.1) and (8.2) for concrete examples). In such a sense theories obtained from different scale fixings are mathematically equivalent. They will be equivalent also physically if identifications of physical and mathematical objects in the theories being compared will be consistent with theirs mathematical equivalence. For example if we assume that $\tilde g$ describes physical metric we cannot assume that this metric is described also by $\tilde{\tilde g}$.
Generation of lepton, quark and vector boson masses
===================================================
We demonstrate now that using the conformal group scale fixing condition (4.13) we can generate the same lepton, quark and vector meson masses as in the conventional SM without however use of any kind of Higgs mechanism and SSB.
In fact inserting the scale fixing condition (4.13) into the Lagrangian (4.2) we obtain $$\tilde L=L^{scaled} = [L_{{G}}+L_{F}+L_{\Phi}^{scaled}+
L_{Y}^{scaled} - {1\over 12}v^2 R + L_{{grav}} ] \sqrt{-g},
\eqno (5.1)$$ in which the condition (4.13) was inserted into $L _{\Phi}$ and $L_{Y}$. We should use the symbol $\tilde\Phi$, $\tilde\Psi$ etc. for the rescaled fields in (5.1), however for the sake of simplicity we shall omit “ $\tilde{}$ ” sign over fields in the following considerations.
The condition (4.13) together with the unitary gauge fixing of $SU(2)_{L}\times U(1)$ gauge group, reduce by (4.8) the Higgs doublet to the form $$\Phi^{{gauge}}={1\over\sqrt{2}}{0\choose v}, \hskip 1cm v>0
\eqno (5.2)$$ and produce the tree level mass terms for leptons, quarks and vector bosons associated with $SU(2)_L$ gauge group. For instance the $\Phi$–lepton Yukawa interaction $L^l_{Y}$ $$L^l_{Y}=-\sum_{i=e,\mu,\tau}G_{i}\bar {l_{i}}_R(\Phi^\ast {l_{i}}_L)
+h.c.$$ passes into $${L^l_{Y}}^{gauged}=-{1\over\sqrt{2}}v(G_{e}\bar
e e + G_\mu \bar\mu \mu + G_\tau \bar\tau \tau) \eqno (5.3)$$ giving the conventional, space–time independent lepton masses
$$m_{e}={1\over\sqrt{2}}G_{e}v, \hskip1cm
m_{\mu}={1\over\sqrt{2}}G_{\mu}v, \hskip1cm
m_{\tau}={1\over\sqrt{2}}G_{\tau}v. \eqno (5.4)$$
Similarly one generates from $\Phi$–quark Yukawa interaction $L^q_{Y}$ the corresponding quark masses. In turn from $L_{\Phi}$-lagrangian (4.5) using the gauge condition (5.2) one obtains $$(D_\mu\tilde\Phi)^\dagger
D^\mu \tilde\Phi={g^2_2 v^2\over4}W_\mu^{+}W^{\mu-}+
{g^2_1+g^2_2\over8}v^2Z^2$$ where $$Z_\mu=-\sin\theta_W B_\mu + \cos\theta_W {W^3}_\mu,
\hskip1cm \cos \theta_W={g_2\over\sqrt{g^2_1+g^2_2}}.$$
Hence one obtains the following vector mesons masses $$m_{W}={v\over 2}g_2, \hskip1cm
m_{Z}={m_W\over \cos\theta_W}. \eqno (5.5)$$
It is remarkable that the analytical form for tree level fermion and vector meson masses in terms of coupling constants and the parameter $v$ is the same as in the conventional SM. We see therefore that the Higgs mechanism and SSB is not indispensable for the fermion and vector mesons mass generation!
We note that the fermion–vector boson interactions in our model are the same as in SM. Hence analogously as in the case of conventional formulation of SM one can deduce the tree level relation between $v$ and $G_F$ – the four–fermion coupling constant of $\beta$–decay: $$v^2=(2G_F)^{-1}\rightarrow v=246GeV. \eqno (5.6)$$ Here we have used the standard decomposition $g^{\mu\nu}\sqrt{-g}=
\eta^{\mu\nu}+\kappa^\prime h^{\mu\nu}$ (see e.g. [@capper]) which reduces the tree level problem for the matter fields to the ordinary flat case task.
We see therefore that the resulting expressions for masses of physical particles are identical as in the conventional SM.
Let us stress that the scale fixing condition like (4.13) does not break $SU(2)_{L}\times U(1)$ gauge symmetry. The symmetry is broken (or rather one of gauge equivalent description is fixed) when (4.13) is combined with unitary gauge condition of electro–weak group leading to (5.2). However, also after imposing of a gauge condition like (5.2) we have a remnant of both the conformal and $SU(3)\times SU(2)_{L}\times U(1)$ initial gauge symmetries: this is reflected in the special, unique relations between couplings and masses in our model
Precision tests of electro–weak interactions and high energy behavior in the present model.
===========================================================================================
Our model represents in fact the gauge field theory model with massive vector mesons and fermions. It is well–known that such models are in general nonrenormalizable [@sigmanonrenormalizable]. We remind however that in nonrenormalizable Fermi model for weak interactions we can make a definite predictions for low energy phenomena e.g. for $\mu$ or neutron decays. Similarly the recent progress with so called Generalized Equivalence Theorem allows to make definite predictions for the scattering operator in nonrenormalizable models like gauged nonlinear $\sigma$–model or other nonrenormalizable gauge field theory models [@equivalence]. Hence in our model we can obtain definite predictions for electro–weak phenomena if we consider processes with energy $\sqrt{s}$ below some ultraviolet (UV) cutoff $\Lambda$. We wish to demonstrate that the cutoff $\Lambda$ is determined by the Higgs mass $m_H$ appearing in the Standard Model. Hence, from this point of view, Higgs mass is nothing else as the UV cutoff which assures that the truncated perturbation series is meaningful. We shall try to elucidate this problem on the example of so called precision tests of electro–weak theory.
One–loop radiative corrections to various electro–weak quantities or processes can be expressed in terms of three quantities $\Delta r$, $\Delta\rho$ and $\Delta k^\prime$. We refer to the recent excellent review by Kniehl for the precise definitions of these quantities and for their analytical expressions [@kniehl]. For an illustration we recall that the expression for W–meson mass, up to one loop order, has the form
$$M_W={M_Z\over\sqrt{2}}\Biggl\{1+\sqrt{1+{2\sqrt{2}\pi\alpha\over M_Z
G_F(1-\Delta r)}}\Biggr\}^{{1\over2}} \eqno (6.1)$$ where $$\alpha={1\over137.036},$$ $$G_F=1.16639\times 10^{-5}GeV^{-2}, \eqno (6.2)$$ $$M_Z=91.1899\pm 0.0044GeV$$ and $\Delta r(m_t, m_H)$ is the one loop correction to $\mu$–decay amplitude which in Standard Model depends on top and Higgs masses. Taking the experimental value for W–mass $M_W=80.21\pm 0.18GeV$ and the recently reported top mass $m_t=174\pm 17GeV$ one gets from (6.1) the central value of Higgs mass $m_H\cong1700Gev$ with the error of several hundreds of GeV [@hioki].
It was suggested by Altarelli [*et.al*]{} [@altarelli] to pass from $\Delta r$, $\Delta\rho$ and $\Delta k^\prime$ to new quantities $\varepsilon_{N1}$, $\varepsilon_{N2}$ and $\varepsilon_{N3}$ such that $\varepsilon_{N2}$ and $\varepsilon_{N3}$ depend on $m_t$ only logaritmically. These parameters characterize the degree of $SU(2)_L\times U(1)$ symmetry breaking and their numerical value significantly different from zero would signal a “new physics” [@schild][@altarelli].
If we calculate these parameters in our model in one–loop approximation we find the specific class of Feynman diagrams with fermion and vector boson loops which contributes to them. Since some vector boson loops will produce divergences, e.g. in the case of fermion – massive vector boson coupling constant, one has to introduce either the new renormalization constant or UV cutoff $\Lambda$ which can be given by the formula [@schild]
$$\log{\Lambda^2\over\mu^2}={2\over 4-D}-\gamma_E+\log{4\pi} \eqno
(6.3)$$ where $\mu$ is the reference mass of dimensional regularization, $D$ is the space–time dimension and $\gamma_E$ is the Euler’s constant.
One obtains the formula for $\varepsilon_{Ni}$ parameters in SM if one adds to the class of Feynman diagrams in our model all appropriate one–loop diagrams with Higgs internal lines. Using the results of [@schild] and [@schildk] one obtains
$$\varepsilon^{SM}_{N1}-\varepsilon^{CSM}_{N1}=
{3\alpha({M_Z}^2)\over16\pi {c_0}^2}
\log{({\Lambda^2\over{m_H}^2})}+O({{M_Z}^2\over{m_H}^2}
\log{({{M_H}^2\over{M_Z}^2})})$$ $$\varepsilon^{SM}_{N2}-\varepsilon^{CSM}_{N2}=
O({{M_Z}^2\over{m_H}^2} \log{({{M_H}^2\over{M_Z}^2})}) \eqno (6.4)$$ $$\varepsilon^{SM}_{N3}-\varepsilon^{CSM}_{N3}=
{\alpha({M_Z}^2)\over48\pi {s_0}^2}
\log{({\Lambda^2\over{m_H}^2})}+O({{M_Z}^2\over{m_H}^2}
\log{({{M_H}^2\over{M_Z}^2})})$$ where $CSM$ index of $\varepsilon_{Ni}$ means that the quantity was calculated in our Conformal Standard Model. Here $\alpha({M_Z}^2)={1\over129}$ and $c_0$ and $s_0$ are defined by the formula
$$s_0^2(1-s_0^2)=s_0^2c_0^2\equiv{\pi\alpha({M_Z}^2)\over
\sqrt{2}G_FM_Z^2}$$.
The above formulas indicate a role which plays in SM the very large Higgs mass: first the term $O({{M_Z}^2\over{m_H}^2}
\log{({{M_H}^2\over{M_Z}^2})})$ for $m_H>1TeV$can be disregarded and second if we take the UV cutoff $\Lambda\simeq m_H$ then the prediction for $\varepsilon_{Ni}$–parameters in the conventional SM and our nonrenormalizable model coincide. Thus the very large Higgs mass preferred by the top mass $m_t=174GeV$ plays in the conventional SM the role of UV cutoff parameter. If the Higgs particle will be not found then our model provides an extremely natural frame–work for the description of electro–weak and strong interactions at least up to TeV energies.
We would like to discuss now the problem of getting predictions from our nonrenormalizable model for electro–weak and strong interactions considered in the flat space–time. Take the process $A+B\rightarrow
C+D$ in our model. This process – up to L–loop order – will be described by the corresponding Feynman diagrams with A, B, C and D external lines and some number of internal fermion, massive vector mesons, gluon and photon lines. Since theory is nonrenormalizable one has to introduce the proper UV cutoff $\Lambda$.
The problem of elaboration of an effective calculational scheme for our model is considerably facilitated by the fact that introducing the suitable Stueckelberger auxiliary fields we can transform our model into the gauged nonlinear $\sigma$–model (GNL$\sigma$M) (see e.g. [@schild], [@sigmaSM] and the discussion in Section 9). It is known that perturbative calculations in GNL$\sigma$M with cutoff $\Lambda$ are well elaborated and lead to interesting physical predictions for various processes [@schild], [@equivalence].
In fact it was recently shown that so called General Equivalence Theorem (GET) holds in gauge field theories irrespectively if they are renormalizable or nonrenormalizable [@equivalence]. This remarkable theorem can be applied in the case of SM for heavy Higgs at high energy where $$m_H, E\gg M_W, m_{f_i}$$ where E is the total energy and $m_{f_i}$ are lepton and quark masses respectively. It was shown that the leading parts coming from the L–loop diagrams are those diagrams for which N defined as
$$N=power\; of\; m_H+power\; of\; E \eqno (6.5)$$ becomes maximal. Using GET one relatively easily determines the leading contribution for any L–loop in SM and obtains high energy limit of a given scattering amplitude [@equivalence]. In the case of Higgsless nonrenormalizable gauge field theory model one introduces cutoff $\Lambda$: in this case at high energy limit defined by inequalities $$\Lambda>E\gg M_W, m_{f_i}$$ the leading diagrams are those for which $$N=power\; of\; \Lambda+power\; of\; E \eqno (6.6)$$ is maximal. Comparing (6.5) with (6.6) we see as in the case of the $\varepsilon_{Ni}$–parameters that the UV cutoff $\Lambda$ in Higgsless gauge models replaces the large mass $m_H$. Using the criterion (6.6) and GET one obtains the high energy limit of scattering amplitude for various processes also in the nonrenormalizable gauge models, like e.g. in the Higgsless GNL$\sigma$M [@equivalence].
We see therefore that nonrenormalizability does not prevent us from getting definite predictions for physical processes in the low or high energy region from our model. Consequently the nonrenormalizable Higgsless models may be as a useful in description of experimental data as the conventional SM.
We considered hence the general variant of our model with $\beta\not=0$ which leads to nonrenormalizable gauge field theory. However the special case of our model with $\beta=0$ discussed in Section 8 gives a renormalizable model for fundamental interactions.
Gravity Sector
==============
Let us impose the scale fixing condition (4.13) on the lagrangian (4.2) and collect all gravitational terms. The lagrangian reads: $$L^{scaled}=[L^{scaled}_{matter}-{1\over12}(1+\beta)v^2R-2\rho(R^{\mu\nu}
R_{\mu\nu}-$$ $${1\over3}R^2)-{\lambda\over4}v^4]\sqrt{-g}
\eqno (7.1)$$ where we have selected the part $L^{scaled}_{matter}$ (describing the matter interacting with gravity) from the remaining purely gravitational terms.
The variation of (7.1) with respect to the metric $g^{\mu\nu}$ leads to the following classical equation of motion: $$\rho[-{2\over3}R_{;\mu;\nu}+2{{R_{\mu\nu}}^{;\eta}}_{;\eta}
-{2\over3}g_{\mu\nu}{R^{;\eta}}_{;\eta}-$$ $$4R^{\eta\lambda}R_{\mu\eta\nu\lambda}+ {4\over3}RR_{\mu\nu}+
g_{\mu\nu}(R^{\eta
\lambda}R_{\eta\lambda}-{1\over3}R^2)]+$$ $${1\over12}(1+\beta)v^2(R_{\mu\nu}
-{1\over2}g_{\mu\nu}R)+{\lambda\over8}v^4g_{\mu\nu}={1\over2}
T_{\mu\nu}. \eqno (7.2)$$
In the empty case $T_{\mu\nu}=0$ this equation is satisfied by all solutions of an empty space Einstein equation with a properly chosen cosmological constant $\Lambda$:
$$R_{\mu\nu}-{1\over2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=0. \eqno
(7.3)$$ In fact (7.3) implies that $$R_{\mu\nu}\sim g_{\mu\nu} \hskip1cm \Rightarrow \hskip1cm
R_{\mu\nu}={1\over4}Rg_{\mu\nu} \eqno (7.4)$$ and then $$R_{\mu\nu}=\Lambda g_{\mu\nu}. \eqno (7.5)$$
Inserting (7.4) into (7.2) we find that the part proportional to $\rho$ vanishes. The remnant can be collected leading to the relation
$${1\over8}v^2g_{\mu\nu}({2\over3}(1+\beta)\Lambda-\lambda v^2)=0 \eqno
(7.6)$$ where the empty space condition $T_{\mu\nu}=0$ were used for the right hand side of (7.6).
It is easy to conclude that (7.6) is satisfied when $$\Lambda={3\over2(1+\beta)}\lambda v^2. \eqno (7.7)$$
Equation (7.7) relates $\lambda$ with a potentially observable cosmological constant $\Lambda$.
Let us go back to the case with the matter. Observe that the term linear in the curvature appears in (7.1) with the coefficient $-{1\over12}(1+\beta)v^2$. In the case of $\beta=0$ we have the old–fashion Standard Model minimally conformally coupled with gravity. In this case, in comparison with the Newtonian constant entering to the ordinary Einstein’s theory (4.12) the coefficient $-{1\over12}v^2$ standing in front of $R$ in (7.1) has an opposite sign and is smaller of many orders of magnitude ($v^2\kappa^2\approx
10^{-38}$). If it would be the only purely gravitational term in the theory it will mean that the geometry in tree approximation is generated by the negative energy and with an extremal strength. This is the price we would have to pay for the positive kinetic term of scalar fields in (4.5), for the gauge invariance and for the renormalizability of the matter sector.
If we want to reproduce the correct gravitational sector already at the classical level rather than preserve renormalizability of the material sector we have to admit for nonzero $\beta$ coupling. This would lead us to a model which is equivalent to the nonrenormalizable gauged nonlinear sigma model in the material sector. Accepting this price we can put
$$-{1\over 12}(1+\beta)v^2=\kappa^{-2} \eqno (7.8)$$ reproducing the Newtonian coupling in front of curvature $R$ in (7.1). This would mean that $\beta\approx-10^{38}$! Notice however that taking the scale fixing condition (4.13) the term $\beta\partial_\mu|\Phi|\partial^\mu|\Phi|$ vanishes. Hence it looks like that the only role of this term is to generate the proper value of Newton constant in the Einstein–Hilbert tree level lagrangian resulting from the Penrose term. We will go back to this point in Section 8.
Towards the renormalizable theory.
==================================
We have shown in Section 6 that the nonrenormalizability of our model does not rise serious calculational problems within the energy range presently accessible in experiments. What more, if Higgs particle will be not found then it cannot be generally excluded that nonrenormalizability will be the indispensable feature of every realistic particle theory model. It would mean that we are compulsed to work with a theory valid for a limited energy regions and even for limited classes of phenomena. Clearly this situation is unsatisfactory and people will always try to find a general description scheme unifying different phenomena and independent on the considered energy range. Hopes for such a universal description are usually set on renormalizable unified models. Being motivated by these hopes let us go back to the problem of renormalizability of our model.
It is easy to see that nonrenormalizability of the matter part of lagrangian (4.2) is connected with the presence of nonlinear interaction (4.6). To see this we can approximate (4.2) demanding that $$g^{\mu\nu}=\eta^{\mu\nu}. \eqno (8.1)$$ This is a conformally flat approximation rather than the flat approximation as we have the scale fixing freedom, and the part of relations (8.1) can be understood as making use of this freedom e.g. in the form of condition (4.14). Putting (4.14) into (4.2) we obtain $$\tilde{\tilde L}=L ^{unimodular}= L_{{G}}+L_{{F}}+L_{Y}+
L_{\Phi} + \beta\partial_\mu|\Phi|\partial^\mu|\Phi| - {1\over 6}
(1+\beta)R\Phi^{\dagger}\Phi + L_{{grav}}\eqno (8.2)$$
Putting in turn (8.1) we obtain the conformally flat approximation lagrangian $L_{cfa}$:
$$L_{cfa} = L_{{G}}+L_{{F}}+L_{Y}+
L_{\Phi} + \beta\partial_\mu|\Phi|\partial^\mu|\Phi|
\eqno (8.3)$$
For $\beta=0$ this is just the renormalizable SM lagrangian (without the negative scalar mass term $-\mu^2\Phi^\dagger\Phi$ however).
The presence of $\beta$–term was justified in Section 7 by the condition of the proper Einsteinian limit of the theory at its classical level. This led us to the rather large value $|\beta|\sim{m_{_{PLANCK}}^2\over v^2}$. Fortunately considerations of Section 7 showed that the predictions in particle sector of our theory are insensitive to this huge value of $\beta$–coupling.
As we have mention in Section 7 for the case with $\beta=0$ after the choice of physical scaling the obtained tree level Newton constant is not correct. It was suggested by various authors that the corrected Newton constant in quantum theories of gravity may be obtained by inclusion of radiative corrections [@zeldowich][@zakharow] (see [@adler] for a pedagogical introduction and [@buchbook] for the recent review of the subject). The authors of [@buchcqg] have discussed a wide class of theories which contains also our model in the case of $\beta=0$. They have shown that taking the proper values for the nonobserved coupling constants like $\rho$ or $\lambda$ and renormalization scale one may generate the induced Newton and cosmological constants with experimental values. However this method is – in our opinion – incomplete since the problem of mass values of elementary particles in the framework in which gravitational constants were determined was not considered. In fact the value of the Newton constant has not an absolute meaning. This constant disappear from the empty space Einstein equations. In the presence of matter the value of Newton constant can be rescaled with simultaneous rescaling of masses and energy levels. Thus the value of induced Newton and cosmological constants must be compared with the effective masses of classical matter fields obtained within the same level of perturbative analysis before going to the final conclusions on induced Einstein lagrangian. According to our knowledge the quantum gravity corrected expressions for the effective masses were not derived so far in the literature.
Until this problem is solved one cannot conclude that that the renormalizable model for fundamental interactions with $\beta=0$ is physically meaningful.
Discussion.
===========
The elementary particle physics is at present at a crossroad. We have in fact three drastically different alternatives:
I$^o$ The Higgs particle exists, its mass will be experimentally determined and will have the value predicted by the radiative corrections of SM. This will confirm the SSB mechanism for mass generation, the validity of SM frame–work and it will represent an extraordinary success of quantum gauge field theory.
II$^o$ The Higgs particle exists but its mass is considerable different from that predicted by the radiative corrections of SM. This would signal some kind of “New Physics” which will imply a reformulation of the present version of SM.
III$^o$ The Higgs particle does not exists. This will lead to a rejection of SM with Higgs sector and it will give preference to Higgsless models for fundamental interactions. In this situation we have two general possibilities:
IIIA$^o$ The physical Higgsless model is renormalizable. The example of such model was discussed by us in Section 8.
IIIB$^o$ The physical Higgsless model is nonrenormalizable. It may be that the renormalizability of Quantum Gravity determined by Einstein–Hilbert action integral coupled with matter fields is not an “accident at work in quantum field theory” but it represents a universal feature that physical fundamental interactions considered simultaneously are nonrenormalizable. In this situation we are compulsed to use the nonrenormalizable models of quantum field theory for a description of fundamental interactions and we have to learn how to deduce predictions for experiments from such models. Several nonrenormalizable models for electroweak interactions were proposed like Schildknecht [*et al.*]{} model, [@schild],GNL$\sigma$M [@sigmaSM], or gauge field theory models with condensates [@condensates]. We have presented in Section 4 a new unified nonrenormalizable model for fundamental interactions based on the gauge and local conformal symmetry.
Our model – in spite of its nonrenormalizability – provides the definite predictions for low and very high energy interactions in terms of the parameters of the model, energy $E$ and the cutoff $\Lambda$. The direct calculations of electro–weak parameters $\varepsilon_{N1}$, $\varepsilon_{N2}$ and $\varepsilon_{N3}$ demonstrate that the Standard Model and the present model results differ by the term proportional to $\log{\Lambda^2\over m_H^2}$: thus it looks like that the very high Higgs mass $m_H$ plays in SM the role of the UV cutoff which in the present model may be replaced by parameter $\Lambda$. We see therefore that the predictive power of our model may be comparable with that of the conventional SM.
In view of the possibility that nonrenormalizable nonabelian massive gauge field theories have to be used for a description of fundamental interactions it seems necessary to develop perturbative and nonperturbative methods for extracting predictions for scattering amplitudes and observables from such models. In particular one should develop the corresponding Generelized Equivalence Theorems and determine explicitly the high energy behavior of cross sections in such models. The comparison of the obtained results with analytic formulas coming from Lipatov calculations [@lipatov] would be very inspiring. It would be also useful to develop systematic two–loop calculus with UV cutoff $\Lambda$ for electro–weak processes. We plan in a near future to present several examples of such calculations.
The present model allows to obtain the Einsteinian form of gravitational interactions in the classical limit. It can be also analyzed by means of effective action for induced gravity [@buchbook].
The authors are grateful to Prof. Iwo Białynicki Birula, Dr. B. Grzadkowski and Dr. M. Kalinowski for interesting discussions and Dr. S.D. Odintsov for sending to us the recent results of his group.
[99]{} CDF Collaboration, FERMILAB-PUB-94/116-E. Z. Hioki, R. Najima, Is the Standard Electroweak Theory Happy with $m_t\sim174GeV$?, preprint TOCUSHIMA 94-02, YCCP-9404. OPAL Collaboration, K. Ahmet, Phys. Lett. [**B251**]{} (1990) 211, ALEPH Collaboration, D. Decamp et al., Phys. Lett. [**B245**]{} (1990) 289, DELPHI Collaboration, P. Abreu et al., Z. Phys. [**C 51**]{} (1991) 25. G. Altarelli, R. Barbieri, F. Caravaglios, Nucl. Phys. [**B405**]{} (1993) 3. M. Pawłowski, R. Raczka, Mass generation in the Standard Model without dynamical Higgs field, [*Bull. Board: hep-ph 9403303*]{}. J. Ellis, G.L. Fogli, E. Lisi, Phys. Lett. [**B318**]{} (1993) 148. G. Montagna [*et al.*]{}, The Top Quark and the Higgs Boson Mass from LEP SLC and CDF Data. [*Bull. Board: hep-ph 9407246*]{}. J. Ellis, G.L. Fogli, E. Lisi, CERN-TH.7261/94 and BARI-TH/177-94. G. Altarelli, R. Barbieri, S. Jadach, Nucl. Phys. [**B309**]{} (1992) 3. B.A. Kniehl, High order corrections to Higgs–boson decays [*Bull. Board: hep-ph 9405317*]{}. S. Dittmaier, C. Grosse-Knetter, D. Schildknecht, On the Role of the Higgs Mechanism in Present Electroweak Precision Tests [*Bull. Board: hep-th 9406378*]{}. D. Caerupeel, M. Leblanc, Mass Generation for Gauge Fields without Scalars [*Bull. Board: hep-ph 9407029*]{}. M.J. Herrero, E.R. Morales, Nucl. Phys. [*B418*]{} (1994) 341. U. Canuto [*et al.*]{}, Phys. Rev. [**D16**]{} (1977) 1643. N.D. Birrell, P.C.W. Davies, [*Quantum fields in curved space*]{} (Cambridge University, Cambridge 1982). R.M. Wald, [*General Relativity*]{} (The Univ. of Chicago Press, Chicago 1984). M.A. Castagnino, J.B. Sztrajman, J. Mat. Phys. [**27**]{} (1986) 1037. R. Penrose, Ann. of Phys. [**10**]{} (1960) 171. K.S. Stelle, Phys. Rev. [**D16**]{} (1977) 953. see e.g. S. Deser, P. von Nieuwenhuizen, Phys. Rev. [**D10**]{} (1974) 401. J.V. Narlikar, [*Introduction to Cosmology*]{} (Jones and Bartlet Pub. 1983). D.M. Capper, G. Leibbrandt, M.R. Medrano, Phys. Rev. [**D8**]{} (1973) 4320. see e.g. i) M. Veltman, Nucl. Phys. [**B7**]{} (1968) 637, ii) D.G. Boulware, Ann. of Phys. [**56**]{} (1970) 140, iii) J.C. Taylor, [*Gauge Theories of Weak Interactions*]{} Ch. 2, Cambridge University Press (Cambridge 1976). C. Grosse–Knetter, The Equivalence Theorem for Heavy-Higgs Standard Model and the Gauged Nonlinear $\sigma$–Model [*Bull. Board: hep-ph 9405412*]{}. A. Kniehl, Phys. Rep. [*240*]{} (1994) 211. S. Dittmaier [*et al.*]{}, On the significance of the electroweak precision data BI-TP 94/03. Ya.B. Zel’dovich, Zh. Exp. Teor. Fiz. Pis’ma Red. [**6**]{} (1967) 883 \[JETP Lett. [**6**]{} (1967) 316\]. A.D. Sakharov, Dok. Akad. Nauk SSSR [**177**]{} (1967) 70 \[Sov. Phys. Dokl. [**12**]{} (1968) 1040\]. S.L. Adler, Rev. Mod. Phys. [**54**]{} (1982) 729. S.D. Buchbinder, S.D. Odintsov, I.L. Shapiro, [*Effective Action in Quantum Gravity*]{} (IOP, Bristol, 1992). S.D. Odintsov, I.L. Shapiro, Class. Quantum Grav. [**9**]{} (1992) 873. i) R. Ma' nka and J. Syska, Boson Condensation in the GSW Electroweak Theory, preprin University of Silesian 1993 (to be published in Phys. Rev. D), ii) G. Pocsik, E. Lendvai, G. Cynolter, Acta Phys. Pol. [**B24**]{} (1993) 1495; see also G. Cynolter, E. Lendvai, G. Pocsik, [*$\rho$-parameter in the vector condensate model of electroweak interactions*]{}, preprint Inst. of Theor. Physics Eotvos Lorad Univ. Budapest. See e.g. L.N. Lipatov, JETF [**90**]{} (1986) 1536; R. Kirschner, L.N. Lipatov, L. Szymanowski, Nucl. Phys. B (Proc. Suppl.) [**29A**]{} (1992) p. 84; L.N. Lipatov, [ibid]{} p. 89; V.S.Fadin and L.N. Lipatov, [ibid]{} p. 93 and references contained therein.
|
---
abstract: 'We formulate the problem of a two-level system in a linearly polarized laser field in terms of a nonlinear Riccati-type differential equation and solve the equation analytically in time intervals much shorter than half the optical period. The analytical solutions for subsequent intervals are then stuck together in an iterative procedure to cover the scale time of the laser pulse. This approach is applicable to pulses of arbitrary (nonrelativistic) strengths, shapes and durations, thus covering the whole region of light-matter couplings from weak through moderate to strong ones. The method allows quick insight into different problems from the field of light–matter interaction. Very good quality of the method is shown by recovering with it a number of subtle effects met in earlier numerically calculated photon-emission spectra from model molecular ions, double quantum wells, atoms and semiconductors. The method presented is an efficient mathematical tool to describe novel effects in the region of, e.g., extreme nonlinear optics, i.e., when two–level systems are exposed to pulses of only a few cycles in duration and strength ensuring the Rabi frequency to approach and even exceed the laser light frequence.'
author:
- 'R. Parzyński, M. Sobczak and A. Plucińska'
title: 'An iterative method for extreme optics of two–level systems'
---
Introduction {#int}
============
In the theory of light-matter interactions there is probably no more fundamental model than the two-level one [@allen]. Over the last decade, for example, the model has succedded in explaining the main features of propagation of strong a few-cycle pulses through atomic and semiconductor media [@ziolk; @casp; @hughes1; @hughes2; @kalosha; @tara; @weg1a; @weg1b; @xiao; @cheng; @weg2], e.g., carrier wave Rabi flopping, third-harmonic generation in disguise of second harmonic and carrier-envelope phase effects, to name a few. It has also made a basis for the description of high-order harmonic generation from a single atom [@mil], a symmetric molecular ion [@zuoa; @zuob; @ivanov1] and a double-well quantum structure [@ivanov2; @bavli; @levi] with emphasis on the strongly non-perturbative picture of the phenomenon and the occurrence of peaks in the spectrum of coherently scattered light at the positions of even harmonics. When applied to double wells, the model turned out to be successful also in explaining the effect of laser control of tunneling [@plata].
Despite its dissemination in atomic, molecular and solid state physics the two-level model of light-matter interaction still suffers from the lack of exact analytical solution covering the whole range of laser intensities as well as pulse shapes and durations. The analytical solutions known hitherto cover only some different limiting cases. For instance, the most celebrated rotating-wave-approximation (RWA) solution [@allen] is restricted to laser intensities ensuring the resonance Rabi frequency, $\Omega_{R}$, to be much smaller than the laser frequency, $\omega $. Beyond RWA, the known analytical solutions include the non-RWA corrections along a perturbative procedure (e.g. [@genkin; @parz1]), either are valid in the so-called multiphoton excitation region [@zuoa; @zuob] ($\omega <<\omega_{21}$ along with $\Omega_{R}<< \omega_{21}$, where $\omega_{21}$ stands for the frequency separation between the two levels) or in the quite opposite strong coupling region [@zuoa; @zuob; @ivanov1; @ivanov2; @bavli; @levi] ($\omega >> \omega_{21}$ and $\Omega_{R}
>> \omega_{21}$). Probably, the only analytical solution covering the whole intensity region is the recent one of Tritschler, Mücke and Wegener [@weg3] for a box-shaped pulse, but obtained within the so-called square-wave approximation (SWA) consisting in replacing the actual time behavior of the field within half the optical cycle by a square of a constant appropriately chosen magnitude. Being approximate, this solution was able to reproduce only qualitatively some features of the exact numerical calculations, especially for the case of the resonant excitation ($\omega = \omega_{21}$), but was less convincing for distinctly off-resonant excitation.
The aim of our paper is to present a quick iterative procedure for the problem of the two-level system in linearly polarized laser field, based on an analytical solution of the Schrödinger equation in very short time intervals. The analytical solution turned out to be possible thanks to defining the problem of level populations in terms of a single nonlinear Riccati-type differential equation in conjunction with dividing each halfcycle of the pulse into a number of narrow slices of equal width and considering constant the electric field within each slice with a value determined by the pulse function at the middle of the slice. The analytically obtained solutions for all slices in the pulse are then stick together by a simple recurrence formula derived relating the boundary conditions in the adjacent slices. This approach offers a simple analytical formula for the ratio of level population amplitudes within each slice, resulting in equally simple analytical formulae for population inversion, induced dipole moment and spectrum of the radiation emitted by this dipole. The photon-emission spectra obtained along the above line reproduce the numerically calculated ones available in literature [@weg1a; @weg1b; @mil; @zuoa; @zuob; @ivanov1; @ivanov2; @bavli; @levi]. Moreover, our iterative method indicates weak points of the square-wave solution of Tritschler at el. [@weg3] and is proved to be particularly useful in the area of extreme nonlinear optics [@weg3], i.e., when a few-cycle, strong pulses stimulate significant population dynamics in a two–level system on a time scale of half the optical cycle.
The analytical solution for short time intervals and iterative method {#anal sol}
=====================================================================
When presenting our analytical solution for short time intervals we start with the standard expansion $\psi (t) = b_{1}(t) |1 \rangle +
b_{2}(t) |2 \rangle $ for the wave function of the two-level system in a laser field, where $|1 \rangle $ and $|2 \rangle $ stands for the time-independent opposite-parity eigenstates of the bare system with eigenfrequencies $\omega_{1}$ and $\omega_{2}$, respectively. The time-dependent population amplitudes of the levels, $b_{1}(t)$ and $b_{2} (t)$, are then governed [@ivanov1] by the equation $$i \frac{d}{dt} b_{k}=\omega_{k}b_{k}- \Omega (t) b_{l}(t),
\label{Jxxa}$$ where both $k$ and $l$ run the values $1, 2$ with the constrain $l
\neq k$, and $\Omega (t) = \Omega_{R} h(t)$ is the instantaneous Rabi frequency with $\Omega_{R} = \mu \epsilon_{0}/\hbar$ being the usual Rabi frequency as determined by the dipole transition matrix element $\mu = \langle 1 | ez |2 \rangle $ and the electric field amplitude $\epsilon_{0}$, while $h(t)=f(t) sin (\omega t + \phi )$ describes the incident-field evolution with $f(t)$ having the sense of pulse shape (for pulses of at least few cycles in duration), $\omega $ the carrier frequency and $\phi $ the carrier-envelope offset phase. The latter is known [@weg1a; @weg1b; @weg2; @morgner; @milos; @paulus; @gurt] to be a relevant quantity determining the response of the system in the regime of few-cycle pulses.
Traditionally, it has been solved in different coupling regimes either a set of two linear differential equations for $b_{k}$ with no RWA applied (e.g [@zuoa; @zuob; @ivanov1; @ivanov2]) or more often (e.g. [@mil; @zuoa; @zuob; @levi; @weg3]) the resulting set of three linear differential equations for the Bloch vector components: $u = 2
\rm{Re}(b_{1}^{\star} b_{2})$, $v= 2 \rm{Im}(b_{1}^{\star} b_{2})$, $w=|
b_{2}|^{2} - | b_{1}|^{2}$. Instead, we prefer to work with only one but nonlinear differential equation for the ratio $r(t) =
b_{2}(t)/b_{1}(t)$ of the population amplitudes. Through the population conservation law, $|b_{1}|^{2}+|b_{2}|^{2}= 1$, the above $r$ determines directly both the population inversion $w
=(|r|^{2}-1)/(|r|^{2}+1)$ and the induced dipole moment $d(t)= \langle
\psi (t) | e z |\psi (t) \rangle = \mu u =2 \mu \rm{Re}(r)/(|r|^{2}+1)$ and, consequently, the spectrum of coherently scattered light as well. After differentiating $r$ over time and then using Eq. (\[Jxxa\]) one obtains [@genkin; @parz1] $r$ to fulfil the following differential equation: $$i \frac{dr}{dt} = (r^{2}-1) \Omega (t) +\omega_{21} r,
\label{Jyxa}$$ where $\omega_{21}$ = $\omega _{2}-\omega _{1}$ is the frequency separation between the bare levels. This equation falls into the family of nonlinear Riccati-type equations and a way for its iterative solution results from the transformation $$r(t)=\frac{1}{2\Omega (t)}\left(\Omega ^{eff}(t) R(t) -\omega _{21}\right),
\label{R2oa}$$ $$\Omega ^{eff}(t)=\sqrt {4\Omega ^2(t)+\omega _{21}^2 },
\label{R3oa}$$ converting Eq. (\[Jyxa\]) into $$i\frac{dR}{dt}=(R^{2}-1)\frac{\Omega ^{eff}}{2}+i\left( {\frac{\omega
_{21} }{\Omega ^{eff}} R-1} \right)\frac{\omega _{21} }{\Omega ^{eff}
\Omega } \frac{d\Omega}{dt}.
\label{R4oa}$$ To avoid the cumbersome second term on the right-hand side, including the derivative $d\Omega /dt$, we divide the time scale of the pulse into a number of sufficiently narrow intervals with $t_{j}^{i} \leq
t_j \leq t_{j}^{f} $ being the running time within the $j$th interval. In each interval of its width much shorter than half an optical cycle we approximate the Rabi frequencies $\Omega (t_{j})$ and $\Omega ^{eff}(t_{j})$ as constants of the values which they actually take in the middle $(t_{j}^{m} )$ of the interval. Under such an approximation, Eq. (\[R4oa\]) when adapted to the $j$th interval looks like $i (dR_{j} /dt_{j} )=(R_{j}^{2} -1) \Omega _{j}^{eff}/2$, where $\Omega _{j}^{eff} =\Omega ^{eff}(t_{j} =t_{j}^{m} )$. The resulting equation has straightforward analytical solution $$R_{j} (t_{j} )=\frac{1-i R_{j}^{in} \cot \left(\Omega _{j}^{eff} (t_{j}
-t_{j}^{i} )/2 \right)}{R_{j}^{in} -i \cot \left(\Omega _{j}^{eff} (t_{j}
-t_{j}^{i} )/2 \right)},
\label{R5oa}$$ where $R_{j}^{in} =R_{j} (t_{j} =t_{j}^{i} )$ is the initial value of $R_{j}$, i.e., that at the beginning of the $j$th interval. For the extreme time in the interval, $t_{j} =t_{j}^{f} $, we have $R_{j}
(t_{j} =t_{j}^{f} )=R_{j+1}^{in} $, resulting in the recurrence formula for the initial conditions $$R_{j+1}^{in} =\frac{1-i R_{j}^{in} \cot \left(\Omega _{j}^{eff} (t_{j}^{f}
-t_{j}^{i} )/2 \right)}{R_{j}^{in} -i \cot \left(\Omega _{j}^{eff} (t_{j}^{f}
-t_{j}^{i} )/2 \right)}.
\label{R6oa}$$ As a consequence of equations (\[R5oa\]) and (\[R6oa\]) we obtain from Eq. (\[R2oa\]) the solution for $r_{j}(t_{j})$ $$r_{j} (t_{j} )=\frac{2\Omega _{j}-\left(\omega _{21} +i\Omega
_{j}^{eff} \cot \left(\Omega _{j}^{eff} (t_{j} -t_{j}^{i} )/2 \right)
\right) r_{j}^{in} }{\omega _{21} -i\Omega _{j}^{eff} \cot
\left(\Omega _{j}^{eff} (t_{j} -t_{j}^{i} )/2\right)+ 2\Omega _{j}
r_{j}^{in} }
\label{R7oa}$$ and also the recurrence formula for the initial conditions, $r_j^{in} $, at the beginnings of subsequent time intervals $$r_{j+1}^{in} =\frac{2\Omega_{j}-\left(\omega _{21} + i\Omega
_{j}^{eff} \cot \left(\Omega _{j}^{eff} (t_{j}^{f} -t_{j}^{i}
)/2\right) \right) r_{j}^{in} }{\omega _{21} - i\Omega _{j}^{eff} \cot
\left(\Omega _{j}^{eff} (t_{j}^{f} -t_{j}^{i} )/2 \right ) + 2\Omega
_{j} r_{j}^{in} },
\label{R8oa}$$ where $\Omega _j^{eff} = \sqrt{4\Omega _{j}^2 +\omega _{21}^2 }$ with $\Omega_{j}=\Omega _{R} h_{j}$ and $h_j = f(t_j^m )\sin (\omega t_j^m
+\phi )$. The solutions in the form of Eqs (\[R7oa\]) and (\[R8oa\]) allow us to obtain population inversion, induced dipole moment and electric field of coherently scattered light within the subsequent time intervals, $t_j^i \leq t_j \leq t_j^f $, and to stick the solutions for the intervals to cover the whole time scale of the incident pulse.
Before writing down the final formulae it is convenient to introduce the dimensionless strength parameter $x=\Omega _{R}/ \omega $, the dimensionless level separation parameter $y =\omega_{21}/\omega $ and the dimensionless time parameter $\tau = \omega t$, where $0 \leq \tau
\leq 2\pi N $ for a $N$-cycle pulse. Then, we divide each halfcycle in the $\tau $ domain into $K$ intervals of width $\pi /K$ each, letting $j$ to fall into the range $1\leq j \leq 2NK$. Within the $j$th interval, the running time covers the range $(j-1) \frac{\pi }{K}=\tau
_{j}^{i} \leq \tau _{j} \leq \tau _{j}^{f} =j \frac{\pi }{K}$ and the middle of the interval occurs at $\tau _{j}^{m} =j \frac{\pi
}{K}-\frac{\pi }{2K}$. Also, we make the replacement $r_{j}^{in}
=I_{j} $ and introduce the normalized effective Rabi frequency within the $j$th interval as $x_{j}^{eff} =\Omega _j^{eff} /\omega =
\sqrt{4x_{j}^{2} +y^{2}}$, where $x_{j}= x h_{j}$ with $h_j =f(\tau
_{j}^{m}) \sin (\tau _{j}^{m} +\phi )$. In this language the recurrence formula of Eq. (\[R8oa\]) reads $$I_{j+1} =\frac{2x_{j} -\left(y + ix_{j}^{eff} \cot \left(\pi
x_{j}^{eff} /2K \right)\right) I_{j} }{y - ix_{j}^{eff} \cot
\left(\pi x_{j}^{eff} /2K \right)+ 2x_{j} I_{j} }.
\label{R9oa}$$ For a given field-system parameters ${x, y, \phi , f(\tau )}$, Eq. (\[R9oa\]) allows us to generate the initial conditions for all subsequent $2NK$ time intervals from the only known initial condition $I_{1}$ for the first interval ($I_{1}=0$ throughout this paper). Having generated the initial conditions we calculate the evolution of population inversion within the $j$th time interval from $$\begin{aligned}
w_{j}(\tau_{j})=\frac{-1}{(1+|I_{j}|^{2}) (x_{j}^{eff})^{2}} \Bigl[ y
\left( y (1-|I_{j}|^{2}) + 4 x_{j} \rm{Re}(I_{j})\right) \nonumber \\
+ 4x_{j} \left( x_{j}(1-|I_{j}|^{2}) - y \rm{Re}(I_{j})\right)
\cos (x_{j}^{eff}(\tau_{j}-\tau_{j}^{i})) - 4x_{j} x_{j}^{eff}
\rm{Im}(I_{j}) \sin (x_{j}^{eff}(\tau_{j}-\tau_{j}^{i}))\Bigr],
\label{A4xa}\end{aligned}$$ while the evolution of the induced dipole moment from $$\begin{aligned}
d_{j}(\tau_{j})=\frac{2 \mu }{(1+|I_{j}|^{2}) (x_{j}^{eff})^{2}}
\Bigl[ x_{j} \left( y (1-|I_{j}|^{2}) +4x_{j} \rm{Re}(I_{j})
\right) \nonumber \\ -y\left( x_{j}(1-|I_{j}|^{2}) -y
\rm{Re}(I_{j})\right) \cos(x_{j}^{eff}(\tau_{j}-\tau_{j}^{i}))
+y x_{j}^{eff} \rm{Im}(I_{j})
\sin(x_{j}^{eff}(\tau_{j}-\tau_{j}^{i}))\Bigr], \label{A5xa}\end{aligned}$$ where $0 \leq \tau_{j}-\tau_{j}^{i} \leq \pi /K$ within each interval. Taking the second derivative of Eq. (\[A5xa\]) with respect to $\tau_{j}$ results in the electric field of the coherently scattered light in the forward direction: $$\begin{aligned}
{\cal \epsilon} _{j}(\tau_{j}) \sim \frac{2 \mu y}{1+|I_{j}|^{2}}
\times \nonumber \\ \left[ \left(
x_{j} (1-|I_{j}|^{2})-y \rm{Re}(I_{j}) \right)
\cos(x_{j}^{eff}(\tau_{j}-\tau_{j}^{i})) -x_{j}^{eff}
\rm{Im}(I_{j}) \sin(x_{j}^{eff}(\tau_{j}-\tau_{j}^{i}))\right].
\label{A6xa}\end{aligned}$$ To study spectra we need to take Fourier transforms ($\tau _{j}
\rightarrow z$, where $z$ is the spectrometer frequency in units of the incident light frequency $\omega $) of equations (\[A5xa\]) and (\[A6xa\]) with the results $$\begin{aligned}
d_{j}(z)=\frac{\mu e^{-iz(j-1)\pi
/K}}{(1+|I_{j}|^{2})(x_{j}^{eff})^{2}} \Bigl[i \Bigl( x_{j} \left(
y(1-|I_{j}|^{2}) + 4x_{j} \text{Re} (I_{j})\right) 2 f_{j}^{0}
\nonumber\\ -y \left( x_{j}(1-|I_{j}|^{2}) -y \text{Re} (I_{j})\right)
(f_{j}^{-1}+f_{j}^{+1})\Bigr) + yx_{j}^{eff} \text{Im} (I_{j})
(f_{j}^{-1}-f_{j}^{+1})\Bigr]
\label{A7xa}\end{aligned}$$ and $$\begin{aligned}
{\cal \epsilon }_{j}(z) \sim \frac{\mu y e^{-iz(j-1)\pi
/K}}{1+|I_{j}|^{2}} \times \nonumber \\ \left[ i \left(
x_{j}(1-|I_{j}|^{2}) - y \text{Re}(I_{j}) \right)
(f_{j}^{-1}+f_{j}^{+1}) - x_{j}^{eff} \text{Im}(I_{j})
(f_{j}^{-1}-f_{j}^{+1})\right], \label{A9xa}\end{aligned}$$ where $$f_{j}^{q}=\frac{e^{-i (z+q x_{j}^{eff}) \pi /K} -1}{z+q x_{j}^{eff}}
\label{A0xa}$$ with $q=0, \pm 1$. Finally, to cover the whole time scale of the pulse one needs to sum up equations (\[A4xa\])-(\[A9xa\]) over $j$, taking into account equations (\[R9oa\]) and (\[A0xa\]).
Quality of the iterative method {#3}
===============================
We have extensively examined the accuracy of the iterative method (equations (\[R9oa\]) – (\[A0xa\])) in wide ranges of pulse shapes $f(\tau )$, pulse strengths $x$, carrier frequencies $y$ and carrier-envelope phases $ \phi $. In any case the method was found to be able to fit the results of direct numerical integrations of the Riccati-type Eq. (\[Jyxa\]), provided that $K$, i.e., the number of intervals into which we divide each optical halfcycle was chosen appropriately. Generally, the higher $K$ the better was the quality of the method, as expected. However, $K$ of the order of only a few units or at most ten appeared to be sufficient to ensure good-quality of the method for not too strong pulses ($x \leq 1$). For extremely strong pulses ($x >> 1$), generating fast population dynamics on a time scale of an optical cycle, an increase in $K$ was needed for the method to reproduce all details of the numerical solution. However, even in the latter case only a little of computer time was consumed to accomplish successfully the iterative procedure with the use of the short–time–interval analytical solutions, i.e., equations (\[R9oa\]) - (\[A0xa\]).
To exemplify the effect of better quality of the iterative method when increasing $K$, let us focus on the one–photon resonance ($y=1$) by a pulse of moderate strength ($x=1$). We intentionally take this case because it is covered neither by the strong-coupling ($y<<1$ and $x>>y$) analytical solution of Ivanov et al. [@ivanov1; @ivanov2] nor by the multiphoton-excitation ($y>>1$ and $x<<y$) analytical solution of Zuo et al. [@zuoa; @zuob]. Moreover, to assess the square-wave-approximation (SWA) solution of Tritschler et al. [@weg3] we choose a box-shaped ($f(\tau )=1$) sine-like ($\phi =0$) pulse. The SWA, originally applied to the system of optical Bloch equations, consisted in replacing the sequence of halfcycles of the electric field by the sequence of identical squares, each of a height ensuring the areas under the halfcycle and square to be equal. In terms of our short–time–interval analytical solution, SWA corresponds to the choice of $K=1$ ($h_{j}= (-1)^{j+1}$) and to rescaling $x
\rightarrow \frac{2}{\pi} x$ resulting in $x_{j}= (-1)^{j+1}
\frac{2}{\pi } x$. In this limit our equations for $w_{j}(\tau_{j})$ and $d_{j}(\tau_{j})$ convert into those of Tritschler et al. obtained by a different analytical approach exploiting the Bloch equations. For the pulse of $N=2$ cycles in duration, now available in the laboratory practice (e.g. [@weg3]), we show in Fig. (\[fig1\]a) the effect of $K$ on the population inversion calculated iteratively with the use of Eq. (\[A4xa\]) (solid lines), and compare this result with that obtained by integrating numerically the Riccati-type Eq. (\[Jyxa\]) for $r$ (dotted line). As seen, the choice of $K=10$ ensures nearly perfect coincidence between the two approaches. On the other hand, Fig. (\[fig1\]b) provides a comparison between our iterative results at $K=10$ and the SWA results (dashed line) leading to a conclusion that the square-wave approximation can be used only to general predictions of qualitative nature.
To prove a good quality of our iterative method we now recover with it some numerically calculated spectra of light coherently scattered by two-level systems, available in the literature. One such a two-level system that has received a lot of attention in the past is the lowest pair of different-symmetry electronic levels of the $H_{2}^{+}$ molecular ion ($1\sigma_{g}$ and $1\sigma_{u}$), a pair being well isolated from other levels particularly at large internuclear distances. In particular, Zuo et al. show in Fig. 6b of their paper [@zuob] the two-level numerically calculated photon-emission spectrum of $H_{2}^{+}$ in the near-resonance region translating into our $y=1.1$ and $x=1.86$. The spectrum was obtained by assuming the $f(\tau
)\cos(\tau )$ electric field with $f(\tau )$ gaussian increasing by $10$ optical cycles ($f(\tau ) = exp[-((\tau -20\pi )/10 \pi
)^{2}]$ for $ 0 \leq \tau \leq 20\pi $) and then keeping a constant value up to $30$ cycles ($f(\tau )=1$ for $20\pi < \tau
\leq 60\pi $). For the above conditions, we apply our Eq. (\[A7xa\]) (with $\mu $ put to $1$) along with Eq. (\[R9oa\]) to present in Fig. (\[fig2\]a) the iteratively calculated spectrum $|d(z)|^{2}= |\sum_{j}^{} d_{j}(z)|^{2}$ with $1 \leq j \leq 2NK = 60K$. To achieve high resolution of our spectrum we chose $K=100$ and we will maintain this choice through all other figures to be presented. Our spectrum of Fig. (\[fig2\]a) consists of Mollow triplets occurring at each odd-order harmonic ($1, 3, 5$ and $7$) with the same sideband separation within the triplets. This iteratively obtained structure is in full agreement with the numerical spectrum of Zuo et al. (Fig. 6b in [@zuob]). In a different paper Zuo et al. [@zuoa] give, for the $f(\tau )\cos\tau = \cos\tau $ field, their numerical spectrum for the same system but under the so-called strong-coupling conditions meaning in our notation $y =
0.445$ and $x = 1.9$ ($x/y = \Omega_{R}/\omega_{21} =
4.27$). Under these conditions our iterative spectrum generated from Eq. (\[A7xa\]) for $N=30$ cycle pulse is shown in Fig. (\[fig2\]b). An interesting feature of the iterative spectrum are (besides the familiar odd-order harmonics $3$, $5$ and $7$) the doublets around the positions of even harmonics caused by the large Rabi splittings of the odd harmonics. This spectrum is a counterpart of the numerical spectrum of Zuo et al. (Fig. 7 in [@zuoa]).
Also Ivanov et al. [@ivanov1; @ivanov2] have calculated the emission spectra from molecular ions but using their analytical formula (Eq. (52) in [@ivanov1]) derived in the limiting case of extremely strong coupling ($y<<1$ and $x>>y$ in our language). We applied our Eq. (\[A7xa\]) to this region and obtained with it the results shown in Fig. (\[fig3\]). This figure presents the heights of the odd-harmonic peaks, $H(n)$, normalized to the third harmonic peak, for $y=0.1$ and two values of $x=14.5$ and $15$, respectively. We have assumed a $30$–cycle pulse of the form $f(\tau )\cos\tau $ with $f(\tau )=1$. Our Fig. (\[fig3\]), obtained along the iterative procedure, coincides perfectly with the appropriate results of Ivanov and Corkum (Fig. 3 in [@ivanov1]).
A different place where two-level approximation has appeared to be reliable is a symmetric double quantum well [@ivanov2; @bavli; @levi; @plata] extensively studied in the context of laser control of tunneling and symmetry breaking with strong short pulses. The latter effect results in the appearance of spectral peaks at the positions of even harmonics from the systems with inversion symmetry. For example, Levinson et al.(Fig. 2 in their paper [@levi]) give the spectra from the double-well structure obtained by integrating numerically the set of three Bloch equations for the $f(\tau )\cos\tau $ pulse with $f(\tau )=1$. The frequency-strength parameters in their numerical calculations fall into the strong-coupling region ($y=0.625$, $x=1.25$ in one case (their Fig. 2a) and $y=0.589$, $x=1.178$ in the other case (their Fig. 2b)). For the above two sets of frequency-strength parameters we show in Fig. (\[fig4\]) our iterative spectra resulting from Eq. (\[A7xa\]) for the $30$-cycle pulse assumed. The asymmetric doublets at the positions of the second and fourth harmonics, formed when using the first set of parameters (Fig. (\[fig4\]a)), are seen to coalesce into single peaks when taking the other set (Fig. (\[fig4\]b)). Moreover, the second set of parameters results in shifting the low-frequency component of the spectrum towards zero. Both behaviours of our iteratively obtained spectra are the same as those in the numerical spectra of Levinson et al. (Fig. 2 in [@levi]) and are connected with approaching the so-called accidental degeneracy of two Floquet states of the system [@ivanov2; @bavli; @plata] at some parameters. The parameter $x$ from the second set does nearly satisfy the condition of the accidental degeneracy, i.e., it ensures for the Bessel function $J_{0}(2x)$ to drop to zero [@ivanov2; @plata].
Mücke et al., [@weg1a] have also used the two-level model to simulate numerically the spectra of light emitted around the third harmonic from $GaAs$ semiconductor exposed to $5$fs pulse of $\text{sech}(\tau /\tau_{0}) \cos\tau $ form, where $\tau_{0}=\tau
_{FWHM}/1.763$. The results of their simulations (Fig. 2 in [@weg1a]) reveal the evolution of the third-harmonic peak into a doublet structure when increasing the envelope pulse area $A$. For the $\text{sech}(\tau /\tau_{0})$ envelope, the area $A$ is related to our $x$ parameter through $A= \pi \tau_{0} x
=(2\pi^{2}/1.763)N_{FWHM}x$, where $N_{FWHM}$ is the full width at half maximum (FWHM) measured in optical cycles ($N_{FWHM}=1.71$ in this case). For the conditions close to those of Mücke et al., we present in Fig. (\[fig5\]) our iterative spectra, obtained from Eq. (\[A9xa\]). Our spectra are a qualitative reproduction of the numerical spectra of Mücke et al. (Fig. 2 in [@weg1a]). A possible source of only qualitative agreement in this case is that our spectra are the pure response of the system, i.e., with no propagation effects included which were naturally taken into account in the simulations of Mücke et al. by coupling the Bloch equations to the Maxwell equations.
Carrier–envelope phase effects {#phase eff}
==============================
We now apply the iterative method to calculate, for a particular case, the dependence of the two–level–system response on the phase difference ($\phi$ ) between carrier wave and the maximum of the pulse envelope. To be specific, we make recurrence to the carrier–wave Rabi flopping originally studied numerically by Hughes [@hughes1] for a resonant ($y=1$) pulse $h(\tau )=\text{sech}(\tau /\tau_{0}) \sin(\tau
)$ of a fixed FWHM ($N_{FWHM}=1.72$) but different pulse envelopes $A=19.24 x$. By coupling the optical Bloch equations to the Maxwell equations, Hughes considered propagation of the $A=2 l \pi $ pulses through a two–level medium, where $l$ was an integer. For the areas $A=6\pi - 14\pi $ of Hughes, the left–hand side column of Fig. \[fig6\] shows population differences, $|b|_{1}^{2}-|b|_{2}^{2}$, versus time obtained by our iterative procedure with the use of Eq. (\[A4xa\]). Our results practically do not differ by nothing from the original numerical results obtained by Hughes just near the front–face of the two–level material, i.e., where the propagation effects were not important yet (the left–side column of Fig. 3 in [@hughes1]). Our graphs confirm the original result of Hughes on incomplete Rabi flops at $A\geq 8\pi $. On the other hand, the right–side column of our Fig. \[fig6\] shows our iteratively obtained population differences but for the $h(\tau )=
\text{sech}(\tau /\tau_{0})\cos(\tau )$ pulse, i.e., the pulse with its carrier–envelope phase $\phi $ shifted by $\pi /2$ with respect to the pulse used by Hughes. Some differences caused by this shift are clearly seen in the middle parts of the population difference curves. These parts correspond to the times for which the pulse intensity has already evolved to its high values. The main differences introduced by changing the carrier–envelope phase $\phi $ consist in either converting the double peaks into single ones (and vice versa) or inverting the asymmetry in double peaks.
The above $\phi $–sensitivity of population inversion produces the dependence of the spectrum of scattered light on carrier–envelope phase. In Fig. 7a, we show the spectrum calculated iteratively with the use of Eq. (\[A9xa\]) for the Hughes pulse $h(\tau )=\text{sech}(\tau /\tau_{0})\sin(\tau + \phi )$, i.e., with $\phi =0$, $N_{FWHM}=1.72$, $y=1$ and $x=1.31$ (this $x$ corresponds to the envelope pulse area $A=8\pi $). Except the spectral peak at the fundamental frequency ($z=1$), one sees a well pronounced peak at the position of second harmonic ($z=2$) because the chosen $x$ is in the vicinity of the value (1.178) ensuring the accidental degeneracy ($J_{0}(2x)=0$) of the Floquet states of the system (compare Fig. \[fig4\] and its discussion). A similar peak around $z=2$ was found by Tritschler et al. (Fig. 1a in [@weg2]) on the basis of their numerical solution of the two–level Bloch equations for a different pulse envelope ($\text{sinc}(\tau /\tau_{0})$ instead of $\text{sech}(\tau /\tau_{0})$) and different light–matter parameters ($N_{FWHM}=1.81$, $y=2$, $x=0.76$). In addition to Fig. 7a, we show in Fig. 7b the dependence (calculated iteratively from Eq. (\[A9xa\])) of the height of the spectral peak at $z=2$ on the carrier–envelope phase $0\leq \phi \leq 2\pi $ in the pulse $h(\tau )=\text{sech}(\tau /\tau_{0})\sin(\tau + \phi )$. The $\phi
$–dependence is well seen and has a period of $\pi $ in agreement with fully numerical calculations of Tritschler et al. [@weg2] and Mücke et al. (Fig. 1b in [@weg1b]) exploiting the optical Bloch equations. The same periodicity is seen in Fig. 7c corresponding to the small peak in Fig. 7a around $z=1.5$.
Summary {#concl}
=======
On the basis of a nonlinear Riccati-type equation, analytically solved in very short time intervals (shorter than half the optical period), we have formulated an effective iterative procedure for the problem of a two-level system exposed to a linearly polarized electromagnetic pulse. For different light-matter couplings (from weak through moderate to strong ones), we have proved very good quality of the procedure by recovering with it a number of subtle effects met in the previous numerically calculated photon-emission spectra and population inversion. We have applied the procedure developed to describe some carrier-envelope phase effects in extreme nonlinear optics, particularly in population inversion and spectrum of coherently scattered light. In the regime of a few–cycle pulses, these carrier–envelope phase effects are of current interest [@morgner; @milos; @paulus; @gurt; @sans; @bra].
If necessary, one could reinterpret the interaction Hamiltonian appropriately thus making the iterative procedure applicable to any form of time–dependent coupling within a two–level system.
[99]{} L. Allen and J. H. Eberly, [*Optical resonance and Two-Level Atoms*]{}, Wiley, New York (1975). R. W. Ziolkowski, J. M. Arnold and D. M. Gogny, Phys. Rev. [**A 52**]{}, 3082 (1995). L. W. Casperson, Phys. Rev. [**A 57**]{}, 609 (1998). S. Hughes, Phys. Rev. Lett. [**81**]{}, 3363 (1998). S. Hughes, Phys. Rev. [**A 62**]{}, 055401 (2000). V. P. Kalosha and J. Herrmann, Phys. Rev.Lett [**83**]{}, 544 (1999). A. V. Tarasishin, S. A. Magnitskii and A. M. Zheltikov, Opt. Commun. [**193**]{}, 187 (2001). O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner and F. X. Kärtner, Phys. Rev. Lett. [**87**]{}, 057401 (2001). O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner and F. X. Kärtner, Phys. Rev. Lett. [**89**]{}, 127401 (2002). J. Xiao, Z. Wang and Z. Xu, Phys. Rev. [**A 65**]{}, 031402 (2002). J. Cheng and J. Zhou, Phys. Rev. [**A 67**]{}, 041404 (2003). T. Tritschler, O. D. Mücke, M. Wegener, U. Morgner and F. X. Kärtner, Phys. Rev. Lett. [**90**]{}, 217404 (2003). B. Sundaram and P. W. Milonni, Phys. Rev. [**A 41**]{}, 6571 (1990). T. Zuo, S. Chelkowski and A. D. Bandrauk, Phys. Rev. [**A 48**]{}, 3837 (1993). T. Zuo, S. Chelkowski and A. D. Bandrauk, Phys. Rev. [**A 49**]{}, 3943 (1994). M. Yu. Ivanov and P. B. Corkum, Phys. Rev. [**A 48**]{}, 580 (1993). M. Yu. Ivanov, P. B. Corkum and P. Dietrich, Laser Phys. [**3**]{}, 375 (1993). R. Bavli and H. Metiu, Phys. Rev. Lett. [**69**]{}, 1986 (1992); Phys. Rev. [**A 48**]{}, 886 (1993); Phys. Rev. [**A 47**]{}, 3299 (1993). A. Levinson, M. Segev, G. Almogy and A. Yariv, Phys. Rev. [**A 49**]{}, 661 (1994). J. M. G. Llorente and J. Plata, Phys. Rev. [**A 45**]{}, 6958 (1992); Phys. Rev. [**E 49**]{}, 3547 (1994). G. M. Genkin, Phys. Rev. [**A 58**]{}, 758 (1998). R. Parzyński and M. Sobczak, Opt. Commun. 228, 111 (2003); J.Phys. [**B**]{}: At. Mol. Opt. Phys. 37, 743 (2004). T. Tritschler, O. D. Mücke and M. Wegener, Phys. Rev. [**A 68**]{}, 033404 (2003). U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus and E. P. Ippen, Phys. Rev. Lett. [**86**]{}, 5462 (2001). D. B. Milošević, G. G. Paulus and W. Becker, Phys. Rev. Lett. [**89**]{}, 153001 (2002). G. G. Paulus, F. Lindner, H. Walther, A. Baltuška, E. Goulielmakis, M. Lezius and F. Krausz, Phys. Rev. Lett. [**91**]{}, 253004 (2003). A. Gürtler, F. Robicheaux, W. J. van der Zande and L. D. Noordam, Phys. Rev. Lett. [**92**]{}, 033002 (2004). G. Sansone, C. Vozzi, S. Stagira, M. Pascolini, L. Poletto, P. Villoresi, G. Tondello, S. De Silvestri and M. Nisoli, Phys. Rev. Lett. [**92**]{}, 113904 (2004). T. Brabec and F. Krausz, Rev. Mod. Phys. [**72**]{}, 545 (2000).
|
---
abstract: |
Understanding physical phenomena is a key competence that enables humans and animals to act and interact under uncertain perception in previously unseen environments containing novel object and their configurations. Developmental psychology has shown that such skills are acquired by infants from observations at a very early stage.
In this paper, we contrast a more traditional approach of taking a model-based route with explicit 3D representations and physical simulation by an [*end-to-end*]{} approach that directly predicts stability and related quantities from appearance. We ask the question if and to what extent and quality such a skill can directly be acquired in a data-driven way—bypassing the need for an explicit simulation.
We present a learning-based approach based on simulated data that predicts stability of towers comprised of wooden blocks under different conditions and quantities related to the potential fall of the towers. The evaluation is carried out on synthetic data and compared to human judgments on the same stimuli.
author:
- 'Wenbin Li^1^, Seyedmajid Azimi^1^, Aleš Leonardis^2^, Mario Fritz^1^'
bibliography:
- 'mybib.bib'
title: |
To Fall Or Not To Fall:\
A Visual Approach to Physical Stability Prediction
---
|
---
abstract: |
Drawdowns measuring the decline in value from [the]{} historical running maxima over a given period of time, are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focus on the side of severity by studying the first drawdown over certain [pre-specified]{} size. In this paper, we extend the discussion by investigating the frequency of drawdowns, and some of their inherent characteristics. We [consider]{} two types of drawdown time sequences depending on whether a historical running maximum [is reset or not]{}. For each type, we study the frequency rate of drawdowns, the Laplace transform of the $n$-th drawdown time, the distribution of the running maximum and the value process at the $n$-th drawdown time, as well as some other quantities of interest. Interesting relationships between these two drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent drawdowns are also proposed and priced.
*Keywords*: Drawdown; Frequency; Brownian motion
*MSC*(2000): Primary 60G40; Secondary 60J65 91B2[4]{}
author:
- 'David Landriault[^1]'
- 'Bin Li[^2]'
- 'Hongzhong Zhang[^3]'
title: On the Frequency of Drawdowns for Brownian Motion Processes
---
15.5pt
Introduction
============
We consider a drifted Brownian motion $X=\{X_{t},t\geq0\}$, defined on a filtered probability space $(\Omega,\{\mathcal{F}_{t},t\geq0\},\mathbb{P})$, with dynamics$$X_{t}=x_{0}+\mu t+\sigma W_{t},$$ where $x_{0}\in\mathbb{R}
$ is the initial value, $\mu\in\mathbb{R}
$, $\sigma>0$, and $\{W_{t},t\geq0\}$ is a standard Brownian motion. The time of the first drawdown over size $a>0$ is denoted by $$\tau_{a}:=\inf\{t>0:M_{t}-X_{t}\geq a\}, \label{tau a}$$ where $M=\left\{ M_{t},t\geq0\right\} $ with $M_{t}:=\sup_{s\in\lbrack
0,t]}X_{t}$ is the running maximum process of $X$. Here and henceforth, we follow the convention that $\inf{\emptyset}=\infty$ and $\sup{\emptyset}=0$.
Drawdown is one of the most frequently quoted path-dependent risk indicators for mutual funds and commodity trading advisers (see, e.g., Burghardt et al. [@DrawdownRisk]). From a risk management standpoint, large drawdowns should be considered as extreme events of which both the severity and the frequency need to be investigated. Considerable attention has been paid to the severity aspect of the problem by [pre-specifying]{} a threshold, namely $a>0$, of the size of drawdowns, and subsequently studying various properties associated to the first drawdown time $\tau_{a}$. In this paper, we extend the discussion by investigating the frequency of drawdowns. To this end, we derive the joint distribution of the $n$-th drawdown time, the running maximum, and the value process at the drawdown time for a drifted Brownian motion. Using the general theory on renewal process, we proceed to characterize the behavior of the frequency of drawdown episodes in a long time-horizon. Finally, we introduce some insurance policies which protect against the risk associated with frequent drawdowns. These policies are similar to the sequential barrier options in over-the-counter (OTC) market (see, e.g., Pfeffer [@Pfef01]). Through Carr’s randomization of maturities, we provide closed-form pricing formulas by making use of the main theoretical results of the paper.
Literature review
-----------------
The first drawdown time $\tau_{a}$ is the first passage time of the drawdown process $\left\{ M_{t}-X_{t},t\geq0\right\} $ to level $a$ or above. It has been extensively studied in the literature of applied probability. The joint Laplace transform of $\tau_{a}$ and $M_{\tau_{a}}$ was first derived by Taylor [@Taylor75] for a drifted Brownian motion. Lehoczky [@Lehoczky77] extended the results to a general time-homogeneous diffusion by a perturbation approximation approach. An infinite series expansion of the distribution of $\tau_{a}$ was derived by Douady et al. [@DSY00] for a standard Brownian motion and the results were generalized to a drifted Brownian motion by Magdon et al. [@MDD04]. [The dual of drawdown, known as drawup, measures the increase in value from the historical running minimum over a given period of time. ]{}The probability that a drawdown precedes a drawup is subsequently studied by Hadjiliadis and Vecer [@HadjVece] and Pospisil et al. [@PospVeceHadj] under the drifted Brownian motion and the general time-homogeneous diffusion process, respectively. Mijatovic and Pistorius [@MijatovicPistorius] derived the joint Laplace transform of $\tau_{a}$ and the last passage time at level $M_{\tau_{a}}$ prior to $\tau_{a}$, associated with the joint distribution of the running maximum, the running minimum, and the overshoot at $\tau_{a}$ for a spectrally negative Lévy process. The probability that a drawdown precedes a drawup in a finite time-horizon is studied under drifted Brownian motions and simple random walks in [@ZhanHadj]. [More recently, [@Zhang13; @ZhanHadj12] studied Laplace transforms of the drawdown time, the so-called speed of market crash, and various occupation times at the first exit and the drawdown time for a general time-homogeneous diffusion process.]{}
In quantitative risk management, drawdowns and its descendants have become an increasingly popular and relevant class of path-dependent risk indicators. A portfolio optimization problem with constraints on drawdowns was explicitly solved by Grossman and Zhou [@GrosZhou] in a Black-Scholes framework. Hamelink and Hoesli [@HameHoes04] used the relative drawdown as a performance measure in optimization of real estate portfolios. Chekhlov et al. [@ChekUryaZaba] proposed a new family of risk measures called conditional drawdown and studied parameter selection techniques and portfolio optimization under constraints on conditional drawdown. Some [novel]{} financial derivatives were introduced by Vecer [@Vece06] to hedge maximum drawdown risk. Pospisil and Vecer [@PospVece] invented a class of Greeks to study the sensitivity of investment portfolios to running maxima and drawdowns. Later, Carr et al. [@CarrZhanHaji] introduced a class of European-style digital drawdown insurances and proposed semi-static hedging strategies using barrier options and vanilla options. The swap type insurances and cancelable insurances against drawdowns were studied in Zhang et al. [@ZhanLeunHadj].
Definitions
-----------
[While sustaining downside risk can be appropriately characterized using the drawdown process and the first drawdown time, economic turmoil and volatile market fluctuations are better described by quantities containing more path-wise information, such as the frequency of drawdowns. ]{}The existing knowledge about the first drawdown time $\tau_{a}$ provides only limited and implicit information about the frequency of drawdowns. For the purpose of tackling the problem of frequency directly and systematically, we define below two types of drawdown time sequences depending on whether the last running maximum needs to be recovered or not.
The first sequence $\{\tilde{\tau}_{a}^{n},n\in\mathbb{N}
\}$ is called the *drawdown times with recovery*, defined recursively as$$\tilde{\tau}_{a}^{n}:=\inf\{t>\tilde{\tau}_{a}^{n-1}:M_{t}-X_{t}\geq
a,M_{t}>M_{\tilde{\tau}_{a}^{n-1}}\}, \label{tau til}$$ where $\tilde{\tau}_{a}^{0}=0$. Note that, after each $\tilde{\tau}_{a}^{n-1}$, the corresponding running maximum $M_{\tilde{\tau}_{a}^{n-1}}$ must be recovered before the next drawdown time $\tilde{\tau}_{a}^{n}$. In other words, the running maximum is reset and updated only when the previous one is revisited. Since the sample paths of $X$ are almost surely (a.s.) continuous, we have that $M_{\tilde{\tau}_{a}^{n}}-X_{\tilde{\tau}_{a}^{n}}=a$ a.s. if $\tilde{\tau}_{a}^{n}<\infty$.
The second sequence $\{\tau_{a}^{n},n\in\mathbb{N}
\}$ is called the *drawdown times without recovery*, defined recursively as$$\tau_{a}^{n}:=\inf\{t>\tau_{a}^{n-1}:M_{[\tau_{a}^{n-1},t]}-X_{t}\geq a\},
\label{tau}$$ where $\tau_{a}^{0}:=0$ and $M_{[s,t]}:=\sup_{s\leq u\leq t}X_{u}$. From definition (\[tau\]), it is implicitly assumed that the running maximum $M_{\tau_{a}^{n}}$ is reset to $X_{\tau_{a}^{n}}$ at the drawdown time $\tau_{a}^{n}$. In fact, $\tau_{a}^{n}$ is the so-called iterated stopping times associated with $\tau_{a}$ defined as$$\tau_{a}^{n}=\left\{
\begin{array}
[c]{ll}\tau_{a}^{n-1}+\tau_{a}\circ\theta_{\tau_{a}^{n-1}}, & \text{when }\tau
_{a}^{n-1}\text{ and }\tau_{a}\circ\theta_{\tau_{a}^{n-1}}\text{ are
finite,}\\
\infty, & \text{otherwise,}\end{array}
\right. \label{iterated}$$ where $\theta$ is the Markov shift operator such that $X_{t}\circ\theta
_{s}=X_{s+t}$ for $s,t\geq0$.
Note that both $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$ are independent of the initial value $x_{0}$ for not only the drifted Brownian motion $X$, but also a general Lévy process. In view of definitions (\[tau\]) and (\[tau til\]), it is clear that the following inclusive relation of the two types of drawdown times holds:$$\{\tilde{\tau}_{a}^{n},n\in\mathbb{N}
\}\subset\{\tau_{a}^{n},n\in\mathbb{N}
\}.$$ In other words, for each $n\in\mathbb{N}
$, there exists a unique positive integer $m\geq n$ such that $\tilde{\tau
}_{a}^{n}=\tau_{a}^{m}$ (if $\tilde{\tau}_{a}^{n}<\infty$).
Our motivation for introducing the two drawdown time sequences are as follows. The drawdown times with recovery $\{\tilde{\tau}_{a}^{n},n\in\mathbb{N}
\}$ are easy to identify from the sample paths of $X$ by searching the running maxima. Moreover, they are consistent with definition (\[tau a\]) of the first drawdown $\tau_{a}$ in the sense that a drawdown can be considered as incomplete if the running maximum has not been revisited. However, there are also some crucial drawbacks of $\{\tilde{\tau}_{a}^{n},n\in\mathbb{N}
\}$ which motivate us to introduce the drawdown times without recovery $\{\tau_{a}^{n},n\in\mathbb{N}
\}$. First, the downside risk during recovering periods is neglected. One or more larger drawdowns may occur in a recovering period. Second, the threshold $a$ needs to be adjusted to gain a more integrated understanding about the severity of drawdowns. In other words, the selection of $a$ becomes tricky. Third, the requirement of recovery is too strong. In real world, a historical high water mark may never be recovered again, [as in the case of a financial bubble [@Bubble11].]{}
The rest of the paper is organized as follows. In Section 2, some preliminaries on exit times and the first drawdown time $\tau_{a}$ of the drifted Brownian motion $X$ are presented. In Section 3, the frequency rate of drawdowns, and the Laplace transform of $\tilde{\tau}_{a}^{n}$ associated with the distribution of $M_{\tilde{\tau}_{a}^{n}}$ and/or $X_{\tilde{\tau}_{a}^{n}}$ are derived. Section 4 is parallel to Section 3 but studies the drawdown times without recovery $\{\tau_{a}^{n},n\in\mathbb{N}
\}$. Interesting connections between the two drawdown time sequences are established. In Section 5, some insurance contracts are introduced to insure against the risk of frequent drawdowns.
Preliminaries
=============
Henceforth, for ease of notation, we write $\mathbb{E}_{x_{0}}[\,\cdot
\,]=\mathbb{E}[\left. \cdot\,\right\vert X_{0}=x_{0}]$ for the conditional expectation, $\mathbb{P}_{x_{0}}\{\,\cdot\,\}$ for the corresponding probability and $\mathbb{E}_{x_{0}}[\,\cdot\,;U]=\mathbb{E}_{x_{0}}[\,\cdot\,{1}_{U}]$ with ${1}_{U}$ denoting the indicator function of a set $U\subset\Omega$. In particular, when $x_{0}=0$, we drop the subscript $x_{0}$ from the conditional expectation and probability.
For $x\in\mathbb{R}
$, let $T_{x}^{+}=\inf\left\{ t\geq0:X_{t}>x\right\} $ and$\ T_{x}^{-}=\inf\left\{ t\geq0:X_{t}<x\right\} $ be the first passage times of $X$ to levels in $\left[ x,\infty\right) $ and $\left( -\infty,x\right] $, respectively. For $a<x<b$ and $\lambda>0$, it is known that$$\mathbb{E}_{x}[\mathrm{e}^{-\lambda T_{a}^{-}}]=\mathrm{e}^{\beta_{\lambda
}^{-}(x-a)}\qquad\text{and}\qquad\mathbb{E}_{x}[\mathrm{e}^{-\lambda T_{b}^{+}}]=\mathrm{e}^{\beta_{\lambda}^{+}(x-b)}, \label{one-sided L}$$ where $\beta_{\lambda}^{\pm}=\frac{-\mu\pm\sqrt{\mu^{2}+2\lambda\sigma^{2}}}{\sigma^{2}}$ (see, e.g., formula 2.0.1 on Page 295 of Borodin and Salminen [@borodin2002handbook]). By letting $\lambda\rightarrow0+$ in (\[one-sided L\]), we have$$\mathbb{P}_{x}\left\{ T_{b}^{+}<\infty\right\} =\mathrm{e}^{\frac{-\mu
+|\mu|}{\sigma^{2}}(x-b)}\qquad\text{and}\qquad\mathbb{P}_{x}\left\{
T_{a}^{-}<\infty\right\} =\mathrm{e}^{\frac{-\mu-|\mu|}{\sigma^{2}}(x-a)}.
\label{one-sided P}$$
From Taylor [@Taylor75] or Equation (17) of Lehoczky [@Lehoczky77], we have the following joint Laplace transform of the first drawdown time $\tau_{a}$ and its running maximum $M_{\tau_{a}}$.
\[lem 1\]For $\lambda,s>0$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}-sM_{\tau_{a}}}\right]
=\frac{c_{\lambda}}{b_{\lambda}+s} \label{JL}$$ where $b_{\lambda}=\frac{\beta_{\lambda}^{+}\mathrm{e}^{-\beta_{\lambda}^{-}a}-\beta_{\lambda}^{-}\mathrm{e}^{-\beta_{\lambda}^{+}a}}{\mathrm{e}^{-\beta_{\lambda}^{-}a}-\mathrm{e}^{-\beta_{\lambda}^{+}a}}$ and $c_{\lambda
}=\frac{\beta_{\lambda}^{+}-\beta_{\lambda}^{-}}{\mathrm{e}^{-\beta_{\lambda
}^{-}a}-\mathrm{e}^{-\beta_{\lambda}^{+}a}}$.
A Laplace inversion of (\[JL\]) with respect to $s$ results in $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}};M_{\tau_{a}}>x]=\frac{c_{\lambda}}{b_{\lambda}}\mathrm{e}^{-b_{\lambda}x},\label{L1}$$ for $x>0$. Furthermore, letting $x\rightarrow0+$ in (\[L1\]), we immediately have $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}}]=c_{\lambda}/b_{\lambda}.\label{lap}$$ A numerical evaluation of the distribution function of $\tau_{a}$ (and more generally $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$) by an inverse Laplace transform method will be given at the end of Section 4. Other forms of infinite series expansion of the distribution of $\tau_{a}$ were derived by Douady et al. [@DSY00] and Magdon et al. [@MDD04] for a standard Brownian motion and a drifted Brownian motion, respectively. By taking the derivative with respect to $\lambda$ in (\[lap\]) and letting $\lambda
\rightarrow0+$, we have $$\mathbb{E}[\tau_{a}]=\frac{\sigma^{2}\mathrm{e}^{2\mu a/\sigma^{2}}-\sigma
^{2}-2\mu a}{2\mu^{2}}.$$ It is straightforward to check that $$\lim_{\lambda\rightarrow0+}b_{\lambda}=\lim_{\lambda\rightarrow0+}c_{\lambda
}=\frac{\gamma}{\mathrm{e}^{\gamma a}-1},\label{bc}$$ where $\gamma=\frac{2\mu}{\sigma^{2}}$. In the risk theory literature, the constant $\gamma$ is known as the *adjustment coefficient*. In particular, when $\mu=0$, the quantity $\frac{\gamma}{\mathrm{e}^{\gamma a}-1}$ is understood as $\lim_{\gamma\rightarrow0}\frac{\gamma}{\mathrm{e}^{\gamma a}-1}=\frac{1}{a}$. It follows from (\[lap\]) and (\[bc\]) that $$\mathbb{P}\left\{ \tau_{a}<\infty\right\} =\lim_{\lambda\rightarrow
0+}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}}\right] =1.$$ Furthermore, we have $$\mathbb{P}\left\{ M_{\tau_{a}}\geq x\right\} =\mathbb{P}\left\{ M_{\tau
_{a}}\geq x,\tau_{a}<\infty\right\} =\lim_{\lambda\rightarrow0+}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}};M_{\tau_{a}}\geq x\right]
=\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}.\label{M}$$ which implies that the running maximum at the first drawdown time $M_{\tau
_{a}}$ follows an exponential distribution with mean $\left( \mathrm{e}^{\gamma a}-1\right) /\gamma$ (see, e.g., Lehoczky [@Lehoczky77]).
The drawdown times with recovery
================================
We begin our analysis with the drawdown times with recovery $\{\tilde{\tau
}_{a}^{n},n\in\mathbb{N}
\}$ given that their structure leads to a simpler analysis than their counterpart ones without recovery.
We first consider the asymptotic behavior of the frequency rate of drawdowns with recovery. Let $\tilde{N}_{t}^{a}=\sum\nolimits_{n=1}^{\infty}1_{\left\{
\tilde{\tau}_{a}^{n}\leq t\right\} }$ be the number of drawdowns with recovery observed by time $t\geq0$, and define $\tilde{N}_{t}^{a}/t$ to be the frequency rate of drawdowns. It is clear that $\left\{ \tilde{N}_{t}^{a},t\geq0\right\} $ is a delayed renewal process where the first drawdown time is distributed as $\tau_{a}$, while the subsequent inter-drawdown times are independent and identically distributed as $T_{X_{\tau_{a}}+a}^{+}\circ\tau_{a}$. From Theorem 6.1.1 of Rolski et al. [@Rolskibook], it follows that, with probability one, $$\lim_{t\rightarrow\infty}\frac{\tilde{N}_{t}^{a}}{t}=\left\{
\begin{array}
[c]{lc}\frac{1}{\mathbb{E}[\tau_{a}]+\mathbb{E}[T_{a}^{+}]}=\frac{2\mu^{2}}{\sigma^{2}\left( \mathrm{e}^{2\mu a/\sigma^{2}}-1\right) }, & \text{if }\mu>0,\\
0, & \text{if }\mu\leq0.
\end{array}
\right.$$ Moreover, one could easily obtain some central limit theorems for $\tilde
{N}_{t}^{a}$ by Theorem 6.1.2 of Rolski et al. [@Rolskibook].
Next, we study the joint Laplace transform of $\tilde{\tau}_{a}^{n}$ and $M_{\tilde{\tau}_{a}^{n}}$. Note that $X_{\tilde{\tau}_{a}^{n}}=M_{\tilde
{\tau}_{a}^{n}}-a$ a.s. whenever $\tilde{\tau}_{a}^{n}<\infty$, and thus the following theorem is sufficient to characterize the triplet $\left(
\tilde{\tau}_{a}^{n},M_{\tilde{\tau}_{a}^{n}},X_{\tilde{\tau}_{a}^{n}}\right)
$.
\[thm M til L\]For $n\in\mathbb{N}
$ and $\lambda,x\geq0$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}};M_{\tilde{\tau
}_{a}^{n}}>x\right] =\left( \frac{c_{\lambda}}{b_{\lambda}}\right)
^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}\sum_{m=0}^{n-1}\frac{(b_{\lambda
}x)^{m}}{m!}\mathrm{e}^{-b_{\lambda}x}. \label{M tau til L}$$
To prove this result, we first condition on the first drawdown time $\tau_{a}$ and subsequently on the time for the process $X$ to recover its running maximum. Using the strong Markov property of $X$ and (\[JL\]), it is clear that $$\begin{aligned}
\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}-sM_{\tilde{\tau
}_{a}^{n}}}\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau
}_{a}^{n}-sM_{\tilde{\tau}_{a}^{n}}};\tilde{\tau}_{a}^{n}<\infty\right]
\nonumber\\
& =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}-sM_{\tau_{a}}}\right]
\mathbb{E}\left[ \mathrm{e}^{-T_{a}^{+}}\right] \mathbb{E}\left[
\mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n-1}-sM\tilde{\tau}_{a}^{n-1}}\right]
\nonumber\\
& =\frac{c_{\lambda}}{b_{\lambda}+s}\mathrm{e}^{-\beta_{\lambda}^{+}a}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n-1}-sM\tilde{\tau
}_{a}^{n-1}}\right] \nonumber\\
& =\left( \frac{c_{\lambda}}{b_{\lambda}+s}\right) ^{n-1}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau
_{a}-sM_{\tau_{a}}}\right] \nonumber\\
& =\left( \frac{c_{\lambda}}{b_{\lambda}+s}\right) ^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}. \label{babu}$$ Given that $\left( b_{\lambda}/\left( b_{\lambda}+s\right) \right) ^{n}$ is the Laplace transform of an Erlang random variable (rv) with mean $n/b_{\lambda}$ and variance $n/\left( b_{\lambda}\right) ^{2}$, a tail inversion of (\[babu\]) wrt $s$ yields (\[M tau til L\]).
In particular, letting $x\rightarrow0+$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}}\right] =\left(
c_{\lambda}/b_{\lambda}\right) ^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a},
\label{L tau til}$$ for $n\in\mathbb{N}
$. Furthermore, letting $\lambda\rightarrow0+$ in (\[L tau til\]), together with (\[bc\]) and $\lim_{\lambda\rightarrow0+}\beta_{\lambda}^{+}=\frac
{-\mu+|\mu|}{\sigma^{2}}$, we have $$\mathbb{P}\left\{ \tilde{\tau}_{a}^{n}<\infty\right\} =\left\{
\begin{array}
[c]{lc}1, & \text{if }\mu\geq0,\\
\mathrm{e}^{(n-1)\gamma a}, & \text{if }\mu<0.
\end{array}
\right. \label{tau til P}$$ In other words, a historical running maximum may never be recovered if the drift $\mu<0$.
For $n\in\mathbb{N}
$ and $x>0$, we have$$\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x,\tilde{\tau}_{a}^{n}<\infty\right\} =\left\{
\begin{array}
[c]{lc}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\sum_{m=0}^{n-1}\frac
{1}{m!}\left( \frac{\gamma x}{\mathrm{e}^{\gamma a}-1}\right) ^{m}, &
\text{if }\mu\geq0,\\
\mathrm{e}^{(n-1)\gamma a}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma
a}-1}}\sum_{m=0}^{n-1}\frac{1}{m!}\left( \frac{\gamma x}{\mathrm{e}^{\gamma
a}-1}\right) ^{m}, & \text{if }\mu<0.
\end{array}
\right. \text{.} \label{M tau til P}$$
Substituting (\[L tau til\]) into (\[M tau til L\]) yields $$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}};M_{\tilde{\tau
}_{a}^{n}}>x\right] =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}}\right] \sum_{m=0}^{n-1}\frac{(b_{\lambda}x)^{m}}{m!}\mathrm{e}^{-b_{\lambda}x}. \label{pep}$$ Taking the limit when $\lambda\rightarrow0+$ in (\[pep\]), and then using (\[bc\]), one arrives at$$\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x,\tilde{\tau}_{a}^{n}<\infty\right\} =\mathbb{P}\left\{ \tilde{\tau}_{a}^{n}<\infty\right\}
\sum_{m=0}^{n-1}\frac{(\frac{\gamma x}{\mathrm{e}^{\gamma a}-1})^{m}}{m!}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\text{.}
\label{pep1}$$ Substituting (\[tau til P\]) into (\[pep1\]) results in (\[M tau til P\]).
Note that (\[pep1\]) indicates $$\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x\left\vert \tilde{\tau}_{a}^{n}<\infty\right. \right\} =\sum_{m=0}^{n-1}\frac{1}{m!}\left(
\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}\right) ^{m}\mathrm{e}^{-\frac{\gamma
x}{\mathrm{e}^{\gamma a}-1}}\text{,} \label{abc}$$ for all $\mu\in\mathbb{R}$. This result can be interpreted probabilistically. Indeed, when $\tilde{\tau}_{a}^{n}<\infty$, $M_{\tilde{\tau}_{a}^{m}}-M_{\tilde{\tau}_{a}^{m-1}}$ follows an exponential distribution with mean $\left( \mathrm{e}^{\gamma a}-1\right) /\gamma$ for $m=1,2,...,n$. From the strong Markov property, the rv’s $M_{\tilde{\tau}_{a}^{m}}-M_{\tilde{\tau}_{a}^{m-1}}$ for all $m=1,2,...,n$ are all independent, and thus $M_{\tilde{\tau}_{a}^{n}}=\sum_{m=1}^{n}\left( M_{\tilde{\tau}_{a}^{m}}-M_{\tilde{\tau}_{a}^{m-1}}\right) $ is an Erlang rv with survival function (\[abc\]).
In particular, when $n\rightarrow\infty$, it is easy to check that $\lim_{n\rightarrow\infty}\mathbb{P}\left\{ M_{\tilde{\tau}_{a}^{n}}>x\right\} =\mathbb{P}\left\{ T_{x}^{+}<\infty\right\} $ which agrees with (\[one-sided P\]). For completeness, we conclude this section with a result that is immediate from (\[M tau til L\]) and the fact that $M_{\tilde{\tau
}_{a}^{n}}-X_{\tilde{\tau}_{a}^{n}}=a$ a.s. whenever $\tilde{\tau}_{a}^{n}<\infty$.
For $n\in\mathbb{N}
$ and $x\geq-a$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tilde{\tau}_{a}^{n}};X_{\tilde{\tau
}_{a}^{n}}>x\right] =\left( \frac{c_{\lambda}}{b_{\lambda}}\right)
^{n}\mathrm{e}^{-(n-1)\beta_{\lambda}^{+}a}\sum_{m=0}^{n-1}\frac{\left(
b_{\lambda}(x+a)\right) ^{m}}{m!}\mathrm{e}^{-b_{\lambda}(x+a)}.$$
Drawdown times without recovery
===============================
In this section, we focus on the drawdown times without recovery which are more challenging to analyze than their counterparts with recovery.
Let $N_{t}^{a}=\sum\nolimits_{n=1}^{\infty}1_{\left\{ \tau_{a}^{n}\leq
t\right\} }$ be the number of drawdowns without recovery by time $t\geq0$. Clearly, $\left\{ N_{t}^{a},t\geq0\right\} $ is a renewal process with independent inter-drawdown times, all distributed as $\tau_{a}$. By Theorem 6.1.1 of Rolski et al. [@Rolskibook], it follows that, with probability one,$$\lim_{t\to\infty}\frac{N_{t}^{a}}{t}=\frac{1}{\mathbb{E}\left[ \tau
_{a}\right] }=\frac{2\mu^{2}}{\sigma^{2}\mathrm{e}^{2\mu a/\sigma^{2}}-\sigma^{2}-2\mu a}\text{,}$$ which is consistent with our intuition based on (\[iterated\]). Here again, one can also obtain some central limit theorems for $N_{t}^{a}$ by an application of Theorem 6.1.2 of Rolski et al. [@Rolskibook].
Next, we characterize the joint distribution of $\left( \tau_{a}^{n},X_{\tau_{a}^{n}}\right) $ by deriving an explicit expression for $\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}^{n}};X_{\tau_{a}^{n}}>x]$.
\[yyz\]For $n\in\mathbb{N}
$ and $\lambda,x>0$, the joint distribution of $\left( \tau_{a}^{n},X_{\tau_{a}^{n}}\right) $ satisfies $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}^{n}};X_{\tau_{a}^{n}}>x]=\left(
\frac{c_{\lambda}}{b_{\lambda}}\right) ^{n}\mathrm{e}^{-b_{\lambda}(x+na)}\sum_{m=0}^{n-1}\frac{\left( b_{\lambda}(x+na)\right) ^{m}}{m!}.
\label{L tau x}$$
Given that $X_{\tau_{a}^{n}}+na$ is a positive rv (and $X_{\tau_{a}^{n}}$ is not), we prove (\[L tau x\]) by first deriving an expression for the joint Laplace transform of $\left( \tau_{a}^{n},X_{\tau_{a}^{n}}+na\right) $. By conditioning on the first drawdown time and its associated value process, and by making use of the strong Markov property and (\[JL\]), it is clear that for all $s\ge0$, $$\begin{aligned}
\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}-s\left( X_{\tau_{a}^{n}}+na\right) }\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau
_{a}-s\left( X_{\tau_{a}}+a\right) }\right] \mathbb{E}\left[
\mathrm{e}^{-\lambda\tau_{a}^{n-1}-s\left( X_{\tau_{a}^{n-1}}+\left(
n-1\right) a\right) }\right] \nonumber\\
& =\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}-sM_{\tau_{a}}}\right]
\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-1}-s\left( X_{\tau
_{a}^{n-1}}+\left( n-1\right) a\right) }\right] \nonumber\\
& =\frac{c_{\lambda}}{b_{\lambda}+s}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-1}-s\left( X_{\tau_{a}^{n-1}}+\left( n-1\right)
a\right) }\right] \nonumber\\
& =\left( \frac{c_{\lambda}}{b_{\lambda}+s}\right) ^{n}\text{.} \label{aaa}$$ The Laplace transform inversion of (\[aaa\]) with respect to $s$ results in$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};\left( X_{\tau_{a}^{n}}+na\right) \in\mathrm{d} y\right] =\left( c_{\lambda}\right) ^{n}\frac{y^{n-1}\mathrm{e}^{-b_{\lambda}y}}{\left( n-1\right) !}\mathrm{d}
y\text{,} \label{aaa1}$$ for $y\geq0$. Integrating (\[aaa1\]) over $y$ from $x+na$ to $\infty$ yields (\[L tau x\]).
Letting $s\rightarrow0+$ in (\[aaa\]), it follows that $$\mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}^{n}}]=\left( c_{\lambda}/b_{\lambda
}\right) ^{n}=\left( \mathbb{E}[\mathrm{e}^{-\lambda\tau_{a}}]\right) ^{n}.
\label{L tau}$$ Note that (\[L tau\]) and (\[bc\]) implies that$$\mathbb{P}\left\{ \tau_{a}^{n}<\infty\right\} =1.$$ It is worth pointing out that the relation $\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}}\right] =\left( \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}}\right] \right) ^{n}$ holds more generally for $X$ a general Lévy process or a renewal risk process (also known as the Sparre Andersen risk model [@AndersenRiskmodel]) given that the inter-drawdown times $\tau_{a}^{1}$, and $\left\{ \tau_{a}^{n}-\tau_{a}^{n-1}\right\}
_{n\geq2}$ form a sequence of i.i.d. rvs.
Similarly, letting $\lambda\rightarrow0+$ in (\[L tau x\]), it follows that $$\mathbb{P}\left\{ X_{\tau_{a}^{n}}\geq x\right\} =\mathrm{e}^{-\frac
{\gamma(x+na)}{\mathrm{e}^{\gamma a}-1}}\sum_{m=0}^{n-1}\frac{\left(
\frac{\gamma(x+na)}{\mathrm{e}^{\gamma a}-1}\right) ^{m}}{m!},
\label{X tau P}$$ for $n\in\mathbb{N}
$ and $x\geq-na$. As expected, (\[X tau P\]) is the survival function of an Erlang rv with mean $n\left( \mathrm{e}^{\gamma a}-1\right) /\gamma$ and variance $n\left( \left( \mathrm{e}^{\gamma a}-1\right) /\gamma\right)
^{2}$, later translated by $-na$ units.
Our objective is now to include $M_{\tau_{a}^{n}}$ in the analysis of the $n$-th drawdown time. A result particularly useful to do so is provided in Lemma \[constLT\] which consider a specific constrained Laplace transform of the first passage time to level $x$.
\[constLT\]For $n\in\mathbb{N}
$ and $x>0$, the constrained Laplace transform of $T_{x}^{+}$ together with this first passage time occurring before $\tau_{a}^{n}$ is given by$$\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{n}\right] =\mathrm{e}^{-b_{\lambda}x}\sum_{j=0}^{n-1}\left( c_{\lambda
}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\frac{x(x+ja)^{j-1}}{j!}\text{.}
\label{cLT}$$
We prove this result by induction on $n$. For $n=1$, we have$$\begin{aligned}
\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{1}\right] & =\mathbb{E}\left[ e^{-\lambda T_{x}^{+}}\right]
-\mathbb{E}\left[ e^{-\lambda T_{x}^{+}};T_{x}^{+}>\tau_{a}^{1}\right] \\
& =\mathrm{e}^{-\beta_{\lambda}^{+}x}-\int_{0}^{x}\mathbb{E}\left[
\mathrm{e}^{-\lambda\tau_{a}^{1}}; M_{\tau_{a}^{1}}\in\mathrm{d} y\right]
\,\mathbb{E}_{y-a}\left[ \mathrm{e}^{-\lambda T_{x}^{+}}\right] \\
& =\mathrm{e}^{-\beta_{\lambda}^{+}x}-\int_{0}^{x}c_{\lambda}\mathrm{e}^{-b_{\lambda}y}\,\mathrm{e}^{-\beta_{\lambda}^{+}\left( x-y+a\right)
}\mathrm{d} y\\
& =\mathrm{e}^{-\beta_{\lambda}^{+}x}-c_{\lambda}\mathrm{e}^{-\beta_{\lambda
}^{+}a}\frac{\mathrm{e}^{-\beta_{\lambda}^{+}x}-\mathrm{e}^{-b_{\lambda}x}}{b_{\lambda}-\beta_{\lambda}^{+}}\text{,}$$ where we used in the third equality.
On the other hand, using the fact that $c_{\lambda}\mathrm{e}^{-\beta
_{\lambda}^{+}a}=b_{\lambda}-\beta_{\lambda}^{+}$, we have $$\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{1}\right] =\mathrm{e}^{-b_{\lambda}x}\text{.}$$
We now assume that (\[cLT\]) holds for $n=1,2,...,k-1$ and shows that (\[cLT\]) also holds for $n=k$. Indeed, by the total probability formula,$$\begin{aligned}
\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{k}\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{1}\right] +\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau_{a}^{1}<T_{x}^{+}<\tau_{a}^{k}\right] \nonumber\\
& =\mathrm{e}^{-b_{\lambda}x}+\int_{0}^{x}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}};M_{\tau_{a}}\in\mathrm{d}y\right] \mathbb{E}_{y-a}\left[
\mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{k-1}\right]
\mathrm{d}y\nonumber\\
& =\mathrm{e}^{-b_{\lambda}x}+\int_{0}^{x}c_{\lambda}\mathrm{e}^{-b_{\lambda
}y}\,\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x-y+a}^{+}};T_{x-y+a}^{+}<\tau_{a}^{k-1}\right] \mathrm{d}y\text{.} \label{parta}$$ Substituting (\[cLT\]) at $n=k-1$ into (\[parta\]) yields$$\begin{aligned}
& \mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};T_{x}^{+}<\tau_{a}^{k}\right] \\
& =\mathrm{e}^{-b_{\lambda}x}+c_{\lambda}\mathrm{e}^{-b_{\lambda}\left(
x+a\right) }\sum_{j=0}^{k-2}\int_{0}^{x}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\frac{\left( x-y+a\right) (x-y+\left(
j+1\right) a)^{j-1}}{j!}\mathrm{d}y\\
& =\mathrm{e}^{-b_{\lambda}x}+c_{\lambda}\mathrm{e}^{-b_{\lambda}\left(
x+a\right) }\left( x+\sum_{j=1}^{k-2}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\int_{0}^{x}\left( \frac{\left( y+\left(
j+1\right) a\right) ^{j}}{j!}-a\frac{\left( y+\left( j+1\right) a\right)
^{j-1}}{\left( j-1\right) !}\right) \mathrm{d}y\right) \\
& =\mathrm{e}^{-b_{\lambda}x}\left( 1+c_{\lambda}\mathrm{e}^{-b_{\lambda}a}x+\sum_{j=2}^{k-1}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right)
^{j}\frac{x\left( x+ja\right) ^{j-1}}{j!}\right) \\
& =\mathrm{e}^{-b_{\lambda}x}\sum_{j=0}^{k-1}\left( c_{\lambda}\mathrm{e}^{-b_{\lambda}a}\right) ^{j}\frac{x(x+ja)^{j-1}}{j!}\text{.}$$ This completes the proof.
In the next theorem, we provide a distributional characterization of the $n$-th drawdown time $\tau_{a}^{n}$ with respect to both $M_{\tau_{a}^{n}}$ and $X_{\tau_{a}^{n}}$.
\[jointd\]For $n\in\mathbb{N}
$ and $x>0$, we have $$\begin{aligned}
& \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] \nonumber\\
& =\left( c_{\lambda}\right) ^{n}\mathrm{e}^{-b_{\lambda}(y+na)}\sum
_{m=0}^{n-1}\frac{x(x+ma)^{m-1}(y-x+(n-m)a))^{n-1-m}\mathrm{1}_{\left\{
y-x+(n-m)a\geq0\right\} }}{m!(n-m-1)!}\mathrm{d}y\text{.} \label{JJ}$$
By conditioning on the drawdown episode during which the drifted Brownian motion process $X$ reaches level $x$ for the first time and subsequently using the strong Markov property, we have $$\begin{aligned}
& \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] \nonumber\\
& =\sum_{m=0}^{n-1}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y,\tau_{a}^{m}<T_{x}^{+}<\tau_{a}^{m+1}\right] \nonumber\\
& =\sum_{m=0}^{n-1}\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau
_{a}^{m}<T_{x}^{+}<\tau_{a}^{m+1}\right] \mathbb{E}_{x}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-m}};X_{\tau_{a}^{n-m}}\in\mathrm{d}y\right] \label{J1}$$ From Lemma \[constLT\], we know that$$\begin{aligned}
\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau_{a}^{m}<T_{x}^{+}<\tau_{a}^{m+1}\right] & =\mathbb{E}\left[ \mathrm{e}^{-\lambda T_{x}^{+}};\tau_{a}^{m}<T_{x}^{+}\right] -\mathbb{E}\left[ \mathrm{e}^{-\lambda
T_{x}^{+}};\tau_{a}^{m+1}<T_{x}^{+}\right] \nonumber\\
& =\left( c_{\lambda}\right) ^{m}\frac{x(x+ma)^{m-1}}{m!}\mathrm{e}^{-b_{\lambda}\left( x+ma\right) }\text{.} \label{J2}$$ By Theorem \[yyz\], we have $$\begin{aligned}
& \mathbb{E}_{x}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n-m}};X_{\tau_{a}^{n-m}}\in\mathrm{d}y\right] \nonumber\\
& =\frac{\left( c_{\lambda}\right) ^{n-m}(y-x+(n-m)a)^{n-m-1}\mathrm{e}^{-b_{\lambda}(y-x+(n-m)a)}1_{\left\{ y-x+(n-m)a\geq0\right\} }}{(n-m-1)!}\mathrm{d}y. \label{J3}$$ Substituting (\[J2\]) and (\[J3\]) into (\[J1\]) and simplifying, one easily obtains (\[JJ\]).
Recall that $\tau_{a}^{1}=\tilde{\tau}_{a}^{1}=\tau_{a}$ and $X_{\tau_{a}}=M_{\tau_{a}}-a$ a.s.. Therefore, by letting $\lambda\rightarrow0+$ and $x=a$ in (\[J2\]), it follows that, for $m=0,1,2,\cdots$,$$\begin{aligned}
\mathbb{P}\left\{ \tilde{\tau}_{a}^{2}=\tau_{a}^{2+m}\right\} &
=\mathbb{P}\{\tau_{a}^{m}<T_{a}^{+}<\tau_{a}^{m+1}\}\nonumber\\
& =\frac{(m+1)^{m-1}}{m!}\left( \frac{\gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}\mathrm{e}^{-\frac{\left( m+1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}, \label{eq:f2}$$ which is the probability mass function of a generalized Poisson rv (see, e.g., Equation (9.1) of Consul and Famoye [@Lagrangian2006] with $\theta
=\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$). For completeness, a rv $Y$ has a generalized Poisson$\left( \theta,\lambda\right) $ distribution if its probability mass function $p_{Y}$ is given by $$p_{Y}\left( m\right) =\frac{\theta\left( \theta+\lambda m\right)
^{m-1}e^{-\theta-\lambda m}}{m!}\text{,\qquad}m=0,1,2,...\text{,}$$ when both $\theta,\lambda>0$.
Note that a generalization of (\[eq:f2\]) will be proposed in Theorem \[thm tt\].
\[rk dd\]Equation (\[eq:f2\]) can be interpreted as follows: the number of drawdowns **without** recovery between two successive drawdowns with recovery follows a generalized Poisson distribution with $\theta
=\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$.
The following result connecting the two drawdown time sequences is provided. It should be noted that the rv $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ represents the number of drawdowns without recovery over the first $k$ drawdowns with recovery. When $k=2$, (\[allo\]) coincides with (\[eq:f2\]).
\[thm tt\]For any $k\in\mathbb{N}
$, $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ follows a generalized Poisson distribution with parameters $\theta=(k-1)\gamma a/($$^{\gamma a}-1)$ and $\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$, i.e., for $m=0,1,2,\ldots,$ we have$$\mathbb{P}\left\{ \tilde{\tau}_{a}^{k}=\tau_{a}^{k+m}\right\} =\mathbb{P}\left\{ N_{\tilde{\tau}_{a}^{k}}^{a}=k+m\right\} =\frac{k-1}{m+k-1}\frac{\left( \frac{\left( m+k-1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}}{m!}\mathrm{e}^{-\frac{\left( m+k-1\right) \gamma
a}{\mathrm{e}^{\gamma a}-1}}. \label{allo}$$
It is clear that $\left\{ \tilde{\tau}_{a}^{k}=\tau_{a}^{k+m}\right\}
\ $corresponds to the event that $m$ drawdowns without recovery will occur over the first $k$ drawdowns with recovery, i.e. $$\left\{ \tilde{\tau}_{a}^{k}=\tau_{a}^{k+m}\right\} =\left\{ N_{\tilde
{\tau}_{a}^{k}}^{a}=k+m\right\} \text{.}$$ Next we prove $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ follows a generalized Poisson distribution. By Remark \[rk dd\] and the strong Markov property of $X$, we know that the numbers of drawdowns without recovery between any two successive drawdowns with recovery are i.i.d. and follow a generalized Poisson distribution with $\theta=\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$. Thus, $$N_{\tilde{\tau}_{a}^{k}}^{a}-k=\sum_{i=2}^{k}\left( N_{\tilde{\tau}_{a}^{i}}^{a}-N_{\tilde{\tau}_{a}^{i-1}}^{a}-1\right) \text{,}$$ corresponds to a sum of i.i.d. rv’s with a generalized Poisson distribution $\theta=\lambda=\gamma a/(\mathrm{e}^{\gamma a}-1)$. Using Theorem 9.1 of Consul and Famoye [@Lagrangian2006], we have that $N_{\tilde{\tau}_{a}^{k}}^{a}-k$ follows a generalized Poisson distribution with parameters $\theta=(k-1)\gamma a/($$^{\gamma a}-1)$ and $\lambda=\gamma
a/(\mathrm{e}^{\gamma a}-1)$.
Next, we propose the following corollary which can be viewed as an extension to Taylor [@Taylor75] and Lehoczky [@Lehoczky77] from the first drawdown case to the $n$-th drawdown without recovery.
\[aab\]For $n\in\mathbb{N}
$ and $x>0$, we have$$\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x\right]
=\left( \frac{c_{\lambda}}{b_{\lambda}}\right) ^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}b_{\lambda}^{m}}{m!}\mathrm{e}^{-b_{\lambda}\left(
ma+x\right) }.$$
Taking the integral of (\[JJ\]) with respect to $y$ in $(-na,\infty)$, we have$$\begin{aligned}
\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x\right]
& =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!(n-m-1)!}\int_{x-(n-m)a}^{\infty}\mathrm{e}^{-b_{\lambda}(y+na)}(y-x+(n-m)a)^{n-m-1}\mathrm{d}y\\
& =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!(n-m-1)!}\int
_{0}^{\infty}\mathrm{e}^{-b_{\lambda}(z+x+ma)}z^{n-m-1}\mathrm{d}z\\
& =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!(n-m-1)!}\mathrm{e}^{-b_{\lambda}(x+ma)}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}z}z^{n-m-1}\mathrm{d}z\\
& =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}}{m!b_{\lambda}^{n-m}}\mathrm{e}^{-b_{\lambda}(x+ma)}.\end{aligned}$$ which completes the proof.
The marginal distribution of $M_{\tau_{a}^{n}}$ can easily be obtained from Corollary \[aab\] by letting $\lambda\rightarrow0+$ and subsequently making use of (\[bc\]). Indeed, $$\mathbb{P}\left\{ M_{\tau_{a}^{n}}>x\right\} =\sum_{m=0}^{n-1}\frac{x(x+ma)^{m-1}\left( \frac{\gamma}{\mathrm{e}^{\gamma a}-1}\right)
^{m}}{m!}\mathrm{e}^{-\frac{\gamma(ma+x)}{\mathrm{e}^{\gamma a}-1}}\text{.}
\label{PP}$$ Rearrangements of (\[PP\]) yields$$\mathbb{P}\left\{ M_{\tau_{a}^{n}}>x\right\} =\sum_{k=0}^{n-1}D_{k,n}\frac{\left( \frac{\gamma x}{\mathrm{e}^{\gamma a}-1}\right) ^{k}}{k!}\mathrm{e}^{-\frac{\gamma x}{\mathrm{e}^{\gamma a}-1}}\text{,}
\label{survivalM}$$ where $D_{0,n}=1$, and$$D_{k,n}=\sum_{m=k}^{n-1}\frac{k\left( \frac{m\gamma a}{\mathrm{e}^{\gamma
a}-1}\right) ^{m-k}}{m\left( m-k\right) !}\mathrm{e}^{-\frac{m\gamma
}{\mathrm{e}^{\gamma a}-1}a}=\sum_{m=0}^{n-1-k}\frac{k\left( \frac{\left(
m+k\right) \gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}}{(m+k)m!}\mathrm{e}^{-\frac{\left( m+k\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}\text{,} \label{D1}$$ for $k=1,2,...,n-1$. Note that by substituting $k$ by $k+1$ in (\[allo\]), it follows that (\[D1\]) can be rewritten as$$D_{k,n}=\sum_{m=0}^{n-1-k}\mathbb{P}\left\{ \tilde{\tau}_{a}^{k+1}=\tau
_{a}^{k+1+m}\right\} \text{,}$$ which is equivalent to $$D_{k,n}=\mathbb{P}\left\{ \tilde{\tau}_{a}^{k+1}\leq\tau_{a}^{n}\right\}
=\mathbb{P}\left\{ \tilde{N}_{\tau_{a}^{n}}^{a}>k\right\} \text{.}$$
Then, $$\mathbb{P}\left\{ M_{\tau_{a}^{n}}\in\mathrm{d}y\right\} =\sum_{k=1}^{n}d_{k,n}\frac{\left( \frac{\gamma a}{\mathrm{e}^{\gamma a}-1}\right)
^{k}y^{k-1}e^{-\frac{\gamma a}{\mathrm{e}^{\gamma a}-1}y}}{\left( k-1\right)
!}\mathrm{d}y\text{,}$$ where $\left\{ d_{k,n}\right\} _{k=1}^{n}$ are given by$$\begin{aligned}
d_{k,n} & \equiv D_{k-1,n}-D_{k,n}\\
& =\sum_{j=k}^{n}\frac{k-1}{j-1}\frac{\left( \frac{\left( j-1\right)
\gamma a}{\mathrm{e}^{\gamma a}-1}\right) ^{j-k}}{\left( j-k\right)
!}\mathrm{e}^{-\frac{\left( j-1\right) \gamma a}{\mathrm{e}^{\gamma a}-1}}\left( 1-\sum_{m=0}^{n-j-1}\frac{(m+1)^{m-1}}{m!}\left( \frac{\gamma
a}{\mathrm{e}^{\gamma a}-1}\right) ^{m}\mathrm{e}^{-\frac{\left( m+1\right)
\gamma a}{\mathrm{e}^{\gamma a}-1}}\right) \text{.}$$ In conclusion, $M_{\tau_{a}^{n}}$ follows a mixed-Erlang distribution which is an important class of distribution in risk management (see, e.g., Willmot and Lin [@WillLin] for an extensive review of mixed Erlang distributions).
\[mE\]Note that the distribution of $M_{\tau_{a}^{n}}$ does not come as a surprise. Indeed, one can obtain the structural form of the distribution of $M_{\tau_{a}^{n}}$ by conditioning on $\tilde{N}_{\tau_{a}^{n}}^{a}$, namely the number of drawdowns with recovery over the first $n$ drawdowns (without recovery). Using the strong Markov property of the process $X$ and Equation (\[M\]), it follows that $M_{\tau_{a}^{n}}\left\vert \tilde{N}_{\tau_{a}^{n}}^{a}=m\right. $ is an Erlang rv with mean $m\frac{\mathrm{e}^{\gamma
a}-1}{\gamma}$ and variance $m\left( \frac{\mathrm{e}^{\gamma a}-1}{\gamma
}\right) ^{2}$ for $m=1,2,...,n$. Thus, in (\[survivalM\]), $D_{k,n}$ can be interpreted as the survival function of $\tilde{N}_{\tau_{a}^{n}}^{a}$, i.e. $$D_{k,n}=\mathbb{P}\left\{ \tilde{N}_{\tau_{a}^{n}}^{a}>k\right\}
=\mathbb{P}\left\{ \tilde{\tau}_{a}^{k+1}\leq\tau_{a}^{n}\right\} .$$
The next corollary investigates the actual drawdown $M_{t}-X_{t}$ at $t=\tau_{a}^{n}$.
For $a\leq x\leq na$, we have $$\begin{aligned}
& \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}\leq x\right] \\
& =(c_{\lambda})^{n}\mathrm{e}^{-b_{\lambda}(na-x)}\sum_{m=0}^{n-1}\left(
\frac{(na-x)^{m}}{b_{\lambda}^{n-m}m!}-\frac{\mathrm{1}_{\left\{
x\leq(n-m)a\right\} }((n-m)a-x)^{n-m-1}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}y}y(y+ma)^{m-1}\mathrm{d}y}{m!(n-m-1)!}\right) .\end{aligned}$$
We have$$\begin{aligned}
& \mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}>x\right] \nonumber\\
& =\int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\in\mathrm{d}y\right]
+\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}-X_{\tau_{a}^{n}}>x,X_{\tau_{a}^{n}}\leq-x\right] \nonumber\\
& =\int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x+y,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] +\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};X_{\tau_{a}^{n}}\leq-x\right]
\nonumber\\
& =\int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x+y,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] +\left(
c_{\lambda}/b_{\lambda}\right) ^{n}\left( 1-\mathrm{e}^{-b_{\lambda}(na-x)}\sum_{m=0}^{n-1}\frac{\left( b_{\lambda}(na-x)\right) ^{m}}{m!}\right) , \label{MX}$$ where the last step is due to (\[L tau x\]). Moreover, by Theorem \[jointd\], the first term of (\[MX\])$$\begin{aligned}
& \int_{-x}^{\infty}\mathbb{E}\left[ \mathrm{e}^{-\lambda\tau_{a}^{n}};M_{\tau_{a}^{n}}>x+y,X_{\tau_{a}^{n}}\in\mathrm{d}y\right] \\
& =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{((n-m)a-x)^{n-m-1}\mathrm{1}_{\left\{ -x+(n-m)a\geq0\right\} }}{m!(n-m-1)!}\int_{-x}^{\infty}\mathrm{e}^{-b_{\lambda}(y+na)}(x+y)(x+y+ma)^{m-1}\mathrm{d}y\\
& =(c_{\lambda})^{n}\sum_{m=0}^{n-1}\frac{((n-m)a-x)^{n-m-1}\mathrm{1}_{\left\{ x\leq(n-m)a\right\} }}{m!(n-m-1)!}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}(z-x+na)}z(z+ma)^{m-1}\mathrm{d}z\\
& =(c_{\lambda})^{n}\mathrm{e}^{-b_{\lambda}\left( na-x\right) }\sum
_{m=0}^{n-1}\frac{((n-m)a-x)^{n-m-1}\mathrm{1}_{\left\{ x\leq(n-m)a\right\}
}}{m!(n-m-1)!}\int_{0}^{\infty}\mathrm{e}^{-b_{\lambda}z}z(z+ma)^{m-1}\mathrm{d}z.\end{aligned}$$ Substituting this back into (\[MX\]), we complete the proof.
To complete the section, we consider a numerical example to compare the distribution of the $n$-th drawdown times $\tilde{\tau}_{a}^{n}$ and $\tau
_{a}^{n}$ whose Laplace transforms are given in (\[L tau til\]) and (\[L tau\]), respectively. We implement a numerical inverse Laplace transform approach proposed by Abate and Whitt [@AbatWhit06]. For ease of notation, we denote the cumulative distribution functions of $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$ by $F_{n}$ and $\tilde{F}_{n}$, respectively.
**Table 4.1** Distribution of the $n$-th drawdown times when $a=0.1$ and $\sigma=0.2$$$\begin{tabular}
[c]{c|c|c|c}\hline
& $\mu=0.1$ & $\mu=0$ & $\mu=-0.1$\\\hline
\multicolumn{1}{l|}{\begin{tabular}
[c]{l}\\
$n=1$\\
$n=2$\\
$n=3$\\
$n=4$\\
$n=5$\\
$n=6$\end{tabular}
} & \multicolumn{1}{|l|}{\begin{tabular}
[c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\
0.9779 & 0.9779\\
0.8759 & 0.4865\\
0.6651 & 0.1024\\
0.4060 & 0.0082\\
0.1942 & 0.0002\\
0.0721 & 0.0000
\end{tabular}
} & \multicolumn{1}{|l|}{\begin{tabular}
[c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\
0.9908 & 0.9908\\
0.9366 & 0.4406\\
0.7926 & 0.0885\\
0.5652 & 0.0070\\
0.3262 & 0.0002\\
0.1492 & 0.0000
\end{tabular}
} & \multicolumn{1}{|l}{\begin{tabular}
[c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\
0.9967 & 0.9967\\
0.9719 & 0.3636\\
0.8874 & 0.0663\\
0.7166 & 0.0050\\
0.4871 & 0.0001\\
0.2696 & 0.0000
\end{tabular}
}\\\hline
\end{tabular}
$$
Table 4.1 presents the probabilities that at least $n$ drawdowns with or without recovery occurs before time $1$ for different values of the drift $\mu$. We observe that $F_{n}(1)>\tilde{F}_{n}(1)$ for $n\geq2$ due to the relation between $\tau_{a}^{n}$ and $\tilde{\tau}_{a}^{n}$ given in (\[allo\]). In addition, it shows that $F_{n}(1)$ increases as $\mu$ decreases. However, we observe the opposite trend for $\tilde{F}_{n}(1)$ when $n\geq2$. This is because the previous running maximum is less likely to be revisited for a smaller $\mu$. Since the drawdown risk is in principle a type of downside risk, we think smaller $\mu$ should lead to higher downside risks. In this sense, we suggest that the drawdown times without recovery are better to capture the essence of drawdown risks.
**Table 4.2** Distribution of drawdown times when $a=0.1$ and $\sigma=0.12$$$\begin{tabular}
[c]{c|c|c|c}\hline
& $\mu=0.1$ & $\mu=0$ & $\mu=-0.1$\\\hline
\multicolumn{1}{l|}{\begin{tabular}
[c]{l}\\
$n=1$\\
$n=2$\\
$n=3$\\
$n=4$\\
$n=5$\\
$n=6$\end{tabular}
} & \multicolumn{1}{|l|}{\begin{tabular}
[c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\
0.5663 & 0.5663\\
0.1592 & 0.0339\\
0.0225 & 0.0002\\
0.0016 & 0.0000\\
0.0001 & 0.0000\\
0.0000 & 0.0000
\end{tabular}
} & \multicolumn{1}{|l|}{\begin{tabular}
[c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\
0.7845 & 0.7845\\
0.3755 & 0.0494\\
0.0986 & 0.0002\\
0.0137 & 0.0000\\
0.0010 & 0.0000\\
0.0000 & 0.0000
\end{tabular}
} & \multicolumn{1}{|l}{\begin{tabular}
[c]{ll}$F_{n}(1)$ & $\tilde{F}_{n}(1)$\\
0.9257 & 0.9257\\
0.6509 & 0.0463\\
0.2891 & 0.0002\\
0.0730 & 0.0000\\
0.0099 & 0.0000\\
0.0007 & 0.0000
\end{tabular}
}\\\hline
\end{tabular}
$$
Table 4.2 is the equivalent of Table 4.1 with a lower volatility $\sigma
=0.12$. We notice that $F_{n}(1)$ and $\tilde{F}_{n}(1)$ decrease as $\sigma$ decreases. We also have an interesting observation that the trend of $\tilde{F}_{2}(1)$ is not monotone in $\mu$. Again, this is because the occurrence of $\tilde{\tau}_{a}^{n}$ for $n\geq2$ necessitates a recovery for the previous running maximum. Smaller drift does imply higher drawdown risk, meanwhile the recovery becomes more difficult.
Insurance of frequent relative drawdowns
========================================
In this section, we consider insurance policies protecting against the risk of frequent drawdowns. We denote the price of an underlying asset by $S=\{S_{t},t\geq0\}$, with dynamics $$\mathrm{d}S_{t}=rS_{t}\mathrm{d}t+\sigma S_{t}\mathrm{d}W_{t}^{\mathbb{Q}
}\text{,\qquad}S_{0}=s_{0}\text{,}$$ where $r>0$ is the risk-free rate, $\sigma>0$ and $\{W_{t}^{\mathbb{Q}
},t\geq0\}$ is a standard Brownian motion under a risk-neutral measure $\mathbb{Q}
$. It is well known that $$S_{t}=s_{0}\mathrm{e}^{X_{t}}, \label{SX}$$ where $X_{t}=(r-\frac{1}{2}\sigma^{2})t+\sigma W_{t}^{\mathbb{Q}
}$.
In practice, drawdowns are often quoted in percentage. For fixed $0<\alpha<1$, we denote the time of the first relative drawdown over size $\alpha$ by $$\eta_{\alpha}(S)=\inf\left\{ t\geq0:M_{t}^{S}-S_{t}\geq\alpha M_{t}^{S}\right\} ,$$ where $M_{t}^{S}=\sup_{0\leq u\leq t}S_{u}$ represents the running maximum of $S$ by time $t$. By (\[SX\]), it is easy to see that the relative drawdown of the geometric Brownian motion $S$ corresponds to the actual drawdown of a drifted Brownian motion $X$, namely$$\eta_{\alpha}(S)=\inf\left\{ t\geq0:M_{t}^{X}-X_{t}\geq-\log(1-\alpha
)\right\} =\tau_{\bar{\alpha}}(X),$$ where $\bar{\alpha}=-\log(1-\alpha)$. Similarly, we denote the relative drawdown times with and without recovery by$$\tilde{\eta}_{\alpha}^{n}(S)=\inf\{t>\tilde{\eta}_{\alpha}^{n-1}(S):M_{t}^{S}-S_{t}\geq\alpha M_{t}^{S},M_{t}^{S}>M_{\tilde{\eta}_{\alpha}^{n-1}(S)}^{S}\},$$ and $$\eta_{\alpha}^{n}(S)=\inf\{t>\eta_{\alpha}^{n-1}(S):M_{[\eta_{\alpha}^{n-1}(S),t]}^{S}-S_{t}\geq\alpha M_{[\eta_{\alpha}^{n-1}(S),t]}^{S}\}\text{,}$$ respectively. Therefore, we have $$\tilde{\eta}_{\alpha}^{n}(S)=\tilde{\tau}_{\bar{\alpha}}^{n}(X)\qquad
\text{and\qquad}\eta_{\alpha}^{n}(S)=\tau_{\bar{\alpha}}^{n}(X). \label{it}$$
Next, we consider two types of insurance policies offering a protection against relative drawdowns. For the first one, we assume that the seller pays the buyer $\$k$ at time $T$ if a total of $k$ relative drawdowns over size $0<\alpha<1$ occurred prior to time $T$ (for all $k$). For the relative drawdown times with and without recovery, by (\[it\]), the risk-neutral prices are given by $$\tilde{V}_{1}(T)=\mathrm{e}^{-rT}\sum_{k=1}^{\infty}k\mathbb{Q}
\left\{ \tilde{N}_{T}^{\bar{\alpha}}(X)=k\right\} =\mathrm{e}^{-rT}\mathbb{E}^{\mathbb{Q}
}[\tilde{N}_{T}^{\bar{\alpha}}(X)],$$ and $$V_{1}(T)=\mathrm{e}^{-rT}\sum_{k=1}^{\infty}k\mathbb{Q}
\left\{ N_{T}^{\bar{\alpha}}(X)=k\right\} =\mathrm{e}^{-rT}\mathbb{E}^{\mathbb{Q}
}[N_{T}^{\bar{\alpha}}(X)],$$ respectively. For the second type of policies, the seller pays the buyer $\$1$ at the time of each relative drawdown time as long as it occurs before maturity $T$. Hence, their risk-neutral prices are $$\tilde{V}_{2}(T)=\sum_{k=1}^{\infty}\mathbb{E}^{\mathbb{Q}
}[\mathrm{e}^{-r\tilde{\tau}_{\bar{\alpha}}^{k}(X)};\tilde{\tau}_{\bar{\alpha
}}^{k}(X)\leq T],$$ and $$V_{2}(T)=\sum_{k=1}^{\infty}\mathbb{E}^{\mathbb{Q}
}[\mathrm{e}^{-r\tau_{\bar{\alpha}}^{k}(X)};\tau_{\bar{\alpha}}^{k}(X)\leq
T]\text{,}$$ respectively.
\[price\] For $\lambda>0$, we have$$\begin{array}
[c]{ll}\int_{0}^{\infty}\mathrm{e}^{-\lambda T}V_{1}(T)\mathrm{d}T=\frac{1}{\lambda+r}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}, & \int_{0}^{\infty}\mathrm{e}^{-\lambda
T}\tilde{V}_{1}(T)\mathrm{d}T=\frac{1}{\lambda+r}\frac{\bar{c}_{\lambda
+r}/\bar{b}_{\lambda+r}}{1-\mathrm{e}^{-\bar{\beta}_{\lambda+r}^{+}a}\bar
{c}_{\lambda+r}/\bar{b}_{\lambda+r}},\\
\int_{0}^{\infty}\mathrm{e}^{-\lambda T}V_{2}(T)\mathrm{d}T=\frac{1}{\lambda
}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar{c}_{\lambda+r}/\bar
{b}_{\lambda+r}}, & \int_{0}^{\infty}\mathrm{e}^{-\lambda T}\tilde{V}_{2}(T)\mathrm{d}T=\frac{1}{\lambda}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\mathrm{e}^{-\bar{\beta}_{\lambda+r}^{+}a}\bar{c}_{\lambda
+r}/\bar{b}_{\lambda+r}},
\end{array}$$ where $\bar{b}_{\lambda}=\frac{\bar{\beta}_{\lambda}^{+}\mathrm{e}^{-\bar{\beta}_{\lambda}^{-}\bar{\alpha}}-\bar{\beta}_{\lambda}^{-}\mathrm{e}^{-\bar{\beta}_{\lambda}^{+}\bar{\alpha}}}{\mathrm{e}^{-\bar{\beta
}_{\lambda}^{-}\bar{\alpha}}-\mathrm{e}^{-\bar{\beta}_{\lambda}^{+}\bar
{\alpha}}}$, $\bar{c}_{\lambda}=\frac{\bar{\beta}_{\lambda}^{+}-\bar{\beta
}_{\lambda}^{-}}{\mathrm{e}^{-\bar{\beta}_{\lambda}^{-}\bar{\alpha}}-\mathrm{e}^{-\bar{\beta}_{\lambda}^{+}\bar{\alpha}}}$ and $\bar{\beta
}_{\lambda}^{\pm}=\frac{-r+\frac{1}{2}\sigma^{2}\pm\sqrt{(r-\frac{1}{2}\sigma^{2})^{2}+2\lambda\sigma^{2}}}{\sigma^{2}}$.
We provide the proof for $\int_{0}^{\infty}V_{1}(T)\mathrm{e}^{-\lambda
T}\mathrm{d}T$ and $\int_{0}^{\infty}V_{2}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T$ only. The other two results can be derived in a similar fashion. From the definition of $N_{T}^{\bar{\alpha}}(X)$, we have the following relation $$\mathbb{E}^{\mathbb{Q}
}\left[ N_{T}^{\bar{\alpha}}(X)\right] =\sum_{k=1}^{\infty}\mathbb{Q}\left\{ N_{T}^{\bar{\alpha}}(X)\geq k\right\} =\sum_{k=1}^{\infty}\mathbb{Q}\left\{ \tau_{\bar{\alpha}}^{k}(X)\leq T\right\} .$$ By (\[L tau\]), it follows that$$\begin{aligned}
\int_{0}^{\infty}V_{1}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T & =\int
_{0}^{\infty}\mathrm{e}^{-(\lambda+r)T}\mathbb{E}^{\mathbb{Q}
}[N_{T}^{\bar{\alpha}}(X)]\mathrm{d}T\\
& =\sum_{k=1}^{\infty}\int_{0}^{\infty}\mathrm{e}^{-(\lambda+r)T}\mathbb{Q}\left\{ \tau_{\bar{\alpha}}^{k}(X)\leq T\right\} \mathrm{d}T\\
& =\frac{1}{\lambda+r}\sum_{k=1}^{\infty}\mathbb{E}^{\mathbb{Q}
}[\mathrm{e}^{-(\lambda+r)\tau_{\bar{\alpha}}^{k}(X)}]\\
& =\frac{1}{\lambda+r}\sum_{k=1}^{\infty}\left( \frac{\bar{c}_{\lambda+r}}{\bar{b}_{\lambda+r}}\right) ^{n}\\
& =\frac{1}{\lambda+r}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}.\end{aligned}$$ For $\int_{0}^{\infty}V_{2}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T$, by Fubini’s theorem and (\[L tau\]), we have$$\begin{aligned}
\int_{0}^{\infty}V_{2}(T)\mathrm{e}^{-\lambda T}\mathrm{d}T & =\sum
_{k=1}^{\infty}\int_{0}^{\infty}\mathbb{E}^{\mathbb{Q}
}[\mathrm{e}^{-r\tau_{\bar{\alpha}}^{k}(X)};\tau_{\bar{\alpha}}^{k}(X)\leq
T]\mathrm{e}^{-\lambda T}\mathrm{d}T\\
& =\sum_{k=1}^{\infty}\int_{0}^{\infty}\int_{0}^{T}\mathrm{e}^{-rt}\mathbb{Q}
\left\{ \tau_{\bar{\alpha}}^{k}(X)\in\mathrm{d}t\right\} \mathrm{e}^{-\lambda T}\mathrm{d}T\\
& =\sum_{k=1}^{\infty}\frac{1}{\lambda}\int_{0}^{\infty}\mathrm{e}^{-(\lambda+r)t}\mathbb{Q}
\left\{ \tau_{\bar{\alpha}}^{n}(X)\in\mathrm{d}t\right\} \\
& =\sum_{k=1}^{\infty}\frac{1}{\lambda}\left( \frac{\bar{c}_{\lambda+r}}{\bar{b}_{\lambda+r}}\right) ^{n}\\
& =\frac{1}{\lambda}\frac{\bar{c}_{\lambda+r}/\bar{b}_{\lambda+r}}{1-\bar
{c}_{\lambda+r}/\bar{b}_{\lambda+r}}.\end{aligned}$$ This completes the proof.
It is worth pointing out that, through expansion of the randomized prices in Corollary \[price\] in terms of exponentials, it is possible to obtain semi-static hedging portfolios as in [@CarrZhanHaji]. Moreover, capped insurance contracts against frequency of drawdowns can also be formulated and priced using Theorems \[thm M til L\], \[yyz\], and Corollary \[aab\].
To conclude, we consider a pricing example for the four types of insurance contracts proposed earlier. The same numerical Laplace transform approach as in the last section is applied.
**Table 5.1** Insurance contracts prices when $\alpha=15\%$ and $r=5\%$$$\begin{tabular}
[c]{ll|l|l|l|l}\hline
& & $V_{1}(T)$ & $\tilde{V}_{1}(T)$ & $V_{2}(T)$ & $\tilde{V}_{2}(T)$\\\hline
$T=1$ & $\sigma=0.1$ & 0.1102 & 0.1091 & 0.1120 & 0.1108\\
$T=2$ & $\sigma=0.1$ & 0.3011 & 0.2769 & 0.3131 & 0.2885\\
$T=3$ & $\sigma=0.1$ & 0.4743 & 0.4031 & 0.5058 & 0.4318\\\hline
$T=1$ & $\sigma=0.2$ & 1.1777 & 0.7873 & 1.2043 & 0.8081\\
$T=2$ & $\sigma=0.2$ & 2.3815 & 1.1842 & 2.4977 & 1.2550\\
$T=3$ & $\sigma=0.2$ & 3.4651 & 1.4519 & 3.7279 & 1.5890\\\hline
\end{tabular}
$$
As expected, Table 5.1 shows that type 2 contracts have higher prices than type 1 contracts because of earlier payments (at the moment of each drawdown time instead of the maturity $T$). It also shows that $\tilde{V}_{1}(T)$ and $\tilde{V}_{2}(T)$ are respectively lower than $V_{1}(T)$ and $V_{2}(T)$ due to $\tau_{a}^{n}\leq\tilde{\tau}_{a}^{n}$. All the prices increase as $T$ increases or $\sigma$ increases. Moreover, we can expect that the prices will decrease as $\alpha$ or $r$ increases. The latter is due to a higher discount rate which is the risk-free rate under the risk-neutral measure $\mathbb{Q}
$.
0.5cm
**Acknowledgments.** The authors would like to thank Professor Gord Willmot and an anonymous referee for their helpful remarks and suggestions. Support for David Landriault from a grant from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged, as is support for Bin Li from a start-up grant from the University of Waterloo.
[99]{} (2006). A unified framework for numerically inverting [L]{}aplace transforms. 408–421.
(1957). On the collective theory of risk in case of contagion between claims. 104–125.
(2002). 2nd ed. Birkhauser.
(2003). Deciphering drawdown. S16–S20.
(2011). Maximum drawdown insurance. 1195–1230.
(2005). Drawdown measure in portfolio optimization. 13–58.
(2006). . Birkhauser.
(2000). On probability characteristics of “downfalls” in a standard [B]{}rownian motion. 29–38.
(1993). Optimal investment strategies for controlling drawdowns. 241–276.
(2006). Drawdowns preceding rallies in a [B]{}rownian motion model. 403–409.
(2004). Maximum drawdown and the allocation to real estate. 5–29.
(2011). How to detect an asset bubble. 839–865.
(1977). Formulas for stopped diffusion processes with stopping times based on the maximum. 601–607.
(2004). On the maximum drawdown of a [B]{}rownian motion. 147–161.
(2012). On the drawdown of completely asymmetric [L]{}évy processes. 3812–3836.
(2001). Sequential barrier options. 65–74.
(2010). Portfolio sensitivities to the changes in the maximum and the maximum drawdown. 617–627.
(2009). Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. 2563–2578.
(1999). 1 ed. John Wiley & Sons, New York.
(1975). A stopped [B]{}rownian motion formula. 234–246.
(2006). Maximum drawdown and directional trading. 88–92.
(2011). Risk modelling with the mixed [E]{}rlang distribution. 2–16.
(2015). Occupation times, drawdowns, and drawups for one-dimensional regular diffusions. Forthcoming. Available at http://arxiv.org/pdf/1304.8093.
(2010). Drawdowns and rallies in a finite time-horizon. 293–308.
(2012). Drawdowns and the speed of market crash. 739–752.
(2013). Stochastic modeling and fair valuation of drawdown insurance. 840–850.
[^1]: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada (dlandria@uwaterloo.ca)
[^2]: Corresponding Author: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada (bin.li@uwaterloo.ca)
[^3]: Department of Statistics, Columbia University, New York, NY, 10027, USA (hzhang@stat.columbia.edu)
|
---
abstract: 'We report on the quantitative determination of the strain map in a strained Silicon-On-Insulator (sSOI) line with a $200\times70\ nm^2$ cross-section. In order to study a single line as a function of time, we used an X-ray nanobeam with relaxed coherence properties as a compromise between beam size, coherence and intensity. We demonstrate how it is possible to reconstruct the line deformation at the nanoscale, and follow its evolution as the line relaxes under the influence of the X-ray nanobeam.'
author:
- 'F. Mastropietro'
- 'J. Eymery'
- 'G. Carbone'
- 'S. Baudot'
- 'F. Andrieu'
- 'V. Favre-Nicolin'
bibliography:
- 'Article-SOI.bib'
title: 'Time-dependent relaxation of strained silicon-on-insulator lines using a partially coherent x-ray nanobeam'
---
New applications in optoelectronic and electronic semiconductor devices have been achieved by a careful control of strain at the nanoscale level. Several physical properties such as charge carrier mobility in transistors and emission wavelength in quantum dots or well heterostructure have been advantageously improved by applying strain fields adapted to the materials band structure, orientation and doping features [@lee_strained_2004; @bhattacharya_quantum_2004; @pryor_band-edge_2005; @jacobsen_strained_2006].
The measurement of these strain fields has required the development of dedicated techniques with adapted spatial and strain resolution. Electronic imaging techniques have seen tremendous developments and outstanding achievements [@hue_direct_2008], but are always limited by the preparation of thin foil that can considerably relieve internal stress in nanostructures. Very recently, X-ray diffraction has taken profit of the highly brilliant and coherent radiation provided by synchrotron sources [@nugent_coherent_2010]. Moreover, the optimization of dedicated focusing optics (compound refractive lenses [@snigirev_focusing_1998], Fresnel Zone Plate (FZP) [@jefimovs_fabrication_2007; @gorelick_high-efficiency_2011], Kirkpatrick-Baez mirrors [@kirkpatrick_formation_1948; @paganin_coherent_2006]) has allowed the use of nanobeams, increasing the spatial resolution of diffraction measurements. This also allowed the use of coherent X-ray diffraction imaging (CXDI) for structure (shape, size) and strain determination of single nano-objects [@schroer_coherent_2008; @robinson_coherent_2009; @newton_three-dimensional_2010; @newton_phase_2010; @jacques_bulk_2011].
In this letter, we illustrate how the strain of a single strained silicon nanostructure changes during irradiation with x-rays, as a function of measurement time using a partially coherent X-ray nanobeam. Strained Silicon-On-Insulator (sSOI) lines are considered due to their strong interest for enhancing the carrier mobility in metal oxide semiconductors field-effect-transistors (MOSFET) devices [@andrieu_impact_2007; @baudot_elastic_2009].
Silicon lines were etched from a $(001)$ oriented sSOI substrate made by a wafer bonding technique from the Si deposition on a SiGe virtual substrate imposing a biaxial strain, as described in [@ghyselen_engineering_2004]. Lines in tensile strain ($\epsilon_{yy}=+0.78\%$) are oriented along the $[1\overline{1}0]$ direction which corresponds to the usual direction of n-MOSFET channels for which electron transport is improved. The strain relaxes elastically along $[110]$, i.e. perpendicularly to the lines [@baudot_elastic_2009]. An in-plane misorientation of about 1$^o$ is used between the strained Si lines and the Si substrate in order to separate the line and substrate Bragg peaks. The sSOI lines have a width W=225 nm and a height H=70 nm (Fig. \[figPartialIllum\]) and lie on a 145 nm SiO$_2$ layer. The distance d between two adjacent lines is about 775 nm. Grazing-incidence X-ray diffraction have been performed on these line gratings [@baudot_elastic_2009] and gave $\epsilon_{xx}=+0.04\%$ (almost fully relaxed along the line direction \[110\]) and $\epsilon_{yy}=+0.74\%$, which is very close to the biaxial strain before etching. Linear elasticity calculations allow estimating $\epsilon_{zz}\approx-0.25(5)\%$ from elastic constants.
As recently proven, the displacement field of this system can be probed using coherent beams [@newton_phase_2010]. However in the present system the sSOI lines period is about one micron, so that a *single* line can only be studied using a highly focused X-ray beam. The experiment has been conducted on the undulator beamline id01 of the European Synchrotron Radiation Facility, using a 8 keV energy beam obtained with a Si(111) channel cut monochromator. The X-ray beam has been focused to the sample position using a gold FZP with a diameter of 200 $\mu$m and a 70 nm outermost zone width [@jefimovs_fabrication_2007]. A beam stop and a pinhole have been used to cut the contribution of the central part of the direct beam and the higher diffraction orders.
The asymmetric ($1\overline{1}3$) Bragg reflection has been probed during the experiment to reduce the contribution of the Si substrate. In addition, in the considered geometry, the detection plane is almost parallel to Ewald’s sphere and thus the information about the displacement fields is contained in a single image. The incoming radiation is inclined with respect to the sample surface by $\alpha_i=52.3^\circ$ and the diffracted beam is collected with a Maxipix detector [@ponchut_maxipix_2011] at $2\theta_f=56.48^\circ$.
![Nano-focused X-ray beam at the sample position: (heavy black line) sketch of the rectangular cross-section of the sSOI line, $225\times70\ nm^2$; (black contour levels) normalized intensity distribution at the sample position, calculated using a point source 49 m from the fully illuminated Fresnel Zone plate; (red contour levels) normalized intensity distribution taking into account the source size ($64\times13.8\ \mu m^2$ r.m.s.) for the id01 beamline. Contour levels are represented at 25, 50 and 75 $\%$ of the maximum intensity. The calculations are made at 150 $\mu m$ from the focal point, i.e. the estimated position for the sample. The resulting illumination of the sSOI line is both partially coherent and inhomogeneous. []{data-label="figPartialIllum"}](figPartialIllum-z150.pdf){width="8.5cm"}
In this experiment the transverse coherence length is estimated to $11(h)\times53(v)\mu m^2$ at the location of the FZP (49 m from the source), due to the combined effects of the undulator source size and the monochromator [@diaz_coherence_2010]. Thus, to achieve a highly coherent X-ray beam, a partial illumination of the FZP can be used [@mastropietro_coherent_2011]. However, this would lead to a significant increase in the horizontal size of the focused beam (horizontal FWHM of the order of 1 $\mu m$). In order to illuminate a *single* sSOI line, it is therefore necessary to use a partially coherent X-ray beam by using the entire FZP rather than a small area corresponding to the transverse coherence length.
The use of partially coherent X-ray beams has recently been at the focus of a number of studies, [@flewett_extracting_2009; @whitehead_diffractive_2009] and it was also demonstrated that quantitative reconstructions using *ab initio* phase retrieval algorithms were possible in such a case [@clark_high-resolution_2012; @huang_three-dimensional_2012]. The present study differs from these previous ones: firstly, the X-ray beam illuminating the FZP is more than 10 times larger than the transverse coherence length in the horizontal direction, thus introducing a significant blur in the diffraction image. Secondly, the size of the focal point ideally produced by a point source placed at 49 m from the FZP is much smaller than the width of the sSOI line. This is illustrated in Fig.\[figPartialIllum\], where the cross-section of the sSOI line is compared to the shapes of ideal and partially coherent beams. As the size of the source increases the focal point by a factor $\approx 3$, the simultaneous reconstruction of the object and the coherence function -as proposed in [@clark_simultaneous_2011]- becomes difficult.
In order to make a quantitative analysis of the recorded diffraction images, we used a direct approach to calculate the effect of partial coherence and inhomogeneous illumination from the characteristics of the undulator source: (i) the undulator source was modeled as a superposition of incoherent point sources[^1], with a Gaussian intensity distribution ($64(h)\times13.8(v)\ \mu m^2$ r.m.s. source size). To limit the computational requirements, a rectangular array of $9\times5$ sources was taken into account. Then, (ii) the complex amplitude, focused by the FZP, was calculated at the sample position (at each atomic position in a 2D layer of the silicon line), for each of the 45 point sources. Finally, (iii) the intensity on the detector was calculated as the incoherent sum of the intensities contributed by all the point sources.
![Diffraction from a sSOI line by a partially coherent X-ray nanobeam ($(1\overline{1}3)$ reflection), calculated from a simulated deformation of the sSOI line, using either (a) a plane wave illumination or (b) taking into account the partially coherent illumination of the FZP.[]{data-label="figPartialCohDiff"}](figFEM-PartialIllum2.pdf){width="8.5cm"}
A time-dependent study has been performed illuminating for 3000 s the same section of the selected line with the aim of inducing radiation damage at the Si/SiO$_2$ interface, and investigate its nature and its characteristic time. In our experimental conditions, the x-ray beam carries in term of photon flux $ \approx 5\times10^4$ ph/s/nm$^2$. Several 2D diffraction patterns of the described Bragg peak have been collected during the experiment with an acquisition time of 100 s. The most relevant detector images are shown in Fig. \[soi\_gaussiansource\](a,c,e,g) corresponding to the times: T= 0, 800, 1600, 2900 s.
![(a, c, e, g)Two dimensional diffraction patterns compared to (b, d, f, h) calculations for the chosen silicon line at different times T= 0, 800, 1600 and 2900 s. Relevant changes are visible in the experimental intensity distribution and well reproduced by calculations. Intensities are expressed using the same logarithmic colour scale, with a total amplitude spanning three orders of magnitude. The horizontal stripe near $l=2.996$ is due to the Si substrate, and was masked during the refinement.[]{data-label="soi_gaussiansource"}](soi_obs_calc.pdf){width="8cm"}
The oscillations along $L$ are directly related to the thickness ($\approx$ 70 nm) of the SOI line. The ’banana-shaped’ intensity distribution observed in the \[110\] direction is a consequence of the a spontaneous bending induced by the free-surface strain relaxation already observed [@moutanabbir_observation_2011]. This strain is relaxed during the X-ray exposure.
In order to determine the deformation field inside the SOI line as a function of time, it was not possible to use standard imaging algorithms based on *ab initio* phase retrieval, due to the high degree of partial coherence. Since the SOI line are perfectly cristalline, we have therefore opted to perform a direct refinement of the 2D displacement field in the plane perpendicular to the direction of the SOI Line. The line was modeled as a perfect silicon crystal, with a $70\times 225\ nm^2$ cross-section, and the displacement of the atoms was described using a polynomial sum: $u_z(x,z)=\sum { a_{n_x,n_z}x^{n_x} z^{n_z}}$, with $0<=n_x<=4$ and $0<=n_z<=2$. This simple model allows the description of asymmetric displacements inside the SOI line. Diffraction near the $(1\overline{1}3)$ reflection is not sensitive to displacement in the horizontal ($x$ in Fig. \[figPartialIllum\]) \[110\] direction (perpendicular to the scattering vector) and the displacement field along the direction of the line ($y$) is assumed to be negligible due to mechanical constraints along this direction.
![Displacement fields $u_z$ obtained from the time-dependent analysis of radiation damage at T= 0, 800, 1600 and 2900 s. The $u_z$ profiles is visibly less curved at T=2900s (d) with respect to the one obtained at (a) T= 0. Scale units expressed in nanometers are represented by colours.[]{data-label="uz_profile"}](figUZ.pdf){width="8cm"}
The scattering is calculated by generating the position of all Si atoms in the $xz$ plane from the $u_z$ displacement field, and uses the PyNX library [@favre-nicolin_fast_2011], which allows fast computing using a graphical processing unit. This method also avoids approximating the scattering as the Fourier transform of the average electronic density multiplied by $exp({2i\pi\vec{s}\cdot \vec{u}})$, which can be incorrect in the presence of a large inhomogeneous displacement field.
The $u_z$ polynomial is minimized using first a parallel simulated annealing algorithm [@falcioni_biased_1999], with 50000 trials, followed by a least squares minimization. The criterion for minimization is $\chi^2=\sum_{i}{\omega_i(I_{obs}^i-\alpha\times I_{calc}^i)^2}$, where $\alpha$ is a scale factor, and the weight $\omega_i$ is equal to $1/I_{obs}^i$ if $I_{obs}^i>0$, and 0 otherwise.
The resulting calculated scattering patterns obtained by fitting the experimental data are shown in Fig. \[soi\_gaussiansource\]. The corresponding displacement fields are depicted in Fig \[uz\_profile\]. $u_z$ visibly changes from T=0 s (a) to T=2900 s (d), where it shows a quite less curved profile. The strain fields $\epsilon_{zx}$ and $\epsilon_{zz}$ can be easily calculated from $u_z$: the standard deviations of $\epsilon_{zx}$ and the mean values of $\epsilon_{zz}$ are summarised in Table \[prl:t1\]. $<\sqrt{\langle \epsilon_{zx}^2 \rangle}>$ decreases continuously during the relaxation process while $<\epsilon_{zz}>$ remains around the same value.
Strain 0 s 800 s 1600 s 2900 s
------------------------------------------ ----------- ----------- ----------- -----------
$\sqrt{\langle \epsilon_{zx}^2 \rangle}$ $0.63\%$ $0.57\%$ $0.45\%$ $0.19\%$
$\langle \epsilon_{zz} \rangle$ $-0.19\%$ $-0.21\%$ $-0.21\%$ $-0.18\%$
: Mean strain values at T=0, 800, 1600 and 2900 s.
\[prl:t1\]
Radiation damage resulting from the highly brilliant beam has been observed both for macromolecular compounds [@marchesini_coherent_2003] and semiconductor structure [@polvino_synchrotron_2008; @favre-nicolin_coherent-diffraction_2009]. A bending of unstrained SOI lines was previously reported and attributed to the underlying oxide structural expansion [@shi_radiation-induced_2012]. In this article we show how it is possible to follow quantitatively the evolution of the 2d strain field.
In the present system, the SOI line remains crystalline during the long X-ray exposure, as demonstrated by the continued presence of a Bragg diffraction spot. Moreover, we have shown that the shape of the diffraction peak can be explained by a simple elastic deformation of the line. The relaxation of the line is due to radiation damage occuring at the interface between the silicon line and the underlying SiO$_2$ [@polvino_synchrotron_2008]. There is no significant change with time of the period of the fringes corresponding to the 70 nm thickness, indicating that any damage at the interface does not extend significantly into the silicon line.
It should be noted that $<\epsilon_{zz}>$ does not vary despite the clear relaxation : this difference in the evolution between $\epsilon_{zz}$ and $\epsilon_{xz}$ is due to the fact that only the part of the SOI line around the incident beam is relaxed, and therefore the line remains stressed (in tension) where the Si/SiO$_{2}$ interface is not damaged. This is a very important result for MOSEFT, as a significant alteration to the Si/SiO$_{2}$ interface would not affect conduction properties. Finally, the shape of the diffraction spot remains unchanged after waiting ten minutes without illumination from an X-ray beam, indicating that direct heating effects (dilatation) are negligible.
To conclude, we have shown that it is possible to retrieve the strain field inside a single SOI line, and follow its evolution as a function of time during irradiation with an intense X-ray beam. This information can be retrieved even though we used a partially coherent beam in order to optimize both the flux and the beam size on the sample. Moreover, the use of a strongly asymmetric reflection allowed the data collection of a scattering plane containing all the relevant information about the two-dimensional relaxation of the line, using single two-dimensional frames.
The intense irradiation only damaged the Si/SiO$_2$ interface and not the crystalline silicon structure, which furthermore keeps the strong normal strain essential to the conduction properties. This demonstration paves the way for the study of complex, working devices such as metal-oxide-semiconductor field-effect transistor (MOS-FET): this has already been conducted on micron-sized FET, [@hrauda_x-ray_2011] but the *in situ* analysis of FET on industrially relevant devices will require very small and intense probes, i.e. X-ray beams exploiting the full intensity of undulator source while avoiding strict coherence constraints.
This work was partially supported by the French ANR XDISPE (ANR-11-JS10-004-01).
[^1]: An incoherent description of the source is possible due to the relatively large source size[@vartanyants_coherence_2010] used on ID01
|
---
abstract: 'In the article some algebraic properties of nonlinear two-dimensional lattices of the form $u_{n,xy} = f(u_{n+1}, u_n, u_{n-1})$ are studied. The problem of exhaustive description of the integrable cases of this kind lattices remains open. By using the approach, developed and tested in our previous works we adopted the method of characteristic Lie-Rinehart algebras to this case. In the article we derived an effective integrability conditions for the lattice and proved that in the integrable case the function $f(u_{n+1}, u_n, u_{n-1})$ is a quasi-polynomial satisfying the following equation $\frac{\partial^2}{\partial u_{n+1}\partial u_{n-1}}f(u_{n+1}, u_n, u_{n-1})=Ce^{{\alpha}u_n-{\frac{{\alpha}m}{2}}u_{n+1}-{\frac{{\alpha}k}{2}}u_{n-1}},$ where $C$ and $\alpha$ are constant parameters and $k,\,m$ are nonnegative integers.'
address:
- '$^1$Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, 112 Chernyshevsky Street, Ufa 450008, Russian Federation'
- '$^2$Bashkir State University, 32 Validy Street, Ufa 450076 , Russian Federation'
- '$^3$State Budgetary Educational Institution, G. Almukhametov Republican Boarding School, 25/1 Zorge Street, Ufa 450059, Russian Federation'
author:
- 'I T Habibullin$^{1,2}$, M N Kuznetsova$^1$ and A U Sakieva$^3$'
title: 'Integrability conditions for two-dimensional lattices'
---
Introduction
============
Multidimensional integrable equations, such as the KP and Davey-Stewartson equations, the two-dimensional Toda lattice and so on have important applications in physics and geometry. In recent years, various methods have been developed to study such kind equations (see, for instance, [@Bogdanov]-[@Zakharov]).
In this paper we consider a class of the lattices of the form $$\label{eq_main}
u_{n,xy} = f(u_{n+1}, u_n, u_{n-1}), \quad -\infty < n < \infty,$$ where the unknown $u_n = u_n(x,y)$ depends on the real $x,y$ and the integer $n$. The function $f$ of three variables is assumed to be analytic in a domain $D\subset \mathbb{C}^3$.
Recall that class (\[eq\_main\]) contains such a famous equation as the two-dimensional Toda lattice, which can be represented as (\[eq\_main\]) in the following three ways $$\begin{aligned}
\label{Toda1}
u_{n,xy}& =& e^{u_{n+1} - 2 u_n + u_{n-1}},\\
v_{n,xy} &=& e^{v_{n+1}-v_n} - e^{v_n - v_{n-1}}, \label{Toda2}\\
\label{Toda3}
w_{n,xy}& =& e^{w_{n+1}} - 2 e^{w_{n}} + e^{w_{n-1}}.\end{aligned}$$ The lattices are related to each other by the linear substitutions $w_n=v_{n+1}-v_n$, $v_n=u_{n}-u_{n-1}$.
An important open problem is to describe all of the integrable cases in the class (\[eq\_main\]). The purpose of the present article is to give an appropriate algebraic formalization of the classification problem and derive effective necessary integrability conditions for (\[eq\_main\]). Our investigation is based on the classification scheme, outlined in [@H2013]-[@HabKuznetsova19]. The scheme realizes the commonly accepted opinion, that any integrable equation in 3D admits a large set of integrable in a sense, two dimensional reductions. The hydrodynamic reduction method, developed in [@Tsarev]-[@Ferapontov2006] is one of the fruitful applications of this idea. According to the method 3D equation is integrable if it admits sufficiently large class of 2D reductions in the form of the integrable hydrodynamic type systems. In our study we observed that existence of a sequence of Darboux integrable reductions might be regarded as a sign of integrability as well.
In the case of lattice (\[eq\_main\]) we use the following
[**Definition 1.**]{} [*Lattice (\[eq\_main\]) is called integrable if there exist locally analytic functions $\varphi$ and $\psi$ of two variables such that for any choice of the integers $N_1$, $N_2$ the hyperbolic type system $$\begin{aligned}
&&u_{N_1,xy} = \varphi(u_{N_1+1},u_{N_1}), \nonumber \\
&&u_{n,xy}=f(u_{n+1}, u_{n},u_{n-1}),\qquad N_1 < n < N_2, \label{finite_sys} \\
&&u_{N_2,xy}=\psi(u_{N_2},u_{N_2-1}). \nonumber \end{aligned}$$ obtained from lattice (\[eq\_main\]) is integrable in the sense of Darboux.*]{}
The above integrable models (\[Toda1\])-(\[Toda3\]) are certainly integrable in the sense of our Definition 1 as well. Recall that Darboux integrability means that the system admits a complete set of nontrivial integrals in both characteristic directions of $x$ and $y$. An effective criterion of such kind of integrability is formulated in terms of the characteristic Lie-Rinehart algebras (see Theorem 1 below in §3). Properties of the characteristic algebra and its application for 1+1 dimensional continuous and discrete Darboux integrable systems are studied in [@ZMHS-UMJ]-[@Sakieva]. In the context of 2+1 dimensional models these algebras for the first time have been considered in [@Sh1995]. In our recent works [@H2013]-[@HabKuznetsova19] we studied the classification problem for two-dimensional lattices of a special kind, where the method of Darboux integrable reductions and the algebraic approach were used as basic implements.
We note that discussion on the alternative approaches to the study of Darboux integrable equations based on the higher symmetries, the Laplace invariants, etc. can be found in [@Anderson]-[@Smirnov].
The article is organized as follows. In §2 we recall necessary definitions, introduce the notion of the characteristic Lie-Rinehart algebra and formulate an algebraic criterion of the Darboux integrability. In the third section we study the connection between the structure of the characteristic algebra and the properties of the function $f(u_{n+1},u_{n},u_{n-1})$ assuming that the reduced system (\[finite\_sys\]) is integrable in the sense of Darboux. We prove that $f$ is a quasi-polynomial with respect to all three arguments and give the complete description of its second order derivative $f_{u_{n+1},u_{n-1}}$. In the last fourth section we derive the main result of the article by formulating the necessary condition of integrability for the lattice (\[eq\_main\]). Some tediously long proofs carried over to the Appendix.
Preliminaries
=============
Now let us recall some basic notions of the integrability theory. A function $I=I(x,\bar u,\bar u_x,\bar u_{xx},...)$ depending on a finite number of the dynamical variables is called an $y$-integral of system (\[finite\_sys\]) if it solves equation $D_yI=0$. Here $\bar u$ is a vector $\bar u=(u_{N_1}, u_{N_1 + 1}, \dots, u_{N_2})$, $\bar u_x$ is its derivative and so on. Similarly, a function $J=J(y,\bar u,\bar u_y,\bar u_{yy},...)$ is a $x$-integral if the equation $D_xJ=0$ holds. Integrals of the form $I=I(x)$ and $J=J(y)$, depending only on $x$ and $y$ are called trivial. A system (\[finite\_sys\]) is called integrable in the sense of Darboux if it admits a complete set of functionally independent nontrivial integrals in both characteristic directions $x$ and $y$. Actually it is required that the number of functionally independent integrals is $N_1 + N_2 - 1$ in each direction.
Let us take a nontrivial $y$-integral $I=I(\bar u,\bar u_x,\bar u_{xx},...)$ of the system (\[finite\_sys\]). Obviously the operator $D_y$ acts on $I$ as a vector field of the following form: $$\label{oper}
D_y=\sum_{j=N_1}^{N_2}\left(u_{j,y}\frac{\partial}{\partial{u_j}}+f_{j}\frac{\partial}{\partial{u_{j,x}}}+D_x(f_j)\frac{\partial}{\partial{u_{j,xx}}}+\cdots \right),$$ where $f_j=f(u_{j+1},u_j,u_{j-1})$. The vector field (\[oper\]) in a natural way splits down into a linear combination of the independent operators $X_j$ and $Z$ $$D_y = \sum_{j=N_1}^{N_2} u_{j,y} X_j +Z,$$ where $$\label{eq6}
X_j = \frac{\partial}{\partial u_j}, \quad Z = \sum_{j=N_1}^{N_2} \left( f_j \frac{\partial}{\partial u_{j,x}} + D_x(f_j) \frac{\partial}{\partial u_{j,xx}}+\cdots\right).$$ Obviously equation $D_yI=0$ right away implies that $X_jI=0$ and $ZI=0$.
We study Darboux integrable systems by using characteristic algebras. Denote by $A$ the ring of locally analytic functions of the dynamical variables $\bar u,\bar u_x,\bar u_{xx},\dots$. Let us introduce the Lie algebra $L_y$ with the usual operation $[W_1, W_2]=W_1 W_2-W_2 W_1$, generated by the differential operators $Z$ and $X_j$ defined in (\[eq6\]) over the ring $ A $. We assume the consistency conditions:
- $[W_1,aW_2]=W_1(a)W_2+a[W_1,W_2]$,
- $(aW_1)b=aW_1(b)$
to be valid for any $W_1,W_2\in L_y$ and $a,b\in A$. In other words we request that, if $W_1\in L_y$ and $a\in A$ then $aW_1\in L_y$. In such a case the algebra $L_y$ is called the Lie-Rinehart algebra [@Rinehart], [@Million]. We call it also the characteristic algebra in the direction of $y$. In a similar way the characteristic algebra $L_x$ in the direction of $x$ is defined.
The algebra $L_y$ is of a finite dimension if it admits a basis consisting of a finite number of the elements $Z_1,Z_2,\dots,Z_k\in L_y$ such that an arbitrary operator $Z\in L_y$ is represented as a linear combination of the form $$Z=a_1Z_1+a_2Z_2+\dots a_kZ_k,$$ where the coefficients are functions $a_1,a_2,\dots,a_k\in A$.
Now we are ready to formulate an algebraic criterion of the integrability of a hyperbolic type system in the sense of Darboux [@ZMHSbook; @ZhiberK], which plays a crucial role in our investigation.
[**Theorem 1.**]{} *System (\[finite\_sys\]) admits a complete set of the $y$-integrals (a complete set of the $x$-integrals) if and only if its characteristic algebra $L_y$ (respectively, its characteristic algebra $L_x$) is of a finite dimension.*
[**Corollary of Theorem 1.**]{} [*System (\[finite\_sys\]) is integrable in the sense of Darboux if and only if both characteristic algebras $L_x$ and $L_y$ are of a finite dimension.*]{}
Investigation of the characteristic algebras
============================================
Assume that lattice (\[eq\_main\]) is integrable in the sense of Definition 1. Then for any pair of the integers $N_1,N_2$ hyperbolic type system (\[finite\_sys\]) has to be Darboux integrable. Therefore according to Theorem 1 the algebras $L_x$ and $L_y$ must be of a finite dimension. Since the elements of the algebras are vector fields with an infinite number of components, the problem of clarifying the linear dependence of a set of elements becomes very non-trivial. To this aim the following lemma provides a very useful implement [@ZMHS-UMJ], [@ZMHSbook].
[**Lemma 1.**]{} *If the vector field of the form $$Z = \sum_{i=N_1}^{N_2} z_{1,i} \frac{\partial}{\partial u_{i,x}} + z_{2,i} \frac{\partial}{\partial u_{i,xx}} + \cdots$$ solves the equation $\left[ D_x, Z \right] = 0$, then $Z=0$.*
[**Lemma 2.**]{} *The following formulas hold:* $$[D_x,X_j]=0,\quad [D_x,Z]=-\sum _{j=N_1}^{N_2} f_j X_j.$$
[**Proof of Lemma 2.**]{} The operator $D_y$ acts on the function $F(\bar u,\bar u_x,\bar u_{xx},\dots)$ by the following rule $$D_y=Z+\sum_{j=N_1}^{N_2}u_{j,y}X_j.$$ Since the operators $D_x$, $D_y$ commute with one another, we have the relation: $$[D_x, D_y]=[D_x,Z+\sum_{j=N_1}^{N_2}u_{j,y}X_j]=0.$$ Using properties of the commutators, we get $$[D_x,Z] + \sum_{j=N_1}^{N_2}f_j X_j + \sum_{j=N_1}^{N_2} u_{j,y} [D_x, X_j] =0.$$ By comparing the coefficients in front of the independent variables $u_{j,y}$, we easily obtain the statement of the Lemma 2.
Let us construct a sequence of the operators $Z_0$, $Z_1$, $Z_2, \ldots$ by setting $$Z_0 = \left[ X_0, Z \right], \quad Z_1 = \left[ X_0, Z_0 \right], \ldots, Z_{j+1} = \left[ X_0, Z_j \right], \ldots$$ It is easy to check that the relations hold $$\label{DxZj}
\left[ D_x, Z_j \right] = -\sum^1_{k = -1} X^{j+1}_0 (f_k) X_k,$$ where $f_0 = f(u_1, u_0, u_{-1})$.
[**Lemma 3.**]{} *Suppose that lattice (\[eq\_main\]) is integrable in the sense of Definiton 1, then the function $f = f(u_{1}, u_0, u_{-1})$ is a quasi-polynomial with respect to any of its arguments $u_{-1}$, $u_0$, $u_1$.*
[**Proof.**]{} We suppose that lattice (\[eq\_main\]) is integrable in the sense of Definition 1, then the characteristic algebra should be finite-dimensional. That is why there exists a natural $M$ such that $Z_{M+1}$ is linearly expressed through the previous members of the sequence: $$\label{mmm}
Z_{M+1}=\sum^{M}_{i=0}{{\lambda}_i{Z_{i}}},$$ where the operators $Z_1, Z_2, \ldots Z_M$ are linearly independent. We commute both sides of equality (\[mmm\]) with the operator $D_x$ and due to (\[DxZj\]) we arrive at: $$-\sum^{1}_{j=-1}X^{M+2}_0(f_j)X_j=\sum^{M}_{i = 0}\left\{D_x({\lambda}_i)Z_{i}-{\lambda}_i\sum^{1}_{k=-1}X^{i+1}_{0}(f_k)X_k\right\}.$$ Comparing the coefficients before linearly independent operators $Z_i$ for $i=0,1,...,M$ one gets $D_x({\lambda}_i)=0$ and therefore ${\lambda}_i=const$. Comparison of the factors before $X_j$ yields for $j=-1,0,1$: $$\label{pro}
(X^{M+1}_{0}-{\lambda}_M{X^M}-{\lambda}_{M-1}X^{M-1}-...-{\lambda}_0)X_0(f_j)=0.$$ Thus all three functions $f_{-1}=f(u_0,u_{-1},u_{-2})$, $f_0=f(u_{1},u_0,u_{-1})$ and $f_1=f(u_2,u_{1},u_0)$ are quasi-polynomials on the variable $u_0$, hence evidently $f(u_{1},u_0,u_{-1})$ is a quasi-polynomial with respect to all of its arguments. Lemma 3 is proved.
[**Lemma 4.**]{} *Operator $W_0=\left[X_{1}, \left[ X_{-1}, Z \right]\right]$ satisfies the condition* $$\label{eq9}
\left[D_x, W_0\right]=-f_{u_1u_{-1}}X_0.$$
[**Proof.**]{} Lemma 4 is easily proved by using the Jacobi identity and formula $$\left[D_x, \left[X_1, Z \right] \right] = -\sum^2_{k =0} X_1(f_j)X_j.$$
Let us now construct a sequence of the form: $$\label{seq_W}
W_0, \quad W_1 = \left[X_0, W_0 \right], \quad W_2 = \left[X_0, W_1 \right], \ldots, W_{k+1} = \left[X_0, W_k\right],...$$ Elements of the sequence satisfy the formulas: $$\label{DxWk}
\left[ D_x, W_k \right] = -X^{k}_0(f_{u_1 u_{-1}})X_0.$$ Since the characteristic algebra is finite-dimensional there exists a natural $M$ such that $W_{M+1}$ is linearly expressed through the previous members: $$W_{M+1} + \lambda_M W_M + \cdots + \lambda_1 W_1 + \lambda_0 W_0 = 0,$$ where $W_0, \ldots, W_M$ are linearly independent. We commute both sides of this equality with the operator $D_x$ and apply formula (\[DxWk\]). Thus we obtain the relations $D_x(\lambda_j)=0$ satisfied for $j=0,1,...,M$ and an equation $$X^{M+1}_0(f_{u_1 u_{-1}}) X_0 + \lambda_M X^M_0(f_{u_1 u_{-1}})X_0 + \cdots + \lambda_0 f_{u_1 u_{-1}} X_0 = 0.$$ Obviously the latter implies: $$\label{char_pol1}
\left(X^{M+1}_0 + \lambda_M X^M_0 + \cdots + \lambda_0 \right) f_{u_1 u_{-1}} = 0.$$ Let us denote through $\Lambda(\lambda)$ the characteristic polynomial of this linear ordinary differential equation, i.e. $$\label{Lambda_00}
\Lambda(\lambda):=\lambda^{M+1} + \lambda_M \lambda^M + \cdots + \lambda_0.$$ Then we have that the differential operator $\Lambda(X_0)$ turns the function $g = f_{u_1 u_{-1}}$ to zero: $$\label{Lambda_0}
\Lambda(X_0)g(u_{1}, u_0, u_{-1}) = 0$$ and there is no any operator of lower order which annulates $g$.
Further it will be convenient to use the following notation for the commutator of two operators: $ad_X Y = \left[X, Y \right]$. In terms of this new notation, members of the sequence (\[seq\_W\]) are written as: $$W_0, \quad ad_{X_0} W_0, \quad ad^2_{X_0} W_0, \quad \ldots, \quad ad^{k+1}_{X_0} W_0.$$ Formula (\[DxWk\]) takes the form: $$\label{DxWk_new}
\left[ D_x, ad^k_{X_0}W_0 \right] = -X^{k}_0(f_{u_1 u_{-1}})X_0.$$
[**Lemma 5.**]{} *Assume that the characteristic polynomial $\Lambda(\lambda)$ admits two different roots $\lambda = \alpha$ and $\lambda = \beta$. Then either a) $\alpha = - \beta$ or b) $\alpha = - 2 \beta$.*
[**Proof.**]{} Let us construct polynomials $\Lambda_{\alpha} (\lambda)$ and $\Lambda_{\beta}(\lambda)$ by the following rule: $$\Lambda_{\alpha} (\lambda)= \frac{\Lambda(\lambda)}{\lambda - \alpha}, \quad \Lambda_{\beta}(\lambda) = \frac{\Lambda(\lambda)}{\lambda - \beta}.$$ Then the operators $$P_{\alpha}=\Lambda_{\alpha}(ad_{X_0}W_0), \quad P_{\beta}=\Lambda_{\beta}(ad_{X_0}W_0)$$ satisfy the relations $$\label{lop}
\left[D_x,P_{\alpha}\right]=A(u_{1},u_{-1})e^{\alpha{u_0}}X_0, \qquad \left[D_x,P_{\beta}\right]=B(u_{1},u_{-1})e^{\beta{u_0}}X_0,$$ where functions $A$ and $B$ don’t vanish identically. These formulas are easily proved, let us begin the first one. The operation of commutation with $D_x$ by virtue of (\[DxWk\_new\]) satisfies the formula: $$\label{ss}
\left[D_x,\Lambda_{\alpha}(ad_{X_0}W_0)\right]=-\Lambda_{\alpha}(X_0)gX_0.$$ Now we have to specify the factor $g_0:=\Lambda_{\alpha}(X_0)g$, that is a solution of the equation $(X_0-\alpha)g_0=0$. Indeed, since $\Lambda(X_0)g=(X_0-\alpha)\Lambda_{\alpha}(X_0)g$ then we get the former equation which implies $g_0=A(u_{1},u_{-1})e^{\alpha{u_0}}$, where $A(u_{1},u_{-1})$ is a nonzero quasi-polynomial on $u_1,u_{-1}$. The second formula of (\[lop\]) is proved in a similar way.
We define a sequence of multiple commutators in such a way $$R_1=\left[P_{\alpha},P_{\beta}\right],\quad R_2=\left[P_{\alpha},R_1\right],\quad...,\quad R_{k+1}=\left[P_{\alpha},R_k\right],\quad...$$ Let us evaluate the commutator $\left[D_x, R_1 \right]$. Due to the Jacobi identity we have $$\begin{aligned}
\fl \left[D_x,R_1\right] = \left[ D_x, \left[P_{\alpha}, P_{\beta} \right]\right] = \left[ P_{\alpha}, \left[ D_x, P_{\beta} \right] \right] - \left[P_{\beta}, \left[ D_x, P_{\alpha} \right] \right] = \nonumber\\
= \left[P_{\alpha},B(u_1, u_{-1})e^{\beta{u_0}}X_0\right]-\left[P_{\beta},A(u_1, u_{-1})e^{\alpha{u_0}}X_0\right]. \label{eq40} \end{aligned}$$ By construction the vector fields $P_{\alpha}$, $P_{\beta}$ do not contain differentiation with respect to the variables $u_1, u_0, u_{-1}$, therefore (\[eq40\]) implies $$\label{eq_41}
\left[ D_x, R_1 \right] = -B(u_1, u_{-1}) e^{\beta u_0} \left[ X_0, P_{\alpha} \right] + A(u_1, u_{-1}) e^{\alpha u_0} \left[ X_0, P_{\beta} \right].$$ It remains to evaluate the commutators $\left[ X_0, P_{\alpha} \right]$ and $\left[X_0, P_{\beta} \right]$. To this aim find their commutators with $D_x$: $$\left[D_x, \left[X_0, P_{\alpha} \right] \right] = \alpha A e^{\alpha u_0} P_{\alpha}, \qquad \left[D_x, \left[X_0, P_{\beta} \right] \right] = \beta B e^{\beta u_0} P_{\beta}.$$ Now due to (\[lop\]) we get $$\left[ D_x, \left[X_0, P_{\alpha} \right] - \alpha P_{\alpha} \right] = 0, \qquad \left[ D_x, \left[X_0, P_{\beta} \right] - \beta P_{\beta} \right] = 0.$$ The last two equations imply in virtue of Lemma 1 the desired relations $\left[X_0, P_{\alpha} \right] = \alpha P_{\alpha}$, $\left[ X_0, P_{\beta} \right] = \beta P_{\beta}$. Thus (\[eq\_41\]) gives rise to $$\left[ D_x, R_1 \right] = -\alpha B e^{\beta u_0} P_{\alpha} + \beta A e^{\alpha u_0} P_{\beta}.$$ By the same way we find $\left[ X_0, R_1 \right] = (\alpha + \beta) R_1$ and then deduce the relation $$\left[ D_x, R_2 \right] = (\alpha + 2 \beta) A e^{\alpha u_0} R_1.$$ It can be proved by induction, that $$\left[D_x,R_m\right]=y_{m}R_{m-1}, \quad \left[X_0,R_{m-1}\right]=z_{m-1}{R_{m-1}}, \quad m \geq 2,$$ where $y_k$ and $z_k$ are solutions to the discrete equations $$\label{yz}
y_{m+1}=y_m+Ae^{\alpha{u_0}}z_m, \quad z_m=z_{m-1}+\alpha, \quad m \geq 2$$ with the following initial data $$\label{bb}
z_1=\alpha+\beta, \quad y_2=(\alpha+2\beta)Ae^{\alpha{u_0}}.$$ The problem (\[yz\]), (\[bb\]) is solved explicitly: $$\label{znyn}
z_n = \alpha n + \beta, \quad y_n = A e^{\alpha u_0} (n + 1) \left( \frac{n}{2} \alpha + \beta \right).$$ Since the characteristic algebra is finite dimensional there exists a natural $N$ such that $R_{N+1}$ is linearly expressed through the previous members of the sequence: $$\label{www}
R_{N+1}={\lambda}_N{R_N}+{\lambda}_{N-1}R_{N-1}+...{\lambda}_1{R_1}+{\lambda}_{\alpha}P_{\alpha}+{\lambda}_{\beta}P_{\beta},$$ where the operators $R_{N},R_{N-1},...R_1,P_{\alpha},P_{\beta}$ are supposed to be linearly independent. Applying the operator $ad_{D_x}$ to both sides of equation (\[www\]), we find $$y_{N+1}R_N=D_x({\lambda}_N)R_N+{\lambda}_N y_N R_{N-1}+...$$ Collecting coefficients before $R_N$, we find the equation $D_x({\lambda}_N)=y_{N+1}$. We concentrate on this equation by representing it in an explicit form $$\sum_{j}\frac{\partial \lambda_N}{\partial u_{j}}u_{j,x}+\sum_{j}\frac{\partial \lambda_N}{\partial u_{j,x}}u_{j,xx}+...=y_{N+1},$$ where $y_{N+1} = A e^{\alpha u_0} (N + 2) \left( \frac{N+1}{2} \alpha + \beta \right)$. Since the r.h.s. does not contain the variables $u_{j,x}$, $u_{j,xx}$,... we get immediately that $D_x({\lambda}_N)=0$. Hence, this equation is satisfied only when ${\lambda}_N=const$ and $y_{N+1}=0$, or when $\frac{N+1}{2} \alpha + \beta = 0$. We can repeat all the reasoning by replacing ${\alpha}\leftrightarrow{\beta}$. Then we arrive at a similar relation with some natural $K$: $\frac{K+1}{2} \alpha + \beta = 0$. In other words the following system of equations $$(N + 1) \alpha + 2 \beta = 0, \quad (K + 1)\beta + 2 \alpha = 0$$ should have solution in natural $N$, $K$. Solving the system we get: $(N + 1) (K + 1) = 4$ when $\alpha\beta\neq0$. Note that if $\alpha\beta=0$ then both of the roots vanish. However that contradicts the requirement $\alpha\neq \beta$. Thus we have either $N = K = 1$ or $K = 0$, $N = 3$. In the first case $\alpha = -\beta$, in the second case $\beta = - 2 \alpha$. This completes the proof of Lemma 5.
These two exceptional cases are studied in the following theorem.
[**Theorem 2.**]{} *If the polynomial (\[Lambda\_00\]) has two different roots $\alpha$ and $\beta=-2\alpha$ (or $\alpha$ and $\beta=-\alpha$) then the characteristic Lie-Rinehart algebra corresponding to the reduced system (\[finite\_sys\]) is of infinite dimension.*
In other words Theorem 2 claimes that in these two cases the reduced system is not integrable in the sense of Darboux. Theorem 2 is proved in Appendix.
Investigation of multiple roots
-------------------------------
Now let’s study the problem of the multiplicity of the roots of the polynomial $\Lambda(\lambda)$ defined by (\[Lambda\_00\]).
[**Lemma 6.**]{} *Polynomial $\Lambda(\lambda)$ does not have any multiple non-zero root.*
[**Proof.**]{} Suppose that $\lambda=\alpha$ is a multiple non-zero root of the polynomial $\Lambda(\lambda)$. Define new polynomials $\Lambda_1(\lambda)=\frac{1}{\lambda-\alpha}\Lambda(\lambda)$ and $\Lambda_2(\lambda)=\frac{1}{(\lambda-\alpha)^2}\Lambda(\lambda)$. Then we consider the equation ${\Lambda}_1(X)g(u_{1},u_0,u_{-1})=y$. Evidently $(X_0-\alpha)y=0$, therefore by solving this equation one can find $y=A(u_1,u_{-1})e^{{\alpha}u_0}$. Similarly we put $\Lambda_2(X)g(u_{1}, u_0, u_{-1})=z$ and then find that $(X_0-\alpha)z={\Lambda}_1(X)g=y$, which implies that $z=e^{{\alpha}u_0}\left(A(u_{1},u_{-1})u_0+B(u_{1},u_{-1})\right)$, where $B=B(u_1,u_{-1})$ is a function.
Due to the formula (\[ss\]) we can obtain that the operators $P={\Lambda}_{1}(ad_{X_0}W_0)$, $T={\Lambda}_{2}(ad_{X_0}W_0) $ satisfy the following commutativity conditions $$\begin{aligned}
&& \left[D_x,P\right]=A(u_{1},u_{-1})e^{{\alpha}u_0}X_0,\label{op1}\\
&& \left[D_x,T\right]=e^{{\alpha}u_0}\left(A(u_{1},u_{-1})u_0+B(u_{1},u_{-1})\right)X_0. \label{op2}
\end{aligned}$$ Let us construct a sequence of the operators due to the formulas: $$K_1=\left[P,T\right], \quad K_2=\left[P,K_1\right], \quad ..., \quad K_{m+1}=\left[P,K_m\right],\quad ...$$ It is easily checked that $\left[X_0,P\right]={\alpha}P$, $\left[X_0,T\right]={\alpha}T+P$. Indeed, let us check the first of these formulas. Evidently we have $\left[D_x,\left[X_0,P\right]\right]=\left[X_0,Ae^{{\alpha}u_0}X_0\right]={\alpha}Ae^{{\alpha}u_0}X_0$. Therefore, $\left[D_x,\left[X_0,P\right]-{\alpha}P\right]=0.$ By virtue of Lemma 1 one obtains the formula desired. In a similar way we prove that $$\begin{aligned}
&&\left[D_x,K_1\right]=e^{{\alpha}u}({\alpha}Au+{\alpha}B-A)P-e^{{\alpha}u_0}{\alpha}AT, \\
&&\left[D_x,K_2\right]=3{\alpha}Ae^{{\alpha}u_0}K_1, \\
&&\left[X_0,K_1\right]=2{\alpha}K_1, \quad \left[X_0,K_2\right]=3{\alpha}K_2.
\end{aligned}$$ It can be proved by induction that for any $m{\geq}2$ $$\left[D_x,K_m\right]=\frac{\alpha}{2}(m+1)mAe^{{\alpha}u_0}K_{m-1}.$$ Since the Lie-Rinehart algebra generated by $P,T$ is supposed to be of a finite dimension then there is an integer $M$ such that $$\label{ser}
K_{M+1}=a_{M}K_{M}+a_{M-1}K_{M-1}+...+a_1{K_1}+{b_1}P+{b_2}T,$$ where $a_j,b_j$ are some functions depending on the dynamical variables $u_j,u_{jx},u_{jxx},...$ and the operators $K_M,K_{M-1},...K_1,P,T$ are linearly independent. By applying the operator $ad_{D_x}$ to equation (\[ser\]) one gets for $M > 0$ $$\frac{1}{2}{\alpha}(M+2)(M+1)Ae^{{\alpha}u_0}K_M=D_x(a_M)K_M+...,$$ where the tail contains the summands with $K_{M-1},K_{M-2},...$. Thus the last equation implies $$\label{cont}
D_x(a_M(\bar{u},\bar{u}_x,\bar{u}_{xx},...))=\frac{\alpha}{2}(M+2)(M+1)A(u_{1},u_{-1})e^{{\alpha}u_0}.$$ Equation (\[cont\]) yields $D_x(a_M)=0$ and $\alpha=0$. The latter contradicts the assumption $\alpha\neq0$.
The case $M=0$ i.e. $K_1={b_1}P+{b_2}T$ should be investigated separately. Here application of the operator $ad_{D_x}$ yields $$\begin{aligned}
&&e^{{\alpha}u_0}({\alpha}Au+{\alpha}B-A)P-{\alpha}e^{{\alpha}u_0}AT=\\
&&=D_x(b_1)P+D_x(b_2)T+{b_1}Ae^{{\alpha}u_0}X_0+{b_2}e^{{\alpha}u_0}(Au_0+B)X_0.
\end{aligned}$$ Comparison of the coefficients before the operators $X_0,P,T$ shows that this equation is contradictory. This completes the proof of Lemma 6.
[**Lemma 7.**]{} *At the point $\lambda = 0$ polynomial $\Lambda(\lambda)$ defined in (\[Lambda\_00\]) might have only a simple root.*
[**Proof.**]{} Assume that $\alpha=0$ is a root the characteristic polynomial ${\Lambda}(\lambda)$ of the multiplicity $k$. Then then due to Theorem 2 we have ${\Lambda}(\lambda)={\lambda}^k$. Let us construct operators: $$\begin{aligned}
&&P_1={\Lambda}_1(ad_{X_0}{W_0})={ad^{k-1}_{X_0}}{W_0},\\
&&P_2={\Lambda}_2(ad_{X_0}{W_0})={ad^{k-2}_{X_0}}{W_0}.\end{aligned}$$ It can be proved that they satisfy relations: $$\begin{aligned}
&&\left[D_x,P_1\right]=-A(u_{1},u_{-1})X_0,\\
&&\left[D_x,P_2\right]=-\left(A(u_{1},u_{-1})u_0+B(u_{1},u_{-1})\right)X_0.
\end{aligned}$$
Let us prove that the Lie-Rinehart algebra generated by $P_1$ and $P_2$, where the coefficient $A(u_{1},u_{-1})$ doesn’t vanish identically, is of infinite dimension. Define a sequence of the multiple commutators in such a way $$P_1, \quad P_2, \quad P_3=\left[P_2, P_1\right], \quad ..., \quad P_m=\left[P_2, P_{m-1}\right], \quad ...$$ It can be proved by induction on $m$ that $$\left[D_x,P_m\right]=A(u_{1},u_{-1})P_{m-1}.$$ If the Lie-Rinehart algebra is of a finite dimension then there exists a natural $M$ such that $$P_{M+1}={\mu}_M{P_M}+{\mu}_{M-1}{P}_{M-1}+...+{\mu}_1P_1.$$ By applying the operator $D_x$ to both sides of the last equality, we obtain: $$\begin{aligned}
A(u_{1},u_{-1})P_{M}&&=D_x({\mu}_M)P_M+{\mu}_MA(u_{1},u_{-1})P_{M}+...+\\
&&+D_x({\mu}_1)P_1+{\mu}_1A(u_{1},u_{-1})P_{1}.\end{aligned}$$ By comparing the coefficients before $P_M$ we get: $$\label{ool}
A(u_{1},u_{-1})=D_x({\mu}_M).$$ Since ${\mu}_M={\mu}_M(\bar{u},\bar{u}_x,\bar{u}_{xx},...)$ depends on a set of the dynamical variable while $A$ depends only on $u_{1}$ and $u_{-1}$ equality (\[ool\]) fails to be true unless $A(u_{1},u_{-1})=0$ that contradicts to our assumption. Therefore the Lie-Rinehart algebra generated by the operators $P_1$ and $P_2$ is of infinite dimension. Lemma 7 is proved.
Thus, summarizing the statements of Lemmas 5–7, we conclude that the polynomial $\Lambda(\lambda)$ defined by (\[Lambda\_00\]) might have only one root and this root is simple. Therefore, equation (\[Lambda\_0\]) has the form $(X_0 - \alpha)f_{u_1 u_{-1}} = 0$, where $\alpha$ is a constant. Thus, we have that the function $f_{u_1 u_{-1}}$ has the following form: $$f_{u_1 u_{-1}} = Q(u_{1}, u_{-1}) e^{\alpha u_0},$$ where $Q(u_1 u_{-1})$ is a function being a quasi-polynomial with respect to any of its arguments $u_{1}$, $u_{-1}$.
Now, let us repeat the reasoning above by changing the operator $X_0$ by $X_1$. For this purpose, we construct a sequence as follows $$H_0 = W_0, \quad H_1 =\left[X_1, H_0\right], \quad H_2 = \left[X_1, H_1\right], \ldots, \quad H_{k+1} = \left[X_1, H_k\right], \ldots$$ Elements of the sequence satisfy the relations: $$\left[ D_x, H_k \right] = -X^{k}_1(f_{u_1 u_{-1}})X_0 = -e^{\alpha u_0}X^{k}_1(Q_{u_1 u_{-1}})X_0 .$$ Since the characteristic algebra is finite-dimensional there exists a natural $K$ such that $H_{K+1}$ is linearly expressed by the previous members: $$H_{K+1} + \lambda_{K} H_K + \cdots + \lambda_1 H_1 + \lambda_0 H_0 = 0,$$ where the operators $H_0, H_1, \ldots, H_k$ are linearly independent. We apply the operator $ad_{D_x}$ to the obtained equation and get relations $D_x(\lambda_j)=0$ for $j=0,1,...K$ and also a relation $\Omega(X_1)Q_{u_1u_{-1}}=0$, where $$\label{3*}
\Omega(\lambda)=\lambda^{K+1} + \lambda_{K} \lambda^{K} + \cdots + \lambda_0$$ is a quasi-polynomial with constant coefficients.
Now we investigate the characteristic polynomial (\[3\*\]) by using the reasonings we applied above to the characteristic polynomial (\[Lambda\_00\]). As a result we prove that $Q_{u_{1} u_{-1}} = \Phi(u_{-1}) e^{\beta u_1}$ and $f_{u_{1} u_{-1}} = \Phi(u_{-1}) e^{\alpha u_0 + \beta u_1}$. Here $\Phi(u_{-1})$ is a quasi-polynomial.
Finally, we repeat this reasoning, replacing $X_{1}$ by $X_{-1}$ and prove the following statement:
[**Theorem 3.**]{} *If lattice (\[eq\_main\]) is integrable in the sense of Definition 1 then the function $f(u_{n+1}, u_n, u_{n-1})$ satisfies the following equation:* $$\label{fu1um1}
f_{u_{n+1}, u_{n-1}}=Ce^{{\alpha}u_n+{\beta}u_{n+1}+{\gamma}u_{n-1}},$$ *where $C,{\alpha},{\beta},{\gamma}$ are constant.*
The necessary integrability conditions
======================================
[**Theorem 4.**]{} *If $\alpha = 0$, $C \neq 0$ in (\[fu1um1\]) then $\beta = 0$ and $\gamma = 0$.* *If $\alpha \neq 0$, $C \neq 0$ then $\beta = -\frac{\alpha}{2}m$, $\gamma = -\frac{\alpha}{2}k$, where $m$, $k$ are nonnegative integers.*
[**Proof.**]{} Suppose that $C \neq 0$ then a minimal order operator (\[Lambda\_0\]) which annulates the function $g = f_{u_1 u_{-1}}$ has the form $\Lambda(X_0) = X_0 - \alpha$. Let us construct the operator $P_0 = \Lambda(ad_{X_0}W_0) / C(\lambda - \alpha)$ and the operator $P_1 = D_nP_0D_n^{-1}$, where $D_n$ stands for the shift operator acting as $D_ny(n)=y(n+1)$. Thus we have the operators $P_0, P_1\in{L}$ of the form $$P_j=\sum_{k}{a_{jk}(1)\frac{\partial}{\partial{u_{kx}}}}+{a_{jk}(2)\frac{\partial}{\partial{u_{kxx}}}}+{a_{jk}(3)\frac{\partial}{\partial{u_{kxxx}}}}+...$$ Due to (\[fu1um1\]) and Lemma 4 these operators satisfy the following commutativity relations $$\label{bvc}
\left[D_x,P_0\right]=e^{\omega}X_0, \quad \left[D_x,P_1\right]=e^{\omega_1}{X_1},$$ where $\omega={\alpha}u_0+{\beta}u_1+{\gamma}u_{-1},{\omega}_1={\alpha}u_1+{\beta}u_2+{\gamma}u_0$. The first relation in (\[bvc\]) is easily proved by the same way as formula (\[lop\]). The second relation follows from the first one by applying the conjugation transformation $X\rightarrow D_nXD_n^{-1}$.
One can easily verify that $$\left[X_0,P_0\right]={\alpha}P_0, \quad \left[X_0,P_1\right]={\gamma}P_1, \quad \left[X_1,P_0\right]={\beta}P_0, \quad \left[X_1,P_1\right]={\alpha}P_1.$$ Let us construct a sequence of the multiple commutators as follows $$K_1=\left[P_0,P_1\right], \quad K_2=\left[P_0,K_1\right],..., \quad K_m=\left[P_0,K_{m-1}\right], ...$$ It is checked by direct calculation that $$\begin{aligned}
&&\left[D_x,K_1\right]=-{\beta}e^{{\omega}_1}P_0+{\gamma}e^{\omega}P_1, \quad \left[X_0,K_1\right]=(\alpha+\gamma)K_1,\\
&&\left[D_x,K_2\right]=(\alpha+2\gamma)e^{\omega}K_1, \quad \left[X_0,K_2\right]=(2\alpha+\gamma)K_2.
\end{aligned}$$ By induction we can prove that for $n\geq{2}$ $$\label{tty}
\left[D_x,K_n\right]=e^{\omega}{\pi}_n K_{n-1},\quad \left[X_0,K_n\right]=y_n{K_n}$$ with $\pi_{n}=\frac{\alpha}{2} n^2+(\gamma-\frac{\alpha}{2})n$, $y_n=\alpha n+\gamma$. Since the characteristic algebra is of a finite dimension then there exists a natural $N$ such that $$\label{pol}
K_{N+1}={\lambda}_N{K_N}+...+{\lambda}_1{K_1}+{\mu}_0{P_0}+{\mu}_1{P_1},$$ where the operators $P_0$, $P_1$, $K_1$,..., $K_N$ are linearly independent. By applying to (\[pol\]) the operator $ad_{D_x}$ one find due to (\[tty\]) that $$\label{vvc}
e^{\omega}{\pi}_{N+1}K_N={D_x}(\lambda_N){K_N}+{\lambda}_N{e^{\omega}}{\pi}_N{K_{N-1}}+...$$ We compare the coefficients before $K_N$ in (\[vvc\]) and we get ${\lambda}_N=0$, ${\pi}_{N+1}=0$. Or the same $$\label{aNg}
\frac{\alpha}{2} N + \gamma = 0.$$ From this formula it follows that if $\alpha = 0$ then $\gamma = 0$. Similarly one can prove that if $\alpha = 0$ then $\beta = 0$.
The case $N=0$ is never realized. Indeed supposing $K_1={\mu}_0{P_0}+{\mu}_1{P_1}$ one obtains a contradictory equation $$-{\beta}e^{{\omega}_1}P_0+{\gamma}e^{\omega}P_1 =D_x({\mu}_0)P_0+D_x({\mu}_1)P_1+{\mu}_0e^{\omega}X_0+{\mu}_1e^{{\omega}_1}X_1$$ unless $\beta=0$, $\gamma=0$.
Thus it follows from (\[aNg\]) that $\gamma = -\frac{\alpha}{2}N$. Formula $\beta = -\frac{\alpha}{2}m$ is proved in a similar way. Moreover, we see that if $\gamma = 0$ then $\beta = 0$. And similarly we obtain that if $\beta = 0$ then $\gamma = 0$. In other words, if $\beta \gamma \neq 0$ then $f_{u_1 u_{-1}}$ has the following form $$f_{u_1 u_{-1}}=Ce^{{\alpha}u_0-{\frac{{\alpha}m}{2}}u_1-{\frac{{\alpha}k}{2}}u_{-1}},$$ where $C \neq 0$, $\alpha \neq 0$. If $\beta = 0$ or $\gamma = 0$ then both $\beta = \gamma = 0$ and $$f_{u_1 u_{-1}}=Ce^{{\alpha}u_0},$$ where $\alpha \neq 0$, $C \neq 0$. Theorem 4 is proved.
The main result is given in
[**Theorem 5.**]{} *Lattice (\[eq\_main\]) which is integrable in the sense of Definition 1, can be reduced by suitable rescalings to one of the following forms:* $$\begin{aligned}
&& u_{n,xy} = e^{\alpha u_n - \frac{\alpha}{2}m u_{n+1} - \frac{\alpha}{2} k u_{n-1}} + a(u_{n+1}, u_n) + b(u_n, u_{n-1}), \\
&& u_{n,xy} = e^{\alpha u_n} u_{n+1} u_{n-1} + a(u_{n+1}, u_n) + b(u_n, u_{n-1}), \\
&& u_{n,xy} = u_{n+1} u_{n-1} + a(u_{n+1}, u_n) + b(u_n, u_{n-1}), \\
&& u_{n,xy} = a(u_{n+1}, u_n) + b(u_n, u_{n-1}); \end{aligned}$$ *here $\alpha \neq 0$ and $m$, $k$ are positive integers.*
Theorem 5 straightforwardly follows from Theorems 3, 4.
Appendix
========
Here we give a complete proof of Theorem 2. We consider the cases $\beta = -2 \alpha$ and $\beta = - \alpha$ separately. Our proof uses the scheme applied earlier in [@ZhMurt], [@Sakieva].
The case $\beta = -2 \alpha$
----------------------------
In this subsection we will prove that if polynomial $\Lambda(\lambda)$ has two different nonzero roots $\alpha$ and $\beta = -2 \alpha$ then the Lie-Rinehart algebra $L$ generated by the operators $X_0$ and $W_0$ is of an infinite dimension.
First we introduce two polynomials according to the rule $${\Lambda}_{\alpha}(\lambda)=\frac{\Lambda(\lambda)}{\lambda-\alpha}, \quad {\Lambda}_{\beta}(\lambda)=\frac{\Lambda(\lambda)}{\lambda+ 2 \alpha}.$$ Then we construct two operators $P_{\alpha}, P_{\beta} \in L$: $$P_{\alpha}={\Lambda}_{\alpha}(ad_{X_0}W_0), \quad P_{\beta}={\Lambda}_{\beta}(ad_{X_0}W_0)$$ and concentrate on the Lie-Rinehart algebra $L_{1} \subset L$ being a subalgebra of $L$ generated by the operators $P_{\alpha}, P_{\beta}$. By construction these operators satisfy the following commutativity relations $$\left[ D_x, P_{\alpha} \right] = A(u_{1}, u_{-1}) e^{\alpha u_0} X_0, \quad \left[ D_x, P_{\beta} \right] = B(u_{1}, u_{-1}) e^{-2 \alpha u_0} X_0,$$ where $A(u_{1}, u_{-1})$, $B(u_{1}, u_{-1})$ are quasi-polynomials in $u_{1}$, $u_{-1}$. We assume that $(u_{-1}, u_1) \in D$, where $D$ is a domain in $\mathbb{C}^2$, where both $A$, $B$ do not vanish.
Let us consider the operators: $$\label{X1X2}
Y_1 = P_{\alpha} + P_{\beta}, \quad Y_2 = \frac{\partial}{\partial u_0}.$$
Assume that $L_i$ stands for the linear space spanned by all possible commutators of the operators $Y_1$ and $Y_2$ of the length less or equal to $i-1$, where $i = 2, 3, \ldots$. We emphasize that the linear combination in this space is taken with coefficients being the functions depending on a finite number of the variables $\bar u,\bar u_x,\bar u_{xx},\dots$. Thus $L_2 = \left\{ Y_1, Y_2 \right\}$ is the linear span of $Y_1$ and $Y_2$, $\mathrm{dim}\, L_2 = 2$. Similarly $L_3$ is the linear envelop of the vector $Y_1$, $Y_2$ and $Y_3 = \left[ Y_1, Y_2 \right]$, i.e. $L_3 = \left\{Y_1, Y_2, Y_3 \right\}$. Therefore, $L_4 = \left\{Y_1, Y_2, Y_3, \left[Y_1, Y_3 \right], \left[Y_2, Y_3 \right] \right\}$ etc.
Let us denote $\delta(i) = \mathrm{dim}\, L_i - \mathrm{dim}\, L_{i-1}$. We also will use the following notations for multiple commutators: $$Y_{i_1,\ldots, i_n} = ad_{Y_{i_1}}\ldots ad_{Y_{i_{n-1}}} Y_{i_n}, \quad \mathrm{where} \quad ad_Y W= \left[ Y, W \right].$$
[**Lemma 8.**]{} *Assume that polynomial $\Lambda(\lambda)$ has two different nonzero roots $\alpha$ and $-2 \alpha$. Then the following formulas are true:* $$\delta(i) = 2, \quad i= 6n + 2, \quad i=6n+4, \quad n = 1,2, \ldots,$$ $$\delta(i) = 1, \quad i = 6n-1, \quad i = 6n, \quad i = 6n+1, \quad i = 6n+3, \quad n=1,2,\ldots,$$ $$\begin{aligned}
L_{6n+2} = L_{6n+1} \oplus \left\{ Y_{1 \ldots 121}, Y_{ 2 1 \ldots 121} \right\},\\
L_{6n+4} = L_{6n+3} \oplus \left\{ Y_{1 \ldots 121}, Y_{ 2 1 \ldots 121} \right\},\\
L_{6n-1} = L_{6n-2} \oplus \left\{ Y_{1 \ldots 121}\right\},\\
L_{6n} = L_{6n - 1} \oplus \left\{Y_{1 \ldots 121}\right\},\\
L_{6n+1} = L_{6n} \oplus \left\{Y_{1 \ldots 121}\right\},\\
L_{6n+3} = L_{6n+2} \oplus \left\{Y_{1 \ldots 121}\right\}.\end{aligned}$$
[**Proof.**]{} We introduced the operators $Y_1$, $Y_2$ by formulas (\[X1X2\]). The following commutation relations are true for these operators: $$\label{DxX1X2}
\left[ D_x, Y_1 \right] = \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_2, \quad \left[ D_x, Y_2 \right] = 0.$$ We introduce the operator of length 2: $Y_3 = \left[ Y_2, Y_1 \right] = Y_{21}$. Then using the Jacobi identity and formulas (\[DxX1X2\]), we get $$\label{DxX3}
\left[ D_x, Y_3 \right] = \alpha \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) Y_2.$$ If we assume that $Y_3$ is linearly expressed by $Y_1$ and $Y_2$, i.e. $$\label{linX3}
Y_3 = \lambda_1 Y_1 + \lambda_2 Y_2,$$ then we get a contradiction. Indeed by commuting both sides of (\[linX3\]) with $D_x$ and then simplifying due to (\[DxX1X2\]), (\[DxX3\]) we obtain $$\fl \alpha \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) Y_2 = D_x(\lambda_1) Y_1 + \lambda_1 \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_2 + D_x(\lambda_2) Y_2.$$ Comparing the coefficients before independent operators $Y_1$, $Y_2$, we get: $D_x(\lambda_1) = 0$ and $$\alpha \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) = \lambda_1 \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) + D_x(\lambda_2).$$ The last equality implies that $D_x(\lambda_2) = 0$ and $\lambda_1 - \alpha = 0$, $\lambda_1 + 2 \alpha = 0$. Obviously a pair of these equations is inconsistent because $\alpha \neq 0$.
We introduce the commutators of length 3: $Y_4 = \left[Y_1, Y_3 \right]$ and $\bar{Y}_4 = \left[Y_2, Y_3 \right]$. Then $$\begin{aligned}
\left[D_x, \bar{Y}_4 \right] = \alpha^2 \left( A e^{\alpha u_0} + 4B e^{-2 \alpha u_0} \right)Y_2, \label{DxX4}\\
\left[D_x, Y_4 \right] = -\alpha \left( 2 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right)Y_3 + 2 \alpha^2 \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_1. \nonumber\end{aligned}$$ One can see that $\left[ D_x, \bar{Y}_4 \right] = 2 \alpha^2 \left[ D_x, Y_1\right] - \alpha \left[D_x, Y_3\right] = \left[D_x, 2 \alpha^2 Y_1 - \alpha Y_3\right]$. Due to Lemma 1 this equality implies that $\bar{Y}_4 = 2 \alpha^2 Y_1 - \alpha Y_3$. The operator $Y_4 = Y_{121}$ is not linearly expressed through the operators of lower order. Thus, we have $L_4 = \left\{ Y_1, Y_2, Y_3, Y_4 \right\}$.
We introduce the commutators of lenght 4: $Y_5 = \left[Y_1, Y_4 \right]$ and $\bar{Y}_5 = \left[Y_2, Y_4 \right]$. Using the Jacobi identity and formulas (\[DxX1X2\]), (\[DxX4\]), we find $$\begin{aligned}
\fl \left[D_x, \bar{Y}_5 \right] = \alpha^2 \left( 2A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right)Y_3 - 2 \alpha^3 \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_1 = \left[ D_x, -\alpha Y_4 \right],\\
\fl \left[ D_x, Y_5 \right] = -\alpha \left( 2 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right) Y_4 - \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_4, Y_2\right]= - 3 \alpha A e^{\alpha u_0} Y_4.\end{aligned}$$ Due to Lemma 1, we conclude that $\bar{Y}_5 = -\alpha Y_4$. The operator $Y_5 = Y_{1121}$ is not linearly expressed through the commutators of lower order. Thus, we have $L_5 = \left\{Y_1, Y_2, Y_3, Y_4, Y_5 \right\}$.
Let us consider the commutators of the length 5: $$Y_6 = \left[Y_1, Y_5\right], \quad \bar{Y}_6 = \left[Y_2, Y_5\right], \quad \left[Y_3, Y_4\right]$$ The following formulas are true: $$\fl \left[ D_x, \bar{Y}_6 \right] = 0, \quad \left[ D_x, \left[Y_3, Y_4 \right]\right] = -3 \alpha^2 A e^{\alpha u_0} Y_4 = \left[D_x, \alpha Y_5 \right], \quad \left[ D_x, Y_6 \right] = - 3 \alpha A e^{\alpha u_0} Y_5.$$ Using Lemma 1, we conclude that $\left[Y_3, Y_4 \right] = \alpha Y_5$. The operator $Y_6 = Y_{11121}$ is not linearly expressed by the operators of lower order. So, we have $L_6 = \left\{ Y_1, Y_2, Y_3, Y_4, Y_5, Y_6 \right\}$.
Now we introduce the operators of the length 6: $$Y_7 = \left[Y_1, Y_6\right] = Y_{111121}, \quad \bar{Y}_7 = \left[Y_2, Y_6\right] = Y_{211121}, \quad \left[Y_3, Y_5\right].$$ One can prove that the following formulas are true: $$\bar{Y}_7 = \alpha Y_6, \quad \left[Y_3, Y_5\right] = \alpha Y_6, \quad \left[ D_x, Y_7 \right] = \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_6.$$ The operator $Y_7 = Y_{111121}$ is not linearly expressed by the operators of lower order and $L_7 = \left\{ Y_1, Y_2, Y_3, Y_4, Y_5, Y_6, Y_7 \right\}$.
Then we consider the operators of the length 7: $$Y_8 = \left[ Y_1, Y_7\right], \quad \bar{Y}_8 = \left[ Y_2, Y_7 \right], \quad \left[ Y_3, Y_6 \right], \quad \left[ Y_4, Y_5 \right].$$ One can prove that $$\begin{aligned}
\left[D_x, \bar{Y}_8 \right] = \alpha^2 \left( -4 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right)Y_6,\\
\left[ D_x, Y_8 \right] = \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_7 + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_8.\end{aligned}$$ The operators $Y_8$, $\bar{Y}_8$ is not expressed through the operators of lower order. It is not difficult to show that $\left[Y_3, Y_6 \right] = \bar{Y}_8 - \alpha Y_7$, $\left[ Y_4, Y_5 \right] = 2 \alpha Y_7 - \bar{Y}_8$. Thus we have that the space $L_8$ is obtained from $L_7$ by adding two linearly independent elements $Y_8 = Y_{1111121}$ and $\bar{Y}_8 = Y_{2111121}$, i.e. $L_8 =\left\{Y_1, Y_2, Y_3, Y_4, Y_5, Y_6, Y_7, Y_8, \bar{Y}_8 \right\}$.
Now let us introduce the operators of the length 8: $$Y_9 = \left[Y_1, Y_8 \right], \quad \bar{Y}_9 = \left[Y_2, Y_8\right], \quad \left[ Y_1, \bar{Y}_8 \right], \quad \left[Y_2, \bar{Y}_8 \right], \quad \left[ Y_3, Y_7 \right], \quad \left[Y_4, Y_6 \right].$$ One can show that $$\begin{aligned}
\left[Y_4, Y_6 \right] = \alpha Y_8, \quad \left[Y_3, Y_7 \right] = -\alpha Y_8, \quad \left[ Y_2, \bar{Y}_8 \right] = 2 \alpha^2 Y_7 + \alpha \bar{Y}_8,\\
\left[ Y_1, \bar{Y}_8 \right] = \alpha Y_8, \quad \bar{Y}_9 = 0,\\
\left[D_x, Y_9 \right] = \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right)Y_8.\end{aligned}$$ So we see that $Y_9 = Y_{11111121}$ is not linearly expressed through the operators of lower order and $L_9 = L_8 \oplus {Y_9}$.
The operators of length 9 are constructed by the following way: $$\fl Y_{10} = \left[Y_1, Y_9\right], \quad \bar{Y}_{10} = \left[Y_2, Y_9\right], \quad \left[Y_3, \bar{Y}_8\right], \left[ Y_3, Y_8 \right], \quad \left[Y_4, Y_7 \right], \quad \left[Y_5, Y_6\right].$$ For these operators the relations hold: $$\begin{aligned}
\left[ Y_3, \bar{Y}_8 \right] = -3 \alpha^2 Y_8, \quad \left[Y_3, Y_8 \right] = \bar{Y}_{10}, \quad \left[Y_4, Y_7\right] = -\alpha Y_9 - \bar{Y}_{10}, \\
\left[Y_5, Y_6\right] = 2 \alpha Y_9 + \bar{Y}_{10}, \\
\left[D_x, Y_{10} \right] = \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0}\right)Y_9 + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{10},\\
\left[ D_x, \bar{Y}_{10} \right] = \alpha^2 \left( -A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right) Y_8.\end{aligned}$$ The operators $Y_{10} = Y_{111111121}$ and $\bar{Y}_{10} = X_{211111121}$ are not linearly expressed through the operators of lower order, $L_{10} = L_9 \oplus \left\{Y_{10}, \bar{Y}_{10} \right\}$.
Now let us introduce the notation: $Y_n = \left[Y_1, Y_{n-1} \right]$, $\bar{Y}_n = \left[Y_2, Y_{n-1} \right]$. We prove Lemma 7 by induction. Assume that for $i = n - 1$ the following formulas are true: $$\begin{aligned}
\fl \left[D_x, Y_{6(n-1)-1} \right]= -\alpha \left( 2 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right) Y_{6(n-1)-2} +\label{Ind_01}\\
\left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \left[Y_2, Y_{6(n-1)-2}\right] \right),\nonumber\\
\fl \left[ D_x, Y_{6(n-1)} \right] = -3 \alpha A e^{\alpha u_0} Y_{6(n-1)-1} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6(n-1)-1} \right],\\
\fl \left[D_x, Y_{6(n-1)+1} \right] = -3 \alpha A e^{\alpha u_0}Y_{6(n-1)} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6(n-1)} \right],\\
\fl \left[D_x, Y_{6(n-1)+2} \right] = \alpha \left( - 2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_{6(n-1)+1} + \nonumber\\
+ \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6(n-1)+1} \right],\\
\fl \left[D_x, Y_{6(n-1)+3} \right] = \alpha \left( - A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6(n-1)+2} +\nonumber \\
+\left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6(n-1)+2} \right],\\
\fl \left[D_x, Y_{6(n-1)+4} \right] = \alpha \left( - A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6(n-1)+3} +\nonumber\\
+ \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6(n-1)+3} \right],\\
\fl \bar{Y}_{6(n-1)} = \left[Y_2, Y_{6(n-1)-1} \right] = 0,\\
\fl \bar{Y}_{6(n-1)-1} = -\alpha Y_{6(n-1)-2},\\
\fl \bar{Y}_{6(n-1)+1} = \alpha Y_{6(n-1)},\\
\fl \bar{Y}_{6(n-1)+3} = 0,\\
\fl \left[Y_1, \bar{Y}_{6(n-1) + 2}\right] = \alpha Y_{6(n-1)+2},\\
\fl \left[Y_2, \bar{Y}_{6(n-1)+2} \right] = 2 \alpha^2 Y_{6(n-1)+1} + \alpha \bar{Y}_{6(n-1)+2},\\
\fl \left[Y_1, \bar{Y}_{6(n-1)+4}\right] = - \alpha Y_{6(n-1)+4},\\
\fl \left[Y_2, \bar{Y}_{6(n-1)+4} \right] = 2 \alpha^2 Y_{6(n-1)+3} - \alpha \bar{Y}_{6(n-1)+4}. \label{Ind_014}\end{aligned}$$ Let us prove formulas (\[Ind\_01\])–(\[Ind\_014\]) for $i = n$. We introduce the commutators of length $6n-2$: $$\begin{aligned}
\fl Y_{6n-1} = Y_{6(n-1)+5} = \left[Y_1, Y_{6(n-1)+4} \right],\qquad
\bar{Y}_{6n-1} = \bar{Y}_{6(n-1)+5} = \left[ Y_2, Y_{6(n-1)+4} \right].\end{aligned}$$ Using the Jacobi identity and formulas (\[Ind\_01\])–(\[Ind\_014\]) we find that the following equalities hold: $$\begin{aligned}
\fl \left[D_x, \bar{Y}_{6n-1} \right] = \alpha^2 \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) Y_{6(n-1)+3}
- \alpha \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0}\right) \bar{Y}_{6(n-1)+4} = \\
= \left[D_x, -\alpha Y_{6(n-1)+4} \right],\end{aligned}$$ $$\begin{aligned}
\fl \left[D_x, Y_{6n-1} \right] = \left[D_x, \left[Y_1, Y_{6(n-1)+4} \right] \right]=
\left[Y_1, \left[D_x, Y_{6(n-1)+4}\right] \right] - \left[ Y_{6(n-1)+4}, \left[D_x, Y_1\right] \right] = \\
\fl =\left[ Y_1, \alpha \left(-A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6(n-1)+3} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6(n-1)+3} \right] \right] -\\
\fl -\left[ Y_{6(n-1)+4}, \left( A e^{\alpha u_0 } + B e^{-2 \alpha u_0}\right) Y_2 \right]= \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6(n-1)+4} + \\
\fl + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_1, \left[Y_2, Y_{6(n-1)+3} \right] \right] + \left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_2, Y_{6(n-1)+4} \right] = \\
\fl= \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6(n-1)+4} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, \bar{Y}_{6(n-1)+4} \right] + \\
\fl + \left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6(n-1)+4} \right] = -3 \alpha A e^{\alpha u_0} Y_{6(n-1)+4} = -3 \alpha A e^{\alpha u_0} Y_{6n-2}.\end{aligned}$$ Using Lemma 1 we conclude that $\bar{Y}_{6n-1} = -\alpha Y_{6(n-1)+4} $. One can see that the operator $Y_{6n-1}$ is not linearly expressed through the operators of less indices. Thus we obtain $L_{6n-1} = L_{6n-2} \oplus \left\{ Y_{6n-1}\right\}$, $\delta(6n-1) = 1$.
Now we consider the commutators of length $6n-1$: $$Y_{6n} = \left[Y_1, Y_{6n-1}\right], \quad \bar{Y}_{6n} = \left[Y_2, Y_{6n-1} \right].$$ The formulas of commutation with the operator $D_x$ are: $$\begin{aligned}
\fl \left[ D_x, \bar{Y}_{6n} \right] = \left[ D_x, \left[Y_2, Y_{6n-1} \right] \right] = \\
=\left[Y_2, -3\alpha A e^{\alpha u_0} Y_{6n-2} \right]
=-3 \alpha^2 A e^{\alpha u_0} Y_{6n-2} - 3 \alpha A e^{\alpha u_0} \left[Y_2, Y_{6n-2} \right] = \\
= -3 A \alpha^2 e^{\alpha u_0} Y_{6n-2} + 3 \alpha^2 A e^{\alpha u_0} Y_{6(n-1)+4} = 0,\end{aligned}$$ $$\begin{aligned}
\fl \left[ D_x, Y_{6n} \right] = \left[D_x, \left[ Y_1, Y_{6n-1} \right] \right] = \left[Y_1, -3 \alpha A e^{\alpha u_0} Y_{6n-2} \right]- \\
-\left[Y_{6n-1}, \left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_2 \right] = -3 \alpha A e^{\alpha u_0} Y_{6n-1} + \\
+\left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n} = -3 \alpha A e^{\alpha u_0} Y_{6n-1}.\end{aligned}$$ According to Lemma 1, we conclude that $\bar{Y}_{6n} = 0$. The operator $Y_{6n}$ is not linearly expressed through the operators of lower order. The equalities $L_{6n} = L_{6n-1} \oplus \left\{ Y_{6n} \right\}$, $\delta(6n) = 1$ are true.
Let us consider the commutators of length $6n$: $$Y_{6n+1} = \left[ Y_1, Y_{6n} \right], \quad \bar{Y}_{6n+1} = \left[Y_2, Y_{6n} \right].$$ The following formulas hold: $$\begin{aligned}
\fl \left[D_x, \bar{Y}_{6n+1} \right] = \left[D_x, \left[Y_2, Y_{6n} \right] \right] = \left[ Y_2, -3 \alpha A e^{\alpha u_0} Y_{6n-1} \right] = \\
= -3 \alpha^2 A e^{\alpha u_0} Y_{6n-1}- 3 \alpha A e^{\alpha u_0} \left[ Y_2, Y_{6n-1} \right] = - 3 \alpha^2 A e^{\alpha u_0} Y_{6n-1}= \left[ D_x, \alpha Y_{6n} \right],\end{aligned}$$ $$\begin{aligned}
\fl \left[D_x, Y_{6n+1}\right] = \left[D_x, \left[Y_1, Y_{6n} \right]\right] = \left[ Y_1, -3 \alpha A e^{\alpha u_0} Y_{6n-1}\right]
-\left[ Y_{6n}, \left( Ae^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_2 \right] =\\
= -3 \alpha A e^{\alpha u_0} Y_{6n} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+1}
= \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_{6n}.\end{aligned}$$ So we have $\left[D_x, \bar{Y}_{6n+1} \right] = \left[ D_x, \alpha Y_{6n} \right]$. Due to Lemma 1 the last equality implies that $\bar{Y}_{6n+1} = \alpha Y_{6n}$. The operator $Y_{6n+1}$ is not linearly expressed through the operators of lower order, $L_{6n+1} = L_{6n} \oplus \left\{ Y_{6n+1} \right\}$, $\delta(6n+1) = 1$.
Let us consider the commutators of length $6n+1$: $$Y_{6n+2} = \left[Y_1, Y_{6n+1} \right], \quad \bar{Y}_{6n+2} = \left[Y_2, Y_{6n+1} \right].$$ For these operators the following formulas are satisfied: $$\begin{aligned}
\fl \left[D_x, \bar{Y}_{6n+2}\right] = \left[ D_x, \left[Y_1, Y_{6n+1} \right] \right] = \left[ Y_1, \alpha \left(-2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_{6n} \right] - \\
-\left[Y_{6n+1}, \left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_2 \right]
= \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_{6n+1} +\\
+ \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+2}.\end{aligned}$$ The operators $Y_{6n+2} = \left[Y_1, Y_{6n+1}\right]$ and $\bar{Y}_{6n+2} = \left[ Y_2, Y_{6n+1} \right]$ are not linearly expressed by operators of lower order. The equalities are true: $L_{6n+2} = L_{6n+1} \oplus \left\{ Y_{6n+2}, \bar{Y}_{6n+2} \right\}$, $\delta(6n+2) = 2$.
We introduce the commutators of length $6n+2$: $$\fl Y_{6n+3} = \left[ Y_1, Y_{6n+2} \right], \quad \bar{Y}_{6n+3} = \left[ Y_2, Y_{6n+2} \right], \quad \left[ Y_1, \bar{Y}_{6n+2} \right] , \quad \left[ Y_2, \bar{Y}_{6n+2} \right].$$ The following formulas are satisfied: $$\begin{aligned}
\fl \left[D_x, \left[ Y_2, \bar{Y}_{6n+2} \right] \right] = \left[Y_2, \alpha^2 \left( -4 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right) Y_{6n} \right] = \\
= \alpha^3 \left( -4 A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6n} + \alpha^2 \left( -4 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6n} \right] = \\
= \alpha^3 \left( -4 A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6n} + \alpha^3 \left( -4 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right)Y_{6n} = \\
=\alpha^3 \left( -8 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_{6n} = \left[D_x, 2 \alpha^2 Y_{6n+1} + \alpha \bar{Y}_{6n+2} \right].\end{aligned}$$ So we have that $\left[D_x, \left[ Y_2, \bar{Y}_{6n+2} \right] \right] = \left[D_x, 2 \alpha^2 Y_{6n+1} + \alpha \bar{Y}_{6n+2} \right]$. Apply Lemma 1 to this equality we conclude that $\left[ Y_2, \bar{Y}_{6n+2} \right] =2 \alpha^2 Y_{6n+1} + \alpha \bar{Y}_{6n+2} $.
Then, we find $$\begin{aligned}
\fl \left[ D_x, \left[Y_1, \bar{Y}_{6n+2}\right] \right] = \left[Y_1, \alpha^2 \left( -4 A e^{\alpha u_0} - B e^{-2 \alpha u_0}\right) Y_{6n} \right] - \left[ \bar{Y}_{6n+2}, \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_2\right] = \\
= \alpha^2 \left( -4 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right) Y_{6n+1} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_2, \bar{Y}_{6n+2} \right] =\\
= \alpha^2 \left( -4 A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right) Y_{6n+1} + \left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left( 2 \alpha^2Y_{6n+1} + \alpha \bar{Y}_{6n+2} \right) = \\
= \alpha^2 \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_{6n+1} + \alpha \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+2} = \\
= \left[ D_x, \alpha Y_{6n+2} \right].\end{aligned}$$ Thus we have $\left[ D_x, \left[Y_1, \bar{Y}_{6n+2}\right] - \alpha Y_{6n+2} \right] $. Due to Lemma 1 this formula gives $\left[Y_1, \bar{Y}_{6n+2}\right] = \alpha Y_{6n+2}$. Then we find $$\begin{aligned}
\fl \left[ D_x, \bar{Y}_{6n+3} \right] = \left[ D_x, \left[Y_2, Y_{6n+2} \right] \right] = \\
= \left[ Y_2, \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)Y_{6n+1} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)\bar{Y}_{6n+2} \right] = \\
= \alpha^2 \left( -2 A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0}\right) Y_{6n+1} + \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6n+1} \right] + \\
+ \alpha \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+2} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)\left[Y_2, \bar{Y}_{6n+2} \right] = \\
\alpha^2 \left( -2 A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0}\right) Y_{6n+1} + \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+2} + \\
+\alpha \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+2} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left( 2\alpha^2 Y_{6n+1} + \alpha \bar{Y}_{6n+2}\right) = 0.\end{aligned}$$ It is clear by Lemma 1 that $\bar{Y}_{6n+3} = 0$. Now we calculate $$\begin{aligned}
\fl \left[ D_x, Y_{6n+3} \right] = \left[ D_x, \left[Y_1, Y_{6n+2} \right] \right] = \left[ Y_1, \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_{6n+1} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0}\right) \bar{Y}_{6n+2} \right] - \\
- \left[ Y_{6n+2}, \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_2\right] = \alpha \left( -2 A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_{6n+2} +\\
+ \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0}\right) \left[Y_1, \bar{Y}_{6n+2} \right] + \left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6n+2} \right] = \\
=\alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6n+2}.\end{aligned}$$ At this step we obtain $L_{6n+3} = L_{6n+2} \oplus \left\{ Y_{6n+3} \right\}$, $\delta(6n+1) = 1$.
Let us consider the commutators of length $6n+3$: $$Y_{6n+4} = \left[ Y_1, Y_{6n+3} \right], \quad \bar{Y}_{6n+4} = \left[ Y_2, Y_{6n+3} \right].$$ The following formulas are satisfied: $$\begin{aligned}
\fl \left[D_x, \bar{Y}_{6n+4} \right] = \left[D_x, \left[Y_2, Y_{6n+3} \right] \right] = \left[Y_2, \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0}\right)Y_{6n+2} \right] = \\
= \alpha^2 \left( -A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right) Y_{6n+2} + \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6n+2} \right] = \\
= \alpha^2 \left( -A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right) Y_{6n+2}.\end{aligned}$$ $$\begin{aligned}
\fl \left[ D_x, Y_{6n+4} \right] = \left[ D_x, \left[Y_1, Y_{6n+3} \right] \right]
=\left[ Y_1, \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6n+2}\right] - \\
-\left[ Y_{6n+3}, \left( A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0 }\right) Y_2 \right] = \\
= \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6n+3} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right)\bar{Y}_{6n+4}.\end{aligned}$$ Therefore, we have $L_{6n+4} = L_{6n+3} \oplus \left\{Y_{6n+3} \right\}$, $\delta(6n+1) = 1$.
We consider the operators of the length $6n+4$: $$Y_{6(n+1)-1} = \left[ Y_1, Y_{6n+4} \right], \quad \bar{Y}_{6(n+1)-1} = \left[Y_2, Y_{6n+4} \right], \quad \left[Y_1, \bar{Y}_{6n+4} \right], \quad \left[Y_2, \bar{Y}_{6n+4} \right].$$ Let us calculate the formulas by which the operator $D_x$ commutes with these operators: $$\begin{aligned}
\fl \left[D_x, \left[ Y_2, \bar{Y}_{6n+4} \right]\right] = \left[Y_2, \alpha^2 \left(-A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right) Y_{6n+2} \right] = \\
= \alpha^3 \left(-A e^{\alpha u_0} + 8 B e^{-2 \alpha u_0} \right)Y_{6n+2} + \alpha^2 \left(-A e^{\alpha u_0} + 8 B e^{-2 \alpha u_0} \right) \left[Y_2, Y_{6n+2} \right] =\\
= \alpha^3 \left( -A e^{\alpha u_0} + 8 B e^{-2 \alpha u_0 }\right) Y_{6n+2} = \left[ D_x, 2 \alpha^2 Y_{6n+3} - \alpha \bar{Y}_{6n+4}\right].\end{aligned}$$ Thus we have: $\left[D_x, \left[ Y_2, \bar{Y}_{6n+4} \right]\right] = \left[ D_x, 2 \alpha^2 Y_{6n+3} - \alpha \bar{Y}_{6n+4}\right]$. Due to Lemma 1 we obtain, that $\left[ Y_2, \bar{Y}_{6n+4} \right] = 2 \alpha^2 Y_{6n+3} - \alpha \bar{Y}_{6n+4}$.
For $\bar{Y}_{6n+4}$ the following formula is satisfied: $$\begin{aligned}
\left[ D_x, \left[ Y_1, \bar{Y}_{6n+4} \right] \right] = \\
=\left[Y_1, \alpha^2 \left( -A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right) Y_{6n+2} \right] -
- \left[ \bar{Y}_{6n+4}, \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_2 \right] =\\
= \alpha^2 \left( -A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right)Y_{6n+3}
+ \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_2, \bar{Y}_{6n+4} \right] =\\
= \alpha^2 \left(-A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right) Y_{6n+3} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0}\right) \left( 2 \alpha^2 X_{6n+3} - \alpha \bar{Y}_{6n+4} \right) = \\
= \alpha^2 \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) Y_{6n+3} - \alpha \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+4} = \left[ D_x, -\alpha Y_{6n+4} \right].\end{aligned}$$ Due to Lemma 1 we have $\left[ Y_1, \bar{Y}_{6n+4} \right] = -\alpha Y_{6n+4}$.
For $\bar{Y}_{6(n+1)-1}$ the following formula is true: $$\begin{aligned}
\left[D_x, \bar{Y}_{6(n+1)-1} \right] = \left[ D_x, \left[Y_2, Y_{6n+4} \right] \right] = \\
=\left[ Y_2, \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right)Y_{6n+3} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+4}\right] = \\
=\alpha^2 \left( -A e^{\alpha u_0} - 4 B e^{-2 \alpha u_0} \right)Y_{6n+3} + \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+4} + \\
+ \alpha \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+4} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_2, \bar{Y}_{6n+4} \right] = \\
= \alpha^2 \left( A e^{\alpha u_0} - 2 B e^{-2 \alpha u_0} \right) Y_{6n+3} + \alpha \left( -A e^{\alpha u_0} - B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+4} = \left[D_x, -\alpha Y_{6n+4} \right].\end{aligned}$$ Due to Lemma 1, we conclude that $\bar{Y}_{6(n+1)-1} = -\alpha Y_{6n+4}$.
For $Y_{6(n+1)-1}$ the following formula is true: $$\begin{aligned}
\fl \left[D_x, Y_{6(n+1)-1} \right] = \left[D_x, \left[Y_1, Y_{6n+4} \right] \right] = \\
= \left[Y_1, \alpha \left(-A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right)Y_{6n+3} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \bar{Y}_{6n+4} \right] - \\
- \left[ Y_{6n+4}, \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) Y_2 \right] = \\
= \alpha \left( -A e^{\alpha u_0} + 2 B e^{-2 \alpha u_0} \right) Y_{6n+4} + \left( A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_1, \bar{Y}_{6n+4} \right] + \\
+ \left(A e^{\alpha u_0} + B e^{-2 \alpha u_0} \right) \left[ Y_2, Y_{6n+4} \right] = -3 \alpha A e^{\alpha u_0}Y_{6n+4}.\end{aligned}$$ It is implies that the operator $Y_{6(n+1)-1} = Y_{1\ldots 121}$ is not linearly expressed through the operators of lower order and $L_{6(n+1)-1} = L_{6n+4} \oplus \left\{Y_{6(n+1)-1} \right\}$. Thus we have $\delta(6(n+1)-1) = 1$ and Lemma 8 is proved. Evidently the Lemma 8 allows to complete the proof or the first part of the Theorem 2.
The case $\alpha = -\beta$
--------------------------
Now we pass to the second part of the Theorem 2. Here we prove that if polynomial $\Lambda(\lambda)$ has two different nonzero roots $\alpha$ and $\beta = -\alpha$ then the Lie-Rinehart algebra $L$ generated by the operators $X_0$ and $W_0$ is of an infinite dimension.
First we introduce two polynomials according to the rule $${\Lambda}_{\alpha}(\lambda)=\frac{\Lambda(\lambda)}{\lambda-\alpha}, \quad {\Lambda}_{\beta}(\lambda)=\frac{\Lambda(\lambda)}{\lambda+ \alpha}.$$ Then we construct two operators $P_{\alpha}, P_{\beta} \in L$: $$P_{\alpha}={\Lambda}_{\alpha}(ad_{X_0}W_0), \quad P_{\beta}={\Lambda}_{\beta}(ad_{X_0}W_0)$$ and concentrate on the Lie-Rinehart algebra $L_{1} \subset L$ being a subalgebra of $L$ generated by the operators $P_{\alpha}, P_{\beta}$. By construction these operators satisfy the following commutativity relations $$\left[ D_x, P_{\alpha} \right] = A(u_{1}, u_{-1}) e^{\alpha u_0} X_0, \quad \left[ D_x, P_{\beta} \right] = B(u_{1}, u_{-1}) e^{- \alpha u_0} X_0,$$ where $A=A(u_{1}, u_{-1})$, $B=B(u_{1}, u_{-1})$ are some quasi-polynomials in $u_{1}$, $u_{-1}$. We assume that $(u_{-1}, u_1) \in D$, here $D$ is a domain in $\mathbb{C}^2$, where both $A$, $B$ do not vanish.
Let us consider the operators: $$Y_1 = P_{\alpha} + P_{\beta}, \quad Y_2 = \frac{\partial}{\partial u}.$$ For these operators the following formulas are true: $$\label{sin_DX1X2}
\left[D_x, Y_1 \right] = \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_2, \quad \left[D_x, Y_2\right] = 0$$
[**Lemma 9.**]{} *Assume that polynomial $\Lambda(\lambda)$ defined by (\[Lambda\_00\]) has two different nonzero roots $\alpha$ and $-\alpha$. Then the following formulas hold:* $$L_{2k+1} = L_{2k} \oplus \left\{Y_{2k+1} \right\}, \quad L_{2k} = L_{2k-1} \oplus \left\{ Y_{2k}, \bar{Y}_{2k} \right\}.$$
[**Proof.**]{} Let us consider the operator $Y_3 = \left[Y_2, Y_1\right]$. Using the Jacobi identity and formulas (\[sin\_DX1X2\]) we prove that $$\begin{aligned}
\fl \left[D_x, Y_3 \right] = \left[ D_x, \left[ Y_2, Y_1 \right] \right] = \left[ Y_2, \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_2 \right] = \\
= \left(A \alpha e^{\alpha u_0} - B \alpha e^{-\alpha u_0} \right) Y_2 = \alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_2.\end{aligned}$$ We can see that $Y_3$ is not linearly expressed through the previous operators. Thus $L_3 = \left\{ Y_1, Y_2, Y_3 \right\}$.
Let us construct the operators of lenght 3: $$Y_4 = \left[ Y_1, Y_3 \right] = Y_{121}, \qquad \bar{Y}_4 = \left[Y_2, Y_3 \right] = Y_{221}.$$ Using the Jakobi identity and formulas for the operators $Y_1$, $Y_2$, $Y_3$ we find: $$\begin{aligned}
\fl \left[ D_x, Y_4 \right] = \left[ D_x, \left[Y_1, Y_3 \right] \right] = \left[ Y_1, \alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) \right] -
\left[Y_3, \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_2 \right] =\\
= -\alpha \left(A e^{\alpha u_0} - B e^{-\alpha u_0}\right) Y_3 + \left( Ae^{\alpha u_0} + B e^{-\alpha u_0} \right) \left[ Y_2, Y_3 \right] = \\
= - \alpha \left(A e^{\alpha u_0} - B e^{-\alpha u_0}\right) Y_3 + \alpha^2 \left( Ae^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_1,\end{aligned}$$ $$\begin{aligned}
\fl \left[ D_x, \bar{Y}_4 \right] = \left[ D_x, \left[Y_2, Y_3 \right] \right] = \left[ Y_2, \alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right)Y_2 \right] = \\
= \alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0}\right)Y_2 = \left[D_x, \alpha^2 Y_1 \right].\end{aligned}$$ Thus we obtain the equality $\left[ D_x, \bar{Y}_4 -\alpha^2 Y_1 \right] = 0 $. Due to Lemma 1 we conclude that $\bar{Y}_4 = \alpha^2 Y_1$. The operator $Y_4$ is not expressed through the operators of lower order. So $L_4 = \left\{ Y_1, Y_2, Y_3, Y_4 \right\}$.
Now we construct the commutators of length 4: $$Y_5 = \left[Y_1, Y_4 \right], \qquad \bar{Y}_5 = \left[Y_2, Y_4 \right].$$ Now we need to calculate the formulas by which the operator $D_x$ commutes with these operators: $$\begin{aligned}
\fl \left[ D_x, \bar{X}_5 \right] = \left[ D_x, \left[X_2, X_4 \right] \right] = \left[ X_2, -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) X_3 + \alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) X_1\right] = \\
-\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) \bar{X}_4 + \alpha^3 \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) X_1 = \\
= -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) \alpha^2 X_1 + \alpha^3 \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) X_1 = 0.\end{aligned}$$ Based on Lemma 1 we claim that $\bar{X}_5 = 0$. We have one more formula: $$\begin{aligned}
\fl \left[D_x, Y_5 \right] = \left[ D_x, \left[Y_1, Y_4 \right] \right] =
= \left[ Y_1, -\alpha \left(A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_3 + \alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_1\right] - \\
-\left[ Y_4, \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) X_2 \right]= \\
= -\alpha \left(A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_4 + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \bar{Y}_5 =\\
= -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_4.\end{aligned}$$ Thus we see that $Y_5$ is not linearly expressed through the operators of lower order and $L_5 = L_4 \oplus \left\{ Y_5 \right\}$.
Now we construct the commutators of length 5: $$Y_6 = \left[Y_1, Y_5 \right], \quad \bar{Y}_6 = \left[ Y_2, Y_5 \right].$$ These operators are satisfied the formulas: $$\begin{aligned}
\left[ D_x, Y_6 \right] = \left[ D_x, \left[ Y_1, Y_5 \right] \right] = \\
= \left[ Y_1, -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_4 \right] - \left[ Y_5, \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_2 \right] = \\
= - \alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_5 + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \bar{Y}_6.\end{aligned}$$ Thus we obtain that $L_6 = L_5 \oplus \left\{Y_6, \bar{Y}_6 \right\}$.
Then we consider the commutators of length 6: $$Y_7 = \left[ Y_1, Y_6 \right], \quad \bar{Y}_7 = \left[ Y_2, Y_6 \right], \quad \left[Y_1, \bar{Y}_6 \right], \quad \left[Y_2, \bar{Y}_6 \right]$$ The following formulas are true: $$\begin{aligned}
\fl \left[D_x, \left[ Y_2, \bar{Y}_6 \right] \right] = \left[ Y_2, -\alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_4 \right] = \\
= - \alpha^3 \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_4 - \alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \left[Y_2, Y_4 \right] = \\
= -\alpha^3 \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_4 = \left[ D_x, \alpha^2 Y_5 \right],\end{aligned}$$ $$\begin{aligned}
\fl \left[ D_x, \left[ Y_1, \bar{Y}_6 \right] \right] = \left[ Y_1, -\alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_4\right] - \left[ \bar{Y}_6, \left(A e^{\alpha u_0} + B e^{-\alpha u_0}\right) X_2 \right] = \\
= -\alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_5 + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \left[Y_2, \bar{Y}_6 \right] = \\
= -\alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0}\right)Y_5 + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \alpha^2 Y_5 = 0,\end{aligned}$$ $$\begin{aligned}
\fl \left[ D_x, \bar{Y}_7 \right] = \left[ D_x, \left[ Y_2, Y_6\right] \right] =\\
= \left[ Y_2, -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right)Y_5 + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \bar{Y}_6 \right] = \\
= -\alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0 } \right) Y_5 + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \left[ Y_2, \bar{Y}_6 \right] = 0,\end{aligned}$$ $$\begin{aligned}
\fl \left[D_x, Y_7 \right] = \left[D_x, \left[Y_1, Y_6 \right] \right] = \\
=\left[Y_1, -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_5 + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right)\bar{Y}_6 \right] - \\
- \left[ Y_6, \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_2 \right] = -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_6 + \\
+ \left(A e^{\alpha u_0} + B e^{-\alpha u_0} \right)\left[ Y_1, \bar{Y}_6 \right] + \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \left[Y_2, Y_6 \right] =\\
= -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_6.\end{aligned}$$ Thus we see that $\bar{Y}_6 = \alpha^2 Y_5$, $\bar{Y}_6 = 0$, $\bar{Y}_7 = 0$, the operator $Y_7$ is not linearly expressed through the operators of lower order and $L_7 = L_6 \oplus \left\{Y_7 \right\}$, $\delta(7) = 1$.
It can be proved by induction that $ \left[Y_2, Y_i \right] = 0$, $$\begin{aligned}
\left[ D_x, Y_{i+1} \right] = -\alpha \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right)Y_i , \quad i=3,4,\ldots,\\
\left[ D_x, \left[ Y_2, Y_{i+1} \right] \right] = -\alpha^2 \left(A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_i.\end{aligned}$$ Then we have $$\begin{aligned}
\fl \left[D_x, \left[ Y_2, \left[Y_2, Y_{i+1} \right] \right] \right] = \left[ Y_2, -\alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_1 \right] = \\
= -\alpha^3 \left( A e^{\alpha u_0} - B e^{-\alpha u_0} \right) Y_i = \alpha^2 \left[ D_x, Y_{i+1} \right].\end{aligned}$$ This equality implies $ \left[D_x, \left[ Y_2, \left[Y_2, Y_{i+1} \right] \right] -\alpha^2 Y_{i+1} \right] = 0$. Using Lemma 1 we conclude that $\left[ Y_2, \left[Y_2, Y_{i+1} \right] \right] = \alpha^2 Y_{i+1}$, $i = 4,6,\ldots,2n$.
The following formula is true $$\begin{aligned}
\fl \left[ D_x, \left[ Y_1, \left[Y_2, Y_{i+1} \right] \right] \right] = \left[ Y_1, -\alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right)Y_i \right] - \\
- \left[ \left[ Y_2, Y_{i+1} \right], \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right)Y_2 \right] = \\
= - \alpha^2 \left( A e^{\alpha u_0} + B e^{-\alpha u_0} \right) Y_{i+1} + \left(A e^{\alpha u_0} + B e^{-\alpha u_0} \right) \left[ Y_2, \left[ Y_2, Y_{i+1} \right] \right] = 0.\end{aligned}$$ Due to Lemma 1 we obtain that $\left[Y_1, \left[Y_2, Y_{i+1} \right] \right] = 0$, $i = 4,6,\ldots 2n$.
Thus we conclude that $L_{2k+1} = L_{2k} \oplus \left\{Y_{2k+1} \right\} $, $L_{2k} = L_{2k-1} \oplus \left\{ Y_{2k}, \bar{Y}_{2k} \right\}$. This completes the proof of the Lemma 9. Now the second part of Theorem 2 immediately follows from the Lemma 9.
Conclusions
===========
In the article we study the problem of integrable classification of a rather specific but important class of two-dimensional lattices. We used to this aim the method of Darboux integrable reductions and the concept of characteristic Lie algebras [@H2013]-[@HabKuznetsova19]. By applying these implements we derived the necessary conditions of integrability for lattices of the form (\[eq\_main\]). Efficiency of these conditions is illustrated in [@FHKN].
References {#references .unnumbered}
==========
[30]{}
Bogdanov L V and Konopelchenko B G 2013 Grassmannians $Gr(N-1,N+1)$, closed differential $N-1$ forms and $N$-dimensional integrable systems [*J. Phys. [A]{}: Math. Theor.*]{} [**46**]{} 085201
Pavlov M V and Popowicz Z 2009 On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems [*SIGMA*]{} [**5**]{} 011
Pogrebkov A K 2008 Commutator identities on associative algebras and the integrability of nonlinear evolution equations [*Theor. Math. Phys.*]{} [**154**]{}:3 405–17
Mañas M, Alonso L M and Álvarez-Fernández C 2009 The multicomponent 2D Toda hierarchy: discrete flows and string equations [*Inverse Problems*]{} [**25**]{} 065007
Zakharov V E and A. B. Shabat A B 1974 A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I [*Funct. Anal. Appl.*]{} [**8**]{} 226–35
Habibullin I T 2013 Characteristic Lie rings, finitely-generated modules and integrability conditions for (2+ 1)-dimensional lattices [*Physica Scripta*]{} [**87**]{}:6 065005
Habibullin I T and Poptsova M N 2017 Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings [*SIGMA*]{} [**13**]{} 073
Habibullin I T and Poptsova M N 2018 Algebraic properties of quasilinear two-dimensional lattices connected with integrability [*Ufa Math. J.*]{} [**10**]{}:3 86–105
Kuznetsova M N 2019 Classification of a subclass of quasilinear two-dimensional lattices by means of characteristic algebras [*Ufa Math. J.*]{} [**11**]{}:3 110–32
Habibullin I T and Kuznetsova M N 2020 On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras [*Theor. Math. Phys.*]{} [**203**]{} 569–581 [*arXiv preprint*]{} arXiv:1907.12269
Gibbons J and Tsarev S P 1996 Reductions of Benney’s equations [*Phys. Lett. [A]{}*]{} [**211**]{}:1 19–24.
Ferapontov E V 1997 Laplace transformations of hydrodynamic-type systems in Riemann invariants [*Theor. Math. Phys.*]{} [**110**]{}:1 68–77
Ferapontov E V and Khusnutdinova K R 2004 On the integrability of (2+1)-dimensional quasilinear systems [*Commun. Math. Phys.*]{} [**248**]{} 187–206
Ferapontov E V, Khusnutdinova K R and Pavlov M V 2005 Classification of Integrable (2+1)-Dimensional Quasilinear Hierarchies [*Theor. Math. Phys.*]{} [**144**]{}:1 907–915
Ferapontov E V, Khusnutdinova K R and Tsarev S P 2006 On a class of three-dimensional integrable Lagrangians [*Commun. Math. Phys.*]{} [**261**]{} 225–243
Zhiber A V, Murtazina R D, Habibullin I T and Shabat A B 2012 Characteristic Lie rings and integrable models in mathematical physics [*Ufa Math. J.*]{} [**4**]{}:3 17–85
Habibullin I, Zheltukhina N and Pekcan A 2008 On the classification of Darboux integrable chains [*J. Math. Phys.*]{} [**49**]{}:10 102702
Habibullin I, Zheltukhina N and Pekcan A 2009 Complete list of Darboux integrable chains of the form $t_{1 x}= t_x+ d (t, t_1)$ [*J. Math. Phys.*]{} [**50**]{}:10 102710
Zheltukhin K, Zheltukhina N and Bilen E 2017 On a class of Darboux-integrable semidiscrete equations [*Adv. Diff. Eq.*]{} [**2017**]{}:182
Zhiber A V, Murtazina R D, Habibullin I T and Shabat A B 2012 [*Characteristic Lie rings and nonlinear integrable equations*]{} (Moscow–Izhevsk: Institute of Computer Science) p 376 (in Russian).
Zhiber A V and Kostrigina O S 2007 Exactly integrable models of wave processes [*Vestnik USATU*]{} [**9**]{}:7(25) 83–89 (in Russian)
Shabat A B and Yamilov R I 1981 Exponential systems of type I and the Cartan matrix [*Preprint*]{} Bashkir branch of AS USSR, Ufa (in Russian).
Zhiber A V and Murtazina R D 2008 On the characteristic Lie algebras for equations $u_{xy}=f(u,u_x)$ [*J. Math. Sci.*]{} [**151**]{}:4 3112–22.
Sakieva A U 2012 Characteristic Lie ring of the Zhiber-Shabat-Tzitzeica equation [*Ufa Math. J.*]{} [**4**]{}:3 153–58.
Shabat A B 1995 Higher symmetries of two-dimensional lattices [*Phys. Lett. [A]{}*]{} [**200**]{}:2 121–33
Anderson I M and Kamran N 1997 The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane [*Duke Math. J*]{} [**87**]{}:2 265–319
Zhiber A V and Sokolov V V 2001 Exactly integrable hyperbolic equations of Liouville type [*Russ. Math. Surv.*]{} [**56**]{} 61–101
Gubbiotti G, Scimiterna C and Yamilov R I 2018 Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations II: General Solutions [*SIGMA*]{} [**14**]{} 008
Smirnov S V 2015 Darboux integrability of discrete two-dimensional Toda lattices [*Theor. Math. Phys.*]{} [**182**]{}:2 189–210
Rinehart G 1963 Differential forms for general commutative algebras [*Trans. Amer. Math. Soc.*]{} [**108**]{} 195–222
Millionshchikov D 2018 Lie Algebras of Slow Growth and Klein-Gordon PDE [*Algebr. Represent. Theor.*]{} [**21**]{} 1037–69
Ferapontov E V, Habibullin I T, Kuznetsova M N and Novikov V S 2020 On a class of 2D integrable lattice equations [*arXiv preprint*]{} arXiv:2005.06738
|
---
abstract: |
In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius $r_n$ of a random quadrangulation with $n$ faces is shown to converge, up to scaling, to the width $r=R-L$ of the support of the one-dimensional ISE, or precisely: $$n^{-1/4}r_n\;\mathop{\longrightarrow}^{\textrm{\emph{law}}}\;(8/9)^{1/4}\,r.$$ More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero.
The first combinatorial ingredient is an encoding of quadrangulations by *trees embedded in the positive half-line*, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the *conjugation of tree principle*, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge.
[From]{} probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks $(e^{(n)},\hat W^{(n)})$ to the Brownian snake description $(e,\hat W)$ of ISE.
Our results suggest the existence of a *Continuum Random Map* describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity.
address:
- 'Philippe Chassaing, Université Henri Poincaré, B.P. 239, 54506 Vand[œ]{}uvre-lès-Nancy'
- 'Gilles Schaeffer, CNRS – LORIA, B.P. 239, 54506 Vand[œ]{}uvre-lès-Nancy'
author:
- Philippe Chassaing
- Gilles Schaeffer
bibliography:
- 'resume.bib'
title: Random Planar Lattices and Integrated SuperBrownian Excursion
---
\#1\#2\#3 ł
=cmex10\#1\#2[==\#1]{}\#1\#2\#3[==\#1]{}\#1\#2\#3
Introduction
============
[From]{} a distant perspective, this article uncovers a surprising, and hopefully deep, relation between two famous models: *random planar maps*, as studied in combinatorics and quantum physics, and *Brownian snakes*, as studied in probability theory and statistical physics. More precisely, our results connect some distance-related functionals of *random quadrangulations* with functionals of Aldous’ *Integrated SuperBrownian Excursion* (ISE) in dimension one.
Quadrangulations {#quadrangulations .unnumbered}
----------------
On the one hand, quadrangulations are finite plane graphs with four-regular faces (see Figure \[fig:aquad\] and Section \[sec:def\] for precise definitions). Random quadrangulations, like random triangulations, random polyhedra, or the $\phi^4$-models of physics, are instances of a general family of random lattices that has received considerable attention both in combinatorics (under the name *random planar maps*, following Tutte’s terminology [@tutte-base]) and in physics (under the name *Euclidean two-dimensional discretised quantum geometry*, or simply *dynamical triangulations* or *fluid lattices* [@ambjorn; @BIPZ; @gross]).
Many probabilistic properties of random planar maps have been studied, that are *local properties* like vertex or face degrees [@bender-canfield; @gao-wormald], or $0-1$ laws for properties expressible in first order logic [@bender-compton]. Other well documented families of properties are related to connectedness and constant size separators [@BaFlScSo01], also known as branchings into baby universes [@phys-baby]. In this article we consider another fundamental aspect of the geometry of random maps, namely *global properties of distances*. The *profile* $\smash{(H^{n}_k)_{k\geq0}}$ and *radius* $r_n$ of a random quadrangulation with $n$ faces are defined in analogy with the classical profile and height of trees: $\smash{H_k^{n}}$ is the number of vertices at distance $k$ from a basepoint, while $r_n$ is the maximal distance reached. The profile was studied (with triangulations instead of quadrangulations) by physicists Watabiki, Ambj[ø]{}rn et *al.* [@ambjorn-watabiki; @watabiki] who gave a consistency argument proving that the only possible scaling for the profile is $\smash{k\sim
n^{1/4}}$, a property which reads in their terminology *the internal Hausdorff dimension is 4*. Independently the conjecture that $\mathbb{E}(r_n)\sim cn^{1/4}$ was proposed by Schaeffer [@these-schaeffer].
![Random quadrangulations, in planar or spherical representation.[]{data-label="fig:aquad"}](\rep quadrangulation.eps){width="11cm"}
Integrated SuperBrownian Excursion {#integrated-superbrownian-excursion .unnumbered}
----------------------------------
On the other hand, ISE was introduced by Aldous as a model of random distributions of masses [@AALD]. He considers random embedded discrete trees as obtained by the following two steps: first an abstract tree $t$, say a Cayley tree with $n$ nodes, is taken from the uniform distribution and each edge of $t$ is given length $1$; then $t$ is embedded in the regular lattice on $\mathbb{Z}^d$, with the root at the origin, and edges of the tree randomly mapped on edges of the lattice. Assigning masses to leaves of the tree $t$ yield a random distribution of mass on $\mathbb{Z}^d$. Upon scaling the lattice to $n^{-1/4}\mathbb{Z}^d$, these random distributions of mass admit, for $n$ going to infinity, a continuum limit $\mathcal{J}$ which is a random probability measure on $\mathbb{R}^d$ called ISE.
Derbez and Slade proved that ISE describes in dimension larger than eight the continuum limit of a model of lattice trees [@DS], while Hara and Slade obtained the same limit for the incipient infinite cluster in percolation in dimension larger than six [@HS]. As opposed to these works, we shall consider ISE in dimension one and our embedded discrete trees should be thought of as folded on a line. The support of ISE is then a random interval $(L,R)$ of $\mathbb{R}$ that contains the origin.
From quadrangulations to ISE {#from-quadrangulations-to-ise .unnumbered}
----------------------------
The purpose of this paper is to draw a relation between, on the one hand, random quadrangulations, and, on the other hand, Aldous’ ISE: upon proper scaling, the profile of a random quadrangulations is described in the limit by ISE translated to have support $(0,R-L)$. This relation implies in particular that the radius $r_n$ of random quadrangulations, again upon scaling, weakly converges to the width of the support of ISE in one dimension, that is the continuous random variable $r=R-L$. We shall indeed prove (Corollary \[radcv\]) that $$n^{-1/4}r_n\;\mathop{\longrightarrow}^{\textrm{law}}\;(8/9)^{1/4}\,r,$$ as well as the convergence of moments. While this proves the conjecture $\mathbb{E}(r_n)\sim cn^{1/4}$, the value of the constant $c$ remains unknown because, as mentioned by Aldous [@AALD], little is known on $R$ or $R-L$.
The path from quadrangulations to ISE consists of three main steps, the first two of combinatorial nature and the last with a more probabilistic flavor. Our first step, Theorem \[thm:welllab\], revisits a correspondence of Cori and Vauquelin [@cori-vauquelin] between planar maps and some *well labelled trees*, that can be viewed as plane trees embedded in the positive half-line. Thanks to an alternative construction [@these-schaeffer Ch. 7], we show that under this correspondence the profile can be mapped to the mass distribution on the half-line. In particular, the radius $r_n$ of a random quadrangulation is equal in law to the maximal label $\mu_n$ of a random well labelled tree.
Safe for the positivity condition, well labelled trees would be constructed exactly according to Aldous’ prescription for embedded discrete trees. Well labelled trees are thus to Aldous’ embedded trees what the Brownian excursion is to the Brownian bridge, and we seek an analogue of Vervaat’s relation. At the discrete level a classical elegant explanation of such relations is based on Dvoretsky and Motzkin’s cyclic shifts and cycle lemma. Our second combinatorial step, Theorem \[thm:coupling\], consists in the adaptation of these ideas to embedded trees. More precisely, via the *conjugation of tree principle* of [@these-schaeffer Chap. 2], we bound the discrepancy between the mass distribution of our conditioned trees on the positive half-line and a translated mass distribution of freely embedded trees. In particular we construct a coupling between well labelled trees and freely embedded trees such that the largest label $\mu_n$, and thus the radius $r_n$, is coupled to the width of the support $(L_n,R_n)$ of random freely embedded trees: $$|r_n-(R_n-L_n)|\leq 3.$$ Since our freely embedded trees are constructed according to Aldous’ prescription, one could expect to be able to conclude directly. However two obstacles still need to be bypassed at this point.
Contour walks and Brownian snakes. {#contour-walks-and-brownian-snakes. .unnumbered}
----------------------------------
The first obstacle is that the construction of ISE as a continuum limit of mass distributions supported by embedded discrete trees was only outlined in Aldous’ original paper. The original mathematical definition is by embedding a continuum random tree (CRT), which amounts to exchanging the embedding and the continuum limit. But Borgs et *al.* proved that indeed ISE is the limit of mass distributions supported by embedded Cayley trees [@BCHS] and their proof could certainly be adapted to other simple classes of trees and in particular to our embedded plane trees.
The second, more important, obstacle is that weak convergence of probability measures is not adequate to our purpose, since we are interested in particular in convergence of the width of the support, *which is not a continuous functional on the space of measures*. In order to circumvent this difficulty, we turn to the description of ISE in terms of superprocesses: ISE can be constructed from the Brownian snake with lifetime $e$, the standard Brownian excursion [@AALD; @LEG].
[From]{} the discrete point of view, we consider the encoding of an embedded plane tree by a pair of contour walks $(x_k,y_k)$, that encode respectively the height of the node visited at time $k$ and its position on the line. Our last result, Theorem \[thm:converge\], is the weak convergence, upon proper scaling, of this pair of walks to the Brownian snake with lifetime $e$: $$\big(e^{(n)}(s),\hat W^{(n)}(s)\big)\mathop{\longrightarrow}^{\textrm{law}}
\big(e(s),\hat W_s\big).$$ As $R=\sup_s \hat W_s$ and $L=\inf_s\hat W_s$ this convergence, together with some deviation bounds obtained in the proof allows us to conclude on the radius. (A similar weak convergence was independently proved by Marckert and Mokkadem [@MM] but without the deviation bounds we need here.)
More generally the joint convergence of the minimum and the mass distribution of discrete embedded trees implies that, upon scaling, the label distribution of well labelled trees converges to ISE translated to have the minimum of its support at the origin. The same then holds for the profile of random quadrangulations.
Dynamical triangulations and a Continuum Random Map {#dynamical-triangulations-and-a-continuum-random-map .unnumbered}
---------------------------------------------------
Although we concentrate in this article on the radius and profile of random quadrangulations, our derivation suggests a much tighter link between random quadrangulations and ISE. We conjecture that a Continuum Random Map (CRM) can be built from ISE that would describe the continuum limit of scaled random quadrangulations, in a similar way as the CRT describes the continuum limit of scaled random discrete trees. From the point of view of physics, the resulting CRM would describe in the limit the geometry of scaled dynamical triangulations as studied in discretised two-dimensional Euclidean pure quantum geometries [@ambjorn; @BIPZ; @gross]. We plan to discuss this connection further in future work.
Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------
Section 2 contains the definition the combinatorial model of random lattice. Sections 3 and 4 are devoted to the first combinatorial steps. Section 5 contains to the definition of probabilistic models, and the statement of the convergence result. Finally in Section 6 we give the proof of this convergence.
![Two distinct planar maps, and a spherical representation of the second.[]{data-label="fig:amap"}](\rep sphere.eps){width="11cm"}
The combinatorial models of random lattice {#sec:def}
==========================================
Planar maps and quadrangulations
--------------------------------
A *planar map* is a proper embedding (without edge crossings) of a connected graph in the plane. Loops and multiple edges are *a priori* allowed. A planar map is *rooted* if there is a *root*, *i.e.* a distinguished edge on the border of the infinite face, which is oriented counterclockwise. The origin of the root is called the *root vertex*. Two rooted planar maps are considered identical if there exists an homeomorphism *of the plane* that sends one map onto the other (roots included).
The difference between planar graphs and planar maps is that the cyclic order of edges around vertices matters in maps, as illustrated by Figure \[fig:amap\]. Observe that planar maps can be equivalently defined on the sphere. In particular Euler’s characteristic formula applies and provides a relation between the numbers $n$ of edges, $f$ of faces and $v$ of vertices of any planar map: $f+v=n+2$.
The *degree* of a face or of a vertex of a map is its number of incidence of edges. A planar map is a *quadrangulation* if all faces have degree four. All (planar) quadrangulations are *bipartite*: their vertices can be colored in black or white so that the root is white and any edge joins two vertices with different colors. In particular a quadrangulation contains no loop but may contain multiple edges. See Figures \[fig:aquad\] and \[fig:etiq\] for examples of quadrangulations.
Let $\mathcal{Q}_n$ denote the set of rooted quadrangulations with $n$ faces. A quadrangulation with $n$ faces has $2n$ edges (because of the degree constraint) and $n+2$ vertices (applying Euler’s formula). The number of rooted quadrangulations with $n$ faces was obtained by W.T. Tutte [@tutte-base]: $$\label{equ:tutte}
|\mathcal{Q}_n|\;\;=\;\;\frac{2}{n+2}\,\frac{3^n}{n+1}{2n\choose n}.$$ Various alternative proofs of this result have been obtained (see *e.g.* [@BIPZ; @cori-vauquelin; @arques-tres-bien-etiquete; @these-schaeffer]). Our treatment will indirectly provide another proof, related to [@cori-vauquelin; @these-schaeffer].
Random planar lattices
----------------------
Let $L_n$ be a random variable with uniform distribution on $\mathcal{Q}_n$. Formally, $L_n$ is the $\mathcal{Q}_n$-valued random variable such that for all $Q\in\mathcal{Q}_n$ $$\Pr(L_n=Q)\;=\;\frac1{|\mathcal{Q}_n|}\;=\;
\frac1{\frac{2}{n+2}\frac{3^n}{n+1}{2n\choose
n}}.$$ The random variable $L_n$ is our *random planar lattice*. To explain this terminology, taken from physics, observe that locally the usual planar square lattice is a planar map whose faces and vertices all have degree four. Our random planar lattice corresponds to a relaxation of the constraint on vertices.
![Labelling by distance from the root vertex and the two possible configurations of labels (top: a simple face; bottom: a confluent face).[]{data-label="fig:etiq"}](\rep etiqter.eps "fig:"){width="6cm"} ![Labelling by distance from the root vertex and the two possible configurations of labels (top: a simple face; bottom: a confluent face).[]{data-label="fig:etiq"}](\rep possible1.eps "fig:"){width="2cm"}
Classical variants of this definition are obtained by replacing quadrangulations with $n$ faces by triangulations with $2n$ triangles, or by (vertex-)4-regular maps with $n$ vertices, or by all planar maps with $n$ edges, *etc.* All these random planar lattices have been considered both in combinatorics (see [@BaFlScSo01] and references therein) and in mathematical physics (see [@ambjorn] and references therein; in the physics literature, definitions are usually phrased using “symmetry weights” instead of rooted objects, but this is strictly equivalent to the combinatorial definition). Although details of local topology vary between families, most probabilistic properties are believed to be “universal”, that is qualitatively analogue for all “reasonable” families. Observe also that random maps in classical families have exponentially small probability to be symmetric, so that all results hold as well as in the model of uniform unrooted maps [@RiWo95].
In this article we focus on quadrangulations because of their combinatorial relation, detailed in Section \[sec:welllab\], to well labelled trees.
The profile of a map
--------------------
The distance $d(x,y)$ between two vertices $x$ and $y$ of a map is the minimal number of edges on a path from $x$ to $y$ (in other terms all edges have abstract length $1$).
The *profile* of a rooted map $M$ is the sequence $(H_k)_{k\geq1}$, where $H_k\equiv H_k^{[M]}$ is the number of vertices at distance $k$ of the root vertex $v_0$. We shall also consider the cumulated profile $\widehat H_k^{[M]}\;=\;\sum_{\ell=1}^k
H_\ell^{[M]}$. By construction the support of the profile of a rooted map is an interval *i.e.* $\{k\mid H_k>0\}=[1,r]$ where $r$ is the *radius* of the map (sometimes also called *eccentricity*). The radius $r$ is closely related to the *diameter*, that is the largest distance between two vertices of a map: in particular $r\leq d\leq 2r$. The quadrangulation of Figure \[fig:etiq\] has radius 3.
The *profile of the random planar lattice $L_n$* is the random variable $(H^{(n)}_k)_{k\geq1}$ that is defined by taking the profile $(H^{[L_n]}_k)_{k\geq1}$ of an instance of $L_n$, while $(\widehat H^{(n)}_k)_{k\geq1}$ denotes the *cumulated* profile of $L_n$. Similarly the radius of a random planar lattice is a positive integer valued random variable $r_n$.
Encoding the profile with well labelled trees {#sec:welllab}
=============================================
Well labelled trees and the encoding result
-------------------------------------------
![A well labelled tree with its label distribution. \[fig:constrained\]](\rep constrained.eps){width="6cm"}
A *plane tree* is a rooted planar map without cycle (and thus with only one face). Equivalently plane trees can be recursively defined as follows:
- the smallest tree is made of a single vertex,
- any other tree is a non-empty sequence of subtrees attached to a root.
In other term, each vertex has a possibly empty sequence of sons, and each vertex but the root has a father. The number of plane trees with $n$ edges is the well known Catalan number $$C(2n)\;=\;\frac1{n+1}{2n\choose n}.$$
A plane tree is *well labelled* if all its vertices have positive integral labels, the labels of two adjacent vertices differ at most by one, and the label of the root vertex is one. Let $\mathcal{W}_n$ denote the set of well labelled trees with $n$ edges.
The *label distribution* of a well labelled tree $T$ is the sequence $(\lambda_k)_{k\geq1}\equiv(\lambda_k^{[T]})_{k\geq1}$ where $\lambda_k^{[T]}$ is the number of vertices with label $k$ in the tree $T$. The cumulated label distribution is defined by $\widehat\lambda_k^{[T]}= \sum_{\ell=1}^{k}\lambda_\ell^{[T]}$. By construction the support of the label distribution is an interval: there exists an integer $\mu$ such that $\{k\mid\lambda_k>0\}=[1,\mu]$. This integer $\mu$ is the maximal label of the tree. These definitions are illustrated by Figure \[fig:constrained\].
The following theorem will serve us to reduce the study of the profile of quadrangulations to the study of the label distribution of well labelled trees.
\[thm:welllab\] There exists a bijection $\mathcal{T}$ between rooted quadrangulations with $n$ faces and well labelled trees with $n$ edges, such that the profile $(H_k^{[Q]})_{k\geq1}$ of a quadrangulation $Q$ is mapped onto the label distribution $(\lambda_k^{[T]})_{k\geq1}$ of the tree $T=\mathcal{T}(Q)$.
Theorem \[thm:welllab\] and Tutte’s formula (\[equ:tutte\]) imply that the number of well labelled trees with $n$ edges equals $$\label{equ:welllab}
|\mathcal{W}_n|\;=\;\frac{2}{n+2}\,\frac{3^n}{n+1}{2n\choose n}.$$ This result was proved already by Cori and Vauquelin [@cori-vauquelin], who introduced well labelled trees to give an encoding of all planar maps with $n$ edges. Because of a classical bijection between the latter maps and quadrangulations with $n$ faces, their result is equivalent to the first part of Theorem \[thm:welllab\]. Their bijection has been extended to bipartite maps by Arquès [@arques-tres-bien-etiquete] and to higher genus maps by Marcus and Vauquelin [@marcus-vauquelin]. All these constructions were recursive and based on encodings of maps with permutations (also known as rotation systems).
However, our interest in well labelled trees lies in the relation between the profile and the label distribution, which does not appear in Cori and Vauquelin’s bijection. The bijection we use here is much simpler and immediately leads to the second part of Theorem \[thm:welllab\]. This approach was extended to non separable maps by Jacquard [@benj-these] and to higher genus by Marcus and Schaeffer [@marcus-schaeffer].
We postpone to Section \[sec:conjugacy\] the discussion of the interesting form of Formula (\[equ:welllab\]) and its relation to Catalan’s numbers. Instead the rest of this part is concerned with the proof of Theorem \[thm:welllab\], which goes in three steps. First some properties of distances in quadrangulations are indicated (Section \[sec:distances\]). This allows in a second step to define the encoding, as a mapping $\mathcal{T}$ from quadrangulations to well labelled trees (Section \[sec:encode\]). A decoding procedure allows then to prove that $\mathcal{T}$ is faithful (Section \[sec:decode\]).
Properties of distances in a quadrangulation {#sec:distances}
--------------------------------------------
![The rules of selection of edges and an example.[]{data-label="exemple"}](\rep regles1.eps "fig:"){width="2cm"} ![The rules of selection of edges and an example.[]{data-label="exemple"}](\rep extrater.eps "fig:"){width="9cm"}
Let $Q$ be a rooted quadrangulation and denote $v_0$ its root vertex. The labelling $\phi$ of the map ${Q}$ is defined by $\phi(x)=d(x,v_0)$ for each vertex $x$, where $d(x,y)$ denote the distance in $Q$ (cf. Figure \[fig:etiq\]). Observe that in the number of label $k$ in the labelling of the map $Q$ is precisely the number of vertices at distance $k$ of $v_0$, that is $\smash{H^{[Q]}_k}=|\{x\mid
\phi(x)=k\}|$. This labelling satisfies the following immediate properties:
If $x$ and $y$ are joined by an edge, $|\phi(x)-\phi(y)|=1$. Indeed the quadrangulation being bipartite, a vertex $x$ is white if and only if $\phi(x)$ is even, black if and only if $\phi(x)$ is odd.
Around a face, four vertices appear: a black $x_1$, a white $y_1$, a black $x_2$ and a white $y_2$. These vertices satisfy at least one of the two equalities $\phi(x_1)=\phi(x_2)$ or $\phi(y_1)=\phi(y_2)$ (cf. Figure \[fig:etiq\]).
A face will be said *simple* when only one equality is satisfied and *confluent* otherwise (see Figure \[fig:etiq\]). It should be noted that one may have $x_1=x_2$ or $y_1=y_2$.
Construction of the encoding $\mathcal{T}$ {#sec:encode}
------------------------------------------
Let $Q$ be a rooted quadrangulation with its distance labelling. The map ${Q}'$ is obtained by dividing all confluent faces ${Q}$ into two triangular faces by an edge joining the two vertices with maximal label. Let us now define a subset $\mathcal{T}(Q)$ of edges of ${Q}'$ by two selection rules:
- In each confluent face of ${Q}$, the edge that was added to form ${Q}'$ is selected.
- For each simple face $f$ of ${Q}$, an edge $e$ is selected: let $v$ be the vertex with maximal label in $f$, then $e$ is the edge leaving $v$ with $f$ on its left.
These two selection rules are illustrated by Figure \[exemple\]. The first selected edge around the endpoint of the root of $Q$ is taken to be the root of $\mathcal{T}(Q)$.
The proof of Theorem \[thm:welllab\] is now completed in two steps. First, in the rest of this section, $\mathcal{T}(Q)$, which is *a priori* only defined as a subset of edges of ${Q}'$ together with their incident vertices, is shown to be in fact a well labelled tree with $n$ edges. Second, in the next section the inverse mapping is described and used to prove that the mapping $\mathcal{T}$ is faithful.
The mapping ${\mathcal{T}}$ sends a quadrangulation $Q$ with $n$ faces on a well labelled trees with $n$ edges.
![Impossibility of cycles.[]{data-label="sans-cycle"}](\rep cycleter.eps){width="7cm"}
If the vertex $x$ is not the root of $Q$, then one of its neighbors in $Q$, say $y$, has a smaller label. The edge $(x,y)$ can be incident to: at least a confluent face; at least a simple face in which $x$ has maximal label; or two simple faces in which $x$ has intermediate label. In all three cases, $x$ is incident to the selected edge of at least one face. Thus all vertices of $Q$ but its root are also vertices of $\mathcal{T}(Q)$: in particular $\mathcal{T}(Q)$ has $n+1$ vertices. Next, the number of edges of $\mathcal{T}(Q)$ is $n$, because this is the number of faces of $Q$ and two faces cannot select the same edge (as immediately follows from inspection of the selection rules). Now the planarity of $Q$ and thus of ${Q}'$ grants that each connected component of $\mathcal{T}(Q)$ is planar. Provided we can rule out cycles, this imply that $\mathcal{T}(Q)$ is a forest of trees with $n$ edges and $n+1$ vertices, *i.e.* a single tree. This tree is then clearly well labelled.
Suppose now that there exists a cycle in $\mathcal{T}(Q)$ and let $e\ge 0$ be the value of the smallest label of a vertex of this cycle. Either all these labels are equal to $e$, or there is in the cycle an edge $(e,e+1)$ and an edge $(e+1,e)$. In both cases the rules of selection of edges imply that each connected component of ${Q}'$ defined by the cycle contains a vertex $x$ (resp. $y$) with label $e-1$, as shown by Figure \[sans-cycle\]. According to Jordan’s theorem, either the shortest path from $x$ to the root or the shortest path from $y$ to the root has to intersect the cycle, leading to a contradiction with the definition of labels by distances. There are thus no cycles and $\mathcal{T}(Q)$ is a tree.
The inverse $\mathcal{Q}$ of the mapping $\mathcal{T}$ {#sec:decode}
------------------------------------------------------
Let $T$ be a well labelled tree with $n$ edges. Recall that the tree $T$ can be viewed as a planar map that has a unique face $F_0$. A *corner* is a sector between two consecutive edges around a vertex. A vertex of degree $k$ defines $k$ corners and the total number of corners of $T$ is $2n$. The label of a corner is by definition the label of the corresponding vertex.
The image $\mathcal{Q}(T)$ is defined in three steps.
1. A vertex $v_0$ with label $0$ is placed in the face $F_0$ and one edge is added between this vertex and each of the $\ell$ corners with label $1$. The new root is taken to be the edge arriving from $v_0$ at the corner before the root of $T$.
After Step (1) a uniquely defined rooted map $T_0$ with $\ell$ faces has been obtained (see Figure \[fig:cords\], with $\ell=5$). The next steps take place independently in each of those faces and are thus described for a generic[^1] face $F$ of $T_0$. Let $k$ be the degree of $F$ (by construction $k\geq3$). Among the corners of $F$ only one belongs to $v_0$ and has label $0$. Let the corners be numbered from $1$ to $k$ in clockwise order along the border, starting right after $v_0$. Let moreover $e_i$ be the label of corner $i$ (so that $e_1=e_{k-1}=1$ and $e_k=0$). In Figure \[fig:cords\] the corners are explicitly represented with their numbering for one of the faces.
![Step (1), and the cords $(i,s(i))$ in one of the faces.[]{data-label="fig:cords"}](\rep cords.eps){width="11cm"}
2. The function successor $s$ is defined for all corners $1,\ldots,k-1$ by $$s(i)= \inf\{j>i\mid e_j=e_i-1\}.$$ For each corner $i\ge 2$ such that $s(i)\neq i+1$, a cord $(i,s(i))$ is added inside the face, in such a way that the various cords do not intersect (Property \[pro:licite\]).
Once this construction has be carried on in each face, a planar map ${T}'$ is obtained.
3. All edges of with labels of the form $(e,e)$ of ${T}'$ are deleted. The resulting map is a quadrangulation $\mathcal{Q}(T)$ with $n$ faces (Property \[pro:isquad\]).
The following proposition ends the proof of Theorem \[thm:welllab\].
\[pro:inverse\] The mapping $\mathcal Q$ is the inverse bijection of the mapping $\mathcal{T}$.
Let us first prove the two properties that validate the preceding construction.
\[pro:licite\] The cords $(i,s(i))$ do not intersect.
Suppose that two cords $(i,s(i))$ and $(j,s(j))$ cross each other. Upon maybe exchanging $i$ and $j$ one has $i<j<s(i)<s(j)$. The first two inequalities imply, together with the definition of $s$, that $e_j> e_{s(i)}$, while the two last inequalities imply $e_{s(i)}\ge e_j$. This is a contradiction.
\[pro:isquad\] The faces of ${T}'$ are of one of the two types of Figure \[fig:deuxfaces\]: either triangular with labels $e,e+1,e+1$, or quadrangular with labels $e,e+1,e+2,e+1$. The faces of $\mathcal{Q}(T)$ are all quadrangular.
Let $f$ be a face of ${T}'$. The face $f$ is included in a face $F$ of $T_0$ so that its corners inherit the numbering and labelling of those of $F$. Let $j$ be the corner with largest number in $f$ and $i_1<i_2<j$ its two neighbors in $f$ (cf. Figure \[fig:deuxfaces\]). Let us compute the label of the corners $i_1$ and $i_2$: the edge $(i_1,j)$ is a cord by construction so that $j=s(i_1)$ and $e_{i_1}=e_j+1$; moreover, as $i_1<i_2<j$, this imply $e_{i_2}\ge
e_{i_1}$ (or $j$ would not be $s(i_1)$) and finally $e_{i_2}=e_{i_1}=e_j+1$.
By construction and planarity, no cord can arrive at $i_1$ between the unique leaving cord $(i_1,j)$ and the edge $(i_1,i_1+1)$ of $F$. The latter edge thus borders the face $f$. There are then two cases, as illustrated by Figure \[fig:deuxfaces\]:
- if $e_{i_1+1}=e_{i_1}$, then $i_2=i_1+1$, and the face is triangular (left hand figure),
- otherwise $e_{i_1+1}=e_{i_1}+1$ (recall $e_{i_1+1}\geq e_{i_1}$ since $i_1<i_1+1<j=s(i_1)$) and the cord leaving $i_1+1$ goes to $i_2$ (otherwise $s(i_1+1)<i_2$ with $e_{s(i_1+1)}=e_{i_1+1}-1=e_{i_2}$ and a cord $(s(i_1+1),j)$ would exclude $i_2$ from the face $f$): the face is quadrangular (right hand side in Figure \[fig:deuxfaces\]).
Observe finally that the deletion of edges with labels of the form $(e,e)$ will join triangular faces two by two so that $\mathcal{Q}(T)$ has only quadrangular faces.
![Two possible sizes for $f$: triangular or quadrangular.\[fig:deuxfaces\]](\rep triquad.eps){width="12cm"}
Given a well labelled tree $T$, faces of its image $\mathcal{Q}(T)$ are as described by Figure \[fig:deuxfaces\]. The selection rules for $\mathcal{T}$ then shows that each face correctly selects an edge of $T$, so that ${\mathcal T}({\mathcal Q}(T))=T$. Thus ${\mathcal T}$ and ${\mathcal Q}$ are inverse bijections between well labelled trees and a subset $\tilde{\mathcal Q}_n$ of the set of quadrangulations. Equality of cardinalities, as granted for instance by the alternative bijection of [@cori-vauquelin], proves that $\tilde {\mathcal Q}_n$ is the full set of quadrangulations with $n$ faces and concludes the proof. However, we provide below a direct proof of the equality ${\mathcal Q}({\mathcal{T}}(Q))=Q$ for any quadrangulation $Q$, for this provides better understanding of the bijection.
Let $Q$ be a quadrangulation, $Q'$ and $T=\mathcal{T}(Q)$ the map and tree as in Section \[sec:encode\], and $T_0$, $T'$ as in the construction of $\mathcal{Q}(T)$. Consider first the selection rules applied around the root to construct $\mathcal
T(Q)$. Each edge (with labels) 1-2 of $\mathcal T(Q)$ forms a directly oriented corner with an edge 1-0 in its face of creation, while each edge 1-1 forms two such corners (one on each side). Hence, in accordance with Step (1) of the reciprocal construction, an edge 1-0 arrives at each corner with label $1$. Thus the submap $T_0$ of $T'$ is also a submap of $Q'$. Moreover $T_0$ covers all vertices of $T'$ (resp. $Q'$), so that edges of $T'$ (resp. $Q'$) not in $T_0$ are cords of faces of $T_0$. Accordingly, $T'=Q'$ if, inside each face of $T_0$, both $T'$ and $Q'$ have the same cords.
The maps ${{Q}'}$ and ${T}'$ have the same vertices, and, due to Property \[pro:isquad\], the same number of faces of degree 4 (that is, the number of edges $i-(i+1)$ in $T$), and the same number of faces of degree 3 (that is, the number of edges $i-i$ in $T$). By Euler’s formula, they thus have also the same number of edges and finally, they are equal as planar maps if, inside all faces of $T_0$, each cord of ${T}'$ is a cord of ${{Q}'}$.
Let us now work inside a face $F$ of $T_0$ (see Figure \[fig:faceinduction\]). By construction of $T_0$ the face $F$ has only one corner with label $0$ (incident to the root) and two corners with label $1$ (since such a corner is incident to an edge $(0,1)$ after Step (1)). If $F$ has degree 3 (resp. 4), the corners around the face, in clockwise order, are labelled 0-1-1 (resp. 0-1-2-1), and there is no cord, neither in $Q'$ nor in $T'$. Let us thus assume that $F$ has degree $k$ larger than 4, and number the $k$ corners of $F$ in clockwise order starting after the root corner. Let $e_i$ be the label of the $i$th corner, so that $e_1=1$, $e_2=2$, $e_{k-2}=2$, $e_{k-1}=1$, $e_k=0$, and $e_i\geq2$ otherwise. This numbering corresponds to the one used to define cords of $T'$: for each corner $i$ with $e_i=2$ but the last one, $s(i)=k-1$ and the cord $(i,k-1)$ appears in $T'$.
In order to check that these cords appear also in $Q'$ we consider corners with label 2 in increasing order: they are numbered $2=i_1<i_2<\ldots<i_p=k-2$. In $Q'$ let $f_1$ be the face that contains the corners numbered $k$, $1$ and $k-1$ (with labels $0$, $1$ and $1$). The face $f_1$ contains a fourth corner, with label $2$: it can be $i_p$ (if the cord $(1,i_p)$ is in $Q'$) or $i_1$ (if the cord $(i_1,k-1)$ is in $Q'$). In the first case the face $f_1$ in $Q'$ has corners $(1,i_p,k-1,k)$ and there is a contradiction with the fact that the edge $(i_p,k-1)$ has not been selected in $F$ by the construction $\mathcal{T}$. Hence the second case hold: the cord $(i_1,k-1)$ is in $Q'$ and the edge $(1,i_1)$ was selected.
![Faces of $T_1$ inside a face $F$ of $T_0$.\[fig:faceinduction\]](\rep face.eps){width="4.2cm"}
Now assume that the edges $(i_{j'},k-1)$ belong to $Q'$ for $j'<j<k-2$ and check that $(i_j,k-1)$ belongs to $Q'$. Consider in $Q'$ the face $f_j$, included in the face $F$ and bordering the edge $(i_j-1,i_j)$ of the cycle $F$. If $e_{i_j-1}=2$ then $i_j-1=i_{j-1}$ and the face $f_j$ is triangular (since the selection rule of confluent faces was applied by $\mathcal{T}$) and contains the cord $(i_j,k-1)$. Otherwise $e_{i_j-1}=3$ and the face $f_j$ is quadrangular and of simple type (since the selection rule of simple faces was applied by $\mathcal{T}$). Therefore there is an edge from $i_j$ to a corner with label $1$, which can only be the cord $(i_j,k-1)$, for a cord $(i_j,1)$ would cross $(i_1,k-1)$.
All cords with labels 1-2 are thus identical in $T'$ and $Q'$. Let $T_1$ be the union of $T_0$ and these cords. In view of the previous discussion, the faces of $T_1$ are exactly the previous subdivisions into faces $f_j$ of all faces $F$ of $T_0$. Moreover each face $f$ of $T_1$ has only one corner with label 1 and two with label 2, all other labels being at least 3. Shifting down all labels by one inside face $f$, the situation is exactly equivalent to that of the face $F$ above (observe that the rules for the construction $\mathcal{T}$ and $\mathcal{Q}$ remain unaffected by the shift since only vertices with label greater or equal to two are considered). The identity between cords of $T'$ and $Q'$ of successively greater orders can thus be checked inductively.
Finally ${{Q}'}$ contains all the edges of ${T}'$, that is ${{Q}'}={T}'$, and since the deletion of edges with labels of the form $(e,e)$ in ${{Q}'}$ (resp. ${T}'$) produces $Q$ (resp. ${\mathcal Q}({\mathcal{T}}(Q))$), we obtain ${\mathcal Q}({\mathcal{T}}(Q))=Q$.
Well labelled and embedded trees {#sec:conjugacy}
================================
Unconstrained well labelled trees as embedded trees
---------------------------------------------------
Formula (\[equ:welllab\]) for the number of well labelled trees with $n$ edges, $$|\mathcal{W}_n|\;=\;\frac{2}{n+2}\,\frac{3^n}{n+1}{2n\choose n}
\;=\;\frac{2}{n+2}\cdot 3^n\cdot C(2n),$$ is remarkably simple and yet not immediately clear from definition. Indeed, even though $C(2n)$ is known to be the number of plane trees, the positivity of labels makes it difficult to count labellings that make a plane tree well labelled.
It is thus natural to work first without this positivity condition: define a plane tree to be an *unconstrained well labelled tree* if its vertices have integral labels, the labels of two adjacent vertices differ at most by one, and the label of the root vertex is one. Let $\mathcal{U}_n$ denote the set of unconstrained well labelled trees with $n$ edges.
The labelling of a labelled tree can be recovered uniquely from the label of its root and the variations of labels along all edges. We shall denote $\kappa(\epsilon)\in\{-1,0,1\}$ the variation of labels along the edge $\epsilon$ when it is traversed away from the root. Since there is no positivity condition on the labels of unconstrained well labelled trees, all $\kappa(\epsilon)$ can be set independently and the number of labellings of a plane tree that yield an unconstrained well labelled tree is just $3^n$. That is, $$|\mathcal{U}_n|\;=\;\frac{3^n}{n+1}\binom{2n}{n}\;=\;3^n\cdot C(2n).$$
The definition of label distribution extends to unconstrained well labelled trees. For $U\in\mathcal{U}_n$ let $(\lambda_k)_{m<k<M}\equiv(\lambda_k^{[U]})_{k\in\mathbb{Z}}$ be the number of vertices with label $k$ in the tree $U$. The label distribution of $U$ is supported by an interval $[m,M]$ with $m\leq
1\leq M$. The cumulated label distribution is defined with respect to the minimum label $m$ by $\widehat\lambda_k^{[U]}=\sum_{\ell=1}^{k}\lambda_{m+\ell-1}^{[U]}$. These definitions are illustrated by Figure \[fig:unconstrained\].
![An unconstrained well labelled tree with its label distribution and a representation of the embedding on the line (the plane order structure of the tree is lost in the latter representation). \[fig:unconstrained\]](\rep unconstrained.eps){width="12cm"}
Observe moreover that similar unconstrained labellings have been considered by D. Aldous [@AALD] with the following interpretation (we restrict to our special one-dimensional case). The tree is folded on the lattice $\mathbb{Z}$ with the root set at position $1$ and each edge mapped on an elementary vector (here $+1$, $0$, or $-1$). The label of a node then describe its position on the line and, upon counting the number of nodes at position $j$, a mass distribution is obtained. More precisely, with our notations, Aldous’ discrete mass distribution associated to a tree $U\in \mathcal{U}_n$ is just the empirical measure of labels $$\mathcal{J}^{[U]}\;=\;\frac{1}n\sum_{k\in\mathbb{Z}}\lambda^{[U]}_k\delta_k,$$ where $\delta_k$ denote the dirac mass at $k$.
In view of this interpretation and for concision’s sake, let us rename *unconstrained well labelled trees* and call them instead *embedded trees*.
Random trees and random quadrangulations
----------------------------------------
Let $W_n$ and $U_n$ be random variables with uniform distribution on $\mathcal{W}_n$ and $\mathcal{U}_n$. More precisely, $$\Pr(W_n=W)\;=\;\frac1{\frac{2}{n+2}\frac{3^n}{n+1}{2n\choose n}},
\quad\textrm{ and }\quad
\Pr(U_n=U)\;=\;\frac1{\frac{3^n}{n+1}{2n\choose n}},$$ for all $W\in\mathcal{W}_n$ and $U\in\mathcal{U}_n$.
The label distribution of the corresponding random trees are two random variables that we shall denote ${(\lambda^{(n)}_k)_{k\geq1}\equiv(\lambda^{[W_n]}_k)_{k\geq1}}$ for random well labelled trees, and ${(\Lambda^{(n)}_k)_{k\in\mathbb{Z}}\equiv
(\lambda^{[U_n]}_k)_{k\in\mathbb{Z}}}$ for random embedded trees. For random well labelled trees we also use the notation $\mu_n$ for the maximal label, and for random embedded trees the notations $m_n$ and $M_n$ for the minimal and maximal label respectively. Finally cumulated profiles $\widehat\lambda_k^{(n)}=\sum_{\ell=1}^k\lambda_\ell^{[W_n]}$ and $\widehat\Lambda_k^{(n)}=\sum_{\ell=1}^k\lambda_{m_n+\ell-1}^{[U_n]}$ are defined accordingly (the minimum $m_n$ in $\widehat\Lambda_k^{(n)}$ is understood for the same realisation $U_n$).
At this point we are given three random variables: random quadrangulations $L_n$, random well labelled trees $W_n$ and random embedded trees $U_n$. On the one hand, according to Theorem \[thm:welllab\], random quadrangulations “are” random well labelled trees, as illustrated by the next corollary.
\[cor:id\] The label distribution of random well labelled trees has the same distribution as the profile of quadrangulations: $$(\lambda^{(n)}_k)_{k\geq1}\;\;\mathop{=\rule{0mm}{2mm}}^{\textrm{law}}\;\;
(H^{(n)}_k)_{k\geq1}.$$ In particular $r_n=\mu_n$.
On the other hand, random embedded trees seem to be a simple variant of well labelled trees that has the great advantage to be defined in accordance with Aldous’ prescription for discrete embedded trees. This leads us to study more precisely the relation between $W_n$ and $U_n$. By definition, $\mathcal{W}_n\subset\mathcal{U}_n$, and according to Tutte’s formula (\[equ:welllab\]), $$\label{equ:UtoW}
|\mathcal{W}_n|\;=\;\frac{2}{n+2}\cdot|\mathcal{U}_n|.$$ For combinatorists, this relation could be reminiscent of the relation between the number of Dyck walks and the number of bilatere Dyck walks (see [@Stanley Ch. 5]).
Equivalently, from a more probabilistic point of view, the relation reads $$\Pr(U_n\in\mathcal{W}_n)\;=\;\frac{2}{n+2},$$ and random well labelled trees are random embedded trees conditioned to positivity. This is exactly similar to Kemperman’s formula for the probability that a simple symmetric walk on $\mathbb{Z}$ starting from $k>0$ and ending at $0$ after $n$ steps remains positive until the last step (see [@pitman]).
Cyclic shifts and the cycle lemma {#sec:cycle}
---------------------------------
The idea to consider cyclic shifts originates in Dvoretsky and Motzkin’s work and was used by Raney to prove Lagrange inversion formula and by Takács to prove and extend Kemperman’s formula for random walks (we refer to [@Stanley Ch. 5] and [@pitman] for these historical references and many more). We shall prove a consequence of this idea to the study of “height distribution” of simple walks, that will be fundamental in the next section.
Let $n$ and $k$ be nonnegative integers and let $\mathcal{B}_{n,k}$ denote the set of walks of length $2n+k$ with $n$ increments $+1$ and $n+k$ increments $-1$, that end with a negative increment $-1$. A walk $w\in\mathcal{B}_{n,k}$ is described either by its sequence of increments $w=(w_1,\ldots,w_{2n+k})$, $w_i\in\{+1,-1\}$, or by the partial sums $w(p)=\sum_{i=1}^pw_i$, $p=0,\ldots,2n+k$. By construction, $w(0)=0$, $w_{2n+k}=-1$, $w(2n+k)=-k$, and $w(2n+k-1)=-k+1$. Finally for $k\geq1$, consider the subset $\mathcal{D}_{n,k}$ of $\mathcal{B}_{n,k}$ of “positive” walks defined by the condition: $w(p)>-k$ for all $0\leq p<2n+k$.
Two walks $w$ and $w'$ of $\mathcal{B}_{n,k}$ belong the same *conjugacy class* if they differ by a cyclic shift, that is, if there exists $s$ such that $$w=(w_1,\ldots,w_{2n+k})\quad\textrm{ and }\quad
w'=(w_s,\ldots,w_{s+2n+k}),$$ where indices are considered modulo $2n+k$.
Define a (left-to-right) record to be a step $p\geq1$ at which a minimum is reached for the first time: $w(q)>w(p)$ for all $q<p$. Since $w(2n+k)=-k$ there are at least $k$ records. Let us denote $p_1<\cdots<p_k$ the $k$ lowest records, so that in particular $w(p_k)$ is the minimum value reached by the walk and $w(p_i)=k-i+w(p_k)$. The steps $p_i$, $i=1,\ldots,k$ are called the *low records* of $w$.
![Two conjugated walks from $\mathcal{D}_{n,2}$ and $\mathcal{B}_{n,2}$ with low records and floor level of Dyck height indicated. \[fig:cyclic\]](\rep cyclic.eps){width="7cm"}
The following immediate properties are illustrated by Figure \[fig:cyclic\].
\[pro:Dyck\] A walk of $\mathcal{B}_{n,k}\!$ belongs to $\mathcal{D}_{n,k}\!$ if and only if its lowest record $p_k\!=\!2n\!+\!k$.
\[pro:records\] Cyclic shifts transport low records: Let $w=(w_1,\ldots,w_{2n+k})$ and $w'=(w_s,\ldots,w_{s+2n+k})$ and assume $\{p_1,\dots,p_k\}$ are the low records of $w$. Then the low records of $w'$ are $\{p_1+s,\ldots,p_k+s\}$ (modulo $2n+k$).
The classical cycle lemma follows from Properties \[pro:Dyck\] and \[pro:records\].
\[lem:cycle\] Let ${C}$ be a conjugacy class of $\mathcal{B}_{n,k}$. Then $$(n+k)\cdot|{C}\cap\mathcal{D}_{n,k}|\;=\;k\cdot|{C}|.$$
Let us apply a double counting argument:
- The left hand side counts walks in ${C}$ with a (low) record at the last position ($w\in\mathcal{D}_{n,k}$) and a down step marked ($n+k$ choices).
- The right hand side counts walks in ${C}$ with a down step at the last position ($w\in\mathcal{B}_{n,k}$) and a low record marked ($k$ choices).
Now a bijection is obtained between these two sets upon sending the marked step to the last position by a cyclic shift and marking the former last step.
Given a walk $w$ with lowest record $p_k$, the *height-to-min* of a step $p$ is $\tilde w(p)=w(p)-w(p_k)$, which is nonnegative by definition. In order to study $\tilde w$, it will be convenient to consider the height of the walk relatively to the $k$ low records: given a walk $w$ with low records $p_1<\cdots<p_k$, let us define the *Dyck height* at step $p$ by $$\bar w(p)=
\left\{
\begin{array}{ll}
w(p)-w(p_1)+1, & \textrm{if }0\leq p<p_1,\\
w(p)-w(p_i), & \textrm{if }p_i\leq p<p_{i+1}, \textrm{ with }1\leq i\leq k-1\\
w(p)-w(p_k), & \textrm{if }p_k\leq p\leq2n+k.
\end{array}
\right.$$ The Dyck height can be understood as the height inside each of the $k$ Dyck factors separated by low records. Let $\hat\ell_i(w)$ (resp. $\hat h_i(w)$) denote the number of down steps of $w$ ending at Dyck height (resp. height-to-min) at most $i$, $$\smash{\hat\ell_i}(w)\;=\;|\{p\mid \bar w(p)\leq i, \; w_p=-1\}|,\;\;
\textrm{ and }\;\;
\smash{\hat h_i}(w)\;=\;|\{p\mid \tilde w(p)\leq i, \; w_p=-1\}|.$$ Then by construction, for all $w$ and $p$, $\bar w(p)\leq\tilde
w(p)\leq\bar w(p)+k$ and, for all $i$, $\hat
h_i(w)\leq\hat\ell_i(w)\leq\hat h_{i+k}(w)$. The following lemma immediately follows from Property \[pro:records\].
\[lem:height\] The Dyck height commutes with cyclic shift. In particular the Dyck height distribution $\hat\ell_i$ is invariant under cyclic shift: $$(\hat\ell_i(w))_{i\geq0}\;=\;(\hat\ell_i(w'))_{i\geq0},\qquad
\textrm{for all $w$ and $w'$ in the same conjugacy class.}$$ The height-to-min distribution thus satisfies the following weaker invariance: $$\hat h_i(w)\;\leq\; \hat h_{i+k}(w'), \qquad \textrm{for all $i\geq 0$
and $w$, $w'$ in the same conjugacy class.}$$
From the probabilistic point of view this result can be understood as a simplified discrete version of Vervaat’s relation between the Brownian excursion and the Brownian bridge and their local times relatively to the minimum.
How to lift the positivity condition for labelled trees
-------------------------------------------------------
In view of Relation (\[equ:UtoW\]) one can expect to apply ideas of the previous section to related well labelled trees to embedded trees. As a matter of fact we shall prove the following theorem.
\[thm:combi\] There exists a partition of $\mathcal{U}_n=\bigcup_{C\in\mathcal{C}_n}C$ into disjoint *conjugacy classes* each of size at most $n+2$ and such that in each class $C\in\mathcal{C}_n$
- well labelled trees are fairly represented: $$2\cdot|{C}|\;=\;(n+2)\cdot|{C}\cap\mathcal{W}_n|,$$
- and for any $W\in\mathcal{W}_n\cap C$, $U\in C$ and $k\geq1$, $$\widehat\Lambda_{k-2}(U)\;\leq\;
\widehat\lambda_k(W)\;\leq\;
\widehat\Lambda_{k+2}(U).$$
\[cor:count\] The number of well labelled trees with $n$ edges, (which is also the number of quadrangulations with $n$ faces), is $$|\mathcal{W}_n|\;=\;\frac{2}{n+2}\cdot|\mathcal{U}_n|
\;=\;\frac{2}{n+2}\cdot\frac{3^n}{n+1}\binom{2n}{n}.$$
The proof of Theorem \[thm:combi\] is presented in the next section. It relies on an encoding of plane trees in terms of another family of trees, called *blossom trees*, and on the *conjugation of trees* principle which is an analogue of the cycle lemma for blossom trees. This principle was introduced in [@these-schaeffer] in order to give a direct combinatorial proof of Corollary \[cor:count\] based on the cycle lemma. However that proof did not rely on well labelled trees and does not provide the link to the profile.
Theorem \[thm:combi\] admits the following probabilistic restatement.
\[thm:coupling\] There is a coupling $(W_n,U_n)$ (*i.e.* a distribution on $\mathcal{W}_n\times\mathcal{U}_n$ such that the marginals are $W_n$ and $U_n$ as previously defined) such that the induced joint distribution $(\lambda^{(n)},\Lambda^{(n)})$ satisfies for all $k$ $$\widehat\Lambda_{k-2}^{(n)}\;\leq\;
\widehat\lambda_k^{(n)}\;\leq\;
\widehat\Lambda_{k+2}^{(n)},$$ and in particular $$|\mu_n-(M_n-m_n)|\leq3.$$
The distribution on $\mathcal{W}_n\times\mathcal{U}_n$ is immediately obtained from the partition $\mathcal{U}_n=\bigcup_{C\in\mathcal{C}_n}C$ as follows: for any $(W,U)$ in $\mathcal{W}_n\times\mathcal{U}_n$, let $$\Pr((W_n,U_n)=(W,U))=\left\{
\begin{array}{cl}
\frac1{2|\mathcal{U}_n|} &
\textrm{if $U$, $W$ are both in $C$ with $|C\cap\mathcal{W}_n|=2$,}\\
\frac1{|\mathcal{U}_n|} &
\textrm{if $U$, $W$ are both in $C$ with $|C\cap\mathcal{W}_n|=1$,}\\
0 & \textrm{if $U\in C_1$ and $W\in C_2$ with $C_1\neq C_2$.}
\end{array}
\right.$$ In view of the first part of Theorem \[thm:combi\], the marginals are uniformly distributed. The second part of Theorem \[thm:combi\] gives the two inequalities.
Blossom trees and the conjugation of trees
------------------------------------------
Theorem \[thm:combi\] is clearly analogous to Lemma \[lem:cycle\] and \[lem:height\]. However we were not able to define directly conjugacy classes on embedded trees. Instead we first construct an encoding of embedded trees in terms of another family, blossom trees, and then define conjugacy classes of blossom trees and prove Theorem \[thm:combi\].
Following [@these-schaeffer], let a *blossom trees* be a plane tree with the following properties:
- Vertices of degree one are of two types: arrows and flags. The root of the a blossom tree is a flag, which is said to be *special*, as opposed to the other *normal* flags.
- All inner nodes have degree four and each of them is adjacent to exactly one arrow.
Let $\mathcal{B}_n$ be the set of blossom trees with $n$ inner nodes. By construction these trees have $n$ vertices of degree four and thus $2n+2$ of degree one, that is $n$ arrows and $n+2$ flags. The labelling of a blossom tree is given by the following *labelling process*:
- Start with *current label* $2$ just after the root.
- Turn around the border of the tree in counterclockwise direction.
- Each time an arrow is reached, the *current label* is increased by $1$.
- Each time a flag is reached, the *current label* is decreased by one and then written on the flag.
- Stop when the root flag is reached again (no label is written there).
This definition is illustrated by Figure \[fig:exofbin\].
![From embedded trees to blossom trees: an example.\[fig:example\]](\rep exofbin1.eps){width="9cm"}
![From embedded trees to blossom trees: an example.\[fig:example\]](\rep tobinary1.eps){width="11cm"}
![From embedded trees to blossom trees: an example.\[fig:example\]](\rep extobin1.eps){width="11cm"}
Embedded trees with $n$ edges are in one-to-one correspondence with blossom trees with $n$ inner nodes with the same label distribution.
In order to prove this lemma we work on a set of decorated blossom trees: in these trees, the root flag is special and any flag with label $e$ (as given by the labelling process) can either be empty or be decorated by an embedded tree with root label $e$ (for $e\neq1$ the immediate generalisation of embedded trees is meant). The *combined label distribution* of a decorated blossom tree counts labels of decorations (embedded trees on flags) and of empty flags. Examples of decorated blossom trees are given in Figure \[fig:example\] (in these figures, labels along edges indicate values taken by the current label during the labelling process).
The first step of the encoding of an embedded tree consists in writing it on the normal flag of the unique blossom tree with two flags and no inner node (Figure \[fig:example\] top-left example). Then the encoding is performed by recursively transforming the decorated blossom tree according to the local rules of Figure \[fig:toblossom\]. Each time the leftmost rule is applied to a flag decorated by an embedded tree reduced to a vertex with label $e$, this vertex is suppressed the flag becomes empty, with label $e$ (by construction, $e$ agrees with the labelling process; the combined label distribution is left unchanged). When one of the other three rules is applied, a new inner node is created while an edge of embedded tree is suppressed. The relation between the position of the created arrow and the root labels of the embedded trees grants that the compatibility with the labelling process is preserved (observe that in the middle rule subtrees have been switched for this purpose).
As long as there are decorated flags a rule can be applied. Once there is no more decorated flag, a blossom tree is obtained. Rules are local so that rules applied in distinct subtrees commute. As a consequence the final blossom tree does not depend on the order in which rules are applied. Each rule is uniquely reversible so that the encoding is bijective.
![The labelling process for two conjugated blossom trees. Three steps $x$, $y$ and $z$ have been distinguished to illustrate the correspondence: up step = arrow, down step = flag. \[fig:process\]](\rep border.eps){width="12cm"}
The partition $\mathcal{U}_n\equiv\mathcal{B}_n=
\bigcup_{C\in\mathcal{C}_n}C$ is the partition of blossom trees in conjugacy classes: two blossom trees $A$ and $B$ are in the same *conjugacy class* $C$ if $B$ is obtained from $A$ by first replacing the root flag of $A$ by a normal flag and then choosing a new special flag. This operation is called a *cyclic shift* of the tree. In other terms each conjugacy class $C$ is the set of blossom trees that can be obtained from a specific *unrooted* blossom tree (the flags of which are all normal) by selecting a special flag (root flag) in all the possible ways.
Given a blossom tree $B$ with $n$ arrows and $n+2$ flags, the evolution of the current label, while performing its labelling process, is a walk $w_B$ with $n$ increments $+1$ and $n+2$ negative increments $-1$, that starts from $2$, and whose last step, when the process reaches again the root flag, is a negative increment. Upon decreasing all labels by two, the walk $w_B$ is thus a walk of $\mathcal{B}_{n,2}$ as defined in Section \[sec:cycle\].
Moreover each cyclic shift of the tree $B$ is equivalent to the corresponding cyclic shift of the walk $w_B$. Finally a blossom tree encodes a well labelled tree if and only if all its labels are positive, that is, if and only if the walk $w_B$ belongs to $\mathcal{D}_{n,2}$ (upon decreasing all label by two). The first statement of Theorem \[thm:combi\] is thus exactly the cycle lemma (Lemma \[lem:cycle\]).
Finally, let us consider label distributions. Given $W$ a well labelled tree and $U$ an embedded tree in the same conjugacy class of trees, the corresponding walks $w_W$ and $w_U$ belong to the same conjugacy class of walks. But the cumulated label distributions satisfies $\widehat\lambda^{[W]}_k=\hat h_k(w_W)$ and $\widehat\Lambda^{[U]}_k=\hat h_k(w_U)$ so that Lemma \[lem:height\] gives the second statement of Theorem \[thm:combi\].
Quadrangulations, Brownian snake and ISE {#sec:proba}
========================================
Encoding embedded trees by pairs of contour walks
-------------------------------------------------
Let $\bar{\mathcal{U}}_n$ be the set of embedded trees with root label zero instead of one. These trees, that are simply obtained from trees of $\mathcal{U}_n$ by [shifting all labels down by one]{}, will be more convenient for our purpose.
Let $U$ be an embedded tree of $\bar{\mathcal{U}}_n$ and consider the following traversal of $U$, where traversing an edge takes unit time:
- At time $t=0$, the traversal arrives at the root.
- If the traversal reaches at time $t$ a vertex $v_t$ having $k$ sons for the $\ell$th time with $\ell\leq k$, its next step is toward the $\ell$th son of $v_t$.
- If the traversal reaches at time $t$ a vertex $v_t$ having $k$ sons for the $(k+1)$th time, its next step is back toward the father of $v_t$.
This traversal is called the *contour traversal* because, as exemplified by Figure \[fig:traversal\], it turns around the tree. In particular every edge is traversed twice (first away from and then toward the root) and the complete traversal takes $2n$ steps. The *contour pair* of $U$ is then defined by the height (*i.e.* distance to the root in the abstract tree), $E^{[U]}(t)$ and label $V^{[U]}(t)$ of vertex $v_t$ traversed at time $t=0,\ldots,2n$. (The path $E$ is often called the *Dyck path* associated to the tree $U$ [@Stanley Ch. 5], or the *contour process* in [@LEG Ch. I.3].)
The following proposition is immediate from the definition of contour pairs.
\[pro:contours\] The contour pair construction is a one-to-one correspondence between $\bar{\mathcal{U}}_n$ (or $\mathcal{U}_n$) and the set $\mathcal{EV}_{2n}$ of pairs of walks of length $2n$ such that:
- the walk $E$ is an excursion with increment $\pm1$ or Dyck path, that is $E(0)=E(2n)=0$, $|E(t)-E(t+1)|=1$ and $E(t)\geq0$ for all $t=0,\ldots,2n-1$;
- the walk $V$ is a bridge with increment $\{-1,0,1\}$ or bilatere Motzkin path, that is $V(0)=V(2n)=0$ and $(V(t)-V(t+1))\in\{-1,0,1\}$ for all $t$;
- and the consistency condition hold:\
$\left(\; E(t)=E(t') \textrm{ and }
E(s)\geq E(t) \textrm{ for all } t<s<t'\;\right)
\;\Rightarrow\;V(t)=V(t').
$
![Contour traversal and contour pair $(E,V)$ of a tree. \[fig:traversal\]](\rep contourwalks.eps){width="12cm"}
The excursion $E$ alone determines a unique unlabelled rooted plane tree, while the walk $V$ describes one of the $3^n$ labelling of the tree encoded by $E$. Recall that for an embedded tree $U$, $\kappa(\epsilon)\in\{-1,0,1\}$ denotes the variation along edge $\epsilon$ when traversed away from the root. In particular if $\epsilon$ is traversed for the first time between time $t$ and $t+1$ and for again between $t'$ and $t'+1$, then $$\kappa(\epsilon)=V(t+1)-V(t)=V(t')-V(t'+1).$$ This local condition is equivalent to the consistency condition of Proposition \[pro:contours\].
Random trees as random contour pairs {#sec:contourpair}
------------------------------------
Endow now $\bar{\mathcal{U}}_n$ with the uniform distribution and let $(E^{(n)},V^{(n)})\equiv(E^{[U_n]},V^{[U_n]})$ denote the contour pair of the random tree $U_n$. According to Proposition \[pro:contours\], the random contour pair $(E^{(n)},V^{(n)})$ is uniformly distributed on $\mathcal{EV}_{2n}$ and $E_n$ is uniformly distributed on $\mathcal{E}_{2n}$, the set of Dyck walks of length $2n$. More precisely, for all $(E,V)\in\mathcal{EV}_{2n}$, $$\Pr((E^{(n)},V^{(n)})=(E,V))\;=\;\frac{1}{\frac{3^n}{n+1}{2n\choose n}},
\qquad
\Pr(E^{(n)}=E)\;=\;\frac{1}{\frac{1}{n+1}{2n\choose n}}.$$
In order to state convergence results, let us now defined scaled version of these random walks: given a random tree $U_n$ and its contour pair $(E^{(n)},V^{(n)})$, let $$\begin{aligned}
e^{(n)}&=&\left(\frac{E^{(n)}(\lfloor2ns\rfloor)}{\sqrt{2n}}
\right)_{0\leq s\leq1}
\quad\textrm{and}\qquad
\hat W^{(n)}\;\;=\;\;
\left(\frac{V^{(n)}(\lfloor2ns\rfloor)}{(8n/9)^{1/4}}
\right)_{0\leq s\leq1}.\end{aligned}$$ The random variables $e^{(n)}$ and $\hat W^{(n)}$ take their values in the Skorohod space $D([0,1],\mathbb{R})$ of càdlàg real functions (right continuous with left limits).
As was proved by Kaigh [@KAIGH], the scaled version $e^{(n)}$ of the contour process converges weakly to the normalised Brownian excursion $e$. Our aim is to state an analogous result for the random variable $$\begin{aligned}
X^{(n)}&\equiv&{\left(e^{(n)},\hat W^{(n)}\right)},\end{aligned}$$ that takes its value in the Skorohod space $D([0,1],\mathbb{R}^2)$.
A Brownian snake
----------------
Let $e$ be the normalised Brownian excursion and $$W={\left(W_s(t)\right)}_{0\le s\le 1,\ 0\le t\le e(s)}$$ be the Brownian snake with lifetime $e$, as studied previously in [@AALD; @BCHS; @DZ; @DS; @LEG; @SERLET].
![Spacial extension of the snake at time $s_1$. \[fig:snake\]](\rep snake-ex.eps){width="10cm"}
More precisely, the process $W$ can be defined as follows:
- for all $0\leq s\leq 1$, $t\rightarrow W_s(t)$ is a standard Brownian motion defined for $0\le t\le e(s)$ (see Figure \[fig:snake\]);
- the application $s\rightarrow W_s(.)$ is a path-valued Markov process with transition function satisfying: for $s_1<s_2$, and for $m=\inf_{s_1\le u\le s_2} \, e(u)$, conditionally given $W_{s_1}(.)$ (see Figure \[fig:snake-consist\]),
- on the one hand we have that $${\left(W_{s_1}(t)\right)}_{0\le t\le m}=
{\left(W_{s_2}(t)\right)}_{0\le t\le m},$$
- and on the other hand ${\left(W_{s_2}(m+t)\right)}_{0\le t\le
e(s_2)-m}$ is a standard Brownian motion starting from $W_{s_2}(m)$, independent of $W_{s_1}(.)$.
![Consistency of the snake between times $s_1$ and $s_2$. \[fig:snake-consist\]](\rep snake-consist.eps){width="10cm"}
The Brownian snake can be viewed as a branching Brownian motion, or as an embedded continuum random tree (see [@AALD]). More precisely the excursion $e$ can be thought of as the contour walk obtained by contour traversal of a continuum random tree, while the snake $W_s(\cdot)$ at times $s$ describes the embedding of the branch to the root at time $s$.
Instead of considering the full Brownian snake $W_s(t)$ we shall concentrate, as we did in the discrete case, on its description by a contour pair (or “head of the snake” description) $X=(X_s)_{0\leq
s\leq1}$, defined by (see also Figure \[fig:contour-snake\]) $$\hat W_s=W_s(e(s)), \quad X_s={\left(e(s),\hat W_s\right)}, \qquad
\textrm{for $0\leq s\leq1$.}$$ In complete analogy with the discrete case, the full Brownian snake can be reconstructed from its contour pair description since $W_s(t)=\hat W_{\sigma(s,t)}$ where $\sigma(s,t)=\sup\{s'\leq s\mid
e(s')=t\}$. However we need only and shall content with results in terms of $X$ (see [@MM] for a complete discussion of the relation between the full snake and its contour description).
Integrated SuperBrownian Excursion
----------------------------------
Let $\mathcal{J}_n$ denote the empirical measure of labels of a random embedded tree: $$\mathcal{J}_n=\frac1n\sum_k\Lambda_k^{(n)}\ \delta_k.$$ Following Aldous [@AALD], for any simple family of trees like our embedded trees, $\mathcal{J}_n$ is expected to converge upon scaling to a random mass distribution $\mathcal{J}$ supported by a random interval $0\in[L,R]\subset\mathbb{R}$. This random measure $\mathcal
J$ is called Integrated SuperBrownian Excursion (ISE) by Aldous, in view of its relation to $W$ through $$\label{bs2ise}
\int g\ d\mathcal{J}=\int_0^1\,g{\left(\hat W_s\right)}\, ds,$$ for any measurable test function $g$, see [@LEG Ch. IV.6]. In [@BCHS] the convergence of $\mathcal{J}_n$ to $\mathcal{J}$ is proved for random embedded Cayley trees. Although these trees are not exactly our random embedded *plane* trees, the proof could easily be adapted.
According to Corollary \[cor:id\] and Theorem \[thm:coupling\], the radius $r_n$ is given by the width of the support of $\mathcal{J}_n$. However the weak convergence of $\mathcal{J}_n$ to $\mathcal{J}$, as obtained in [@BCHS] is not sufficient for our purpose since $r=R-L$, the width of the support of $\mathcal J$, is not a continuous functional of the measure $\mathcal J$.
Convergence of snakes
---------------------
Instead of weak convergence of $\mathcal J_n$ to $\mathcal J$, we shall thus prove in Section \[sec:conv\] the following stronger result.
![The contour description $(e,\hat W_s)$: the excursion $e$ encodes the extension of the snake, the second walk describes the horizontal position of its head. \[fig:contour-snake\]](\rep snake-head.eps){width="10cm"}
\[thm:converge\] The scaled contour pair $X^{(n)}$ converges weakly to $X$ in $D([0,1],\mathbb{R}^2)$.
This theorem establishes weak convergence of the scaled contour (or head of the snake) description of embedded trees to the head of the snake description of the Brownian snake with lifetime $e$. We moreover obtain a deviation bound for the maximal extension of the snake $\hat W^{(n)}_s$.
\[pro:devbound\] There exists $y_0>0$ such that for all $y>y_0$ and $n$, $$\mathbb{P}\left(\sup_{0\leq s\leq1}\hat
W^{(n)}_s\,>\,(8/9)^{1/4}y\right)\;\leq e^{-y}.$$
Theorem \[thm:converge\] was independently obtained by Marckert and Mokkadem [@MM]. They extend the convergence result to the explicit full description $(W_s(t))_{s,t}$ but their alternative proof does not provide the exponential bound of Proposition \[pro:devbound\].
The radius of a random quadrangulation and the width of ISE
-----------------------------------------------------------
According to Corollary \[cor:id\] and to Theorem \[thm:coupling\], the radius $r_n$ of the quadrangulation corresponding to $U_n$ satisfies $$\left\vert
{\left(8/9\right)}^{1/4}{\left(\sup_{0\le s\le 1} \hat W^{(n)}_s
-\inf_{0\le s\le 1} \hat W^{(n)}_s\right)}-\,n^{-1/4}\,r_n\right\vert\le
3n^{-1/4}.$$ Theorem \[thm:converge\] and Proposition \[pro:devbound\] thus prove the conjecture $\smash{\mathbb{E}(r_n)=\Theta(n^{1/4})}$ and lead to a much more precise characterization:
\[radcv\] The random variable $n^{-1/4}\ r_n$ converges weakly to $(8/9)^{1/4}\ r$, in which $$r=\sup_{0\le s\le 1}\hat W_s-\inf_{0\le s\le 1}\hat W_s.$$ Furthermore, convergence of all moments holds true.
In view of Relation (\[bs2ise\]) the random variable $r$ is also the width of ISE process $\mathcal J$.
The profile and a CRM
---------------------
Actually, Theorems \[thm:welllab\] and \[thm:coupling\] suggest that not only the scaled radius but the full scaled profile converges (at least in distribution) to the ISE mass distribution. More precisely, define the distribution function $F(x)$ of the translated ISE by $$W_{\min}=\inf_{0\le s\le 1}\hat W_s,
\hspace{0,5cm}F(x)=\mathcal J{\left((-\infty,W_{\min}+x]\right)}
=\mathcal{J}{\left([W_{\min},W_{\min}+x)]\right)},$$ and the scaled distribution function of the profile of random quadrangulations by $$F_n(x)
=\frac 1{n+1}\ \widehat \lambda^{(n)}_{\lfloor(8n/9)^{1/4}x\rfloor}
=\frac 1{n+1}\ \widehat H^{(n)}_{\lfloor(8n/9)^{1/4}x\rfloor}.$$ where $\widehat\lambda^{(n)}_k$ is the cumulated distribution of labels of a random well labelled tree (as defined in Section \[sec:welllab\]) and $\widehat H^{(n)}_k$ is the cumulated profile of a random quadrangulation (as defined in Section \[sec:def\]).
Then we prove the following corollary of Theorems \[thm:welllab\], \[thm:coupling\], \[thm:converge\] and Corollary \[radcv\].
\[profile\] The scaled profile $F_n$ converges weakly to $F$ in $D([0,+\infty),\mathbb{R})$.
A natural conjecture is that there is a continuum analogue to Theorem \[thm:welllab\] that allows to define from ISE a Continuum Random Map (CRM), such that the properties of scaled distances in random quadrangulations (distances between arbitrary pairs of points, not only with respect to a basepoint) would be described by the properties of distance in the CRM. In view of the interpretation of random quadrangulations as 2d Euclidean pure quantum geometries, this CRM might be considered as a natural candidate model of continuum 2d pure quantum geometry. We plan to discuss this connection further in a subsequent paper.
Random embedded trees and the Brownian snake {#sec:conv}
============================================
In this section we prove Theorem \[thm:converge\]. Finite dimensional density functions are first calculated (Section \[sec:fidis\]). Section \[sec:bound\] then provides the deviation bound for the maximum of the label walk. Finally tightness is proved using the previous bound (Section \[sec:tight\]). The theorem then follows from standard results on weak convergence in the space $D([0,1],\mathbb{R}^2)$ [@BILL].
Finite dimensional density functions {#sec:fidis}
------------------------------------
From now on in this section, $p$ and ${\tau}=(\tau_1,\dots,\tau_p)$ are fixed with $0<\tau_1<\cdots<\tau_p<1$, and we prove the following finite dimensional density convergence result.
\[pro:fidis\] The sequence of random variables $$X^{(n)}(\tau)=
\left(\frac{E^{(n)}(\lfloor2\tau_in\rfloor)}{(2n)^{1/2}},\;
\frac{V^{(n)}(\lfloor2\tau_in\rfloor)}{(8n/9)^{1/4}}
\right)_{1\le i\le p}=\left(e^{(n)}(\tau_i),\;
\hat W^{(n)}_{\tau_i}\right)_{1\le i\le p}$$ weakly converges to $X(\tau)=\left(e(\tau_i),\;
\hat W_{\tau_i}\right)_{1\le i\le p}$, that is, $$\lim_n\mathbb{E}\left[\Phi\left(X^{(n)}(\tau)\right)\right]=
\mathbb{E}\left[\Phi\left(X(\tau)\right)\right]$$ or any bounded continuous function $\Phi$ on $\mathbb{R}^{2p}$.
For this aim it will be convenient to prove first the weak convergence of $$Y^{(n)}(\tau)=\left(\left(e^{(n)}(\tau_i)\right)_{1\le i\le p},
\Big(
\inf_{[\tau_i,\tau_{i+1}]}e^{(n)}
\Big)_{1\le i\le p-1}\right),$$ then that of $$Z^{(n)}(\tau)=\left(Y^{(n)}(\tau),\;\left(
\hat W^{(n)}_{\tau_i}\right)_{1\le i\le p},
\left(\mu^{(n)}_{i}\right)_{1\le i\le p-1}\right),$$ in which $\mu^{(n)}_i$ is defined by $$\mu^{(n)}_i\;=\;\hat W^{(n)}_{\tau'_i} \qquad \textrm{ for any }\quad
\tau'_i\in\mathop{\mathrm{arginf}}_{[\tau_i,\tau_{i+1}]}e^{(n)}.$$ (The consistency condition of Proposition \[pro:contours\] grants that $\mu^{(n)}_i$ is indeed independent of the exact choice of $\tau'_i$ in $\mathop{\mathrm{arginf}}_{[\tau_i,\tau_{i+1}]}e^{(n)}$.)
The weak convergence of $Y^{(n)}(\tau)$ follows from [@KAIGH], as a special case, but does not fill our needs. In the next section we prove a local limit theorem for finite dimensional distributions of $e^{(n)}$ which is not a consequence of [@KAIGH]. Once the local limit theorem for $Y^{(n)}(\tau)$ is proved, we recall its interpretation in terms of trees, using the key notion of *shape*. This leads us to split $\mathbb{R}^{4p-2}$ in $(p-1)!$ regions $(R_\sigma)_{\sigma\in\mathbb{S}_{p-1}}$ and to prove weak convergence of $Z^{(n)}(\tau)$ separately on each region. Finally we identify the limit and the weak convergence of $X^{(n)}(\tau)$ follows from that of $Z^{(n)}(\tau)$.
### A local limit theorem for $Y^{(n)}(\tau)$.
The characteristics of the walk $e^{(n)}$ we are interested in are contained in the sequence of successive heights at (resp. minima between) the $\tau_i$. Let $(x,m)={\left((x_i)_{1\le i\le p},(m_i)_{1\le i\le p-1}\right)}$ be a typical value of $Y^{(n)}(\tau)$, that is, $x_i\sqrt{2n}$ and $m_i\sqrt{2n}$ are integers, and they satisfy $$\inf(x_i,x_{i+1}) \geq m_i, \hspace{1cm}1\le i\le p-1.$$ For the first result, we need a more handy parametrisation, by the successive up and down relative variations: let $\gamma_0=x_1$, $\beta_p=x_p$, $$\label{rel:bgxm}
\gamma_i=x_{i+1}-m_{i},\qquad \beta_i=x_i-m_i, \qquad
\textrm{for $i=1,\ldots,p-1$},$$ and by convention $\beta_0=\gamma_p=0$.
\[pro:fdd-shape\] Let $K$ be a compact subset of $$\left\{(x,m)\in\mathbb R^p\times \mathbb R^{p-1}
\ \left\vert\ 0<m_i<\inf(x_i,x_{i+1}),1\le i\le p-1\right.\right\},$$ that is, of the domain of definition of coherent values of the $x_i$ and $m_i$. Then, uniformly for $(x,m)\in K\cap \big((2n)^{-1/2}\mathbb{Z}\big)^{2p-1}$, $$\begin{aligned}
\mathbb{P}\left(Y^{(n)}(\tau)=\left(
x,m\right)\right)
\,=\,(2n)^{-2\delta}\cdot \zeta(x,m)\cdot
{\left(1+\mathcal{O}_K(n^{-1/2})\right)},\end{aligned}$$ where $\delta=(2p-1)/4$, and $\zeta(x,m)$ reads, in terms of the $\beta_i$ and $\gamma_i$ as given by relations (\[rel:bgxm\]): $$\begin{aligned}
\zeta(x,m)&=&2^{2p}\,(2\pi)^{-p/2}
\prod_{i=0}^p\frac{(\beta_i+\gamma_i)
e^{-\frac{(\beta_i+\gamma_i)^2}{2(\tau_{i+1}-\tau_{i})}}}
{(\tau_{i+1}-\tau_{i})^{3/2}}.\end{aligned}$$
The notation $\mathcal{O}_K$ is used to stress the fact that the error term is uniform for fixed $K$.
Let, for all $n$ and $a$ non negative integers, $$\begin{aligned}
\nonumber C(n;a)&=&\frac{a+1}{n+1}{n+1\choose (n-a)/2}\\
\label{rel:asympt}&=&\frac{2^{n+1}}{\sqrt{2\pi} n}\
\frac{(a+1)}{\sqrt{n}}\,e^{-\frac{a^2}{2n}}
\left(1+O({\textstyle\frac{a}n})
\right),\end{aligned}$$ where the error term is uniform for $a=O(n^{1/2})$.
The Catalan numbers $C(2n;0)$ are well known to give the cardinality of $\mathcal{E}_{2n}$. More generally, the reflection principle proves that $C(n;a)$ is the number of meanders with increments $\pm1$ (aka left factors of Dyck walks), that have length $n$ and end at height $a$. Given non negative numbers $b$ and $c$, $C(n; b+c)$ is also the number of walks with increments $\pm1$ that have length $n$, minimum $-b$ and final height $c-b$, as follows from a decomposition at first and last passage at the minimum.
We thus have $$\mathbb{P}\left(Y^{(n)}(\tau)=(x,m)\right)\;=\;
\frac{\prod_{i=0}^pC\left(\lfloor2n\tau_{i+1}\rfloor-
\lfloor2n\tau_{i}\rfloor\,;\;(2n)^{1/2}
\beta_i+(2n)^{1/2}\gamma_i\right)}{C(2n;0)}.$$ Combined with (\[rel:asympt\]), it yields Proposition \[pro:fdd-shape\].
The expression $$\label{def:zeta}
\zeta(x,m)\;=\;2^{2p}(2\pi)^{-p/2}\
\prod_{i=0}^p\ \ \frac{(\beta_i+\gamma_i)
e^{-\frac{(\beta_i+\gamma_i)^2}
{2(\tau_{i+1}-\tau_i)}}}{(\tau_{i+1}-\tau_{i})^{3/2}}\
\mathbf{1}_{\beta_i\ge0}
\mathbf{1}_{\gamma_i\ge0}
\mathbf{1}_{m_i\ge0}$$ where the $\beta_i$ and $\gamma_i$ are given by relations (\[rel:bgxm\]), is the expected limit density of probability for the $x_i$ and $m_i$: in particular it is coherent with the density of evaluations of the normalized Brownian excursion at $p$ points, and of the $p-1$ minima between them, as given in [@SERLET].
### Shapes {#subs:shapes}
Let us now define a *shape* to be a rooted plane tree $T$ with $q$ edges, that we call *superedges* to distinguish them from edges of embedded trees. We shall endow each superedge with a length: let $\eta_1$, …, $\eta_q$ denote the edges of $T$ in *prefix order* (i.e. the order induced by first visits in contour traversal) and $\mathcal{L}(\eta_i)$ the length of $\eta_i$.
![A random tree $U_n$, the decomposition of its normalised contour $e^{(n)}$ at times $(\tau_1,\tau_2,\tau_3)$ and at minima in between, and the extracted shape (prefix order shown on edges and vertices). In particular: $\mathcal{L}(\eta_4)=7=(x_2-m_1)\sqrt{2n}$. \[fig:shape\]](\rep shape.eps){width="13cm"}
Given the $p$ normalised times $0<\tau_1<\cdots<\tau_p<1$ and $U$ an embedded tree of $\bar{\mathcal{U}}_n$, let us extract a shape $T$ and, for each superedge $\eta$ of $T$, a length $\mathcal{L}(\eta)$. In order to do this, let $t_i=\lfloor2n\tau_i\rfloor$ and consider $v_{t_i}$ the vertex visited at time $t_i$ by the contour traversal. The $p$ *fixed vertices* $v_{t_i}$ and the root $r$ span a minimal subtree of $U$ (the union of the branches from $v_{t_i}$ to $r$, see Figure \[fig:shape\], left hand side). Apart from the $v_{t_i}$, this subtree contains vertices of two types: *branchpoints* that have at least two sons in the subtree (black vertices in Figure \[fig:shape\]), and *smooth vertices* that have exactly one son in the subtree (grey vertices). Let us define the shape $T$ by taking the fixed vertices, the root and the branchpoints as vertex set, and the paths connecting these vertices as superedges (Figure \[fig:shape\], right hand side). A superedge $\eta$ of $T$ is thus by definition a set of edges of $U$ and we let $\mathcal{L}(\eta)=|\eta|$, which is just the length of the path $\eta$ in the tree $U$ (for instance, superedge $\eta_4$ in Figure \[fig:shape\] is made of $7$ edges).
Observe now that the shape and superedge lengths extracted from a random tree $U_n$ completely determine $Y^{(n)}(\tau)$. To check this assertion, let us assume that the extraction yields a shape $T$ and superedges $\eta_i$ of normalised length $\ell(\eta_i)$ with $\mathcal{L}(\eta_i)=\ell(\eta_i)\sqrt{2n}$, $i=1,\dots, q$. Let $A_i$ be the set of superedges on the unique path from $v_{t_i}$ to the root, so that $\sum_{\eta\in A_i}\ell(\eta)=e^{(n)}(\tau_i) =
x_i$. The contour traversal of the shape $T$ starts from the root $v_{t_0}=v_{t_{p+1}}$ and reaches successively each leaf $v_{t_1}$, …, $v_{t_p}$. From $v_{t_i}$ to $v_{t_{i+1}}$ a set $B_i$ of superedges are first traversed toward the root, followed by a set $C_i$ of superedges that are traversed away from the root (so that $B_i=A_i\setminus A_{i+1}$ while $C_i=A_{i+1}\setminus A_i$). Then $\smash{\sum_{\eta\in B_i}\ell(\eta)}=\beta_i$ and $\smash{\sum_{\eta\in C_i}\ell(\eta)}=\gamma_i$ are the lengths of these journeys, with $\beta_i$ and $\gamma_i$ given by Relations (\[rel:bgxm\]). In particular the normalised lengths $\ell(\eta)$ of superedges are all of the form $$(x_i-m_i),\hspace{7mm}(x_{i+1}-m_i)\hspace{7mm}\textrm{or}\hspace{1cm}
\vert m_j-m_i\vert \ \ \textrm{with} \ \ j< i.$$ In the latter case, $m_j$ has to be a record, that is for $j<k<i$, $m_k>\sup(m_i,m_j)$. These relations are exemplified by Figure \[fig:shape\].
Conversely, as explained in [@LEG Ch. 3] or [@SERLET Section 2], the $x_i$ and $m_i$ (or the $\beta_i$ and $\gamma_i$ as defined by Relations (\[rel:bgxm\])) exactly determine the shape and superedge lengths of $U_n$.
![A shape $T$ with prefix ordering of superedges and vertices, and the matrix $M_T$ mapping superedge lengths onto $(x,m)$. \[fig:matrix\]](\rep matrixprefix.eps){width="5.5cm"}
[$
\begin{array}{c|ccccccc}
&\eta_1\!&\!\eta_2\!&\!\eta_3\!&\!\eta_4\!&\!\eta_5\!&
\!\eta_6\!&\!\eta_7\\\hline
m_3& 1&0&0&0&0&0&0\\
m_1& 1&1&0&0&0&0&0\\
x_1& 1&1&1&0&0&0&0\\
m_2& 1&1&0&1&0&0&0\\
x_2& 1&1&0&1&1&0&0\\
x_3& 1&1&0&1&0&1&0\\
x_4& 1&0&0&0&0&0&1
\end{array}
$ ]{}
>From the limit density $\zeta(x,m)$ of the previous section, the probability that $m_i=m_j$ for some $i\neq j$ is seen to tend to zero as $n$ goes to infinity. This implies that, with probability tending to one as $n$ goes to infinity, the shape is a binary tree and $q=2p-1$ (indeed the existence of a branchpoint of larger degree corresponds to the equality of two minima $m_i$ and $m_j$). From now on we thus restrict our attention to binary shapes $T$.
Associated to any binary shape $T$, there is a matrix $M_T$ with size $2p-1\times 2p-1$ and entries zero or one, that sends the $2p-1$ normalized lengths of superedges $(\ell(\eta_i))$ on the $(x_i,m_i)$s. Moreover, provided the lengths of superedges on the one hand, and the $x_i$ and $m_i$ on the other hand, are sorted according to the prefix order, the matrix $M_T$ is lower triangular with ones on the diagonal. As a consequence, $M_T$ is *Lebesgue measure preserving*. Indeed the $k$th vertex in the prefix order is reached by the $k$th edge and all other edges on the path to the root have already been visited.
Finally let us consider labels of $U_n$: the variation of label along edges extends to superedges, upon setting for any syperedge $\eta$ $$\kappa(\eta)=\sum_{\epsilon\in\eta}\kappa(\epsilon).$$ Let $\tilde M_T$ be the double action of $M_T$ on $\mathbb R^{4p-2}$. Since the matrix $M_T$ also sends the normalized increments $(k_i)_{1\leq i\leq 2p-1}$ of labels along the $2p-1$ superedges onto the normalized labels $(w_i,\mu_i)$ of the $2p-1$ vertices of the tree, $\tilde M_T$ describes the one-to-one correspondance between shapes equiped with superedges’ lengths and label variations on the one hand, and $Z^{(n)}(\tau)$ on the other hand. More precisely, $\tilde M_T$ describes the restriction of this correspondance to shape $T$.
### A local limit theorem for $Z^{(n)}(\tau)$ in a fixed region. {#subs:Zfidis}
As already observed, the limit law of $Y^{(n)}(\tau)$ charges only the region $$R=\left\{(x,m,w,\mu)\in(\mathbb R^p\times \mathbb R^{p-1})^2
\ \left\vert\ 0<m_i<\inf(x_i,x_{i+1}),1\le i\le p-1\right.\right\}.$$ For a given permutation $\sigma$ on $p-1$ symbols, define $$R_{\sigma}=\left\{\left.(x,m,w,\mu)\in R
\ \right\vert\ 0<m_{\sigma(1)}<m_{\sigma(2)}
<\dots<m_{\sigma(p-1)}\right\}.$$ Observe that the shape is constant on $R_{\sigma}$ and denote it $T_\sigma$. (Conversely a given shape may appear in many different regions $R_{\sigma}$, as there are only $C(p-1)$ different shapes.) Let $$R_{\sigma,\varepsilon}=\left\{\left.(x,m,w,\mu)\in R_{\sigma}
\ \right\vert\
d\left((x,m,w,\mu),\partial R_{\sigma}\right)
\ge\varepsilon \right\}.$$ (The distance is the usual distance in $\mathbb{R}^{4p-2}$ and $\partial R_\sigma$ denote the boundary of $R_\sigma$, which is clearly a finite union of closed $(4p-3)$-dimensional cones.)
Let $\Delta(n,K)$ denote the set of possible values of $Z^{(n)}(\tau)$ that belong to a compact subset $K$ of $\mathbb R^{4p-2}$, that is $$\Delta(n,K)\;=\;K\cap{\left({\left((2n)^{-1/2}\mathbb Z\right)}^{2p-1}
\times{\left({\left(8n/9\right)}^{-1/4}\mathbb Z\right)}^{2p-1}\right)}.$$ Furthermore, let $\xi$ (resp. $f$) be defined, on $\mathbb R^{2p-1}\times \mathbb R^{2p-1}$ (resp. $(\mathbb R^p\times \mathbb R^{p-1})^2$), by $$\begin{aligned}
\label{equ:defxi}
\xi(\ell,k)&=&(2\pi)^{-2\delta}
\prod_{i=1}^{2p-1}\;\ell_i^{-1/2}\;
\exp\left(-\frac{k_i^2}{2\,\ell_i}\right),\\
\nonumber
\xi_T&=&\xi\circ \tilde M_T^{-1},\\
\nonumber f(x,m,w,\mu)&=&\zeta(x,m)\ \sum_{\sigma}\xi_{T_\sigma}(x,m,w,\mu)
\;\cdot\; \mathbf{1}_{R_{\sigma}}(x,m).\end{aligned}$$ The function $f$ is exactly the density of $Z(\tau)$, that is, the density of the evaluation of $X$ at the $p$ points $\tau_i$ and at the $p-1$ minima of $e$ between them. This density was described in [@SERLET Propositions 2 & 3]. The function $\xi_{T_\sigma}$ is the conditional density of the labels $(w,\mu)$ given $(x,m)$. We shall prove the following local limit law for $Z^{(n)}(\tau)$.
\[lem:locZ\] Let $K$ be a compact subset of $R_{\sigma,\varepsilon}$, $\epsilon>0$. Then, uniformly for $(x,m,w,\mu)\in \Delta(n,K)$, $$\begin{aligned}
\lefteqn{\mathbb{P}\left(Z^{(n)}(\tau)=(x,m,w,\mu)\right)}\\
&=&
(2n)^{-2\delta}(8n/9)^{-\delta}
\ \ f(x,m,w,\mu)\ \ {\left(1+\mathcal
O_K{\left(n^{-1/2}\right)}\right)},\end{aligned}$$
\[cor:weakZ\] Let $K$ denote a compact subset of $R_{\sigma,\varepsilon}$. For any uniformly continuous function $\Phi$ with support $K$, $$\begin{aligned}
\lim_n\mathbb{E}\left[\Phi\left(Z^{(n)}(\tau)\right)\right]&=&
\mathbb{E}\left[\Phi\left(Z(\tau)\right)\right].\end{aligned}$$
Our aim is to compute for $(x,m,w,\mu)\in \Delta(n,K)$ the probability $$\begin{aligned}
\lefteqn{\mathbb{P}\left(Z^{(n)}(\tau)=(x,m,w,\mu)\right)}\\
&=&
\mathbb{P}\left(Y^{(n)}(\tau)=(x,m)\right)\cdot
\mathbb{P}\left(Z^{(n)}(\tau)=(x,m,w,\mu)
\mid Y^{(n)}=(x,m)\right).\end{aligned}$$ >From Proposition \[pro:fdd-shape\], we already have $$\begin{aligned}
\mathbb{P}\left(Y^{(n)}(\tau)=(x,m)\right)
&=&
(2n)^{-2\delta}\ \ \zeta(x,m)\ {\left(1+O{\left(n^{-1/2}\right)}\right)}.\end{aligned}$$ As discussed in the previous section, the normalised lengths $\ell_i=(2n)^{-1/2}\mathcal{L}_i$ of superedges, obtained from $(x,m)$ through $M_T$, are of the form $$(x_i-m_i),\hspace{7mm}(x_{i+1}-m_i)\hspace{7mm}\textrm{or}\hspace{1cm}
\vert m_j-m_i\vert \ \ \textrm{with} \ \ j< i.$$ In particular the fact that $(x,m,\cdot,\cdot)\in K\subset
R_{\sigma,\varepsilon}$ for some $\epsilon>0$ grants that these normalized lengths are uniformly bounded away from 0. In turn the variation $\kappa^{(n)}(\eta)$ of labels along any superedge $\eta$ is the sum of at least $\mathcal
O_{\varepsilon}(\sqrt{n})$ i.i.d. uniform random variables on $\{1,0,-1\}$. Therefore we can apply uniform bounds for the local limit theorem [@PET pages 189–197]: if $S_n$ denotes the sum of $n$ i.i.d. random variables uniform on $\{+1,0,-1\}$, we have $$\mathbb{P}\left(S_n=\kappa\right)
=\ \sqrt{\frac{3}{4\pi n}}\ e^{-3\kappa^2/4n}+O\left(\frac 1n\right).$$ This allows us to calculate the probability that the variations of labels along superedges $\kappa^{(n)}(\eta_i)$ are equal to $\kappa_i=k_i{\left(8n/9\right)}^{1/4}$, $i=1,\ldots,2p-1$: uniformly for $(x,m,w,\mu)\in\Delta(n,K)$, $$\begin{aligned}
\lefteqn{\mathbb{P}\left(\kappa^{(n)}(\eta_i)=\kappa_i,\;{1\leq i\leq
2p-1}\;\Big\vert\; Y^{(n)}(\tau)=(x,m) \right)}\\
&=&\ \prod_{i=1}^{2p-1}\left(\sqrt{\frac{3}{4\pi \,\mathcal{L}_i}}
\ e^{-3\kappa_i^2/4\,\mathcal{L}_i}
+O\left(\frac{1}{\varepsilon\sqrt{n}}\right)
\right),\\
&=&
\ {\left(\frac 9{8n}\right)}^{\delta}\xi(\ell,k)
\ \ \left(1+\mathcal O_K\left(n^{-1/2}\right)\right),\end{aligned}$$ in which the sequence of normalised variations of labels along superedges, $k=(k_i)_{1\leq i\leq2p-1}$ is the inverse image of $(w,\mu)$ through $M_T$. In other terms, $$(\ell,k)=\tilde M_T^{-1}(x,m,w,\mu),$$ and the previous relation can be written $$\begin{aligned}
\lefteqn{\mathbb{P}\left(Z^{(n)}(\tau)=(x,m,w,\mu)\mid
Y^{(n)}=(x,m)\right)}\\
&=&{\left(8n/9\right)}^{-\delta}
\, \xi_{T_\sigma}(x,m,w,\mu)
\, \big(1+\mathcal O_K\big(n^{-1/2}\big)\big),\end{aligned}$$ leading to the desired result, through Proposition \[pro:fdd-shape\].
In view of the measure preserving property of $\tilde M_T$, we have $$\mathbb{E}(\Phi(Z(\tau)))\,=
\int\Phi\circ \tilde M_T(\ell,k)\;f\circ \tilde M_T(\ell,k)\,d\ell\,dk\,.$$ Therefore, the lemma follows upon proving that $$\lim_n\left|\mathbb{E}\left(\Phi(Z^{(n)}(\tau))\right)-
\int\Phi\circ \tilde M_T(\ell,k)\;f\circ \tilde M_T(\ell,k)\,d\ell\,dk\,\right|
=0.$$ Set $$\phi_T=\Phi\circ \tilde M_T,\hspace{15mm}f_T=f\circ \tilde M_T.$$ The function $\phi_T$ has a compact support $\tilde
K=\tilde M_T^{-1}K$ that is included in $\tilde
M_T^{-1}K_{\sigma,\epsilon}
\subset(\epsilon,\infty)^{2p-1}\times\mathbb{R}^{2p-1}$. Since $\tilde K$ is compact, there exists an $\epsilon'>0$ and a compact $K'$ such that $\tilde K\subset
K'\subset(\epsilon,\infty)^{2p-1}\times\mathbb{R}^{2p-1}$ and $d(\partial \tilde K,\partial K')>\epsilon'$. We shall use the fat boundary $K'\setminus\tilde K$, on which $\phi_T$ is identically zero, to deal with boundary effects.
Let finally $\tilde\Delta(n,K)=\tilde M_T^{-1}\Delta(n,K)$ be the discretized version of $\tilde K$ and similarly $\Delta'(n,K)=\tilde
M_T^{-1}\Delta(n,M_TK')$ that of $K'$; by construction $\tilde\Delta(n,K)\subset\Delta'(n,K)$ are finite sets with $\mathcal O_K(n^{3\delta})$ elements, and $\phi_T$ is identically zero over $\Delta'(n,K)\setminus\tilde\Delta(n,K)$.
First, we have $$\begin{aligned}
{\mathbb{E}\left(\Phi(Z^{(n)}(\tau))\right)}
&=&
\sum_{(\ell,k)\in\Delta'(n,K)}\phi_T(\ell,k)
\,\mathbb{P}\left((\ell^{(n)},k^{(n)})=(\ell,k)\right)\\
&=&
(2n)^{-2\delta}(8n/9)^{-\delta}
\!\!\sum_{(\ell,k)\in\Delta'(n,K)}\!\!\phi_T(\ell,k)\,f_T(\ell,k)\\
&&\;\;+\;\;\Vert\Phi\Vert_\infty\cdot\mathcal O_K(n^{-1/2}),\end{aligned}$$ the second equality due to the local limit convergence (Lemma \[lem:locZ\]). Next, difference between the discrete summation and the integral is bounded in terms of the modulus of continuity $\omega(\phi_T\cdot f_T,K',\cdot)$ of $\phi_T\cdot f_T$ on $\tilde K$ (recall that the modulus of continuity of a function $g$ on a compact $K$ is $\omega(g,K,\epsilon)=\sup_{0<d(x,y)<\epsilon}|g(x)-g(y)|$, and that if $g$ is uniformly continuous on $K$, it satisfies $\omega(g,K,\epsilon)=O(\epsilon)$ as $\epsilon$ tends to zero). This yields $$\begin{aligned}
&&\left|
(2n)^{-2\delta}(8n/9)^{-\delta}
\!\!\sum_{(\ell,k)\in\Delta'(n,K)}\!\!\phi_T(\ell,k)\,f_T(\ell,k)
-
\int\phi_T(\ell,k)\,f_T(\ell,k)\,d\ell\,dk
\right|\\
&&\qquad\leq\;\mathrm{measure}(K')\;\cdot\;
\omega(\phi_T\cdot f_T,K', n^{-1/4}).\end{aligned}$$ Observe that in this summation the compact $K'$ has been approximated by a union of boxes of diameter $O(n^{-1/4})$ and boundary effect should be considered. However $\phi_T$ is identically zero on a region $K'\setminus K$ with $d(\delta K,\delta
K')>\varepsilon'$, that contains all boxes intersecting the boundary for $n$ large enough. The boundary effect is thus null.
### Weak convergence of $Z^{(n)}(\tau)$.
According to the Porte-Manteau Theorem [@BILL Ch. 1], we need $$\lim_n \mathbb{E}\ \left[\Phi{\left(Z^{(n)}(\tau)\right)}\right]=\int\Phi\
f =\mathbb{E}{\left[\Phi{\left(Z(\tau)\right)}\right]}$$ to hold for any bounded uniformly continuous $\Phi$. Now consider $$K_m\ =\ \overline B(0,\rho_m)\bigcap{\left(\build{\bigcup}{\sigma}{}
R_{\sigma,\varepsilon_m}\times\mathbb{R}^{2p-1}\right)},$$ in which $\overline B(0,\rho_m)$ is the closed ball, in $\mathbb R^{4p-2}$, with radius $\rho_m$ and let simultaneously $\rho_m$ increase to $+\infty$ and $\varepsilon_m$ decrease to $0$. We obtain a increasing sequence of compacts $K_m$, each of these compacts having $(p-1)!$ connected components. As the limit of this sequence has a Lebesgue–negligible complement in $\mathbb R^{4p-2}$, we can choose the sequences $(\rho_m)_{m>1}$ and $(\varepsilon_m)_{m>1}$ in such a way that $$\mathbb{P}{\left(Z(\tau)\in K_m\right)}\ge 1-\frac 1m.$$ There exist uniformly continuous functions $\Psi_m:\ \ \mathbb{R}^{4p-2}
\longrightarrow [0,1]$ , such that $$\Psi_m\vert_{K_m}
\equiv 1, \hspace{1cm}\Psi_m\vert_{K_{m+1}^c}
\equiv 0.$$ By construction, $$\left|
\mathbb{E}\Big(\Phi(Z(\tau))\Big)\;-\;
\mathbb{E}\Big(\Psi_m\cdot\Phi\;(Z(\tau))\Big)
\right|
\;\leq\; \frac{\Vert\Phi\Vert_\infty}m.,$$ Moreover the product $\Psi_m\cdot\Phi$ is now a finite sum of functions satisfying the assumptions of Corollary \[cor:weakZ\]: this yields $$\begin{aligned}
\lim_n\ \mathbb{E}\left(\Psi_m\cdot\Phi\;(Z^{(n)}(\tau))\right)&=&\mathbb{E}\
\left(\Psi_m\cdot\Phi\;(Z(\tau))\right).\end{aligned}$$ Next observe that, by definition of $\Psi_m$, $$\mathbb{P}\left(Z^{(n)}(\tau)\in K_m\right)\;\geq\;
\mathbb{E}\left(\Psi_{m-1}(Z^{(n)}(\tau))\right),$$ and $$\mathbb{E}\big(\Psi_{m-1}(Z(\tau))\big)\;\geq\;
\mathbb{P}\big(Z(\tau)\in K_{m-1}\big).$$ Therefore, applying Corollary \[cor:weakZ\] to $\Psi_{m-1}$, $$\begin{aligned}
\liminf_n\ \mathbb{P}\left(Z^{(n)}(\tau)\in K_m\right)&\geq&
\mathbb{P}\left(Z(\tau)\in K_{m-1}\right)\;\ge\; 1-\frac 1{m-1}.\end{aligned}$$ Moreover $$\begin{aligned}
\mathbb{E}\left((1-\Psi_m)\cdot\Phi\;(Z^{(n)}(\tau))\right)
&\leq& \mathbb{P}\left(\Psi_m(Z^{(n)}(\tau))<1\right)\cdot
\Vert\Phi\Vert_\infty,\\
&\leq& \mathbb{P}\left(Z^{(n)}(\tau)\not\in K_m\right)\cdot
\Vert\Phi\Vert_\infty.\end{aligned}$$ Thus, taking limit for $n$ going to infinity and applying the previous lower bound, $$\begin{aligned}
\limsup_n\mathbb{E}\left((1-\Psi_m)\cdot\Phi\;(Z^{(n)}(\tau))\right)
&\leq&\frac{\Vert\Phi\Vert_\infty}{m-1}.\end{aligned}$$ Finally, the decomposition $$\begin{aligned}
\mathbb{E}\left(\Phi(Z^{(n)}(\tau))\right)
&=&
\mathbb{E}\left(\Psi_m\cdot\Phi\;(Z^{(n)}(\tau))\right)
\;+\;
\mathbb{E}\left((1-\Psi_m)\cdot\Phi\;(Z^{(n)}(\tau))\right),\end{aligned}$$ yields $$\begin{aligned}
\limsup_n\left|\mathbb{E}\left(\Phi(Z^{(n)}(\tau))\right)
-\mathbb{E}\big(\Phi(Z(\tau))\big)\right|
\leq \frac{2\Vert\Phi\Vert_\infty}{m-1}.\end{aligned}$$ Letting $m$ go to infinity gives the weak convergence of $Z^{(n)}(\tau)$ to $Z(\tau)$ as claimed. The convergence of $X^{(n)}(\tau)$ is a by-product.
A deviation bound for the largest label {#sec:bound}
---------------------------------------
In this section, a rough but exponential deviation bound for the value of the largest label in a forest of $k$ embedded trees with $n$ edges is obtained. For $k=1$, Proposition \[pro:devbound\] is exactly obtained.
Let $\mathcal{EV}_{k,2n}$ denote the set of $k$-uples of element of $\mathcal{EV}$ of total length $2n$: an element $[(E_1,V_1),\ldots,(E_k,V_k)]$ of $\mathcal{EV}_{k,2n}$ codes for a forest of $k$ embedded trees (each $(E_i,V_i)$ codes for a tree, according to Proposition \[pro:contours\]). Equivalently, one may concatenate the $k$ pairs and view any element of $\mathcal{EV}_{k,2n}$ as a pair $(E,V)=(E_1\cdots E_k,V_1\cdots
V_k)\in\mathcal{EV}_{2n}$ together with a set of concatenation times $0=t_0\leq t_1\leq\cdots\leq t_k=2n$, subject to the conditions $E(t_i)=V(t_i)=0$ for all $i=1,\ldots,k$. In this identification, $E_i$ has length $2n_i=t_i-t_{i-1}$.
Let $(E^{k,n},V^{k,n})$ denote a random forest of $\mathcal{EV}_{k,2n}$ under the uniform distribution.
\[pro:loosebound\] There exists $y_0>0$ such that, for all $n$, $k$ and $y\ge y_0$, $$\label{equ:loosebound}
{\textstyle\mathbb{P}}
\left(\sup_{0\leq t\leq 2n}V^{k,n}(t)
\,>\, yn^{1/4}\right)\;<\;e^{-y}.$$
This is the key fact in the proof of tightness of the sequence $X^{(n)}$, given in the last subsection, and it also leads to the convergence of moments in Corollary \[radcv\] through the following weaker formulation (with $k=1$): for $y>y_0$, and for all $n$, $$\mathbb{P}\left(\sup_{0\leq s\leq1}\hat
W^{(n)}_s\,>\,(8/9)^{1/4}y\right)\;\leq e^{-y},$$ which is exactly Proposition \[pro:devbound\].
The proof is based on a *branch decomposition*, that is discussed in the next paragraph. Then, after two preliminary results on parameters of the middle branch of a random tree (Paragraphs \[sec:middlebranchlength\] and \[sec:middlebranchlargest\]), Proposition \[pro:loosebound\] is proved by induction (Paragraphs \[sec:induct3\], \[sec:induct4\] and \[sec:induct5\]). At the price of more technical details in these latter paragraphs, the bound could be improved to $e^{-c_\epsilon y^{4/3-\epsilon}}$ for any fixed $\epsilon>0$.
### The branch decomposition at time $t$ {#subs:brandw}
Let $(E,V)\in\mathcal{EV}_{k,2n}$ with concatenation times $0\leq
t_1\leq\ldots\leq t_k\leq2n$ and let $t\in(0,2n)$. Suppose moreover that $t_{p-1}<t<t_p$, that is $t$ occurs during the contour traversal of $(E_p,V_p)$, the $p$th component of the forest encoded by $(E,V)$ and let $U$ be the tree encoded by $(E_p,V_p)$.
To any vertex $v$ of $U$ is associated the set of edges in the unique simple path from $v$ to the root of $U$, denoted $\mathrm{br}(v)$ (for the *branch* of $v$).
![Decomposition of a forest at time $t$. Arrows under the walks indicate concatenation times. \[fig:branch\]](\rep middlebranch.eps){width="11cm"}
Recall that $v_t$ denote the vertex visited at time $t$ of the contour traversal of $U$. Observe that the height $E(t)$ is $|\textrm{br}(v_t)|$, the length of the branch from the root $r$ to $v_t$, while the label $V(t)$ of $v_t$ is given by $$\label{iidonbranches}
V(t)=\sum_{\epsilon\in\textrm{br}(v_t)}\kappa(\epsilon).$$ Let $\ell=E(t)=|\mathrm{br}(v_t)|$, and call $\epsilon_i$ the edge of $\mathrm{br}(v_t)$ between heights $i-1$ and $i$, for $i=1,\ldots,\ell$. The maximal label on the branch $\textrm{br}(v_t)$ is $$\bar H(t)\;=\;\sup_{0\leq j\leq \ell}\;\sum_{i=1}^j \kappa(\varepsilon_i)$$ where $\varepsilon_i$ is the $i$th edge of $\textrm{br}(v_t)$.
The branch of $v_t$ induces a decomposition of $U$ into two forests of trees, the *branch decomposition*, which we now phrase in terms of $(E,V)$.
- From time $t_{p-1}$ to $t$, the edges $\epsilon_1$, …, $\epsilon_{\ell}$ are successively traversed away from the root on the branch $\mathrm{br}(v_t)$, at times $t'_1<\cdots<t'_{\ell}<t$. Let $(E'_i,V'_i)$ be the part of the contour walks $(E,V)$ between $t'_i$ and $t'_{i+1}$ (with the convention that $t'_0=t_{p-1}$ and $t'_{\ell+1}=t$).
- From time $t$ to $t_p$, the edges $\epsilon_\ell$, …, $\epsilon_1$ are successively traversed back toward the root on the branch $\mathrm{br}(v_t)$, at times $t''_{\ell+1}<\cdots<t''_{1}$. Let $(E''_i,V''_i)$ be the part of the contour walks $(E,V)$ between $t''_{i+1}$ and $t''_i$ (with the convention that $t''_{\ell+1}=t$ and $t''_0=t_{p}$).
The contour walks $(E'_i,V'_i)$ (resp. $(E''_i,V''_i)$) encodes for the left (resp. right) subtree attached at the $i$th vertex of the branch $\textrm{br}(v_t)$. One can see $t'_i$ as the time of the last upcrossing of heights $(i-1,i)$ before time $t$, and $t''_i$ as the time of the first downcrossing of heights $(i,i-1)$ after time $t$.
Upon shifting the walks $E'_i$ and $E''_i$ down by $i$ so that they start from zero, and also shifting the walks $V'_i$ and $V''_i$ by $V(t'_i)$ (resp. $V(t''_{i+1})$): the two forests $$\begin{aligned}
\textrm{left}(t)&=&[(E_1,V_1),\ldots,(E_{p-1},V_{p-1}),
(E'_0,V'_0),\ldots,(E'_\ell,V'_\ell)]
\\
\textrm{right}(t)&=&[(E''_\ell,V''_\ell),\ldots,(E''_0,V''_0),
(E_{p+1},V_{p+1}),\ldots,(E_k,V_k)]\end{aligned}$$ belong respectively to $\mathcal{EV}_{k',n'}$ and $\mathcal{EV}_{k'',n''}$ with $k'=p+\ell$, $n'=t-\ell$, $k''=k-p+1+\ell$, and $n''=2n-t-\ell$.
Let us apply this branch decomposition to a random forest $(E^{k,n},V^{k,n})$. In view of expression (\[iidonbranches\]), conditionally on $E^{(k,n)}(t)=\ell$, $$V^{(k,n)}(t)\;\mathop{=}^{\textrm{law}}\;S_\ell,$$ where $S_k$ denote the sum of $k$ i.i.d. random variables uniform on $\{-1,0,1\}$. Moreover, again conditionally on $E^{(k,n)}(t)=\ell$, $$\textrm{left}^{(k,n)}(t)
\;\mathop{=}^{\textrm{law}}\;(E^{k',n'},V^{k',n'}),
\qquad
\textrm{right}^{(k,n)}(t)
\;\mathop{=}^{\textrm{law}}\;(E^{k'',n''},V^{k'',n''}).$$
### The middle branch length in a random forest {#sec:middlebranchlength}
The first step of the proof is a bound for the tail probability of $E^{k,n}(n)$, the length of the middle branch ($t=n$).
For $4\ell^2>27\cdot n$, $$\begin{aligned}
\Pr(E^{k,n}(n)>\ell)&<&A\cdot\frac{\ell}{n^{1/2}}
\exp\left({-\frac{4\ell^2}{27n}}\right).\end{aligned}$$
With the notations of Subsection \[subs:brandw\], for $1\le p\le k$, let $t^{k,n}_p$ denote the $p$th concatenation time of $E^{k,n}$. Then $$\Pr{\left(E^{k,n}(n)=\ell\,\left\vert\,
t^{k,n}_{p+1}-t^{k,n}_p=2m,\, 0\le n-t^{k,n}_p=a\le 2m \right.\right)}
=\Pr{\left(E^{(m)}(a)=\ell\right)},$$ so the proof reduces to bound $\Pr{\left(E^{(m)}(a)>\ell\right)}$ uniformly on pairs $(a,m)$ such that $0\le a\le 2m\le 2n$. We have $$\begin{aligned}
\Pr{\left(E^{(m)}(a)=\ell\right)}&=&\frac{C(a;\ell)\,C(2m-a;\ell)}{C(2m;0)}\\
&=&\frac{C(\alpha;\ell)\,C(\beta;\ell)}{C(2m;0)},\end{aligned}$$ with an obvious change of variables. From [@bolobas Ch. I.3, Th 2], for $k>n/2+h$ and $1\leq h\leq
n/6$ $${\sqrt{\frac{\pi n}2}}\binom{n}{k}\;<\;2^n
\exp\left({-\frac{2h^2}{n}+\frac{2h}n+\frac{4h^3}{n^2}}\right).$$ With $n=\alpha+1$, $k=(\alpha+\ell)/2+1$, $1\leq
h=\ell/6\leq \alpha/6\leq n/6$, this bound yields: $$\begin{aligned}
C(\alpha;\ell)&=&\frac{\ell+1}{\alpha+1}{\alpha+1\choose (\alpha+\ell)/2+1}\\
&<&\sqrt{\frac 8{\pi}}\cdot\frac\ell{n^{3/2}}\;2^n
\exp\left({-\frac{\ell^2}{18n}
+\frac{\ell}{3n}+\frac{\ell^3}{54n^2}}\right),\\
&<&e^{1/3}\sqrt{\frac 8{\pi}}\cdot\frac\ell{n^{3/2}}\;2^n
\exp\left({-\frac{\ell^2}{27n}}\right)\\
&<&2e^{10/27}\sqrt{\frac 8{\pi}}\cdot\frac\ell{\alpha^{3/2}}\;2^{\alpha}
\exp\left({-\frac{\ell^2}{27\alpha}}\right),\end{aligned}$$ Thus, $$\begin{aligned}
\Pr{\left(E^{(m)}(a)=\ell\right)}
&<&B_1\cdot\frac{\ell^{2}m^{3/2}}{(\alpha\beta)^{3/2}}
\exp\left({-\frac{\ell^2}{27\alpha}
{-\frac{\ell^2}{27\beta}}}\right),\\
&<&B_2\cdot\frac{\ell^{2}}{m^{3/2}}
\exp\left({-\frac{4\ell^2}{27m}}\right),\end{aligned}$$ the latter inequality since the maximum of the function $(xy)^{-3/2}\exp{\left(-x^{-1}-y^{-1}\right)}$, subject to $x+y=27m/\ell^2<1$, is obtained for $x=y$. The last inequality entails that, for $4\ell^2>27m$, $$\begin{aligned}
\Pr{\left(E^{(m)}(a)>\ell\right)}
&<&B_3\cdot\frac{\ell}{m^{1/2}}
\exp\left({-\frac{4\ell^2}{27m}}\right)\\
&<&B_3\cdot\frac{\ell}{n^{1/2}}
\exp\left({-\frac{4\ell^2}{27n}}\right).\end{aligned}$$
### The largest label on the middle branch {#sec:middlebranchlargest}
\[lem:branch\] Let $\bar H^{k,n}$ be the largest label on the branch between the root and the vertex reached at $t=n$ by the contour traversal of $(E^{k,n},V^{k,n})$. Then there exists $c_0$ such that for all $k$, $n$ and $h$, $$\mathbb{P}(\bar H^{k,n}>h)\leq c_0\ e^{-\frac1{9}(hn^{-1/4})^{4/3}}.$$
As already discussed the conditional probability that the largest label is $h$ knowing that the branch has length $\ell$ is exactly the probability that a random walk with steps $\{1,0,-1\}$ of length $\ell$ has maximal value $h$. Using the reflection principle, Azuma’s inequality [@AS Th. 2.1, p. 85] then reads $$\begin{aligned}
\mathbb{P}(\bar H^{k,n}>h\mid E^{k,n}(n)=\ell)&\leq&
2\ e^{-h^2/2\ell}.\end{aligned}$$ Next, as previously calculated, for $4\ell^2\geq 27n$, $$\begin{aligned}
\mathbb{P}(E^{k,n}(n)>\ell)\;<\;A \cdot(\ell/n^{1/2})\;e^{-4\ell^2/27n}\end{aligned}$$ Finally, assuming $h>(27/4)^{3/4}n^{1/4}$, so that the previous inequality holds, $$\begin{aligned}
\mathbb{P}(\bar H^{k,n}>h)
&\leq&\mathbb{P}(\bar H^{k,n}>h\mid E^{k,n}(n)\leq h^{2/3}n^{1/3})\;+\;
\mathbb{P}(E^{k,n}(n)>h^{2/3}n^{1/3})\\
&\leq&
2 e^{-h^{4/3}/2n^{1/3}}\;+\;A\cdot(h^{2/3}n^{-1/6})\,e^{-4h^{4/3}/27n^{1/3}}.\end{aligned}$$ The lemma follows for $h>(27/4)^{3/4}n^{1/4}$, taking $c_0$ large enough, and also for $h\le(27/4)^{3/4}n^{1/4}$, taking $c_0\ge e^{3/4}$.
### Conditional induction in the case $E(n)>0$ {#sec:induct3}
\[lem:induct1\] Assume the bound (\[equ:loosebound\]) holds true for some $y_0$ for $V^{k,m}$ with $m<n$. Then, for all $\ell>0$ and $k\geq p\geq0$, the probability $$\begin{aligned}
\lefteqn{p_{k,n}(y;h,\ell,p)\;=}\\
&&\mathbb{P}\Big(\sup_{0\leq t\leq n}V^{k,n}(t)\;>\;yn^{1/4}\;\Big|\;
\bar H^{k,n}=h,\, E^{k,n}(n)=\ell,\, t^{k,n}_{p-1}<n<t^{k,n}_{p}\,\Big),\end{aligned}$$ satisfies $$\label{ind-bound}
p_{k,n}(y;h,\ell,p)\leq 4e^{-2^{1/4}(y-hn^{-1/4})},
\quad\textrm{ provided $h\le n^{1/4}(y-y_0/2^{1/4})$.}$$
Assume $(E^{k,n},V^{k,n})=(E_1\cdots E_k,V_1\cdots V_k)$ is such that $E^{k,n}(n)=\ell>0$ and $t^{k,n}_{p-1}<n<t^{k,n}_{p}$, so that $t=n$ occurs inside $(E_p,V_p)$. Apply the branch decomposition (see Section \[subs:brandw\]) at $t=n$ to $(E^{k,n},V^{k,n})$ and let $$\begin{aligned}
(\bar E^{k,n},\bar V^{k,n})&=&\textrm{left}^{k,n}(n)
\;=\;[(E_1,V_1),\ldots,(E_{p-1},V_{p-1}),
(E'_0,V'_0),\ldots,(E'_\ell,V'_\ell)],\\
(\bar{\bar E}^{k,n},\bar{\bar V}^{k,n})&=&\textrm{right}^{k,n}(n)
\;=\;[(E''_\ell,V''_\ell),\ldots,(E''_0,V''_0),
(E_{p+1},V_{p+1}),\ldots,(E_k,V_k)].\end{aligned}$$ Upon taking $k'=p+\ell$, $k''=k-p+\ell+1$ and $n'=(n-\ell)/2$, $$(\bar E^{k,n},\bar V^{k,n})\;\mathop{=}^{\textrm{law}}\;
(E^{k',n'},V^{k',n'})
\quad\textrm{ and }\quad
(\bar{\bar E}^{k,n},\bar{\bar V}^{k,n})\;\mathop{=}^{\textrm{law}}\;
(E^{k'',n'},V^{k'',n'})$$ Observe now that, in the previous decomposition, $$\sup_{0\leq t\leq 2n}V(t)\;\leq\;
\sup\Big(\bar H+\sup_{0\leq t\leq 2n'}\bar V(t)\,,\;
\bar H+\sup_{0\leq t\leq 2n'}\bar{\bar V}(t)\Big),$$ so that $$\begin{aligned}
\lefteqn{p_{k,n}(y;h,\ell,p)\leq}\\
&&\mathbb{P}\Big(
\sup_{0\leq t\leq 2n'}\bar V^{k,n}(t)>yn^{1/4}-h\;
\Big|\;\bar H^{k,n}=h,\, E^{k,n}(n)=\ell,\,
t^{k,n}_{p-1}<n<t^{k,n}_{p}\,\Big)\\
&+&
\mathbb{P}\Big(
\sup_{0\leq t\leq 2n'}\bar{\bar V}^{k,n}(t)>yn^{1/4}-h\;
\Big|\;\bar H^{k,n}=h,\, E^{k,n}(n)=\ell,\,
t^{k,n}_{p-1}<n<t^{k,n}_{p}\,\Big).\end{aligned}$$ Hence, in view of the preceding identities in law, $$\begin{aligned}
p_{k,n}(y;h,\ell,p)&\leq&
\mathbb{P}\Big(
\sup_{0\leq t\leq 2n'}V^{k',n'}(t)>yn^{1/4}-h\;\Big)\\
&&\!\!\!\!\!+\;\;
\mathbb{P}\Big(
\sup_{0\leq t\leq 2n'}V^{k'',n'}(t)>yn^{1/4}-h\;\Big).\end{aligned}$$ Observe that $2n'\le n$. Hence $$\begin{aligned}
p_{k,n}(y;h,\ell,p)&\leq&
\mathbb{P}\Big(
\sup_{0\leq t\leq 2n'}V^{k',n'}(t)>2^{1/4}(y-hn^{-1/4})n'{}^{1/4}\;\Big)\\
&&\!\!\!\!\!+\;\;
\mathbb{P}\Big(
\sup_{0\leq t\leq 2n'}V^{k'',n'}(t)>2^{1/4}(y-hn^{-1/4})n'{}^{1/4}\;\Big).\end{aligned}$$ The induction hypothesis now implies, for $2^{1/4}(y-hn^{-1/4})\ge y_0$, that is, for all $h\le n^{1/4}(y-y_0/2^{1/4})$, $$p_{k,n}(y;h,\ell,p)\leq 2e^{-2^{1/4}(y-hn^{-1/4})},$$ which is exactly the lemma, up to a factor 2 added for later convenience.
### Conditional induction in the case $E(n)=0$ {#sec:induct4}
\[lem:induct2\] Assume the bound (\[equ:loosebound\]) holds true for some $y_0$ for $V^{k,m}$ with $m<n$. Then, provided $y\geq y_0/2^{1/4}$, $$\begin{aligned}
\mathbb{P}\Big(\sup_{0\leq t\leq 2n}V^{k,n}(t)\;>\;yn^{1/4}\;\Big|\;
E^{k,n}(n)=0\,\Big)\;\leq\; 4e^{-2^{1/4}y}.\end{aligned}$$ The bound (\[ind-bound\]) thus remain valid in the case $\ell=0$.
In this case the decomposition at $t=n$ is even simpler. Let $p=\sup\{i\mid t_i< n\}$, and $q=\inf\{i\mid t_{i}> n\}$, and consider a decomposition in four parts, cutting at times $t_p$, $n$ and $t_q$. The two contour walks for $0\leq t\leq t_p$ and $t_q\leq t\leq 2n$ are uniform on $\mathcal{EV}_{p,t_p}$ and $\mathcal{EV}_{k-q,2n-t_q}$. The other two contour walks are uniform on $\mathcal{EV}_{n-t_p}$ and $\mathcal{EV}_{t_q-n}$.
The result then follows using the induction hypothesis on the 4 parts.
### Complete induction and proof of Proposition \[pro:loosebound\] {#sec:induct5}
Observe that the bounds of Lemmas \[lem:induct1\] and \[lem:induct2\] do not depend on $\ell$ or $p$, so that, assuming (\[equ:loosebound\]) holds for some $y_0$ for $V^{k,m}$ with $m<n$, $$\begin{aligned}
\mathbb{P}\Big(\sup_{0\leq t\leq 2n}V^{k,n}(t)\;>\;yn^{1/4}\;\Big|\;
\bar H^{k,n}=h\,\Big)
&\leq& 4e^{-2^{1/4}(y-hn^{-1/4})},\end{aligned}$$ provided $h\le n^{1/4}(y-y_0/2^{1/4})=h_0$. Using this bound, $$\begin{aligned}
f_{k,n}(y)&=&
{\mathbb{P}}
\Big(\sup_{0\leq t\leq 2n}V^{k,n}(t)\,>\,yn^{1/4}\Big)
\\&\leq&
4e^{-2^{1/4}y}\sum_{h=0}^{h_0}
e^{2^{1/4}hn^{-1/4}}\mathbb{P}(\bar H^{k,n}=h)
\;+\;\mathbb{P}{\left(\bar H^{k,n}>h_0\right)},\\
&\leq&4e^{-2^{1/4}y}{\left(1+\sum_{h=1}^\infty
(e^{2^{1/4}hn^{-1/4}}-e^{2^{1/4}(h-1)n^{-1/4}})
\mathbb{P}(\bar H^{k,n}\geq h)\right)}\\
&&+\;\mathbb{P}(\bar H^{k,n}>{n^{1/4}(y-y_0/2^{1/4})}).\end{aligned}$$ In view of Lemma \[lem:branch\], the summation is bounded by a convergent integral that evaluates to a constant $c_1$. Lemma \[lem:branch\] allows also to dispose of the second term: $$\begin{aligned}
f_{k,n}(y)
&\leq& 4e^{-2^{1/4}y}(1+c_1)+c_0e^{-\frac1{9}(y-y_0/2^{1/4})^{4/3}}\end{aligned}$$ Now observe that $$\begin{aligned}
e^{-\frac1{9}(y-y_0/2^{1/4})^{4/3}}
&=&
e^{-y}\ e^{y_0/2^{1/4}}\ e^{(y-y_0/2^{1/4})
-\frac1{9}(y-y_0/2^{1/4})^{4/3}}\\
&\leq&
e^{-y}\ e^{y_0/2^{1/4}}\ e^{(y_0-y_0/2^{1/4})
-\frac1{9}(y_0-y_0/2^{1/4})^{4/3}}\\
&=&
e^{-y}\ e^{y_0 -\frac1{9}(1-1/2^{1/4})^{4/3}y_0^{4/3}}\end{aligned}$$ for $y\ge y_0$, as soon as $x\rightarrow x
-\frac1{9}x^{4/3}$ is decreasing on the interval $\left[y_0-y_0/2^{1/4}, +\infty\right[$, that is, for $y_0\ge 1933$. The bound can thus be rewritten as $$\begin{aligned}
f_{k,n}(y)
&\leq&e^{-y}\left(4e^{-(2^{1/4}-1)y_0}(1+c_1)+
c_0e^{y_0 -\frac19(1-1/2^{1/4})^{4/3}y_0^{4/3}}
\right)\end{aligned}$$ which is smaller than $e^{-y}$ for $y_0$ large enough, so that induction can be carried on (Recall that $c_0$ and $c_1$ do not depend on $n$). The case $n=1$ holds true for $y_0\ge 1$, and the proof of Proposition \[pro:loosebound\] is complete.
Tightness {#sec:tight}
---------
Tightness for the bidimensional path follows from the tightness of the two projections. The tightness of the first projection was proved by Kaigh [@KAIGH]. Thus we only have to prove the following proposition (cf. [@BILL Ch. 3]).
For all $\varepsilon>0$ and $\delta>0$ there exists $m$ such that for $n$ large enough $$\mathbb{P}{\left(\sup_{0\le s\le 1}\left\vert \hat W^{(n)}_s
-\hat W^{(n)}_{\lfloor ms\rfloor/m} \right\vert\,\ge\,\delta\right)}\le
\varepsilon.$$
We proceed by bounding, for $m$ large enough and all $i$ with $1\leq
i\leq m$, $$\begin{aligned}
p_{m,i}(n) &=& \mathbb{P}{\left(\sup_{(i-1)/m\le s\le i/m}\left\vert
\hat W^{(n)}_s
-\hat W^{(n)}_{(i-1)/m} \right\vert\,\ge\,\delta\right)}\\
&=&
\mathbb{P}{\left(\sup_{t_1\leq t\leq t_2}\left\vert
V^{(n)}(t)-V^{(n)}(t_1)\right\vert\,\ge\,\delta'n^{1/4}\right)},\end{aligned}$$ where $t_1=\lfloor 2(i-1)n/m\rfloor$, $t_2=\lfloor 2in/m\rfloor$ and $\delta'=(8/9)^{1/4}\delta$.
The proposition is an immediate consequence of the following lemma, upon taking $m$ large enough and summing on $i$.
For all $\varepsilon>0$ and $\delta>0$ there exist $m_0$ such that for all $m>m_0$ and $1\leq i\leq m$, $$\exists n_0,\;\forall n>n_0,\qquad p_{m,i}(n)\;\leq\;
\frac{\varepsilon}{2m}+C_\delta(m),$$ where $C_\delta(m)$ is exponentially decreasing in a positive power of $m$.
For simplicity of notations let us assume $t_2-t_1=2n/m$. (The general case is identical but obscured by a collection of $\lfloor n/m\rfloor$.)
Let us consider the shape $T$ associated with the times $t_1$ and $t_2$, as defined in Section \[subs:shapes\]. The branches $\textrm{br}(v_{t_1})$ from $v_{t_1}$ to the root and $\textrm{br}(v_{t_2})$ from $v_{t_2}$ to the root meet at the unique branchpoint $v$ of the shape $T$. Let $B=\textrm{br}(v_{t_1})\setminus\textrm{br}(v_{t_2})$ be the branch between $v_{t_1}$ and $v$ and $C=\textrm{br}(v_{t_2})\setminus\textrm{br}(v_{t_1})$ the branch between $v_{t_2}$ and $v$. Between $t_1$ and $t_2$, a total of $k$ edges $\epsilon_i$, $i=1,\ldots,k$ of $U$ are successively traversed on the branch from $v_{t_1}$ to $v_{t_2}$ by the contour walk. The $k'$ first are along the branch $B$ and traversed toward the root, while the $k''$ next are along the branch $C$ and traversed away from the root. By construction, $$\begin{aligned}
k'&=&E^{(n)}(t_1)-\inf_{[t_1,t_2]}E^{(n)}
\;=\;\sqrt{2n}\ {\left[e^{(n)}
{\left(\frac {i-1}{m}\right)}-\inf_{[\frac{i-1}{m},\frac{i}m]}e^{(n)}
\right]},\\
k''&=&E^{(n)}(t_2)-\inf_{[t_1,t_2]}E^{(n)}
\;=\;\sqrt{2n}\ {\left[e^{(n)}
{\left(\frac {i}{m}\right)}-\inf_{[\frac{i-1}{m},\frac{i}m]}e^{(n)}
\right]},\end{aligned}$$ and the total length $\Delta^{(n)}_i=k=k'+k''$ of the branch from $v_{t_1}$ to $v_{t_2}$ is $$\Delta^{(n)}_i\;=\;E^{(n)}(t_1)+E^{(n)}(t_2)-2\inf_{[t_1,t_2]}E^{(n)}
\;=\;\sqrt{2n}\;\Delta_{m,i}\,e^{(n)},$$ with the notation $$\Delta_{m,i} f=\left\vert f
{\left(\frac {i-1}{m}\right)}+f
{\left(\frac {i}{m}\right)}-2\inf_{[\frac{i-1}{m},\frac{i}m]}f
\right \vert.$$ The walk $(E^{(n)}(t))_{t_1\leq t\leq t_2}$ is decomposed along the branch from $v_{t_1}$ to $v_{t_2}$ into a sequence $$E_0,\epsilon_1,E_1,\epsilon_2,\ldots,E_{k-1},\epsilon_{k},E_k,$$ where the $\Delta^{(n)}_i=k$ passages on the branch separate $\Delta^{(n)}_i$ subtrees (each coded by a $E_i$) of total size (number of edges) $N^{(n)}_i=(2n/m-\Delta^{(n)}_i)/2$.
In this decomposition, conditionally given $\Delta^{(n)}_i=k$ (and $N^{(n)}_i=n'=n/m-k/2$), the forest satisfies $$[(E_0\cdots E_{k},V_0\cdots V_{k})]\;
\mathop{=}^{\textrm{law}}\;
(E^{k+1,n'},V^{k+1,n'}),$$ upon resetting all walks to start at zero. Under the same conditions, the variation of labels on edges of $B$ and $C$ are i.i.d. random variables $\zeta(\epsilon)$, uniform on $\{-1,0,+1\}$: the random variable $$\bar H^{(n)}_i\;=\;\sup_{0\leq \ell\leq
\Delta^{(n)}_i}\;\sum_{j=0}^{\ell} \zeta(\epsilon_j)$$ has tail distribution bounded by Azuma’s inequality.
Our strategy is to bound $\Delta^{(n)}_i$ and, conditionally given $\Delta^{(n)}_i$, to bound separately the variations on the forest and on the branch:
- First fix $\alpha>0$ and let $a_1$ denote the tail probability for $\Delta^{(n)}_i$: $$\begin{aligned}
a_1&=&\mathbb{P}{\left(\Delta^{(n)}_i\geq m^{-\alpha} \sqrt{2n}\right)}
\;=\;\mathbb{P}{\left(\Delta_{m,i}e^{(n)}\geq m^{-\alpha}\right)}.\end{aligned}$$
- On the complementary event for $\Delta^{(n)}_i$ we consider the variation in a forest, $$\begin{aligned}
a_2&=&\mathbb{P}{\left(\sup_{0\leq t\leq 2N^{(n)}_i}
\left|V^{\Delta^{(n)}_i+1,N^{(n)}_i}(t)\right|\,\ge\,\frac{\delta'}4\, n^{1/4}\
\textrm{ and }\;\Delta_{m,i} e^{(n)}\le m^{-\alpha}\right)},
\\
&\leq&2\mathbb{P}{\left(\sup_{0\leq
t\leq2N^{(n)}_i}{V^{\Delta^{(n)}_i+1,N^{(n)}_i}(t)}\,\ge\,\frac{\delta'}4\,
n^{1/4}\
\textrm{ and }\;\Delta_{m,i} e^{(n)}\le m^{-\alpha}\right)},\end{aligned}$$
- and the variations on the branch from $t_1$ to $t_2$, $$\begin{aligned}
a_3&=&\mathbb{P}{\left(
\sup_{0\leq \ell\leq \Delta^{(n)}_i}\;\Big|\smash{\sum_{j=0}^{\ell}}
\zeta(\epsilon_j)\Big|\;\ge\;\frac{\delta'}4\,n^{1/4}\;
\textrm{ and }\;\Delta^{(n)}_i\le
m^{-\alpha}n^{1/4}\right)}, \\
&\leq&2\mathbb{P}{\left(\bar H_i^{(n)}\,\ge\,\frac{\delta'}4\,n^{1/4}\;
\textrm{ and }\;\Delta^{(n)}_i\le m^{-\alpha}n^{1/4}\right)}.\end{aligned}$$
In view of the decomposition we have $$p_{m,i}(n)\; \le \;a_1+a_2+a_3.$$
We now bound separately each term $a_1$, $a_2$, and $a_3$. Let us start with $a_2$ and write $$a_2\;=\;\sum_{k=0}^{m^{-\alpha}\sqrt {2n}}\mathbb{P}{\left(
\sup_{0\leq t\leq2n'}{V^{k+1,n'}(t)}\,
\ge\,\frac{\delta'}4\,n^{1/4}\
\right)}\mathbb{P}{\left(\Delta^{(n)}_i=k\right)},$$ where $n'=n/m-k/2$. In order to apply the tail estimate of the previous subsection (Proposition \[pro:loosebound\]), we need $$y\;:=\;\frac{\delta'}4(n/n')^{1/4}\;\ge\;y_0.$$ Since $n'\le n/m$ this condition is satisfied as soon as $$\frac{\delta'}4\ m^{1/4}\ \ge y_0,
\quad\textrm{ that is }\quad
m\ge(4y_0/\delta')^4.$$ Then, $$\begin{aligned}
\mathbb{P}{\left(
\sup_{0\leq t\leq2n'}{V^{k+1,n'}(t)}
\,\ge\,\frac{\delta'}4\,n^{1/4}\ \right)}
\;\le\; \exp{\left(-\delta' m^{1/4}/4\right)}.\end{aligned}$$ The latter bound being independent of $k$ we obtain $$a_2\;\le\; 2\exp{\left(-\delta' m^{1/4}/4\right)}.$$
Let us now turn to $a_3$. Conditionally given that $\Delta^{(n)}_i=
k$, the maximum on the branch $\bar H_i^{(n)}$ is distributed as the maximum of a random walk with $k$ steps that are independent increments uniform on $\{-1,0,1\}$. Thus, applying Azuma’s inequality and the reflection principle, we have, for $k\le m^{-\alpha}\sqrt
{2n}$, $$\mathbb{P}{\left(
\left.\bar H_i^{(n)}
\,\ge\,\frac{\delta'}4\,n^{1/4}\ \right\vert
\ \Delta^{(n)}_i= k\right)}
\;\le\; 2\, \exp{\left(-\frac{{{\delta}'}^2}{32\sqrt 2}\ m^{\alpha}\right)}.$$ Again the latter bound is independent of $k$ so that $$a_3\
\le\ \exp{\left(-\frac{{{\delta}'}^2}{32\sqrt 2}\ m^{\alpha}\right)}.$$
Finally let us deal with $a_1$. For $n$ large enough, the convergence of $e^{(n)}$ to the normalised excursion $e$ entails $$\left\vert a_1\ -\ \mathbb{P}{\left(\Delta_{m,i} e\ge
m^{-\alpha}\right)}\right\vert\ \le\ \frac{\varepsilon }{2m}.$$ Let us thus consider, for $1\leq i\leq m$, the probability $ \pi_{m,i}=
\mathbb{P}{\left(\Delta_{m,i}\, e\ge m^{-\alpha}\right)}$, and restrict the choice of $\alpha$ to $0<\alpha<1/2$. The finite dimensional distribution of the value of $e$ at two points and the minimum between them was considered in Section \[sec:fidis\] (take $p=2$ in the continuum limit of Proposition \[pro:fidis\] or see [@SERLET]). For $i\neq 1,m$ this entails, $$\pi_{m,i}=\frac{m^{3/2}}{\pi(\tau_1\tau_2')^{3/2}}\int_{
\begin{subarray}{c}
x>0,y>0,\\
\beta>m^{-\alpha},\
y+x>\beta>\vert x-y\vert
\end{subarray} }
xy\beta\ e^{-y^2/2\tau_1\ -x^2/2\tau_2'}
e^{-m\beta^2/2}dx dy d\beta,$$ in which $$\tau_1= \frac{i-1}m, \qquad
\tau_2= \frac{i}m, \qquad\textrm{and}\quad \tau'_2=1-\tau_2.$$ Thus $$\begin{aligned}
\pi_{m,i}&\le&\frac{m^{3/2}}{\pi(\tau_1\tau_2')^{3/2}}\int_{
\begin{subarray}{c}
x>0,y>0,\\
\beta>m^{-\alpha}
\end{subarray} }
xy\beta\ e^{-y^2/2\tau_1\ -x^2/2\tau_2'}
e^{-m\beta^2/2}dx dy d\beta
\\
&\le&\frac{m^{1/2}}{\pi(\tau_1\tau_2')^{1/2}}\ e^{-m^{1-2\alpha}/2}\\
&\le&\frac{m^{3/2}}{\pi}\ e^{-m^{1-2\alpha}/2}.\end{aligned}$$ For the remaining two cases $i=1$ and $i=m$, we have with $\tau_1=\frac{m-1}m$, $$\pi_{m,1}=\pi_{m,m}=\sqrt{\frac{2m^3}{\pi \tau_1^3}}\int y^2
e^{-m\,y^2/2} e^{-y^2/2\tau_1}
1_{y>m^{-\alpha}} dy,$$ Now $x\rightarrow xe^{-x^2/a}$ is bounded by $\sqrt{\frac a{2e}}$, so that $$\begin{aligned}
\pi_{m,1}
&\le& \sqrt{\frac{2m}{\pi \tau_1^2 e}}\int u e^{-u^2/2}
\ 1_{u>m^{0.5-\alpha}}du,\\
&\le& m^{1/2}\ e^{-m^{1-2\alpha}/2}.\end{aligned}$$ Thus for all $i$, $\pi_{m,i}$ is bounded by an exponentially descreasing function of $m$.
The proof of the lemma is then concluded by summing the contribution of $a_1$, $a_2$ and $a_3$: for all $m$ there exists $n_0$ such that for all $n>n_0$, $$\sum_{i=1}^m\ p_{m,i}(n)\le \frac{\varepsilon }{2}
+C_\delta(m),$$ in which $C_\delta(m)$ is exponentially small in a power of $m$.
Taking $m$ and then $n$ large enough the tightness is proved, and together with Lemma \[pro:fidis\] this concludes the proof of Theorem \[thm:converge\].
Convergence of the profile {#sec:profile}
--------------------------
In view of the Skorohod representation theorem [@ROW II.86.1], we may assume the joint existence, on some probabilistic triple $(\Omega,A,\mathbb P)$, of a sequence of copies of $X^{(n)}$, and of a copy of $X$ (and we keep the same notation $X^{(n)}$ and $X$, as for the original), such that, for almost any $\omega \in \Omega$, ${\left(X^{(n)}_t(\omega)\right)}_{0\le t\le 1}$ converges to ${\left(X_t(\omega)\right)}_{0\le t\le 1}$ in the Skorohod topology of $D([0,1],\mathbb{R}^2)$. In this section we build copies of $F_n$ and $F$, such that, almost surely, $F_n$ converges to $F$.
First, the set $$\Omega_1=\left\{\omega\left\vert
\ t\longrightarrow X_t(\omega)\textrm{ is continuous}\right.\right\}$$ has probability 1, so that uniform convergence of ${\left(X^{(n)}_t(\omega)\right)}_{0\le t\le 1}$ to ${\left(X_t(\omega)\right)}_{0\le t\le 1}$ holds almost surely. Set $$\begin{aligned}
W^{(n)}_{\min}&=&\inf_{0\le s\le 1}\
W^{(n)}_s(\omega),\\
\delta_n( \omega)&=&\sup_{0\le s\le 1}\left\vert
W^{(n)}_s(\omega)-\hat W_s(\omega)\right\vert,\\
\Psi_n(x, \omega)&=&\int_0^1 \mathbf{1}_{\hat W^{(n)}_s(\omega)\le x}\ ds,\\
\tilde F_n(x)&=&\Psi_n{\left(W^{(n)}_{\min}+x, \omega\right)},\\
\Psi(x, \omega)&=&\int_0^1 \mathbf{1}_{\hat W_s(\omega)\le x}\ ds\\
&=&\mathcal J{\left((-\infty,x]\right)}.\end{aligned}$$ It follows from general results on superprocessus that $\Phi$, the distribution function of the random measure ISE, is almost surely continuous [@LeGallPerso]. Now from $$\Psi(x-\delta_n)\le\Psi_n(x)\le\Psi(x+\delta_n)$$ and the almost sure continuity of $\Psi$, it follows that the set $$\Omega_3=\{\omega\ \vert\ \forall x,
\ \ \lim_n\Psi_n(x, \omega)=\Psi(x, \omega)\textrm{ and }
x\rightarrow\Psi(x, \omega)
\textrm{ is continuous}\}$$ has probability 1. In $\Omega_3$, as we deal with increasing functions, *uniform* convergence of $\Psi_n$ to $\Psi$ holds true. Hence, on the set $$\Omega_4=\{\omega\in\Omega_3\ \vert\
\ \ \lim_nW^{(n)}_{\min}(\omega)=W_{\min}(\omega)\},$$ i.e. almost surely, uniform convergence of $\tilde F_n$ to $F$ holds true.
On the other hand, we explain below why $\tilde F_n$ is close to some copy of $F_n$, that we shall denote $F_n$ too. As in section \[sec:contourpair\], from $X^{(n)}$ one recovers a random uniform contour pair $(E^{(n)},V^{(n)})$, and a random uniform embedded tree $U_n\in\bar{\mathcal{U}}_n$. Now we can choose at random a well labelled tree $W_n$ in the conjugacy class of $U_n$ in such a way that Theorem \[thm:coupling\] holds for $(W_n,U_n)$. Theorem \[thm:coupling\] entails that $\widehat \lambda^{(n)}$ and $\widehat \Lambda^{(n)}$ have the same asymptotic behavior, and $F_n$ is just $\widehat \lambda^{(n)}$ suitably normalised. So we have now to establish the relation between $\widehat \Lambda^{(n)}$ and $\tilde F_n(x)$. First, set $$\widehat f^{(n)}_y=2n\ \tilde F_n{\left(y(8n/9)^{-1/4}, \omega\right)}.$$ As we have $$m_n=(8n/9)^{1/4}\ W^{(n)}_{\min},$$ $\widehat f^{(n)}_y$ is the number of visits of the contour traversal of $U_n$ to a node whose label is not larger than $m_n+y$, but the number of visits, by the contour traversal, of a given node, is exactly 1 plus the number of children of this node, so that $$\widehat \Lambda^{(n)}_y+\widehat \Lambda^{(n)}_{y-1}-1\le
\widehat f^{(n)}_y\le\widehat \Lambda^{(n)}_y+\widehat \Lambda^{(n)}_{y+1}.$$ Hence, due to Theorem \[thm:coupling\], $$2\widehat \lambda^{(n)}_{y-3}-1\le
\widehat f^{(n)}_y\le 2\widehat \lambda^{(n)}_{y+3},$$ or, equivalently, $$\frac n{n+1}\ \tilde F_n{\left(x-cn^{-1/4}, \omega\right)}\le
F_n{\left(x, \omega\right)}\le\frac n{n+1}\ \tilde F_n{\left(x+cn^{-1/4}, \omega\right)}
+\frac 1{2n+2},$$ $c$ being a constant. That is, on $\Omega_4$, uniform convergence of $F_n$ to $F$ holds true.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank David Aldous, Jean-François Le Gall and Balint Virag for stimulating discussions.
[99]{}
D. J. Aldous. Tree-based models for random distribution of mass. , 73(3-4):625–641, 1993.
N. Alon and J. H. Spencer. [*The probabilistic method.*]{} With an appendix by Paul Erdös. John Wiley & Sons, Inc., New York, 1992.
J. Ambjørn, B. Durhuus and T. Jónsson. Cambridge Monographs on Mathematical Physics, 1997.
J. Ambjørn and Y. Watabiki, Scaling in Quantum Gravity. , 445:129–144, 1995.
D. Arquès. Les hypercartes planaires sont des arbres très bien étiquetés. , 58:11–24, 1986.
C. Banderier, P. Flajolet, G. Schaeffer and M. Soria. Random Maps, Coalescing Saddles, Singularity Analysis, and Airy Phenomena. 19:194-246, 2001.
E. A. Bender and E. R. Canfield. Face Sizes of 3-polytopes. , 46:58–65, 1989.
E. A. Bender, K. J. Compton and L. B. Richmond. 0–1 laws for maps. , 14:215–237, 1999.
P. Billingsley. John Wiley & Sons, 1968.
E. Brezin, C. Itzykson, G. Parisi and J.-B. Zuber. Planar Diagrams. , 59:35–47, 1978.
B. Bollobás, Academic Press, 1985.
C. Borgs, J. Chayes, R. van der Hofstad and G. Slade. Mean-field lattice trees. On combinatorics and statistical mechanics. , 3(2-4):205–221, 1999.
R. Cori and B. Vauquelin. Planar maps are well labeled trees. , 33(5):1023–1042, 1981.
A. Dembo and O. Zeitouni. Large deviations for random distribution of mass. , 45–53, IMA Vol. Math. Appl., 76, Springer, New York, 1996.
E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions. , 193(1):69–104, 1998.
Z. Gao and N. C. Wormald. The Distribution of the Maximum Vertex Degree in Random Planar Maps. , 89:201–230, 2000.
D. Gross, T. Piran and S. Weinberg. World Scientific, 1992.
T. Hara and G. Slade. The incipient infinite cluster in high-dimensional percolations. , 4:48-55, 1998.
B. Jacquard. PhD thesis, [École Polytechnique]{}, 1997, Palaiseau.
S. Jain and S. D. Mathur. World sheet geometry and baby universes in 2-d quantum gravity. , 305:208–213, 1993.
W. D. Kaigh. An invariance principle for random walk conditioned by a late return to zero. , 4(1):115–121, 1976.
J.-F. Le Gall, [*Spatial branching processes, random snakes and partial differential equations.*]{} Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1999.
J.-F. Le Gall, personnal communication.
M. Marcus and G. Schaeffer. Une bijection simple pour les cartes orientables. Manuscript, 10pp, july 2001.
M. Marcus and B. Vauquelin. Un codage des cartes de genre quelconque. In B. Leclerc and J.Y. Thibon, ed., [*Séries formelles et combinatoire algébrique, 7ème colloque*]{}, 399–416. Université de Marne-la-Vallée, 1995.
J.-F. Marckert and A. Mokkadem. States spaces of the snake and of its tour – Convergence of the discrete snake. Manuscript, 2002.
V. V. Petrov. . Springer, 1975.
J. Pitman. Enumerations of Trees and Forests related to Branching Processes and Random Walks. Tech. Report 482, Dept. Statistics, University of California at Berkeley.
Almost all maps are asymmetric. , 63(1):1–7, 1995.
L. C. G. Rogers, D. Williams, [*Diffusions, Markov processes, and martingales. Vol. 1. Foundations.*]{} Second edition. Wiley Series in Probability and Mathematical Statistics, 1994.
G. Schaeffer. PhD. thesis, [ Université Bordeaux I]{}, 1998, Bordeaux.
L. Serlet. A large deviation principle for the Brownian snake, , 67(1):101–115, 1997.
R. Stanley. volume II, Cambridge Series in Advanced Mathematics, 1999.
W.T. Tutte. A census of planar maps. , 15:249–271, 1963.
Y. Watabiki. Construction of noncritical string field theory by transfert matrix formalism in dynamical triangulations. , 441:119–166, 1995.
[^1]: The infinite face is only apparently different from the others: imagine the map on the sphere.
|
---
abstract: '\[sec:abstract\] Characterizing the fundamental tradeoffs for maximizing *energy efficiency* (EE) versus *spectrum efficiency* (SE) is a key problem in wireless communication. In this paper, we address this problem for a point-to-point additive white Gaussian noise (AWGN) channel with the transmitter powered solely via *energy harvesting* from the environment. In addition, we assume a practical on-off transmitter model with *non-ideal* circuit power, i.e., when the transmitter is on, its consumed power is the sum of the transmit power and a constant circuit power. Under this setup, we study the optimal transmit power allocation to maximize the average throughput over a finite horizon, subject to the time-varying energy constraint and the non-ideal circuit power consumption. First, we consider the *off-line* optimization under the assumption that the energy arrival time and amount are *a priori* known at the transmitter. Although this problem is non-convex due to the non-ideal circuit power, we show an efficient optimal solution that in general corresponds to a *two-phase* transmission: the first phase with an *EE-maximizing* on-off power allocation, and the second phase with a *SE-maximizing* power allocation that is non-decreasing over time, thus revealing an interesting result that both the EE and SE optimizations are unified in an energy harvesting communication system. We then extend the optimal off-line algorithm to the case with multiple parallel AWGN channels, based on the principle of *nested optimization*. Finally, inspired by the off-line optimal solution, we propose a new *online* algorithm under the practical setup with only the past and present energy state information (ESI) known at the transmitter.'
author:
- 'Jie Xu and Rui Zhang [^1] [^2]'
title: 'Throughput Optimal Policies for Energy Harvesting Wireless Transmitters with Non-Ideal Circuit Power'
---
Energy harvesting, power control, energy efficiency, spectrum efficiency, circuit power.
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
Introduction {#sec:introduction}
============
or energy efficient wireless communication has recently drawn significant attention due to the growing concerns about the operator’s cost as well as the global environmental cost of using fossil fuel based energy to power cellular infrastructures [@Chen:NetworkES; @Han:GreenRadio; @Niu:CellZooming]. To achieve the optimal energy usage efficiency for cellular networks, various innovative “green” techniques across different layers of communication protocol stacks have been proposed [@Bhargava:GreenCellular; @YChenComMag]. Among others, how to maximize the bits-per-Joule [*energy efficiency*]{} (EE) for the point-to-point wireless link has received a great deal of interest recently [@Energy_EfficientPacketTransmission; @Miao; @fractionalprogramming].
=1 ![Tradeoff between EE and SE for the ideal circuit power case of $\alpha = 0$ (the left sub-figure) and the non-ideal circuit power case of $\alpha > 0$ (the right sub-figure).[]{data-label="fig:EESE"}](EESE.eps "fig:"){width="7cm"}
Besides maximizing EE, another key design objective in wireless communication is to maximize the [*spectrum efficiency*]{} (SE) or the number of transmitted bits-per-second-per-Hz (bps/Hz), due to the explosive growth of wireless devices and applications that require high data rates. In order to design wireless communication systems both energy and spectrum efficiently, the fundamental EE-SE relationship needs to be examined carefully. For the simple additive white Gaussian noise (AWGN) channel with bandwidth $W$ and noise power spectral density $N_0$, by applying the Shannon’s capacity formula, the SE and EE are expressed as $\xi_{\rm SE} =
\log_2(1+\frac{P}{WN_0})$ and $\xi_{\rm EE} =
W\log_2(1+\frac{P}{WN_0})/P$, respectively, with $P$ denoting the transmit power. It thus follows that the optimal EE-SE tradeoff is characterized by $\xi_{\rm EE}=\frac{\xi_{\rm SE}}{(2^{\xi_{\rm
SE}}-1)N_0}$, where $\xi_{\rm EE}$ is a monotonically decreasing function of $\xi_{\rm SE}$, as shown in the left sub-figure of Fig. \[fig:EESE\]. In this case, any SE increment will inevitably result in a decrement in EE. However, in practical wireless transmitters, besides the direct transmit power $P$, there also exists [*non-ideal*]{} circuit power consumed when $P>0$, which accounts for the power consumptions at e.g. the AC/DC converter and the analog radio frequency (RF) amplifier, and amounts to a significant part of the total consumed power at the transmitter. Moreover, when there is no data transmission, i.e., $P=0$, the transmitter can turn into a [*micro-sleep*]{} mode [@Blume:ES], by switching off the power amplifier to reduce the circuit power consumption. For the ease of description, in this paper the transmitter status with $P>0$ and $P=0$ are referred to as the *on* and *off* modes, respectively. Denote the non-ideal circuit power during an “on” mode as $\alpha\geq 0$ in Watt, the efficiency of the RF chain as $0<\eta\leq 1$, and the power consumed during an “off” mode as $\beta\geq 0$ in Watt. A practical power consumption model for the wireless transmitter is given by [@Kim] $$\begin{aligned}
\label{eq1}
P_{\rm{total}}=\left\{\begin{array}{ll} \frac{P}{\eta} + \alpha, &
P>0 \\ \beta, & P=0, \end{array} \right.\end{aligned}$$ where $P_{\rm{total}}$ denotes the total power consumed at the transmitter. In practice, $\beta$ is generally much smaller as compared to $\alpha$ and thus can be ignored for simplicity [@YChenComMag; @Miao; @fractionalprogramming]. In this paper, we assume $\beta = 0$. Therefore, in (\[eq1\]) without loss of generality we can further assume $\eta = 1$ since $\eta$ is only a scaling constant.[[^3]]{} With the above simplifications, the EE can be re-expressed as $\xi_{\rm EE} =
W\log_2(1+\frac{P}{WN_0})/(P+\alpha)$ for $P>0$ and the resulting new EE-SE tradeoff is shown in the right sub-figure of Fig. \[fig:EESE\] for a given $\alpha>0$, from which it is observed that the non-ideal circuit power drastically changes the behavior of the EE-SE tradeoff as compared to the ideal case of $\alpha=0$.
Recently, a new design paradigm for achieving green wireless communication has drawn a great deal of attention, in which wireless terminals are powered primarily or even solely by harvesting the energy from environmental sources such as solar and wind, thereby reducing substantially the energy cost in traditional wireless systems [@Bhargava:GreenCellular; @Kansal]. With the embedded energy harvesting device and rechargeable battery, wireless transmitters can replenish energy from the environment without the need of replacing battery or drawing power from the main grid. Thus, communication utilizing energy harvesting nodes can promisingly achieve a jointly spectrum and energy efficiency maximization goal. However, there are new challenges in designing energy harvesting powered wireless communication, which are not present in traditional systems. For example, the intermittent nature of most practical energy harvesting sources causes random power availability at the transmitter, due to which a new type of transmitter-side power constraint, namely [*energy harvesting constraint*]{}, is imposed, i.e., the energy accumulatively consumed up to any time cannot exceed that accumulatively harvested. As a result, existing EE-SE tradeoffs (cf. Fig. \[fig:EESE\]) revealed for conventional wireless systems assuming a given constant power supply are not directly applicable to an energy harvesting system, with or without the non-ideal circuit power. It is worth noting that some prior work in the literature has investigated the throughput-optimal power control policies for the energy harvesting wireless transmitter assuming an ideal circuit-power model (i.e., $\alpha=0, \eta=1$, and $\beta=0$ in (\[eq1\])), in which useful structural properties of the optimal solution were obtained (see e.g. [@Yang; @Zhang; @Yener] and references therein). However, there is very limited work on studying the effects of the non-ideal circuit power with $\alpha>0$ on the throughput-optimal power allocation for energy harvesting communication systems. To our best knowledge, only [@Bai2011] has proposed a calculus-based approach to address this problem; however, it does not reveal the structure of the optimal solution. Motivated by the known result that the non-ideal circuit power modifies the EE-SE tradeoff considerably in the conventional case with constant power supply as shown in Fig. \[fig:EESE\], we expect that it should also play an important role in the EE-SE tradeoff characterization under the new setup with random power supply due to energy harvesting, which motivates our work.
In this paper, we study the throughput maximization problem for a point-to-point AWGN channel with an energy harvesting powered transmitter over a finite horizon. For the purpose of exposition, we assume that the receiver has a constant power supply (e.g. battery). We also assume that at the transmitter, the renewable energy arrives at a discrete set of time instants with variable energy amount. Under this setup, we investigate the effects of the non-ideal circuit power with $\alpha>0$ on the throughput-optimal power allocation as well as the resulting new EE-SE tradeoff. The main contributions of this paper are summarized as follows.
- First, we consider the *off-line* optimization under the assumption that the energy arrival time and amount for harvesting are *a priori* known at the transmitter. We show that the optimal power allocation to maximize the average throughput under this setup is a [*non-convex*]{} optimization problem, due to the non-ideal circuit power. Nevertheless, we derive an efficient optimal solution for this problem, which is shown to correspond to a novel *two-phase* transmission structure: the first phase with an *EE-maximizing* on-off power allocation, and the second phase with a *SE-maximizing* power allocation that is non-decreasing over time. Thus, we reveal an interesting result that both the EE and SE optimizations are unified in an energy harvesting powered wireless system.
- We then extend the optimal off-line policy for the single-channel case to the general case with multiple parallel AWGN channels, subject to a total energy harvesting power constraint. Using tools from [*nested optimization*]{}, we transform this problem with multi-dimensional (vector) power optimization to an equivalent one with only one-dimensional (scalar) power optimization, which can then be efficiently solved by the algorithm derived for the single-channel case.
- Furthermore, inspired by the off-line optimal solution, we propose a heuristic *online* algorithm under the practical setup where only the causal (past and present) energy state information (ESI) for harvesting is assumed to be known at the transmitter. It is shown by simulations that the proposed online algorithm achieves a small performance gap from the throughput upper bound by the optimal off-line solution, and also outperforms other heuristically designed online algorithms.
The rest of this paper is organized as follows. Section \[sec:system model\] introduces the system model and presents the problem formulation. Section \[sec:offline\] derives the optimal off-line power allocation policy for the single-channel case. Section \[sec:multi-channel\] extends the result to the multi-channel case based on the nested optimization. Section \[sec:online\] presents the proposed online algorithm and Section \[sec:simulation\] evaluates its throughput performance by simulations. Finally, Section \[sec:conclusion\] concludes the paper.
System Model and Problem Formulation {#sec:system model}
====================================
In this paper, we consider the point-to-point transmission over an AWGN channel with constant channel and coherent detection at the receiver. The transmitter is assumed to replenish energy from an energy harvesting device that collects energy over time from a renewable source (e.g. solar or wind). We consider the block-based energy scheduling with each block spanning over $T$ seconds (secs). We assume that the renewable energy arrives during each block at $N-1$ time instants given by $0<t_{1}<\cdots< t_{N-1}<T$, and the energy values collected at these time instants are denoted by $E_1,\ldots,E_{N-1}$, respectively. In general, $N\geq 1$, $t_i$, and $E_i>0$, $i=1,\ldots,N-1$, are modeled by an appropriate random process for the given energy source. For convenience, we assume $t_0
= 0$ and denote $E_0$ as the initial energy stored in the energy storage device at the beginning time of each block. For the purpose of exposition, we assume that the energy storage device has an infinite capacity in this paper. Moreover, we refer to the time interval between two consecutive energy arrivals as an [*epoch*]{}, and denote the length of the $i$th epoch as $L_i = t_{i} - t_{i-1},
i=1,\ldots,N$; for convenience, we denote $t_N = T$.
Suppose that the transmit power over time in each block is denoted by $P(t)\geq 0, t\in (0,T]$. Assume that the maximum transmission rate that can be reliably decoded at the receiver at any time $t$ is a function of $P(t)$, given by $C(t) = R(P(t))$, which satisfies the following properties:
1. $R(P(t)) \ge 0$, $\forall P(t)\geq 0$, and $R(0)=0$;
2. $R(P(t))$ is a strictly concave function over $P(t)\geq 0$;
3. $R(P(t))$ is a monotonically increasing function over $P(t)\geq 0$.
For example, if adaptive modulation and coding (AMC) is applied at the transmitter, then the achievable rate $C(t)$ is denoted by [@GoldsmithBook] $$\label{sys1}
\begin{array}{l}
\displaystyle R(P(t)) = W\log_2\left(1+\frac{hP(t)}{\Gamma WN_0}\right)
\end{array}$$ in bits-per-sec (bps), where $\Gamma$ accounts for the gap from the channel capacity due to a practical coding and modulation scheme used; $h>0$ denotes the constant channel power gain.
As discussed in Section \[sec:introduction\], we assume an on-off transmitter power model given in (\[eq1\]) with $\beta = 0$ and $\eta =1$; thus, we rewrite (\[eq1\]) as $$\label{eq102}
\begin{array}{l}{P_{{\rm{total}}}}(t) = \left\{ {\begin{array}{*{20}{c}}
\displaystyle {{P(t)} + {\alpha},}\\
{{0},}
\end{array}} \right.\begin{array}{*{20}{l}}
{P(t) > 0}\\
{P(t) = 0}.
\end{array}\end{array}$$
Since the accumulatively consumed energy up to any time at the transmitter cannot exceed the energy accumulatively harvested, the energy harvesting constraint on the total consumed power is given by $$\label{eq103}\begin{array}{l}\displaystyle \int_0^{t_i} {{P_{{\rm{total}}}}\left( t \right){\rm{d}}t} \le \sum\limits_{j = 0}^{i - 1} {{E_j}}, ~ i = 1,\ldots,N. \end{array}$$ Thus, the throughput maximization problem over a finite horizon $T$ can be formulated as follows. $$\begin{aligned}
\label{eq105}
\mathop{\max }\limits _{P(t)\geq 0} ~ &
\int_0^T {R\left( P(t) \right)} {\rm{d}}t \nonumber \\
{\rm s.t.} ~ & \int_0^{t_i} {{P_{{\rm{total}}}}\left( t \right)}
{\rm{d}}t \le \sum\limits_{j = 0}^{i - 1} {{E_j}}, ~i = 1,\ldots,N.\end{aligned}$$
The optimal online solution for the above problem with the causal ESI, i.e., for any given $t$, only $E_i$’s with $t_i\leq t$ are known at the transmitter, can be numerically solved by the technique of dynamic programming similar to [@Zhang]. However, such a solution is of high computational complexity due to “the curse of dimensionality” for dynamic programming. In addition, the resulting solution will not provide any insight to the structure of the optimal power allocation for an energy harvesting transmitter. Therefore, in this paper, we take an alternative approach by first solving the off-line optimization for (\[eq105\]), assuming that all the energy arrival time $t_i$’s and amount $E_i$’s are [*a priori*]{} known at the transmitter in each block transmission, and then based on the structure of the off-line optimal solution, devising online algorithms for the practical setup with only causal ESI known at the transmitter.
For the off-line optimization of (\[eq105\]), it is easy to see that the objective function is concave; however, the constraint is non-convex in general since $P_{\rm{total}}(t)$ in (\[eq102\]) is a concave function of $P(t)$ if $\alpha>0$. As a result, the problem is in general non-convex and thus cannot be solved by standard convex optimization techniques. In the next section, we will propose an efficient solution for this problem by exploiting its special structure.
It is worth noting that for the off-line optimization, (\[eq105\]) can be shown to be convex if $\alpha=0$. In this case, similar problems to (\[eq105\]) have been studied in the literature [@Yang; @Zhang], in which the throughput-optimal power allocation $P(t)$ was shown to follow a non-decreasing piecewise-constant (staircase) function over $t$. This power allocation can be interpreted as maximizing the SE of the point-to-point AWGN channel subject to the new energy harvesting power constraint. As will be shown later in this paper, the non-ideal circuit power with $\alpha>0$ will change the optimal power allocation for this problem considerably.
Off-Line Optimization {#sec:offline}
=====================
In this section, we solve the off-line optimization problem in (\[eq105\]) with the non-ideal circuit power, i.e., $\alpha>0$.
Reformulated Problem
--------------------
First, we give the following lemma.[[^4]]{}
\[Lemma:1\] During any $i$th epoch $(t_{i-1}, t_{i}]$, $i=1,\ldots,N$, the optimal solution for (\[eq105\]) is given by $P(t)=P_i>0$ for the portion of time ${\mathcal T}_i^{\rm{on}} \subseteq (t_{i-1},
t_{i}]$, and $P(t)=0$ for the remaining time ${\mathcal
T}_i^{\rm{off}} \subseteq (t_{i-1}, t_{i}]$, where ${\mathcal
T}_i^{\rm{on}} \cap {\mathcal T}_i^{\rm{off}} = \phi$ and ${\mathcal
T}_i^{\rm{on}} \cup {\mathcal T}_i^{\rm{off}} = (t_{i-1}, t_{i}]$.
See Appendix \[appendix:proof Lemma 1\].
According to Lemma \[Lemma:1\] and by denoting the constant transmit power $P_i>0$ for the “on” period with length $0 \le
l_i^{{\rm{on}}} \le L_i$ in the $i$th epoch, (\[eq105\]) can be reformulated as $$\begin{aligned}
\label{eq4}
\mathop {\max }\limits_{\{P_i\},~\{l_i^{{\rm{on}}}\}} ~& \sum \limits _{i=1}^{N}l_i^{{\rm{on}}}R(P_i) \nonumber \\
{\rm{s.t.}} ~ & P_i>0, ~ i=1,\ldots, N \nonumber \\
& 0\le l_i^{{\rm{on}}} \le L_i, ~ i=1,\ldots, N \nonumber \\
& \sum \limits_{j=1}^{i}{(P_j + \alpha)l_j^{{\rm{on}}}} \le \sum
\limits _{j=0}^{i-1} E_j, ~ i=1,\ldots, N.\end{aligned}$$ However, the above problem is still non-convex due to the coupling between $P_i$’s and $l_i^{\rm{on}}$’s. In the following, we first solve this problem for the special case of $N=1$ and then generalize the solution to the case with $N\geq1$.
Single-Epoch Case with $N=1$
----------------------------
In the single-epoch case with $N=1$, the problem in (\[eq4\]) is reduced to $$\begin{aligned}
\label{eq105oneepoch}
\mathop{\max }\limits_{l_1^{\rm{on}},P_1} ~&
{l_1^{\rm{on}}}R(P_1) \nonumber \\
{\rm s.t.} ~& P_1>0 \nonumber \\
& 0 \le {l_1^{\rm{on}}} \le T \nonumber \\
& {l_1^{\rm{on}}}(P_1+\alpha) \le E_0.\end{aligned}$$ The solution of the above problem is given in the following proposition.
\[Proposition:1\] The optimal solution $P_1^*$ and $l_1^{\rm{on}*}$ for (\[eq105oneepoch\]) is expressed as $$\label{SinEpoch:2}
\displaystyle P_1^* = \max\left(P_{ee},\frac{E_0}{T}-\alpha\right)$$ $$\label{SinEpoch:3}
\displaystyle l_1^{\rm{on}*} = \frac{E_0}{P_1^* + \alpha}$$ where $P_{ee}$ is given by $$\label{optimalEE}
\begin{array}{l}\displaystyle
P_{ee}=\mathop{\arg\max }\limits_{P_1> 0} \frac{R(P_1)}{P_1+\alpha}.
\end{array}$$
See Appendix \[appendix:proof Proposition 1\].
It is worth noting that $P_{ee}$ given in (\[optimalEE\]) is the optimal power allocation that maximizes the EE of the AWGN channel under the non-ideal circuit power model as shown in [@Miao]. From Proposition \[Proposition:1\], it follows that if $P_{ee}
> \frac{E_0}{T}-\alpha$, we have $P_1^* = P_{ee}$, $l_1^{\rm{on}*} < T$ and $l_1^{\rm{off}*} = T- l_1^{\rm{on}*}
> 0$, which corresponds to an on-off transmission. However, if $P_{ee}
\leq \frac{E_0}{T}-\alpha$, we have $P_1^* =\frac{E_0}{T}-\alpha$, $l_1^{\rm{on}*} =T$ and $l_1^{\rm{off}*} = 0$, which corresponds to a continuous transmission. We will see in the next subsection that the EE-maximizing power allocation $P_{ee}$ plays an important role in the general case with $N\geq 1$. Also note that the right-hand side (RHS) of (\[optimalEE\]) is a quasi-concave function of $P_1$ since it is concave-over-linear [@Boydbook]; thus, $P_{ee}$ can be efficiently obtained by a simple bisection search [@Boydbook].
Multi-Epoch Case with $N\geq 1$
-------------------------------
Inspired by the solution for the single-epoch case, we derive the optimal solution for (\[eq4\]) in the general case with $N\geq 1$, as given by the following theorem.
\[theorem:1\] The optimal solution of (\[eq4\]), denoted by $[P_1^*,
\ldots, P_N^*]$ and $[l_1^{{\rm{on}}*},\ldots,l_N^{{\rm{on}}*}]$, is obtained as follows. Denoting $$\begin{aligned}
i_{ee,0} &= 0,\nonumber\\
i_{ee,j} &= \min \bigg\{i \bigg| \frac{\sum \nolimits
_{k=i_{ee,j-1}}^{i-1} E_k}{\sum \nolimits _{k=i_{ee,j-1}+1}^{i}L_k} - \alpha \le P_{ee},\bigg.\nonumber\\
& ~~~~~~~~~~~~~~\bigg. i=i_{ee,j-1}+1,\ldots,N \bigg \}, j\ge 1,\nonumber\\
J &= \arg \max\limits_{j\ge 0}i_{ee,j},\nonumber\\
i_{ee} &= i_{ee,J}, \label{SinEpoch:12}\end{aligned}$$ the optimal transmit power for epochs $1,\ldots,i_{ee}$ is given by $$\label{optimal01}
\begin{array}{l}
\displaystyle P_i^* = P_{ee}, ~ i = 1,\ldots, i_{ee}
\end{array}$$ and the optimal on-period ${l_i^{{\rm{on}}*}}, i = 1,\ldots,i_{ee}$, is any set of non-negative values satisfying $$\label{optimal02}
\begin{array}{l}
\displaystyle \left( {{P_{ee}} + \alpha } \right)\sum\nolimits_{i =
1}^{{i_{ee}}} {l_i^{{\rm{on}}*}} = \sum\nolimits_{i = 0}^{{i_{ee}}
- 1} {{E_i}}
\end{array}$$ $$\label{optimal03}
\begin{array}{l}
\displaystyle \left( {{P_{ee}} + \alpha } \right)\sum\nolimits_{i =
1}^{j} {l_i^{{\rm{on}}*}} \leq \sum\nolimits_{i = 0}^{j - 1}
{{E_i}}, ~ j=1,\ldots,i_{ee}.
\end{array}$$ Moreover, for epochs $i_{ee} + 1,\ldots,N$, the optimal solution is given by $$\label{optimal104}
\begin{array}{l}
{l_i^{{\rm{on}}*}} = L_i, ~ i = i_{ee}+1,\ldots, N
\end{array}$$ $$\label{optimal05}
\begin{array}{l}
\displaystyle P_i^* = \frac{\sum \nolimits _{k={n_{i-1}}}^{n_i-1}
E_k}{\sum \nolimits _{k={n_{i-1}}+1}^{n_i}L_k} - \alpha, ~ i =
i_{ee}+1,\ldots, N
\end{array}$$ where $$\begin{aligned}
\label{optimal06}
n_{i_{ee}} & = i_{ee},\nonumber\\
n_i & = \arg \mathop {\min }\limits_{j: ~{n_{i-1}+1}
\leq j \leq N} \left\{ {\frac{{\sum\nolimits_{k = {n_{i-1}} }^{j -
1} {{E_k}} }}{{\sum\nolimits_{k = {n_{i-1}+1}}^j {{L_k}} }} - \alpha
} \right\}, \nonumber\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~i = i_{ee}+1,\ldots, N.\end{aligned}$$
See Appendix \[appendix:proof theorem 1\].
It is interesting to take note that the optimal transmission policy given in Theorem \[theorem:1\] has a [*two-phase*]{} structure, which is explained as follows in more details.
- $0<t \leq t_{i_{ee}}$: In the first phase, the optimal transmission is an on-off one with a constant power $P_{ee}$ for all the on-periods. Note that $P_{ee}$ is the EE-maximizing power allocation given in (\[optimalEE\]). Also note that the optimal on-periods ${l_i^{{\rm{on}}*}}, i = 1,\ldots,i_{ee}$, may not be unique provided that they satisfy the conditions given in (\[optimal02\]) and (\[optimal03\]). Without loss of generality, we assume that in each epoch, the transmitter chooses to be on initially with power $P_{ee}$ provided that its stored energy is not used up, i.e., ${l_1^{{\rm{on}}*}}=\min(E_0/(P_{ee}+\alpha),L_1)$, ${l_2^{{\rm{on}}*}}=\min\left((E_1+E_0-(P_{ee}+\alpha)l_1^{{\rm{on}}*})/(P_{ee}+\alpha),L_2\right)$, and so on.
- $t_{i_{ee}}<t\leq T$: In the second phase, a continuous transmission is optimal. Since ${l_i^{{\rm{on}}*}} = L_i, ~ i =
i_{ee}+1,\ldots, N$, the problem in (\[eq4\]) for $i = i_{ee}+1,\ldots,
N$, is reduced to $$\begin{aligned}
\mathop {\max }\limits_{\{P_i\}} ~& \sum \limits _{i=i_{ee}+1}^{N}L_iR(P_i) \nonumber \\
{\rm{s.t.}} ~ & P_i>0, ~ i=i_{ee}+1,\ldots, N \nonumber \\
& \sum \limits_{j=i_{ee}+1}^{i}L_jP_j \le \sum \limits
_{j=i_{ee}}^{i-1} E_j- \sum \limits_{j=i_{ee}+1}^{i}L_j\alpha, \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~
i=i_{ee}+1,\ldots, N.\end{aligned}$$ The optimal solution for the above problem has been shown in [@Yang; @Zhang] to follow a non-decreasing piecewise-constant (staircase) function, which is given in (\[optimal05\]). It is worth noting that the staircase power allocation achieves the maximum SE for an equivalent AWGN channel subject to a sequence of energy harvesting power constraints (modified to take into account the circuit power $\alpha$) for $i = i_{ee}+1,\ldots, N$.
From the above discussion, it is revealed that for the throughput maximization in an energy harvesting transmission system subject to the non-ideal circuit power, the optimal transmission unifies both the EE and SE maximization policies independently developed in [@Miao] and [@Yang; @Zhang], respectively. To summarize, one algorithm for solving the problem in (\[eq4\]) for the general case of $N\geq 1$ is given in Table I.
[|p[3.2in]{}|]{} **Algorithm**\
1) Calculate $P_{ee}$ and obtain $i_{ee}$ as [ $$\begin{aligned}
i_{ee,0} &= 0,\nonumber\\
i_{ee,j} &= \min \bigg\{i \bigg| \frac{\sum \nolimits
_{k=i_{ee,j-1}}^{i-1} E_k}{\sum \nolimits _{k=i_{ee,j-1}+1}^{i}L_k} - \alpha \le P_{ee},\bigg.\nonumber\\&\bigg.~~~~~~~~~~~~~~~~~~
i=i_{ee,j-1}+1,\ldots,N \bigg \}, j\ge 1,\nonumber\\
J &= \arg \max\limits_{j\ge 0}i_{ee,j},~~~i_{ee} = i_{ee,J}.\nonumber\end{aligned}$$ ]{} 2) For the first $i_{ee}$ epochs, set $$P_i^* = P_{ee}, i=1, \ldots, i_{ee},$$ $$l_{1}^{\rm on*} = \min\left(\frac{{E_0}}{P_{ee}+\alpha},L_1\right)$$ $$l_{i}^{\rm on*} = \min\left(\frac{\sum\nolimits_{j=1}^{i-1}{E_j}}{P_{ee}+\alpha} - \sum\nolimits_{j=1}^{i-1}{l_{j}^{\rm on*}},L_i\right), i=2, \ldots, i_{ee}.$$
3\) If $i_{ee} = N$, algorithm ends; otherwise, set $$l_{i}^{\rm on*} = L_i, i=i_{ee}+1, \ldots, N.$$
4\) Reset $$\begin{array}{l}
\displaystyle E_0' \gets 0, T' \gets T- \sum \nolimits_{j=1}^{i_{ee}} L_j, N' \gets N -i_{ee},\\
L_j' \gets L_{j+i_{ee}}, E_{j}' \gets E_{j+i_{ee}}, j=1,\ldots,N'.\\
\end{array}$$ 5) Determine $$i_{\min} = \arg \mathop {\min }\limits_{j} \left\{ {\frac{{\sum\nolimits_{k = 0 }^{j - 1} {{E_k'}} }}{{\sum\nolimits_{k = 0}^j {{L_k'}} }} - \alpha } \right\}$$ $$P_t = \left\{ {\frac{{\sum\nolimits_{k = 0 }^{i_{\min} - 1} {{E_k'}} }}{{\sum\nolimits_{k = 0}^{i_{\min}} {{L_k'}} }} - \alpha } \right\},$$ and set transmit power as $P_t$ in the next $i_{\min}$ epochs.\
6) If $i_{\min} = N'$, algorithm ends; otherwise, reset the parameters as follows, and go to 5). $$\begin{array}{l}
\displaystyle E_0' \gets 0, T' \gets T'- \sum \nolimits_{j=1}^{i_{\min}} L_j', N' \gets N' -i_{\min},\\
L_j' \gets L_{j+i_{\min}}', E_{j}' \gets E_{j+i_{\min}}', j=1,\ldots,N'.\\
\end{array}$$\
\[remark:3.1\] We discuss some implementation issues on energy harvesting communication systems with the optimal transmit power allocation given in Theorem \[theorem:1\]. It is worth noting that in practical wireless systems, the duration of a communication block is usually on the order of millisecond, while the energy harvesting process evolves at a much slower speed, e.g., solar and wind power typically remains constant over windows of seconds. As a result, each epoch between any two consecutive energy arrivals in our model (during which the optimal power policy in Theorem \[theorem:1\] allocates a constant power) can be assumed to be sufficiently long, thus containing many communication blocks. In each communication block, pilot signals can be transmitted to help estimate the signal power at the receiver, which may change from one epoch to another due to transmit power adaptation; thus, the transmission rate in (\[sys1\]) is practically achievable with AMC at the transmitter and coherent detection at the receiver.
\[example:1\] To illustrate the optimal two-phase transmission given in Theorem \[theorem:1\], we consider an example of a band-limited AWGN channel with bandwidth $W = 1$MHz and the noise power spectral density $N_0
= 10^{-16}$Watts-per-Hz (W/Hz). We assume that the attenuation power loss from the transmitter to the receiver is $h=-$80dB. Considering the channel capacity with $\Gamma=1$, we thus have $R(P) =
W\log_2(1+\frac{Ph}{\Gamma N_0W}) = \log_2(1+100P)$Mbps. It is assumed that the energy arrives at time instants $[0, 4, 6, 11, 14, 16, 18]$sec, and the corresponding energy values are \[0.5, 0.5, 0.5, 1, 0.5, 0.75, 0.5\]Joule, as shown in Fig.\[fig:offline\]. It is also assumed that $T=20$secs and the circuit power is $\alpha = $115.9mW. Under this setup, we compute $P_{ee} = 79.2$mW. In Fig. \[fig:offline\], we compare the optimal allocation of the total consumed transmitter power by the algorithm in Table I with that obtained by the algorithm given in [@Yang; @Zhang]. Note that the algorithm in [@Yang] or [@Zhang] solves (\[eq4\]) in the special case with the ideal circuit power $\alpha=0$. Here, we apply this algorithm to obtain a suboptimal power allocation with $\alpha>0$, by assuming that the transmitter is always on, i.e., ${l_i^{{\rm{on}}}} = L_i, ~ i = 1,\ldots, N$. As observed in Fig. \[fig:offline\](a), the optimal power allocation has a two-phase structure, i.e., an on-off transmission with transmit power $P_{ee}$ when $0< t\le t_3$ followed by a continuous transmission with non-decreasing staircase power allocation when $t_3< t\le T$, which is in accordance with Theorem \[theorem:1\]. In contrast, as observed in Fig. \[fig:offline\](b), the suboptimal power allocation by the algorithm in [@Yang] or [@Zhang] with the transmitter always on results in a continuous transmission with non-decreasing staircase power allocation during the entire block i.e. $0< t\leq T$. In addition, it can be shown that the proposed optimal solution achieves the total throughput 63.14Mbits, while the suboptimal solution achieves only 55.80Mbits, over $T=20$secs.
=1 ![Power allocation by off-line policies: (a) the optimal off-line policy; and (b) the off-line policy in [@Yang; @Zhang].[]{data-label="fig:offline"}](offline.eps "fig:"){width="8cm"}
Multi-Channel Optimization {#sec:multi-channel}
==========================
In this section, we extend the optimal off-line power allocation for the single-channel case to the general case with multiple parallel AWGN channels subject to a total energy harvesting power constraint and the non-ideal circuit power consumption at the transmitter. The multi-channel setup is applicable when the communication channel is decomposable into orthogonal channels by joint transmitter and receiver signal processing such as OFDM (orthogonal frequency division multiplexing) and/or MIMO (multiple-input multiple-output).
Without loss of generality, we assume a power vector ${\bf Q}(t) =
[Q_1(t),\ldots, Q_K(t)]\succeq 0, t\in(0,T]$, with each element denoting the power allocation over time in one of a total $K$ parallel AWGN channels, where ${\bf Q}(t) \succeq 0$ denotes that ${\bf Q}(t)$ is elementwise no smaller than zero. We also assume a sum-throughput over the $K$ channels, denoted by $C(t) = R({\bf Q}(t))$, which satisfies
1. $R({\bf Q}(t)) \ge 0, \forall {\bf Q}(t)\succeq 0$, and $R({\bf{0}})=0$;
2. $R({\bf Q}(t))$ is a strictly joint concave function over ${\bf
Q}(t)\succeq 0$;
3. $R({\bf Q}(t))$ is a monotonically increasing function with respect to each argument in ${\bf
Q}(t)$ i.e. $Q_k(t)\geq 0, k=1,\ldots,K$.
An example of the above multi-channel sum-throughput is the sum-rate over $K$ parallel AWGN channels achieved by joint AMC, which is given by $$\label{MultiChannel:1}
\begin{array}{l}
\displaystyle R({\bf Q}(t)) = W \sum \limits_{k=1}^K
\log_2\left(1+\frac{h_k Q_k(t)}{\Gamma WN_0}\right),
\end{array}$$ where $h_k\geq 0$ denotes the channel power gain of the $k$th channel. Similar to (\[eq102\]) in the single-channel case, by taking into account the non-ideal circuit power $\alpha$, the total power consumed at the transmitter for the multi-channel case is modeled by $$\begin{aligned}
\label{Gene:1}
Q_{\rm{total}}(t) = \left\{ \begin{array}{ll}
\sum \limits_{k=1}^{K} Q_k(t) + \alpha, & \sum \limits _{k=1}^{K} Q_k(t) > 0 \\
0 & \sum \limits _{k=1}^{K} Q_k(t) = 0. \end{array} \right.\end{aligned}$$ Then the throughput maximization problem over a finite horizon $T$ in the multi-channel case is formulated as $$\begin{aligned}
\label{Gene:2}
\mathop {\max }\limits_{{\bf{Q}}(t)\succeq 0} ~&
\int_0^T R({\bf{Q}}(t)) {\rm{d}}t \nonumber \\
{\rm s.t.} ~& \int_0^{t_i} {{Q_{{\rm{total}}}}\left( t \right)}
{\rm{d}}t \le \sum\limits_{j = 0}^{i - 1} {{E_j}}, ~i=1,\ldots,N.\end{aligned}$$ Similar to (\[eq105\]), the above problem is non-convex with $\alpha>0$ and thus cannot be solved by standard convex optimization techniques. In the following, we will apply the principle of [*nested optimization*]{} to convert this problem with multi-dimensional (vector) power optimization to an equivalent problem with only one-dimensional (scalar) power optimization, which is then optimally solvable by the algorithm in Table I for the single-channel case.
To apply the nested optimization, we first introduce an auxiliary variable $P(t) = \sum \limits_{k=1}^K Q_k(t)$, and rewrite the objective function of (\[Gene:2\]) equivalently as $$\begin{aligned}
\label{Gene:2:object:1}
& \mathop {\max }\limits_{ P \left( t \right)\geq 0} \max
\limits_{{\bf{Q}}(t):{Q}_k(t)\geq 0, \forall k, \sum \limits_{k=1}^K
Q_k(t) \leq P(t)} \int_0^T R({\bf{Q}}(t)) {\rm{d}}t \nonumber \\
=& \mathop {\max }\limits_{ P\left( t \right)\geq 0} \int_0^T \max
\limits_{{\bf{Q}}(t):{Q}_k(t)\geq 0, \forall k, \sum \limits_{k=1}^K
Q_k(t) \leq P(t)} R({\bf{Q}}(t)) {\rm{d}}t.\end{aligned}$$ Define the auxiliary function $$\label{Gene:3}
\begin{array}{l}
\displaystyle \bar R(P(t)) = {\mathop {\max }\limits_{{\bf
Q}(t):{Q}_k(t)\geq 0, \forall k, \sum \limits_{k=1}^K Q_k(t) \leq
P(t)}} R({\bf{Q}}(t))
\end{array}$$ for which it can be easily verified that the maximum is attained when $\sum \limits_{k=1}^K Q_k(t) = P(t)$. Thus, without loss of generality, we can rewrite (\[Gene:2\]) equivalently as $$\begin{aligned}
\label{Gene:4}
\mathop {\max }\limits_{ P\left( t \right)\geq 0} ~&
\int_0^T \bar R( P(t)) {\rm{d}}t \nonumber \\
{\rm s.t.} ~& \displaystyle\int_0^{t_i} {{P_{{\rm{total}}}}\left( t
\right)} {\rm{d}}t \le \sum\limits_{j = 0}^{i - 1} {{E_j}},
~i=1,\ldots,N
\end{aligned}$$ where $$\label{gene06}\begin{array}{l}{
P_{{\rm{total}}}}(t) = \left\{ {\begin{array}{*{20}{c}}
\displaystyle { P(t) + {\alpha},}\\
{{0},}
\end{array}} \right.\begin{array}{*{20}{l}}
{ P(t) > 0}\\
{ P(t) = 0}.
\end{array}\end{array}$$ Thus, the original problem with vector power optimization is converted by the nested optimization to an equivalent problem with only scalar power optimization. Thereby, we can first solve (\[Gene:4\]) to get the optimal solution of $P(t)$, and then with the obtained $P(t)$ solve (\[Gene:3\]) to find the optimal solution of ${\bf Q}(t)$ for (\[Gene:2\]). Since the problem in (\[Gene:3\]) is a convex optimization problem, it can be solved by standard techniques e.g. the Lagrange duality method [@Boydbook] (in the special case of the sum-rate given in (\[MultiChannel:1\]), the optimal solution can be obtained by the well-known “water-filling” algorithm [@Boydbook]).
In order to solve (\[Gene:4\]), we first give the following proposition.
\[proposition:multichannel\] The function $\bar R(P(t))$ satisfies the following properties:
1. $\bar R(P(t)) \ge 0, \forall P(t)\geq 0$, and $\bar{R}(0)=0$;
2. $\bar R(P(t))$ is a strictly concave function over $P(t)\geq 0$;
3. $\bar R(P(t))$ is a monotonically increasing function over $P(t)\geq 0$.
See Appendix \[appendix:proof propostion multichannel\].
Since $\bar R(P(t))$ satisfies the same conditions as $R(P(t))$ for the single-channel case, it follows that (\[Gene:4\]) can be similarly solved by the algorithm in Table I, with one minor modification: the EE-maximizing power allocation in the multi-channel case needs to be obtained as $$\begin{aligned}
\label{Gene:8}
P_{ee}&= \arg \mathop {\max }\limits_{{P}> 0} \frac{\bar
R(P)}{P+\alpha}\nonumber\\& = \arg \mathop {\max }\limits_{ {P}> 0}
\frac{\mathop {\max }\limits_{{\bf{Q}}:Q_k \ge 0, \forall k,
\sum\limits_{k = 1}^K {{Q_k}} \le P} R({\bf{Q}})}{P+\alpha}.\end{aligned}$$ Since the maximum in the above problem is attained by $\sum\limits_{k = 1}^K {{Q_k}}=P$, the optimal solution of ${\bf Q}$ is obtained as $$\label{Gene:10}
\begin{array}{l}\displaystyle
{\bf{Q}}^{ee}= \arg \mathop {\max }\limits_{{\bf{Q}}:Q_k \ge 0,
\forall k, \sum\limits_{k = 1}^K {{Q_k}} \leq P} \frac{
R({\bf{Q}})}{\sum\limits_{k = 1}^K {{Q_k}}+\alpha}
\end{array}$$ with ${\bf{Q}}^{ee}= [Q^{ee}_1, \ldots, Q^{ee}_K]$, and $$\label{Gene:11}\begin{array}{l}\displaystyle
P_{ee}= \sum\limits_{k = 1}^K {{Q^{ee}_k}}.
\end{array}$$ Since the RHS of (\[Gene:10\]) is a quasi-concave function, this problem is quasi-convex and thus can be efficiently solved by the bisection method [@Boydbook]. Here we omit the detail for brevity.
To summarize, the algorithm for solving (\[Gene:2\]) for the multi-channel case is given in Table II.
**Algorithm**
-----------------------------------------------------------------------------------------------------------------------------------------------------------
1\) Obtain $P_{ee}$ by solving (\[Gene:10\]) and (\[Gene:11\]); apply the algorithm in Table \[table1\] to obtain the solution $P^*(t)$ for (\[Gene:4\]).
2\) With the obtained $P^*(t)$, solve (\[Gene:3\]) to obtain the solution ${\bf{Q}}^*(t)$ for (\[Gene:2\]).
: Optimal Off-Line Policy for Multi-Channel Case[]{data-label="table3"}
Online Algorithm {#sec:online}
================
In the previous two sections, we have studied the optimal off-line policies assuming the non-causal ESI at the transmitter, which provide the throughput upper bound for all online policies. In this section, we will address the practical online case with only the causal (past and present) ESI assumed to be known at the transmitter. In particular, we will propose an online policy based on the structure of the optimal off-line policy obtained previously in Section \[sec:offline\]. Due to the space limitation, we will only consider the single-channel case for the study of online algorithms, while similar results can be obtained for the general multi-channel case, based on the optimal off-line policy given in Section \[sec:multi-channel\].
Proposed Online Algorithm
-------------------------
For the purpose of exposition, we assume that the harvested energy is modeled by a compound Poisson process, where the number of energy arrivals over a horizon $T$ follows a Poisson distribution with mean $\lambda_eT$ and the energy amount in each arrival is independent and identically (i.i.d.) distributed with mean $\bar{E}$. It is assumed that $\lambda_e$ and $\bar{E}$ are known at the transmitter.
We propose an online power allocation algorithm based on the structure of the optimal off-line solution revealed in Theorem \[theorem:1\]. Specifically, considering the start time of each block, from Theorem \[theorem:1\], we obtain the closed-form solution for the optimal off-line power allocation at $t=0$ in the following proposition.
\[proposition:online\] Suppose there are $N-1$ energy arrivals in $(0,T)$ with $N\geq 1$, the optimal off-line power allocation solution for (\[eq4\]) at $t=0$ is given by $$\label{Online1}
\begin{array}{l}
\displaystyle {P^*}(0) =
\max\left(\min_{i=1,\ldots,N}\left(\frac{\sum
\nolimits_{k=0}^{i-1}E_k}{\sum \nolimits_{k=1}^{i}L_k} -
\alpha\right), P_{ee}\right).
\end{array}$$
See Appendix \[appendix:proof propostion online\].
Note that in (\[Online1\]), $E_0$ is available at the transmitter at $t=0$, while $N$, $E_i, i=1,\ldots,N-1$, and $L_i, i=1,\ldots,N$, are all unknown at the transmitter due to the causal ESI. As a result, we cannot compute ${P^*}(0)$ in (\[Online1\]) at $t=0$ for the online policy. Nevertheless, we can approximate the expression of ${P^*}(0)$ based on the statistical knowledge of the energy arrival process, i.e., $\lambda_e$ and $\bar{E}$, as follows.
Denote $$\displaystyle \frac{{\sum \nolimits_{k=0}^{i-1}E_k}}{{\sum
\nolimits_{k=1}^{i}L_k}} = \frac{E_0 + {\sum
\nolimits_{k=1}^{i-1}E_k}}{{\sum \nolimits_{k=1}^{i}L_k}} =
\frac{E_0}{{\sum \nolimits_{k=1}^{i}L_k}} + \frac{{\sum
\nolimits_{k=1}^{i-1}E_k}}{{\sum \nolimits_{k=1}^{i}L_k}}.$$ For any $i\le N$, ${\sum \nolimits_{k=1}^{i-1}E_k}$ is the total energy harvested during $(0,t_i)$ and ${\sum \nolimits_{k=1}^{i}L_k}=t_i$. We thus have $$\label{Online2}
\frac{{\sum \nolimits_{k=1}^{i-1}E_k}}{{\sum
\nolimits_{k=1}^{i}L_k}} \approx
\frac{\lambda_et_i\bar{E}}{t_i}=\lambda_e\bar{E}, \ \forall 1< i\le N,$$ where the approximation becomes exact when $t_i\rightarrow \infty$.
Using (\[Online2\]), we can approximate $\displaystyle \frac{{\sum \nolimits_{k=0}^{i-1}E_k}}{{\sum
\nolimits_{k=1}^{i}L_k}}$ as $\displaystyle
\frac{E_0}{{\sum \nolimits_{k=1}^{i}L_k}} + \lambda_e\bar{E},$ and $$\begin{aligned}
&\min_{i=1,\ldots,N}\left(\frac{\sum
\nolimits_{k=0}^{i-1}E_k}{\sum \nolimits_{k=1}^{i}L_k} -
\alpha\right) \\ \approx &\min_{i=1,\ldots,N}\left(\frac{E_0}
{{\sum \nolimits_{k=1}^{i}L_k}} + \lambda_e\bar{E} -
\alpha\right) = \frac{E_0}{T}
+ \lambda_e \bar{E} - \alpha,\end{aligned}$$ and then obtain $$\label{Online:4}
\begin{array}{l}
\displaystyle {P^*}(0) \approx \displaystyle \max\left(\frac{E_0}{T}
+ \lambda_e \bar{E} - \alpha, P_{ee}\right).
\end{array}$$ Since $E_0$, $\lambda_e$ and $\bar{E}$ are all known at the transmitter at $t=0$, (\[Online:4\]) can be computed in real time.
For any $0<t<T$, by denoting the stored energy as $E_s(t)$ with $E_s(0) = E_0$, we can view the online throughput maximization at time $t$ over the remaining time $T-t$ to have an initial stored energy $E_0=E_s(t)$. Therefore, by replacing $T$ and $E_0$ in (\[Online:4\]) as $T-t$ and $E_s(t)$, respectively, we obtain the following online transmit power allocation policy: $$\begin{aligned}
\label{Online:5}
{P}_{\rm online}(t) = \left\{ \begin{array}{ll}
\max\left(\frac{E_s(t)}{T-t} + \lambda_e\bar{E} - \alpha,
P_{ee}\right), & E_s(t)>0 \\ 0, & E_s(t)=0 \end{array} \right.\end{aligned}$$ for any $t\in[0,T)$.
The online policy in (\[Online:5\]) provides some useful insights. Note that $\frac{E_s(t)}{T-t} + \lambda_e\bar{E} - \alpha$ can be viewed as the “expected” available transmit power for the remaining time in each block, which can be negative for some $t$ if $\frac{E_s(t)}{T-t} + \lambda_e\bar{E} < \alpha$. Thus, if this value is less than the EE-maximizing power allocation $P_{ee}$, the transmitter should transmit with $P_{ee}$ to save energy; however, if the inequality is reversed, the transmitter should transmit more power to maximize the SE. Moreover, as compared to the optimal off-line power allocation for the single-epoch case given in Proposition \[Proposition:1\], we see that the online policy (\[Online:5\]) bears a similar structure, by noting that $E_0/T-\alpha$ in (\[SinEpoch:2\]) for the single-epoch case is also the available transmit power for the remaining time in each block. Last, it is worth remarking that the online power allocation policy in (\[Online:5\]) is expressed as a function of the continuous time for convenience; however, in practice, this policy needs to be implemented in discrete time steps by properly quantizing the continuous-time function. The time step needs to be carefully chosen in implementation: On one hand, it is desirable to use smaller step values to achieve higher quantization accuracy for energy saving, while on the other hand, the time step needs to be sufficiently large, i.e., at least larger than one communication block (cf. Remark \[remark:3.1\]) so that the receiver can have a timely estimate of any transmit power adjustment.
=1 ![Power allocation by the proposed online policy.[]{data-label="fig:online"}](online.eps "fig:"){width="8cm"}
To illustrate the proposed online power allocation policy in (\[Online:5\]), we consider the same channel setup and harvested energy process for the off-line case in Example \[example:1\] (cf. Fig. \[fig:offline\]). In Fig. \[fig:online\], we show the total transmitter power consumption by the proposed online policy assuming that the exact average harvested power $\lambda_e\bar{E} =
(\sum_{i=1}^{N-1}E_i)/T= 187.5$mW is known at the transmitter. It is observed that the online power allocation is no more piecewise-constant like the optimal off-line power allocation in Fig. \[fig:offline\](a). Nevertheless, it is also observed that these two policies result in some similar power allocation patterns, i.e., starting with an on-off power allocation followed by a non-decreasing (in the sense of average power profile for the online policy case) power allocation. This suggests that the proposed online policy captures the essential features of the optimal off-line policy. Moreover, it can be shown that the proposed online policy achieves the total throughput 61.61Mbits over $T=20$secs, which is only 1.53Mbits from 63.14Mbits of the optimal off-line policy. In addition, it can be verified that the total throughput obtained by the proposed online policy is very robust to the assumed average harvested power value $\lambda_e\bar{E}$. For example, by setting $\lambda_e \bar{E}$ to be 150mW or 200mW, the proposed online policy obtains the throughput 61.38Mbits and 61.60Mbits, respectively, which is a very small loss in either case.
Simulation Results {#sec:simulation}
==================
In the section, we compare the performance of the proposed online policy with the performance upper bound achieved by the optimal off-line policy under a stochastic energy harvesting setup modeled by the compound Poisson process. The amount of energy in each energy arrival is assumed to be independent and uniformly distributed between 0 and $2\bar{E}$. For the purpose of comparison, we also consider two alternative heuristically designed online power allocation policies given as follows.
- [**Energy Efficient Policy (EEP)**]{}: In this online policy, the transmitter transmits with the EE-maximizing power allocation $P_{ee}$ given in (\[optimalEE\]) provided that there is a non-zero stored energy, i.e., $$\begin{aligned}
\label{Online:6}
{P}_{\rm EEP}(t) = \left\{ \begin{array}{ll} P_{ee}, & E_s(t)>0 \\
0, & E_s(t)=0 \end{array} \right.\end{aligned}$$ for any $t\in[0,T)$.
- [**Energy Neutralization Policy (ENP)**]{}: This online policy transmits with a constant power that satisfies the long-term energy consumption constraint if there is available stored energy, i.e., $$\begin{aligned}
\label{Online:7}
{P}_{\rm ENP}(t) = \left\{ \begin{array}{ll} \lambda_e \bar{E} - \alpha, & E_s(t)>0 \\
0, & E_s(t)=0 \end{array} \right.\end{aligned}$$ for any $t\in[0,T)$. Note that in the above we have assumed that $\lambda_e \bar{E}>\alpha$.
=1 ![Average throughput versus the energy arrival rate $\lambda_e$ with $\bar E = 0.5$J and $T=$20secs.[]{data-label="fig:lambda"}](lambda.eps "fig:"){width="8cm"}
=1 ![Average throughput versus the block duration $T$ with $\bar
E = $0.5J.[]{data-label="fig:horizon"}](horizon.eps "fig:"){width="8cm"}
=1 ![Average throughput in a single-cell OFDMA downlink system with renewable powered BS versus the energy arrival rate $\lambda_e$ with $\bar E = 200$J and $T=$20secs.[]{data-label="fig:OFDMA"}](OFDMA.eps "fig:"){width="8cm"}
First, we consider a single AWGN channel in Figs. \[fig:lambda\] and \[fig:horizon\] with the same channel parameters as for Example \[example:1\]. In Fig. \[fig:lambda\], we show the average throughput over $T=20$secs versus $\lambda_e$, with $\bar{E}=0.5$Joule (J). It is observed that when $\lambda_e$ is small, the proposed online policy and EEP obtain similar performance as the optimal off-line policy. Since the average harvested energy is small when $\lambda_e$ is small, it is more likely that $P_{ee}$ is greater than both $\frac{E_s(t)}{T-t} + \lambda_e E - \alpha$ (c.f. (\[Online:5\])) and $\frac{\sum
\nolimits _{k=0}^{i-1} E_k}{\sum \nolimits _{k=1}^{i}L_k} - \alpha, \forall i=1,\ldots,N$ (c.f. (\[Online1\])). Thus, both the proposed online and off-line policies choose to transmit with $P_{ee}$ during the “on” periods to save energy. However, for ENP, the transmit power level deviates from $P_{ee}$ and thus significant amount of energy is consumed due to the non-ideal circuit power; as a result, the achievable throughput is almost zero. As $\lambda_e$ increases, the throughput gap between the optimal off-line policy and all online policies enlarges, and the performance of EEP degenerates severely. Moreover, ENP is observed to obtain a similar performance as the proposed online policy. This is because in this case, $\frac{E_s(t)}{T-t}$ is negligibly small as compared with $\lambda_e E$, and as a result the proposed online policy degenerates to ENP.
Fig. \[fig:horizon\] shows the average throughput versus $T$, for two different values of $\lambda_e=0.3$/sec and 1/sec, with $\bar{E}=0.5$J. In both cases of $\lambda_e$, the proposed online policy is observed to perform close to the optimal off-line policy, for all values of $T$. With small value of $\lambda_e$, i.e., $\lambda_e=0.3$/sec, EEP performs much better than ENP since it is more energy efficient, while with a larger value of $\lambda_e$, i.e., $\lambda_e=1$/sec, the reverse becomes true, which can be similarly explained as for Fig. \[fig:lambda\].
Furthermore, for the multi-channel scenario, in Fig. \[fig:OFDMA\] we evaluate the average throughput of a single-cell downlink system with the base station (BS) powered by energy harvesting. It is assumed that the BS covers a circular area with radius 1000 meters, and serves $K$ users whose locations are generated following a spatial homogeneous Poisson point process (HPPP) with density $10^{-6}$ users/m$^2$. Consider a simplified channel model without fading, in which the channel power gain of each user from the BS is determined by a pathloss model $c_0\left(\frac{r}{r_0}\right)^{-\zeta}$, where $c_0 = -60$dB is a constant equal to the pathloss at a reference distance $d_0 = 10$m, and $\zeta = 3$ is the pathloss exponent. Assuming an OFDMA (orthogonal frequency division multiplexing access) based user multiple access, a total bandwidth $W = 5$MHz is equally allocated to the $K$ users. The noise power spectral density at each user receiver is set as $N_0 = -174$dBm/Hz and $\Gamma=1$ is assumed. The circuit power at the BS is set as $\alpha = 60$Watt. Fig. \[fig:OFDMA\] shows the average throughput versus $\lambda_e$ with $\bar E = 200$J and an energy scheduling period $T=$20secs. It is observed that the proposed online policy always performs better than EEP. With small values of $\lambda_e$, i.e., $\lambda_e\le0.8$/sec, ENP preforms clearly worse than the proposed online algorithm, while when $\lambda_e\ge0.95$/sec, both schemes perform similarly. This observation is expected as can be similarly explained for Fig. \[fig:lambda\].
Concluding Remarks {#sec:conclusion}
==================
In this paper, we studied the throughput-optimal transmission policies for energy harvesting wireless transmitters with the non-ideal circuit power. We first obtained the optimal off-line solution in the single-channel case, which is shown to have a new two-phase transmission structure by unifying existing results on separately maximizing energy efficiency and spectrum efficiency. We then extended the optimal off-line solution to the general case with multiple AWGN channels subject to a total energy harvesting power constraint, by the technique of nested optimization. Finally, we proposed an online algorithm based on a closed-form off-line solution. It is shown by simulations that the proposed online algorithm has a very close performance to the upper bound achieved by the optimal off-line solution, and also outperforms other heuristically designed online algorithms.
After submission of this manuscript, we become aware of one interesting related work [@Devillers2012] that is worth mentioning. In [@Devillers2012], the throughput optimization in a single-channel energy harvesting communication system with battery leakage is introduced. The impact of battery leakage is very similar to that of the non-ideal circuit power considered in this paper; as a result, the optimal transmission policy developed in [@Devillers2012] is similar to the one proposed in Section \[sec:offline\] of this paper. One difference is that the optimal off-line policy in [@Devillers2012] is developed by decoupling the general multi-epoch problem into multiple equivalent single-epoch subproblems, while the same optimal policy in this paper is derived by decoupling the problem into only two subproblems (c.f. Appendix \[appendix:proof theorem 1\]). Compared with the solution in [@Devillers2012], the off-line policy in this paper reveals the optimal two-phase structure in which EE and SE optimizations are unified, and thus motivates our online policies, which are not given in [@Devillers2012].
Proof of Lemma \[Lemma:1\] {#appendix:proof Lemma 1}
==========================
Denote the length of the on-period ${\mathcal T}_i^{\rm{on}}$ as $l_{i}^{\rm{on}}$ and that of the off-period ${\mathcal
T}_i^{\rm{off}}$ as $l_{i}^{\rm{off}}$, where $l_{i}^{\rm{on}} +
l_{i}^{\rm{off}} = L_i$. Without loss of generality, we only need to consider the case with on-period ${\mathcal T}_i^{\rm{on}} = (t_{i-1},
t_{i-1}+l_{i}^{\rm{on}}]$ and off-period ${\mathcal T}_i^{\rm{off}} =
(t_{i-1}+l_{i}^{\rm{on}},t_{i}]$, since exchanging the power allocation at two different time instants in an epoch does not change the throughput and the energy constraint. Next, we prove that the transmit power should be constant during the on-period ${\mathcal T}_i^{\rm{on}}$ in an epoch by contradiction.
Suppose that the optimal allocated transmit power $\bar P(t)$, where $\bar P(t) > 0, t \in (t_{i-1}, t_{i-1}+l_{i}^{\rm{on}}]$ is not constant. Since $R(P(t))$ is a strictly concave function, based on Jensen’s inequality, we have $$\label{SinEpoch:6}
R\left(\frac{\int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \bar P(t){\rm{d}}t}{l_i^{\rm{on}}}\right) > \int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \frac{ R( \bar P(t))}{l_i^{\rm{on}} }{\rm{d}}t,$$ and then $$\begin{aligned}
\label{SinEpoch:7}
& \int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}}
R\left(\frac{\int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \bar
P(t){\rm{d}}t}{l_i^{\rm{on}}}\right){\rm{d}}t \nonumber\\= &{l_i^{\rm{on}} }
R\left(\frac{\int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \bar
P(t){\rm{d}}t}{l_i^{\rm{on}}}\right) \nonumber\\ >&
\int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} { R( \bar P(t))}{\rm{d}}t.\end{aligned}$$ Thus, if we construct a new transmit power allocation $\hat P(t)$ as $\hat P(t) = P_i = \frac{\int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \bar
P(t){\rm{d}}t}{l_i^{\rm{on}}}>0, t \in (t_{i-1},t_{i-1}+l_i^{\rm{on}}]$, we can achieve a larger throughput than that achieved by $\bar P(t)$. Moreover, we verify that $\hat P(t)$ consumes the same total energy as $\bar
P(t)$ in the $i$th epoch, i.e., $$\begin{aligned}
\label{reformulation:10}
&\int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \left(\hat
P(t)+\alpha\right) {\rm{d}}t = l_i^{\rm{on}} (P_i+\alpha) \nonumber\\ =&
\int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \bar P(t){\rm{d}}t + l_i^{\rm{on}}
\alpha = \int_{t_{i-1}}^{t_{i-1}+l_i^{\rm{on}}} \left(\bar
P(t)+\alpha\right){\rm{d}}t.\end{aligned}$$ Therefore, based on (\[SinEpoch:7\]) and (\[reformulation:10\]), we conclude that $\bar P(t)$ cannot be optimal and thus Lemma \[Lemma:1\] is proved.
Proof of Proposition \[Proposition:1\] {#appendix:proof Proposition 1}
======================================
To solve (\[eq105oneepoch\]), we note that the third inequality constraint must be met with equality by the optimal solution, since otherwise the throughput can be further improved by increasing $P_1$. Thus, by substituting ${l_1^{\rm{on}}}=
\frac{E_0}{P_1+\alpha}$ into the objective function as well as the constraint $l_1^{\rm on}\leq T$, the problem becomes equivalent to finding $$\begin{aligned}
\label{eq105oneepochequ} P_1^* &= \arg
\mathop { \max }\limits_{P_1>0, P_1\geq E_0/T-\alpha}
\frac{E_0}{P_1+\alpha} R(P_1) \nonumber\\&= \arg \mathop { \max }\limits_{P_1>0,
P_1 \geq E_0/T-\alpha} \frac{R(P_1)}{P_1+\alpha}.\end{aligned}$$ Consider first the following problem with the relaxed power constraint: $$\begin{array}{l}\displaystyle
\mathop {\max }\limits_{P_1> 0} \frac{R(P_1)}{P_1+\alpha}.
\end{array}$$ This problem has been studied in [@Miao], where the globally optimal solution is known as the EE-maximizing power allocation, denoted by $P_{ee}$. It was also shown in [@Miao] that given $\alpha>0$, $\frac{R(P_1)}{P_1+\alpha} $ is monotonically increasing with $P_1$ if $0\leq P_1 <P_{ee}$, and monotonically decreasing with $P_1$ if $P_1>P_{ee}$. Thus, the solution of (\[eq105oneepochequ\]) is obtained as $$\label{SinEpoch:8}
P_1^* = \max\left(P_{ee}, \frac{E_0}{T}-\alpha\right).$$ Accordingly, the optimal on-period is given by $$\label{SinEpoch:9}
\displaystyle l_1^{\rm{on}*} = \frac{E_0}{P_1^* + \alpha}.$$ Proposition \[Proposition:1\] is thus proved.
Proof of Theorem \[theorem:1\] {#appendix:proof theorem 1}
==============================
To prove Theorem \[theorem:1\], we construct the following two sub-problems ${\mathbb{P}}_1$ and ${\mathbb{P}}_2$ for the power allocation optimization in the first $i_{ee}$ epochs and the last $N-i_{ee}$ epochs, respectively. $$\begin{aligned}
\label{equ:AppC:1}
{\mathbb{P}}_1: \mathop {\max }\limits_{\{P_i\},\{l_i^{{\rm{on}}}\}} ~ & \sum\limits_{i = 1}^{{i_{ee}}} {l_i^{{\rm{on}}}} R({P_i}), \nonumber \\
{\rm{s}}{\rm{.t}}{\rm{. }} ~& 0 \le l_i^{{\rm{on}}} \le {L_i},i = 1, \ldots ,{i_{ee}}, \nonumber \\
~& \sum\limits_{j = 1}^i {({P_j} + \alpha )l_j^{{\rm{on}}}} \le \sum\limits_{j = 0}^{i - 1} {{E_j}} ,i = 1, \ldots
,{i_{ee}}.\end{aligned}$$ $$\begin{aligned}
\label{equ:AppC:2}
{\mathbb{P}}_2: \mathop {\max }\limits_{\{{P_i}\},\{l_i^{{\rm{on}}}\}} ~ & \sum\limits_{i = {i_{ee}} + 1}^N {l_i^{{\rm{on}}}} R({P_i}), \nonumber \\
{\rm{s}}{\rm{.t}}{\rm{. }}~ & 0 \le l_i^{{\rm{on}}} \le {L_i},i = {i_{ee}} + 1, \ldots ,N \nonumber \\
~ & \sum\limits_{j = {i_{ee}} + 1}^i {({P_j} + \alpha
)l_j^{{\rm{on}}}} \le \sum\limits_{j = {i_{ee}}}^{i - 1} {{E_j}} ,\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~i
= {i_{ee}} + 1, \ldots ,N.\end{aligned}$$ We will first prove that the solution given in Theorem \[theorem:1\] is optimal for both ${\mathbb{P}}_1$ and ${\mathbb{P}}_2$, and then prove that the optimal solutions for ${\mathbb{P}}_1$ and ${\mathbb{P}}_2$ are also optimal for (\[eq4\]).
First, we prove that the solution given in (\[optimal01\]), (\[optimal02\]) and (\[optimal03\]) is optimal for ${\mathbb{P}}_1$.
Consider the throughput maximization problem ${\mathbb{P}}_1$, with the arrived energy $E_0, \ldots, E_{i_{ee}-1}$ at time $t_0, \ldots,
t_{i_{ee}-1}$ over the horizon $T_1={\sum \nolimits _{k=1}^{i_{ee}}
L_k}$. We construct an auxiliary throughput maximization problem $\bar{\mathbb{P}}_1$ with the energy arrival ${\sum \nolimits
_{k=0}^{i_{ee}-1} E_k}, 0, \ldots, 0$ at time $t_0, \ldots,
t_{i_{ee}-1}$ over the same horizon $T_1$ as follows. $$\begin{aligned}
\label{equ:AppC:3}
\bar{\mathbb{P}}_1: \mathop {\max }\limits_{\{{P_i}\},\{l_i^{{\rm{on}}}\}} ~ & \sum\limits_{i = 1}^{{i_{ee}}} {l_i^{{\rm{on}}}} R({P_i}), \nonumber \\
{\rm{s}}{\rm{.t}}{\rm{. }} ~& 0 \le l_i^{{\rm{on}}} \le {L_i},i = 1, \ldots ,{i_{ee}}, \nonumber \\
~& \sum\limits_{j = 1}^{i_{ee}} {({P_j} + \alpha )l_j^{{\rm{on}}}} \le \sum\limits_{j = 0}^{i_{ee} - 1} {{E_j}}.\end{aligned}$$ It is clear that the optimal throughput of $\bar{\mathbb{P}}_1$ is an upper bound on that of ${\mathbb{P}}_1$ since any feasible solution of ${\mathbb{P}}_1$ is also feasible for $\bar{\mathbb{P}}_1$. Note that there is no energy arrived in $t_1, \ldots, t_{i_{ee}-1}$ for $\bar{\mathbb{P}}_1$, so $\bar{\mathbb{P}}_1$ is indeed equivalent to a throughput maximization problem for the single-epoch case studied in Section III-B over a horizon $T_1$. It can be easily verified based on (\[SinEpoch:12\]) that $\displaystyle \frac{
\sum\nolimits_{j = 0}^{{i_{ee}} - 1} {{E_j}}}{T_1} - \alpha \le P_{ee}$; thus, it follows after some simple manipulation that the optimal value of $\bar{\mathbb{P}}_1$ is $\displaystyle \frac{
\sum\nolimits_{j = 0}^{{i_{ee}} - 1} {{E_j}}}{P_{ee} + \alpha} \cdot R(P_{ee})$ and is attained by $\bar P_1^* = \cdots =
\bar P_{i_{ee}}^* = P_{ee}$ and $\displaystyle \sum_{j=1}^{i_{ee}}\bar l_j^{{\rm{on}}*} = \frac{
\sum\nolimits_{j = 0}^{{i_{ee}} - 1} {{E_j}}}{P_{ee} + \alpha}$. Meanwhile, for ${\mathbb{P}}_1$ we can always construct a feasible solution based on (\[optimal01\]), (\[optimal02\]) and (\[optimal03\]) by setting $$\begin{aligned}
P_j^* & =P_{ee},\ \forall j=1,\ldots,i_{ee},\nonumber \\
l_k^{{{\rm{on}}}*} &= L_k,\ \forall k\neq i_{ee,j}, \forall j=1,\ldots,J,\nonumber \\
l_{i_{ee,j}}^{{\rm{on}}*} &= \frac{\sum_{k=i_{ee,j-1}}^{i_{ee,j}-1}E_k}{P_{ee}+\alpha}- \sum_{k=i_{ee,j-1}+1}^{i_{ee,j}}L_k,\ \forall j=1,\ldots,J.\label{equ:revision:1}\end{aligned}$$ It can be verified from (\[SinEpoch:12\]) that the solution satisfying (\[equ:revision:1\]) is feasible for ${\mathbb{P}}_1$, and attains an objective value of $\displaystyle \frac{
\sum\nolimits_{j = 0}^{{i_{ee}} - 1} {{E_j}}}{P_{ee} + \alpha} \cdot R(P_{ee})$, which is the same as the optimal value of $\bar{\mathbb{P}}_1$. The gap between ${\mathbb{P}}_1$ and $\bar{\mathbb{P}}_1$ is thus zero, and accordingly, the solution given in (\[optimal01\]), (\[optimal02\]) and (\[optimal03\]) is optimal for ${\mathbb{P}}_1$.
Second, we prove that the solution in (\[optimal104\]), (\[optimal05\]) and (\[optimal06\]) is optimal for Problem ${\mathbb P}_2$ over the last $N-i_{ee}$ epochs. First, we show ${l_i^{{\rm{on}}*}} = L_i, i = i_{ee}+1,\ldots, N$ by contradiction as follows. Suppose that the optimal solution $\hat P(t)$ contains an “off” period with $(\hat t^{\rm{off}}, \hat t^{\rm{off}}+\Delta
\hat t^{\rm{off}}) \subset (t_{i_{ee}},T]$, i.e., $\hat P(t) = 0, t
\in (\hat t^{\rm{off}}, \hat t^{\rm{off}}+\Delta \hat
t^{\rm{off}})$.
Due to the definition of (\[SinEpoch:12\]), it follows immediately that $$\label{Proof101}
\begin{array}{l}\displaystyle \frac{\sum \nolimits _{k=i_{ee}}^{i-1} E_k}{\sum \nolimits _{k=i_{ee}+1}^{i}L_k} - \alpha > P_{ee}, \forall i > i_{ee}\end{array}$$ Therefore, we can always find a time duration with $\hat P(t) = \hat P^{\rm{on}} > P_{ee}, t \in (\hat
t^{\rm{on}}, \hat t^{\rm{on}}+\Delta \hat t^{\rm{on}}) \subset
(t_{i_{ee}},T]$, and construct a new policy $\displaystyle \bar
P(t)$ with $\displaystyle \bar P(t) = \bar P^{\rm{on}} = \frac{(\hat
P^{\rm{on}} + \alpha)\Delta \hat t^{\rm{on}}}{\Delta \hat
t^{\rm{on}} + \delta} - \alpha, t\in (\hat t^{\rm{on}}, \hat
t^{\rm{on}}+\Delta \hat t^{\rm{on}}) \cup (\hat t^{\rm{off}}, \hat
t^{\rm{off}}+\delta) $. Note that we have chosen $\delta$ to be sufficiently small so that $(\hat t^{\rm{off}}, \hat
t^{\rm{off}}+\delta) \subseteq (\hat t^{\rm{off}}, \hat
t^{\rm{off}}+\Delta \hat t^{\rm{off}})$ and $\bar P^{\rm{on}} >
P_{ee}$. The energy consumed by $\bar P(t)$ during $(\hat
t^{\rm{on}}, \hat t^{\rm{on}}+\Delta \hat t^{\rm{on}}) \cup (\hat
t^{\rm{off}}, \hat t^{\rm{off}}+\delta)$ is $\hat E = (\bar
P^{\rm{on}} + \alpha)(\Delta \hat t^{\rm{on}}+\delta) = (\hat
P^{\rm{on}} + \alpha) \Delta \hat t^{\rm{on}}$, which is same as the initial policy $\hat P(t)$. The throughput for the newly constructed policy $\bar P(t)$ and initial policy $\hat P(t)$ during $(\hat t^{\rm{on}}, \hat t^{\rm{on}}+\Delta \hat t^{\rm{on}}) \cup
(\hat t^{\rm{off}}, \hat t^{\rm{off}}+\delta) $ are $$\displaystyle B_1 = R(\bar P^{\rm{on}})(\Delta \hat t^{\rm{on}}+\delta) =
R(\bar P^{\rm{on}}) \frac{\hat E}{\bar P^{\rm{on}} + \alpha}$$ and $$\displaystyle B_2 = R(\hat P^{\rm{on}})\Delta \hat
t^{\rm{on}} = R(\hat P^{\rm{on}}) \frac{\hat E}{\hat
P^{\rm{on}} + \alpha}$$ respectively. Since $\hat P^{\rm{on}} > \bar P^{\rm{on}} > P_{ee}$, and $ \frac{R(x)}{x + \alpha}$ is monotonically decreasing as a function of $x$ when $x>P_{ee}$, we conclude that $B_1 > B_2$. Therefore, the new policy achieves a higher throughput than the initial policy.
Moreover, we need to check that the new policy also satisfies the energy constraint as follows. If $\hat t^{\rm{off}} > \hat
t^{\rm{on}}+\Delta \hat t^{\rm{on}}$, or $(\hat t^{\rm{on}}, \hat
t^{\rm{on}}+\Delta \hat t^{\rm{on}}) $ and $ (\hat t^{\rm{off}},
\hat t^{\rm{off}}+\delta) $ are in the same epoch, it is evident that the energy constraint is satisfied. If we cannot find an interval $(\hat t^{\rm{off}}, \hat t^{\rm{off}}+\Delta \hat
t^{\rm{off}})$ latter than or in the same epoch as $(\hat
t^{\rm{on}}, \hat t^{\rm{on}}+\Delta \hat t^{\rm{on}})$, i.e., all the allocated power prior to $\hat t^{\rm{off}}$ is smaller than $P_{ee}$, in this case based on (\[Proof101\]), there must be some energy left at time $\hat t^{\rm{off}}$. Since we selected $\delta$ to be sufficiently small, we can still guarantee that the energy constraint is satisfied. Therefore, we prove that ${l_i^{{\rm{on}}*}} = L_i, i = i_{ee}+1,\ldots, N$.
After determining ${l_i^{{\rm{on}}*}} = L_i, i = i_{ee}+1,\ldots,
N$, we realize that Problem ${\mathbb P}_2$ for the last $N-i_{ee}$ epochs has the same structure as that studied in [@Yang], and thus [@Yang Theorem 1] is applicable here. After some change of notation, we can prove that the solution in (\[optimal104\]), (\[optimal05\]) and (\[optimal06\]) is optimal for ${\mathbb
P}_2$ over the last $N-i_{ee}$ epochs.
Next, we show that the optimal solutions of ${\mathbb P}_1$ and ${\mathbb P}_2$ are optimal for (\[eq4\]) to complete the proof for Theorem \[theorem:1\]. First, we can verify by following the similar contradiction proof in the above that any energy harvested during $[0,t_{i_{ee}})$ should be used up in the first $i_{ee}$ epochs, i.e., $\sum\limits_{j = 1}^{{i_{ee}}} {({P_j^*} +
\alpha )l_j^{{\rm{on*}}}} = \sum\limits_{j = 0}^{{i_{ee}} - 1}
{{E_j}}$. Therefore, the energy constraint for (\[eq4\]) is equivalent to $$\label{Appendix2:01}
\begin{array}{l}
\sum \limits _{j=1}^{i}{(P_j + \alpha)l_j^{{\rm{on}}}} \le \sum \limits _{j=0}^{i-1} E_j, i=1,\ldots, i_{ee}\\
\sum \limits _{j=i_{ee}+1}^{i}{(P_j + \alpha)l_j^{{\rm{on}}}} \le
\sum \limits _{j=i_{ee}}^{i-1} E_j, i=i_{ee}+1,\ldots, N.
\end{array}$$ Since the energy constraint is decoupled before and after $t_{i_{ee}}$, solving (\[eq4\]) is equivalent to optimizing ${P_i}$ and $l_i^{{\rm{on}}}$ over $i = {1, \ldots ,i_{ee}}$ and $i ={i_{ee}} +
1, \ldots ,N$ separately. Thus, the optimal solutions for ${\mathbb{P}}_1$ and ${\mathbb{P}}_2$ are also optimal for (\[eq4\]). Theorem \[theorem:1\] is thus proved.
Proof of Proposition \[proposition:multichannel\] {#appendix:proof propostion multichannel}
=================================================
The first and third properties of $\bar R(P(t))$ can be directly verified by the first and third properties of $R({\bf Q}(t))$, respectively. Thus, to complete the proof of Proposition \[proposition:multichannel\], we only need to show the second property of $\bar R(P(t))$, i.e., it is a strictly concave function of $P(t)$. Similar to [@ZhangCR Appendix B], we show the proof of this result as follows.
Since $\bar R(P(t))$ is obtained as the optimal value of (\[Gene:3\]), which is a convex optimization problem and satisfies the Slater’s condition [@Boydbook]. Thus, the duality gap for this problem is zero. As a result, $\bar R(P(t))$ can be equivalently obtained as the optimal value of the following min-max optimization problem: $$\begin{aligned}
\label{equ:appendixB:1}
\bar R(P) &= \min\limits_{\mu \ge 0} \max\limits_{Q_k \geq 0}
R({\bf{Q}}) - \mu\left(\sum \limits_{k=1}^K Q_k - P\right)
\\ \label{equ:appendixB:2}
& = \min\limits_{\mu \ge 0} R({\bf{Q}}^{(\mu)}) - \mu\sum
\limits_{k=1}^K Q_k^{(\mu)} + \mu P \\
\label{equ:appendixB:3} & = R({\bf{Q}}^{(\mu^{(P)})}) - \mu^{(P)}
\sum \limits_{k=1}^K Q_k^{(\mu^{(P)})} +\mu^{(P)} P\end{aligned}$$ where we have removed $t$ for brevity, and in (\[equ:appendixB:2\]) ${\bf{Q}}^{(\mu)} =
[{Q}_1^{(\mu)},\ldots,{Q}_K^{(\mu)}]$ is the optimal solution for the maximization problem with a given $\mu$, while in (\[equ:appendixB:3\]) $\mu^{(P)}$ is the optimal solution for the minimization problem with a given $P$. Since $R({\bf{Q}}(t))$ is a strictly joint concave function, the optimal solutions in the above must be unique. Denote $\omega$ as any constant in $[0,1]$. Let $\mu^{(P_1)}$, $\mu^{(P_2)}$ and $\mu^{(P_3)}$ be the optimal $\mu$ for $\bar R(P_1)$, $\bar R(P_2)$ and $\bar R(P_3)$ with $P_3=\omega
P_1 + (1-\omega)P_2$, respectively. For $j=1,2$, we have $$\begin{aligned}
\label{equ:appendixB:4}
\bar R(P_j) &= R({\bf{Q}}^{(\mu^{(P_j)})}) - \mu^{(P_j)} \sum
\limits_{k=1}^K Q_k^{(\mu^{(P_j)})} + \mu^{(P_j)} P_j \\
\label{equ:appendixB:5} & \le R({\bf{Q}}^{(\mu^{(P_3)})}) -
\mu^{(P_3)} \sum \limits_{k=1}^K Q_k^{(\mu^{(P_3)})} + \mu^{(P_3)}
P_j\end{aligned}$$ where [*strict inequality*]{} holds for (\[equ:appendixB:5\]) if $P_j \neq P_3$ since the optimal $\mu^{(P_j)}, j=1,2$, are unique. Thus, we have $$\begin{aligned}
\label{equ:appendixB:6}
& \omega \bar R(P_1) + (1-\omega) \bar R(P_2)
\\ \le& R({\bf{Q}}^{(\mu^{(P_3)})}) - \mu^{(P_3)} \sum \limits_{k=1}^K Q_k^{(\mu^{(P_3)})} + \mu^{(P_3)} P_3\label{equ:appendixB:6-1}
\\ \label{equ:appendixB:7}
=& \bar R(P_3) \\ =& \bar R(\omega P_1 + (1-\omega)P_2),\end{aligned}$$ where [*strict inequality*]{} holds for (\[equ:appendixB:6-1\]) if $\omega\in(0,1)$. Therefore, $\bar R(P(t))$ is a strictly concave function over $P(t)\geq 0$. The proof of Proposition \[proposition:multichannel\] is thus completed.
Proof of Proposition \[proposition:online\] {#appendix:proof propostion online}
===========================================
We prove this proposition by considering the following two cases.
First, consider the case when $i_{ee}$ exists, i.e., $i_{ee}\in\{1,\ldots,N\}$. In this case, setting $P_{ee}$ as the transmit power at time 0 is optimal according to Table \[table1\]. Furthermore, we have $\frac{\sum \nolimits_{k=0}^{i_{ee,1}-1}E_k}{\sum
\nolimits_{k=1}^{i_{ee,1}}L_k} - \alpha < P_{ee}$ based on (\[SinEpoch:12\]), and thus $$\displaystyle \min_{i=1,\ldots,N}\left(\frac{\sum
\nolimits_{k=0}^{i-1}E_k}{\sum \nolimits_{k=1}^{i}L_k} -
\alpha\right) \leq \frac{\sum \nolimits_{k=0}^{i_{ee,1}-1}E_k}{\sum
\nolimits_{k=1}^{i_{ee,1}}L_k} - \alpha < P_{ee}.$$ Thus, (\[Online1\]) is equivalent to $P^*(0) = P_{ee}$, which is the optimal solution.
Next, consider the case when $i_{ee}$ does not exist, which implies that $\frac{\sum \nolimits_{k=0}^{j-1}E_k}{\sum
\nolimits_{k=1}^{j}L_k} - \alpha > P_{ee}, \forall j=1,\ldots,N$. Thus, we have $$\displaystyle \min_{i=1,\ldots,N}\left(\frac{\sum
\nolimits_{k=0}^{i-1}E_k}{\sum \nolimits_{k=1}^{i}L_k} -
\alpha\right) > P_{ee}$$ and (\[Online1\]) is equivalent to $P^*(0) = \min \limits
_{i=1,\ldots,N}(\frac{\sum \nolimits_{k=0}^{i-1}E_k}{\sum
\nolimits_{k=1}^{i}L_k} - \alpha)$, which is the optimal solution according to Table \[table1\].
From the above two cases, (\[Online1\]) coincides with the optimal solution for $P^*(0)$. Thus, Proposition \[proposition:online\] is proved.
[1]{}
T. Chen, Y. Yang, H. Zhang, H. Kim, and K. Horneman, “Network energy saving technologies for green wireless access networks,” [*IEEE Wireless Commun.*]{}, vol. 18, no. 5, pp. 30-38, Oct. 2011.
C. Han [*et al.*]{}, “Green radio: Radio techniques to enable energy-efficient wireless networks,” [*IEEE Commun. Mag.*]{}, vol. 49, no. 6, pp. 46-54, Jun. 2011.
Z. Niu, Y. Wu, J. Gong, and Z. Yang, “Cell zooming for cost-efficient green cellular networks,” [*IEEE Commun. Mag.*]{}, vol. 48, no. 11, pp. 74-78, Nov. 2010.
Z. Hasan, H. Boostanimehr, and V. Bhargava, “Green cellular networks: A survey, some research issues and challenges,” [*IEEE Commun. Surveys & Tutorials*]{}, vol. 13, no. 4, pp. 524-540, 2011.
Y. Chen, S. Zhang, S. Xu, and G. Y. Li, “Fundamental trade-offs on green wireless networks,” [*IEEE Commun. Mag.*]{}, vol. 49, no. 6, pp. 30-37, Jun. 2011.
E. Uysal-Biyikoglu, B. Prabhakar, and A. Gamal, “Energy-efficient packet transmission over a wireless link,” [*IEEE/ACM Trans. Network.*]{}, vol. 10, no. 4, pp. 487-499, Aug. 2002.
G. W. Miao, N. Himayat, and G. Y. Li, “Energy-efficient link adaptation in frequency-selective channels,” [*IEEE Trans. Commun.*]{}, vol. 58, no. 2, pp. 545-554, Feb. 2010.
C. Isheden, Z. Chong, E. Jorswieck, and G. Fettweis, “Framework for link-level energy efficiency optimization with informed transmitter,” [*IEEE Trans. Wireless Commun.*]{}, vol. 11, no. 8, pp. 2946-2957, Aug. 2012.
O. Blume, H. Eckhardt, S. Klein, E. Kuehn, and W. M. Wajda, “Energy savings in mobile networks based on adaptation to traffic statistics,” [*Bell Labs Tec. J.*]{}, vol. 15, no. 2, pp. 77-94, Sep. 2010.
H. Kim and G. Veciana, “Leveraging dynamic spare capacity in wireless system to conserve mobile terminals energy,” [*IEEE/ACM Trans. Network.*]{}, vol. 18, no. 3, pp. 802-815, Jun. 2010.
A. Kansal, J. Hsu, S. Zahedi, and M. B. Srivastava, “Power management in energy harvesting sensor networks,” [*ACM Trans. Embed. Comput. Syst.*]{}, vol. 6, no. 4, Sep. 2007.
J. Yang and S. Ulukus, “Optimal packet scheduling in an energy harvesting communication system,” [*IEEE Trans. Commun.*]{}, vol. 60, no. 1, pp. 220-230, Jan. 2012.
C. K. Ho and R. Zhang, “Optimal energy allocation for wireless communications with energy harvesting constraints,” [*IEEE Trans. Sig. Process.*]{}, vol. 60, no. 9, pp. 4808-4818, Sep. 2012.
K. Tutuncuoglu and A. Yener, “Optimum transmission policies for battery limited energy harvesting nodes,” [*IEEE Trans. Wireless Commun.*]{}, vol. 99, no. 2, pp. 1-10, Feb. 2012.
Q. Bai, J. Li, and J. A. Nossek, “Throughput maximizing transmission strategy of energy harvesting nodes,” in [*Proc. IWCLD,*]{} Rennes, France, Nov. 2011.
A. Goldsmith, [*Wireless Communications.*]{} Cambridge, U.K.: Cambridge Univ. Press, 2005.
S. Boyd and L. Vandenberghe, [*Convex Optimization*]{}, Cambridge University Press, 2004.
R. Zhang, F. Gao, and Y.-C. Liang, “Cognitive beamforming made practical: Effective interference channel and learning-throughput tradeoff,” [*IEEE Trans. Commun.*]{}, vol. 58, no. 2, pp. 706-718, Feb. 2010.
B. Devillers and D. Gunduz, “A general framework for the optimization of energy harvesting communication systems with battery imperfections,” [*Journal of Commun. and Netw.*]{}, vol. 14, no. 2, pp. 130-139, Apr. 2012.
[^1]: J. Xu is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail:elexjie@nus.edu.sg).
[^2]: R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail: elezhang@nus.edu.sg). He is also with the Institute for Infocomm Research, A\*STAR, Singapore.
[^3]: Note that the results of this paper can be readily extended to the case with $\eta < 1$ by appropriately scaling the obtained solutions.
[^4]: We thank the anonymous reviewer who brought our attention to [@Bai2011], in which an alterative proof for Lemma \[Lemma:1\] is given based on a calculus approach.
|
---
abstract: 'We derive the phase space density of bosons from a general boson interferometry formula. We find that the phase space density is connected with the two-particles and the single particle density distribution functions. If the boson density is large, the two particles density distribution function can not be expressed as a product of two single particle density distributions. However, if the boson density is so small that two particles density distribution function can be expressed as a product of two single particle density distributions, then Bertsch’s formula is recovered. For a Gaussian model, the effects of multi-particles Bose-Einstein correlations on the mean phase space density are studied.'
address: |
$^1$Physics Department, McGill university, Montreal QC H3A 2T8, Canada\
$^2$ Computer Science Department, Concordia University, QC H3G 1M8, Montreal, Canada
author:
- 'Q. H. Zhang$^{1,2}$, J. Barrette$^1$ and C. Gale$^1$'
title: ' Phase-space density in heavy-ion collisions revisited'
---
PACS number(s): 13.60.Le, 13.85.Ni, 25.75.Gz
Introduction
============
The principal aim of the study of relativistic heavy-ion collisions is the search for evidences of the state of a quark gluon plasma (QGP) in the early stage of the reactions[@review; @rev1]. A quantity of great interest for the study of the QGP is maximum energy density that has been reached in the experiment. This maximum energy density is connected with the final state energy density and phase space volume. For ultra high energy collisions, more than eighty percent of final state particles are pions; therefore, it is very important to measure the phase space density of pions which can be used to infer the energy density in the early stage of the collisions[@Zhang01]. Besides this, it is very important to estimate the phase space density also for the following reasons: (1) If the boson phase space density is very large, then pions will tend to stay in the same state and pion condensate may occur. (2): if the density of pions is very large, then the mean free-length among pions will be small; therefore, the evolution of pions in the final state should be described by the hydrodynamical equation[@Shur].
Bertsch suggested a method which uses pion interferometry measurements[@HBT] to calculate the mean pion phase space density several years ago[@Bertsch]. Since then several calculations have been done at AGS energy[@Ba97] and SPS energy[@Bertsch; @HC99; @WX; @TWH99; @Murry; @TH99]. It has been found that the phase space density of pions is very low at AGS and SPS energies[@Bertsch; @Ba97; @HC99; @TWH99; @TH99]; however this may be not the case at RHIC energies.
Bose-Einstein (BE) correlation effects on the pion multiplicity distribution, on the single pion distribution, and on two-pion interferometry have been studied by many authors for a Gaussian source distribution[@multi; @Z98; @ZPH98; @Zhang99; @HPZ00]. It has been shown by Bialas and Zalewski that those results are valid for a wide class of models[@BZ; @Bialas; @KZ]. Fialkowski and Wit have implemented multi-particle Bose-Einstein correlations in Monte-Carlo generators and have studied Bose-Einstein correlation effects on the $W$ mass shift, pion multiplicity distribution[@FW; @FWW]. But the effects of multi-particle Bose-Einstein correlations on the mean phase space density have never been studied before. Theoretical studies[@HPZ00; @Zhang99] have shown that pion interferometry depends strongly on the pion multiplicity distribution which was overlooked in previous HBT analyses. Thus it is interesting for us to study the effects of the general pion interferometry formula on the mean phase space density.
This paper is arranged in the following way. In the Section II, we re-derive Bertsch’s formula and point out its implicit assumption. In Sec. III, we derive a phase space density formula from the general pion interferometry and we find that if the phase space density becomes small this new expression will be the same as the Bertsch’s except a extra normalization factor. Unfortunately this simple relationship does not hold when the phase space density becomes large. In Sec. IV, multi-pion Bose-Einstein correlation effects on the mean phase space density are studied. We find that multi-particle BE correlations will increase the mean phase space density. Finally we give our conclusions in Sec. V.
Bertsch’s formula and implicit assumptions
==========================================
The two-pion interferometry formula can be written as $$\begin{aligned}
C_2^{I}(p_1,p_2)&=&\frac{P_2(p_1,p_2)}{P_1(p_1)P_1(p_2)} =1+
\nonumber\\
&&\frac{\int d^4x d^4 y g^I(x,k)g^I(y,k)\exp(iq(x-y))}
{\int d^4x d^4 y g^I(x,p_1)g^I(x,p_2)}.
\label{e1}\end{aligned}$$ Here $k=(p_1+p_2)/2$ and $q=p_1-p_2$ are two-pions average momentum and relative momentum respectively. $g^I(x,k)$ is a Wigner function which can be interpreted as the probability of finding a pion at position $x$ with momentum $k$. $P_2(p_1,p_2)$ and $P_1(p)$ are two-particle and single-particle inclusive distributions which are defined by $$\begin{aligned}
P_2(p_1,p_2)&=&\frac{d^6n}{d^3p_1d^3p_2}=
P_1(p_1)P_1(p_2)+
\nonumber\\
&&\int g^{I}(x,k)g^{I}(y,k)\exp(iq(x-y))d^4xd^4y,
\nonumber\\
P_1(p)&=&\frac{d^3n}{d^3p}=\int d^4x g^{I}(x,p),
\label{e2}\end{aligned}$$ with $$\begin{aligned}
\int d^3 p P_1(p)&=&\langle n\rangle,
\nonumber\\
\int d^3 p_1 d^3p_2 P_2(p_1,p_2)&=&\langle n(n-1)\rangle.
\label{e3}\end{aligned}$$ From Eqs. (\[e1\], \[e2\]), it is easily checked that if[@Bertsch; @TWH99] $$g^{I}(x,k)\rightarrow \delta(x_0-\tau)f^{I}(x,k)\frac{1}{(2\pi)^3},
\label{e41}$$ then $$\begin{aligned}
P_1(p)=\frac{1}{(2\pi)^3}\int f^{I}(x,p)d^3 x,
\label{e4}\end{aligned}$$ and $$\begin{aligned}
P_2(p_1,p_2)-P_1(p_1)P_1(p_2)
&=&\frac{1}{(2\pi)^6}\int d^3x d^3 y f^{I}(x,k)
\nonumber\\
&& f^{I}(y,k) \exp(iq(x-y).
\label{e5}\end{aligned}$$ The reason that we put $\frac{1}{(2\pi)^3}$ in Eq. (\[e41\]) is that in statistical physics for a infinite volume[@ZP00] $$P_1(p)=\frac{V}{(2\pi)^3}f(p).
\label{e6}$$ Here $V$ is the volume and $f(p)$ is the Bose-Einstein distribution.
Integrate Eq.(\[e5\]) over $q$ we have $$\begin{aligned}
&&\int d^3 q [P_2(p_1,p_2)-P_1(p_1)P_1(p_2)]
\nonumber\\
&&=\frac{1}{(2\pi)^3}\int f^{I}(x,k)^2 d^3x.
\label{e7}\end{aligned}$$
The average phase space density $\langle f^{I}\rangle_{k}$ can be defined as $$\begin{aligned}
&&\langle f^{I}\rangle_k=\frac{\int f^{I}(x,k)^{2}d^3x}{\int f^{I}(x,k)d^3 x}
\nonumber\\
&&=\frac{\int d^3 q[P_2(k+\frac{q}{2},k-\frac{q}{2})-
P_1(k+\frac{q}{2})P_1(k-\frac{q}{2})]}
{P_1(k)}
\nonumber\\
&&
=\frac{\int d^3 q[C_2^I(q,k)-1]
P_1(k+\frac{q}{2})P_1(k-\frac{q}{2})}
{P_1(k)}
\label{e8}\end{aligned}$$ Using the smooth approximation, $p_1\sim p_2\sim k$, which has been shown to be valid in heavy-ion collisions for its large phase space[@CGZ94], we have $$\langle f^{I}\rangle_k=P_1(k)\int d^3 q[C_2^{I}(k,q)-1].
\label{e9}$$ In Refs.[@Ba97; @HC99; @TWH99], the authors have calculated the phase space density by assuming that $$\begin{aligned}
&&C_2^{I}(p_1,p_2)=1+
\nonumber\\
&&\lambda \exp(-\frac{1}{2}q_o^2R_o^2-
\frac{1}{2}q_s^2R_s^2-\frac{1}{2}q_l^2R_l^2-2R_oR_lq_oq_l).
\label{e11}\end{aligned}$$
But the above parameterization of two-pion interferometry (Eqs. (\[e11\])) is not general, as in practice the two-pion correlation is fitted using function[@Zhang99; @ZPH98] $$C^{ex}_2(p_1,p_2)=AC_2^{I}(p_1,p_2).
\label{e15}$$ Here $A$ is a normalization factor which exists in the two-pion interferometry formula[@Zhang99; @ZPH98]. If we use $C_2^{ex}(q,k)$ to take the place of $C_2^I(q,k)$ in Eq. (\[e9\]), the phase space density will be $$\langle f\rangle^{ex}_{k}=(A-1)P_1(k)+A\times \langle f^I\rangle_{k}.
\label{e17}$$ Thus the phase space density will increase if $A$ is larger than one or decrease if $A$ is smaller than one. In the latter part of this paper we will show that this extra factor $A$ though has been used in the data analyses will not appear in the phase space density formula on the condition that the phase space is large and the density is small. This guarantee that the application of Bertsch formula for heavy-ion collisions system is appropriate if the phase space density is small.
Phase space density from the general pion interferometry formula
================================================================
It has been shown in Refs. [@HPZ00; @Zhang99] that the single particle spectrum, the two-particles spectrum and the two-pion interferometry formula read[@note1] $$P_1(p)=\sum_{i=1}^{N_{max}}h_i G_i(p,p),
\label{e18}$$ $$\begin{aligned}
P_2(p_1,p_2)&=&\sum_{i=1}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}h_{i+j}
[G_i(p_1,p_1)G_i(p_2,p_2)
\nonumber\\
&&+G_i(p_1,p_2)G_i(p_2,p_1)].
\label{e19}\end{aligned}$$ Where the definitions of $h_i$ and $G_i(p,q)$ can be found in Ref.[@HPZ00]. $N_{max}$ is the maximum multiplicity in the experiment. If $N_{max}=\infty$, we will obtain the formula of $P_2(p_1,p_2)$ and $P_1(p)$ given in Refs. [@HPZ00; @Zhang99].
The two-pion correlation function is[@HPZ00] $$\begin{aligned}
C_2(p_1,p_2)&=&\frac{P_2(p_1,p_2)}{P_1(p_1)P_1(p_2)}
\nonumber\\
&=&
C_2^{res}(p_1,p_2)[1+R_2(p_1,p_2)]
\label{e351}\end{aligned}$$ with $$\begin{aligned}
&&C_2^{res}(p_1,p_2)=
\nonumber\\
&&
\frac{\sum_{i}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}
h_{i+j}G_i(p_1,p_1)G_j(p_2,p_2)}
{\sum_{i,j=1}^{N_{max}}h_ih_jG_i(p_1,p_1)G_j(p_2,p_2)},
\label{e34}\end{aligned}$$ and $$\begin{aligned}
&&R_2(p_1,p_2)=
\nonumber\\
&& \frac{\sum_{i=1}^{N_{max}-1}
\sum_{j=1}^{N_{max}-i}h_{i+j}G_i(p_1,p_2)G_j(p_2,p_1)}
{\sum_{i=1}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}h_{i+j}G_i(p_1,p_1)G_j(p_2,p_2)}.
\label{e35}\end{aligned}$$ In Ref. [@HPZ00], we have shown that $R_2(k,q)|_{q=\infty}=0$. So if the two-pion correlation function is expressed as Eq. (\[e15\]), then $$A=C_2^{res}(p_1,p_2),$$ which is a function of $q$ and $k$. We can always define a Wigner function $S(x,k)$ which fulfils the following equation[@Z98; @ZPH98; @HPZ00] $$\sum_{i=1}^{N_{max}}h_iG_i(p_1,p_2)=\int S(x,k)\exp(iqx) d^4x.
\label{e28}$$ Thus, $$P_1(p)=\sum_{i=1}^{N_{max}}h_iG_i(p,p)=\int S(x,p)d^4x.
\label{e40}$$ It has been shown in Refs. [@HPZ00; @Z98; @ZPH98] that for a special multiplicity distribution $p_n=\frac{\omega(n)}{\sum_{n=0}^{n=\infty} \omega(n)}$ or for a small phase space density and $p_n$ is a Poisson form, we have $h_{i+j}=h_ih_j$. Then Eqs. (\[e19\], \[e351\]) change to (for $N_{max}=\infty$)[@Z98; @ZPH98; @HPZ00] $$\begin{aligned}
P_2(p_1,p_2)&=&P_1(p_1)P_1(p_2)+
\nonumber\\
&&\int S(x,k)S(y,k)\exp(iq(x-y))dx dy,
\label{e411}\end{aligned}$$ and $$\begin{aligned}
&&C_2(p_1,p_2)=1+
\nonumber\\
&&\frac{\int S(x,k)S(y,k)\exp(iq(x-y))dxdy}
{\int dx dy S(x,p_1)S(y,P_2)}.
\label{e42}\end{aligned}$$ So we obtain Eq. (\[e1\]). Thus all the derivations given in Ref.[@Bertsch] are valid. This implies that the normalization factor $A$ must be one and this can be shown from Eq. (\[e34\]) under the condition that $N_{max}=\infty$ and $h_{i+j}=h_ih_j$. From Eq.(\[e17\]), we have $$\langle f\rangle^{ex}_{k}=\langle f^I\rangle_{k}.~~~$$ This verifies the rightness of the application of Bertsch’s formula for the case of small boson densities. However, the relationship $h_{i+j}=h_ih_j$ does not hold for all cases. For the following four kinds of multiplicity distributions: $$\begin{aligned}
p_n&=&\frac{\langle n\rangle^n}{n!}\exp(-\langle n\rangle)
\nonumber\\
&& ~~(Poisson~~~ distribution),
\nonumber\\
p_n&=&\frac{\langle n\rangle^n}{(1+\langle n\rangle)^{n+1}}~~~
\nonumber\\
&&~(Bose-Einstein~~~ distribution),
\nonumber\\
p_n&=&\frac{(n+k-1)!}{n!(k-1)!}
\frac{(\langle n\rangle/k)^n}{(1+\frac{\langle n\rangle}{k})^{n+k}}
\nonumber\\
&&(negative~~ binomial~~ distribution),
\nonumber\\
p_n&=&\frac{1}{n\Gamma(k)}(\frac{kn}{\langle n\rangle})^{k}\exp(-kn/\langle n\rangle)~~
\nonumber\\
&&~~(Gamma~~ distribution),\end{aligned}$$ We can prove that[@HPZ00] that if the phase space volume is large, $h_2/h_1^2\sim h_3/(h_1h_2)\sim 1$ for the Poisson distribution, $h_2/h_1^2=2$ and $h_3/(h_1h_2)=3$ for the Bose-Einstein distribution.
If there are strong BE correlations among bosons, then it is impossible to express the two-particles distribution $S(x,p_1;y,p_2)$ as $S(x,p_1)S(y,p_2)$. However, one can find a real function $S_i(x,k)$ which fulfils the following equation $$G_i(p_1,p_2)=\int S_i(x,k)\exp(iqx)d^4x.
\label{e33}$$ Then we define $S(x,k;y,k)$ and $S(x,p_1;y,p_2)$ as $$\begin{aligned}
S(x,k;y,k)&=&\sum_{i=1}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}h_{i+j}S_i(x,k)S_j(y,k),
\label{e37}\end{aligned}$$ and $$\begin{aligned}
S(x,p_1;y,p_2)&=&\sum_{i=1}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}
\nonumber\\
&&
h_{i+j}S_i(x,p_1)S_j(y,p_2),
\label{e38}\end{aligned}$$ which satisfies $$\begin{aligned}
&&\sum_{i=1}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}h_{i+j}G_i(p_1,p_2)G_j(p_2,p_1)
=
\nonumber\\
&&\int S(x,k;y,k)\exp(iq(x-y))dxdy,
\label{e36}\end{aligned}$$ and $$\begin{aligned}
&&\sum_{i=1}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}
h_{i+j}G_i(p_1,p_1)G_j(p_2,p_2)
=
\nonumber\\
&&\int S(x,p_1;y,p_2)dxdy.\end{aligned}$$ Because $\rho^{*}(p_1,p_2)=\rho(p_2,p_1)$, then $G_i^{*}(p,p)=G_i(p,p)$ and $G_i^{*}(p,q)=G_i(q,p)$. Thus $S(x,p_1;y,p_2)$ must be a real function which fulfils the requirement of the Wigner function. If $h_{i+j}=h_ih_j$ and $N_{max}\rightarrow \infty$, from Eq. (\[e38\]), Eq. (\[e33\]) and Eq. (\[e28\]), we have $S(x,p_1;y,p_2)=S(x,p_1)S(y,p_2)$, thus Eqs. (\[e40\],\[e411\],\[e42\]) are obtained. So we can identify $S(x,p_1,y,p_2)$ as a two pion distribution function which represents the probability of two pions emitted from point $x$ with momentum $p_1$ and from point $y$ with momentum $p_2$ respectively. Two-particle spectrum distributions can be written as $$\begin{aligned}
P_2(p_1,p_2)&=&\int S(x,p_1;y,p_2)dx dy +
\nonumber\\
&&
\int S(x,k;y,k)\exp{[iq(x-y)]}dxdy.\end{aligned}$$ Then $R_2(p_1,p_2)$ reads $$\begin{aligned}
R_2(p_1,p_2)=\frac{\int S(x,k;y,k)\exp(iq(x-y))dx dy}
{\int S(x,p_1;y,p_2) dx dy}.\end{aligned}$$ If[@Bertsch] $$\begin{aligned}
S(x,k;y,k)&\rightarrow &
\delta(x_0-\tau)\delta(y_0-\tau)f(x,k;y,k)\frac{1}{(2\pi)^6}
\label{e41x}\end{aligned}$$ and $$\begin{aligned}
S(x,k)&\rightarrow &\delta(x_0-\tau)\frac{1}{(2\pi)^3}f(x,k),
\label{e4211}\end{aligned}$$ then $$\begin{aligned}
\langle f\rangle_{k}&=&
\frac{\int f(x,k;x,k) d^3x}{\int f(x,k) d^3x}
\nonumber\\
&=&\frac{\int d^3q R_2(p_1,p_2)C_2^{res}(p_1,p_2)P_1(p_1)P_1(p_2)}
{P_1(k)}.
\label{f40}\end{aligned}$$ Using the smoothness approximation, $P_1(p_1)\sim P_1(p_2)\sim P_1(k)$, we have $$\begin{aligned}
\langle f\rangle_{k}&=&P_1(k)\int d^3q [\frac{C_2(p_1,p_2)}{C_2^{res}(p_1,p_2)}-1]
C_2^{res}(p_1,p_2)
\nonumber\\
&=&AP_1(k)\int d^3q[\frac{C_2(q,k)}{A}-1]
\nonumber\\
&=&A\langle f^{I}\rangle_{k}.
\label{e49}\end{aligned}$$ In the above we have used the approximation $A\sim C_2^{res}(q,k)\sim const$. It is interesting to point out that normally $A$ should be a function of $q$ and $k$; however, in heavy-ion collisions, the practice is to normally fit it as a constant if the phase space density is small[@HPZ00]. If the phase space density is high, it has been suggested in Ref.[@HPZ00] to fit data using the function $$\begin{aligned}
C_2(p_1,p_2)&=&C_2^{res}(p_1,p_2)[1+R_2(p_1,p_2)].
\label{e51}\end{aligned}$$ Here $C_2^{res}(q,k)={\cal{N}}[1+B(k)\cdot exp(-q^2 R_{res}^2(k))$ and $R_2=\lambda(k) \exp(-q^2 R^2(k))$. Thus it is not the best choice to use a constant $A$ in Eq.(\[e15\]).
multi-particle BE correlation effects on the mean phase space density
=====================================================================
In the following we will study the effects of multi-pion BE correlations on the mean phase space density. We assume $g^I(x,p)$ to be[@multi; @PGG90] $$g^I(x,p)=\delta(x_0)\frac{n_0}{(2\pi R\Delta )^3}
\exp(-\frac{x^2}{2R^2}-\frac{p^2}{2\Delta^2}),
\label{e31}$$ then $f^{I}(x,p)$ reads $$f^{I}(x,p)=\frac{(2\pi)^3n_0}{(2\pi R\Delta)^3}\exp(-\frac{x^2}{2R^2}-\frac{p^2}{2\Delta^2}).
\label{e32}$$ Due to Eq. (5), we immediately come to the conclusion that $n_0$ is the mean pion multiplicity observed in the experiment. It is easily checked that $$\langle f^{I}\rangle_{k}=\frac{\int d^3x f^{I}(x,p)^2}{\int d^3x f^{I}(x,p)}=
\frac{n_0}{(\sqrt{2}R\Delta)^3}\exp(-\frac{p^2}{2\Delta^2}).
\label{e32y}$$ However, we have neglected the high-order BE correlation effects to get this $\langle f^I \rangle_k$. If we keep only the leading terms in Eqs. (19,20)(correspondingly, this implies that we have assumed that the phase space volume is large), then $$P_1(p)=h_1G_1(p,p),$$ and $$\begin{aligned}
P_2(p_1,p_2)&=&h_2[G_1(p_1,p_1)G_1(p_2,p_2)+
\nonumber\\
&&G_1(p_1,p_2)G_1(p_2,p_1)].\end{aligned}$$ Thus the corresponding two-pion correlation function becomes $$C_2(p_1,p_2)=\frac{h_2}{h_1^2}[1+\frac{G_1(p_1,p_2)G_1(p_2,p_1)}{
G_1(p_1,p_1)G_1(p_2,p_2)}] .
\label{ext51}$$ Bring Eq. (\[ext51\]) into Eq. (\[e8\]) and ignore the extra normalization factor, $\frac{h_2}{h_1^2}$, we get Eq. (\[e32y\]); furthermore, if we bring Eq. (\[ext51\]) into Eq. (\[f40\]) and ignore the extra normalization factor, Eq. (\[e32y\]) will be regained. This results demonstrate that the Bertsch formula will give the correct result on the condition that the phase space volume is large. In the following we will study the multi-particles correlation effects on the mean phase space density due to the large phase space density.
We define a function $r_1(k)$ as $$r_1(k)=\frac{\langle f\rangle_{k}}{\langle f^{I}\rangle_{k}}.$$
In the Fig.\[f5\], $r_1(k)$ is shown as a function of $k/\Delta$. It is clear that for a large phase space, $r_1(k) \sim constant$; on the other hand, if the phase space is small, $\langle f\rangle_{k}$ and $\langle f^{I}\rangle_{k}$ have big differences at small momentum and small differences at large momentum. This is easily understood since quantum effects are big for small momentum particles. It is interesting to notice that when the phase space is large, $r_1\sim h_2/h_1^2(\sim C^{res}(q,k))$, which is two for the Bose-Einstein distribution, one for the Poisson distribution. From Eq. (\[e41x\]) and Eq. (\[e36\])(taking $q=0$), we have $$\begin{aligned}
&&\int f(x,k;y,k)d^3x d^3y=(2\pi)^6\int S(x,k;y,k)d^4x d^4y
\nonumber\\
&=&(2\pi)^6\sum_{i=1}^{N_{max}-1}\sum_{j=1}^{N_{max}-i}
h_{i+j}G_i(k,k)G_j(k,k).
\label{e51z}\end{aligned}$$ From Eq. (\[e4211\]) and Eq. (\[e28\])(taking $q=0$), we get $$\begin{aligned}
\int f(x,k)dx&=&(2\pi)^3\int S(x,k) d^4x
\nonumber\\
&=&(2\pi)^3\sum_{i=1}^{N_{max}}h_iG_i(k,k).
\label{e51y}\end{aligned}$$ From the definition of $C_2^{res}(p_1,p_2)$(Eq. (\[e34\])) and Eqs. (\[e51z\], \[e51y\]), we have $$C_2^{res}(q,k)_{q=0}=\frac{\int f(x,k;y,k) d^3x d^3y}{\int d^3x d^3y f(x,k)f(y,k)}.$$
In Fig. \[f41\], $C_2^{res}(q,k)_{q=0}$ vs $k/\Delta $ is shown. We notice that when phase space is large, $C_2^{res}(q,k)_{q=0}$ is a constant. In this case, $\int d^3x d^3y f(x,k;y,k) = 2 \int d^3 d^3 y f(x,k) f(y, k)$ for the Bose-Einstein distribution and $\int d^3x d^3y f(x,k;y,k) = \int d^3 d^3 y f(x,k) f(y, k)$ for the Poisson multiplicity distribution. But these relationships do not hold anymore for a small phase space volume.
In Eqs. (\[e8\],\[e49\]), we find that $\langle f\rangle_{k}$ is $A$ times larger than $\langle f^{I}\rangle_{k}$ when the phase space density is small and the function form of $\langle f\rangle_k$ and $\langle f^I\rangle_k$ are different in the numerator. In the following we would like to show the relationship between $f(x,k;x,k)$ and $f(x,k)$. It is found that $$\begin{aligned}
&&\int f(x,k;x,k)d^3x=
\nonumber\\
&&\frac{1}{(2\pi)^3}\int f(x,k;y,k)e^{i{\bf q}({\bf x-y})}d^3q d^3xd^3y
\nonumber\\
&=&(2\pi)^3\int S(x,k;y,k)e^{iq(x-y)} d^4x d^4y d^3q
\nonumber\\
&=&(2\pi)^3\int d^3q
\sum_{i=1}^{N_{max}-1}
\sum_{j=1}^{N_{max}-i}
\nonumber\\
&&h_{i+j}G_i(p_1,p_2)G_j(p_2,p_1)\end{aligned}$$ and $$\begin{aligned}
\int f^2(x,k)d^3x &=&(2\pi)^3
\int d^3q
\sum_{i=1}^{N_{max}}
\sum_{j=1}^{N_{max}}
\nonumber\\
&&
h_{i}h_{j}G_i(p_1,p_2)G_j(p_2,p_1).\end{aligned}$$ In the above derivation, we have used Eqs. (\[e28\], \[e33\], \[e36\], \[e41x\], \[e4211\]). We define a function $r_2(k)$ as $$r_2(k)= \frac{\int f(x,k;x,k)d^3x }{\int d^3x f^2(x,k)}.$$ Similar to Ref.[@HPZ00], we can prove that $$r_2(k)\sim C_2^{res}(q,k)\sim \frac{h_2}{h_1^2}=constant~~~~v\rightarrow\infty~ .
\label{e51x}$$
In the following, we will discuss the effects of multi-boson correlations on the distribution function of pions in the momentum space. If the multi-boson symmetrization effects are small, the distribution function is $f^{I}(x,p)$. If the multi-boson correlations are strong, the distribution function is $f(x,k)$. We define a function $$r_3(k)=\frac{\int d^3x f(x,k)}{\int f^{I}(x,k)d^3x}
=\frac{(2\pi)^3 P_1(k)}{\int f^{I}(x,k) d^3x},$$ which reflects the effects of multi-boson correlations on the distribution function. In Fig.\[f7\], $r_3(k)$ vs. $k/\Delta$ is shown. It is interesting to notice that when phase space is large, $$r_3(k)=1, v\rightarrow \infty
\label{e56}$$ for all distributions. On the other hand, when phase space is small, boson density becomes large at small momentum region. We define $\langle f\rangle^I_{k}$ as $$\langle f\rangle^{I}_{k}=\frac{\int f^2(x,k)dx }{\int f(x,k)dx}.$$ This definition is similar to Eq. (\[e8\]) but with $f(x,k)$ taking the place of $f^I(x,k)$. From Eq. (\[e8\]), Eq. (\[e15\]), Eq. (\[f40\]), Eq. (\[e49\]), and Eq. (\[e51x\]), we get $$\begin{aligned}
\langle f\rangle^{I}_{k}&= & \frac{\langle f\rangle_k}{r_2(k)}
\nonumber\\
&\sim &P(k)\int d^3 q[\frac{C_2(q,K)}{A}-1]~~v\rightarrow \infty.
\label{e52}\end{aligned}$$ This is one of the main results of this paper. This result guarantees that the application of Bertsch’s formula for a large system and dilute gas is appropriate. On the other hand, Bertsch formula is incomplete when the phase space density is high. This comes from the fact that we can not calculate $\int f^2(x,k) d^3x$ from two-pion interferometry formula though we could calculate $\int f(x,k)d^3x$ from the single particle spectrum. If the phase space density is high, $C_2^{res}(p_1,p_2)$ is not a constant anymore, we can not find the approximation formula as Eq. (\[e52\]); however Eq. (\[f40\]) can still be used to find the ratio of the number of particles pairs which are emitted from the same phase space cell to the average number of particles. The relation among $\langle f\rangle_{k}$, $\langle f\rangle^I_k$ and $\langle f^I\rangle_k$ will be very complex when the phase space density is large. Then the physical meaning of the Bertsch formula is no longer clear. Multi-pion BE symmetrization will affect the current formalism and we believe that those effects will be similar to the effects of multi-pion BE on the Wigner function $g(x,p)$, which have been presented in Ref.[@ZPH98] (or $f^I(x,p)$ presented here).
Conclusions
===========
In this paper, the mean phase space density distribution of bosons is derived from the general pion interferometry formula. We find that when the phase space is small and thus the boson density is high, the two particle source distribution can not be expressed as a product of two single particles source distributions. On the other hand, when the phase space is large and thus the boson density is small, Bertsch’s formula is recovered. Thus Bertsch’s formula can be used for the heavy-ion system if the freezeout pion phase space density is small. Multi-pion BE correlation effects on the mean phase space density distribution are studied, it is found that when the phase space density is large, bosons are concentrated in small momentum region and this effect is connected with the pion multiplicity distributions.
[**Acknowledgement**]{}
This work was supported in part by the Natural Science and Engineering Research Council of Canada, and part by the Fonds Nature et Technologies of Quebec.
[29]{} Proc. of Quark Matter 20001, Jan. 14-20 (2001), SUNY, New York. Nucl. Phys. A [**698**]{}, (2002). Quark Gluon Plasma, edited by R. C. Hwa (World Scientific, Singapore),1990; Quark Gluon Plasma II, edited by R. C. Hwa (World Scientific, Singapore),1995. Q. H. Zhang, hep-ph/0106242. E. Shuryak, Phys. Rev. D[**42**]{}, 1764 (1990). W. A. Zajc, in Hadronic Multi-particle production, ed. by P. Carruthers and D. Strottman ( World Scientific, Singapore, 1986), p43; U. Wiedemann and U. Heinz, Phys. Rep. [**319**]{}, 145 (1999); R. M. Weiner, Phys. Rep. [**327**]{}, 327 (2000); T. Csorgo, hep-ph/0001233. G. Bertsch, Phys. Rev. Lett. [**72**]{}, 2349 (1994); Errata [**77**]{}, 789 (1996). J. Barrette et.al; Phys. Rev. Lett. [**78**]{}, 2916 (1997). D. Ference, U. Heinz, B. Tomasik, U. A. Wiedemann and J. G. Cramer, Phys. Lett. B[**457**]{}, 347 (1999). F. Q. Wang and N. Xu, Phys. Rev. C[**61**]{}, 021904 (2000); though this paper is about the baryon phase space density, the idea is the same as in Ref.[@Bertsch]. B. Tomasik, U. A. Wiedemann and U. Heinz, nucl-th/9907096. M. Murray, nucl-ex/0201006; nucl-ex/0102009. B. Tomasik, and U. Heinz, nucl-th/0108051. W. A. Zajc, Phys. Rev. D[**35**]{}, 3396 (1987); S. Pratt, Phys. Lett. B[**301**]{}, 159 (1993); W. Q. Chao, C. S. Gao and Q. H. Zhang, J. Phys. G[**21**]{}, 847 (1995); T. Csorgo and J. Zimanyi, Phys. Rev. Lett. [**80**]{}, 916 (1998). Q. H. Zhang, Phys. Rev. C[**57**]{} 877 (1998); Phys. Lett. B[**406**]{} 366 (1997). Q. H. Zhang, P. Scotto and U. Heinz, Phys. Rev. C[**58**]{}, 3757 (1998). Q. H. Zhang, Phys. Rev. C[**59**]{}, 1646 (1999); Phys. Rev. C[**58**]{}, 18 (1998). U. Heinz, P. Scotto and Q.H. Zhang, Ann. Phys.[**288**]{}, 325 (2001) and references cited there. A. Bialas and K. Zalewski, Phys. Lett. B[**436**]{}, 153 (1998); Euro. Phys. J. C[**6**]{}, 349 (1999); Acta. Phys. Slov. [**49**]{}, 145 (1999); Phys. Rev. D. [**59**]{}, 097502-1 (1999); A. Bialas, hep-ph/9812457. K. Zalewski, hep-ph/0202213; Acta Phys. Polon. B[**32**]{} 3983 (2001). K. Fialkowski and R. Wit, Euro. Phys. J. C[**2**]{}, 691 (1998); Phys. Lett. B[**438**]{}, 154 (1998); Z. Phys. C[**74**]{}, 145 (1997); Euro. Phys. J. C[**13**]{}, 133 (2000); hep-ph/0012356. K. Fialkowski, R. Wit and J. Wosiek, Phys. Rev. D[**58**]{} 094013 (1998); J. Wosiek, Phys. Lett. B[**399**]{},130 (1997). In previous writting, one normally sums the terms in Eqs.(\[e18\],\[e19\]) from one to $\infty$. But, in pratice, one can change the summation from $N_{min}$ to $N_{max}$. In this paper, we choose $N_{min}=1$ for simplicity and this does not affect our conclusions. Q. H. Zhang and S. Padula, Phys. Rev. C[**62**]{}, 024902 (2000). W. Q. Chao, C. S. Gao and Q. H. Zhang, Phys. Rev. C[**49**]{}, 3224 (1994); S. Pratt, Phys. Rev. C[**56**]{}, 1095 (1997); If there are strong correlations between coordinate and momentum spaces, then this smooth approximation may be not valid. See for example, Q. H. Zhang, U. A. Wiedemann, C. Slotta, U. Heinz, Phys. Lett. B [**407**]{} 33 (1997). S. S. Padula, M. Gyulassy and S. Gavin, Nucl. Phys. B[**329**]{}, 357 (1990).
|
---
abstract: |
Two different simple measurements of galaxy star formation rate with different timescales are compared empirically on $156,395$ fiber spectra of galaxies with $r<17.77$ mag taken from the Sloan Digital Sky Survey in the redshift range $0.05<z<0.20$: a ratio ${\ensuremath{\textsf{A}}}/
{\ensuremath{\textsf{K}}}$ found by fitting a linear sum of an average old stellar poplulation spectrum ([$\textsf{K}$]{}) and average A-star spectrum ([$\textsf{A}$]{}) to the galaxy spectrum, and the equivalent width (EW) of the ${\ensuremath{\mathrm{H}\alpha}}$ emission line. The two measures are strongly correlated, but there is a small clearly separated population of outliers from the median correlation that display excess ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ relative to [$\mathrm{H}\alpha$]{}EW. These “K+A” (or “E+A”) galaxies must have dramatically decreased their star-formation rates over the last $\sim 1$ Gyr. The K+A luminosity distribution is very similar to that of the total galaxy population. The K+A population appears to be bulge-dominated, but bluer and higher surface-brightness than normal bulge-dominated galaxies; it appears that K+A galaxies will fade with time into normal bulge-dominated galaxies. The inferred rate density for K+A galaxy formation is $\sim 10^{-4}\,h^3~\mathrm{Mpc^{-3}\,Gyr^{-1}}$ at redshift $z\sim 0.1$. These events are taking place in the field; K+A galaxies don’t primarily lie in the high-density environments or clusters typical of bulge-dominated populations.
author:
- 'Alejandro D. Quintero, David W. Hogg, Michael R. Blanton, David J. Schlegel, Daniel J. Eisenstein, James E. Gunn, J. Brinkmann, Masataka Fukugita, Karl Glazebrook, and Tomotsugu Goto'
bibliography:
- 'apj-jour.bib'
- 'ccpp.bib'
title: Selection and photometric properties of K+A galaxies
---
Introduction
============
Cosmic star formation activity appears to be coming to an end. All indicators of star formation show that the cosmic mean comoving density of star formation has been declining since redshifts near unity [[[[e.g.]{}]{}]{}, @hogg01sfr and references therein]. The bulk of stars in the local Universe are found in old stellar populations [@fukugita98a; @hogg02a]. One consequence of this global decline is that galaxies in the local Universe ought to be shutting down their individual star formation activities. It is not understood whether the changes in star formation in individual galaxies are expected to be gradual or dramatic.
Our extremely local neighborhood in the disk of the Milky Way has been forming stars at a relatively constant rate for the last few billion years [@barry88a; @pardi94; @prantzos98; @rocha-pinto00; @gizis02a], but it appears that it can’t continue for much longer. When the total available gas content is compared to the star formation rate, it appears that the Milky Way will run out of gas in the next few Gyr [[[[e.g.]{}]{}]{}, @mooney88a; @blitz96a]. This is also true for the majority of local disk galaxies [@kennicutt94a]. In detail these calculations do not account for gas recycling, gas flux from plasma resevoirs, or new gas accretion, but the trend they imply for cosmic star formation activity is consistent with that found in cosmological observations.
Here we begin a comparison of different optical star-formation rate measurements in a large sample of galaxies with spectroscopic observations in the Sloan Digital Sky Survey [SDSS; [[[e.g.]{}]{}]{}, @york00a]. One star-formation rate measurement makes use of the strength of the [$\mathrm{H}\alpha$]{} emission line, which is emitted in ionization regions around O and B stars, with lifetimes of $\sim
10^7~\mathrm{yr}$. Another uses the fraction of the galaxy light emitted by A stars, with lifetimes of $\sim 10^9~\mathrm{yr}$. These star-formation indicators are sensitive to stars of different masses and lifetimes.
Star-formation indicators tracking stars with different lifetimes can be compared to flag galaxies with strong star-formation rate derivatives. In particular, comparison of our [$\mathrm{H}\alpha$]{} and A-star measurements locates galaxies that have changed their star-formation rates within the last $\sim 10^9~\mathrm{yr}$. If star formation is triggered by mergers or interactions, such galaxies might be found to be undergoing or recovering from an interaction, and their statistics might provide a measure of the merger rate. If star-formation efficiency is a strong function of local environment, then galaxies with negative star-formation derivatives might be tracers of the environmental conditions at the transition between those that are conducive to star-formation and those that aren’t.
One population of galaxies with strong negative star-formation rate derivatives is already known: “K+A” or “E+A” galaxies [@dressler83a; @zabludoff96a; @goto03a]. These are galaxies combining the stellar absorption lines of A stars with those of old stellar populations, but, at the same time, showing little sign of current star-formation. K+A galaxies are exceedingly rare—they comprise between $10^{-2}$ and $\sim 10^{-4}$ of the galaxy population (depending on definition)—but they are expected to provide important information about the conditions under which stars form and galaxies evolve.
For example, if it is true that elliptical or bulge-dominated galaxies are formed by the mergers of spiral or disk-dominated galaxies, then the central concentration of stars, the alpha-enhanced chemical abundances [[[[e.g.]{}]{}]{}, @worthey98a], and the lack of cold gas, dust and star formation [[[[e.g.]{}]{}]{}, @roberts94a] all require large, brief, centrally concentrated starbursts during or immediately following disk–disk mergers. Note that the abundance ratios provide the evidence that the starbursts must be brief. These merger events therefore likely go through a K+A phase for $\sim 10^9~\mathrm{yr}$; in this scenario, it should be possible to use the abundances of K+A galaxies to constrain merger rates. Indeed, it has already been shown that the A stars tend to be centrally concentrated in K+A galaxies and that some have “kinematically hot” stellar populations like bulge-dominated galaxies [@norton01a]. On the other hand, in some K+A galaxies, there is residual gas, implying that the star formation event does not always exhaust the entire gas reservoir [@Chang01a].
Although any scenario in which bulge-dominated galaxies are created by mergers virtually requires the remnants to be K+A events, an apparent excess of A stars over O and B stars can be produced in heavily dust-enshrouded star-formation regions from which young stars escape on Gyr timescales, or in mergers in which an old galaxy captures a smaller, younger galaxy, shutting off star formation in the process.
In what follows, a cosmological world model with $(\Omega_\mathrm{M},\Omega_\mathrm{\Lambda})=(0.3,0.7)$ is adopted, and the Hubble constant is parameterized $H_0=100\,h~\mathrm{km\,s^{-1}\,Mpc^{-1}}$, for the purposes of calculating distances and volumes [[[[e.g.]{}]{}]{}, @hogg99cosm].
Data
====
The SDSS is taking $ugriz$ CCD imaging of $10^4~\mathrm{deg^2}$ of the Northern Galactic sky, and, from that imaging, selecting $10^6$ targets for spectroscopy, most of them galaxies with $r<17.77~\mathrm{mag}$ [[[[e.g.]{}]{}]{}, @gunn98a; @york00a; @stoughton02a].
All the data processing, including astrometry [@pier03a], source identification, deblending and photometry [@lupton01a], calibration [@fukugita96a; @smith02a], spectroscopic target selection [@eisenstein01a; @strauss02a; @richards02a], and spectroscopic fiber placement [@blanton03a] are performed with automated SDSS software.
Every spectral “plate” of fiber positions includes several faint ($15.5$ to $18.5$ mag) F stars. The spectra are calibrated with these F-star spectra; ie, they are multiplied by the function of wavelength that makes the F-star spectra match F-star spectrophotometry. Although this calibration procedure produces consistent calibration at the 10-percent level [as measured by comparisons of old stellar populations at various redshifts, @eisenstein03b], it does not deal carefully with the diversity of F-star spectra, or the possibility that some of the F stars, even at inferred distances $>2$ kpc, may be inside some part of the Galactic extinction. Inasmuch as the calibration stars are beyond the Galactic extinction, however, the spectra are effectively extinction-corrected.
Redshifts are measured on the reduced spectra by an automated system, which models each galaxy spectrum as a linear combination of stellar eigenspectra (Schlegel, in preparation). The eigenspectra are constructed over the rest-frame wavelength range $4100<\lambda<6800$ Å from a high-resolution spectroscopic library [@prugniel01a]. The central velocity dispersion $\sigma_v$ is determined by fitting the detailed spectral shape as a velocity-smoothed sum of stellar spectra (Schlegel & Finkbeiner, in preparation).
It must be emphasized that the 3 arcsec diameter spectroscopic fibers of the SDSS spectrographs do not obtain all of each galaxy’s light because at redshifts of $0.05<z<0.2$ they reprsent apertures of between $2$ and $7~h^{-1}\,\mathrm{kpc}$ diameter. For this reason, in what follows, SDSS imaging data rather than spectroscopy are used to infer the global properties of the galaxies.
Galaxy luminosities and colors are computed in fixed bandpasses, using Galactic extinction corrections [@schlegel98a] and $K$ corrections [computed with `kcorrect v1_11`; @blanton03b]. They are $K$ corrected not to the redshift $z=0$ observed bandpasses but to bluer bandpasses $^{0.1}g$, $^{0.1}r$ and $^{0.1}i$ “made” by shifting the SDSS $g$, $r$, and $i$ bandpasses to shorter wavelengths by a factor of 1.1 [[[[c.f.]{}]{}]{}, @blanton03b; @blanton03d]. This means that galaxies at redshift $z=0.1$ (typical of the sample used here) have trivial $K$ corrections.
To the azimuthally averaged radial profile of every galaxy in the observed-frame $i$ band, a seeing-convolved Sérsic model is fit, as described elsewhere [@blanton03d]. The Sérsic model has surface brightness $I$ related to angular radius $r$ by $I\propto
\exp[-(r/r_0)^{(1/n)}]$, so the parameter $n$ (Sérsic index) is a measure of radial concentration (seeing-corrected). At $n=1$ the profile is exponential, and at $n=4$ the profile is de Vaucouleurs.
To every best-fit Sérsic profile, the @petrosian76a photometry technique is applied, with the same parameters as used in the SDSS survey. This supplies seeing-corrected Petrosian magnitudes and radii. A $K$-corrected surface-brightness $\mu_{^{0.1}i}$ in the $^{0.1}i$ band is computed by dividing half the $K$-corrected Petrosian light by the area of the Petrosian half-light circle.
For the purposes of computing large-scale structure statistics, we have assembled a complete subsample of SDSS galaxies known as the NYU LSS `sample12`. This subsample is described elsewhere [[[[e.g.]{}]{}]{}, @blanton03d]; it is selected to have a well-defined window function and magnitude limit. In addition, the sample of galaxies used here was selected to have apparent magnitude in the range $14.5<r<17.77~\mathrm{mag}$, redshift in the range $0.05<z<0.20$, and absolute magnitude $M_{^{0.1}i}>-24~\mathrm{mag}$. These cuts left 156,395 galaxies.
For each galaxy, a selection volume $V_\mathrm{max}$ is computed, representing the total volume of the Universe (in $h^{-3}~\mathrm{Mpc^3}$) in which the galaxy could have resided and still made it into the sample. The calculation of these volumes is described elsewhere [@blanton03c; @blanton03d]. For each galaxy, the quantity $1/V_\mathrm{max}$ is that galaxy’s contribution to the cosmic number density, and the quantity $L/V_\mathrm{max}$, where $L$ is the luminosity in some bandpass, is that galaxy’s contribution to the luminosity density in that bandpass.
Around each galaxy in a complete subsample, `sample10`, of `sample12`, there is a measure of the overdensity $\delta_1$ inside a spherical Gaussian window of radius $1~h^{-1}\,\mathrm{Mpc}$, made by deprojecting the angular distribution of nearby galaxies detected in the imaging at fluxes corresponding to luminosities within 1 mag of $L^\ast$ at the target galaxy’s redshift, as described elsewhere [@eisenstein03a; @hogg03b]. The individual overdensity estimates are low in signal-to-noise, but they are unbiased in the mean when averaged over sets of galaxies. A galaxy in an environment with the cosmic mean density has $\delta_1=0$.
There is another measure $\delta_8$ on an $8\,h^{-1}~\mathrm{Mpc}$ scale, that is a measure of the three-dimensional comoving number density excess around each galaxy in a sphere of $8\,h^{-1}$ Mpc comoving radius, with no correction for peculiar velocities, as described elsewhere [@hogg03b; @blanton03d]. Neighbor galaxies are counted, the result is divided by the prediction made from the galaxy luminosity function [@blanton03c], and unity is subtracted to produce the overdensity estimate. The estimates $\delta_8$ are higher in signal-to-noise than the estimates $\delta_1$, but they are slightly biased because they do not account for peculiar velocities.
Method
======
An average old stellar population spectrum (hereafter called “K” because it is dominated by K-type stars) was made by averaging the spectra of bulge-dominated galaxies with luminosities near $L^{\ast}$ and in typical environments, as described previously [@eisenstein03b]. An average A-star spectrum (“A”) was obtained by scaling and taking the mean of calibrated SDSS spectra of stars with types determined by eye to be near A0. These two spectra were normalized by scaling them to have equal total flux within the wavelength range $3800<\lambda<5400$ Å.
A linear sum of the K and A spectra was fit to every SDSS spectrum in the wavelength range $3800<\lambda<5400$ Å, weighting by the inverse square uncertainty in the observed spectrum. A $280~\mathrm{km\,s^{-1}}$ region was masked out around [$\mathrm{H}\beta$]{} and the [$\mathrm{[O\,III]}$]{} lines; this mask size was determined by observing that it would mask out the great majority of the emission-line flux in the great majority of spectra. Before fitting, the A spectrum was smoothed to the appropriate velocity dispersion for each spectrum. [@dressler83a] pioneered this method by showing that post-starburst galaxies can be modeled well by an A dwarf spectrum and a K0 spectrum over the range of 3400 to 5400 angstroms. They found, as we do, that this model simultaneously describes the broad spectral continuum and the depths of narrow features.
The fitting procedure returns two numbers, the amplitudes [$\textsf{K}$]{} and [$\textsf{A}$]{} of the K and A spectral components in the best fit.
The [$\mathrm{H}\alpha$]{} line flux is measured in a 20 Å width interval centered on the line. Before the flux is computed, the best-fit model A+K spectrum is scaled to have the same flux continuum as the data in the vicinity of the emission line and subtracted to leave a continuum-subtracted line spectrum. This method fairly accurately models the [$\mathrm{H}\alpha$]{} absorption trough in the continuum, although in detail it leaves small negative residuals, as will be shown below. The flux is converted to a rest-frame equivalent width (EW) with a continuum found by taking the inverse-variance-weighted average of two sections of the spectrum about 150 Å in size and on either side of the emission line.
Some general caveats apply to our measurements. The first is that the K+A fit is not always good; in particular, galaxies that have undergone a very recent ($<10^{8}~\mathrm{yr}$) starburst can have the blue end of their spectra dominated by O and B stars, not A stars. These galaxies will be assigned high ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$, because the A template fits better than the K, even though the A stars are not dominating the spectra. Of course, most of these galaxies also show strong nebular [$\mathrm{H}\alpha$]{} emission, and, indeed, they do contain young stars.
We rely heavily on the flux-calibration of the spectra, because ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ is sensitive to the color or tilt of the spectrum. If there were large ($>20~\mathrm{percent}$) absolute calibration deviations among the galaxy spectra, we might introduce some spurious interlopers. However, the peak of the ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ distribution of galaxies with low [$\mathrm{H}\alpha$]{}EW at ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}=0$ is narrow and symmetrical about zero. This is a strong indicator that the spectrophotometric calibration is well-behaved. Note that we rely on the consistency of SDSS spectrophotometric calibration, but not on its absolute accuracy. The typical standard linear fitting errors in ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ are 0.03 to 0.1, roughly consistent with the scatter around ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}=0$.
We are looking at fiber spectra, inside the central 3 arcsec diameter of each galaxy. The physical size of this aperture is between $2$ and $7~h^{-1}\,\mathrm{kpc}$ in diameter, thereby excluding light from the outer parts of the galaxies. Some of our galaxies classified as K+A could, in principle, have active star formation in their outskirts. Strictly speaking, the K+A classification is a classification of the star formation indicators in the inner regions of galaxies. For this reason, in what follows the SDSS imaging rather than spectroscopy is used to measure global properties of the galaxies.
The fitting procedure doesn’t take dust exctinction into account. The ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ ratio is sensitive to the shape of the blue end of the spectrum, so it will be reduced in a galaxy in which the A stars are behind significant amounts of dust. This is also true of the [$\mathrm{H}\alpha$]{}EW measurements, although less so because [$\mathrm{H}\alpha$]{} is redward of the K+A fitting region. For these reasons, there may be some truly post-starburst galaxies not classified as K+A galaxies because their young stars are dust-enshrouded.
The K+A population defined by this project does not contain any significant AGN because the selection permits so little [$\mathrm{H}\alpha$]{}emission. There may be some truly post-starburst galaxies not classified as K+A galaxies because they contain central AGN and therefore emit significant [$\mathrm{H}\alpha$]{} emission.
Model spectra
=============
The [$\textsf{K}$]{}, [$\textsf{A}$]{}, and [$\mathrm{H}\alpha$]{} measurements were made also on model spectra made by convolving PEGASE [@fioc97a] instantaneous starburst models with different star formation histories.
The original instantaneous starburst history was made with solar metallicity. Although nebular emission is included in the model, there was no modeling of galactic winds, infall, or internal extinction. For each timestep, the [$\textsf{K}$]{} and [$\textsf{A}$]{} measurements were computed with exactly the same least-square-fitting procedure as was performed on the data. The continuum around the [$\mathrm{H}\alpha$]{} line was also computed in exactly the same way as in the data, but the flux of the [$\mathrm{H}\alpha$]{} line was taken directly from the PEGASE outputs, because PEGASE returns integrated flux values for the emission lines rather than flux densities.
Three model spectra and associated evolutionary tracks were computed by convolving the instantaneous PEGASE burst model: A galaxy forming stars at a fixed rate for 14 Gyr; a galaxy forming stars at a fixed rate for 10 Gyr, shutting off, and fading quiescently for 4 Gyr; and a galaxy forming stars at a fixed rate for 3 Gyr, shutting off, and fading quiescently for 11 Gyr.
In the end, because of small differences between the old model spectra and old observed galaxies, the final [$\textsf{K}$]{} and [$\textsf{A}$]{} values were transformed (by a linear transformation equivalent to a flux-conserving shear in the [$\textsf{K}$]{} [[[vs.]{}]{}]{} [$\textsf{A}$]{} plane) so that an old model galaxy shows ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}=0$, just like an old real galaxy. The linear transformation affects the model outputs by less than 15 percent.
The measurements ${\ensuremath{\textsf{A}}}/ {\ensuremath{\textsf{K}}}$ and [$\mathrm{H}\alpha$]{}EW are plotted as a function of cosmic time for the three models in Figure \[fig:Poster\_timemodels\].
Results
=======
The measured ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ values for the galaxies in the sample are plotted against the measured [$\mathrm{H}\alpha$]{}EW values in Figure \[fig:Poster\_AE\_vs\_Halpha\]. The two star-formation indicators are highly correlated, as expected. Overplotted on the figure are models of galaxies with star formation histories consisting of constant star-formation rates followed by periods of total quiescence.
It is worthy of note that most of the galaxies appear more quiescent ([[[i.e.]{}]{}]{}, have lower ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ and lower [$\mathrm{H}\alpha$]{}EW) than the model (labeled “14”) in which stars form at a constant rate over the entire lifetime of the Universe. In other words, the present-day star-formation rate appears much lower than the cosmic time-averaged mean.
The models in which star formation has “shut off” drop precipitously in [$\mathrm{H}\alpha$]{}EW (which indicates stars with lifetimes of $\sim 10^7$ yr) and then slowly in ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ (which indicates stars with lifetimes of $\sim 10^9$ yr. The models suggest that galaxies whose star-formation activities have shut off will evolve into the zero-star-formation population along the zero [$\mathrm{H}\alpha$]{}EW line. Indeed, Figure \[fig:Poster\_AE\_vs\_Halpha\] shows a clear “spur” of galaxies along this line. These are the “K+A galaxies”.
Figure \[fig:Poster\_histograms\] shows the ratio of [$\mathrm{H}\alpha$]{}EW to ${\ensuremath{\textsf{A}}}/ {\ensuremath{\textsf{K}}}$ for samples cut in ${\ensuremath{\textsf{A}}}/ {\ensuremath{\textsf{K}}}$. The histograms are bimodal, showing a population with anomalously low [$\mathrm{H}\alpha$]{}EW. These are the K+A galaxeis. The figure shows that K+A galaxies form an identifiable separate population. This figure also suggests that the line that separates the “K+A galaxies” from the rest of the population has a slope of $({\ensuremath{\mathrm{H}\alpha}}\mathrm{EW})/({\ensuremath{\textsf{A}}}/ {\ensuremath{\textsf{K}}})=5$ on Figure \[fig:Poster\_AE\_vs\_Halpha\]. This figure also shows that by defining “K+A galaxies” to have ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}> 0.2$ we are excluding a large number of transition objects. The cut was put at 0.2 to avoid interlopers with true values of ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}\approx
0.0$ but scattered high by calibration or statistical errors.
The selection of K+A galaxies for what follows is indicated by the green lines on Figure \[fig:Poster\_AE\_vs\_Halpha\]. There are 1194 K+A galaxies within these selection boundaries in `sample12`. Spectra of a randomly selected subsample of this population is shown in Figure \[fig:Poster\_catalog\].
The astute reader will notice that the mode [$\mathrm{H}\alpha$]{}EW is less than zero; this offset is due to the crudeness of the continuum model subtracted during [$\mathrm{H}\alpha$]{} line measurement. This offset does not affect any of our conclusions.
It is interesting to compare our selection criteria with those of @zabludoff96a and @goto03a. @zabludoff96a considered galaxies in the redshift range $0.05<z<0.13$ with [$\mathrm{[O\,II]}$]{}emission $EW<2.5$ Å and [$\mathrm{H}\gamma$]{}, [$\mathrm{H}\delta$]{}, and [$\mathrm{H}\beta$]{} absorption $EW>5.5$ Å. They found 21 K+A galaxies out of a sample of 11113 (roughly 0.2 percent). @goto03a also worked with an SDSS sample of galaxies, but in the redshift range $0.05<z<1.0$ and with ${\ensuremath{\mathrm{H}\delta}}>4$ Å. They found 3340 HDS galaxies out of a sample of 95479 (roughly 3.4 percent).
Figure \[fig:Poster\_lumweight\] shows the luminosity function of the K+A galaxies compared with the luminosity function of the entirety of `sample12`. The shapes are very similar. Interestingly, the shape of the K+A luminosity function appears closer to the shape of the total luminosity function than it does to the early-type galaxy or red galaxy luminosity functions [[[[e.g.]{}]{}]{}, @blanton03c]. In detail, quantitative comparisons of the luminosity functions ought to involve the fading with time of the K+A galaxy light.
Figure \[fig:Poster\_aehist\] shows the distribution of ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ for the K+A galaxies, in number-density units (the “ratio function”?). Overplotted are the expected distributions of ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ under the assumption of steady-state creation of K+A galaxies along evolutionary tracks like those of the “10” model ([[[i.e.]{}]{}]{}, 10 Gyr of constant star formation followed by nothing) or the “3” model ([[[i.e.]{}]{}]{}, 3 Gyr of constant star formation followed by nothing). Neither of these models is a good description, suggesting that there is probably a large diversity to the star-formation histories that end in a K+A phase according to the selection criteria used here.
Figure \[fig:manyd\_gmr\_n\] shows the luminosity-density weighted distribution of all galaxies in the plane of $^{0.1}(g-r)$ color and Sérsic index $n$ (seeing-corrected concentration). As discussed elsewhere [@blanton03d], the general galaxy population is bimodal in this plane, showing a population of blue, exponential (disk-dominated) galaxies and a separate population of red, concentrated (bulge-dominated) galaxies. Figure \[fig:manyd\_gmr\_n\] also shows that galaxies chosen to have high star-formation rates, by either ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ or [$\mathrm{H}\alpha$]{}EW, lie predominantly in the blue, exponential class. Galaxies chosen to have low star-formation rates lie predominantly in the red, concentrated class. Galaxies chosen to have high ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ but low [$\mathrm{H}\alpha$]{}EW—the K+A galaxies—have some members clearly in the blue, exponential class, and some in an outlier class that appears concentrated, but blue. If star formation has truly been shut off in these galaxies, they are expected to fade and redden and become part of the red, concentrated class.
Figure \[fig:manyd\_gmr\_mu\] is similar to Figure \[fig:manyd\_gmr\_n\] but shows the distribution of objects in the color-surface-brightness plane. The population of objects in this plane is not clearly bimodal, however the bulge-dominated galaxies form a strong peak at high surface brightnesses and red colors. Just like Figure \[fig:manyd\_gmr\_n\], this shows that galaxies with high A/K or high [$\mathrm{H}\alpha$]{}EW appear to be disk-dominated and galaxies with low A/K or low [$\mathrm{H}\alpha$]{}EW appear to be bulge-dominated. The A+K population shows even higher surface brightnesses than the bulge-dominated galaxies, suggesting that as they age, the A+K galaxies will fade and redden into the bulge-dominated part of the diagram.
Three-color images made from the SDSS imaging were inspected by eye for a randomly selected sample of 160 of the K+A galaxies in the sample. The vast majority of them ($\sim90$ percent) appear bulge dominated. The $\sim10$ percent that appear to be disk dominated do also show significant bulges. About $30$ percent either show tidal features or lie extremely close on the sky to nearby neighbors, indicating possible past or current interactions.
In Table \[tab:odensity\], the mean overdensities $\left<\delta_1\right>$ and $\left<\delta_8\right>$ on the $1\,h^{-1}$ and $8\,h^{-1}$ Mpc scales are given for the different galaxy populations mentioned above. Error bars on these averages were computed by splitting the sample into three disjoint sky regions. This technique produces conservative error estimates because it effectively includes contributions from variations in calibration and large-scale structure. The numbers of galaxies used for mean overdensities are smaller than used in the Figures because overdensities have only been calculated on an unbiased subsample of the sample we are using. Table \[tab:odensity\] shows that K+A galaxies, on average, live in the lower-density regions more typical of spiral galaxies, and not the higher-density regions typical of bulge-dominated galaxies.
Discussion
==========
Two star-formation indicators, [$\mathrm{H}\alpha$]{}EW and ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$, with two different timescales, $\sim 10^7$ yr and $\sim 10^9$ yr, have been measured in a sample of 156,395 SDSS (optically selected) galaxies in the redshift range $0.05<z<0.20$. The two star-formation indicators are strongly correlated, as expected. Comparison with models shows that the average levels of star formation in these galaxies is below the average over cosmic time. This fits in with many observations of more distant galaxies that have implied a rapid decrease in the global comoving star formation rate with cosmic time to the present day [@hogg01sfr and references therein].
A clearly separated, distinct population of “K+A galaxies” is found, with very weak or no [$\mathrm{H}\alpha$]{}EW, but significant ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$. The star-formation indicators [$\mathrm{H}\alpha$]{}EW and ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$ have very different timescales over which, in effect, they average recent star formation rates. Galaxies that abruptly (ie, on timescales $<10^9$ yr) shut off star formation will fade rapidly in [$\mathrm{H}\alpha$]{}EW, but slowly in ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$. This timescale mismatch explains the presence of the distinctly visible K+A population shown in \[fig:Poster\_AE\_vs\_Halpha\]; we expect these galaxies to indicate abruptly terminated star formation.
Bulge-dominated and disk-dominated galaxies can be separated by their colors, concentrations (Sérsic indices), and surface brightnesses. By these measures, most of the K+A galaxies appear to be bluer, higher surface-brightness counterparts of bulge-dominated galaxies. The offsets in color and surface brightness are consistent with the fading and redenning expected in the evolution of stellar populations. Taken on face value, this is evidence that K+A galaxies are plausibly the post-starburst progenitors of typical bulge-dominated galaxies, as might be expected in a scenario in which disk-dominated galaxies, for instance, merge to form bulge-dominated galaxies.
Interestingly, if that conclusion is correct, then these K+A galaxies are the progenitors of the bulge-dominated galaxies of the field, not of rich clusters, because the K+A galaxies live in the typical overdensities of spirals, not ellipticals. Past work on K+A galaxies has implied that they form a special cluster population [eg, @dressler83a], but this is presumably a consequence of the selection in those studies.
In order to put the steady-state predictions onto Figure \[fig:Poster\_aehist\], it was necessary to multiply the time spent at each value of ${\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}}$, or $|\mathrm{d}t/\mathrm{d}({\ensuremath{\textsf{A}}}/{\ensuremath{\textsf{K}}})|$ by a comoving rate density to match the total abundance. This rate density is 1.8 or 1.0 times $10^{-4}\,h^3~\mathrm{Mpc^{-3}\,Gyr^{-1}}$ for the “3” or “10” model respectively. In other words, the comoving K+A galaxy creation rate density is on this order at redshifts $z\sim 0.1$. This suggests that of order 1 percent of galaxies are currently K+A, and of order 10 percent have been through a K+A phase since redshift unity.
The A and K spectra used for this study will be made available with the electronically published version of this article.
We thank Douglas Finkbeiner, Michel Fioc, Robert Lupton, Bob Nichol, Jim Peebles, Don Schneider, David Spergel, Michael Strauss, and Christy Tremonti for useful ideas, conversations, and software. This research made use of the NASA Astrophysics Data System. ADQ, DWH, and MRB are partially supported by NASA (grant NAG5-11669) and NSF (grant PHY-0101738). DJE is supported by NSF (grant AST-0098577) and by an Alfred P. Sloan Research Fellowship.
Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy, the Max-Planck-Institute for Astrophysics, New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.
[lcccc]{}
|
---
author:
- 'Henri Anciaux[^1], Konstantina Panagiotidou[^2]'
title: 'Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms'
---
****
[The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by $J$ be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field $N$ is said to be *Hopf *if the tangent vector field $JN$ is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the *Hopf *curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.****]{}
*2010 MSC: 53C42, 53C40, 53B25 ***
Introduction {#introduction .unnumbered}
============
The study of real hypersurfaces in complex space forms, i.e. the complex projective space ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ and the complex hyperbolic space ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{H}^{}} }}^n$, have attracted a lot of attention in the last decades (see [@NR] for a survey of the subject and references therein). The complex structure $J$ of a complex space form induces a rich structure on real hypersurface; in particular, on an arbitrary oriented hypersurface ${{\cal S}}$ of ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ or ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{H}^{}} }}^n$ with unit vector normal field $N$, a canonical tangent field, called *the structure vector field *or *the Reeb vector field, *is defined by $\xi :=-JN$. If $\xi$ is a principal direction on ${{\cal S}}$, i.e. an eigenvector of the shape operator, ${{\cal S}}$ is called a *Hopf hypersurface*. It turns out that the principal curvature associated to the structure vector $\xi$ (the *Hopf principal curvature*) of a connected, Hopf hypersurface must be constant (this was proved in [@Ma] in the projective case and in [@KS] in the hyperbolic case). Moreover, in [@CR], Hopf hypersurfaces in ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ are locally characterized as tubes over complex submanifolds, while in [@Mo], the same statement is proved for Hopf hypersurfaces of ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{H}^{}} }}^n$ whose Hopf principal curvature $a$ satisfies $|a|>2$. Recently Hopf hypersurfaces of ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{H}^{}} }}^n$ with small Hopf principal curvature, i.e. satisfying $|a| \leq 2$, have been studied through a kind of generalized Gauss map in [@IR] and [@Iv], while in [@Ki] a unified approach is proposed, relating Hopf hypersurfaces to totally complex (or para-complex) submanifolds of some natural quaternionic manifold.********
The purpose of this paper is to address the study of real hypersurfaces in *pseudo-complex space forms *${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_p$, which are the pseudo-Riemannian generalizations of the complex space forms, and in *para-complex space form *${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$. The latter space is the para-complex analog of ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ and is equipped with both a pseudo-Riemannian metric and a *para-complex *structure, still denoted by $J$, which satisfies $J^2=Id$. Furthermore, given a real hypersurface in ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ with non-degenerate induced metric, the Hopf field is defined exactly as in the complex case. We refer to the next section for the precise definition of ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ and a brief description of its geometry. Since both the pseudo-complex and the para-complex case will be studied simultaneously, we define ${\epsilon}$ in such way that $J^2=-{\epsilon}Id$, i.e. ${\epsilon}=1$ corresponds to the complex case and ${\epsilon}=-1$ to the para-complex case. Moreover, ${{\cal M}}$ will denote the pseudo-Riemannian complex space form ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_p$ or the para-complex space form ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$, with holomorphic or para-holomorphic curvature $4c$, where $c: = \pm 1$.******
Our results are:
\[one\] There exist no umbilic real hypersurface, nor real hypersurface with parallel shape operator, in ${{\cal M}}$.
\[constant\] Let ${{\cal S}}$ be a connected, non-degenerate hypersurface of ${{\cal M}}$ which is Hopf, i.e. its structure vector $\xi$ is a principal direction of ${{\cal S}}$. Then the corresponding principal curvature $a$, i.e. defined by $A\xi = a\xi$, is constant.
\[tubes\]
Let ${{\cal S}}$ be a connected, non-degenerate hypersurface of ${{\cal M}}$ with unit normal $N$. Assume that ${{\cal S}}$ is Hopf and denote by $a$ the corresponding principal curvature, i.e. $A\xi = a\xi.$ Then if $c \, {\epsilon}{\langle}N,N{\rangle}=1$, or if $c \, {\epsilon}{\langle}N,N{\rangle}=-1$ and $|a|>2$, then ${{\cal S}}$ is, locally, a tube over a complex or para-complex submanifold.
*In the case $c=1, {\epsilon}=1$ and $p=0$, ${{\cal M}}$ is the complex projective space ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$, and if $c=-1, {\epsilon}=1$ and $p=n$, we have ${{\cal M}}={\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{H}^{}} }}^n$, the complex hyperbolic space. Hence Theorem \[tubes\] generalizes [@CR] and [@Mo]. Observe that in these two cases, the metric being positive, we have ${\langle}N,N{\rangle}=1$. ***
This paper is organized as follows: in Section \[s1\] the geometry of the pseudo-Riemannian complex and the para-complex space forms is described. Section \[s2\] contains basic relations about the geometry of real hypersurfaces in ${{\cal M}}$ and the proof of Theorem \[one\]. In Section \[s3\] four Lemmas about real hypersurfaces and the proof of Theorem \[constant\] are presented. Finally, in Section \[s4\] the proof of Theorem \[tubes\] is given and at the end of the Section some open problems are proposed for further research on this area.
The ambient spaces: pseudo-Riemannian complex and para-complex space forms {#s1}
==========================================================================
The abstract structures
-----------------------
All along the paper the ambient space will be a $2n$-dimensional pseudo-Riemannian manifold $({{\cal M}},{\langle}\cdot,\cdot{\rangle}, J)$ endowed with is a complex or para-complex structure $J$, i.e. a $(1,1)$ tensor field satisfying $J^2 = -{\epsilon}Id$ which is compatible with respect to ${\langle}\cdot,\cdot{\rangle}$, i.e.$${\langle}J \cdot,J\cdot{\rangle}= {\epsilon}{\langle}\cdot,\cdot{\rangle}.$$ In other words, $J$ is an isometry in the complex case and an anti-isometry in the para-complex case. This assumption implies that the signature of ${\langle}\cdot,\cdot{\rangle}$ must be even in the complex case and neutral in the para-complex case.
The bilinear map $\omega(X,Y) := {\langle}JX,Y{\rangle}$ is alternate and non-degenerate. Furthermore, the 2-form $\omega$ is closed, hence symplectic. Therefore, the triple $({\langle}\cdot,\cdot{\rangle},J, \omega)$ is a *pseudo-Kähler *or *para-Kähler *structure.****
We assume furthermore that the curvature ${R}$ of ${\langle}\cdot,\cdot{\rangle}$ satisfies $${R}(X,Y) ={c} \left( {\epsilon}X \wedge Y + J X \wedge JY + 2 {\langle}X,JY{\rangle}J \right),$$ where the notation $X\wedge Y$ denotes the operator $Z \to (X \wedge Y) Z= {\langle}Y,Z{\rangle}X -{\langle}X,Z{\rangle}Y$ and where $c$ is a real constant. Observe that if $X$ is a non-null vector, we have ${\langle}R(X,JX)JX,X{\rangle}=4c, $ i.e. any complex or para-complex $2$-plane $Span(X,JX)$ has sectional curvature $4c$. The constant $4c$ is called the *holomorphic *or *para-holomorphic *curvature of $({{\cal M}}, {\langle}\cdot,\cdot{\rangle},J)$.****
Observe that the rescaled ${\lambda}{\langle}\cdot,\cdot{\rangle}$, where ${\lambda}$ is a positive constant has holomorphic curvature ${\lambda}^{-2} c$. On the other hand, replacing the metric ${\langle}\cdot,\cdot{\rangle}$ by its opposite $-{\langle}\cdot,\cdot{\rangle}$ leaves invariant the curvature operator ${R}$. It follows that if $({{\cal M}},{\langle}\cdot,\cdot{\rangle},J)$ has (para-)holomorphic curvature $4c$, then $({{\cal M}},-{\langle}\cdot,\cdot{\rangle},J)$ has (para-)holomorphic curvature $-4c$.
In the next two sections instances of such manifolds will be described explicitly.
Pseudo-Riemannian complex space forms
-------------------------------------
We consider the space ${\ensuremath{{\mathbb{C}^{}} }}^{n+1}$ endowed with the pseudo-Hermitian form: $${\langle}{\langle}\cdot,\cdot{\rangle}{\rangle}_p = - \sum_{j=1}^p dz_j d\bar{z}_j + \sum_{j=p+1}^{n+1} dz_j d\bar{z}_j$$ The corresponding metric ${\langle}\cdot,\cdot{\rangle}_{2p} := \Re{\langle}{\langle}\cdot,\cdot{\rangle}{\rangle}_p$ has signature $(2p, 2(n+1-p)).$ We define the hyperquadrics $${\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{p,c}:=\{ z \in {\ensuremath{{\mathbb{C}^{}} }}^{n+1} | \, {\langle}z,z{\rangle}_{2p} =c \},$$ For example, ${\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{0,1} = {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}$ is the round unit sphere; The pseudo-Riemannian complex space forms are the quotients of these hyperquadrics by the natural ${\ensuremath{{\mathbb{S}^{}} }}^1$-action: $${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{p,c} := {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{2p,c} \slash \sim,$$ where $ z \sim z' $ if there exists ${\theta}\in {\ensuremath{{\mathbb{R}^{}} }}$ such that $z' = (\cos {\theta}, \sin {\theta}) .z$. In particular
- ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^{n}_{0,1} = {\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ is the complex projective space;
- ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{n,-1} ={\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{H}^{}} }}^n$ is the complex hyperbolic space;
We denote by $\pi$ the canonical projection $\pi: {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{2p,c} \to {\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{p,c}. $ We endow ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_p$ with the metric ${\langle}\cdot,\cdot{\rangle}$ that makes the projection $\pi$ a pseudo-Riemannian submersion. The projection $\pi: {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{2p,c} \to {\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{p,c}$ also induces a natural complex structure in ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{p,c}$. It is easy to check that $({\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{p, c},J,{\langle}\cdot,\cdot{\rangle})$ is a pseudo-Kähler manifold and that its curvature tensor satisfies $$R(X,Y) ={c} \big( X \wedge Y + J X \wedge JY + 2 {\langle}X,JY{\rangle}J \big).$$ In particular, ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{p,c}$ has constant holomorphic curvature $4c$.
Observe that the involutive map $ (z_1, \ldots , z_n ) \mapsto (z_{p+1}, \ldots , z_n, z_1, \ldots z_p)$ is an anti-isometry between ${\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{p, c} $ and ${\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{n+1-p, -c}$. It follows that the spaces ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{p,c}$ and ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_{n+1-p, -c}$ are anti-isometric.
Para-complex space forms
------------------------
The set of *para-complex *(or *split-complex, *or *double*) numbers ${\ensuremath{{\mathbb{D}^{}} }}$ is the two-dimensional real vector space ${\ensuremath{{\mathbb{R}^{}} }}^2$ endowed with the commutative algebra structure whose product rule is given by $$(x , y) . (x' , y')= (xx'+yy', xy'+x'y).$$ The para-complex projective plane is the set of para-complex lines of ${\ensuremath{{\mathbb{D}^{}} }}^{n+1}$. We consider the neutral metric $${\langle}\cdot,\cdot{\rangle}_*:= \sum_{j=1}^n dx_j^2- dy_j^2$$ and the hyperquadric $${\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{n+1,-1}:= \{ z \in {\ensuremath{{\mathbb{D}^{}} }}^{n+1} | \, {\langle}z,z{\rangle}_* =-1 \} .$$ Then we define: $${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n := {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{n+1,-1} \slash \sim,$$ where $ z \sim z' $ if there exists ${\theta}\in {\ensuremath{{\mathbb{R}^{}} }}$ such that $z' = (\cosh {\theta}, \sinh {\theta}) .z$. We endow ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ with the metric $g$ that makes the projection $\pi: {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{n,-1} \to {\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ a pseudo-Riemannian submersion. The metric $g$ has neutral signature $(n,n)$. For technical reasons it is convenient to introduce the “polar” space $\overline{{\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n}$ of ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ by $$\overline{{\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n} := {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{n+1,1} \slash \sim.$$ The anti-isometry $J$ of ${\ensuremath{{\mathbb{D}^{}} }}^{n+1}$ induces canonically an anti-isometry between ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ and $ \overline{{\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n}$.******
According to [@GM], the curvature operator of ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ is given by $$R(X,Y) =-X \wedge Y + J X \wedge JY + 2 {\langle}X,JY{\rangle}J .$$ In particular, ${\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ has constant para-holomorphic curvature $4$ (but it is not characterized by this property). On the other hand, $\overline{{\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n}$ has constant para-holomorphic curvature $-4$.
Auxiliary relations about real hypersurfaces and proof of Theorem \[one\] {#s2}
==========================================================================
In this section let ${{\cal S}}$ be an immersed real hypersurface in ${{\cal M}}$, whose induced metric is non-degenerate. This implies the local existence of a unit normal vector field $N$. After a possible change of metric ${\langle}\cdot,\cdot{\rangle}\to -{\langle}\cdot,\cdot{\rangle},$ there is no loss of generality in assuming that ${\langle}N,N{\rangle}=1$, and we will do so in the remainder of the paper. Observe that reversing the metric has the effecting of reversing its curvature $c$. Hence, without loss of generality, we could alternatively assume that $c=1$ and let ${\langle}N,N{\rangle}$ take the two possible values $\pm 1$. However the first choice seems more natural.
The structure of a real hypersurface in [[M]{}]{}
-------------------------------------------------
The *structure vector field* $\xi$ is given by $$\begin{aligned}
\label{a1}
\xi:=-{\epsilon}JN.\end{aligned}$$ It follows that $N=J\xi$ and that ${\langle}\xi,\xi{\rangle}={\epsilon}$. The orthogonal complement $\xi^\perp:= {\mbox{Hor}}$, a $(2n-2)$-dimensional subspace of $T {{\cal S}}$, will be refered as to the *horizontal distribution. *Given a vector $X$ tangent to ${{\cal S}}$, the vector $JX$ is not necessarily tangent to ${{\cal S}}$ but its tangential part, that we denote by $\varphi$, is horizontal. Introducing the one-form $\eta:={\langle}J \cdot,N{\rangle}$, we have $$\begin{aligned}
\label{a3}
JX&=&\varphi X+ {\langle}JX,N {\rangle}N \\
&=&\varphi X + \eta(X)N. \nonumber\end{aligned}$$ Observe also that $$\eta(X)={\langle}JX,N{\rangle}=-{\langle}X,JN{\rangle}={\epsilon}{\langle}X,\xi{\rangle}$$ and $$\begin{aligned}
\label{a2}
\eta(\xi)={\langle}J\xi,N{\rangle}={\langle}N,N{\rangle}=1.\end{aligned}$$ On the other hand, doing $X=\xi$ in Equation (\[a3\]), we get $$\begin{aligned}
\label{a4}
\varphi\xi=0.\end{aligned}$$**
Now, we have $$\begin{aligned}
-{\epsilon}X&=&J^{2}X \nonumber \\
&=&J \big(\varphi X+ \eta(X)N \big)\nonumber\\
&=&J(\varphi X) + \eta(X) JN \nonumber\\
&=&\varphi (\varphi X) + \eta(\varphi X)N+ \eta(X)JN.\end{aligned}$$ Considering the tangent and normal parts of this equation, we get that $\eta \circ \varphi =0$ and $$-{\epsilon}X = \varphi^2 X -{\epsilon}\eta(X)\xi,$$ so that $$\begin{aligned}
\label{a5}
\varphi^2 = -{\epsilon}Id +{\epsilon}\eta ( \cdot) \xi.\end{aligned}$$ Finally, denoting by $g$ the induced metric on ${{\cal S}}$, we have the following relation: $$\begin{aligned}
\label{a6}
{\langle}\varphi X, \varphi Y{\rangle}&=&{\langle}JX- \eta(X)N,JY- \eta(Y)N{\rangle}\nonumber\\
&=&{\langle}JX,JY{\rangle}-\eta(X) {\langle}N,JY{\rangle}- \eta(Y){\langle}N,JX{\rangle}+\eta(X)\eta(Y) {\langle}N,N{\rangle}\nonumber\\
&=&{\epsilon}{\langle}X,Y{\rangle}- 2 \eta(X)\eta(Y) +\eta(X)\eta(Y) \nonumber\\
&=&{\epsilon}{\langle}X,Y{\rangle}- \eta(X)\eta(Y).\end{aligned}$$
We conclude that, according to Relations (\[a1\]), (\[a2\]), (\[a4\]), (\[a5\]) and (\[a6\]), the quadruple $(\varphi,\eta,\xi,{\langle}\cdot,\cdot{\rangle})$ defines an *almost contact metric structure* on ${{\cal S}}$ when ${\epsilon}=1$ and an *almost para-contact metric structure* on ${{\cal S}}$ when ${\epsilon}=-1$.
The Gauss and the Weingarten formulas are respectively given by the equations $$\begin{aligned}
{\nabla}_{X}Y&=&\overline{\nabla}_{X}Y+ {\langle}AX,Y{\rangle}N \label{Gauss-formula}\\
{\nabla}_{X}N&=&-AX,\label{Weingarten-formula}\end{aligned}$$ where $\nabla$ and $\overline{\nabla}$ are the Levi-Civita connection on ${{\cal M}}$ and ${{\cal S}}$ respectively and $A$ is the shape operator of ${{\cal S}}$ with respect to $N$. Denoting by $\overline{R}$ and $R$ the curvature of $\overline{\nabla}$ and $\nabla$ respectively, the Gauss equation takes the form: $${\langle}{R}(X,Y)Z, W{\rangle}= {\langle}\overline{R}(X,Y)Z,W{\rangle}+ {\langle}AX, Z{\rangle}{\langle}AY, W{\rangle}-{\langle}AX, W{\rangle}{\langle}AY,Z{\rangle}$$ for $X,Y,Z$ and $W$ tangent to ${{\cal S}}$. Hence $$\begin{aligned}
\overline{R}(X,Y)Z &=&c\Big( {\epsilon}X \wedge Y + J X \wedge JY + 2 {\langle}X,JY{\rangle}\Big)Z + {\langle}AY,Z{\rangle}AX - {\langle}AX,Z{\rangle}AY\\
&=&c \Big( {\epsilon}X \wedge Y + \varphi X \wedge \varphi Y + 2 {\langle}X, \varphi Y{\rangle}\varphi \Big) Z + (AX \wedge AY)Z,\end{aligned}$$ so that $$\overline{R}(X,Y)= AX \wedge AY + c\left( {\epsilon}X \wedge Y + \varphi X \wedge \varphi Y + 2 {\langle}X, \varphi Y{\rangle}\varphi \right).$$ We now deal with Codazzi equation: for $X,Y$ and $Z$ tangent to ${{\cal S}}$, we have $${\langle}R(X,Y)Z, N{\rangle}= {\langle}(\overline{\nabla}_X A)Y -(\overline{\nabla}_Y A)X, Z {\rangle}.$$ Using the expression of ${R}$, we have $$\begin{aligned}
{\langle}R(X,Y)Z, N{\rangle}&=&c\Big( {\epsilon}{\langle}(X \wedge Y)Z,N{\rangle}+{\langle}( J X \wedge JY )Z,N{\rangle}+
2 {\langle}X,JY{\rangle}{\langle}JZ,N{\rangle}\Big) \\
&=&c\Big( {\epsilon}{\langle}Y,Z{\rangle}{\langle}X,N{\rangle}- {\epsilon}{\langle}X,Z{\rangle}{\langle}Y, N{\rangle}+ {\langle}JY,Z{\rangle}{\langle}JX,N{\rangle}- {\langle}JX,Z{\rangle}{\langle}JY, N{\rangle}\\
& & - 2{\langle}X, \varphi Y{\rangle}{\langle}Z, JN{\rangle}\Big)\\
&=&c \ \Big( {\langle}\varphi Y,Z{\rangle}\eta(X)- {\langle}\varphi X,Z{\rangle}\eta(Y) + 2{\epsilon}{\langle}X, \varphi Y{\rangle}{\langle}Z, \xi{\rangle}\Big)\end{aligned}$$ to get $$\begin{aligned}
\label{Codazzi-equation}
(\overline{\nabla}_X A)Y - (\overline{\nabla}_Y A)X = c \Big( \eta(X) \varphi Y - \eta(Y) \varphi X +
2 {\epsilon}{\langle}X, \varphi Y{\rangle}\xi \Big).
\end{aligned}$$
Proof of Theorem \[one\]
------------------------
The proof is an easy consequence of the Codazzi equation.
Assume first that ${{\cal S}}$ is umbilic, i.e. there exists $\lambda \in C^{\infty}({{\cal S}})$ such that $A=\lambda Id$. Then the Codazzi equation (\[Codazzi-equation\]) becomes: $$(X\cdot\lambda)Y-(Y\cdot\lambda)X=c \Big( \eta(X)\varphi Y- \eta(Y)\varphi X +2{\epsilon}{\langle}X,\varphi Y{\rangle}\xi \Big),$$ Taking $X$ horizontal and non-vanishing, and $Y=\xi$ yields $$(X\cdot\lambda)\xi-(\xi\cdot\lambda)X=c \varphi X.$$ The inner product of the above relation with $\varphi X$ implies $c=0$, which is a contradiction.
Assume now that ${{\cal S}}$ has parallel shape operator, i.e. $(\overline{\nabla}_{X}A)Y=0$, for any tangent vectors $X,Y$. Then the Codazzi equation becomes $$0 = c \Big( \eta(X)\varphi Y-\eta(Y)\varphi X +2 {\epsilon}{\langle}X,\varphi Y{\rangle}\xi \Big).$$ Taking $X$ horizontal and non-vanishing, and $Y=\xi$ yields $c \varphi X=0.$ Since $\varphi X$ does not vanish, we get $c=0$, a contradiction.
Proof of Theorem \[constant\] {#s3}
=============================
Before providing the proof of Theorem some basic Lemmas which hold for real hypersurfaces in ${{\cal M}}$ are given.
Basic Lemmas
------------
\[Prop13\] \[l1\] Let [[S]{}]{}be a real hypersurface in ${{\cal M}}$. Then: $$\begin{aligned}
\label{b1}
\overline{\nabla}_X \xi = {\epsilon}\varphi A X\end{aligned}$$ and $$\begin{aligned}
\label{b2}
(\overline{\nabla}_X \varphi)Y = \eta(Y) AX-{\epsilon}{\langle}AX,Y {\rangle}\xi .\end{aligned}$$
Using successively Gauss equation, Weingarten equation and Equation (\[a3\]), we first calculate $$\begin{aligned}
\overline{\nabla}_X \xi &=&{\nabla}_{X} \xi - {\langle}AX, \xi{\rangle}N\\
&=&-{\epsilon}{\nabla}_X JN - {\langle}AX, \xi{\rangle}N\\
&=& -{\epsilon}J \nabla_X N - {\langle}AX, \xi{\rangle}N \\
&=& {\epsilon}J A X- {\langle}AX,\xi{\rangle}N\\
&=&{\epsilon}\varphi AX + \eta(AX) - {\langle}AX,\xi{\rangle}N.\end{aligned}$$ Taking the tangential part of this yields Equation (\[b1\]).
As for Equation (\[b2\]), using again the Gauss, Weingarten equation and Equation (\[a3\]), we have $$\begin{aligned}
(\overline{\nabla}_X \varphi ) Y&=& \overline{\nabla}_X (\varphi Y) -\varphi (\overline{\nabla}_X Y)\\
&=& \nabla_X (\varphi Y) - {\langle}AX, \varphi Y{\rangle}N -\varphi (\overline{\nabla}_X Y)\\
&=&\nabla_X \big(JY -{\epsilon}\eta(Y)N \big) - {\langle}AX, \varphi Y{\rangle}N -\varphi (\overline{\nabla}_X Y)\\
&=&J \nabla_X Y -{\epsilon}\nabla_X (\eta(Y)N ) - {\langle}AX, \varphi Y{\rangle}N -\varphi (\overline{\nabla}_X Y)\\
&=&J \big( \overline{\nabla}_X Y + {\langle}AX,Y{\rangle}N \big)
-{\epsilon}\big( {\langle}\nabla_X Y, \xi{\rangle}N +{\langle}Y,\nabla_X \xi{\rangle}N + {\langle}Y,\xi{\rangle}\nabla_X N \big) \\
&& \quad \quad - {\langle}AX, \varphi Y{\rangle}N -\varphi (\overline{\nabla}_X Y)\\
&=& \eta(\overline{\nabla}_X Y) N + {\langle}AX,Y{\rangle}JN
- {\epsilon}\big( {\langle}\overline{\nabla}_X Y, \xi{\rangle}N +{\langle}Y,\overline{\nabla}_X \xi{\rangle}N - {\langle}Y,\xi{\rangle}AX \big) \\
&& \quad \quad - {\langle}AX, \varphi Y{\rangle}N \\
&=& -{\epsilon}{\langle}AX,Y{\rangle}\xi - {\langle}Y, \varphi AX{\rangle}N + {\epsilon}{\langle}Y,\xi{\rangle}AX + {\langle}\varphi AX,Y{\rangle}N\\
&=& \big( \eta(Y) AX - {\epsilon}{\langle}AX,Y{\rangle}\xi\big).\end{aligned}$$
\[l2\] The following two relations hold on a hypersurface ${{\cal S}}$ of ${{\cal M}}$: $$\begin{aligned}
{\langle}(\overline{\nabla}_X A)Y-(\overline{\nabla}_Y A)X,\xi {\rangle}=2c {\langle}X,\varphi Y {\rangle},\label{b3}\\
{\langle}(\overline{\nabla}_{X}A)\xi, \xi{\rangle}={\langle}(\overline{\nabla}_{\xi}A)X, \xi{\rangle}={\langle}(\overline{\nabla}_{\xi}A)\xi,X{\rangle}.\label{b4}\end{aligned}$$
Taking the inner product of Codazzi equation (\[Codazzi-equation\]) with $\xi$, recalling that ${\langle}\xi,\xi{\rangle}=~{\epsilon}$, implies (\[b3\]).
The first equality in (\[b4\]) is a particular case of (\[b3\]) making $Y=\xi$. For the second equality in (\[b4\]) we have $$\begin{aligned}
{\langle}(\overline{\nabla}_{\xi}A)\xi,X{\rangle}&=&{\langle}\overline{\nabla}_\xi(A\xi), X{\rangle}-{\langle}A\overline{\nabla}_{\xi}\xi,X{\rangle}\\
&=&\xi\cdot {\langle}A\xi,X{\rangle}-{\langle}A\xi,\overline{\nabla}_{\xi}X{\rangle}-{\langle}\overline{\nabla}_{\xi}\xi,AX{\rangle}\\
&=&\xi\cdot {\langle}A\xi,X{\rangle}-{\langle}A\xi,\overline{\nabla}_\xi X{\rangle}-\xi \cdot {\langle}\xi, AX{\rangle}+{\langle}\xi,\overline{\nabla}_\xi (AX){\rangle}\\
&=&-{\langle}\xi,A\overline{\nabla}_\xi X{\rangle}+{\langle}\xi,\overline{\nabla}_\xi (AX){\rangle}\\
&=&{\langle}\xi, (\overline{\nabla}_\xi A)X{\rangle}.\end{aligned}$$
\[l3\] Let ${{\cal S}}$ be a Hopf hypersurface in ${{\cal M}}$ and $a$ the Hopf curvature, i.e. $A\xi = a\xi.$ Then the following relations hold on [[S]{}]{}$$\begin{aligned}
&&grad\; a={\epsilon}(\xi\cdot a)\xi, \label{b5}\\
&&A\varphi A-\frac{a}{2}(A\varphi+\varphi A)-c {\epsilon}\varphi=0, \label{b6}\\
&&(\xi\cdot a)(\varphi A+A\varphi)=0 \label{b7}.\end{aligned}$$
— Proof of (\[b5\]): we first calculate, using several times Equation (\[b1\]), $$\begin{aligned}
(\overline{\nabla}_X A)\xi &=& \overline{\nabla}_X (A\xi) - A \overline{\nabla}_X \xi \\
&=& (X \cdot a ) \xi + a \overline{\nabla}_X \xi -{\epsilon}A \varphi A X\\
&=& (X \cdot a ) \xi + {\epsilon}a\varphi AX - {\epsilon}A \varphi A X.
$$ Hence we obtain $$\begin{aligned}
\label{c1}
(\overline{\nabla}_X A)\xi &=& (X \cdot a ) \xi + {\epsilon}(a Id - A) \varphi A X.\end{aligned}$$ Taking the inner product of (\[c1\]) with $\xi$ yields (taking into account that ${\langle}\xi, \xi{\rangle}=~\!{\epsilon}$) $$\begin{aligned}
{\langle}(\overline{\nabla}_X A)\xi, \xi {\rangle}= {\epsilon}(X \cdot a )={\epsilon}{\langle}grad \; a , X{\rangle}. \label{c2}\end{aligned}$$ On the other hand, making $X=\xi$ and recalling that $\varphi \xi$ vanishes, we get $$\begin{aligned}
(\overline{\nabla}_\xi A )\xi = (\xi \cdot a ) \xi.\end{aligned}$$ Putting together these last two equations, we conclude, using Lemma \[l2\], $$\begin{aligned}
{\langle}grad \; a , X{\rangle}&=& {\epsilon}{\langle}(\overline{\nabla}_X A)\xi, \xi {\rangle}\\
&=& {\epsilon}{\langle}(\overline{\nabla}_\xi A)\xi, X {\rangle}\\
&=& {\epsilon}(\xi \cdot a ) {\langle}\xi,X{\rangle},\end{aligned}$$ from which Equation (\[b5\]) follows.
— Proof of (\[b6\]): first, by an easy calculation, $${\langle}(\overline{\nabla}_X A)Y,\xi{\rangle}={\langle}(\overline{\nabla}_X A)\xi,Y{\rangle}$$ Then, using Equations (\[c1\]), (\[c2\]) and Lemma \[l2\], we get $$\begin{aligned}
{\langle}(\overline{\nabla}_X A)Y,\xi{\rangle}&=&{\langle}(\overline{\nabla}_X A)\xi,Y{\rangle}\\
&=& (X \cdot a){\langle}\xi,Y{\rangle}+ {\epsilon}{\langle}(aId - A)\varphi AX,Y{\rangle}\\
&=&{\epsilon}{\langle}(\overline{\nabla}_ \xi A)\xi, X{\rangle}{\langle}\xi,Y{\rangle}+{\epsilon}{\langle}(a Id - A)\varphi AX,Y{\rangle}\\
&=& {\epsilon}(\xi \cdot a) {\langle}\xi,X{\rangle}{\langle}\xi,Y{\rangle}+{\epsilon}{\langle}(a Id - A)\varphi AX,Y{\rangle}.\end{aligned}$$ Interchanging $X$ and $Y$ and substracting, we calculate $$\begin{aligned}
{\langle}(\overline{\nabla}_X A)Y-(\overline{\nabla}_Y A)X,\xi{\rangle}&=&
{\epsilon}\big( {\langle}(a Id - A)\varphi AX,Y{\rangle}- {\langle}(a Id - A)\varphi AY,X{\rangle}\big).\end{aligned}$$ Now, from (\[b3\]) (Lemma \[l2\]) this implies $$\begin{aligned}
{\epsilon}\big( {\langle}(a Id - A)\varphi AX,Y{\rangle}- {\langle}(a Id - A)\varphi AY,X{\rangle}\big)&=&2 c {\epsilon}{\langle}X,\varphi Y{\rangle}{\langle}\xi,\xi{\rangle}\\
&=& 2 c {\langle}X,\varphi Y{\rangle}.\end{aligned}$$ It follows, using the facts that $A$ is self-adjoint (and therefore $aId -A$ as well) and that $\varphi$ is skew-symmetric, that $$\begin{aligned}
2{\epsilon}c {\langle}X,\varphi Y{\rangle}&=& {\langle}(a Id - A)\varphi AX,Y{\rangle}- {\langle}(a Id - A)\varphi AY,X{\rangle}\\
&=& -{\langle}X, A \varphi (aId -A) Y{\rangle}- {\langle}X, (a Id - A)\varphi AY{\rangle},\end{aligned}$$ which implies that $$\begin{aligned}
2{\epsilon}c \varphi &=&- A \varphi (aId -A) - (a Id - A)\varphi A\\
&=& -a(A\varphi + \varphi A) +2 A\varphi A, \end{aligned}$$ from which Equation (\[b6\]) follows.
— Proof of (\[b7\]): setting ${\beta}:= {\epsilon}\xi \cdot a$ (so in particular $\mbox{grad}\;a = {\beta}\xi$), we have $$\begin{aligned}
{\langle}\overline{\nabla}_X (\mbox{grad}\;a) , Y {\rangle}-{\langle}\overline{\nabla}_Y (\mbox{grad}\;a) , X {\rangle}&=& X \cdot{\langle}\mbox{grad}\;a , Y {\rangle}- {\langle}\mbox{grad}\;a , \overline{\nabla}_X Y {\rangle}\\
&&\; \; \; \quad -Y \cdot {\langle}\mbox{grad}\;a , X {\rangle}+ {\langle}\mbox{grad}\;a , \overline{\nabla}_Y X {\rangle}\\
&=&X \cdot (Y \cdot a ) - Y \cdot (X \cdot a ) + {\langle}\mbox{grad}\;a , \overline{\nabla}_X Y - \overline{\nabla}_Y X {\rangle}\\
&=& \left([X,Y]-\overline{\nabla}_X Y - \overline{\nabla}_Y X {\rangle}\right) \cdot a \\
&=&0.\end{aligned}$$ It follows that $$\begin{aligned}
\label{c3}
0&=&{\langle}\overline{\nabla}_X {\beta}\xi , Y {\rangle}-{\langle}\overline{\nabla}_Y {\beta}\xi , X {\rangle}\nonumber \\
&=& (X \cdot {\beta}) {\langle}\xi,Y{\rangle}- {\beta}{\langle}\varphi A X,Y{\rangle}-(Y \cdot {\beta}) {\langle}\xi,X{\rangle}- {\beta}{\langle}\varphi A Y,X{\rangle}\nonumber \\
&=& (X \cdot {\beta}) {\langle}\xi,Y{\rangle}-(Y \cdot {\beta}) {\langle}\xi,X{\rangle}- {\beta}{\langle}(\varphi A +A \varphi )X,Y{\rangle}.\end{aligned}$$ Making $Y=\xi$ yields $$\begin{aligned}
0&=& X \cdot {\beta}{\langle}\xi,\xi{\rangle}-(\xi \cdot {\beta}) {\langle}\xi,X{\rangle}- {\beta}{\langle}(\varphi A +A \varphi )X,\xi{\rangle}\\
&=&{\epsilon}X \cdot {\beta}-(\xi \cdot {\beta}) {\langle}\xi,X{\rangle}- {\beta}{\langle}A \varphi X,\xi{\rangle}\\
&=&{\epsilon}X \cdot {\beta}- (\xi \cdot {\beta}) {\langle}\xi,X{\rangle}- {\beta}{\langle}\varphi X,a\xi{\rangle}\\
&=&{\epsilon}X \cdot {\beta}- (\xi \cdot {\beta}) {\langle}\xi,X{\rangle}.\end{aligned}$$ Hence we have $X \cdot {\beta}={\epsilon}(\xi \cdot {\beta}) {\langle}\xi,X{\rangle}$, which implies that $ (X \cdot {\beta}) {\langle}\xi,Y{\rangle}=(Y \cdot {\beta}) {\langle}\xi,X{\rangle}.$ Combining with (\[c3\]) yields $${\beta}\, {\langle}(\varphi A +A \varphi )X,Y{\rangle},$$ which implies the desired identity.
\[l4\] If $X$ is an principal vector of $A$ with principal curvature ${\kappa}$, then $\varphi X$ is a principal vector with principal curvature $$\bar{{\kappa}}:=\frac{ {\kappa}a+2 c {\epsilon}}{2{\kappa}-a}.$$ In particular the principal subspace $E_{\kappa}:=\{ X \in T{{\cal S}}| \, AX ={\kappa}X\}$ is $\varphi$-invariant if and only if ${\kappa}^2 - a{\kappa}- c {\epsilon}=0$.
We write Equation (\[b6\]) in the case when $A X={\kappa}X$: $${\kappa}A\varphi X-\frac{a}{2}(A\varphi X+ {\kappa}\varphi X)-c {\epsilon}\varphi X=0,$$ so that $$({\kappa}-\frac{a}{2}) A\varphi X = (c {\epsilon}+ \frac{a {\kappa}}{2}) \varphi X,$$ i.e. $$A \varphi X= \frac{ {\kappa}a+2 c{\epsilon}}{2{\kappa}-a} \varphi X,$$ so we get the required expression for $\bar{{\kappa}}$ satisfying $A (\varphi X) = \bar{{\kappa}} (\varphi X)$. Finally, is $E_{\kappa}$ is $\varphi$-stable, we must have ${\kappa}= \frac{ {\kappa}a+2 c {\epsilon}}{2{\kappa}-a}$, which implies the last claim of the Lemma.
*Proof of Theorem \[constant\].*
We proceed by contradiction. By Equation (\[b5\]) if $a$ is not constant, then $\xi\cdot a \neq0$. Consider $Z$ a horizontal vector. So $\eta(Z)=0$ and, by Lemma \[l1\], we have $(\overline{\nabla}_\xi A) Z=0$. It follows that $$\begin{aligned}
\label{r1}
0 &=& \overline{\nabla}_{\xi} \big( (\varphi A+A\varphi)Z \big) \nonumber \\
&=&
\varphi(\overline{\nabla}_{\xi}A)Z+(\overline{\nabla}_{\xi}A)\varphi Z.\end{aligned}$$ We now write the Codazzi equation (\[Codazzi-equation\]) with $X=\xi$ and $Y=Z$, yields, using (\[c1\]): $$\begin{aligned}
(\overline{\nabla}_{\xi}A)Z&=&(\overline{\nabla}_{Z}A)\xi + c \Big( \eta(\xi) \varphi Z - \eta(Z) \varphi \xi +
2{\epsilon}{\langle}\xi, \varphi Z{\rangle}\xi \Big)\\
&=& (\overline{\nabla}_{Z}A)\xi + c \varphi Z \\
&=& (Z\cdot a)\xi+ {\epsilon}( aId -A)\varphi AZ+c \varphi Z\\
&=& {\epsilon}( aId -A)\varphi AZ+c \varphi Z.\end{aligned}$$ (Equation (\[b5\]) implies that $ (Z\cdot a)$ vanishes). It follows that $$\begin{aligned}
\varphi(\overline{\nabla}_{\xi}A)Z&=& {\epsilon}\varphi ( aId -A)\varphi AZ+ c \varphi^2 Z\\
&=&{\epsilon}a \varphi^2 AZ -{\epsilon}\varphi A\varphi A Z - c {\epsilon}Z\\
&=& -a AZ - A^2 Z -c {\epsilon}Z.
\end{aligned}$$ Analogously, the Codazzi equation with $X=\xi$ and $Y=\varphi Z$ $$\begin{aligned}
(\overline{\nabla}_{\xi}A)\varphi Z&=&(\overline{\nabla}_{\varphi Z}A)\xi + c \Big( \eta(\xi) \varphi^2 Z - \eta(\varphi Z) \varphi \xi +
2 {\epsilon}{\langle}\xi, \varphi^2 Z{\rangle}\xi \Big)\\
&=& (\overline{\nabla}_{\varphi Z}A)\xi - c{\epsilon}Z \\
&=& (\varphi Z\cdot a)\xi+ {\epsilon}( aId -A)\varphi A \varphi Z - c {\epsilon}Z\\
&=&{\epsilon}( aId -A)\varphi A \varphi Z - c {\epsilon}Z\\
&=&- a AZ +A^2 Z - c {\epsilon}Z.\end{aligned}$$ By (\[r1\]) we deduce that $$a AZ = -c {\epsilon}Z.$$ This implies $a $ does not vanish, and moreover that the restriction of $A$ to the horizontal space is $-c {\epsilon}a^{-1} Id.$ Since the horizontal space is $\varphi$-stable, it follows from Lemma \[l4\], that $$\big(-c {\epsilon}a^{-1}\big)^2 -a \big(- c {\epsilon}a^{-1}\big) -c {\epsilon}=0,$$ i.e. $$\frac{c^2}{a^2} + c {\epsilon}-c {\epsilon}=0.$$ This implies $c=0$, a contradiction.
Proof of the Theorem \[tubes\] {#s4}
==============================
We recall that ${{\cal M}}:= {\ensuremath{{\mathbb{D}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n$ or ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{P}^{}} }}^n_p$ and that $\widetilde{{{\cal M}}}:= {\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{n,-c}$ or ${\ensuremath{{\mathbb{S}^{}} }}^{2n+1}_{2p, c}$ is a bundle over ${{\cal M}}$ with projection $\pi.$ We still denote by ${\langle}\cdot,\cdot{\rangle}$ the metric on $\widetilde{{{\cal M}}}$. We shall denote by $\widetilde{\nabla}$ the Levi-Civita connection on $\widetilde{{{\cal M}}}$. This is nothing but the tangential part of the flat connection of ${\ensuremath{{\mathbb{R}^{}} }}^{2n+2}$. We denote by $\widetilde{J}$ the complex (resp. para-complex) structure of ${\ensuremath{{\mathbb{C}^{}} }}^{n+1}$ (resp. ${\ensuremath{{\mathbb{D}^{}} }}^{n+1}$).
Let $F: U \to {{\cal M}}$ be a local parametrization of ${{\cal S}}$ and $\widetilde{F}: {{\cal S}}\to \widetilde{{{\cal M}}}$ a lift of $F$, i.e. $\pi \circ \widetilde{F} = F$. In particular ${\langle}\widetilde{F},\widetilde{F}{\rangle}=c{\epsilon}$. Observe that the choice of $\widetilde{F}$ is not unique, but none of them satisfies ${\langle}d\widetilde{F}(\cdot),\widetilde{J} \tilde{F}{\rangle}=0$, because the integral submanifolds of the hyperplane distribution $\widetilde{J}^\perp$ have at most dimension $n$ (Legendrian submanifolds).
By a slight abuse of notation, we still denote by $N$ the composition of the unit normal vector field on $F(U)$ with $F$. In other words, $N : U \to T{{{\cal M}}}$. Let $\widetilde{N}$ be a lift of $N$, i.e. $\widetilde{N} : U \to T\widetilde{{{\cal M}}}$ such that $$\begin{aligned}
d \pi_{\widetilde{F}(x)} \circ \widetilde{N}(x) &=&N(x), \, \forall x \in U.\end{aligned}$$ In particular $\widetilde{N} \in \widetilde{{{\cal M}}}$ (resp. $\overline{\widetilde{{{\cal M}}}}$) if ${\epsilon}=1$ (resp. ${\epsilon}=-1$) and ${\langle}\widetilde{N},\widetilde{N}{\rangle}=1$. Moreover, we have $${\langle}d\widetilde{F}(\cdot), \widetilde{N}{\rangle}=0.$$ Since the immersion $d\widetilde{F}$ has co-dimension 2, the choice of $\widetilde{N}$ is not unique. Since moreover $d\widetilde{F}$ is tranverse to the vector field $\widetilde{J} \widetilde{F}$, we may choose $\widetilde{N}$ in such a way that $${\langle}\widetilde{N}, \widetilde{J} \widetilde{F}{\rangle}=0.$$
We denote by $\xi$ the tangent vector field on $U$ such that $\Xi := dF(\xi) = -JN$. We also set $\widetilde{\Xi}:= -\widetilde{J}\widetilde{N}.$ Of course $\widetilde{\Xi}$ is the lift of $\Xi.$
We now assume that $F(U)$ is Hopf, i.e. $A \xi = a \xi$. By Theorem \[constant\], $a $ is some real constant. We need some extra notation: we set ${\epsilon}':= c {\epsilon}$ and $$({{\texttt{cos}\varepsilon}}', {{\texttt{sin}\varepsilon}}' ) := \left\{ \begin{array}{ll} (\cos ,\sin ) & \mbox{ if } {\epsilon}'=1,\\
(\cosh ,\sinh ) & \mbox{ if } {\epsilon}'=-1.
\end{array} \right.$$ We also set the obvious notations ${{\texttt{tan}\varepsilon}}':=\frac{{{\texttt{sin}\varepsilon}}'}{{{\texttt{cos}\varepsilon}}'}$ and ${{\texttt{cot}\varepsilon}}':=\frac{{{\texttt{cos}\varepsilon}}'}{{{\texttt{sin}\varepsilon}}'}$. Finally, we introduce $f:= \pi \circ \widetilde{f} : U \to {{\cal M}}$, where $$\widetilde{f}:= {{\texttt{cos}\varepsilon}}'({\theta}) \widetilde{F} + {{\texttt{sin}\varepsilon}}' ({\theta}) \widetilde{N}$$ and ${\theta}$ is some real constant to be determined later.
By Proposition 10, p. 97 of [@An1] (see also [@ON]), we have
$$\begin{aligned}
\widetilde{\nabla}_{\widetilde{\Xi}} \widetilde{N} &=& \widetilde{\nabla_{\Xi} N} +
{\langle}\widetilde{J} \widetilde{F},\widetilde{J} \widetilde{F}{\rangle}{\langle}\widetilde{\Xi}, \widetilde{J} \widetilde{N}{\rangle}\widetilde{J} \widetilde{F}\\
&=& \widetilde{\nabla_{\Xi} N} - c \widetilde{J} \widetilde{F}\\
&=&- a \widetilde{\Xi} -{\epsilon}{\epsilon}' \widetilde{J} \widetilde{F}.\end{aligned}$$
It follows that $$\begin{aligned}
d\widetilde{f}(\xi) &=&{{\texttt{cos}\varepsilon}}'({\theta}) d\widetilde{F}(\xi) + {{\texttt{sin}\varepsilon}}'({\theta})d \widetilde{N} (\xi)\\
&= &{{\texttt{cos}\varepsilon}}'({\theta}) \widetilde{\Xi}+ {{\texttt{sin}\varepsilon}}'({\theta})\widetilde{\nabla}_{\widetilde{\Xi}} \widetilde{N}\\
&=&-{\epsilon}{{\texttt{cos}\varepsilon}}'({\theta}) \widetilde{J} \widetilde{N}+{{\texttt{sin}\varepsilon}}' ({\theta}) \left( a{\epsilon}\widetilde{J} \widetilde{N} - {\epsilon}{\epsilon}' \widetilde{J} \widetilde{F}\right)\\
& = & -{\epsilon}\Big({\epsilon}' {{\texttt{sin}\varepsilon}}'({\theta}) \widetilde{J} \widetilde{F} + ({{\texttt{cos}\varepsilon}}' ({\theta})- a {{\texttt{sin}\varepsilon}}'({\theta}) )\widetilde{J} \widetilde{N}\Big).\end{aligned}$$ We now choose ${\theta}$ in order to have $df(\xi)=0$, i.e. $d \widetilde{f}(\xi) \in \widetilde{J} \widetilde{f}\, {\ensuremath{{\mathbb{R}^{}} }}$. This is equivalent to the existence of ${\lambda}\in {\ensuremath{{\mathbb{R}^{}} }}$ such that $$\left\{ \begin{array}{lll} {\epsilon}' {{\texttt{sin}\varepsilon}}'({\theta}) &=& {\lambda}{{\texttt{cos}\varepsilon}}'({\theta}), \\
{{\texttt{cos}\varepsilon}}'({\theta})- a {{\texttt{sin}\varepsilon}}'({\theta}) &=& {\lambda}{{\texttt{sin}\varepsilon}}'({\theta}).
\end{array} \right.$$ It follows that ${\lambda}={\epsilon}' {{\texttt{tan}\varepsilon}}' ({\theta})$ and $$a = \frac{{{\texttt{cos}\varepsilon}}' ({\theta})}{{{\texttt{sin}\varepsilon}}' ({\theta})}-{\lambda}= \frac{({{\texttt{cos}\varepsilon}}'({\theta}))^2-{\epsilon}' ({{\texttt{sin}\varepsilon}}'({\theta}))^2 }{{{\texttt{cos}\varepsilon}}' ({\theta}) {{\texttt{sin}\varepsilon}}' ({\theta})}= 2 {{\texttt{cot}\varepsilon}}'(2 {\theta})$$ so that, taking $${\theta}:=\frac{1}{2} {{\texttt{cot}\varepsilon}}'^{-1} (a/2),$$ we get $df(\xi)=0$. This is possible for all real number $a$ if ${\epsilon}'=1$, and if $|a|>2$ if ${\epsilon}'=-1$.[^3] Hence $f$ is constant along the integral lines of $\xi$. In particular, the rank of $f$ is strictly less than $2n-1$.
We now claim that the rank of $f $ is even and that its image is a complex submanifold of ${{\cal M}}$. We first calculate, for a horizontal vector $v \in {\mbox{Hor}}$: $$\begin{aligned}
d \widetilde{f}(v) &=&{{\texttt{cos}\varepsilon}}'({\theta}) d\widetilde{F} (v)+ {{\texttt{sin}\varepsilon}}'({\theta})d \widetilde{N} (v)\\
&=& {{\texttt{cos}\varepsilon}}'({\theta}) d\widetilde{F} (v) + {{\texttt{sin}\varepsilon}}'({\theta}) \widetilde{\nabla}_{ d\widetilde{F} (v)} \widetilde{N}\\
&=&{{\texttt{cos}\varepsilon}}'({\theta}) d\widetilde{F} (v) + {{\texttt{sin}\varepsilon}}'({\theta}) \Big( -d\widetilde{F} (Av)
+ {\langle}\widetilde{J} \widetilde{F}, \widetilde{J} \widetilde{F} {\rangle}{\langle}d\widetilde{F} (v), \widetilde{J} \widetilde{N}{\rangle}\widetilde{J}\widetilde{F} \Big)\\
&=& {{\texttt{cos}\varepsilon}}'({\theta}) d\widetilde{F} (v) - {{\texttt{sin}\varepsilon}}'({\theta}) d\widetilde{F} (Av) \\
&=& d\widetilde{F} ({{\texttt{cos}\varepsilon}}' ({\theta})v -{{\texttt{sin}\varepsilon}}'({\theta})Av).\end{aligned}$$ Hence $Ker ( df) = Ker ({{\texttt{cos}\varepsilon}}' ({\theta}) Id - {{\texttt{sin}\varepsilon}}'({\theta}) A)$ and therefore $$rank (f) = 2n - {\rm dim} Ker ({{\texttt{cos}\varepsilon}}'({\theta}) Id - {{\texttt{sin}\varepsilon}}'({\theta}) A)$$ Moreover, $v \in Ker(df)$ if and only if $Av = {{\texttt{cot}\varepsilon}}'({\theta}) v$, i.e. ${{\texttt{cot}\varepsilon}}'({\theta})$ is a principal curvature of $F.$ Observe that ${{\texttt{cot}\varepsilon}}'^2 ({\theta})- \frac{{{\texttt{cot}\varepsilon}}' ({\theta})}{2} a -{\epsilon}'=0$, so by Lemma \[l4\] of Section \[s2\], the corresponding eigenspace is $J$-invariant. In particular the rank of $f$ is even.
If $v$ does not belong to $Ker ({{\texttt{cos}\varepsilon}}'({\theta}) Id - {{\texttt{sin}\varepsilon}}'({\theta}) A),$ we claim that there exists $w $ such that $Jdf(v)= df(w).$ Since $$\widetilde{J}d \widetilde{f}(v)= d\widetilde{F}( {{\texttt{cos}\varepsilon}}'({\theta}) \varphi v - {{\texttt{sin}\varepsilon}}' ({\theta}) \varphi Av)$$ this is equivalent to $$\varphi ({{\texttt{cos}\varepsilon}}' ({\theta}) Id - {{\texttt{sin}\varepsilon}}' ({\theta}) A) v =( {{\texttt{cos}\varepsilon}}'({\theta}) Id -{{\texttt{sin}\varepsilon}}' ({\theta}) A)w.$$ Hence, we get the required relation setting $$w := ( {{\texttt{cos}\varepsilon}}'({\theta}) Id -{{\texttt{sin}\varepsilon}}' ({\theta}) A)^{-1} \varphi({{\texttt{cos}\varepsilon}}' ({\theta}) Id - {{\texttt{sin}\varepsilon}}' ({\theta}) A) v.$$ This proves that $df(T U)$ is stable with respect to $J$, i.e. $f(U)$ is a complex submanifold. The easy task to check that $F(U)$ is the tube of radius ${\theta}$ over $f(U)$ is left to the reader.
Open Problems
-------------
Summarizing, in this paper some basic results are presented and a characterization of real hypersurfaces with Hopf curvature satisfying $|\alpha|>2$ in pseudo-Riemannian complex space forms and para-complex space forms is given. Therefore, a first question which is raised in a natural way is:
*Are there real hypersurfaces in pseudo-Riemannian complex space forms or para-complex space forms whose Hopf curvature is small, i.e. $|\alpha|\leq2$?*
Following similar steps to those which have been done in the study of real hypersurfaces in the cases of complex space forms, complex two-plane Grassmannians, etc., a great amount of questions concerning real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms come up. For instance, it would be interesting to answer the following:
*Are there real hypersurfaces in pseudo-Riemannian complex space forms or para-complex space forms whose shape operator commutes with $\varphi$, i.e. $A\varphi=\varphi A$?*
[XXXXX]{}
H. Anciaux, *Minimal submanifolds in Pseudo-Riemannian geometry, *World Scientific, 2010**
H. Anciaux, *Surfaces with one constant principal curvature in three-dimensional space forms, *arXiv:1307.6735**
A. Bejancu, K. L. Duggal, *Real hypersurfaces of indefinite Kaehler manifolds*, Internat. J. Math. and Math. Sci. **16** no. 3, (1993), 545–556
T. Cecil, P. Ryan, *Focal sets and real hypersurfaces in complex projective space, *Trans. Amer. Math. Soc. [**269**]{} (1982), 481–499**
P. M. Gadea, A. M. Montesinos Amilibia, *Spaces of constant para-holomorphic curvature, *Pacific J. of Maths. [**136**]{} no. 1, (1989), 85–101**
T. Ivey, P. Ryan, *Hopf Hypersurfaces of Small Hopf Principal Curvature in ${\ensuremath{{\mathbb{C}^{}} }}{\ensuremath{{\mathbb{H}^{}} }}^2$, *Geom. Dedicata [**141**]{} (2009), 147–161**
T. Ivey, *A d’Alembert Formula for Hopf Hypersurfaces, *Results in Maths. [**60**]{} (2011), 293–309**
U.-H. Ki, Y.-J. Suh, *On real hypersurfaces of a complex space form, *Math. J. Okayama Univ. [**32**]{} (1990), 207–221**
M. Kimura, *Hopf hypersurfaces in nonflat complex space forms, *Proceedings of The Sixteenth International Workshop on Diff. Geom. **16** (2012) 25–34**
R. Niebergall, P. Ryan, *Real Hypersurfaces in Complex Space Forms, *Tight and Taut Submanifolds MSRI Publications Volume 32, 1997**
Y. Madea, *On real hypersurfaces of a complex projective space, *J. Math. Soc. Japan [**28**]{} (1976), 529–540**
S. Montiel, *Real hypersurfaces of a complex hyperbolic space, *J. Math. Soc. Japan **37** (1985), no. 3, 515–535**
O’Neill, *The fundamental equations of a submersion, *Michigan Math. J., **13** (1966), 459–469**
[2]{} Henri Anciaux\
Universidade de São Paulo, IME\
1010 Rua do Matão,\
Cidade Universitária\
05508-090 São Paulo, Brazil\
henri.anciaux@gmail.com\
Konstantina Panagiotidou\
Faculty of Engineering\
Aristotle University of Thessaloniki\
Thessaloniki 54124, Greece\
kapanagi@gen.auth.gr
[^1]: Universidade de São Paulo; supported by CNPq (PDE 211682/2013-6)
[^2]: Faculty of Engeneering, Aristotle University of Thessaloniki, Greece
[^3]: In the case ${\epsilon}'=-1$, if we instead set $$\widetilde{f}':=\sinh({\theta}) \widetilde{F} + \cosh ({\theta}) \widetilde{N},$$ which is valued in $\overline{\widetilde{{{\cal M}}}}$, we get again $a=2 \coth (2{\theta})$. The map $f':=\pi \circ \widetilde{f}'$ is the *polar *of $f := \pi \circ \widetilde{f}.$**
|
---
abstract: |
An algorithm to simulate the dynamics of a quantum state over a three-site lattice interacting with classical harmonic oscillators has been devised.
The oscillators are linearly coupled to the quantum state and are acted upon by a fluctuation-dissipation process to take the equilibrium thermal environment into account, thus allowing to investigate how stochastic forces may affect the quantum dynamics.
The implementation of the algorithm has been written in .
author:
- 'Paolo M. Pumilia'
bibliography:
- '/home/pol/Notes/Bibliography/Physics/physics.bib'
date: 'august 19th, 2002'
title: 'Three-site quantum lattice with thermal bath '
---
paolo.pumilia@acm.org
The physical model
==================
The Hosltein hamiltonian for a lattice connected to independent site oscillators can be distinguished into three terms $$\begin{aligned}
h & = & h_{a}+h_{b}+h_{ab}\label{h}\end{aligned}$$ corresponding to the *primary* system $a$, representing those states allowed to the quantum particle, the *secondary* system $b$, representing the vibrating lattice masses ($m$) and the interaction energy $ab$ [@Holstein-59a].
In a three-site lattice, assuming the nearest neighbours approximation hold, the quantum states are described by the hamiltonian $$h_{a}=V\left(\left|1\right\rangle \left\langle 2\right|+\left|2\right\rangle \left\langle 1\right|+\left|2\right\rangle \left\langle 3\right|+\left|3\right\rangle \left\langle 2\right|\right)\label{h_a}$$ where $\left|n\right\rangle $ are the eigenstates of the isolated sites and $V$ is the overlap integral between neighbour sites.
Vibrations are described by harmonic oscillators of equal mass $m$ and frequency $\omega $, with respective position $r_{n}$ and conjugated variable $p_{n}$, (where $n$ indicates the location on the lattice):$$h_{b}=\frac{1}{2m}\left(p_{1}^{2}+m^{2}\omega ^{2}r_{1}^{2}\right)+\frac{1}{2m}\left(p_{2}^{2}+m^{2}\omega ^{2}r_{2}^{2}\right)+\frac{1}{2m}\left(p_{3}^{2}+m^{2}\omega ^{2}r_{3}^{2}\right)\label{h_b}$$ Interaction terms are assumed to be linear with respect to $r_{n}$$$\begin{aligned}
h_{ab} & = & \epsilon _{1}\left|1\right\rangle \left\langle 1\right|+\epsilon _{2}\left|2\right\rangle \left\langle 2\right|+\epsilon _{3}\left|3\right\rangle \left\langle 3\right|\label{h_ab}\end{aligned}$$ $\epsilon _{n}=\chi r_{n}$ , with $\chi $ real positive, being the coupling energy at site $n$. Such physical model has been studued by Hennig [@hennig-91] and Kenkre & Andersen [@kenkre-dimer-86], but without thermal environment.
Lattice with free boundary conditions
-------------------------------------
When loose ends boundary conditions apply to the primary system, we are lead to the following equation for the density matrix: $$\begin{aligned}
\dot{\rho }_{11} & = & i\omega _{o}\left(\rho _{21}-\rho _{12}\right)\nonumber \\
\dot{\rho }_{12} & = & -i\omega _{12}\rho _{12}+i\omega _{o}\left(\rho _{22}-\rho _{11}-\rho _{13}\right)\nonumber \\
\dot{\rho }_{13} & = & -i\omega _{13}\rho _{13}+i\omega _{o}\left(\rho _{23}-\rho _{12}\right)\nonumber \\
\dot{\rho }_{21} & = & i\omega _{12}\rho _{21}+i\omega _{o}\left(\rho _{11}+\rho _{31}-\rho _{22}\right)\nonumber \\
\dot{\rho }_{22} & = & i\omega _{o}\left(\rho _{12}+\rho _{32}-\rho _{21}-\rho _{23}\right)\label{density-matrix-free}\\
\dot{\rho }_{23} & = & -i\omega _{23}\rho _{23}+i\omega _{o}\left(\rho _{13}+\rho _{33}-\rho _{22}\right)\nonumber \\
\dot{\rho }_{31} & = & i\omega _{13}\rho _{31}+i\omega _{o}\left(\rho _{21}-\rho _{32}\right)\nonumber \\
\dot{\rho }_{32} & = & i\omega _{23}\rho _{32}+i\omega _{o}\left(\rho _{22}-\rho _{31}-\rho _{33}\right)\nonumber \\
\dot{\rho }_{33} & = & i\omega _{o}\left(\rho _{23}-\rho _{32}\right)\nonumber \end{aligned}$$ where $\omega _{o}=V/h$ is the frequency for the oscillation of the quasi-particle between two neighbour sites and $\omega _{kn}=-\omega _{nk}=\left(\epsilon _{n}-\epsilon _{k}\right)/\hbar =\chi \left(r_{n}-r_{k}\right)/\hbar $.
Using operators $$\begin{aligned}
u_{1} & = & \rho _{11}-\rho _{22}\nonumber \\
u_{2} & = & \rho _{22}-\rho _{33}\nonumber \\
v_{1} & = & i\left(\rho _{12}-\rho _{21}\right)\nonumber \\
v_{2} & = & i\left(\rho _{32}-\rho _{23}\right)\label{density-operators}\\
v_{3} & = & i\left(\rho _{31}-\rho _{13}\right)\nonumber \\
w_{1} & = & \rho _{12}+\rho _{21}\nonumber \\
w_{2} & = & \rho _{23}+\rho _{32}\nonumber \\
w_{3} & = & \rho _{13}+\rho _{31}\nonumber \end{aligned}$$ a new set of equations, that will be used to perform numerical simulations, can be obtained for the primary system.
As a fluctuation-dissipation process has been finally attached to each oscillator, the complete dynamical model can be written:$$\begin{aligned}
\dot{u}_{1} & = & -\omega _{o}\left(2v_{1}+v_{2}\right)\nonumber \\
\dot{u}_{2} & = & \omega _{o}\left(v_{1}+2v_{2}\right)\nonumber \\
\dot{v}_{1} & = & \omega _{12}w_{1}+\omega _{o}\left(2u_{1}+w_{3}\right)\nonumber \\
\dot{v}_{2} & = & -\omega _{23}w_{2}-\omega _{o}\left(2u_{2}-w_{3}\right)\nonumber \\
\dot{v}_{3} & = & -\omega _{13}w_{3}-\omega _{o}\left(w_{1}-w_{2}\right)\nonumber \\
\dot{w}_{1} & = & -\omega _{12}v_{1}+\omega _{o}v_{3}\nonumber \\
\dot{w}_{2} & = & \omega _{23}v_{2}-\omega _{o}v_{3}\nonumber \\
\dot{w}_{3} & = & \omega _{13}v_{3}-\omega _{o}\left(v_{1}+v_{2}\right)\label{sch eq with noise}\\
\dot{r}_{1} & = & p_{1}/m\nonumber \\
\dot{r}_{2} & = & p_{2}/m\nonumber \\
\dot{r}_{3} & = & p_{3}/m\nonumber \\
\dot{p}_{1} & = & -m\omega ^{2}r_{1}-\frac{\chi }{3}\left(c+u_{2}+2u_{1}\right)-\gamma _{1}p_{1}+f_{1}\left(t\right)\nonumber \\
\dot{p}_{2} & = & -m\omega ^{2}r_{2}-\frac{\chi }{3}\left(c+u_{2}-u_{1}\right)-\gamma _{2}p_{2}+f_{2}\left(t\right)\nonumber \\
\dot{p}_{3} & = & -m\omega ^{2}r_{3}-\frac{\chi }{3}\left(c-u_{1}-2u_{2}\right)-\gamma _{3}p_{3}+f_{3}\left(t\right)\nonumber \end{aligned}$$ where $\gamma _{n}$ are the damping coefficients and $f_{n}\left(t\right)$ models gaussian noise, with $\delta $-shaped time correlation, satisfying the following relations:$$\begin{aligned}
\left\langle f_{n}(t)\right\rangle & = & 0\label{zero mean f}\\
\left\langle f_{n}(t)f_{n}(t')\right\rangle & = & 2\gamma \theta \delta _{\left(t-t'\right)}\label{delta corr f}\end{aligned}$$ having defined thermal energy$$\theta =k_{B}T\label{thermal energy}$$ ($k_{B}$being the Boltzmann constant); hence $$\left\langle f_{n}^{2}(t)\right\rangle ^{1/2}=\left(2\gamma \theta \right)^{1/2}\label{variance f}$$
It can be readily verified that $$c=\rho _{11}+\rho _{22}+\rho _{33}\label{const 1}$$ is a constant of motion. A further constant, that will be used to monitor numerical simulations, is given by $$K=\frac{4}{3}\left(u_{1}^{2}+u_{2}^{2}+u_{1}u_{2}\right)+v_{1}^{2}+v_{2}^{2}+v_{3}^{2}+w_{1}^{2}+w_{2}^{2}+w_{3}^{2}\label{const 2}$$
The dynamical problem is described by a system of stochastic real valued, first order, differential equations (\[sch eq with noise\]) that must be numerically solved.
Notice that, restricting the number of sites to $n=1,2$ we are left with the spin-boson equation with noise: $$\begin{aligned}
\dot{u}_{1} & = & -2\omega _{o}v_{1}\nonumber \\
\dot{v}_{1} & = & \omega _{12}w_{1}+2\omega _{o}u_{1}\nonumber \\
\dot{w}_{1} & = & -\omega _{12}v_{1}\nonumber \\
\dot{r}_{1} & = & p_{1}/m\label{spin-boson with noise}\\
\dot{r}_{2} & = & p_{2}/m\nonumber \\
\dot{p}_{1} & = & -m\omega ^{2}r_{1}-\frac{\chi }{3}\left(c+u_{2}+2u_{1}\right)+f_{1}\left(t\right)\nonumber \\
\dot{p}_{2} & = & -m\omega ^{2}r_{2}-\frac{\chi }{3}\left(c+u_{2}-u_{1}\right)+f_{2}\left(t\right)\nonumber \end{aligned}$$ that has been studied by many researchers; in particular, the author’s work has been inspired the papers[@kenkre-dimer-86; @grigo-dimer-91], referenced to in the bibliography.
Numerical integration method
============================
Stochastic differential equations of first order in time for a vector variabile $\mathbf{x}(t)$ take the from, $$\dot{\mathbf{x}}=\mathbf{F}+\mathbf{gf}\label{eq_diff}$$ where $\mathbf{F=}\left(F_{i}\right)_{1,n}=\left(F_{i}(\mathbf{x},t)\right)_{1,n}$ is the vector of deterministic fields, $\mathbf{f=}\left(f_{i}\right)_{1,n}=\left(f_{i}(t)\right)_{1,n}$ the gaussian stochastic force, satisfying the conditions:$$\begin{aligned}
\left\langle f(t)\right\rangle & = & 0\nonumber \\
\left\langle f(t)f(t')\right\rangle & = & \delta (t-t')\label{gaussian_noise}\end{aligned}$$ while th parameters set $\mathbf{g=}\left(g_{i}\right)_{1,n}=\left(g_{i}\left(\mathbf{x},t\right)\right)_{1,n}$ represents the interaction of the system with the thrmal bath.
Equations given in (\[eq\_diff\]) are *entirely coupled,* since, in general, every components $F_{i}$ and $g_{i}$ are functions of the whole variable $\mathbf{x}$,
The formal solution of (\[eq\_diff\]), assuming the *implicit* time dependence $F=\mathbf{F}(\mathbf{x}(t))$ e $g=\mathbf{g}(\mathbf{x}(t))$, is given by [@spde]$$\begin{aligned}
x_{i}(t)-x_{i}(0)=\int _{0}^{t}dt'F_{i}+\int _{0}^{t}dt'g_{i}f_{i} & \, \, \, \, \, \, \, \, \, & i=1,\ldots n\label{formal_int}\end{aligned}$$
Given a time interval $\left[0,h\right]$ that can be considered infinitely small for $F_{i}$ e $g_{i}$ functions, their value at the arbitrary instant $t'\in \left[0,h\right]$, can be approximated by the first $\kappa $ terms of a Taylor expansion, centered at $t=0$:$$\begin{aligned}
F_{i}\left(t\right) & = & F_{i}^{o}+\delta ^{\kappa }F_{i}\label{F series}\\
g_{i}\left(t\right) & = & g_{i}^{o}+\delta ^{\kappa }g_{i}\label{g series}\end{aligned}$$ The index $\kappa $ in (\[F series\],\[g series\]) indicates the truncation order of the series: $$\delta F_{i}=\sum _{j}\left[F_{i}\right]_{j}\cdot \delta x_{j}(t')+\frac{1}{2}\sum _{jk}\left[F_{i}\right]_{jk}\cdot \delta x_{j}(t')\delta x_{k}(t')+\frac{1}{3!}\sum _{jk}\left[F_{i}\right]_{jkl}\cdot \delta x_{j}(t')\delta x_{k}(t')\delta x_{l}+\ldots \label{dF_series}$$ $$\delta g_{i}=\sum _{j}\left[g_{i}\right]_{j}\cdot \delta x_{j}(t')+\frac{1}{2}\sum _{jk}\left[g_{i}\right]_{jk}\cdot \delta x_{j}(t')\delta x_{k}(t')+\frac{1}{3!}\sum _{jk}\left[g_{i}\right]_{jkl}\cdot \delta x_{j}(t')\delta x_{k}(t')\delta x_{l}+\ldots \label{dg_series}$$ in which $F_{i}$ and $g_{i}$, time derivatives evaluated at $t=0$, have been represented by the following notation:$$\left[F_{i}\right]_{j}\equiv \frac{\partial F_{i}}{\partial x_{j}}(0)\, \, \, \, \, \, \left[F_{i}\right]_{jk}\equiv \frac{\partial ^{2}F_{i}}{\partial x_{j}\partial x_{k}}(0)\, \, \, \, \, \, \left[F_{i}\right]_{jkl}\equiv \frac{\partial ^{3}F_{i}}{\partial x_{j}\partial x_{k}\partial x_{l}}(0)\label{dF_coeff}$$ $$\left[g_{i}\right]_{j}\equiv \frac{\partial g_{i}}{\partial x_{j}}(0)\, \, \, \, \, \, \left[g_{i}\right]_{jk}\equiv \frac{\partial ^{2}g_{i}}{\partial x_{j}\partial x_{k}}(0)\, \, \, \, \, \, \left[g_{i}\right]_{jkl}\equiv \frac{\partial ^{3}g_{i}}{\partial x_{j}\partial x_{k}\partial x_{l}}(0)\label{dg_coeff}$$
Truncation at $\kappa =0$ the general solution obtained after substitution of (\[F series\],\[g series\]) into (\[formal\_int\]):$$x_{i}(h)-x_{i}(0)=\int _{0}^{h}dt'\left(F_{i}^{o}+\delta ^{\kappa }F_{i}\right)+\int _{0}^{h}dt'\left(g_{i}^{o}+\delta ^{\kappa }g_{i}\right)f_{i}\label{formal_int_series}$$ yields$$x_{i}(h)-x_{i}(0)=F_{i}^{o}h+g_{i}^{o}\int _{0}^{h}dt'f_{i}\label{zero_int}$$ where $F_{i}=\left[F_{i}\right]=F_{i}^{o}$ and $g_{i}=\left[g_{i}\right]=g_{i}^{o}$ have been defined.
Since the gaussian integral $$Z_{1}(h)=\int _{0}^{h}dt'f(t')\label{Z1}$$ is of $h^{1/2}$ order, the second term in (\[zero\_int\]) is the lowest order approximation of the trajectory (\[formal\_int\_series\]):$$\delta x_{i}^{(1/2)}=g_{i}^{o}Z_{1}\label{order 1/2}$$ having defined $\delta x_{i}=x_{i}(h)-x_{i}(0)$.
Substituting (\[order 1/2\]) into (\[dF\_series\],\[dg\_series\]), hence into (\[formal\_int\]), and retaining only contribution of the order $h$ in (\[formal\_int\_series\]), we obtain the first order correction
$$\begin{aligned}
\delta x_{i}^{(1)} & = & F_{i}^{o}h+\int _{0}^{h}dt'\sum _{j}\left[g_{i}\right]_{j}\cdot \delta x_{j}^{(1/2)}f_{i}(t')\nonumber \\
& = & F_{i}^{o}h+\sum _{j}\left[g_{i}\right]_{j}\cdot g_{j}^{o}\int _{0}^{h}dt'Z_{1}(t')f_{i}(t')\nonumber \\
& = & F_{i}^{o}h+\frac{1}{2}\sum _{j}\left[g_{i}\right]_{j}\cdot g_{j}^{o}Z_{1}^{2}\label{order 1 correction}\end{aligned}$$
The corresponding displacement on the trajectory is thus given by$$\delta x_{i}=\delta x_{i}^{1/2}+\delta x_{i}^{1}\label{order 1}$$ Again substituting the first order correction (\[order 1 correction\]) into the series (\[dF\_series\],\[dg\_series\]), hence into (\[formal\_int\]), then retaining only terms of the order $h^{3/2}$ in (\[formal\_int\_series\]), we get:$$\begin{aligned}
\delta x_{i}^{(3/2)} & = & \sum _{j}\left[F_{i}\right]_{j}\cdot g_{j}^{o}Z_{1}h+\int _{0}^{h}dt'\sum _{j}\left[g_{i}\right]_{j}\cdot \delta x_{j}^{(1)}f_{i}(t')\nonumber \\
& = & \sum _{j}\left[F_{i}\right]_{j}\cdot g_{j}^{o}Z_{1}h+\int _{0}^{h}dt'\sum _{j}\left[g_{i}\right]_{j}\cdot \left(F_{i}^{o}h+\frac{1}{2}\sum _{k}\left[g_{i}\right]_{k}\cdot g_{k}^{o}Z_{1}^{2}(t')f_{i}(t')\right)\nonumber \\
& = & \sum _{j}\left[F_{i}\right]_{j}\cdot g_{j}^{o}Z_{1}h+\sum _{j}\left[g_{i}\right]_{j}\cdot \left(F_{i}^{o}h+\frac{1}{2}\sum _{k}\left[g_{i}\right]_{k}\cdot g_{k'}^{o}\int _{0}^{h}dtZ_{1}^{2}(t')f_{i}(t')\right)\nonumber \\
& = & \sum _{j}\left[F_{i}\right]_{j}\cdot g_{j}^{o}Z_{1}h+\sum _{j}\left[g_{i}\right]_{j}\cdot \left(F_{i}^{o}h+\frac{1}{3!}\sum _{k}\left[g_{i}\right]_{k}\cdot g_{k'}^{o}Z_{1}^{3}\right)\label{order 3/2}\end{aligned}$$
In the simplified case in which $g$ does not depend on $\mathbf{x}$, the numerical algorithm to reach the order $h^{2}$ in approximating the exact solution of (\[eq\_diff\]) can then be written$$x_{i}(h)-x_{i}(0)=g_{i}^{o}Z_{1}+F_{i}^{o}h+\sum _{j}\left[F_{i}\right]_{j}\cdot g_{j}^{o}Z_{2}+\frac{1}{2}\sum _{j}\left[F_{i}\right]_{j}\cdot \left[F_{j}\right]h^{2}\, \, \, \, \, \, \, \, \, i=1,\ldots n\label{order 2 solution}$$ or, in vectorial notation:$$\mathbf{x}(h)-\mathbf{x}(0)=\mathbf{g}^{o}\mathbf{Z}_{1}+\mathbf{F}^{o}\mathbf{h}+\nabla \mathbf{FgZ}_{2}+\frac{1}{2}\nabla \mathbf{F}\cdot \mathbf{F}h^{2}\label{order 2 vect sol}$$
The algorithm for the three-site lattice
----------------------------------------
Equations (\[sch eq with noise\]) belong to the class (\[eq\_diff\]); therefore the approximation method that has been recalled in the previous chapter can be implemented.
To better expose our procedure, it is convenient switching to the vectorial notation; the system variable $\mathbf{x}$, taking the form:$$\mathbf{x}=\left[\begin{array}{c}
u_{1}\\
u_{2}\\
v_{1}\\
v_{2}\\
v_{3}\\
w_{1}\\
w_{2}\\
w_{3}\\
r_{1}\\
r_{2}\\
r_{3}\\
p_{1}\\
p_{2}\\
p_{3}\end{array}
\right]\label{variables}$$ evolves in time according to the equation$$\dot{\mathbf{x}}=\mathbf{F}+\mathbf{G}\label{vectorial dyn eq.}$$ where $\mathbf{F}$ is the vector of the deterministic forces $$\begin{aligned}
\mathbf{F} & = & \left[\begin{array}{c}
-\omega _{o}\left(2v_{1}+v_{2}\right)\\
\omega _{o}\left(v_{1}+2v_{2}\right)\\
\omega _{12}w_{1}+\omega _{o}\left(2u_{1}+w_{3}\right)\\
-\omega _{23}w_{2}-\omega _{o}\left(2u_{2}-w_{3}\right)\\
-\omega _{13}w_{3}-\omega _{o}\left(w_{1}-w_{2}\right)\\
-\omega _{12}v_{1}+\omega _{o}v_{3}\\
\omega _{23}v_{2}-\omega _{o}v_{3}\\
\omega _{13}v_{3}-\omega _{o}\left(v_{1}+v_{2}\right)\\
p_{1}/m\\
p_{2}/m\\
p_{3}/m\\
-m\omega ^{2}r_{1}-\frac{\chi }{3}\left(c+u_{2}+2u_{1}\right)-\gamma _{1}p_{1}\\
-m\omega ^{2}r_{2}-\frac{\chi }{3}\left(c+u_{2}-u_{1}\right)-\gamma _{2}p_{2}\\
-m\omega ^{2}r_{3}-\frac{\chi }{3}\left(c-u_{1}-2u_{2}\right)-\gamma _{3}p_{3}\end{array}
\right]\label{deterministic forces}\end{aligned}$$ while **$\mathbf{G}$** contains the average values of the stochastic forces:$$\mathbf{G}=\left[\begin{array}{c}
0\\
0\\
0\\
0\\
0\\
0\\
0\\
0\\
0\\
0\\
0\\
\left(2\gamma _{1}\theta _{1}\right)^{1/2}\\
\left(2\gamma _{2}\theta _{2}\right)^{1/2}\\
\left(2\gamma _{3}\theta _{3}\right)^{1/2}\end{array}
\right]\label{averaged stochastic forces}$$
The approximated solution up to the second order in time can be computed according to the formula (\[order 2 vect sol\]):$$\mathbf{x}(h)\simeq \mathbf{x}(0)+\mathbf{g}\cdot \mathbf{Z}_{1}+\mathbf{F}\cdot \mathbf{h}+\nabla \mathbf{F}\cdot \mathbf{g}\cdot \mathbf{Z}_{2}+\frac{1}{2}\nabla \mathbf{F}\cdot \mathbf{F}\cdot \mathbf{h}^{2}\label{order 2 integration}$$ where $Z_{1}$ and $Z_{2}$ are gaussian integrals and the force gradient is given by:$$\nabla \mathbf{F}=\left[\begin{array}{cccccccccccccc}
0 & 0 & -2\omega _{o} & -\omega _{o} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & \omega _{o} & 2\omega _{o} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
2\omega _{o} & 0 & 0 & 0 & 0 & \omega _{12} & 0 & \omega _{o} & -\chi _{1}w_{1} & \chi _{2}w_{1} & 0 & 0 & 0 & 0\\
0 & -2\omega _{o} & 0 & 0 & 0 & 0 & -\omega _{23} & \omega _{o} & 0 & \chi _{2}w_{2} & -\chi _{3}w_{2} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & -\omega _{o} & \omega _{o} & -\omega _{13} & \chi _{1}w_{3} & 0 & -\chi _{3}w_{3} & 0 & 0 & 0\\
0 & 0 & -\omega _{12} & 0 & \omega _{o} & 0 & 0 & 0 & \chi _{1}v_{1} & -\chi _{2}v_{1} & 0 & 0 & 0 & 0\\
0 & 0 & 0 & \omega _{23} & -\omega _{o} & 0 & 0 & 0 & 0 & -\chi _{1}v_{2} & \chi _{3}v_{2} & 0 & 0 & 0\\
0 & 0 & -\omega _{o} & -\omega _{o} & \omega _{13} & 0 & 0 & 0 & -\chi _{1}v_{3} & 0 & \chi _{3}v_{3} & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \nicefrac 1m_{1} & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \nicefrac 1m_{2} & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \nicefrac 1m_{2}\\
-2\chi _{1}/3 & -\chi _{1}/3 & 0 & 0 & 0 & 0 & 0 & 0 & -m_{1}\omega _{1}^{2} & 0 & 0 & -\gamma _{1} & 0 & 0\\
\chi _{2}/3 & -\chi _{2}/3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -m_{2}\omega _{2}^{2} & 0 & 0 & -\gamma _{2} & 0\\
\chi _{3}/3 & 2\chi _{3}/3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -m_{3}\omega _{3}^{2} & 0 & 0 & -\gamma _{3}\end{array}
\right]$$
The implementation of the algorithm according to the rule (\[order 2 integration\]) is displayed in the last section . language has been the language of choiche, since it is maintained that a rather easy and dependable parallel code, that could be required to perform long lasting simulations can be worked out in that environment, .
Marsaglia & Tsang [@marsaglia] fortran routine for random numbers generation has been used to compute $Z_{1}$ and $Z_{2}$ integrals.
Results and Conclusions
=======================
First run of the program allowed to plot variables [:]{} $u_{1}=\rho _{11}-\rho _{22}$ and $u_{2}=\rho _{22}-\rho _{33}$, using the set of physical parameters:
- inter-site interaction $V=0.09$
- free oscillators frequencies $\omega ^{2}=2x10^{-3}$
- vibration-quantum state coupling $\chi =4x10^{-2}$ ;
- $\gamma =0.2$ that imply $\omega ^{2}/2\gamma ^{2}=0.025$ and $\chi ^{2}/2m\omega ^{2}=0.1$
Time integration step has been set to $10^{-3}$ and the reliability of the computation has been monitored through the constant (\[const 2\]), whose changes have kept below $10^{-7}$.
Picture of the site occupation differences $u_{1}=\rho _{11}-\rho _{22}$ (lower) and $u_{2}=\rho _{22}-\rho _{33}$ (upper) evolving in time (arbitrary units) shows the localized motion of the quantum state.
The possibilty to model the dynamics of a three-site lattice coupled to stochastic oscillators allows one to study how a quantum state propagates in a real lattice, when vibrational motion can be described classically, then to investigate the duration of out of equilibrium states and the conditions under which energy transfer without dissipation may occur.
Code listing
============
Here included are the main tokens of code from the procedure, in which the numerical integration at time step `dh` is performed:
[procedure iterator (x : in out coordinates ; xh : out coordinates ; time\_step : in double ; -- time integration step inner\_cycle : in integer ; -- number of steps between next -- print outs]{}
[ outer\_cycle : in integer ; -- total number of trajectory -- points to print out ]{}
[ v,am1,am2,am3,gam1,gam2,gam3,om1,om2,om3: in double; chi1,chi2,chi3,t1,t2,t3,e1b,e2b,e3b : in double; ]{}
[ output\_file : in string ) ]{}
[is ]{}
[function rnor(dum : in Fortran.Fortran\_Integer) return Fortran.double\_precision; ]{}
[pragma Import (fortran, rnor, ‘"rnor\_‘"); ]{}
[dh : double := time\_step ; -- time integratin step ]{}
[dh2 : double := time\_step [\*]{} time\_step; -- squared time integration step ]{}
[dh25 := dh2 [\*]{} 0.5; ]{}
[v2: double := v [\*]{} v ; -- ]{}
[ak0 := c1[\*]{}c1 + c2[\*]{}c2 + c3[\*]{}c3; ]{}
[qom1 := om1[\*]{}om1; ]{}
[qom2 := om2[\*]{}om2; ]{}
[qom3 := om3[\*]{}om3;]{}
[qgam1 := gam1 [\*]{} gam1; ]{}
[qgam2 := gam2 [\*]{} gam2; ]{}
[qgam3 := gam3 [\*]{} gam3;]{}
[norm := (1.0/3.0)[\*]{}( x(1)[\*]{}x(1) + x(2)[\*]{}x(2) + x(1)[\*]{}x(2) ); ]{}
[norm := norm + x(3)[\*]{}x(3) + x(4)[\*]{}x(4) + x(5)[\*]{}x(5);]{}
[norm := norm + x(6)[\*]{}x(6) + x(7)[\*]{}x(7) + x(8)[\*]{}x(8); ]{}
[norm := (3.0/4.0) [\*]{} norm;]{}
[for i in 1..outer\_cycle loop ]{}
[ for j in 1..inner\_cycle loop ]{}
[ time := time + time\_step;]{}
[ w11 := double(rnor(dum)); ]{}
[ w12 := double(rnor(dum)); ]{}
[ w21 := double(rnor(dum)); ]{}
[ w22 := double(rnor(dum)); ]{}
[ w31 := double(rnor(dum)); ]{}
[ w32 := double(rnor(dum)); ]{}
[ z11 := a11[\*]{}w11 ; ]{}
[ z12 := a21[\*]{}z11 + a22[\*]{}w12; ]{}
[ z21 := a11[\*]{}w21 ; ]{}
[ z22 := a21[\*]{}z21 + a22[\*]{}w22; ]{}
[ z31 := a11[\*]{}w31 ; ]{}
[ z32 := a21[\*]{}z31 + a22[\*]{}w32;]{}
[ om12 := de12 - chi2[\*]{}x(10) + chi1[\*]{}x(9); ]{}
[ om23 := de23 - chi3[\*]{}x(11) + chi2[\*]{}x(10); ]{}
[ om13 := de13 - chi3[\*]{}x(11) + chi1[\*]{}x(9); ]{}
[ qom12 := om12 [\*]{} om12; ]{}
[ qom13 := om13 [\*]{} om13; ]{}
[ qom23 := om23 [\*]{} om23; ]{}
[ akey21 := - chi2[\*]{}x(13)/am2 + chi1[\*]{}x(12)/am1 ; ]{}
[ akey23 := - chi2[\*]{}x(13)/am2 + chi3[\*]{}x(14)/am3 ; ]{}
[ akey31 := - chi3[\*]{}x(14)/am3 + chi1[\*]{}x(12)/am1 ;]{}
[ ajay1 := chi1[\*]{}(ak0+x(1)/2.0+x(2))/3.0; ]{}
[ ajay2 := chi2[\*]{}(ak0+x(1)/2.0-x(2)/2.0)/3.0; ]{}
[ ajay3 := chi3[\*]{}(ak0-x(1)-x(2)/2.0)/3.0; ]{}
[ omx3 := om23 + om13; ]{}
[ omx1 := om12 + om13; ]{}
[ xh(1) := x(1) + 2.0[\*]{}v[\*]{}( 2.0[\*]{}x(7) + x(6) ) [\*]{} dh + ]{}
[ v2[\*]{}(x(2)-2.0[\*]{}x(1)+3.0[\*]{}x(5)) [\*]{} dh2 + ]{}
[ v[\*]{}(2.0[\*]{}om23[\*]{}x(4) - om12[\*]{}x(3)) [\*]{} dh2;]{}
[ xh(2) := x(2) - 2.0[\*]{}v[\*]{}( x(7) + 2.0[\*]{}x(6) ) [\*]{} dh + ]{}
[ v2[\*]{}(x(1)-2.0[\*]{}x(2)-3.0[\*]{}x(5)) [\*]{} dh2 + ]{}
[ v[\*]{}(2.0[\*]{}om12[\*]{}x(3) - om23[\*]{}x(4)) [\*]{} dh2;]{}
[ xh(3) := x(3) + ( om12[\*]{}x(6) + v[\*]{}x(8) ) [\*]{} dh - ]{}
[ qom12[\*]{}x(3) [\*]{} dh25 + v2[\*]{}(x(4)-x(3)) [\*]{} dh25 + ]{}
[ v[\*]{}(om12[\*]{}x(2) + omx1[\*]{}x(5)) [\*]{} dh25 + ]{}
[ akey21[\*]{}x(6) [\*]{} dh25 ;]{}
[ xh(4) := x(4) - ( om23[\*]{}x(7) + v[\*]{}x(8) ) [\*]{} dh - ]{}
[ qom23[\*]{}x(4) [\*]{} dh25 + v2[\*]{}(x(3)-x(4)) [\*]{} dh25 + ]{}
[ v[\*]{}(om23[\*]{}x(1) - omx3[\*]{}x(5)) [\*]{} dh25 + ]{}
[ akey23[\*]{}x(7) [\*]{} dh25 ;]{}
[ xh(5) := x(5) - ( om13[\*]{}x(8) + v[\*]{}x(6) + v[\*]{}x(7) ) [\*]{} dh - ]{}
[ qom13[\*]{}x(5) [\*]{} dh25 + v2[\*]{}(x(1)-x(2) - 2.0[\*]{}x(5)) [\*]{} dh25 + ]{}
[ v[\*]{}(omx1[\*]{}x(3) - omx3[\*]{}x(4)) [\*]{} dh25 - ]{}
[ akey31[\*]{}x(8) [\*]{} dh25 ;]{}
[ xh(6) := x(6) - ( om12[\*]{}x(3) - v[\*]{}x(2) - v[\*]{}x(5) ) [\*]{} dh -]{}
[ qom12[\*]{}x(6) [\*]{} dh25 - v2[\*]{}(5.0[\*]{}x(6)+3.0[\*]{}x(7)) [\*]{} dh25 - ]{}
[ v[\*]{}omx1[\*]{}x(8) [\*]{} dh25 - ]{}
[ akey21[\*]{}x(3) [\*]{} dh25 ;]{}
[ xh(7) := x(7) + ( om23[\*]{}x(4) - v[\*]{}x(1) + v[\*]{}x(5) ) [\*]{} dh - ]{}
[ qom23[\*]{}x(7) [\*]{} dh25 - v2[\*]{}(3.0[\*]{}x(6)+5.0[\*]{}x(7)) [\*]{} dh25 - ]{}
[ v[\*]{}omx3[\*]{}x(8) [\*]{} dh25 - ]{}
[ akey23[\*]{}x(4) [\*]{} dh25 ;]{}
[ xh(8) := x(8) + ( om13[\*]{}x(5) - v[\*]{}x(3) + v[\*]{}x(4) ) [\*]{} dh - ]{}
[ qom13[\*]{}x(8) [\*]{} dh25 - 2.0[\*]{}v2[\*]{}x(8) [\*]{} dh25 - ]{}
[ v[\*]{}(omx1[\*]{}x(6) + omx3[\*]{}x(7)) [\*]{} dh25 + ]{}
[ akey31[\*]{}x(5) [\*]{} dh25 ;]{}
[ xh(9) := x(9) + (x(12)/am1) [\*]{} dh - ]{}
[ qom1[\*]{}x(9) [\*]{} dh25 - ]{}
[ dh25 [\*]{} (ajay1 + gam1[\*]{}x(12))/am1 + ]{}
[ (D1/am1) [\*]{} z12 ;]{}
[ xh(10) := x(10) + (x(13)/am2) [\*]{} dh - ]{}
[ qom2[\*]{}x(10) [\*]{} dh25 - ]{}
[ dh25 [\*]{} (ajay2 + gam2[\*]{}x(13))/am2 + ]{}
[ (D2/am2) [\*]{} z22 ;]{}
[ xh(11) := x(11) + (x(14)/am3) [\*]{} dh - ]{}
[ qom3[\*]{}x(11) [\*]{} dh25 - ]{}
[ dh25 [\*]{} (ajay3 + gam3[\*]{}x(14))/am3 + ]{}
[ (D3/am3) [\*]{} z32 ;]{}
[ xh(12) := x(12) + D1[\*]{}z11 - ]{}
[ ( am1[\*]{}qom1[\*]{}x(9) + ajay1 + gam1[\*]{}x(12) ) [\*]{} dh + ]{}
[ qom1[\*]{}(am1[\*]{}gam1[\*]{}x(9) - x(12)) [\*]{} dh25 + ]{}
[ (chi1[\*]{}v[\*]{}x(6) + gam1[\*]{}ajay1 + qgam1[\*]{}x(12)) [\*]{} dh25 - ]{}
[ gam1[\*]{}D1[\*]{}z12;]{}
[ xh(13) := x(13) + D2[\*]{}z21 - ]{}
[ ( am2[\*]{}qom2[\*]{}x(10) + ajay2 + gam2[\*]{}x(13) )[\*]{}dh + ]{}
[ qom2[\*]{}(am2[\*]{}gam2[\*]{}x(10) - x(13)) [\*]{} dh25 + ]{}
[ (-chi2[\*]{}v[\*]{}(x(7)+x(6)) + gam2[\*]{}ajay2 + qgam2[\*]{}x(13)) [\*]{} dh25 - ]{}
[ gam2[\*]{}D2[\*]{}z22;]{}
[ xh(14) := x(14) + D3[\*]{}z31 - ]{}
[ ( am3[\*]{}qom3[\*]{}x(11) + ajay3 + gam3[\*]{}x(14) )[\*]{}dh + ]{}
[ qom3[\*]{}(am3[\*]{}gam3[\*]{}x(11) - x(14)) [\*]{} dh25 + ]{}
[ (chi3[\*]{}v[\*]{}x(7) + gam3[\*]{}ajay3 + qgam3[\*]{}x(14)) [\*]{} dh25 - ]{}
[ gam3[\*]{}D3[\*]{}z32;]{}
[ for ii in 1..config\_space loop ]{}
[ x(ii) := xh(ii); ]{}
[ end loop; ]{}
[ end loop;]{}
[end loop;]{}
|
---
abstract: 'We present a study of axial charges of baryon ground and resonant states with relativistic constituent quark models. In particular, the axial charges of octet and decuplet $N$, $\Sigma$, $\Xi$, $\Delta$, $\Sigma^*$, and $\Xi^*$ baryons are considered. The theoretical predictions are compared to existing experimental data and results from other approaches, notably from lattice quantum chromodynamics and chiral perturbation theory. The relevance of axial charges with regard to $\pi$-dressing and spontaneous chiral-symmetry breaking is discussed.'
author:
- 'Ki-Seok Choi, W. Plessas, and R.F. Wagenbrunn'
title: Axial charges of octet and decuplet baryons
---
Introduction
============
The axial charges $g_A$ of baryon states are essential quantities for the understanding of both the electroweak and strong interactions within the Standard Model of elementary-particle physics. They do not only govern weak processes, such as the $\beta$ decay, but also intertwine the weak and strong interactions. This is most clearly reflected by the Goldberger-Treiman relation, which in case of the $N$ reads $g_A=f_{\pi}g_{\pi NN}/M_N$ [@Goldberger:1958tr]. Given the $\pi$ decay constant $f_{\pi}$ and the nucleon mass $M_N$, the $\pi NN$ coupling constant $g_{\pi NN}$ just turns out to be proportional to $g_A$. Thus the relevance of $\pi$ degrees of freedom in (low- and intermediate-energy) hadronic physics is intimately tied to the axial charges: Whenever $g_A$ becomes sizable, $\pi$ degrees of freedom should matter sensibly. Therefore $g_A$ can also be viewed as an indicator of the phenomenon of spontaneous breaking of chiral symmetry (SB$\chi$S) of non-perturbative quantum chromodynamics (QCD), which is manifested by the non-vanishing value of the light-flavor chiral condensate $\left<0|q\bar q|0\right>^{1/3}\approx -235$ MeV. The axial charges thus constitute important parameters for low-energy effective theories. Any reasonable model of non-perturbative QCD should yield the $g_A$ of correct sizes. In fact, the axial charges may be considered as benchmark observables for the nucleon, and more comprehensively the baryon, structures.
Best known is, of course, the axial charge of the $N$, as its experimental value can be deduced from the ratio of the axial to the vector coupling constants $g_A/g_V=1.2695\pm0.0029$ [@Amsler:2008zzb]; usually this is done under the assumption of conserved vector currents (CVC), which implies $g_V=1$. The deviation of $g_A$ from 1, the axial charge of a point-like particle, can be attributed, according to the Adler-Weisberger sum rule [@Adler:1965ka; @Weisberger:1965hp], to the differences between the $\pi^+ N$ and $\pi^- N$ cross sections in pion-nucleon scattering. Unfortunately, axial charges of other baryon (ground) states are not known directly from experiment.
The axial charges are also ’measured’ in lattice QCD. An increasing number of results has recently become available, even from full-QCD lattice calculations. A recent summary of the lattice-QCD results for the $g_A$ of the nucleon is presented in ref. [@Renner_2009]. The axial charges of hyperons have been studied by Lin et al. [@Lin:2007ap] as well as Erkol et al. [@Erkol:2009ev] and Engel et al. [@Engel:2009nh] in (2+1)- and 2-flavor lattice QCD, respectively. There have also been a number of other attempts to explain the axial charges of the $N$ and the other baryons. We mention only the more modern ones through chiral perturbation theory ($\chi$PT) (see the recent review by Bernard [@Bernard:2007zu] or, for example, ref. [@Jiang:2009sf]), chiral unitary approaches [@Nacher:1999vg], and relativistic constituent quark models (RCQM) [@Glozman:2001zc; @Boffi:2001zb; @Merten:2002nz].
Recently, also the axial charges of the $N$ resonances have come into the focus of interest, because of the question of restoration of chiral symmetry higher in the baryon (as well as meson) spectra. Specifically, it has been argued that the magnitudes of $g_A$ should become small for almost degenerate parity-partner $N$ resonances, indicating the onset of chiral-symmetry restoration with higher excitation energies [@Glozman:2007ek; @Glozman:2008vg]. As the $g_A$ values of $N$ resonances are not known from phenomenology and can hardly be measured experimentally, this remains a highly theoretical question. However, the problem can be explored with the use of lattice QCD. Corresponding first results have already become available, but only for two of the $N$ resonances, namely, $N$(1535) and $N$(1650) [@Takahashi:2008fy]. Both of them have spin $J=\frac{1}{2}$ and parity $P=-1$. Since there is not yet any lattice-QCD result for positive-parity states, the above issue relating to parity-doubling remains unresolved from this side.
The problem of $g_A$ of the $N$ resonances has most recently also been studied within the RCQM [@Choi2010]. The axial charges of all the $N$ resonances up to $\sim$1.9 GeV and $J^P=\frac{1}{2}^\pm, \frac{3}{2}^\pm, \frac{5}{2}^\pm$ have been calculated with $N$ resonance wave functions stemming from realistic RCQM with Goldstone-boson-exchange (GBE) as well as one-gluon-exchange (OGE) hyperfine interactions. One has found the remarkable result that, especially in case of the GBE RCQM, the magnitudes of the axial charges need not be small, even if the energy levels of the opposite-parity partners become (almost) degenerate at increased excitation energies, e.g. the $J^P=\frac{5}{2}^\pm$ resonances $N$(1680) and $N$(1675). Thus the issue of possible chiral-restoration phenomena reflected by the axial charges remains tantalizing until further insights become available (e.g. from lattice QCD or alternative attempts).
Another question related to the axial charges of the $N$ resonances concerns the role of $\{QQQQ\bar Q\}$ components. It has been argued that sizable admixtures of $\{QQQQ\bar Q\}$ are needed in order to reproduce an almost vanishing $g_A$ of the $N$(1535) resonance [@An:2008tz; @Yuan:2009st]. However, these results are usually obtained in a simplistic non-relativistic approach. Meanwhile it is known that a RCQM with realistic $\{QQQ\}$ wave functions can easily explain a practically vanishing $g_A$ of $N$(1535) [@Choi2010], in perfect congruency with the predictions obtained from lattice QCD [@Takahashi:2008fy], and there is no need for considerable $\{QQQQ\bar Q\}$ admixtures in this case. Moreover, the correct sizes of the axial charges of the $N$ ground state and the $N$(1535) as well as $N$(1650) resonances can simultaneously and consistently be reproduced within a RCQM with only $\{QQQ\}$ configurations [@Choi2010].
In the context of hyperons the axial charges are also important to learn about the role of $SU(6)$ flavor-symmetry breaking. In particular, in the case of conserved $SU(3)_F$ the axial charges of the $N$, $\Sigma$, and $\Xi$ ground states are connected by the following simple relations [@Gaillard:1984ny; @Dannbom:1996sh] $$g_A^N=F+D\,, \hspace{3mm} g_A^{\Sigma}=\sqrt{2}F\,, \hspace{3mm} g_A^{\Xi}=F-D\,,
\label{axcharges}$$ which follow through $SU(3)$ Clebsch-Gordan coefficients in the decomposition of the axial form factor into the functions $F$ and $D$ relating to the octet components in $SU(3)$ [@de; @Swart:1963gc]. Note that we adopt the convention of $g_A/g_V$ being positive for the $N$ (like in ref. [@Gaillard:1984ny]) contrary to the PDG [@Amsler:2008zzb]; this then determines also the signs of all other baryon axial charges according to Eq. (\[axcharges\]).
In the present paper we present results from a comprehensive study of the axial charges of octet and decuplet ground states $N$, $\Sigma$, $\Xi$, $\Delta$, $\Sigma^*$, and $\Xi^*$ as well as their resonances along RCQMs. In particular, we employ the RCQMs whose quark-quark hyperfine interactions derive from OGE [@Theussl:2000sj] and GBE dynamics [@Glozman:1997fs]; in the latter case we consider both the version with only the spin-spin interaction from pseudoscalar exchange (psGBE) [@Glozman:1997ag] as well as the extended version that includes all force components (i.e. central, tensor, spin-spin, and spin-orbit) from pseudoscalar, scalar, and vector exchanges (EGBE) [@Glantschnig:2004mu]. The calculations are performed in the framework of Poincarè-invariant quantum mechanics. In order to keep the numerical computations manageable, we have to restrict the axial current operator to the so-called spectator model (SM). It means that the weak-interaction gauge boson couples only to one of the constituent quarks in the baryon. This approximation has turned out to be very reasonable already in a previous study of the axial and induced pseudoscalar form factors of the nucleon [@Glozman:2001zc], where the SM was employed specifically in the point form (PF) of relativistic quantum mechanics [@Melde:2004qu]. It has also been used in studies of the electromagnetic structure of the $N$, reproducing both the proton and neutron form factors in close agreement with the experimental data [@Boffi:2001zb; @Wagenbrunn:2000es; @Berger:2004yi; @Melde:2007zz].
In the following chapter we explain the formalism for the calculation of the matrix elements of the axial current operator and give the definition of the axial charges for the different baryon ground and resonant states. Subsequently we present the results and compare them to experimental data as well as to results from other approaches, notably from lattice QCD and $\chi PT$. In the final chapter we draw our conclusions.
Formalism
=========
In hadronic physics the (diagonal) baryon axial charges $g_A^B$ govern such processes like $n\rightarrow pe^-\bar \nu_e$, $\Sigma^-\rightarrow \Sigma^0 e^-\bar \nu_e$, $\Xi^-\rightarrow \Xi^0 e^-\bar \nu_e$ etc. They can generally be calculated through semileptonic decays $B_1 \rightarrow B_2 \ell\bar \nu$ with strangeness change $\Delta S=0$. The axial charge is conveniently defined through the value of the axial form factor $G_A(Q^2)$ at $Q^2=0$, where $Q^2=-q^2$ is the four-momentum transfer. The axial form factor $G_A(Q^2)$ can be deduced from the relativistically invariant matrix element of the axial current operator $\hat A^\mu_+ (Q^2)$ sandwiched between the eigenstates of baryons $B_1$ and $B_2$. Here, the subscript $+$ refers to the isospin-raising ladder operator $\tau_+ = \frac{1}{2}\left(\tau_1 + i\tau_2\right)$, with $\tau_i$ being the usual Pauli matrices. In the specific case of the neutron $\beta$ decay the matrix element of $\hat A^\mu_+(Q^2=0)$ reads $$\left<p\left|\hat A^\mu_+\right|n\right>=
g_A^N \bar U_p(P,J'_3)\gamma^\mu \gamma_5 \frac{\tau_+}{2}
U_n(P,J_3) \,,
\label{n-p}$$ where $U_n$ and $U_p$ are the neutron and proton spinors, depending on the four-momentum $P$ and helicities $J_3$ and $J'_3$, respectively; $\gamma^\mu$ and $\gamma_5$ are the usual Dirac matrices. Alternatively, the matrix element in Eq. (\[n-p\]) can also be expressed as $$\left<p\left|\hat A^\mu_3\right|p\right>=
g_A^N \bar U_p(P,J'_3)\gamma^\mu \gamma_5 \frac{\tau_3}{2}
U_p(P,J_3)
\label{p-p}$$ or $$\left<n\left|\hat A^\mu_3\right|n\right>=
g_A^N \bar U_n(P,J'_3)\gamma^\mu \gamma_5 \frac{\tau_3}{2}
U_n(P,J_3) \,.
\label{n-n}$$
In the spirit of the latter relations we may express the axial charge $g_A^B$ of any baryon $B=N, \Delta, \Sigma, \Xi, ...$ and its resonances more generally. Let us denote the baryon states by $\left|B;P,J,J_3\right>$, i.e. as simultaneous eigenstates of the four-momentum operator $\hat P^\mu$, the intrinsic-spin operator $\hat J$ and its $z$-projection $\hat J_3$. Since $\hat P^\mu$ and the invariant mass operator $\hat M$ commute, these eigenstates can be obtained by solving the eigenvalue equation of $\hat M$ $$\hat M \left|B;P,J,J_3\right>=M \left|B;P,J,J_3\right> \, ,$$
Then the axial charge $g_A^B$ of any baryon state $B$ with $J=\frac{1}{2}, \frac{3}{2}, \frac{5}{2}$ is given by the matrix elements of the axial current operator $\hat A_3^{\mu}$ for zero momentum transfer $Q^2$ as $$\begin{aligned}
&&\left<B;P,\frac{1}{2},J_3'\left|{\hat A}^{\mu}_3\right|B;P,\frac{1}{2},J_3\right>=\nonumber\\
&&\hspace{1.1cm}C_B\bar U_B(P,J_3')g_A^B \gamma^{\mu}\gamma_5 \frac{\tau_3}{2} U_B(P,J_3) \, ,\nonumber\\
&&\left<B;P,\frac{3}{2},J_3'\left|{\hat A}^{\mu}_3\right|B;P,\frac{3}{2},J_3\right>=\nonumber\\
&&\hspace{1.1cm}C_B\bar U_B^\nu(P,J_3')g_A^B \gamma^{\mu}\gamma_5 \frac{\tau_3}{2}
U_{B;\nu}(P,J_3) \, ,\nonumber\\
&&\left<B;P,\frac{5}{2},J_3'\left|{\hat A}^{\mu}_3\right|B;P,\frac{5}{2},J_3\right>=\nonumber\\
&&\hspace{1.1cm}C_B\bar U_B^{\nu\lambda}(P,J_3')g_A^B \gamma^{\mu}\gamma_5 \frac{\tau_3}{2}
U_{B;\nu\lambda}(P,J_3) \, , \end{aligned}$$ where the coefficients $C_B$ are specified by $$C_N=2C_{\Delta}=\frac{1}{\sqrt{2}}C_{\Sigma}=C_{\Xi}=1 \,.$$ Here, $U_B(P,J_3)$ are the usual Dirac spinors for $J=\frac{1}{2}$ baryons and $U_{B;\nu}(P,J_3)$ as well as $U_{B;\nu\lambda}(P,J_3)$ are the Rarita-Schwinger spinors [@Rarita:1941mf] for $J=\frac{3}{2}$ and $J=\frac{5}{2}$ baryons, respectively, with the normalizations as given in Appendix A.
Omitting from now on the denotation after $B$ we can write the matrix elements of $\hat A^\mu_3$ for any ground and resonance states as $$\begin{aligned}
&&\left<P,J,J_3'\right|{\hat A^\mu_3 (Q^2=0)}\left|P,J,J_3\right>=
\nonumber \\
&&
2M\sum_{\sigma_i\sigma'_i}{\int{
d^3{\vec k}_1 d^3{\vec k}_2 d^3{\vec k}_3}}
\frac{\delta^3(\vec k_1+\vec k_2+\vec k_3)}{2\omega_1 2\omega_2 2\omega_3} \nonumber \\
&&
\times\Psi^\star_{PJJ_3'}\left({\vec k}_1,{\vec k}_2,{\vec k}_3;
\sigma'_1,\sigma'_2,\sigma'_3\right) \nonumber \\
&&
\times\left<k_1,k_2,k_3;\sigma'_1,\sigma'_2,\sigma'_3\right|\hat{A}^{\mu}_3
\left|k_1,k_2,k_3;\sigma_1,\sigma_2,\sigma_3\right>
\nonumber\\
&&
\times\Psi_{PJJ_3}\left({\vec k}_1,{\vec k}_2,{\vec k}_3;
\sigma_1,\sigma_2,\sigma_3\right) \, .
\label{transampl}\end{aligned}$$ The $\Psi$’s are the momentum-space representations of the baryon eigenstates for $\vec P=0$, i.e. the rest-frame wave functions of the baryon ground and resonance states with corresponding mass $M$ and total angular momentum $J$ and $z$-projections $J_3$ as well as $J_3'$. Here they are expressed as functions of the individual quark three-momenta $\vec k_i$, which sum up to $\vec P=\vec k_1+\vec k_2+\vec k_3=0$; $\omega_i=\sqrt{m^2_i+\vec k^2_i}$ is the energy of quark $i$ with mass $m_i$, and the individual-quark spin orientations are denoted by $\sigma_i$.
The integral on the r.h.s. of Eq. (\[transampl\]) is evaluated along the SM what amounts to the matrix element of the axial current operator $\hat{A}^{\mu}_a$ between (free) three-particle states $\left|k_1,k_2,k_3;\sigma_1,\sigma_2,\sigma_3\right>$ to be assumed in the form $$\begin{gathered}
\left<k_1,k_2,k_3;\sigma'_1,\sigma'_2,\sigma'_3\right|
{\hat A}^{\mu}_3
\left|k_1,k_2,k_3;\sigma_1,\sigma_2,\sigma_3\right>
= \\
3\left<k_1,\sigma'_1\right|\hat{A}^{\mu}_{3,{\rm SM}}
\left|k_1,\sigma_1\right>2\omega_2 2\omega_3
\delta_{\sigma_{2}\sigma'_{2}}\delta_{\sigma_{3}\sigma'_{3}}
\label{eq:axcurr1}\, .\end{gathered}$$ For point-like quarks this matrix element involves the axial current operator of the active quark 1 (with quarks 2 and 3 being the spectators) in the form $$%\begin{multline}
\left<k_1,\sigma'_1\right|\hat{A}^{\mu}_{3,{\rm SM}}
\left|k_1,\sigma_1\right>=
%\\
{\bar u}\left(k_1,\sigma'_1\right)g_A^q \gamma^\mu
\gamma_5 \frac{{\tau}_3}{2} u\left(k_1,\sigma_1\right) \, ,
\label{eq:axcurr2}
%\end{multline}$$ where $u\left(k_1,\sigma_1\right)$ is the spinor of a quark with flavor $u$ or $d$ and $g_A^q=1$ its axial charge. A pseudovector current analogous to the one in Eq. (\[eq:axcurr2\]) was recently also used in the calculation of $g_{\pi NN}$ and the strong $\pi NN$ vertex form factor in ref. [@Melde:2008dg].
For the calculation of the axial charges $g_A$ we can use either one of the components $\mu=i=1, 2, 3$ of the axial current operator $\hat{A}^{\mu}_{3,{\rm SM}}$ in Eq. (\[eq:axcurr2\]). The expression on the r.h.s. then specifies to $$\begin{aligned}
&&{\bar u}\left(k_1,\sigma'_1\right) \gamma^i\gamma_5 \frac{{\tau}_3}{2} u\left( k_1,\sigma_1\right)=\nonumber\\
&&
2\omega_1\chi^*_{\frac{1}{2},\sigma'_1}
\Biggl\{\left[1-\frac{2}{3}\left(1-\kappa\right) \right]\sigma^i \nonumber\\
&&
+\sqrt{\frac{5}{3}}\frac{\kappa^2}{1+\kappa}
\left[\,\left[\vec{v}_{1}\otimes\vec{v}_{1}\right]_2\otimes\vec{\sigma}\right]_1^i\Biggl\}
\frac{{\tau}_3}{2} \chi_{\frac{1}{2},\sigma_1}\,,
\label{eq:ga}\end{aligned}$$ where $\kappa=1/\sqrt{1+v_1^2}$ and $\vec v_1=\vec k_1/m_1$. Herein $\sigma^i$ is the $i$-th component of the usual Pauli matrix $\vec \sigma$ and $v_1$ the magnitude of the three-velocity $\vec v_1$. The symbol $\left[.\otimes .\right]_k^i$ denotes the $i$-th component of a tensor product $\left[.\otimes .\right]_k$ of rank $k$. We note that a similar formula was already published before by Dannbom et al. [@Dannbom:1996sh], however, restricted to the case of total orbital angular momentum $L=0$. Our expression holds for any $L$, thus allowing to calculate $g_A$ for the most general wave function of a baryon ground or resonances state specified by intrinsic spin and parity $J^P$.
Results
=======
Exp EGBE psGBE OGE LO EOT JT NR
-------------- ------------------- ------- ------- ------- --------------------------- ----------------------------- ------------------ -------
N 1.2695$\pm$0.0029 1.15 1.15 1.11 1.18$\pm$0.10 1.314$\pm$0.024 1.18 1.65
$\Sigma$ - 0.65 0.65 0.65 0.636$\pm$0.068$^\dagger$ 0.686$\pm$0.021$^\dagger$ 0.73 0.93
$\Xi$ - -0.21 -0.22 -0.22 -0.277$\pm$0.034 -0.299$\pm$0.014$^\ddagger$ -0.23$^\ddagger$ -0.32
$\Delta$ - -4.48 -4.47 -4.30 - - $\sim\,$-4.5 -6.00
$\Sigma^{*}$ - -1.06 -1.06 -1.00 - - - -1.41
$\Xi^{*}$ - -0.75 -0.75 -0.70 - - - -1.00
\
$^\dagger$ Due to another definition of $g_A^{\Sigma}$ this numerical value is different by a $\sqrt{2}$ from the one quoted in the original paper.\
$^\ddagger$ Due to another definition of $g_A^{\Xi}$ this value has a sign opposite to the one in the original paper.
\[EGBE\]
In Table \[EGBE\] we present the RCQM results from our calculations for the axial charges $g_A^B$ of the octet and decuplet ground states $B=N$, $\Sigma$, $\Xi$, $\Delta$, $\Sigma^*$, and $\Xi^*$. Except for the $N$ there are no direct experimental data for $g_A$ (from $\Delta S=0$ decays). The predictions for $g_A^N$ by all three RCQMs come close to the experimental value, with all of them falling slightly below it. This is also the trend of present-day lattice-QCD calculations [@Renner_2009]; only the EOT result seems to represent a notable exception, even if we take the theoretical uncertainties into account (in Table \[EGBE\] we have chosen to quote the EOT result corresponding to their calculation with the smallest quark mass of 35 MeV). In addition, also the JT prediction obtained from $\chi$PT remains below the experimental value. There might be a variety of reasons why the different approaches underestimate the $g_A^N$. However, one should also bear in mind that the phenomenological value of $g_A^N\sim\,$1.27 is supposed under the conjecture of conserved vector currents. What concerns the RCQMs, and in particular the [psGBE]{} RCQM, interestingly, it has recently been found [@Melde:2004qu] that also the $\pi NN$ coupling constant turns out to be too small, namely $\frac{f^2_{\pi NN}}{4\pi}=$ 0.0691, as compared to the phenomenological value of about 0.075 [@Bugg:2004cm]. It remains to be clarified if in case of the RCQMs these undershootings of both the $g_A^N$ and $f^2_{\pi NN}$, which are related by the Goldberger-Treiman relation, have to be interpreted as lacking $\pi-$dressing effects.
In the last column of Table \[EGBE\] we quote also the nonrelativistic limit of the prediction by the EGBE RCQM (i.e. for the limit $\kappa \rightarrow$ 1 in Eq. (\[eq:ga\])). It deviates grossly from the relativistic result, indicating that a consideration of axial charges within a nonrelativistic approach is unreliable. This conclusion is further substantiated by considering the axial charges of $N$ resonances, for which indeed no experimental data are available but lattice-QCD results have recently been produced. While the relativistic predictions of especially the EGBE RCQM agree very well with the lattice-QCD data in case of both the $N$(1535) and $N$(1650) resonances, the nonrelativistic limits deviate here too [@Choi2010].
For the $g_A^B$ of the octet and decuplet ground states the RCQMs yield very similar results. While the predictions of the psGBE and the EGBE are essentially the same, differences occur only for the OGE RCQM, but they remain within at most $\sim\,$6%. In case of the octet states $\Sigma$ and $\Xi$ we can also compare to lattice-QCD as well as $\chi$PT results. The comparison of the RCQM predictions to the former is quite satisfying, as the figures agree rather well. Except for $g_A^{\Sigma}$ practically the same is true with regard to the $\chi$PT results of JT. Again, the results from the nonrelativistic limit of the EGBE RCQM fall short; as in the case of the $N$ the corresponding values are always bigger (in absolute value) than all of the other results.
For the decuplet ground states $\Delta$, $\Sigma^*$, and $\Xi^*$ there are neither experimental data nor lattice-QCD results. Only for the $\Delta$ we may compare with a $\chi$PT prediction, showing again a striking similarity. For the other cases of $\Sigma^*$ and $\Xi^*$ we have here produced first predictions and one has still to await results from other approaches.
[lcp[3mm]{}crp[3mm]{}crp[3mm]{}cr]{}\
&& && && State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\
$N$(939) & $\frac{1}{2}^{+}$ && 939 & 1.15 && 939& 1.15 && 939 & 1.11
$N$(1440) & $\frac{1}{2}^{+}$ && 1464 & 1.16 && 1459 & 1.13 && 1578 & 1.10
$N$(1520) & $\frac{3}{2}^{-}$ && 1524 & -0.64 && 1519 & -0.21 && 1520 & -0.15
$N$(1535) & $\frac{1}{2}^{-}$ && 1498 & 0.02 && 1519 & 0.09 && 1520 & 0.13
$N$(1650) & $\frac{1}{2}^{-}$ && 1581 & 0.51 && 1647 & 0.46 && 1690 & 0.44
$N$(1700) & $\frac{3}{2}^{-}$ && 1608 & -0.10 && 1647 & -0.50 && 1690 & -0.47
$N$(1710) & $\frac{1}{2}^{+}$ && 1757 & 0.35 && 1776 & 0.37 && 1860 & 0.32
$N$(1720) & $\frac{3}{2}^{+}$ && 1746 & 0.35 && 1728 & 0.34 && 1858 & 0.25
$N$(1675) & $\frac{5}{2}^{-}$ && 1676 & 0.84 && 1647 & 0.83 && 1690 & 0.80
$N$(1680) & $\frac{5}{2}^{+}$ && 1689 & 0.89 && 1728& 0.83 && 1858 & 0.70
\[N\]
Next we come to discuss the axial charges of nucleon and other baryon resonances. As mentioned in the Introduction, especially the nucleon resonances have recently attracted interest, mainly because the issue of chiral-symmetry restoration higher in the baryon spectra has been raised [@Glozman:2007ek; @Glozman:2008vg] and because first lattice-QCD calculations have been performed [@Takahashi:2008fy]. Certainly the results of the latter have still to be taken with care, as they correspond to relatively high quark masses. For the case of the nucleon we have presented resonance axial charges from RCQMs already in a previous paper [@Choi2010]; for completeness we repeat them here in Table \[N\]. While for details we refer to ref. [@Choi2010], we mention as the main characterization of these results that\
$i$) the RCQM predictions perfectly agree with the lattice-QCD results for the $N$(1535) and $N$(1650) resonances, i.e. in the two cases for which lattice-QCD calculations have so far become available,\
$ii$) the small, practically vanishing, $g_A$ of $N$(1535) can be reproduced with $\{QQQ\}$ configurations alone,\
$iii$) the predictions of different RCQMs are generally very similar except for the $J^P=\frac{3}{2}^-$ resonances $N$(1520) and $N$(1700),\
$iv$) a relativistic description is necessary and a simple $SU(6)\times O(3)$ nonrelativistic quark model is not reliable, and\
$v$) there is no tendency of the axial charges of high-lying parity partners to assume particularly small values.
[lcp[3mm]{}crp[3mm]{}crp[3mm]{}cr]{}\
&& && && State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\
$\Sigma$(1193) & $\frac{1}{2}^{+}$ && 1194 & 0.65 && 1182 & 0.65 && 1121 & 0.65
$\Sigma$(1560) & $\frac{1}{2}^{-}$ && 1672 & -0.15 && 1678& -0.07&& 1655 & 0.01
$\Sigma$(1620) & $\frac{1}{2}^{-}$ && 1740 & 0.62 && 1736& 0.58&& 1770 & 0.54
$\Sigma$(1660) & $\frac{1}{2}^{+}$ && 1664 & 0.69 && 1619 & 0.64 && 1755 & 0.64
$\Sigma$(1670) & $\frac{3}{2}^{-}$ && 1681 & -0.92 && 1678& -0.48 && 1655 & -0.24
$\Sigma$(1775) & $\frac{5}{2}^{-}$ && 1765 & 1.06 && 1736& 1.03&& 1770 & 0.97
$\Sigma$(1880) & $\frac{1}{2}^{+}$ && 1903 & 0.38 && 1912 & 0.42 && 1980 & 0.17
$\Sigma$(1940) & $\frac{3}{2}^{-}$ && 1725 & -0.45 && 1736& -0.83&& 1770 & -0.78
$\Sigma^*$(1385) & $\frac{3}{2}^{+}$ && 1365 & -1.06 && 1389& -1.06&& 1311 & -1.00
$\Sigma^*$(1690) & $\frac{3}{2}^{+}$ && 1812 & -1.05 && 1865& -1.03&& 1932 & -0.99
$\Sigma^*$(1750) & $\frac{1}{2}^{-}$ && 1761 & -0.08 && 1759 & -0.13 && 1718 & -0.18
\[Sigma\]
The RCQM predictions for the axial charges of the octet $\Sigma$ and decuplet $\Sigma^*$ resonances are quoted in Table \[Sigma\]. The gross pattern of the results is like the one of the $N$ resonances. First of all the different [RCQMs]{} yield similar values for the axial charges except for the cases of $\Sigma$(1670) and $\Sigma$(1940), which are again $J^P=\frac{3}{2}^-$ resonances and are to be assigned to the same octets as $N$(1520) and $N$(1700), respectively, according to a recent identification of baryon resonances [@Melde:2008yr] (see also [@PDG2010]). For both $\Sigma$(1670) and $\Sigma$(1940) the differences among the predictions prevail also in the comparison between the EGBE and the psGBE RCQMs hinting to considerable influences from tensor and/or spin-orbit forces, just as in the corresponding two $N$ resonances. All of the RCQMs produce very small values for the axial charges of $\Sigma$(1560). Notably, this state falls into the same octet as $N$(1535) [@Melde:2008yr; @PDG2010], whose $g_A$ is also extremely small (see Table \[N\]). A similar small axial charge is found for the decuplet $\Sigma^*$(1750). In general, however, we do not observe the axial charges to become small as we go up to higher resonances.
The results for the axial charges of the octet $\Xi$ and decuplet $\Xi^*$ resonances are collected in Table \[Xi\]. Here, all the RCQMs produce similar predictions, where the axial charges of the octet resonances remain rather small with values ranging from -0.2 to -0.4.
[lcp[3mm]{}crp[3mm]{}crp[3mm]{}cr]{}\
&& && && State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\
$\Xi$(1318) & $\frac{1}{2}^{+}$ && 1355 & -0.21 && 1348 & -0.22 && 1193 & -0.22
$\Xi$(1690) & $\frac{1}{2}^{+}$ && 1813 & -0.23 && 1806 & -0.22 && 1826 & -0.22
$\Xi$(1820) & $\frac{3}{2}^{-}$ && 1807 & -0.38 && 1792 & -0.40 && 1751 & -0.23
$\Xi^*$(1530) & $\frac{3}{2}^{+}$ && 1512 & -0.75 && 1528 & -0.75 && 1392 & -0.70
\[Xi\]
Finally, in Table \[Delta\] the axial charges of the $\Delta$ resonances are given. Again, all of the RCQMs yield similar predictions. Only it is remarkable that the axial charges especially of the $J^P=\frac{3}{2}^+$ states are rather big in absolute value. If we consider the $\Delta$(1232) ground state, its $g_A$ is at least three times larger than the one of the $N$ ground state. Remarkably a ratio of about the same size has recently been found between the $\pi N\Delta$ and the $\pi NN$ strong coupling constants [@Melde:2008dg]. The smallest $g_A$ is found for $\Delta$(1620). It should be noted that it falls into the same decuplet as the $\Sigma^*$(1750), whose $g_A$ was also seen as the smallest among the $\Sigma^*$ resonances (cf. Table \[Sigma\]).
[lcp[3mm]{}crp[3mm]{}crp[3mm]{}cr]{}\
&& && && State & $J^{p}$ && Mass & $g_{A}$ && Mass & $g_{A}$ && Mass & $g_{A}$\
$\Delta$(1232) & $\frac{3}{2}^{+}$ && 1231 & -4.48 && 1240 & -4.47 && 1231 & -4.30
$\Delta$(1600) & $\frac{3}{2}^{+}$ && 1686 & -4.41 && 1718 & -4.33 && 1855 & -4.20
$\Delta$(1620) & $\frac{1}{2}^{-}$ && 1640 & -0.76 && 1642 & -0.75 && 1621 & -0.74
$\Delta$(1700) & $\frac{3}{2}^{-}$ && 1639 & -1.68 && 1642 & -1.66 && 1621 & -1.58
\[Delta\]
Conclusions
===========
We have presented results from a comprehensive and consistent study of axial charges $g_A^B$ of octet and decuplet baryon ground and resonant states with RCQMs. The dynamical models differ mainly with regard to their hyperfine $Q$-$Q$ interactions, which stem from OGE and psGBE as well as EGBE. Whenever a comparison is possible with either experimental data or established theoretical results (especially from lattice QCD and $\chi$PT), the RCQM predictions turn out to be quite reasonable. The values deduced from a nonrelativistic approximation in general differ grossly, indicating that a relativistic approach to the axial charges is mandatory. The RCQMs considered here rely on $\{QQQ\}$ configurations only. Nevertheless, the $g_A^B$ results never fall short but rather produce a consistent picture. Already for the ground states one finds a scatter of $g_A^B$ values from small to large. Through the Goldberger-Treiman relation one may thus expect smaller or larger $\pi$-dressing effects depending on the baryon state. Particularly big are the axial charges of the $\delta$ ground and first excited states, much in congruency with the relatively large $\pi N\Delta$ coupling constant. The axial charges of some baryon resonances are rather sensitive to tensor and/or spin-orbit forces in the hyperfine interaction. These resonances fall into the same flavor multiplets. In general the pattern observed from the predictions for $g_A^B$ is congruent with the classification of baryons into flavor multiplets as found recently. From the RCQMs predictions presented here, no particular trend is observed for the axial charges of $N$ and other baryon resonances to become small, when the excitation energy is increased. Certainly, the consideration of baryon axial charges remains an exciting field, and one is eager to see additional experimental data as well as more theoretical results from different approaches to QCD.
The authors are grateful to L.Ya. Glozman for valuable incentives regarding specific aspects of this work and to the Graz lattice-QCD group for several clarifying discussions about respective calculations. This work was supported by the Austrian Science Fund, FWF, through the Doctoral Program on [*Hadrons in Vacuum, Nuclei, and Stars*]{} (FWF DK W1203-N08).
Rarita-Schwinger spinors
========================
For the $J=\frac{1}{2}$, $\frac{3}{2}$, and $\frac{5}{2}$ baryons with four-momentum $P$ and energy $E=\sqrt{M^2+{\vec P}^2}$ we employ Dirac and Rarita-Schwinger spinors, similar as in ref. [@Choi:2007gy], as follows:
$\bullet \;\; J=\frac{1}{2}, \;\; U\left(P,J_3=\pm\frac{1}{2}\right)\,:$ $$\begin{aligned}
U\left(P,\frac{1}{2}\right)=\left(\begin{array}{c}
\sqrt{E+M}\\0\\ \frac{{\vec\sigma} \cdot{\vec P}}{\sqrt{E+M}}\\0\end{array}\right) ,
\hspace{7mm} U\left(P,-\frac{1}{2}\right)=\left(\begin{array}{c}
0\\\sqrt{E+M}\\ 0\\ \frac{{\vec\sigma} \cdot{\vec P}}{\sqrt{E+M}}\end{array}\right) ,\end{aligned}$$ where $\vec\sigma$ are the Pauli matrices. These Dirac spinors are normalized as $$\bar{U}(P,J_3') U(P,J_3)=\delta_{J_3', J_3}2M\, .$$ $\bullet \;\; J=\frac{3}{2}, \;\;
U^\mu \left(P,J_3=\pm\frac{1}{2},\pm\frac{3}{2}\right)\,:$ $$\begin{aligned}
U^{\mu}\left(P,\frac{3}{2}\right)&=&e^{\mu}_{+}(P)U\left(P,\frac{1}{2}\right)\, ,
\nonumber\\
U^{\mu}\left(P,\frac{1}{2}\right)&=&\sqrt{\frac{2}{3}}e^{\mu}_{0}(P)U\left(P,\frac{1}{2}\right)
+\sqrt{\frac{1}{3}}e^{\mu}_{+}(P)U\left(P,-\frac{1}{2}\right)\, ,
\nonumber\\
U^{\mu}\left(P,-\frac{1}{2}\right)&=&\sqrt{\frac{1}{3}}e^{\mu}_{-}(P)U\left(P,\frac{1}{2}\right)
+\sqrt{\frac{2}{3}}e^{\mu}_{0}(P)U\left(P,-\frac{1}{2}\right)\, ,
\nonumber\\
U^{\mu}\left(P,-\frac{3}{2}\right)&=&e^{\mu}_{-}(P)U\left(P,-\frac{1}{2}\right)\, .\end{aligned}$$ $\bullet \;\; J=\frac{5}{2}, \;\;
U^{\mu\nu}\left(P,J_3=\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{5}{2}\right)\,:$ $$\begin{aligned}
U^{\mu\nu}\left(P,\frac{5}{2}\right)&=&e^{\mu}_{+}e^{\nu}_{+}U\left(P,\frac{1}{2}\right)\, ,
\nonumber\\
U^{\mu\nu}\left(P,\frac{3}{2}\right)&=&\sqrt{\frac{2}{5}}e^{\mu}_{+}
e^{\nu}_{0}U\left(P,\frac{1}{2}\right)+\sqrt{\frac{1}{5}}e^{\mu}_{+}e^{\nu}_{+}
U\left(P,-\frac{1}{2}\right)
+\sqrt{\frac{2}{5}}e^{\mu}_{0}e^{\nu}_{+}U\left(P,\frac{1}{2}\right)\, ,\nonumber\\
U^{\mu\nu}\left(P,\frac{1}{2}\right)&=&\sqrt{\frac{1}{10}}e^{\mu}_{+}
e^{\nu}_{-}U\left(P,\frac{1}{2}\right)+\sqrt{\frac{1}{5}}e^{\mu}_{+}e^{\nu}_{0}
U\left(P,-\frac{1}{2}\right)
+\sqrt{\frac{2}{5}}e^{\mu}_{0}e^{\nu}_{0}U\left(P,\frac{1}{2}\right)\nonumber\\
&&+\sqrt{\frac{1}{5}}e^{\mu}_{0}e^{\nu}_{+}U\left(P,-\frac{1}{2}\right)
+\sqrt{\frac{1}{10}}e^{\mu}_{-}e^{\nu}_{+}U\left(P,\frac{1}{2}\right)\, ,
\nonumber\\
U^{\mu\nu}\left(P,-\frac{1}{2}\right)&=&\sqrt{\frac{1}{10}}e^{\mu}_{+}
e^{\nu}_{-}U\left(P,-\frac{1}{2}\right)+\sqrt{\frac{1}{5}}e^{\mu}_{0}e^{\nu}_{-}
U\left(P,\frac{1}{2}\right)
+\sqrt{\frac{2}{5}}e^{\mu}_{0}e^{\nu}_{0}U\left(P,-\frac{1}{2}\right)\nonumber \\
&&+\sqrt{\frac{1}{5}}e^{\mu}_{-}e^{\nu}_{0}
U\left(P,\frac{1}{2}\right)
+\sqrt{\frac{1}{10}}e^{\mu}_{-}e^{\nu}_{+}
U\left(P,-\frac{1}{2}\right)\, ,\nonumber\\
U^{\mu\nu}\left(P,-\frac{3}{2}\right)&=&\sqrt{\frac{2}{5}}e^{\mu}_{0}
e^{\nu}_{-}U\left(P,-\frac{1}{2}\right)+\sqrt{\frac{1}{5}}e^{\mu}_{-}
e^{\nu}_{-}U\left(P,-\frac{1}{2}\right)
+\sqrt{\frac{2}{5}}e^{\mu}_{-}e^{\nu}_{0}U\left(P,-\frac{1}{2}\right)\, ,
\nonumber\\
U^{\mu\nu}\left(P,-\frac{5}{2}\right)&=&e^{\mu}_{-}e^{\nu}_{-}
U\left(P,-\frac{1}{2}\right)\, .\end{aligned}$$ In the latter equation we have suppressed the arguments in the polarization vectors $e^{\mu}_{\lambda}(P)$ defined by $$e^{\mu}_{\lambda}(P)=\left(\frac{\hat{e}_{\lambda}\cdot
{\vec P}}{M},\, \hat{e}_{\lambda}
+\frac{(\hat{e}_{\lambda}\cdot{\vec P}){\vec P}}{M(E+M)}\right),$$ where for $\lambda=+, 0, -$ the unit vectors $\hat e_\lambda$ are written as $$\hat{e}_{+}=-\frac{1}{\sqrt{2}}(1,i,0),\,\hat{e}_{0}=(0,0,1),\,
\hat{e}_{-}
=\frac{1}{\sqrt{2}}(1,-i,0)\, .$$
[10]{}
M. L. Goldberger and S. B. Treiman, Phys. Rev. [**110**]{}, 1178 (1958).
C. Amsler [*et al.*]{} \[Particle Data Group\], Phys. Lett. B [**667**]{}, 1 (2008).
S. L. Adler, Phys. Rev. Lett. [**14**]{}, 1051 (1965).
W. I. Weisberger, Phys. Rev. Lett. [**14**]{}, 1047 (1965).
D. Renner, PoS(LATTICE 2009), 18 (2009).
H. W. Lin and K. Orginos, Phys. Rev. D [**79**]{}, 034507 (2009).
G. Erkol, M. Oka, and T. T. Takahashi, Phys. Lett. B [**686**]{}, 36 (2010).
G. Engel, C. Gattringer, L. Y. Glozman, C. B. Lang, M. Limmer, D. Mohler, and A. Schäfer, PoS(LATTICE 2009), 135 (2009); arXiv:0910.4190.
V. Bernard, Prog. Part. Nucl. Phys. [**60**]{}, 82 (2008).
F. J. Jiang and B. C. Tiburzi, Phys. Rev. D [**80**]{}, 077501 (2009).
J. C. Nacher, A. Parreno, E. Oset, A. Ramos, A. Hosaka, and M. Oka, Nucl. Phys. A [**678**]{}, 187 (2000).
L. Y. Glozman, M. Radici, R. F. Wagenbrunn, S. Boffi, W. Klink, and W. Plessas, Phys. Lett. B [**516**]{}, 183 (2001).
S. Boffi, L. Y. Glozman, W. Klink, W. Plessas, M. Radici, and R. F. Wagenbrunn, Eur. Phys. J. A [**14**]{}, 17 (2002).
D. Merten, U. Löring, K. Kretzschmar, B. Metsch, and H. R. Petry, Eur. Phys. J. A [**14**]{}, 477 (2002).
L. Y. Glozman, Phys. Rep. [**444**]{}, 1 (2007).
L. Y. Glozman and A. V. Nefediev, Nucl. Phys. A [**807**]{}, 38 (2008).
T. T. Takahashi and T. Kunihiro, Phys. Rev. D [**78**]{}, 011503 (2008).
Ki-Seok Choi, W. Plessas, and R .F. Wagenbrunn, Phys. Rev. C [**81**]{}, 028201 (2010).
C. S. An and D. O. Riska, Eur. Phys. J. A [**37**]{}, 263 (2008).
S. G. Yuan, C. S. An, and J. He, arXiv:0908.2333.
J. M. Gaillard and G. Sauvage, Ann. Rev. Nucl. Part. Sci. [**34**]{}, 351 (1984).
K. Dannbom, L. Y. Glozman, C. Helminen, and D. O. Riska, Nucl. Phys. A [**616**]{}, 555 (1997).
J. J. de Swart, Rev. Mod. Phys. [**35**]{}, 916 (1963) \[Erratum-ibid. [**37**]{}, 326 (1965)\].
L. Theussl, R. F. Wagenbrunn, B. Desplanques, and W. Plessas, Eur. Phys. J. A [**12**]{}, 91 (2001).
L. Y. Glozman, Z. Papp, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. C [**57**]{}, 3406 (1998).
L. Y. Glozman, W. Plessas, K. Varga, and R. F. Wagenbrunn, Phys. Rev. D [**58**]{}, 094030 (1998).
K. Glantschnig, R. Kainhofer, W. Plessas, B. Sengl, and R. F. Wagenbrunn, Eur. Phys. J. A [**23**]{}, 507 (2005).
T. Melde, L. Canton, W. Plessas, and R. F. Wagenbrunn, Eur. Phys. J. A [**25**]{}, 97 (2005).
R. F. Wagenbrunn, S. Boffi, W. Klink, W. Plessas, and M. Radici, Phys. Lett. B [**511**]{}, 33 (2001).
K. Berger, R. F. Wagenbrunn, and W. Plessas, Phys. Rev. D [**70**]{}, 094027 (2004).
T. Melde, K. Berger, L. Canton, W. Plessas, and R. F. Wagenbrunn, Phys. Rev. D [**76**]{}, 074020 (2007).
W. Rarita and J. Schwinger, Phys. Rev. [**60**]{}, 61 (1941).
T. Melde, L. Canton, and W. Plessas, Phys. Rev. Lett. [**102**]{}, 132002 (2009).
F. J. Jiang and B. C. Tiburzi, Phys. Rev. D [**78**]{}, 017504 (2008).
D. V. Bugg, Eur. Phys. J. C [**33**]{}, 505 (2004).
T. Melde, W. Plessas, and B. Sengl, Phys. Rev. D [**77**]{}, 114002 (2008).
Online version of the review on the [*Quark Model*]{} by the Particle Data Group available from http://pdg.lbl.gov/2009/reviews/rpp2009-rev-quark-model.pdf (print version to appear in 2010).
K. S. Choi, S. I. Nam, A. Hosaka, and H. C. Kim, J. Phys. G [**36**]{}, 015008 (2009).
|
---
bibliography:
- 'auto\_generated.bib'
title: Search for the standard model Higgs boson produced in association with a or a boson and decaying to bottom quarks
---
=1
$Revision: 216721 $ $HeadURL: svn+ssh://svn.cern.ch/reps/tdr2/papers/HIG-13-012/trunk/HIG-13-012.tex $ $Id: HIG-13-012.tex 216721 2013-11-16 03:46:34Z alverson $
Introduction {#sec:hbb_Introduction}
============
At the Large Hadron Collider (LHC), the ATLAS and CMS collaborations have reported the discovery of a new boson [@Chatrchyan:2012ufa; @Aad:2012tfa] with a mass, [$m_\PH$]{}, near 125and properties compatible with those of the standard model (SM) Higgs boson [@Englert:1964et; @Higgs:1964ia; @Higgs:1964pj; @Guralnik:1964eu; @Higgs:1966ev; @Kibble:1967sv]. To date, significant signals have been observed in channels where the boson decays into $\gamma \gamma$, $\cPZ\cPZ$, or $\PW\PW$. The interaction of this boson with the massive and vector bosons indicates that it plays a role in electroweak symmetry breaking. The interaction with the fermions and whether the Higgs field serves as the source of mass generation in the fermion sector, through a Yukawa interaction, remains to be firmly established.
At ${\ensuremath{m_\PH}\xspace}\approx 125$the standard model Higgs boson decays predominantly into a bottom quark-antiquark pair () with a branching fraction of ${\approx}58\%$ [@Dittmaier:2011ti]. The observation and study of the [$\PH\to\bbbar$]{} decay, which involves the direct coupling of the Higgs boson to down-type quarks, is therefore essential in determining the nature of the newly discovered boson. The measurement of the [$\PH\to\bbbar$]{} decay will be the first direct test of whether the observed boson interacts as expected with the quark sector, as the coupling to the top quark has only been tested through loop effects.
In their combined search for the SM Higgs boson [@PhysRevD.88.052014], the CDF and D0 collaborations at the Tevatron collider have reported evidence for an excess of events in the 115–140mass range, consistent with the mass of the Higgs boson observed at the LHC. In that search, the sensitivity below a mass of 130is dominated by the channels in which the Higgs boson is produced in association with a weak vector boson and decaying to [@PhysRevLett.109.071804]. The combined local significance of this excess is reported to be 3.0 standard deviations at ${\ensuremath{m_\PH}\xspace}=125$, while the expected local significance is 1.9 standard deviations. At the LHC, a search for $\PH\to \bbbar$ by the ATLAS experiment using data samples corresponding to an integrated luminosity of $4.7$at $\sqrt{s}=7\TeV$ resulted in exclusion limits on Higgs boson production, at the 95% confidence level (CL), of 2.5 to 5.5 times the standard model cross section in the 110–130mass range [@Aad:2012gxa].
This article reports on a search at the Compact Muon Solenoid (CMS) experiment for the standard model Higgs boson in the $\Pp\Pp\to {\ensuremath{{{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH}\xspace}$ production mode, where [$\cmsSymbolFace{V}$]{}is either a or a boson and $\PH\to \bbbar$. The previous Higgs boson search in this production mode at CMS used data samples corresponding to integrated luminosities of up to $5.1$at $\sqrt{s}=7\TeV$ and up to $5.3$at $\sqrt{s}=8\TeV$ [@Chatrchyan:2013lba]. The results presented here combine the analysis of the 7data sample in Ref. [@Chatrchyan:2013lba] with an updated analysis of the full 8data sample corresponding to a luminosity of up to $18.9$.
The following six channels are considered in the search: $\PW(\mu\nu)\PH$, $\PW(\Pe\nu)\PH$, $\PW(\tau\nu)\PH$, $\cPZ(\mu\mu)\PH$, $\cPZ(\Pe\Pe)\PH$, and $\cPZ(\nu\nu)\PH$, all with the Higgs boson decaying to . Throughout this article the term “lepton” refers only to charged leptons and the symbol $\ell$ is used to refer to both muons and electrons, but not to taus. For the $\PW(\tau\nu)\PH$ final state, only the 8data are included and only taus with 1-prong hadronic decays are explicitly considered; the $\tau$ notation throughout this article refers exclusively to such decays. The leptonic decays of taus in [$\PW\PH$]{} processes are implicitly accounted for in the $\PW(\mu\nu)\PH$ and $\PW(\Pe\nu)\PH$ channels. Backgrounds arise from production of and bosons in association with jets (from gluons and from light- or heavy-flavor quarks), singly and pair-produced top quarks (), dibosons, and quantum chromodynamics (QCD) multijet processes.
Simulated samples of signal and background events are used to provide guidance in the optimization of the analysis. Control regions in data are selected to adjust the event yields from simulation for the main background processes in order to estimate their contribution in the signal region. These regions also test the accuracy of the modeling of kinematic distributions in the simulated samples.
Upper limits at the 95% CL on the $\Pp\Pp\to {\ensuremath{{{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH}\xspace}$ production cross section times the $\PH\to
\bbbar$ branching fraction are obtained for Higgs boson masses in the 110–135range. These limits are extracted by fitting the shape of the output distribution of a boosted-decision-tree (BDT) discriminant [@Roe:2004na; @Hocker:2007ht]. The results of the fitting procedure allow to evaluate the presence of a Higgs boson signal over the expectation from the background components. The significance of any excess of events, and the corresponding event yield, is compared with the expectation from a SM Higgs boson signal.
Detector and simulated samples {#sec:hbb_Simulations}
==============================
A detailed description of the CMS detector can be found elsewhere [@Chatrchyan:2008aa]. The momenta of charged particles are measured using a silicon pixel and strip tracker that covers the pseudorapidity range $\abs{\eta} < 2.5$ and is immersed in a 3.8 axial magnetic field. The pseudorapidity is defined as $\eta = -\ln[\tan(\theta/2)]$, where $\theta$ is the polar angle of the trajectory of a particle with respect to the direction of the counterclockwise proton beam. Surrounding the tracker are a crystal electromagnetic calorimeter (ECAL) and a brass/scintillator hadron calorimeter (HCAL), both used to measure particle energy deposits and consisting of a barrel assembly and two endcaps. The ECAL and HCAL extend to a pseudorapidity range of $\abs{\eta} < 3.0$. A steel/quartz-fiber Cherenkov forward detector extends the calorimetric coverage to $\abs{\eta} < 5.0$. The outermost component of the CMS detector is the muon system, consisting of gas-ionization detectors placed in the steel return yoke of the magnet to measure the momenta of muons traversing through the detector. The two-level CMS trigger system selects events of interest for permanent storage. The first trigger level, composed of custom hardware processors, uses information from the calorimeters and muon detectors to select events in less than 3.2. The high-level trigger software algorithms, executed on a farm of commercial processors, further reduce the event rate using information from all detector subsystems. The variable $\Delta R = \sqrt {(\Delta\eta)^2 +(\Delta\phi)^2}$ is used to measure the separation between reconstructed objects in the detector, where $\phi$ is the angle (in radians) of the trajectory of the object in the plane transverse to the direction of the proton beams.
Simulated samples of signal and background events are produced using various Monte Carlo (MC) event generators, with the CMS detector response modeled with [@GEANT4]. The Higgs boson signal samples are produced using the [@POWHEG] event generator. The [5.1]{} [@Alwall:2011uj] generator is used for the diboson, +jets, +jets, and samples. The single-top-quark samples, including the -, $t$-, and $s$-channel processes, are produced with and the QCD multijet samples with [6.4]{} [@Sjostrand:2006za]. The production cross sections for the diboson and samples are rescaled to the cross sections from the next-to-leading-order (NLO) generator [@Campbell:2010ff], while the cross sections for the +jets and +jets samples are rescaled to next-to-next-to-leading order (NNLO) cross sections calculated using the <span style="font-variant:small-caps;">fewz</span> program [@Gavin:2010az; @Li:2012wna; @Gavin:2012sy]. The default set of parton distribution functions (PDF) used to produce the NLO samples is the NLO [MSTW2008]{} set [@Martin:2009iq], while the leading-order (LO) CTEQ6L1 set [@Pumplin:2002vw] is used for the other samples. For parton showering and hadronization the and samples are interfaced with [++]{} [@Bahr:2008pv] and , respectively. The parameters for the underlying event description are set to the Z2 tune for the 7samples and to the Z2$^*$ tune for the 8samples [@Chatrchyan:2011id]. The [@Jadach1991275] library is used to simulate tau decays.
During the 2011 data-taking period the LHC instantaneous luminosity reached up to $3.5\times 10^{33}\percms$ and the average number of $\Pp\Pp$ interactions per bunch crossing was approximately nine. During the 2012 period the LHC instantaneous luminosity reached $7.7\times 10^{33}\percms$ and the average number of $\Pp\Pp$ interactions per bunch crossing was approximately twenty-one. Additional simulated $\Pp\Pp$ interactions overlapping with the event of interest in the same bunch crossing, denoted as pileup events, are therefore added in the simulated samples to reproduce the pileup distribution measured in data.
Triggers {#sec:hbb_Triggers}
========
Several triggers are used to collect events consistent with the signal hypothesis in the six channels under consideration.
For the [$\PW(\Pgm\cPgn)\PH$]{} and [$\PW(\Pe\cPgn)\PH$]{} channels, the trigger paths consist of several single-lepton triggers with tight lepton identification. Leptons are also required to be isolated from other tracks and calorimeter energy deposits to maintain an acceptable trigger rate. For the [$\PW(\Pgm\cPgn)\PH$]{} channel and for the 2011 data, the trigger thresholds for the muon transverse momentum, , are in the range of 17 to 24. The higher thresholds are used for the periods of higher instantaneous luminosity. For the 2012 data the muon trigger threshold for the single-isolated-muon trigger is set at 24. For both the 2011 and 2012 data, a single-muon trigger with a 40 threshold, but without any isolation requirements, is also used for this channel. The combined single-muon trigger efficiency is ${\approx}$90% for [$\PW(\Pgm\cPgn)\PH$]{} events that pass all offline requirements that are described in Section \[sec:hbb\_Event\_Selection\].
For the [$\PW(\Pe\cPgn)\PH$]{} channel and for the 2011 data, the electron threshold ranges from 17 to 30. To maintain acceptable trigger rates during the periods of high instantaneous luminosity, the lower-threshold triggers also require two central ($\abs{\eta}<2.6$) jets, with a threshold in the 25–30range, and a minimum requirement on the value of an online estimate of the missing transverse energy, , in the 15–25range. is defined online as the magnitude of the vector sum of the transverse momenta of all reconstructed objects identified by a particle-flow algorithm [@CMS-PAS-PFT-09-001; @CMS-PAS-PFT-10-002]. This algorithm combines the information from all CMS subdetectors to identify and reconstruct online individual particles emerging from the proton-proton collisions: charged hadrons, neutral hadrons, photons, muons, and electrons. These particles are then used to reconstruct jets, and hadronic $\tau$-lepton decays, and also to quantify the isolation of leptons and photons. For the 2012 data, the electron trigger uses a 27threshold on the and no other requirements on jets or are made. The combined efficiency for these triggers for [$\PW(\Pe\cPgn)\PH$]{} events to pass the offline selection criteria is $>$95%.
For the [$\PW(\Pgt\cPgn)\PH$]{} channel trigger, a 1-prong hadronically-decaying tau is required. The of the charged track candidate coming from the tau decay is required to be above 20and the of the tau (measured from all reconstructed charged and neutral decay products) above 35. Additionally, the tau is required to be isolated inside an annulus with inner radius $\Delta R = 0.2$ and outer radius $\Delta R = 0.4$, where no reconstructed charged candidates with $\pt > 1.5$must be found. A further requirement of a minimum of 70is placed on the . The efficiency of this trigger for [$\PW(\Pgt\cPgn)\PH$]{} events that pass the offline selection criteria is $>$90%.
The [$\cPZ(\Pgm\Pgm)\PH$]{} channel uses the same single-muon triggers as the [$\PW(\Pgm\cPgn)\PH$]{}channel. For the [$\cPZ(\Pe\Pe)\PH$]{} channel, dielectron triggers with lower thresholds, of 17 and 8, and tight isolation requirements are used. These triggers are nearly 100% efficient for all [$\cPZ(\ell\ell)\PH$]{} signal events that pass the final offline selection criteria.
For the [$\cPZ(\cPgn\cPgn)\PH$]{} channel, combinations of several triggers are used, all requiring to be above a given threshold. Extra requirements are added to keep the trigger rates manageable as the instantaneous luminosity increased and to reduce the thresholds in order to improve signal acceptance. A trigger with $\MET >150$is used for the complete data set in both 2011 and 2012. During 2011 additional triggers that require the presence of two central jets with $\pt >20\GeV$ and thresholds of 80 or 100, depending on the instantaneous luminosity, were used. During 2012 an additional trigger that required two central jets with $\pt >30\GeV$ and $\MET>80$was used. This last trigger was discontinued when the instantaneous luminosity exceeded $3\times 10^{33}\percms$ and was replaced by a trigger that required $\MET>100$, at least two central jets with vectorial sum $\pt>100\GeV$ and individual above 60 and 25, and no jet with $\pt>40\GeV$ closer than 0.5 in azimuthal angle to the direction. In order to increase signal acceptance at lower values of , triggers that require jets to be identified as coming from quarks are used. For these triggers, two central jets with above 20 or 30, depending on the luminosity conditions, are required. It is also required that at least one central jet with above 20be tagged by the online combined secondary vertex (CSV) -tagging algorithm described in Section \[sec:hbb\_Event\_Reconstruction\]. This online b-tagging requirement has an efficiency that is equivalent to that of the tight offline requirement, $\mathrm{CSV} >0.898$, on the value of the output of the CSV discriminant. The is required to be greater than 80for these triggers. For [$\cPZ(\cPgn\cPgn)\PH$]{} events with $\MET >130$, the combined trigger efficiency for [$\cPZ(\cPgn\cPgn)\PH$]{} signal events is near 100% with respect to the offline event reconstruction and selection, described in the next sections. For events with between 100 and 130the efficiency is 88%.
Event reconstruction {#sec:hbb_Event_Reconstruction}
====================
The characterization of [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}events, in the channels studied here, requires the reconstruction of the following objects, all originating from a common interaction vertex: electrons, muons, taus, neutrinos, and jets (including those originating from quarks). The charged leptons and neutrinos (reconstructed as ) originate from the vector boson decays. The -quark jets originate from the Higgs boson decays.
The reconstructed interaction vertex with the largest value of $\sum_i {\pt}_i^2$, where ${\pt}_i$ is the transverse momentum of the $i$th track associated with the vertex, is selected as the primary event vertex. This vertex is used as the reference vertex for all relevant objects in the event, which are reconstructed with the particle-flow algorithm. The pileup interactions affect jet momentum reconstruction, missing transverse energy reconstruction, lepton isolation, and -tagging efficiencies. To mitigate these effects, all charged hadrons that do not originate from the primary interaction are identified by a particle-flow-based algorithm and removed from consideration in the event. In addition, the average neutral energy density from pileup interactions is evaluated from particle-flow objects and subtracted from the reconstructed jets in the event and from the summed energy in the isolation cones used for leptons, described below [@Cacciari:subtraction]. These pileup-mitigation procedures are applied on an event-by-event basis.
Jets are reconstructed from particle-flow objects using the anti-clustering algorithm [@antikt], with a distance parameter of 0.5, as implemented in the <span style="font-variant:small-caps;">fastjet</span> package [@Cacciari:fastjet1; @Cacciari:fastjet2]. Each jet is required to lie within $\abs{\eta} < 2.5$, to have at least two tracks associated with it, and to have electromagnetic and hadronic energy fractions of at least 1%. The last requirement removes jets originating from instrumental effects. Jet energy corrections are applied as a function of pseudorapidity and transverse momentum of the jet [@Chatrchyan:2011ds]. The missing transverse energy vector is calculated offline as the negative of the vectorial sum of transverse momenta of all particle-flow objects identified in the event, and the magnitude of this vector is referred to as in the rest of this article.
Muons are reconstructed using two algorithms [@Chatrchyan:2012xi]: one in which tracks in the silicon tracker are matched to signals in the muon detectors, and another in which a global track fit is performed, seeded by signals in the muon systems. The muon candidates used in the analysis are required to be successfully reconstructed by both algorithms. Further identification criteria are imposed on the muon candidates to reduce the fraction of tracks misidentified as muons. These include the number of measurements in the tracker and in the muon systems, the fit quality of the global muon track and its consistency with the primary vertex. Muon candidates are considered in the $\abs{\eta} < 2.4$ range.
Electron reconstruction requires the matching of an energy cluster in the ECAL with a track in the silicon tracker [@CMS-PAS-EGM-10-004]. Identification criteria based on the ECAL shower shape, matching between the track and the ECAL cluster, and consistency with the primary vertex are imposed. Electron identification relies on a multivariate technique that combines observables sensitive to the amount of bremsstrahlung along the electron trajectory, the geometrical and momentum matching between the electron trajectory and associated clusters, as well as shower-shape observables. Additional requirements are imposed to remove electrons produced by photon conversions. In this analysis, electrons are considered in the pseudorapidity range $\abs{\eta} < 2.5$, excluding the $1.44 < \abs{\eta}< 1.57$ transition region between the ECAL barrel and endcap, where electron reconstruction is suboptimal.
Charged leptons from the and boson decays are expected to be isolated from other activity in the event. For each lepton candidate, a cone is constructed around the track direction at the event vertex. The scalar sum of the transverse momentum of each reconstructed particle compatible with the primary vertex and contained within the cone is calculated, excluding the contribution from the lepton candidate itself. If this sum exceeds approximately 10% of the candidate , the lepton is rejected; the exact requirement depends on the lepton $\eta$, , and flavor. Including the isolation requirement, the total efficiency to reconstruct muons is in the 87–91% range, depending on and $\eta$. The corresponding efficiency for electrons is in the 81–98% range.
The hadronically-decaying taus are reconstructed using the hadron plus strips (HPS) algorithm [@Chatrchyan:2012zz] which uses charged hadrons and neutral electromagnetic objects (photons) to reconstruct tau decays. Reconstructed taus are required to be in the range $\abs{\eta} < 2.1$. In the first step of reconstruction, charged hadrons are reconstructed using the particle-flow algorithm. Since neutral pions are often produced in hadronic tau decays, the HPS algorithm is optimized to reconstruct neutral pions in the ECAL as objects called “strips”. The strip reconstruction starts by centering one strip on the most energetic electromagnetic particle and then looking for other particles in a window of 0.05 in $\eta$ and 0.20 in $\phi$. Strips satisfying a minimum transverse momentum of $\pt(\text{strip})>1$are combined with the charged hadrons to reconstruct the hadronic tau candidate. In the final step of reconstruction, all charged hadrons and strips are required to be contained within a narrow cone size of $\Delta R$ = 2.8/$\pt(\tau)$, where $\pt(\tau)$ is measured from the reconstructed hadronic tau candidate and is expressed in . Further identification criteria are imposed on the tau candidate to reduce the fraction of electron and muons misidentified as taus. These include the tau candidate passing an anti-electron discriminator and an anti-muon discriminator. The isolation requirement for taus is that the sum of transverse momenta of particle-flow charged hadron and photon candidates, with $\pt > 0.5$and within a cone of $\Delta R<0.5$, be less than 2. The tau reconstruction efficiency is approximately 50% while the misidentification rate from jets is about 1%.
Jets that originate from the hadronization of quarks are referred to as “ jets”. The CSV -tagging algorithm [@Chatrchyan:2012jua] is used to identify such jets. The algorithm combines the information about track impact parameters and secondary vertices within jets in a likelihood discriminant to provide separation between jets and jets originating from light quarks, gluons, or charm quarks. The output of this CSV discriminant has values between zero and one; a jet with a CSV value above a certain threshold is referred to as being “ tagged”. The efficiency to tag jets and the rate of misidentification of non- jets depend on the threshold chosen, and are typically parameterized as a function of the and $\eta$ of the jets. These performance measurements are obtained directly from data in samples that can be enriched in jets, such as $\ttbar$ and multijet events (where, for example, requiring the presence of a muon in the jets enhances the heavy-flavor content of the events). Several thresholds for the CSV output discriminant are used in this analysis. Depending on the threshold used, the efficiencies to tag jets originating from quarks, quarks, and light quarks or gluons are in the 50–75%, 5–25%, and 0.15–3.0% ranges, respectively.
Events from data and from the simulated samples are required to satisfy the same trigger and event reconstruction requirements. Corrections that account for the differences in the performance of these algorithms between data and simulations are computed from data and used in the analysis.
Event selection {#sec:hbb_Event_Selection}
===============
The background processes to [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production with [$\PH\to\bbbar$]{} are the production of vector bosons in association with one or more jets ([[$\cmsSymbolFace{V}$]{}]{}+jets), production, single-top-quark production, diboson production ([[$\cmsSymbolFace{V}$]{}]{}[[$\cmsSymbolFace{V}$]{}]{}), and QCD multijet production. Except for dibosons, these processes have production cross sections that are several orders of magnitude larger than Higgs boson production. The production cross section for the ${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\cPZ$ process, where $\cPZ\to\bbbar$, is only a few times larger than the [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production cross section, and given the nearly identical final state this process provides a benchmark against which the Higgs boson search strategy can be tested.
The event selection is based on the reconstruction of the vector bosons in their leptonic decay modes and of the Higgs boson decay into two -tagged jets. Background events are substantially reduced by requiring a significant boost of the of the vector boson, [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{}, or of the Higgs boson [@PhysRevLett.100.242001]. In this kinematic region the [$\cmsSymbolFace{V}$]{}and $\PH$ bosons recoil away from each other with a large azimuthal opening angle, [$\Delta\phi({\ensuremath{\cmsSymbolFace{V}}\xspace},\PH)$]{}, between them. For each channel, different [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{} boost regions are selected. Because of different signal and background content, each [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{} region has different sensitivity and the analysis is performed separately in each region. The results from all regions are then combined for each channel. The [low-,]{} , and high-boost regions for the [$\PW(\Pgm\cPgn)\PH$]{} and [$\PW(\Pe\cPgn)\PH$]{} channels are defined by $100<{\ensuremath{\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})}\xspace}<130$, $130<{\ensuremath{\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})}\xspace}<180$, and ${\ensuremath{\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})}\xspace}>180$. For the [$\PW(\Pgt\cPgn)\PH$]{} a single ${\ensuremath{\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})}\xspace}>120$region is considered. For the [$\cPZ(\ell\ell)\PH$]{} channels, the low- and high-boost regions are defined by $50<{\ensuremath{\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})}\xspace}<100$and ${\ensuremath{\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})}\xspace}>100$. For the [$\cPZ(\cPgn\cPgn)\PH$]{} channel is used to define the low-, intermediate-, and high-boost [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{} regions as $100<\MET<130$, $130<\MET<170$, and $\MET>170$, respectively. In the rest of the article the term “boost region” is used to refer to these [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{} regions.
Candidate [$\PW\to\ell\cPgn$]{} decays are identified by requiring the presence of a single-isolated lepton and additional missing transverse energy. Muons are required to have $\pt>20$; the corresponding thresholds for electrons and taus are 30 and 40, respectively. For the [$\PW(\ell\cPgn)\PH$]{} and [$\PW(\Pgt\cPgn)\PH$]{} channels, is required to be $>$45 and $>$80, respectively, to reduce contamination from QCD multijet processes. To further reduce this contamination, it is also required for the [$\PW(\ell\cPgn)\PH$]{}channels that the azimuthal angle between the direction and the lepton be ${<}\pi/2$, and that the lepton isolation for the low-boost region be tighter.
Candidate [$\cPZ\to\ell\ell$]{} decays are reconstructed by combining isolated, oppositely-charged pairs of electrons or muons and requiring the dilepton invariant mass to satisfy $75<m_{\ell\ell}<105$. The for each lepton is required to be $>$20.
The identification of [$\cPZ\to\cPgn\cPagn$]{} decays requires the in the event to be within the boost regions described above. The QCD multijet background is reduced to negligible levels in this channel when requiring that the does not originate from mismeasured jets. To that end three event requirements are made. First, for the high-boost region, a ${\ensuremath{\Delta\phi(\MET, \text{jet})}}>0.5$ radians requirement is applied on the azimuthal angle between the direction and the closest jet with $\abs{\eta}<2.5$ and $\pt>20$for the 7analysis or $\pt>25$for the 8analysis (where more pileup interactions are present). For the low- and intermediate-boost regions the requirement is tightened to ${\ensuremath{\Delta\phi(\MET, \text{jet})}}>0.7$ radians. The second requirement is that the azimuthal angle between the missing transverse energy direction as calculated from charged tracks only (with $\pt>0.5\GeV$ and $\abs{\eta}<2.5$) and the direction, [$\Delta\phi(\MET,\MET\text{(tracks)})$]{}, should be smaller than 0.5 radians. The third requirement is made for the low-boost region where the significance (defined as the ratio between the and the square root of the total transverse energy in the calorimeter, measured in ) should be larger than 3. To reduce background events from and [$\PW\cPZ$]{} production in the [$\PW(\ell\cPgn)\PH$]{}, [$\PW(\Pgt\cPgn)\PH$]{}, and [$\cPZ(\cPgn\cPgn)\PH$]{} channels, events with an additional number of isolated leptons, ${\ensuremath{N_{\mathrm{a}\ell}}}>0$, with $\pt>20\GeV$ are rejected.
The reconstruction of the [$\PH\to\bbbar$]{} decay proceeds by selecting the pair of jets in the event, each with $\abs{\eta}<2.5$ and above a minimum threshold, for which the value of the magnitude of the vectorial sum of their transverse momenta, [${\pt}(\mathrm{jj})$]{}, is the highest. These jets are then also required to be tagged by the CSV algorithm, with the value of the CSV discriminator above a minimum threshold. The background from [[$\cmsSymbolFace{V}$]{}]{}+jets and diboson production is reduced significantly when the -tagging requirements are applied and processes where the two jets originate from genuine quarks dominate the final selected data sample.
After all event selection criteria described in this section are applied, the dijet invariant-mass resolution of the two jets from the Higgs decay is approximately 10%, depending on the of the reconstructed Higgs boson, with a few percent shift on the value of the mass peak. The Higgs boson mass resolution is further improved by applying multivariate regression techniques similar to those used at the CDF experiment [@1107.3026]. An additional correction, beyond the standard CMS jet energy corrections, is computed for individual jets in an attempt to recalibrate to the true -quark energy. For this purpose, a specialized BDT is trained on simulated [$\PH\to\bbbar$]{} signal events with inputs that include detailed jet structure information which differs in jets from quarks from that of jets from light-flavor quarks or gluons. These inputs include variables related to several properties of the secondary vertex (when reconstructed), information about tracks, jet constituents, and other variables related to the energy reconstruction of the jet. Because of semileptonic -hadron decays, jets from quarks contain, on average, more leptons and a larger fraction of missing energy than jets from light quarks or gluons. Therefore, in the cases where a low-lepton is found in the jet or in its vicinity, the following variables are also included in the BDT regression: the of the lepton, the $\Delta R$ distance between the lepton and the jet directions, and the transverse momentum of the lepton relative to the jet direction. For the [$\cPZ(\ell\ell)\PH$]{} channels the in the event and the azimuthal angle between the and each jet are also considered in the regression. The output of the BDT regression is the corrected jet energy. The average improvement on the mass resolution, measured on simulated signal samples, when the corrected jet energies are used is $\approx$15%, resulting in an increase in the analysis sensitivity of 10–20%, depending on the specific channel. This improvement is shown in Fig. \[fig:regression\_VV\_VH\] for simulated samples of [$\cPZ(\ell\ell)\PH(\cPqb\cPqb)$]{} events where the improvement in resolution is $\approx$25%. The validation of the regression technique in data is done with samples of $\cPZ\to\ell\ell$ events with two -tagged jets and in $\ttbar$-enriched samples in the lepton+jets final state. In the $\cPZ\to\ell\ell$ case, when the jets are corrected by the regression procedure, the balance distribution, between the $\cPZ$ boson, reconstructed from the leptons, and the -tagged dijet system is improved to be better centered at zero and narrower than when the regression correction is not applied. In the $\ttbar$-enriched case, the reconstructed top-quark mass distribution is closer to the nominal top-quark mass and also narrower than when the correction is not applied. In both cases the distributions for data and the simulated samples are in very good agreement after the regression correction is applied.
![Dijet invariant mass distribution for simulated samples of [$\cPZ(\ell\ell)\PH(\cPqb\cPqb)$]{} events ($m_{H} = 125\GeV$), before (red) and after (blue) the energy correction from the regression procedure is applied. A Bukin function [@Verkerke:2003ir] is fit to the distribution and the fitted width of the core of the distribution is displayed on the figure. []{data-label="fig:regression_VV_VH"}](Hmass_Regression_Simulation_Oct08.pdf){width="48.00000%"}
The signal region is defined by events that satisfy the vector boson and Higgs boson reconstruction criteria described above together with the requirements listed in Table \[tab:BDTsel\]. In the final stage of the analysis, to better separate signal from background under different Higgs boson mass hypotheses, an event BDT discriminant is trained separately at each mass value using simulated samples for signal and all background processes. The training of this BDT is performed with all events in the signal region. The set of event input variables used, listed in Table \[tab:BDTvars\], is chosen by iterative optimization from a larger number of potentially discriminating variables. Among the most discriminant variables for all channels are the dijet invariant mass distribution ([$m(\mathrm{jj})$]{}), the number of additional jets ([$N_{\mathrm{aj}}$]{}), the value of CSV for the Higgs boson daughter with the second largest CSV value (CSV$_{\text{min}}$), and the distance between Higgs boson daughters ([$\Delta R(\mathrm{jj})$]{}). It has been suggested that variables related to techniques that study in more detail the substructure of jets could help improve the sensitivity of the [$\PH\to\bbbar$]{} searches [@PhysRevLett.100.242001]. In this analysis, several combinations of such variables were considered as additional inputs to the BDT discriminant. However they did not yield significant gains in sensitivity and are not included in the final training used.
A fit is performed to the shape of the output distribution of the event BDT discriminant to search for events resulting from Higgs boson production. Before testing all events through this final discriminant, events are classified based on where they fall in the output distributions of several other background-specific BDT discriminants that are trained to discern signal from individual background processes. This technique, similar to the one used by the CDF collaboration [@Aaltonen:2012id], divides the samples into four distinct subsets that are enriched in , [[$\cmsSymbolFace{V}$]{}]{}+jets, dibosons, and [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}. The increase in the analysis sensitivity from using this technique in the [$\cPZ(\cPgn\cPgn)\PH$]{} and [$\PW(\ell\cPgn)\PH$]{} channels is 5–10%. For the [$\cPZ(\ell\ell)\PH$]{} channel the improvement is not as large and therefore the technique is not used for that case. The technique is also not used in the [$\PW(\Pgt\cPgn)\PH$]{} channel because of the limited size of the simulated event samples available for training multiple BDT discriminants. The first background-specific BDT discriminant is trained to separate from [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}, the second one is trained to separate [[$\cmsSymbolFace{V}$]{}]{}+jets from [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}, and the third one separates diboson events from [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}. The output distributions of the background-specific BDTs are used to separate events in four subsets: those that fail a requirement on the BDT are classified as -like events, those that pass the BDT requirement but fail a requirement on the [[$\cmsSymbolFace{V}$]{}]{}+jets BDT are classified as [[$\cmsSymbolFace{V}$]{}]{}+jets-like events, those that pass the [[$\cmsSymbolFace{V}$]{}]{}+jets BDT requirement but fail the requirement on the diboson BDT are classified as diboson-like events and, finally, those that pass all BDT requirements are considered [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}-enriched events. The events in each subset are then run through the final event BDT discriminant and the resulting distribution, now composed of four distinct subsets of events, is used as input to the fitting procedure.
As a validation of the multivariate approach to this analysis, these BDT discriminants are also trained to find diboson signals ($\cPZ\cPZ$ and , with $\cPZ\to \bbbar$) rather than the [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}signal. The event selection used in this case is identical to that used for the [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}search.
\[tab:BDTsel\]
\[tab:BDTvars\]
As a cross-check to the BDT-based analysis, a simpler analysis is done by performing a fit to the shape of the dijet invariant mass distribution of the two jets associated with the reconstructed Higgs boson, [$m(\mathrm{jj})$]{}. The event selection for this analysis is more restrictive than the one used in the BDT analysis and is optimized for sensitivity in this single variable. Table \[tab:MjjSel\] lists the event selection of the [$m(\mathrm{jj})$]{} analysis. Since the diboson background also exhibits a peak in the [$m(\mathrm{jj})$]{} distribution from bosons that decay into b quark pairs, the distribution is also used to measure the consistency of the diboson rate with the expectation from the standard model. A consistent rate measurement would support the validity of the estimate of the background processes in the Higgs boson search.
\[tab:MjjSel\]
Background control regions {#sec:hbb_Background_Control_Regions}
==========================
Appropriate control regions are identified in data and used to validate the simulation modeling of the distributions used as input to the BDT discriminants, and to obtain scale factors used to adjust the simulation event yield estimates for the most important background processes: production of and bosons in association with jets and production. For the and backgrounds the control regions are defined such that they are enriched in either heavy-flavor (HF) or light-flavor (LF) jets. Furthermore, these processes are split according to how many of the two jets selected in the Higgs boson reconstruction originate from quarks, and separate scale factors are obtained for each case. The notation used is: [${\ensuremath{\cmsSymbolFace{V}}\xspace}+\cPqu\cPqd\cPqs\cPqc\Pg$]{} for the case where none of the jets originate from a quark, [${\ensuremath{\cmsSymbolFace{V}}\xspace}+\cPqb$]{} for the case where only one of the jets is from a quark, and [${\ensuremath{\cmsSymbolFace{V}}\xspace}+\bbbar$]{} for the case where both jets originate from quarks.
To obtain the scale factors by which the simulated event yields are adjusted, a set of binned likelihood fits is simultaneously performed to CSV distributions of jets for events in the control regions. These fits are done separately for each channel. Several other distributions are also fit to verify consistency. These scale factors account not only for cross section discrepancies, but also for potential residual differences in physics object selection. Therefore, separate scale factors are used for each background process in the different channels. The uncertainties in the scale factor determination include two components: the statistical uncertainty due to the finite size of the samples and the systematic uncertainty. The latter is obtained by subtracting, in quadrature, the statistical component from the full uncertainty which includes the effect of various sources of systematic uncertainty such as -tagging, jet energy scale, and jet energy resolution.
Tables \[tab:WlnControl\]–\[tab:ZnnControl\] list the selection criteria used to define the control regions for the [$\PW(\ell\cPgn)\PH$]{}, [$\cPZ(\ell\ell)\PH$]{}, and [$\cPZ(\cPgn\cPgn)\PH$]{} channels, respectively. Because of the limited size of the simulated event samples the scale factors obtained for the [$\PW(\ell\cPgn)\PH$]{} channels are applied to the [$\PW(\Pgt\cPgn)\PH$]{} channel. Table \[tab:SFs2012\] summarizes the fit results for all channels for the 8data. The scale factors are found to be close to unity for all processes except for [${\ensuremath{\cmsSymbolFace{V}}\xspace}+\cPqb$]{} for which the scale factors are consistently found to be close to two. In this case, most of the excess occurs in the region of low CSV$_{\text{min}}$ values in which events with two displaced vertices are found relatively close to each other, within a distance $\Delta{\mathrm R}<0.5$ defined by the directions of their displacement trajectories with respect to the primary vertex. This discrepancy is interpreted as arising mainly from mismodeling in the generator parton shower of the process of gluon-splitting to -quark pairs. In this process the dominant contribution typically contains a low- quark that can end up not being reconstructed as a jet above the threshold used in the analysis, or that is merged with the jet from the more energetic quark. These discrepancies are consistent with similar observations in other studies of the production of vector bosons in association with heavy-flavor quarks by the ATLAS and CMS experiments [@Aad:2013vka; @Chatrchyan:2013zja; @Chatrchyan:2012vr].
Figures \[fig:control\_regions\_ex\] and \[fig:control\_regions\_BDT\] show examples of distributions for variables in the simulated samples and in data for different control regions and for different channels. The scale factors described above have been applied to the corresponding simulated samples.
\[tab:WlnControl\]
[cccc]{} Variable & +LF & & +HF\
$\pt(\mathrm{j}_1)$ & $>$30 & $>$30 & $>$30\
$\pt(\mathrm{j}_2)$ & $>$30 & $>$30 & $>$30\
[${\pt}(\mathrm{jj})$]{}& $>$100 & $>$100 & $>$100\
${\ensuremath{m(\mathrm{jj})}}$ & $<$250 & $<$250 &$<$250, $\notin$\[90-150\]\
CSV$_{\text{max}}$ & $\in$\[0.244–0.898\] & $>$0.898 & $>$0.898\
[$N_{\mathrm{aj}}$]{}& $<$2 & $>$1 & $=$0\
[$N_{\mathrm{a}\ell}$]{}& $=$0 & $=$0 & $=$0\
& $>$45 & $>$45 & $>$45\
significance & ${>}2.0 (\mu)\, {>}3.0(\Pe)$& – & –\
\[tab:ZllControl\]
[ccc]{} Variable & +jets &\
$m_{\ell\ell}$ & \[75–105\] & $\notin$\[75–105\]\
$\pt(\mathrm{j}_1)$ & $>$20 & $>$20\
$\pt(\mathrm{j}_2)$ & $>$20 & $>$20\
[$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{}& $>$50 & \[50–100\]\
${\ensuremath{m(\mathrm{jj})}}$ & $<$250, $\notin$\[80–150\] & $<$250, $\notin$\[80–150\]\
CSV$_{\text{max}}$ & $>$0.244 & $>$0.244\
CSV$_{\text{min}}$ & $>$0.244 & $>$0.244\
\[tab:ZnnControl\]
\[tab:SFs2012\]
[cccccc]{} Process & [$\PW(\ell\cPgn)\PH$]{}& [$\cPZ(\ell\ell)\PH$]{}& [$\cPZ(\cPgn\cPgn)\PH$]{}\
Low [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{}\
[$\PW+\cPqu\cPqd\cPqs\cPqc\Pg$]{} & $1.03 \pm 0.01 \pm 0.05$ & – & $0.83 \pm 0.02 \pm 0.04$\
[$\PW+\cPqb$]{} & $2.22 \pm 0.25 \pm 0.20$ & – & $2.30 \pm 0.21 \pm 0.11$\
[$\PW+\bbbar$]{} & $1.58 \pm 0.26 \pm 0.24$ & – & $0.85 \pm 0.24 \pm 0.14$\
[$\cPZ+\cPqu\cPqd\cPqs\cPqc\Pg$]{} & – & $1.11 \pm 0.04 \pm 0.06$ & $1.24 \pm 0.03 \pm 0.09$\
[$\cPZ+\cPqb$]{} & – & $1.59 \pm 0.07 \pm 0.08$ & $2.06 \pm 0.06 \pm 0.09$\
[$\cPZ+\bbbar$]{} & – & $0.98 \pm 0.10 \pm 0.08$ & $1.25 \pm 0.05 \pm 0.11$\
& $1.03 \pm 0.01 \pm 0.04$ & $1.10 \pm 0.05 \pm 0.06$ & $1.01 \pm 0.02 \pm 0.04$\
Intermediate [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{}\
[$\PW+\cPqu\cPqd\cPqs\cPqc\Pg$]{} & $1.02 \pm 0.01 \pm 0.07$ & – & $0.93 \pm 0.02 \pm 0.04$\
[$\PW+\cPqb$]{} & $2.90 \pm 0.26 \pm 0.20$ & – & $2.08 \pm 0.20 \pm 0.12$\
[$\PW+\bbbar$]{} & $1.30 \pm 0.23 \pm 0.14$ & – & $0.75 \pm 0.26 \pm 0.11$\
[$\cPZ+\cPqu\cPqd\cPqs\cPqc\Pg$]{} & – & – & $1.19 \pm 0.03 \pm 0.07$\
[$\cPZ+\cPqb$]{} & – & – & $2.30 \pm 0.07 \pm 0.08$\
[$\cPZ+\bbbar$]{} & – & – & $1.11 \pm 0.06 \pm 0.12$\
& $1.02 \pm 0.01 \pm 0.15$ & – & $0.99 \pm 0.02 \pm 0.03$\
High [$\pt({\ensuremath{\cmsSymbolFace{V}}\xspace})$]{}\
[$\PW+\cPqu\cPqd\cPqs\cPqc\Pg$]{} & $1.04 \pm 0.01 \pm 0.07$ & – & $0.93 \pm 0.02 \pm 0.03$\
[$\PW+\cPqb$]{} & $2.46 \pm 0.33 \pm 0.22$ & – & $2.12 \pm 0.22 \pm 0.10$\
[$\PW+\bbbar$]{} & $0.77 \pm 0.25 \pm 0.08$ & – & $0.71 \pm 0.25 \pm 0.15$\
[$\cPZ+\cPqu\cPqd\cPqs\cPqc\Pg$]{} & – & $1.11 \pm 0.04 \pm 0.06$ & $1.17 \pm 0.02 \pm 0.08$\
[$\cPZ+\cPqb$]{} & – & $1.59 \pm 0.07 \pm 0.08$ & $2.13 \pm 0.05 \pm 0.07$\
[$\cPZ+\bbbar$]{} & – & $0.98 \pm 0.10 \pm 0.08$ & $1.12 \pm 0.04 \pm 0.10$\
& $1.00 \pm 0.01 \pm 0.11$ & $1.10 \pm 0.05 \pm 0.06$ & $0.99 \pm 0.02 \pm 0.03$\
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
{width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"}
Uncertainties {#sec:hbb_Uncertainties}
=============
The systematic uncertainties that affect the results presented in this article are listed in Table \[tab:syst\] and are described in more detail below.
The uncertainty in the CMS luminosity measurement is estimated to be 2.2% for the 2011 data [@CMS-PAS-SMP-12-008] and 2.6% for the 2012 data [@CMS-PAS-LUM-12-001]. Muon and electron trigger, reconstruction, and identification efficiencies are determined in data from samples of leptonic -boson decays. The uncertainty on the event yields resulting from the trigger efficiency estimate is 2% per lepton and the uncertainty on the identification efficiency is also 2% per lepton. The parameters describing the [$\cPZ(\cPgn\cPgn)\PH$]{} trigger efficiency turn-on curve have been varied within their statistical uncertainties and also estimated for different assumptions on the methods used to derive the efficiency. This results in an event yield uncertainty of about 3%.
The jet energy scale is varied within its uncertainty as a function of jet and $\eta$. The efficiency of the analysis selection is recomputed to assess the variation in event yields. Depending on the process, a 2–3% yield variation is found. The effect of the uncertainty on the jet energy resolution is evaluated by smearing the jet energies according to the measured uncertainty. Depending on the process, a 3–6% variation in event yields is obtained. The uncertainties in the jet energy scale and resolution also have an effect on the shape of the BDT output distribution. The impact of the jet energy scale uncertainty is determined by recomputing the BDT output distribution after shifting the energy scale up and down by its uncertainty. Similarly, the impact of the jet energy resolution is determined by recomputing the BDT output distribution after increasing or decreasing the jet energy resolution. An uncertainty of 3% is assigned to the event yields of all processes in the [$\PW(\ell\cPgn)\PH$]{} and [$\cPZ(\cPgn\cPgn)\PH$]{} channels due to the uncertainty related to the missing transverse energy estimate.
Data/MC -tagging scale factors are measured in heavy-flavor enhanced samples of jets that contain muons and are applied consistently to jets in signal and background events. The measured uncertainties for the -tagging scale factors are: 3% per -quark tag, 6% per charm-quark tag, and 15% per mistagged jet (originating from gluons and light , , or quarks) [@Chatrchyan:2012jua]. These translate into yield uncertainties in the 3–15% range, depending on the channel and the specific process. The shape of the BDT output distribution is also affected by the shape of the CSV distributions and an uncertainty is assigned according to a range of variations of the CSV distributions.
The total [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}signal cross section has been calculated to NNLO accuracy, and the total theoretical uncertainty is $\approx$4% [@Dittmaier:2012vm], including the effect of scale variations and PDF uncertainties [@Alekhin:2011sk; @Botje:2011sn; @Lai:2010vv; @Martin:2009iq; @Ball:2011mu]. This analysis is performed in the boosted regime, and differences in the spectrum of the [$\cmsSymbolFace{V}$]{}and $\PH$ bosons between data and MC introduce systematic effects in the signal acceptance and efficiency estimates. Two calculations are available that evaluate the NLO electroweak (EW) [@HAWK1; @HAWK2; @HAWK3] and NNLO QCD [@Ferrera:2011bk] corrections to [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production in the boosted regime. Both the electroweak and QCD corrections are applied to the signal samples. The estimated uncertainties of the NLO electroweak corrections are 2% for both the [$\cPZ\PH$]{}and [$\PW\PH$]{}production processes. The estimate for the NNLO QCD correction results in an uncertainty of $5\%$ for both the [$\cPZ\PH$]{}and [$\PW\PH$]{}production processes.
The uncertainty in the background event yields estimated from data is approximately 10%. For [[$\cmsSymbolFace{V}$]{}]{}+jets, the difference between the shape of the BDT output distribution for events generated with the and the [++]{} Monte Carlo generators is considered as a shape systematic uncertainty. For the differences in the shape of the BDT output distribution between the one obtained from the nominal samples and those obtained from the and [@Frixione:2002ik] generators are considered as shape systematic uncertainties.
An uncertainty of 15% is assigned to the event yields obtained from simulation for single-top-quark production. For the diboson backgrounds, a 15% cross section uncertainty is assumed. These uncertainties are consistent with the CMS measurements of these processes [@Chatrchyan:2012ep; @Chatrchyan:2013oev]. The limited number of MC simulated events is also taken into account as a source of uncertainty.
The combined effect of the systematic uncertainties results in an increase of about 15% on the expected upper limit on the Higgs boson production cross section and in a reduction of 15% on the expected significance of an observation when the Higgs boson is present in the data at the predicted standard model rate.
\[tab:syst\]
Results {#sec:hbb_Results}
=======
Results are obtained from combined signal and background binned likelihood fits to the shape of the output distribution of the BDT discriminants. These are trained separately for each channel and for each Higgs boson mass hypothesis in the 110–135range. In the simultaneous fit to all channels, in all boost regions, the BDT shape and normalization for signal and for each background component are allowed to vary within the systematic and statistical uncertainties described in Section \[sec:hbb\_Uncertainties\]. These uncertainties are treated as independent nuisance parameters in the fit. All nuisance parameters, including the scale factors described in Section \[sec:hbb\_Background\_Control\_Regions\], are adjusted by the fit.
In total 14 BDT distributions are considered. Figure \[fig:mBDT\_Znn\_example\] shows an example of these distributions after the fit for the high-boost region of the [$\cPZ(\cPgn\cPgn)\PH$]{} channel, for the ${\ensuremath{m_\PH}\xspace}=125$mass hypothesis. The four partitions in the left panel correspond to the subsets enriched in , [[$\cmsSymbolFace{V}$]{}]{}+jets, diboson, and [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production, as described in Section \[sec:hbb\_Event\_Selection\]. The right panel shows the right-most, [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}-enriched, partition in more detail. For completeness, all 14 BDT distributions used in the fit are shown in Figs. \[fig:BDTWln8TeV\_mu\]–\[fig:BDTZnn8TeV\] in Appendix \[sec:hbb\_Appendix\]. Table \[table:3bin\_yields\] lists, for partial combinations of channels, the total number of events in the four highest bins of their corresponding BDT for the expected backgrounds, for the 125SM Higgs boson signal, and for data. An excess compatible with the presence of the SM Higgs boson is observed. Figure \[fig:BDT\_S\_over\_B\_all\] combines the BDT outputs of all channels where the events are gathered in bins of similar expected signal-to-background ratio, as given by the value of the output of their corresponding BDT discriminant (trained with a Higgs boson mass hypothesis of 125). The observed excess of events in the bins with the largest signal-to-background ratio is consistent with what is expected from the production of the standard model Higgs boson.
![Post-fit BDT output distributions for [$\cPZ(\cPgn\cPgn)\PH$]{} in the high-boost region for 8data (points with error bars), all backgrounds, and signal, after all selection criteria have been applied. The event BDT discriminant values for events in the four different subsets are rescaled and offset to assemble a single BDT output variable. This leads to the four equally-sized partitions shown in the panel. The partitions correspond, starting from the left, to the event subsets enriched in , [[$\cmsSymbolFace{V}$]{}]{}+jets, diboson, and [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production. The panel shows the right-most, [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}-enriched, partition in more detail. The bottom inset in each figure shows the ratio of the number of events observed in data to that of the Monte Carlo prediction for signal and backgrounds.[]{data-label="fig:mBDT_Znn_example"}](BDT_Znn_HighPt_ZnunuHighPt_8TeV_PostFit_s "fig:"){width="49.00000%"} ![Post-fit BDT output distributions for [$\cPZ(\cPgn\cPgn)\PH$]{} in the high-boost region for 8data (points with error bars), all backgrounds, and signal, after all selection criteria have been applied. The event BDT discriminant values for events in the four different subsets are rescaled and offset to assemble a single BDT output variable. This leads to the four equally-sized partitions shown in the panel. The partitions correspond, starting from the left, to the event subsets enriched in , [[$\cmsSymbolFace{V}$]{}]{}+jets, diboson, and [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production. The panel shows the right-most, [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}-enriched, partition in more detail. The bottom inset in each figure shows the ratio of the number of events observed in data to that of the Monte Carlo prediction for signal and backgrounds.[]{data-label="fig:mBDT_Znn_example"}](BDT_Znn_HighPt_Last_ZnunuHighPt_8TeV_PostFit_s "fig:"){width="49.00000%"}
\[table:3bin\_yields\]
![Combination of all channels into a single distribution. Events are sorted in bins of similar expected signal-to-background ratio, as given by the value of the output of their corresponding BDT discriminant (trained with a Higgs boson mass hypothesis of 125). The two bottom insets show the ratio of the data to the background-only prediction (above) and to the predicted sum of background and SM Higgs boson signal with a mass of 125(below).[]{data-label="fig:BDT_S_over_B_all"}](PRD_BDT_Summary_Oct10){width="55.00000%"}
The results of all channels, for all boost regions and for the 7 and 8data, are combined to obtain 95% confidence level (CL) upper limits on the product of the [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production cross section times the [$\PH\to\bbbar$]{}branching fraction, with respect to the expectations for a standard model Higgs boson ($\sigma/\sigma_{\mathrm{SM}}$). At each mass point the observed limit, the median expected limit, and the 1 and 2 standard deviation bands are calculated using the modified frequentist method CL$_\mathrm{s}$ [@Read:2002hq; @junkcls; @LHC-HCG]. Figure \[fig:Limits\] displays the results.
For a Higgs boson mass of 125the expected limit is 0.95 and the observed limit is 1.89. Given that the resolution for the reconstructed Higgs boson mass is $\approx$10%, these results are compatible with a Higgs mass of 125. This is demonstrated by the red dashed line in the left panel of Fig. \[fig:Limits\], which is the expected limit obtained from the sum of expected background and the signal of a SM Higgs boson with a mass of 125.
{width="48.00000%"} {width="48.00000%"}
For all channels an excess of events over the expected background contributions is indicated by the fits of the BDT output distributions. The probability (p-value) to observe data as discrepant as observed under the background-only hypothesis is shown in the right panel of Fig. \[fig:Limits\] as a function of the assumed Higgs boson mass. For ${\ensuremath{m_\PH}\xspace}=125$, the excess of observed events corresponds to a local significance of 2.1 standard deviations away from the background-only hypothesis. This is consistent with the 2.1 standard deviations expected when assuming the standard model prediction for Higgs boson production.
The relative sensitivity of the channels that are topologically distinct is demonstrated in Table \[tab:limits\_by\_mode\] for ${\ensuremath{m_\PH}\xspace}=125$. The table lists the expected and observed limits and local significance for the [$\PW(\ell\cPgn)\PH$]{} and [$\PW(\Pgt\cPgn)\PH$]{} channels combined, for the [$\cPZ(\ell\ell)\PH$]{}channels combined, and for the [$\cPZ(\cPgn\cPgn)\PH$]{} channel.
\[tab:limits\_by\_mode\]
[ccccc]{} ${\ensuremath{m_\PH}\xspace}=125$& $\sigma /\sigma_{\mathrm{SM}}$ (95% CL) & $\sigma/\sigma_{\mathrm{SM}}$ (95% CL) & Significance & Significance\
& median expected & observed & expected & observed\
$\PW(\ell\cPgn,\Pgt\cPgn)\PH$ & 1.6 & 2.3 & 1.3 & 1.4\
[$\cPZ(\ell\ell)\PH$]{} & 1.9 & 2.8 & 1.1 & 0.8\
[$\cPZ(\cPgn\cPgn)\PH$]{} & 1.6 & 2.6 & 1.3 & 1.3\
All channels & 0.95 & 1.89 & 2.1 & 2.1\
The best-fit values of the production cross section for a 125Higgs boson, relative to the standard model cross section (signal strength, $\mu$), are shown in the left panel of Fig. \[fig:mu-values\] for the [$\PW(\ell\cPgn)\PH$]{} and [$\PW(\Pgt\cPgn)\PH$]{} channels combined, for the [$\cPZ(\ell\ell)\PH$]{} channels combined, and for the [$\cPZ(\cPgn\cPgn)\PH$]{} channel. The observed signal strengths are consistent with each other, and the value for the signal strength for the combination of all channels is $1.0\pm 0.5$. In the right panel of Fig. \[fig:mu-values\] the correlation between the signal strengths for the separate [$\PW\PH$]{} and [$\cPZ\PH$]{} production processes is shown. The two production modes are consistent with the SM expectation, within uncertainties. This figure contains slightly different information than the one on the left panel as some final states contain signal events that originate from both [$\PW\PH$]{} and [$\cPZ\PH$]{} production processes. The [$\PW\PH$]{}process contributes approximately 20% of the Higgs boson signal event yields in the [$\cPZ(\cPgn\cPgn)\PH$]{} channel, resulting from events in which the lepton is outside the detector acceptance, and the [$\cPZ(\ell\ell)\PH$]{} process contributes less than 5% to the [$\PW(\ell\cPgn)\PH$]{} channel when one of the leptons is outside the detector acceptance. The dependency of the combined signal strength on the value assumed for the Higgs boson mass is shown in the left panel of Fig. \[fig:mu-values1\].
In the right panel of Fig. \[fig:mu-values1\] the best-fit values for the $\kappa_{\ensuremath{\cmsSymbolFace{V}}\xspace}$ and $\kappa_\cPqb$ parameters are shown. The parameter $\kappa_{\ensuremath{\cmsSymbolFace{V}}\xspace}$ quantifies the ratio of the measured Higgs boson couplings to vector bosons relative to the SM value. The parameter $\kappa_\cPqb$ quantifies the ratio of the measured Higgs boson partial width into relative to the SM value. They are defined as: ${\kappa_{\ensuremath{\cmsSymbolFace{V}}\xspace}}^2 =
\left.\sigma_{{\ensuremath{{{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH}\xspace}}\middle/\sigma^{\mathrm{SM}}_{{\ensuremath{{{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH}\xspace}}\right. $ and $ {\kappa_\cPqb}^2 = \left.\Gamma_{\bbbar}\middle/\Gamma^{\mathrm{SM}}_{\bbbar}\right. $, with the SM scaling of the total width [@Heinemeyer:2013tqa]. The measured couplings are consistent with the expectations from the standard model, within uncertainties.
{width="48.00000%"} {width="48.00000%"}
{width="48.00000%"} {width="48.00000%"}
Results for the dijet mass cross-check analysis
-----------------------------------------------
The left panel of Fig. \[fig:MJJ-combined\] shows a weighted dijet invariant mass distribution for the combination of all channels, in all boost regions, in the combined 7 and 8data, using the event selection for the [$m(\mathrm{jj})$]{}cross-check analysis described in Section \[sec:hbb\_Event\_Selection\]. For each channel, the relative event weight in each boost region is obtained from the ratio of the expected number of signal events to the sum of expected signal and background events in a window of [$m(\mathrm{jj})$]{} values between 105 and 150. The expected signal used corresponds to the production of the SM Higgs boson with a mass of 125. The weight for the highest-boost region is set to 1.0 and all other weights are adjusted proportionally. Figure \[fig:MJJ-combined\] also shows the same weighted dijet invariant mass distribution with all backgrounds, except diboson production, subtracted. The data are consistent with the presence of a diboson signal from $\cPZ\cPZ$ and [$\PW\cPZ$]{}channels, with $\cPZ\to \bbbar$), with a rate consistent with the standard model prediction from the generator, together with a small excess consistent with the production of the standard model Higgs boson with a mass of 125. For the [$m(\mathrm{jj})$]{} analysis, a fit to the dijet invariant mass distribution results in a measured Higgs boson signal strength, relative to that predicted by the standard model, of $\mu = 0.8\pm 0.7$, with a local significance of 1.1 standard deviations with respect to the background-only hypothesis. For a Higgs boson of mass 125, the expected and observed 95% CL upper limits on the production cross section, relative to the standard model prediction, are 1.4 and 2.0, respectively.
{width="48.00000%"} {width="48.00000%"}
Diboson signal extraction
-------------------------
As a validation of the multivariate technique, BDT discriminants are trained using the diboson sample as signal, and all other processes, including [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production (at the predicted standard model rate for a 125Higgs mass), as background. This is done for the 8dataset only. The observed excess of events for the combined $\PW\cPZ$ and $\cPZ\cPZ$ processes, with $\cPZ\to\bbbar$, differs by over 7 standard deviations from the event yield expectation from the background-only hypothesis. The corresponding signal strength, relative to the prediction from the diboson generator mentioned in Section \[sec:hbb\_Simulations\], and rescaled to the cross section from the NLO generator, is measured to be $\mu_{{\ensuremath{\cmsSymbolFace{V}}\xspace}{\ensuremath{\cmsSymbolFace{V}}\xspace}} = {1.19}_{-0.23}^{+0.28}$.
Summary {#sec:hbb_Conclusions}
=======
A search for the standard model Higgs boson when produced in association with an electroweak vector boson and decaying to is reported for the $\PW(\mu\nu)\PH$, $\PW(\Pe\nu)\PH$, $\PW(\tau\nu)\PH$, $\cPZ(\mu\mu)\PH$, $\cPZ(\Pe\Pe)\PH$, and $\cPZ(\nu\nu)\PH$ channels. The search is performed in data samples corresponding to integrated luminosities of up to 5.1at $\sqrt{s}=7\TeV$ and up to 18.9at $\sqrt{s}=8\TeV$, recorded by the CMS experiment at the LHC.
Upper limits, at the 95% confidence level, on the [${{\ensuremath{\cmsSymbolFace{V}}\xspace}}\PH$]{}production cross section times the $\PH \to \bbbar$ branching fraction, with respect to the expectations for a standard model Higgs boson, are derived for the Higgs boson in the mass range 110–135. For a Higgs boson mass of 125the expected limit is 0.95 and the observed limit is 1.89.
An excess of events is observed above the expected background with a local significance of 2.1 standard deviations. The expected significance when taking into account the production of the standard model Higgs boson is also 2.1 standard deviations. The sensitivity of this search, as represented by the expected significance, is the highest for a single experiment thus far. The signal strength corresponding to this excess, relative to that of the standard model Higgs boson, is $\mu = 1.0\pm 0.5$. The measurements presented in this article represent the first indication of the $\PH \to \bbbar$ decay at the LHC.
Acknowledgments {#acknowledgments .unnumbered}
===============
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: the Austrian Federal Ministry of Science and Research and the Austrian Science Fund; the Belgian Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport; the Research Promotion Foundation, Cyprus; the Ministry of Education and Research, Recurrent financing contract SF0690030s09 and European Regional Development Fund, Estonia; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucléaire et de Physique des Particules / CNRS, and Commissariat à l’Énergie Atomique et aux Énergies Alternatives / CEA, France; the Bundesministerium für Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Office for Research and Technology, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Republic of Korea; the Lithuanian Academy of Sciences; the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Business, Innovation and Employment, New Zealand; the Pakistan Atomic Energy Commission; the Ministry of Science and Higher Education and the National Science Centre, Poland; the Fundação para a Ciência e a Tecnologia, Portugal; JINR, Dubna; the Ministry of Education and Science of the Russian Federation, the Federal Agency of Atomic Energy of the Russian Federation, Russian Academy of Sciences, and the Russian Foundation for Basic Research; the Ministry of Education, Science and Technological Development of Serbia; the Secretaría de Estado de Investigación, Desarrollo e Innovación and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the National Science Council, Taipei; the Thailand Center of Excellence in Physics, the Institute for the Promotion of Teaching Science and Technology of Thailand, Special Task Force for Activating Research and the National Science and Technology Development Agency of Thailand; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.
Individuals have received support from the Marie-Curie programme and the European Research Council and EPLANET (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of Czech Republic; the Council of Science and Industrial Research, India; the Compagnia di San Paolo (Torino); the HOMING PLUS programme of Foundation for Polish Science, cofinanced by EU, Regional Development Fund; and the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF.
Post-fit BDT distributions {#sec:hbb_Appendix}
==========================
Figures \[fig:BDTWln8TeV\_mu\]–\[fig:BDTZnn8TeV\] show all the post-fit BDT distributions, for the [$m_\PH$]{}= 125training, for all channels and for all boost regions. In order to better display the different shapes of the signal and background BDT distributions, Fig. \[fig:BDT\_norm\] shows these distributions for the highest-boost region in each channel, normalized to unity. See Section \[sec:hbb\_Results\] for more details.
{width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"}
{width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"}
![Post-fit BDT output distributions for [$\PW(\Pgt\cPgn)\PH$]{} for 8data (points with error bars), all backgrounds, and signal, after all selection criteria have been applied. The bottom inset shows the ratio of the number of events observed in data to that of the Monte Carlo prediction for signal and backgrounds.[]{data-label="fig:BDTWln8TeV_t"}](BDT_Wtn_Wtn_PostFit_s){width="49.00000%"}
{width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"}
{width="49.00000%"} {width="49.00000%"} {width="49.00000%"} {width="49.00000%"}
{width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"}
The CMS Collaboration \[app:collab\]
====================================
=5000=500=5000
|
---
abstract: |
Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted ${\mathcal{G}}(S)$, is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and $ab=ba$. Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal of constant transformations on $X$. We prove that if $|X|\geq4$, then, with a few exceptions, the diameter of ${\mathcal{G}}(J)$ is $5$. On the other hand, we prove that for every positive integer $n$, there exists a semigroup $S$ such that the diameter of ${\mathcal{G}}(S)$ is $n$.
We also study the left paths in ${\mathcal{G}}(S)$, that is, paths $a_1-a_2-\cdots-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all $i\in \{1,\ldots ,m\}$. We prove that for every positive integer $n\geq2$, except $n=3$, there exists a semigroup whose shortest left path has length $n$. As a corollary, we use the previous results to solve a purely algebraic old problem posed by B.M. Schein.
*$2010$ Mathematics Subject Classification*. 05C25, 05C12, 20M20.
*Keywords*: Commuting graph, path, left path, diameter, transformation semigroup, ideal.
author:
- |
João Araújo[^1]\
[Universidade Aberta, R. Escola Politécnica, 147]{}\
[1269-001 Lisboa, Portugal]{}\
[&]{}\
[Centro de Álgebra, Universidade de Lisboa]{}\
[1649-003 Lisboa, Portugal, jaraujo@ptmat.fc.ul.pt]{}\
[(+351)21 790 47 00 Fax (+351) 21 795 42 88]{}
- |
Michael Kinyon\
[Department of Mathematics, University of Denver]{}\
[Denver, Colorado 80208, mkinyon@math.du.edu]{}
- |
Janusz Konieczny\
[Department of Mathematics, University of Mary Washington]{}\
[Fredericksburg, Virginia 22401, jkoniecz@umw.edu]{}
title: Minimal Paths in the Commuting Graphs of Semigroups
---
Introduction
============
The commuting graph of a finite non-abelian group $G$ is a simple graph whose vertices are all non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Commuting graphs of various groups have been studied in terms of their properties (such as connectivity or diameter), for example in [@BaBu03], [@Bu06], [@IrJa08], and [@Se01]. They have also been used as a tool to prove group theoretic results, for example in [@Be83], [@RaSe01], and [@RaSe02].
The concept of the commuting graph carries over to semigroups. Let $S$ be a finite non-commutative semigroup with center $Z(S)=\{a\in S:ab=ba\mbox{ for all $b\in S$}\}$. The *commuting graph* of $S$, denoted ${\mathcal{G}}(S)$, is the simple graph (that is, an undirected graph with no multiple edges or loops) whose vertices are the elements of $S-Z(S)$ and whose edges are the sets $\{a,b\}$ such that $a$ and $b$ are distinct vertices with $ab=ba$.
This paper initiates the study of commuting graphs of semigroups. Our main goal is to study the lengths of minimal paths. We shall consider two types of paths: ordinary paths and so called left paths.
We first investigate the semigroup $T(X)$ of full transformations on a finite set $X$, and determine the diameter of the commuting graph of every ideal of $T(X)$ (Section \[stx\]). We find that, with a few exceptions, the diameter of ${\mathcal{G}}(J)$, where $J$ is an ideal of $T(X)$, is $5$. This small diameter does not extend to semigroups in general. We prove that for every $n\geq2$, there is a finite semigroup $S$ whose commuting graph has diameter $n$ (Theorem \[tald\]). To prove the existence of such a semigroup, we use our work on the [*left paths*]{} in the commuting graph of a semigroup.
Let $S$ be a semigroup. A path $a_1-a_2-\cdots-a_m$ in ${\mathcal{G}}(S)$ is called a *left path* (or $l$-path) if $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for every $i\in\{1,\ldots,m\}$. If there is any $l$-path in ${\mathcal{G}}(S)$, we define the *knit degree* of $S$, denoted $\operatorname{kd}(S)$, to be the length of a shortest $l$-path in ${\mathcal{G}}(S)$.
For every $n\geq2$ with $n\ne3$, we construct a band (semigroup of idempotents) of knit degree $n$ (Section \[smlp\]). It is an open problem if there is a semigroup of knit degree $3$. The constructions presented in Section \[smlp\] also give a band $S$ whose commuting graph has diameter $n$ (for every $n\geq4$). As another application of our work on the left paths, we settle a conjecture on bands formulated by B.M. Schein in 1978 (Section \[ssch\]). Finally, we present some problems regarding the commuting graphs of semigroups (Section \[spro\]).
Commuting Graphs of Ideals of $T(X)$ {#stx}
====================================
Let $T(X)$ be the semigroup of full transformations on a finite set $X$, that is, the set of all functions from $X$ to $X$ with composition as the operation. We will write functions on the right and compose from left to right, that is, for $a,b\in T(X)$ and $x\in X$, we will write $xa$ (not $a(x)$) and $x(ab)=(xa)b$ (not $(ba)(x)=b(a(x))$). In this section, we determine the diameter of the commuting graph of every ideal of $T(X)$. Throughout this section, we assume that $X=\{1,\ldots,n\}$.
Let $\Gamma$ be a simple graph, that is, $\Gamma=(V,E)$, where $V$ is a finite non-empty set of vertices and $E\subseteq\{\{u,v\}:u,v\in V, u\ne v\}$ is a set of edges. We will write $u-v$ to mean that $\{u,v\}\in E$. Let $u,w\in V$. A *path* in $\Gamma$ from $u$ to $w$ is a sequence of pairwise distinct vertices $u=v_1,v_2,\ldots,v_m=w$ ($m\geq1$) such that $v_i-v_{i+1}$ for every $i\in\{1,\ldots,m-1\}$. If ${\lambda}$ is a path $v_1,v_2,\ldots,v_m$, we will write ${\lambda}=v_1-v_2-\cdots-v_m$ and say that ${\lambda}$ has *length* $m-1$. We say that a path ${\lambda}$ from $u$ to $w$ is a *minimal path* if there is no path from $u$ to $w$ that is shorter than ${\lambda}$.
We say that the *distance* between vertices $u$ and $w$ is $k$, and write $d(u,w)=k$, if a minimal path from $u$ to $w$ has length $k$. If there is no path from $u$ to $w$, we say that the distance between $u$ and $w$ is infinity, and write $d(u,w)=\infty$. The maximum distance $\max\{d(u,w):u,w\in V\}$ between vertices of $\Gamma$ is called the *diameter* of $\Gamma$. Note that the diameter of $\Gamma$ is finite if and only if $\Gamma$ is connected.
If $S$ is a finite non-commutative semigroup, then the commuting graph ${\mathcal{G}}(S)$ is a simple graph with $V=S-Z(S)$ and, for $a,b\in V$, $a-b$ if and only if $a\ne b$ and $ab=ba$.
For $a\in T(X)$, we denote by $\operatorname{im}(a)$ the image of $a$, by $\operatorname{ker}(a)=\{(x,y)\in X\times X:xa=ya\}$ the kernel of $a$, and by $\operatorname{rank}(a)=|\operatorname{im}(a)|$ the rank of $a$. It is well known (see [@ClPr64 Section 2.2]) that in $T(X)$ the only element of $Z(T(X))$ is the identity transformation on $X$, and that $T(X)$ has exactly $n$ ideals: $J_1,J_2,\ldots,J_n$, where, for $1\leq r\leq n$, $$J_r=\{a\in T(X):\operatorname{rank}(a)\leq r\}.$$ Each ideal $J_r$ is principal and any $a\in T(X)$ of rank $r$ generates $J_r$. The ideal $J_1$ consists of the transformations of rank $1$ (that is, constant transformations), and it is clear that ${\mathcal{G}}(J_1)$ is the graph with $n$ isolated vertices.
Let $S$ be a semigroup. We denote by ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(S)$ the subgraph of ${\mathcal{G}}(S)$ induced by the non-central idempotents of $S$. The graph ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(S)$ is said to be the *idempotent commuting graph* of $S$. We first determine the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$. This approach is justified by the following lemma.
\[lad1\] Let $2\leq r<n$ and let $a,b\in J_r$ be such that $ab\ne ba$. Suppose $a-a_1-a_2-\cdots-a_k-b$ ($k\geq1$) is a minimal path in ${\mathcal{G}}(J_r)$ from $a$ to $b$. Then there are idempotents $e_1,e_2,\ldots,e_k\in J_r$ such that $a-e_1-e_2-\cdots-e_k-b$ is a minimal path in ${\mathcal{G}}(J_r)$ from $a$ to $b$.
Since $J_r$ is finite, there is an integer $p\geq1$ such that $e_1=a_1^p$ is an idempotent in $J_r$. Note that $e_1\notin Z(J_r)$ since for any $x\in X-\operatorname{im}(e_1)$, $e_1$ does not commute with $c_x\in J_r$, where $c_x$ is the constant transformation with $\operatorname{im}(c_x)=\{x\}$. Since $a_1$ commutes with $a$ and $a_2$, the idempotent $e_1=a_1^p$ also commutes with $a$ and $a_2$, and so $a-e_1-a_2-\cdots-a_k-b$. Repeating the foregoing argument for $a_2,\ldots,a_k$, we obtain idempotents $e_2,\ldots,e_k$ in $J_r$ such that $a-e_1-e_2-\cdots-e_k-b$. Since the path $a-a_1-a_2-\cdots-a_k-b$ is minimal, it follows that $a,e_1,e_2,\ldots,e_k,b$ are pairwise distinct and the path $a-e_1-e_2-\cdots-e_k-b$ is minimal.
It follows from Lemma \[lad1\] that if $d$ is the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$, then the diameter of ${\mathcal{G}}(J_r)$ is at most $d+2$.
Idempotent Commuting Graphs {#sside}
---------------------------
In this subsection, we assume that $n\geq3$ and $2\leq r<n$. We will show that, with some exceptions, the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$ is $3$ (Theorem \[tdia\]).
Let $e\in T(X)$ be an idempotent. Then there is a unique partition $\{A_1,A_2,\ldots,A_k\}$ of $X$ and unique elements $x_1\in A_1, x_2\in A_2,\ldots, x_k\in A_k$ such that for every $i$, $A_ie=\{x_i\}$. The partition $\{A_1\ldots,A_k\}$ is induced by the kernel of $e$, and $\{x_1,\ldots,x_k\}$ is the image of $e$. We will use the following notation for $e$: $$\label{eide}
e=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_k,x_k{\rangle}.$$ Note that $(X,x{\rangle}$ is the constant idempotent with image $\{x\}$. The following result has been obtained in [@ArKo03] and [@Ko02] (see also [@ArKo04]).
\[lcen\] Let $e=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_k,x_k{\rangle}$ be an idempotent in $T(X)$ and let $b\in T(X)$. Then $b$ commutes with $e$ if and only if for every $i\in\{1,\ldots,k\}$, there is $j\in\{1,\ldots,k\}$ such that $x_ib=x_j$ and $A_ib\subseteq A_j$.
We will use Lemma \[lcen\] frequently, not always mentioning it explicitly. The following lemma is an immediate consequence of Lemma \[lcen\].
\[ljo1\] Let $e,f\in J_r$ be idempotents and suppose there is $x\in X$ such that $x\in\operatorname{im}(e)\cap\operatorname{im}(f)$. Then $e-(X,x{\rangle}-f$.
\[ljo2\] Let $e,f\in J_r$ be idempotents such that $\operatorname{im}(e)\cap\operatorname{im}(f)=\emptyset$. Suppose there is $(x,y)\in\operatorname{im}(e)\times\operatorname{im}(f)$ such that $(x,y)\in\operatorname{ker}(e)\cap\operatorname{ker}(f)$. Then there is an idempotent $g\in J_r$ such that $e-g-f$.
Let $e=(A_1,x_1{\rangle}\ldots(A_k,x_k{\rangle}$ and $f=(B_1,y_1{\rangle}\ldots(B_m,y_m{\rangle}$. We may assume that $x=x_1$ and $y=y_1$. Since $(x,y)\in\operatorname{ker}(e)\cap\operatorname{ker}(f)$, we have $y\in A_1$ and $x\in B_1$. Let $g=(\operatorname{im}(e),x{\rangle}(X-\operatorname{im}(e),y{\rangle}$. Then $g$ is in $J_r$ since $\operatorname{rank}(g)=2$ and $r\geq2$. By Lemma \[lcen\], we have $eg=ge$ (since $y\in A_1$) and $fg=gf$ (since $\operatorname{im}(f)\subseteq X-\operatorname{im}(e)$ and $x\in B_1$). Hence $e-g-f$.
\[lja1\] Let $e,f\in J_r$ be idempotents such that $\operatorname{im}(e)\cap\operatorname{im}(f)=\emptyset$. Then there are idempotents $g,h\in J_r$ such that $e-g-h-f$.
Let $e=(A_1,x_1{\rangle}\ldots(A_k,x_k{\rangle}$ and $f=(B_1,y_1{\rangle}\ldots(B_m,y_m{\rangle}$. Since $\{A_1,\ldots,A_k\}$ is a partition of $X$, there is $i$ such that $y_1\in A_i$. We may assume that $y_1\in A_1$. Let $g=(X-\{y_1\},x_1{\rangle}(\{y_1\},y_1{\rangle}$ and $h=(X,y_1{\rangle}$. Then $g$ and $h$ are in $J_r$ (since $r\geq2$). By Lemma \[lcen\], $eg=ge$, $gh=hg$, and $hf=fh$. Thus $e-g-h-f$.
\[lja2\] Let $m$ be a positive integer such that $2m\leq n$, $\sigma$ be an $m$-cycle on $\{1,\ldots,m\}$, and $$e=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_m,x_m{\rangle}\mbox{ and }f=(B_1,y_1{\rangle}(B_2,y_2{\rangle}\ldots(B_m,y_m{\rangle}$$ be idempotents in $T(X)$ such that $x_1,\ldots,x_m,y_1,\ldots,y_m$ are pairwise distinct, $y_i\in A_i$, and $x_{i\sigma}\in B_i$ ($1\leq i\leq m)$. Suppose that $g$ is an idempotent in $T(X)$ such that $e-g-f$. Then:
- $x_jg=x_j$ and $y_jg=y_j$ for every $j\in\{1,\ldots,m\}$.
- If $1\leq i,j\leq m$ are such that $A_i=\{x_i,y_i,z\}$, $B_j=\{y_j,x_{j\sigma},z\}$ and $A_i\cap B_j=\{z\}$, then $zg=z$.
Since $eg=ge$, $x_1g=x_i$ for some $i$. Then $x_ig=x_i$ (since $g$ is an idempotent). Thus, $e-g-f$ and Lemma \[lcen\] imply that $y_ig=y_i$. Since $x_i=x_{(i\sigma^{-1})\sigma}\in B_{i\sigma^{-1}}$ and $g$ commutes with $f$, we have $y_{i\sigma^{-1}}g=y_{i\sigma^{-1}}$. But now, since $y_{i\sigma^{-1}}\in A_{i\sigma^{-1}}$ and $g$ commutes with $e$, we have $x_{i\sigma^{-1}}g=x_{i\sigma^{-1}}$. Continuing this way, we obtain $x_{i\sigma^{-k}}g=x_{i\sigma^{-k}}$ and $y_{i\sigma^{-k}}g=y_{i\sigma^{-k}}$ for every $k\in\{1,\ldots,m-1\}$. Since $\sigma$ is an $m$-cycle, it follows that $x_jg=x_j$ and $y_jg=y_j$ for every $j\in\{1,\ldots,m\}$. We have proved (1).
Suppose $A_i=\{x_i,y_i,z\}$, $B_j=\{y_j,x_{j\sigma},z\}$, and $A_i\cap B_j=\{z\}$. Then $zg\in\{x_i,y_i,z\}$ (since $x_ig=x_i$ and $eg=ge$) and $zg\in\{y_j,x_{j\sigma},z\}$ (since $y_jg=y_j$ and $fg=gf$). Since $A_i\cap B_j=\{z\}$, we have $zg=z$, which proves (2).
\[lja3\] Let $n\geq4$. If $n\ne5$ or $r\ne4$, then for some idempotents $e,f\in J_r$, there is no idempotent $g\in J_r$ such that $e-g-f$.
Let $n\ne5$ or $r\ne4$. Suppose that $r<n-1$ or $n$ is even. Then there is an integer $m$ such that $m\leq r$ and $r<2m\leq n$. Let $e$ and $f$ be idempotents from Lemma \[lja2\]. Then $e,f\in J_r$ since $m\leq r$. But every idempotent $g\in T(X)$ such that $e-g-f$ fixes at least $2m$ elements, and so $g\notin J_r$ since $r<2m$.
Suppose that $r=n-1$ and $n=2m+1$ is odd. Then $n\geq7$ since we are working under the assumption that $n\ne5$ or $r\ne4$. We again consider idempotents $e$ and $f$ from Lemma \[lja2\], which belong to $J_r$ since $m<n-1=r$. Note that $X=\{x_1,\ldots,x_m,y_1,\ldots,y_m,z\}$. We may assume that $z\in A_m$ and $z\in B_1$. Since $n\geq7$, we have $m\geq3$. Thus, the intersection of $A_m=\{x_m,y_m,z\}$ and $B_1=\{y_1,x_2,z\}$ is $\{z\}$, and so $zg=z$ by Lemma \[lja2\]. Hence $g=\operatorname{id}_{X}\notin J_r$, which concludes the proof.
\[tdia\] Let $n\geq3$ and let $J_r$ be an ideal in $T(X)$ such that $2\leq r<n$. Then:
- If $n=3$ or $n=5$ and $r=4$, then the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$ is $2$.
- In all other cases, the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$ is $3$.
Suppose $n=3$ or $n=5$ and $r=4$. In these special cases, we obtained the desired result using GRAPE [@So06], which is a package for GAP [@Scel92].
Let $n\geq4$ and suppose that $n\ne5$ or $r\ne4$. By Lemmas \[ljo1\] and \[lja1\], the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$ is at most $3$. By Lemma \[lja3\], the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$ is at least $3$. Thus the diameter of ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$ is $3$, which concludes the proof of (2).
Commuting Graphs of Proper Ideals of $T(X)$ {#ssprop}
-------------------------------------------
In this subsection, we determine the diameter of every proper ideal of $T(X)$. The ideal $J_1$ consists of the constant transformations, so ${\mathcal{G}}(J_1)$ is the graph with $n$ isolated vertices. Thus $J_1$ is not connected and its diameter is $\infty$. Therefore, for the remainder of this subsection, we assume that $n\geq3$ and $2\leq r<n$.
It follows from Lemma \[lad1\] and Theorem \[tdia\] that the diameter of ${\mathcal{G}}(J_r)$ is at most $5$. We will prove that this diameter is in fact $5$ except when $n=3$ or $n\in\{5,6,7\}$ and $r=4$. It also follows from Lemma \[lad1\] that if $e$ and $f$ are idempotents in $J_r$, then the distance between $e$ and $f$ in ${\mathcal{G}}(J_r)$ is the same as the distance between $e$ and $f$ in ${\mathcal{G}_{{\mbox{\tiny $E$}}}}(J_r)$. So no ambiguity will arise when we talk about the distance between idempotents in $J_r$.
For $a\in T(X)$ and $x,y\in X$, we will write $x{\stackrel{a}{{\rightarrow}}}y$ when $xa=y$.
\[lad2\] Let $a,b\in T(X)$. Then $ab=ba$ if and only if for all $x,y\in X$, $x{\stackrel{a}{{\rightarrow}}}y$ implies $xb{\stackrel{a}{{\rightarrow}}}yb$.
Suppose $ab=ba$. Let $x,y\in X$ with $x{\stackrel{a}{{\rightarrow}}}y$, that is, $y=xa$. Then, since $ab=ba$, we have $yb=(xa)b=x(ab)=x(ba)=(xb)a$, and so $xb{\stackrel{a}{{\rightarrow}}}yb$.
Conversely, suppose $x{\stackrel{a}{{\rightarrow}}}y$ implies $xb{\stackrel{a}{{\rightarrow}}}yb$ for all $x,y\in X$. Let $x\in X$. Since $x{\stackrel{a}{{\rightarrow}}}xa$, we have $xb{\stackrel{a}{{\rightarrow}}}(xa)b$. But this means that $(xb)a=(xa)b$, which implies $ab=ba$.
Let $a\in T(X)$. Suppose $x_1,\ldots,x_m$ are pairwise distinct elements of $X$ such that $x_ia=x_{i+1}$ ($1\leq i<m$) and $x_ma=x_1$. We will then say that $a$ contains a cycle $(x_1\,x_2\ldots\,x_m)$.
\[lad0\] Let $a\in J_r$ be a transformation containing a unique cycle $(x_1\,x_2\ldots\,x_m)$. Let $e\in J_r$ be an idempotent such that $ae=ea$. Then $x_ie=x_i$ for every $i\in\{1,\ldots,m\}$.
Since $a$ contains $(x_1\,x_2\ldots\,x_m)$, we have $x_1{\stackrel{a}{{\rightarrow}}}x_2{\stackrel{a}{{\rightarrow}}}\cdots{\stackrel{a}{{\rightarrow}}}x_m{\stackrel{a}{{\rightarrow}}}x_1$. Thus, by Lemma \[lad2\], $$x_1e{\stackrel{a}{{\rightarrow}}}x_2e{\stackrel{a}{{\rightarrow}}}\cdots{\stackrel{a}{{\rightarrow}}}x_me{\stackrel{a}{{\rightarrow}}}x_1e.$$ Thus $(x_1e\,x_2e\ldots\,x_me)$ is a cycle in $a$, and is therefore equal to $(x_1\,x_2\ldots\,x_m)$. Hence, for every $i\in\{1,\ldots,m\}$, there exists $j\in\{1,\ldots,m\}$ such that $x_i=x_je$, and so $x_ie=(x_je)e=x_j(ee)=x_je=x_i$.
To construct transformations $a,b\in J_r$ such that the distance between $a$ and $b$ is $5$, it will be convenient to introduce the following notation.
\[ndi5\] [Let $x_1,\ldots,x_m,z_1,\ldots,z_p$ be pairwise distinct elements of $X$, and let $s$ be fixed such that $1\leq s<p$. We will denote by $$\label{e1ndi5}
a=(*\,z_s{\rangle}(z_p\,z_{p-1}\ldots\,z_1\,x_1{\rangle}(x_1\,x_2\ldots\,x_m)$$ the transformation $a\in T(X)$ such that $$\begin{aligned}
z_pa&=z_{p-1},\,z_{p-1}a=z_{p-2},\ldots,z_2a=z_1,\,z_1a=x_1,\notag\\
x_1a&=x_2,\,x_2a=x_3,\ldots,\,x_{m-1}a=x_m,\,x_ma=x_1,\notag\end{aligned}$$ and $ya=z_s$ for all other $y\in X$. Suppose $w\in X$ such that $w\notin\{x_1,\ldots,x_m,z_1,\ldots,z_p\}$ and $1\leq t<p$ with $t\ne s$. We will denote by $$\label{e2ndi5}
b=(*\,z_s{\rangle}(w\,z_t{\rangle}(z_p\,z_{p-1}\ldots\,z_1\,x_1{\rangle}(x_1\,x_2\ldots\,x_m)$$ the transformation $b\in T(X)$ that is defined as $a$ in (\[e1ndi5\]) except that $wb=z_t$. ]{}
\[lad3\] Let $a\in J_r$ be the transformation defined in [(\[e1ndi5\])]{} such that $m+p>r$. Let $e\in J_r$ be an idempotent such that $ae=ea$. Then:
- $x_ie=x_i$ for every $i\in\{1,\ldots,m\}$.
- $z_je=x_{m-j+1}$ for every $j\in\{1,\ldots,p\}$.
- $ye=x_{m-s}$ for every $y\in X-\{x_1,\ldots,x_m,z_1,\ldots,z_p\}$.
(We assume that for every integer $u$, $x_u=x_v$, where $v\in\{1,\ldots,m\}$ and $u\equiv v\pmod m$.)
Statement (1) follows from Lemma \[lad0\]. By the definition of $a$, we have $$z_p{\stackrel{a}{{\rightarrow}}}z_{p-1}{\stackrel{a}{{\rightarrow}}}\cdots{\stackrel{a}{{\rightarrow}}}z_1{\stackrel{a}{{\rightarrow}}}x_1.$$ Thus, by Lemma \[lad2\], $$z_pe{\stackrel{a}{{\rightarrow}}}z_{p-1}e{\stackrel{a}{{\rightarrow}}}\cdots{\stackrel{a}{{\rightarrow}}}z_1e{\stackrel{a}{{\rightarrow}}}x_1e=x_1.$$ Since $z_1e{\stackrel{a}{{\rightarrow}}}x_1$, either $z_1e=x_m$ or $z_1e\notin\{x_1,\ldots,x_m\}$. We claim that the latter is impossible. Indeed, suppose $z_1e\notin\{x_1,\ldots,x_m\}$. Then $z_je\notin\{x_1,\ldots,x_m\}$ for every $j\in\{1,\ldots,p\}$. Thus the set $\{x_1,\ldots,x_m,z_1e,\ldots,z_pe\}$ is a subset of $\operatorname{im}(e)$ with $m+p$ elements. But this implies that $e\notin J_r$ (since $m+p>r$), which is a contradiction. We proved the claim. Thus $z_1e=x_m$. Now, $z_2e{\stackrel{a}{{\rightarrow}}}z_1e=x_m$, which implies $z_2e=x_{m-1}$. Continuing this way, we obtain $z_3e=x_{m-2},\,z_4e=x_{m-3},\ldots$. (A special argument is required when $j=qm+1$ for some $q\geq1$. Suppose $q=1$, that is, $j=m+1$. Then $z_je{\stackrel{a}{{\rightarrow}}}z_{j-1}e=z_me=x_1$, and so either $z_je=x_m$ or $z_je=z_1$. But the latter is impossible since we would have $x_m=z_1e=z_j(ee)=z_je=z_1$, which is a contradiction. Hence, for $j=m+1$, we have $z_je=x_m$. Assuming, inductively, that $z_je=x_m$ for $j=qm+1$, we prove by a similar argument that $z_je=x_m$ for $j=(q+1)m+1$.) This concludes the proof of (2).
Let $y\in X-\{x_1,\ldots,x_m,z_1,\ldots,z_p\}$. Then $y{\stackrel{a}{{\rightarrow}}}z_s$, and so $ye{\stackrel{a}{{\rightarrow}}}z_se=x_{m-s+1}$. Suppose $s$ is not a multiple of $m$. Then $x_{m-s+1}\ne x_1$, and so $ye{\stackrel{a}{{\rightarrow}}}x_{m-s+1}$ implies $ye=x_{m-s}$. Suppose $s$ is a multiple of $m$. Then $ye{\stackrel{a}{{\rightarrow}}}x_{m-s+1}=x_1$, and so either $ye=x_m$ or $ye=z_1$. But the latter is impossible since we would have $x_m=z_1e=y(ee)=ye=z_1$, which is a contradiction. Hence, for $s$ that is a multiple of $m$, we have $ye=x_m$, which concludes the proof of (3).
The proof of the following lemma is almost identical to the proof of Lemma \[lad3\].
\[lad4\] Let $b\in J_r$ be the transformation defined in [(\[e2ndi5\])]{} such that $m+p>r$. Let $e\in J_r$ be an idempotent such that $be=eb$. Then:
- $x_ie=x_i$ for every $i\in\{1,\ldots,m\}$.
- $z_je=x_{m-j+1}$ for every $j\in\{1,\ldots,p\}$.
- $we=x_{m-t}$.
- $ye=x_{m-s}$ for every $y\in X-\{x_1,\ldots,x_m,z_1,\ldots,z_p,w\}$.
\[lad6\] Let $n\in\{5,6,7\}$ and $r=4$. Then there are $a,b\in J_4$ such that the distance between $a$ and $b$ in ${\mathcal{G}}(J_4)$ is at least $4$.
Let $a=(*\,4{\rangle}(3\,4\,1{\rangle}(1\,2)$ and $b=(*\,1{\rangle}(2\,1\,3{\rangle}(3\,4)$ (see Notation \[ndi5\]). Suppose $e$ and $f$ are idempotents in $J_4$ such that $a-e$ and $f-b$. Then, by Lemma \[lad3\], $e=(\{\ldots,3,1\},1{\rangle}(\{4,2\},2{\rangle}$ and $f=(\{\ldots,2,3\},3{\rangle}(\{1,4\},4{\rangle}$, where “$\ldots$” denotes “$5$” (if $n=5$), “$5,6$” (if $n=6$), and “$5,6,7$” (if $n=7$). Then $e$ and $f$ do not commute, and so $d(e,f)\geq2$. Thus $d(a,b)\geq4$ by Lemma \[lad1\].
\[lad7\] Let $n\in\{6,7\}$ and $r=4$. Let $a\in J_4$ be a transformation that is not an idempotent. Then there is an idempotent $e\in J_4$ commuting with $a$ such that $\operatorname{rank}(e)\ne3$ or $\operatorname{rank}(e)=3$ and $ye^{-1}=\{y\}$ for some $y\in\operatorname{im}(e)$.
If $a$ fixes some $x\in X$, then $a$ commutes with $e=(X,x{\rangle}$ of rank $1$. Suppose $a$ has no fixed points. Let $p$ be a positive integer such that $a^p$ is an idempotent. If $a$ contains a unique cycle $(x_1\,x_2)$, then $e=a^p$ has rank $2$. If $a$ contains a unique cycle $(x_1\,x_2\,x_3\,x_4)$ or two cycles $(x_1\,x_2)$ and $(y_1\,y_2)$ with $\{x_1,x_2\}\cap\{y_1,y_2\}=\emptyset$, then $e=a^p$ has rank $4$.
Suppose $a$ contains a unique cycle $(x_1\,x_2\,x_3)$. Define $e\in T(X)$ as follows. Set $x_ie=x_i$, $1\leq i\leq3$.
Suppose there are $y,z\in X-\{x_1,x_2,x_3\}$ such that $ya=z$ and $za=x_i$ for some $i$. We may assume that $za=x_1$. Define $ze=x_3$ and $ye=x_2$. Let $u$ and $w$ be the two remaining elements in $X$ (only $u$ remains when $n=6$). Since $\operatorname{rank}(a)\leq4$, we have $\{u,w\}a\subseteq\{z,x_1,x_2,x_3\}$. Suppose $ua=wa=z$. Define $ue=x_2$ and $we=x_2$. Then $e$ is an idempotent of rank $3$ such that $ae=ea$ and $x_1e^{-1}=\{x_1\}$. Suppose $ua$ or $wa$ is in $\{x_1,x_2,x_3\}$, say $ua\in\{x_1,x_2,x_3\}$. Define $ue=u$, and $we=x_{i-1}$ (if $wa=x_i$), where $x_{i-1}=x_3$ if $i=1$, or $we=x_2$ (if $wa=z$). Then $e$ is an idempotent of rank $4$ such that $ae=ea$.
Suppose that for every $y\in X-\{x_1,x_2,x_3\}$, $ya\in\{x_1,x_2,x_3\}$. Select $z\in X-\{x_1,x_2,x_3\}$ and define $ze=z$. For every $y\in X-\{z,x_1,x_2,x_3\}$, define $ye=x_{i-1}$ if $ya=x_i$. Then $e$ is an idempotent of rank $4$ such that $ae=ea$.
Since $a\in J_4$, we have exhausted all possibilities, and the result follows.
\[lad8\] Let $n\in\{6,7\}$ and $r=4$. Then for all $a,b\in J_4$, the distance between $a$ and $b$ in ${\mathcal{G}}(J_4)$ is at most $4$.
Let $a,b\in J_4$. If $a$ or $b$ is an idempotent, then $d(a,b)\leq4$ by Lemma \[lad1\] and Theorem \[tdia\]. Suppose $a$ and $b$ are not idempotents. By Lemma \[lad7\], there are idempotents $e,f\in J_4$ such that $ae=ea$, $bf=fb$, if $\operatorname{rank}(e)=3$, then $ye^{-1}=\{y\}$ for some $y\in\operatorname{im}(e)$, and if $\operatorname{rank}(f)=3$, then $yf^{-1}=\{y\}$ for some $y\in\operatorname{im}(f)$. We claim that there is an idempotent $g\in J_4$ such that $e-g-f$. If $\operatorname{im}(e)\cap\operatorname{im}(f)\ne\emptyset$, then such an idempotent $g$ exists by Lemma \[ljo1\]. Suppose $\operatorname{im}(e)\cap\operatorname{im}(f)=\emptyset$. Then, since $n\in\{6,7\}$, both $\operatorname{rank}(e)+\operatorname{rank}(f)\leq7$. We may assume that $\operatorname{rank}(e)\leq\operatorname{rank}(f)$. There are six possible cases. [**Case 1.**]{} $\operatorname{rank}(e)=1$. Then $e=(X,x{\rangle}$ for some $x\in X$. Let $y=xf$. Then $(x,y)\in\operatorname{im}(e)\times\operatorname{im}(f)$ and $(x,y)\in\operatorname{ker}(e)\cap\operatorname{ker}(f)$. Thus, by Lemma \[ljo2\], there is an idempotent $g\in J_4$ such that $e-g-f$. [**Case 2.**]{} $\operatorname{rank}(e)=2$ and $\operatorname{rank}(f)=2$. We may assume that $e=(A_1,1{\rangle}(A_2,2{\rangle}$ and $f=(B_1,3{\rangle}(B_2,4{\rangle}$. If $\{1,2\}\subseteq B_i$ or $\{3,4\}\subseteq A_i$ for some $i$, then we can find $(x,y)\in\operatorname{im}(e)\times\operatorname{im}(f)$ such that $(x,y)\in\operatorname{ker}(e)\cap\operatorname{ker}(f)$, and so a desired idempotent $g$ exists by Lemma \[ljo2\]. Otherwise, we may assume that $3\in A_1$ and $4\in A_2$. If $1\in B_1$ or $2\in B_2$, then Lemma \[ljo2\] can be applied again. So suppose $1\in B_2$ and $2\in B_1$. Now we have $$e=(\{\ldots,3,1\},1{\rangle}(\{\ldots,4,2\},2{\rangle}\mbox{ and }
f=(\{\ldots,2,3\},3{\rangle}(\{\ldots,1,4\},4{\rangle}.$$ We define $g\in T(X)$ as follows. Set $xg=x$ for every $x\in\{1,2,3,4\}$. Let $x\in\{5,6,7\}$ ($x\in\{5,6\}$ if $n=6$). If $x\in A_1\cap B_1$, define $xg=3$; if $x\in A_1\cap B_2$, define $xg=1$; if $x\in A_2\cap B_1$, define $xg=2$; finally, if $x\in A_2\cap B_2$, define $xg=4$. Then $g$ is an idempotent of rank $4$ and $e-g-f$. [**Case 3.**]{} $\operatorname{rank}(e)=2$ and $\operatorname{rank}(f)=3$. We may assume that $e=(A_1,1{\rangle}(A_2,2{\rangle}$ and $f=(B_1,3{\rangle}(B_2,4{\rangle}(B_3,5{\rangle}$. If $\{3,4,5\}\subseteq A_1$ or $\{3,4,5\}\subseteq A_2$, then Lemma \[ljo2\] applies. Otherwise, we may assume that $3,4\in A_1$ and $5\in A_2$. If $1\in B_1\cup B_2$ or $2\in B_3$, then Lemma \[ljo2\] applies again. So suppose $1\in B_3$ and $2\in B_1\cup B_2$. We may assume that $2\in B_1$. Note that if $z\in\{6,7\}$, then $z$ cannot be in $B_2$ since $z\in B_2$ would imply that there is no $y\in\operatorname{im}(f)$ such that $yf^{-1}=\{y\}$. So now $$e=(\{\ldots,3,4,1\},1{\rangle}(\{\ldots,5,2\},2{\rangle}\mbox{ and }
f=(\{\ldots,2,3\},3{\rangle}(\{4\},4{\rangle}(\{\ldots,1,5\},5{\rangle}.$$ We define $g\in T(X)$ as follows. Set $xg=x$ for every $x\in\{1,2,3,5\}$ and $4g=3$. Let $z\in\{6,7\}$. If $z\in A_1\cap B_1$, define $zg=3$; if $z\in A_1\cap B_3$, define $zg=1$; if $z\in A_2\cap B_1$, define $zg=2$; finally, if $z\in A_2\cap B_3$, define $zg=5$. Then $g$ is an idempotent of rank $4$ and $e-g-f$. [**Case 4.**]{} $\operatorname{rank}(e)=2$ and $\operatorname{rank}(f)=4$. We may assume that $e=(A_1,1{\rangle}(A_2,2{\rangle}$ and $f=(B_1,3{\rangle}(B_2,4{\rangle}(B_3,5{\rangle}(B_4,6{\rangle}$. If $\{3,4,5,6\}\subseteq A_1$ or $\{3,4,5,6\}\subseteq A_2$, then Lemma \[ljo2\] applies. Otherwise, we may assume that $3,4,5\in A_1$ and $6\in A_2$ or $3,4\in A_1$ and $5,6\in A_2$.
Suppose $3,4,5\in A_1$ and $6\in A_2$. If $1\in B_1\cup B_2\cup B_3$ or $2\in B_4$, then Lemma \[ljo2\] applies. So suppose $1\in B_4$, and we may assume that $2\in B_1$. Now we have $$\begin{aligned}
e&=(\{\ldots,3,4,5,1\},1{\rangle}(\{\ldots,6,2\},2{\rangle},\notag\\
f&=(\{\ldots,2,3\},3{\rangle}(\{\ldots,4\},4{\rangle}(\{\ldots,5\},5{\rangle}(\{\ldots,1,6\},6{\rangle}.\notag\end{aligned}$$ We define $g\in T(X)$ as follows. Set $xg=x$ for every $x\in\{1,2,3,6\}$, $4g=3$, and $5g=3$. Define $7g=3$ if $7\in A_1$ and $7\in B_1\cup B_2\cup B_3$; $7g=1$ if $7\in A_1$ and $7\in B_4$; $7g=2$ if $7\in A_2$ and $7\in B_1\cup B_2\cup B_3$; and $7g=6$ if $7\in A_2$ and $7\in B_4$. Then $g$ is an idempotent of rank $4$ and $e-g-f$. The argument in the case when $3,4\in A_1$ and $5,6\in A_2$ is similar. [**Case 5.**]{} $\operatorname{rank}(e)=3$ and $\operatorname{rank}(f)=3$. Since both $e$ and $f$ have an element in their range whose preimage is the singleton, we may assume that $e=(A_1,1{\rangle}(A_2,2{\rangle}(\{3\},3{\rangle}$ and $f=(B_1,4{\rangle}(B_2,5{\rangle}(\{6\},6{\rangle}$. If $\{1,2\}\subseteq B_i$ or $\{4,5\}\subseteq A_i$ for some $i$, then Lemma \[ljo2\] applies. Otherwise, we may assume that $4\in A_1$ and $5\in A_2$. If $1\in B_1$ or $2\in B_2$, then Lemma \[ljo2\] applies again. So suppose $1\in B_2$ and $2\in B_1$. So now $$e=(\{\ldots,4,1\},1{\rangle}(\{\ldots,5,2\},2{\rangle}(\{3\},3{\rangle}\mbox{ and }
f=(\{\ldots,2,4\},4{\rangle}(\{\ldots,1,5\},5{\rangle}(\{6\},6{\rangle}.$$ We define $g\in T(X)$ as follows. Set $xg=x$ for every $x\in\{1,2,4,5\}$, $3g=1$, and $6g=4$. Define $7g=4$ if $7\in A_1$ and $7\in B_1$; $7g=1$ if $7\in A_1$ and $7\in B_2$; $7g=2$ if $7\in A_2$ and $7\in B_1$; and $7g=5$ if $7\in A_2$ and $7\in B_2$. Then $g$ is an idempotent of rank $4$ and $e-g-f$. [**Case 6.**]{} $\operatorname{rank}(e)=3$ and $\operatorname{rank}(f)=4$. We may assume that $e=(A_1,1{\rangle}(A_2,2{\rangle}(\{3\},3{\rangle}$ and $f=(B_1,4{\rangle}(B_2,5{\rangle}(B_3,6{\rangle}(\{7\},7{\rangle}$. If $\{4,5,6\}\subseteq A_1$ or $\{4,5,6\}\subseteq A_2$, then Lemma \[ljo2\] applies. So we may assume that $4,5\in A_1$ and $6\in A_2$. If $1\in B_1\cup B_2$ or $2\in B_3$, then Lemma \[ljo2\] applies again. So we may assume that $1\in B_3$ and $2\in B_1$. So now $$\begin{aligned}
e&=(\{\ldots,4,5,1\},1{\rangle}(\{\ldots,6,2\},2{\rangle}(\{3\},3{\rangle},\notag\\
f&=(\{\ldots,2,4\},4{\rangle}(\{\ldots,5\},5{\rangle}(\{\ldots,1,6\},6{\rangle}(\{7\},7{\rangle}.\notag\end{aligned}$$ We define $g\in T(X)$ as follows. Set $xg=x$ for every $x\in\{1,2,4,6\}$ and $5g=4$. Define $7g=4$ if $7\in A_1$; $7g=6$ if $7\in A_2$; $3g=3$ if $3\in B_1\cup A_2$; and $3g=1$ if $3\in B_3$. Then $g$ is an idempotent of rank $4$ and $e-g-f$.
\[tdia2\] Let $n\geq3$ and let $J_r$ be an ideal in $T(X)$ such that $2\leq r<n$. Then:
- If $n=3$ or $n\in\{5,6,7\}$ and $r=4$, then the diameter of ${\mathcal{G}}(J_r)$ is $4$.
- In all other cases, the diameter of ${\mathcal{G}}(J_r)$ is $5$.
Let $n=3$. Then the diameter of ${\mathcal{G}}(J_2)$ is at most $4$ by Lemma \[lad1\] and Theorem \[tdia\]. On the other hand, consider $a=(3\,1{\rangle}(1\,2)$ and $b=(2\,1{\rangle}(1\,3)$ in $J_2$. Suppose $e$ and $f$ are idempotents in $J_2$ such that $a-e$ and $f-b$. By Lemma \[lad3\], $e=(\{1\},1{\rangle}(\{3,2\},2{\rangle}$ and $f=(\{1\},1{\rangle}(\{2,3\},3{\rangle}$. Then $e$ and $f$ do not commute, and so $d(e,f)\geq2$. Thus $d(a,b)\geq4$ by Lemma \[lad1\], and so the diameter of ${\mathcal{G}}(J_2)$ is at least $4$.
Let $n\in\{5,6,7\}$ and $r=4$. If $n=5$, then the diameter of ${\mathcal{G}}(J_4)$ is at least $4$ (by Lemma \[lad6\]) and at most $4$ (by Lemma \[lad1\] and Theorem \[tdia\]). If $n\in\{6,7\}$, then the diameter of ${\mathcal{G}}(J_4)$ is at least $4$ (by Lemma \[lad6\]) and at most $4$ (by Lemma \[lad8\]). We have proved (1).
Let $n\geq4$ and suppose that $n\notin\{5,6,7\}$ or $r\ne4$. Then the diameter of ${\mathcal{G}}(J_r)$ is at most $5$ by Lemma \[lad1\] and Theorem \[tdia\]. It remains to find $a,b\in J_r$ such that the distance between $a$ and $b$ in ${\mathcal{G}}(J_r)$ is at least $5$. We consider four possible cases. [**Case 1.**]{} $r=2m-1$ for some $m\geq2$. Then $2\leq m<r<2m\leq n$. Let $x_1,\ldots,x_m,y_1,\ldots,y_m$ be pairwise distinct elements of $X$. Let $$a=(*\,y_2{\rangle}(y_1\,y_2\ldots\,y_m\,x_1{\rangle}(x_1\,x_2\ldots\,x_m)\mbox{ and }
b=(*\,x_3{\rangle}(x_2\,x_3\ldots\,x_{m-1}\,x_1\,y_1{\rangle}(y_1\,y_2\ldots\,y_m)$$ (see Notation \[ndi5\]) and note that $a,b\in J_r$ and $ab\ne ba$. Then, by Lemma \[lad1\], there are idempotents $e_1,\ldots,e_k\in J_r$ ($k\geq1$) such that $a-e_1-\cdots-e_k-b$ is a minimal path in ${\mathcal{G}}(J_r)$ from $a$ to $b$. By Lemma \[lad3\], $$e_1=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_m,x_m{\rangle}\mbox{ and }e_k=(B_1,y_1{\rangle}(B_2,y_2{\rangle}\ldots(B_m,y_m{\rangle},$$ where $y_i\in A_i$ ($1\leq i\leq m$), $x_{i+1}\in B_i$ ($1\leq i<m)$, and $x_1\in B_m$. Let $g\in T(X)$ be an idempotent such that $e_1-g-e_k$. By Lemma \[lja2\], $x_jg=x_j$ and $y_jg=y_j$ for every $j\in\{1,\ldots,m\}$. Hence $\operatorname{rank}(g)\geq 2m>r$, and so $g\notin J_r$. It follows that the distance between $e_1$ and $e_k$ is at least $3$, and so the distance between $a$ and $b$ is at least $5$. [**Case 2.**]{} $r=2m$ for some $m\geq3$. Then $3\leq m<r=2m<n$. Let $x_1,\ldots,x_m,y_1,\ldots,y_m,z$ be pairwise distinct elements of $X$. Let $$\begin{aligned}
a&=(*\,y_2{\rangle}(z\,y_1\,y_2\ldots\,y_m\,x_1{\rangle}(x_1\,x_2\ldots\,x_m),\notag\\
b&=(*\,x_1{\rangle}(z\,x_3{\rangle}(x_2\,x_3\ldots\,x_m\,x_1\,y_1{\rangle}(y_1\,y_2\ldots\,y_m)\notag\end{aligned}$$ (see Notation \[ndi5\]) and note that $a,b\in J_r$ and $ab\ne ba$. Then, by Lemma \[lad1\], there are idempotents $e_1,\ldots,e_k\in J_r$ ($k\geq1$) such that $a-e_1-\cdots-e_k-b$ is a minimal path in ${\mathcal{G}}(J_r)$ from $a$ to $b$. By Lemma \[lad3\], $$e_1=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_m,x_m{\rangle}\mbox{ and }e_k=(B_1,y_1{\rangle}(B_2,y_2{\rangle}\ldots(B_m,y_m{\rangle},$$ where $y_i\in A_i$ ($1\leq i\leq m$), $x_{i+1}\in B_i$ ($1\leq i<m)$, $x_1\in B_m$, $A_m=\{x_m,y_m,z\}$, and $B_1=\{y_1,x_2,z\}$. Let $g\in T(X)$ be an idempotent such that $e_1-g-e_k$. By Lemma \[lja2\], $x_jg=x_j$ and $y_jg=y_j$ for every $j\in\{1,\ldots,m\}$, and $zg=z$. Hence $\operatorname{rank}(g)\geq 2m+1>r$, and so $g\notin J_r$. It follows that the distance between $e_1$ and $e_k$ is at least $3$, and so the distance between $a$ and $b$ is at least $5$. [**Case 3.**]{} $r=4$. Since we are working under the assumption that $n\notin\{5,6,7\}$ or $r\ne4$, we have $n\notin\{5,6,7\}$. Thus $n\geq8$ (since $r\leq n-1$). Let $$a=\begin{pmatrix}
1&2&3&4&5&6&7&8&9&\!\!\!\!\ldots\, n\\2&3&4&1&2&3&4&1&1&\!\!\!\!\ldots\, 1
\end{pmatrix}\mbox{ and }
b=\begin{pmatrix}
1&2&3&4&5&6&7&8&9&\!\!\!\!\ldots\, n\\5&6&7&8&6&7&8&5&1&\!\!\!\!\ldots\, 1
\end{pmatrix}.$$ Note that $a,b\in J_4$, $ab\ne ba$, $(1\,2\,3\,4)$ is a unique cycle in $a$, and $(5\,6\,7\,8)$ is a unique cycle in $b$. By Lemma \[lad1\], there are idempotents $e_1,\ldots,e_k\in J_4$ ($k\geq1$) such that $a-e_1-\cdots-e_k-b$ is a minimal path in ${\mathcal{G}}(J_4)$ from $a$ to $b$. By Lemma \[lad0\], $ie_1=i$ and $(4+i)e_k=4+i$ for every $i\in\{1,2,3,4\}$. By Lemma \[lad2\], $5e_1=1$ or $5e_1=5$. But the latter is impossible since with $5e_1=5$ we would have $\operatorname{rank}(e_1)\geq5$. Similarly, we obtain $6e_1=2$, $7e_1=3$, $8e_1=4$, $2e_k=5$, $3e_k=6$, $4e_k=7$, and $1e_k=8$. Let $g\in T(X)$ be an idempotent such that $e_1-g-e_k$. By Lemma \[lja2\], $jg=j$ for every $j\in\{1,\ldots,8\}$. Hence $\operatorname{rank}(g)\geq 8>r$, and so $g\notin J_4$. It follows that the distance between $e_1$ and $e_k$ is at least $3$, and so the distance between $a$ and $b$ is at least $5$. [**Case 4.**]{} $r=2$. In this case we let $$a=\begin{pmatrix}
1&2&3&4&5&\!\!\!\!\ldots\, n\\2&1&2&1&1&\!\!\!\!\ldots\, 1
\end{pmatrix}\mbox{ and }
b=\begin{pmatrix}
1&2&3&4&5&\!\!\!\!\ldots\, n\\3&4&4&3&3&\!\!\!\!\ldots\, 3
\end{pmatrix}.$$ Note that $a,b\in J_2$, $ab\ne ba$, $(1\,2)$ is a unique cycle in $a$, and $(3\,4)$ is a unique cycle in $b$. By Lemma \[lad1\], there are idempotents $e_1,\ldots,e_k\in J_2$ ($k\geq1$) such that $a-e_1-\cdots-e_k-b$ is a minimal path in ${\mathcal{G}}(J_2)$ from $a$ to $b$. By Lemma \[lad0\], $1e_1=1$, $2e_1=2$, $3e_k=3$, and $4e_k=4$. By Lemma \[lad2\], $3e_1=1$ or $3e_1=3$. But the latter is impossible since with $3e_1=3$ we would have $\operatorname{rank}(e_1)\geq3$. Again By Lemma \[lad2\], $4e_1=2$ or $4e_1=y$ for some $y\in\{4,5,\ldots,n\}$. But the latter is impossible since we would have $ye_1=y$ and again $\operatorname{rank}(e_1)$ would be at least $3$. Similarly, we obtain $2e_k=3$, and $1e_k=4$. Let $g\in T(X)$ be an idempotent such that $e_1-g-e_k$. By Lemma \[lja2\], $jg=j$ for every $j\in\{1,\ldots,4\}$. Hence $\operatorname{rank}(g)\geq 4>r$, and so $g\notin J_2$. It follows that the distance between $e_1$ and $e_k$ is at least $3$, and so the distance between $a$ and $b$ is at least $5$.
Thus the diameter of ${\mathcal{G}}(J_r)$ is at least $5$, which concludes the proof of (2).
The Commuting Graph of $T(X)$ {#ssctx}
-----------------------------
Let $X$ be a finite set with $|X|=n$. It has been proved in [@IrJa08 Theorem 3.1] that if $n$ and $n-1$ are not prime, then the diameter of the commuting graph of $\operatorname{Sym}(X)$ is at most $5$, and that the bound is sharp since the diameter of ${\mathcal{G}}(\operatorname{Sym}(X))$ is $5$ when $n=9$. In this subsection, we determine the exact value of the diameter of the commuting graph of $T(X)$ for every $n\geq2$.
Throughout this subsection, we assume that $X$ is a finite set with $n\geq2$ elements.
\[ltx\] Let $n\geq4$ be composite. Let $a,f\in T(X)$ such that $a,f\ne\operatorname{id}_X$, $a\in\operatorname{Sym}(X)$, and $f$ is an idempotent. Then $d(a,f)\leq4$.
Fix $x\in\operatorname{im}(f)$ and a cycle $(x_1\ldots x_m)$ of $a$ such that $x\in\{x_1,\ldots,x_m\}$. Consider three cases. [**Case 1.**]{} $a$ has a cycle $(y_1\ldots y_k)$ such that $k$ does not divide $m$. Then $a^m$ is different from $\operatorname{id}_X$ and it fixes $x$. Thus $a-a^m-(X,x{\rangle}-f$, and so $d(a,f)\leq3$. [**Case 2.**]{} $a$ has at least two cycles and for every cycle $(y_1\ldots y_k)$ of $a$, $k$ divides $m$. Suppose there is $z\in\operatorname{im}(f)$ such that $z\in\{y_1,\ldots,y_k\}$ for some cycle $(y_1\ldots y_k)$ of $a$ different from $(x_1\ldots x_m)$. Since $k$ divides $m$, there is a positive integer $t$ such that $m=tk$. Define $e\in T(X)$ by: $$\label{eltx1}
x_1e=y_1,\ldots,x_ke=y_k,\,x_{k+1}e=y_1,\ldots,x_{2k}e=y_k,\ldots,x_{(t-1)k+1}e=y_1,\ldots,x_{tk}e=y_k,$$ and $ye=y$ for all other $y\in X$. Then $e$ is an idempotent such that $ae=ea$ and $z\in\operatorname{im}(e)$. Thus, by Lemma \[ljo1\], $a-e-(X,z{\rangle}-f$, and so $d(a,f)\leq3$.
Suppose that $\operatorname{im}(f)\subseteq\{x_1,\ldots,x_m\}$. Consider any cycle $(y_1\ldots y_k)$ of $a$ different from $(x_1\ldots x_m)$. Since $\operatorname{im}(f)\subseteq\{x_1,\ldots,x_m\}$, $y_1f=x_i$ for some $i$. We may assume that $y_1f=x_1$. Define an idempotent $e$ exactly as in (\[eltx1\]). Then $\operatorname{im}(e)\cap\operatorname{im}(f)=\emptyset$, $(y_1,x_1)\in\operatorname{im}(e)\times\operatorname{im}(f)$, and $(y_1,x_1)\in\operatorname{ker}(e)\cap\operatorname{ker}(f)$. Thus, by Lemma \[ljo2\], there is an idempotent $g\in T(X)-\{\operatorname{id}_X\}$ such that $e-g-f$. Hence $a-e-g-f$, and so $d(a,f)\leq3$. [**Case 3.**]{} $a$ is an $n$-cycle. Since $n$ is composite, there is a divisor $k$ of $n$ such that $1<k<n$. Then $a^k\ne\operatorname{id}_X$ is a permutation with $k\geq2$ cycles, each of length $m=n/k$. By Case 2, $d(a^k,f)\leq3$, and so $d(a,f)\leq4$.
\[ltx1\] Let $n\geq4$ be composite. Let $a,b\in T(X)$ such that $a,b\ne\operatorname{id}_X$ and $a\in\operatorname{Sym}(X)$. Then $d(a,b)\leq5$.
Suppose $b\notin\operatorname{Sym}(X)$. Then $b^k$ is an idempotent different from $\operatorname{id}_X$ for some $k\geq1$. By Lemma \[ltx\], $d(a,b^k)\leq4$, and so $d(a,b)\leq5$.
Suppose $b\in\operatorname{Sym}(X)$. Suppose $n-1$ is not prime. Then, by [@IrJa08 Theorem 3.1], there is a path from $a$ to $b$ in ${\mathcal{G}}(\operatorname{Sym}(X))$ of length at most $5$. Such a path is also a path in ${\mathcal{G}}(T(X))$, and so $d(a,b)\leq5$. Suppose $p=n-1$ is prime. Then the proof of [@IrJa08 Theorem 3.1] still works for $a$ and $b$ unless $a^p=\operatorname{id}_X$ or $b^p=\operatorname{id}_X$. (See also [@IrJa08 Lemma 3.3] and its proof.) Thus, if $a^p\ne\operatorname{id}_X$ and $b^p\ne\operatorname{id}_X$, then there is a path from $a$ to $b$ in ${\mathcal{G}}(\operatorname{Sym}(X))$ of length at most $5$, and so $d(a,b)\leq5$. Suppose $a^p=\operatorname{id}_X$ or $b^p=\operatorname{id}_X$. We may assume that $b^p=\operatorname{id}_X$. Then $b$ is a cycle of length $p$, that is, $b=(x_1\ldots x_p)(x)$. Thus $b$ commutes with the constant idempotent $f=(X,x{\rangle}$. By Lemma \[ltx\], $d(a,f)\leq4$, and so $d(a,b)\leq5$.
\[ltx2\] Let $X=\{x_1,\ldots,x_m,y_1,\ldots,y_k\}$, $a\in\operatorname{Sym}(X)$, and $b=(y_1\ldots y_k\,x_1{\rangle}(x_1\ldots x_m)$. If $ab=ba$ then $a=\operatorname{id}_X$.
Suppose $ab=ba$. By Lemma \[lad2\], $$\label{e1ltx2}
x_1a{\stackrel{b}{{\rightarrow}}}x_2a{\stackrel{b}{{\rightarrow}}}\cdots{\stackrel{b}{{\rightarrow}}}x_ma{\stackrel{b}{{\rightarrow}}}x_1a\quad\mbox{and}\quad y_1a{\stackrel{b}{{\rightarrow}}}y_2a{\stackrel{b}{{\rightarrow}}}\cdots{\stackrel{b}{{\rightarrow}}}y_ka{\stackrel{b}{{\rightarrow}}}x_1a.$$ Since $(x_1\,x_2\ldots\,x_m)$ is a unique cycle in $b$, (\[e1ltx2\]) implies that $$\label{e2ltx2}
x_1a=x_q,\, x_2a=x_{q+1},\ldots,\, x_ma=x_{q+m-1},$$ where $q\in\{1,\ldots,m\}$ ($x_{q+i}=x_{q+i-m}$ if $q+i>m$). Thus $x_1a=x_j$ for some $j$. Since $y_k{\stackrel{b}{{\rightarrow}}}x_1$ and $x_m{\stackrel{b}{{\rightarrow}}}x_1$, we have $y_ka{\stackrel{b}{{\rightarrow}}}x_1a=x_j$ and $x_ma{\stackrel{b}{{\rightarrow}}}x_1a=x_j$. Suppose $j\geq2$. Then $x_jb^{-1}=\{x_{j-1}\}$, and so $y_ka=x_{j-1}=x_ma$. But this implies $y_k=x_m$ (since $a$ is injective), which is a contradiction. Hence $j=1$, and so $x_1a=x_1$. But then $x_ia=x_i$ for all $i$ by (\[e2ltx2\]).
Since $y_ka{\stackrel{b}{{\rightarrow}}}x_1a=x_1$, we have $y_ka=y_k$ since $x_1b^{-1}=\{y_k,x_m\}$. Let $i\in\{1,\ldots,k-1\}$ and suppose $y_{i+1}a=y_{i+1}$. Then $y_ia=y_i$ since $y_ia{\stackrel{b}{{\rightarrow}}}y_{i+1}a=y_{i+1}$ and $y_{i+1}b^{-1}=\{y_{i+1}\}$. It follows that $y_ia=y_i$ for all $i\in\{1,\ldots,k\}$.
\[ltx3\] Let $m$ be a positive integer such that $2m\leq n$, $\sigma$ be an $m$-cycle on $\{1,\ldots,m\}$, $a\in\operatorname{Sym}(X)$, and $$e=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_m,x_m{\rangle}\mbox{ and }f=(B_1,y_1{\rangle}(B_2,y_2{\rangle}\ldots(B_m,y_m{\rangle}$$ be idempotents in $T(X)$ such that $x_1,\ldots,x_m,y_1,\ldots,y_m$ are pairwise distinct, $y_i\in A_i$, and $x_{i\sigma}\in B_i$ ($1\leq i\leq m)$. Then:
- Suppose $X=\{x_1,\ldots,x_m,y_1,\ldots,y_m,z\}$ and $z\in A_i\cap B_j$ such that $A_i\cap B_j=\{z\}$. If $e-a-f$, then $a=\operatorname{id}_X$.
- Suppose $X=\{x_1,\ldots,x_m,y_1,\ldots,y_m,z,w\}$, $z\in A_i\cap B_j$ such that $A_i\cap B_j=\{z\}$, and $w\in A_s\cap B_t$ such that $A_s\cap B_t=\{w\}$, where $s\ne i$ and $t\ne j$. If $e-a-f$, then $a=\operatorname{id}_X$.
To prove (1), suppose $e-a-f$ and note that $A_i=\{x_i,y_i,z\}$ and $B_j=\{y_j,x_{j\sigma},z\}$. By Lemma \[lcen\], there is $p\in\{1,\ldots,m\}$ such that $x_ia=x_p$ and $A_ia\subseteq A_p$. Suppose $p\ne i$. Then $A_p=\{x_p,y_p\}$, and so $A_ia$ cannot be a subset of $A_p$ since $a$ is injective. It follows that $p=i$, that is, $x_ia=x_i$ and $A_ia\subseteq A_i$. Similarly, $y_ja=y_j$ and $B_ja\subseteq B_j$. Thus $za\in A_i\cap B_j=\{z\}$, and so $za=z$. Hence, since $a$ is injective, $y_ia=y_i$.
We have proved that $x_ia=x_i$, $y_ia=y_i$, and $za=z$. We have $B_i=\{y_i,x_{i\sigma}\}$ or $B_i=\{y_i,x_{i\sigma},z\}$ Since $y_ia=y_i$, we have $B_ia\subseteq B_i$ by Lemma \[lcen\]. Since $za=z$ and $a$ is injective, it follows that $x_{i\sigma}a=x_{i\sigma}$. By the foregoing argument applied to $A_{i\sigma}=\{x_{i\sigma},y_{i\sigma}\}$, we obtain $y_{i\sigma}a=y_{i\sigma}$. Continuing this way, we obtain $x_{i\sigma^k}a=x_{i\sigma^k}$ and $y_{i\sigma^k}a=y_{i\sigma^k}$ for every $k\in\{1,\ldots,m-1\}$. Since $\sigma$ is an $m$-cycle, it follows that $x_ja=x_j$ and $y_jg=y_j$ for every $j\in\{1,\ldots,m\}$. Hence $a=\operatorname{id}_X$. We have proved (1). The proof of (2) is similar.
\[tdia3\] Let $X$ be a finite set with $n\geq2$ elements. Then:
- If $n$ is prime, then ${\mathcal{G}}(T(X))$ is not connected.
- If $n=4$, then the diameter of ${\mathcal{G}}(T(X))$ is $4$.
- If $n\geq6$ is composite, then the diameter of ${\mathcal{G}}(T(X))$ is $5$.
Suppose $n=p$ is prime. Consider a $p$-cycle $a=(x_1\,x_2\ldots x_p)$ and let $b\in T(X)$ be such that $b\ne\operatorname{id}_X$ and $ab=ba$. Let $x_q=x_1b$. Then, by Lemma \[lad2\], $x_ib=x_{q+i}$ for every $i\in\{1,\ldots,p\}$ (where $x_{q+i}=x_{q+i-m}$ if $q+i>m$). Thus $b=a^q$, and so, since $p$ is prime, $b$ is also a $p$-cycle. It follows that if $c$ is a vertex of ${\mathcal{G}}(T(X))$ that is not a $p$-cycle, then there is no path in ${\mathcal{G}}(T(X))$ from $a$ to $c$. Hence ${\mathcal{G}}(T(X))$ is not connected. We have proved (1).
We checked the case $n=4$ directly using GRAPE [@So06] through GAP [@Scel92]. We found that, when $|X|=4$, the diameter of ${\mathcal{G}}(T(X))$ is $4$.
Suppose $n\geq6$ is composite. Let $a,b\in T(X)$ such that $a,b\ne\operatorname{id}_X$. If $a\in\operatorname{Sym}(X)$ or $b\in\operatorname{Sym}(X)$, then $d(a,b)\leq5$ by Lemma \[ltx1\]. If $a,b\notin\operatorname{Sym}(X)$, then $a,b\in J_{n-1}$, and so $d(a,b)\leq5$ by Theorem \[tdia2\]. Hence the diameter of ${\mathcal{G}}(T(X))$ is at most $5$. It remains to find $a,b\in T(X)-\{\operatorname{id}_X\}$ such that $d(a,b)\geq5$.
For $n\in\{6,8\}$, we employed GAP [@Scel92]. When $n=6$, we found that the distance between the $6$-cycle $a=(1\,2\,3\,4\,5\,6)$ and $b=\begin{pmatrix}
1&2&3&4&5&6\\2&3&5&1&2&4\end{pmatrix}$ in ${\mathcal{G}}(T(X))$ is at least $5$. And when $n=8$, the distance between the $8$-cycle $a=(1\,2\,3\,4\,5\,6\,7\,8)$ and $b=\begin{pmatrix}
1&2&3&4&5&6&7&8\\2&3&1&1&4&8&6&5\end{pmatrix}$ in ${\mathcal{G}}(T(X))$ is at least $5$.
To verify this with GAP, we used the following sequence of arguments and computer calculations:
1. By Lemma \[lad1\], if there exists a path $a-c_1-c_2-\ldots-c_k-b$, then there exists a path $a-e_1-e_2-\ldots-e_k-b$, where each $e_i$ is either an idempotent or a permutation;
2. Let $E$ be the set idempotents of $T(X)-\{\operatorname{id}_X\}$ and let $G=\operatorname{Sym}(X)-\{\operatorname{id}_X\}$. For $A\subseteq T(X)$, let $C(A)=\{f\in E\cup G:(\exists_{a\in A}) af=fa\}$;
3. Calculate $C(C(\{a\}))$ and $C(\{b\})$;
4. Verify that for all $c\in C(C(\{a\}))$ and all $d\in C(\{b\})$, $cd\neq dc$;
5. If there were a path $a-c_1-c_2-c_3-b$ from $a$ to $b$, then we would have $c_2\in C(C(\{a\}))$, $c_3 \in C(\{b\})$, and $c_2c_3=c_3c_2$. But, by 4., there are no such $c_2$ and $c_3$, and it follows that the distance between $a$ and $b$ is at least $5$.
Let $n\geq9$ be composite. We consider two cases. [**Case 1.**]{} $n=2m+1$ is odd ($m\geq4$). Let $X=\{x_1,\ldots,x_m,y_1,\ldots,y_m,z\}$. Consider $$a=(z\,y_1\,y_2\ldots\,y_m\,x_1{\rangle}(x_1\,x_2\ldots\,x_m)\mbox{ and }
b=(x_2\,x_3\ldots\,x_m\,x_1\,z\,y_2{\rangle}(y_1\,y_2\ldots\,y_m).$$ Let ${\lambda}$ be a minimal path in ${\mathcal{G}}(T(X))$ from $a$ to $b$. By Lemma \[ltx2\], there is no $g\in\operatorname{Sym}(X)$ such that $g\ne\operatorname{id}_X$ and $ag=ga$ or $bg=gb$. Thus, by the proof of Lemma \[lad1\], ${\lambda}=a-e_1-\cdots-e_k-b$, where $e_1$ and $e_k$ are idempotents. By Lemma \[lad3\], $$e_1=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_m,x_m{\rangle}\mbox{ and }e_k=(B_1,y_1{\rangle}(B_2,y_2{\rangle}\ldots(B_m,y_m{\rangle},$$ where $y_i\in A_i$ ($1\leq i\leq m$), $x_{i+1}\in B_i$ ($1\leq i<m)$, $x_1\in B_m$, $A_m=\{x_m,y_m,z\}$, and $B_1=\{y_1,x_2,z\}$. Since $m\geq4$, $A_m\cap B_1=\{z\}$. Thus, by Lemma \[ltx3\], there is no $g\in\operatorname{Sym}(X)$ such that $g\ne\operatorname{id}_X$ and $e_1-g-e_k$. Hence, if ${\lambda}$ contains an element $g\in\operatorname{Sym}(X)$, then the length of ${\lambda}$ is at least $5$. Suppose ${\lambda}$ does not contain any permutations. Then ${\lambda}$ is a path in $J_{n-1}$ and we may assume that all vertices in ${\lambda}$ except $a$ and $b$ are idempotents (by Lemma \[lad3\]). By Lemma \[lja2\], there is no idempotent $f\in J_{n-1}$ such that $e_1-f-e_k$. (Here, the $m$-cycle that occurs in Lemmas \[lja2\] and \[ltx3\] is $\sigma=(1\,2\ldots m)$.) Hence the length of ${\lambda}$ is at least $5$.
[**Case 2.**]{} $n=2m+2$ is even ($m\geq4$). Let $X=\{x_1,\ldots,x_m,y_1,\ldots,y_m,z,w\}$. Consider $$a=(z\,y_1\,y_2\ldots\,y_m\,w\,x_2{\rangle}(x_1\,x_2\ldots\,x_m)\mbox{ and }
b=(w\,x_2\,x_3\ldots\,x_{m-2}\,x_m\,x_1\,x_{m-1}\,y_2{\rangle}(y_1\,y_2\ldots\,y_m).$$ Let ${\lambda}$ be a minimal path in ${\mathcal{G}}(T(X))$ from $a$ to $b$. By Lemma \[ltx2\], there is no $g\in\operatorname{Sym}(X)$ such that $g\ne\operatorname{id}_X$ and $ag=ga$ or $bg=gb$. Thus, by the proof of Lemma \[lad1\], ${\lambda}=a-e_1-\cdots-e_k-b$, where $e_1$ and $e_k$ are idempotents. By Lemma \[lad3\], $$e_1=(A_1,x_1{\rangle}(A_2,x_2{\rangle}\ldots(A_m,x_m{\rangle}\mbox{ and }e_k=(B_1,y_1{\rangle}(B_2,y_2{\rangle}\ldots(B_m,y_m{\rangle},$$ where $y_i\in A_i$ ($1\leq i\leq m$), $x_{i+1}\in B_i$ ($1\leq i\leq m-3)$, $x_m\in B_{m-2}$, $x_1\in B_{m-1}$, $x_{m-1}\in B_m$, $A_1=\{x_1,y_1,w\}$, $A_m=\{x_m,y_m,z\}$, $B_1=\{y_1,x_2,z\}$, and $B_m=\{y_m,x_{m-1},w\}$. Since $m\geq4$, $A_m\cap B_1=\{z\}$ and $A_1\cap B_m=\{w\}$. Thus, by Lemma \[ltx3\], there is no $g\in\operatorname{Sym}(X)$ such that $g\ne\operatorname{id}_X$ and $e_1-g-e_k$. Hence, as in Case 1, the length of ${\lambda}$ is at least $5$. (Here, the $m$-cycle that occurs in Lemmas \[lja2\] and \[ltx3\] is $\sigma=(1,2\ldots, m-3,m-2,m,m-1)$.)
Hence, if $n\geq6$ is composite, then the diameter of ${\mathcal{G}}(T(X))$ is $5$. This concludes the proof.
Minimal Left Paths {#smlp}
==================
In this section, we prove that for every integer $n\geq4$, there is a band $S$ with knit degree $n$. We will show how to construct such an $S$ as a subsemigroup of $T(X)$ for some finite set $X$.
Let $S$ be a finite non-commutative semigroup. Recall that a path $a_1-a_2-\cdots-a_m$ in ${\mathcal{G}}(S)$ is called a *left path* (or $l$-path) if $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for every $i\in\{1,\ldots,m\}$. If there is any $l$-path in ${\mathcal{G}}(S)$, we define the *knit degree* of $S$, denoted $\operatorname{kd}(S)$, to be the length of a shortest $l$-path in ${\mathcal{G}}(S)$. We say that an $l$-path ${\lambda}$ from $a$ to $b$ in ${\mathcal{G}}(S)$ is a *minimal $l$-path* if there is no $l$-path from $a$ to $b$ that is shorter than ${\lambda}$.
The Even Case {#sseven}
-------------
In this subsection, we will construct a band of knit degree $n$ where $n\geq4$ is even. For $x\in X$, we denote by $c_x$ the constant transformation with image $\{x\}$. The following lemma is obvious.
\[lcon\] Let $c_x,c_y,e\in T(X)$ such that $e$ is an idempotent. Then:
- $c_xe=ec_x$ if and only if $x\in\operatorname{im}(e)$.
- $c_xe=c_ye$ if and only if $(x,y)\in\operatorname{ker}(e)$.
Now, given an even $n\geq4$, we will construct a band $S$ such that $\operatorname{kd}(S)=n$. We will explain the construction using $n=8$ as an example. The band $S$ will be a subsemigroups of $T(X)$, where $$X=\{y_0,y_1,y_2,y_3,y_4=v_0,v_1,v_2,v_3,v_4,x_1,x_2,x_3,x_4,u_1,u_2,u_3,u_4,r,s\},$$ and it will be generated by idempotent transformations $a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4,e_1$, whose images are defined by Table 1.
$$\begin{tabular}{|c|ccc|}\hline
$\operatorname{im}(a_1)$ & $y_0$ & $x_1$ & $y_1$ \\\hline
$\operatorname{im}(a_2)$ & $y_1$ & $x_2$ & $y_2$ \\\hline
$\operatorname{im}(a_3)$ & $y_2$ & $x_3$ & $y_3$ \\\hline
$\operatorname{im}(a_4)$ & $y_3$ & $x_4$ & $y_4$ \\\hline
$\operatorname{im}(b_1)$ & $y_4$ & $u_1$ & $v_1$ \\\hline
$\operatorname{im}(b_2)$ & $v_1$ & $u_2$ & $v_2$ \\\hline
$\operatorname{im}(b_3)$ & $v_2$ & $u_3$ & $v_3$ \\\hline
$\operatorname{im}(b_4)$ & $v_3$ & $u_4$ & $v_4$ \\\hline
$\operatorname{im}(e_1)$ & $v_4$ & $r$ & $s$ \\\hline
\end{tabular}$$
We will define the kernels in such a way that the generators with the same subscript will have the same kernel. For example, $\operatorname{ker}(a_1)=\operatorname{ker}(b_1)=\operatorname{ker}(e_1)$ and $\operatorname{ker}(a_2)=\operatorname{ker}(b_2)$. Let $i\in\{2,3,4\}$. The kernel of $a_i$ will have the following three classes (elements of the partition $X/\operatorname{ker}(a_i)$): $$\begin{aligned}
\mbox{Class-1}&=\operatorname{im}(a_{i+1})\cup\ldots\cup\operatorname{im}(a_4)\cup\operatorname{im}(b_1)\cup\ldots\cup\operatorname{im}(b_{i-1}),\notag\\
\mbox{Class-2}&=\operatorname{im}(b_{i+1})\cup\ldots\cup\operatorname{im}(b_4)\cup\operatorname{im}(e_1)\cup\operatorname{im}(a_1)\cup\ldots\cup\operatorname{im}(a_{i-1}),\notag\\
\mbox{Class-3}&=\{x_i,u_i\}.\notag\end{aligned}$$ For example, $\operatorname{ker}(a_2)$ has the following classes: $$\begin{aligned}
\mbox{Class-1}&=\{y_2,x_3,y_3,x_4,y_4,u_1,v_1\},\notag\\
\mbox{Class-2}&=\{v_2,u_3,v_3,u_4,v_4,r,s,y_0,x_1,y_1\},\notag\\
\mbox{Class-3}&=\{x_2,u_2\}.\notag\end{aligned}$$ We define the kernel of $a_1$ as follows: $$\begin{aligned}
\mbox{Class-1}&=\operatorname{im}(a_2)\cup\operatorname{im}(a_3)\cup\operatorname{im}(a_4)\cup\{s\}=\{y_1,x_2,y_2,x_3,y_3,x_4,y_4,s\},\notag\\
\mbox{Class-2}&=\operatorname{im}(b_2)\cup\operatorname{im}(b_3)\cup\operatorname{im}(b_4)\cup\{y_0\}=\{v_1,u_2,v_2,u_3,v_3,u_4,v_4,y_0\},\notag\\
\mbox{Class-3}&=\{x_1,u_1,r\}.\notag\end{aligned}$$
Now the generators are completely defined since $\operatorname{ker}(b_i)=\operatorname{ker}(a_i)$, $1\leq i\leq 4$, and $\operatorname{ker}(e_1)=\operatorname{ker}(a_1)$. Order the generators as follows: $$\label{elist1}
a_1,\,a_2,\,a_3,\,a_4,\,b_1,\,b_2,\,b_3,\,b_4,\,e_1.$$ Let $S$ be the semigroup generated by the idempotents listed in (\[elist1\]). Since the idempotents with the same subscript have the same kernel, they form a right-zero subsemigroup of $S$. For example, $\{a_1,b_1,e_1\}$ is a right-zero semigroup: $a_1a_1=b_1a_1=e_1a_1=a_1$, $a_1b_1=b_1b_1=e_1b_1=b_1$, and $a_1e_1=b_1e_1=e_1e_1=e_1$. The product of any two generators with different indices is a constant transformation. For example, $a_2a_4=c_{y_3}$, $a_4a_2=c_{y_2}$, and $a_1b_3=c_{v_3}$. The semigroup $S$ consists of the nine generators listed in (\[elist1\]) and $10$ constants: $$S=\{a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4,e_1,c_{y_0},c_{y_1},c_{y_2},c_{y_3},c_{y_4},c_{v_1},c_{v_2},c_{v_3},c_{v_4},c_s\},$$ so $S$ is a band. Note that $Z(S)=\emptyset$. Each idempotent in (\[elist1\]) commutes with the next idempotent, so $a_1-a_2-a_3-a_4-b_1-b_2-b_3-b_4-e_1$ is a path in ${\mathcal{G}}(S)$. Moreover, it is a unique $l$-path in ${\mathcal{G}}(S)$, so $\operatorname{kd}(S)=8$.
We will now provide a general construction of a band $S$ such that $\operatorname{kd}(S)=n$, where $n$ is even.
\[dco1\]
Let $k\geq2$ be an integer. Let $$X=\{y_0,y_1,\ldots,y_k=v_0,v_1,\ldots,v_k,x_1,\ldots,x_k,u_1,\ldots,u_k,r,s\}.$$ We will define idempotents $a_1,\ldots,a_k,b_1,\ldots,b_k,e_1$ as follows. For $i\in\{1,\ldots,k\}$, let $$\begin{aligned}
\operatorname{im}(a_i)&=\{y_{i-1},x_i,y_i\},\notag\\
\operatorname{im}(b_i)&=\{v_{i-1},u_i,v_i\},\notag\\
\operatorname{im}(e_1)&=\{v_k,r,s\}.\notag\end{aligned}$$
For $i\in\{2,\ldots,k\}$, define the $\operatorname{ker}(a_i)$-classes by: $$\begin{aligned}
\mbox{Class-1}&=\operatorname{im}(a_{i+1})\cup\ldots\cup\operatorname{im}(a_k)\cup\operatorname{im}(b_1)\cup\ldots\cup\operatorname{im}(b_{i-1}),\notag\\
\mbox{Class-2}&=\operatorname{im}(b_{i+1})\cup\ldots\cup\operatorname{im}(b_k)\cup\operatorname{im}(e_1)\cup\operatorname{im}(a_1)\cup\ldots\cup\operatorname{im}(a_{i-1}),\notag\\
\mbox{Class-3}&=\{x_i,u_i\}.\notag\end{aligned}$$ (Note that for $i=k$, $\mbox{Class-1}=\operatorname{im}(b_1)\cup\ldots\cup\operatorname{im}(b_{k-1})$ and $\mbox{Class-2}=\operatorname{im}(e_1)\cup\operatorname{im}(a_1)\cup\ldots\cup\operatorname{im}(a_{i-1})$.)
Define the $\operatorname{ker}(a_1)$-classes by: $$\begin{aligned}
\mbox{Class-1}&=\operatorname{im}(a_2)\cup\ldots\cup\operatorname{im}(a_k)\cup\{s\},\notag\\
\mbox{Class-2}&=\operatorname{im}(b_2)\cup\ldots\cup\operatorname{im}(b_k)\cup\{y_0\},\notag\\
\mbox{Class-3}&=\{x_1,u_1,r\}.\notag\end{aligned}$$ Let $\operatorname{ker}(b_i)=\operatorname{ker}(a_i)$ for every $i\in\{1,\ldots,k\}$, and $\operatorname{ker}(e_1)=\operatorname{ker}(a_1)$. Now, define the subsemigroup $S_0^k$ of $T(X)$ by: $$\label{edco1}
S_0^k=\mbox{the semigroup generated by $\{a_1,\ldots,a_k,b_1,\ldots,b_k,e_1\}$.}$$
We must argue that the idempotents $a_1,\ldots,a_k,b_1,\ldots,b_k,e_1$ are well defined, that is, for each of them, different elements of the image lie in different kernel classes. Consider $a_i$, where $i\in\{2,\ldots,k\}$. Then $\operatorname{im}(a_i)=\{y_{i-1},x_i,y_i\}$. Then $y_i$ lies in Class-1 (see Definition \[dco1\]) since $y_i\in\operatorname{im}(a_{i+1})$ (or $y_i\in\operatorname{im}(b_1)$ if $i=k$), $y_{i-1}$ lies in Class-2 since $y_{i-1}\in\operatorname{im}(a_{i-1})$, and $x_i$ lies in Class-3. Arguments for the remaining idempotents are similar.
For the remainder of this subsection, $S_0^k$ will be the semigroup (\[edco1\]). Our objective is to prove that $S_0^k$ is a band such that $\pi=a_1-\cdots-a_k-b_1-\cdots-b_k-e_1$ is a shortest $l$-path in $S_0^k$. Since $\pi$ has length $2k=n$, it will follow that $S_0^k$ is a band with knit degree $n$.
We first analyze products of the generators of $S_0^k$.
\[lev2\] Let $1\leq i<j\leq k$. Then:
- $a_ib_i=b_i$, $b_ia_i=a_i$, $a_1e_1=b_1e_1=e_1$, $e_1a_1=b_1a_1=a_1$, and $e_1b_1=a_1b_1=b_1$.
- $a_ia_j=c_{y_{j-1}}$ and $a_ja_i=c_{y_i}$.
- $a_ib_j=c_{v_j}$ and $a_jb_i=c_{v_{i-1}}$.
- $b_ia_j=c_{y_j}$ and $b_ja_i=c_{y_{i-1}}$.
- $b_ib_j=c_{v_{j-1}}$ and $b_jb_i=c_{v_i}$.
- $e_1a_j=c_{y_{j-1}}$ and $a_je_1=c_s$.
- $e_1b_j=c_{v_j}$ and $b_je_1=c_{v_k}$.
Statement (1) is true because the generators of $S_0^k$ are idempotents and the ones with the same subscript have the same kernel. By Definition \[dco1\], Class-2 of $\operatorname{ker}(a_j)$ contains both $\operatorname{im}(a_{j-1})=\{y_{j-2},x_{j-1},y_{j-1}\}$ and $\operatorname{im}(a_i)$ (since $i<j$). Since $y_{j-1}\in\operatorname{im}(a_j)=\{y_{j-1},x_j,y_j\}$, $a_j$ maps all elements of Class-2 to $y_{j-1}$. Hence $a_ia_j=c_{y_{j-1}}$. Similarly, since $i<j$, Class-1 of $\operatorname{ker}(a_i)$ contains both $\operatorname{im}(a_{i+1})=\{y_i,x_{i+1},y_{i+1}\}$ and $\operatorname{im}(a_j)$. Since $y_i\in\operatorname{im}(a_i)=\{y_{i-1},x_i,y_i\}$, $a_i$ maps all elements of Class-1 to $y_i$. Hence $a_ja_i=c_{y_i}$. We have proved (2). Proofs of (3)-(7) are similar. For example, $b_je_1=c_{v_k}$ because Class-2 of $\operatorname{ker}(e_1)=\operatorname{ker}(a_1)$ contains both $\operatorname{im}(b_j)$ and $\operatorname{im}(b_k)=\{v_{k-1},u_k,v_k\}$, and $v_k\in\operatorname{im}(e_1)$.
The following corollaries are immediate consequences of Lemma \[lev2\].
\[cev2\] The semigroup $S_0^k$ is a band. It consists of $2k+1$ generators from Definition [\[dco1\]]{} and $2k+2$ constant transformations: $$S_0^k=\{a_1,\ldots,a_k,b_1,\ldots,b_k,e_1,c_{y_0},c_{y_1},\ldots,c_{y_k},c_{v_1},\ldots,c_{v_k},c_s\}.$$
\[cev2a\] Let $g,h\in S_0^k$ be generators from the list $$\label{ecev2}
a_1,\ldots,a_k,b_1,\ldots,b_k,e_1.$$ Then $gh=hg$ if and only if $g$ and $h$ are consecutive elements in the list.
Lemma \[lev2\] gives a partial multiplication table for $S_0^k$. The following lemma completes the table.
\[lev2a\] Let $1\leq p\leq k$ and $1\leq i<j\leq k$. Then:
- $c_{y_p}a_p=c_{y_p}$, $c_{y_p}b_p=c_{v_{p-1}}$, $c_{y_i}a_j=c_{y_{j-1}}$, $c_{y_j}a_i=c_{y_i}$, $c_{y_i}b_j=c_{v_j}$, $c_{y_j}b_i=c_{v_{i-1}}$, $c_{y_p}e_1=c_s$, $c_{y_0}a_p=c_{y_{p-1}}$, $c_{y_0}b_p=c_{v_p}$, and $c_{y_0}e_1=c_{v_k}$.
- $c_{v_p}a_p=c_{y_{p-1}}$, $c_{v_p}b_p=c_{v_p}$, $c_{v_i}a_j=c_{y_j}$, $c_{v_j}a_i=c_{y_{i-1}}$, $c_{v_i}b_j=c_{v_{j-1}}$, $c_{v_j}b_i=c_{v_i}$, and $c_{v_p}e_1=c_{v_k}$.
- $c_sa_j=c_{y_{j-1}}$, $c_sb_j=c_{v_j}$, $c_sa_1=c_{y_1}$, $c_sb_1=c_{v_0}$, and $c_se_1=c_s$.
We have $c_{y_p}a_p=c_{y_p}$ since $y_p\in\operatorname{im}(a_p)$. By Definition \[dco1\], Class-1 of $\operatorname{ker}(b_p)$ contains both $\operatorname{im}(a_{p+1})$ and $\operatorname{im}(b_{p-1})$. Since $y_p\in\operatorname{im}(a_{p+1})$ and $v_{p-1}\in\operatorname{im}(b_{p-1})$, both $y_p$ and $v_{p-1}$ are in Class-1. Hence $y_pb_p=v_{p-1}b_p=v_{p-1}$, where the last equality is true because $v_{p-1}\in\operatorname{im}(b_p)$. Thus $c_{y_p}b_p=c_{v_{p-1}}$. By Definition \[dco1\], $y_p$ and $s$ belong to Class-1 of $\operatorname{ker}(e_1)$, and $s\in\operatorname{im}(e_1)$. It follows that $c_{y_p}e_1=c_s$. Again by Definition \[dco1\], $y_0$ and $y_{p-1}$ belong to Class-2 of $\operatorname{ker}(a_p)$, and $y_{p-1}\in\operatorname{im}(a_p)$. Hence $c_{y_0}a_p=c_{y_{p-1}}$. Similarly, $c_{y_0}b_p=c_{v_p}$ and $c_{y_0}e_1=c_{v_k}$. By Lemma \[lev2\], $$\begin{aligned}
c_{y_i}a_j&=(c_{y_i}a_i)a_j=c_{y_i}(a_ia_j)=c_{y_i}c_{y_{j-1}}=c_{y_{j-1}},\notag\\
c_{y_j}a_i&=(c_{y_j}a_j)a_i=c_{y_j}(a_ja_i)=c_{y_j}c_{y_i}=c_{y_i},\notag\\
c_{y_i}b_j&=(c_{y_i}a_i)b_j=c_{y_i}(a_ib_j)=c_{y_i}c_{v_j}=c_{v_j},\notag\\
c_{y_j}b_i&=(c_{y_j}a_j)b_i=c_{y_j}(a_jb_i)=c_{y_j}c_{v_{i-1}}=c_{v_{i-1}}.\notag\end{aligned}$$ We have proved (1). Proofs of (2) and (3) are similar.
Table 2 presents the Cayley table for $S_0^2$.
$$\begin{tabular}{c|ccccccccccc}
& $a_1$ & $a_2$ & $b_1$ & $b_2$ & $e_1$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\\hline
$a_1$ & $a_1$ & $c_{y_1}$ & $b_1$ & $c_{v_2}$ & $e_1$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$a_2$ & $c_{y_1}$ & $a_2$ & $c_{y_2}$ & $b_2$ & $c_s$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$b_1$ & $a_1$ & $c_{y_2}$ & $b_1$ & $c_{v_1}$ & $e_1$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$b_2$ & $c_{y_0}$ & $a_2$ & $c_{v_1}$ & $b_2$ & $c_{v_2}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$e_1$ & $a_1$ & $c_{y_1}$ & $b_1$ & $c_{v_2}$ & $e_1$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$c_{y_0}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{v_1}$ & $c_{v_2}$ & $c_{v_2}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$c_{y_1}$ & $c_{y_1}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_2}$ & $c_{s}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$c_{y_2}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{s}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$c_{v_1}$ & $c_{y_0}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_1}$ & $c_{v_2}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$c_{v_2}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{v_1}$ & $c_{v_2}$ & $c_{v_2}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
$c_s$ & $c_{y_1}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_2}$ & $c_{s}$ & $c_{y_0}$ & $c_{y_1}$ & $c_{y_2}$ & $c_{v_1}$ & $c_{v_2}$ & $c_s$ \\
\end{tabular}$$
\[lev3\] Let $g,h,c_z\in S_0^k$ such that $c_z$ is a constant and $g-c_z-h$ is a path in ${\mathcal{G}}(S_0^k)$. Then $gh=hg$.
Note that $g,h$ are not constants since different constants do not commute. Thus $g$ and $h$ are generators from list (\[ecev2\]). We may assume that $g$ is to the left of $h$ in the list. Since $c_z$ commutes with both $g$ and $h$, $z\in\operatorname{im}(g)\cap\operatorname{im}(h)$ by Lemma \[lcon\]. Suppose $g=a_i$, where $1\leq i\leq k-1$. Then $h=a_{i+1}$ since $a_{i+1}$ is the only generator to the right of $a_i$ whose image is not disjoint from $\operatorname{im}(a_i)$. Similarly, if $g=a_k$ then $h=b_1$; if $g=b_i$ ($1\leq i\leq k-1$) then $h=b_{i+1}$; and if $g=b_k$ then $h=e_1$. Hence $gh=hg$ by Corollary \[cev2a\].
\[lev4\] The paths
- $\tau_1=c_{y_0}-a_1-\cdots-a_k-b_1-\cdots-b_k-c_{v_k}$,
- $\tau_2=c_{y_1}-a_2-\cdots-a_k-b_1-\cdots-b_k-e_1-c_s$
are the only minimal $l$-paths in ${\mathcal{G}}(S_0^k)$ with constants as the endpoints.
We have that $\tau_1$ and $\tau_2$ are $l$-paths by Lemmas \[lev2\] and \[lev2a\]. Suppose that ${\lambda}=c_z-\cdots-c_w$ is a minimal $l$-path in ${\mathcal{G}}(S_0^k)$ with constants $c_z$ and $c_w$ as the endpoints. Recall that $z,w\in\{y_0,y_1,\ldots,y_k,v_1,\ldots,v_k,s\}$. We may assume that $z$ is to the left of $w$ in the list $y_0,y_1,\ldots,y_k,v_1,\ldots,v_k,s$. Since ${\lambda}$ is minimal, Lemma \[lev3\] implies that ${\lambda}$ does not contain any constants except $c_z$ and $c_w$. There are five cases to consider.
- ${\lambda}=c_{y_i}-\cdots-c_{y_j}$, where $0\leq i<j\leq k$.
- ${\lambda}=c_{y_i}-\cdots-c_{v_j}$, where $0\leq i\leq k$, $1\leq j\leq k$.
- ${\lambda}=c_{y_i}-\cdots-c_s$, where $0\leq i\leq k$.
- ${\lambda}=c_{v_i}-\cdots-c_{v_j}$, where $1\leq i<j\leq k$.
- ${\lambda}=c_{v_i}-\cdots-c_s$, where $1\leq i\leq k$.
Suppose (a) holds, that is, ${\lambda}=c_{y_i}-\cdots-h-c_{y_j}$, $0\leq i<j\leq k$. Since $hc_{y_j}=c_{y_j}h$, either $h=a_j$ or $h=a_{j+1}$ (where $a_{k+1}=b_1$) (since $a_j$ and $a_{j+1}$ are the only generators that have $y_j$ in their image). Suppose $h=a_{j+1}$. Then, by Corollary \[cev2a\], either ${\lambda}=c_{y_i}-\cdots-a_j-a_{j+1}-c_{y_j}$ or ${\lambda}=c_{y_i}-\cdots-a_{j+2}-a_{j+1}-c_{y_j}$ (where $a_{j+2}=b_1$ if $j=k-1$, and $a_{j+2}=b_2$ if $j=k$). In the latter case, $${\lambda}=c_{y_i}-\cdots-a_1-e_1-b_k-\cdots-b_1-a_k-\cdots-a_{j+2}-a_{j+1}-c_{y_j},$$ which is a contradiction since $a_1$ and $e_1$ do not commute. Thus either ${\lambda}=c_{y_i}-\cdots-a_j-c_{y_j}$ or ${\lambda}=c_{y_i}-\cdots-a_j-a_{j+1}-c_{y_j}$. In either case, ${\lambda}$ contains $a_j$, and so $c_{y_i}a_j=c_{y_j}a_j$ (since ${\lambda}$ is an $l$-path). But, by Lemma \[lev2a\], $c_{y_i}a_j=c_{y_{j-1}}$ and $c_{y_j}a_j=c_{y_j}$. Hence $c_{y_{j-1}}=c_{y_j}$, which is a contradiction.
Suppose (b) holds, that is, ${\lambda}=c_{y_i}-g-\cdots-h-c_{v_j}$, $0\leq i\leq k$ and $1\leq j\leq k$. Then $g$ is either $a_i$ or $a_{i+1}$ ($g=a_{i+1}$ if $i=0$) and $h$ is either $b_j$ or $b_{j+1}$ (where $b_{k+1}=e_1$). In any case, ${\lambda}=c_{y_i}-g-\cdots-a_k-b_1-\cdots-h-c_{v_j}$. Suppose $i\geq1$. Then, by Lemma \[lev2a\] and the fact that ${\lambda}$ is an $l$-path, $c_{v_0}=c_{y_i}b_1=c_{v_j}b_1=c_{v_1}$, which is a contradiction. If $i=0$ and $j<k$, then $c_{y_{k-1}}=c_{y_0}a_k=c_{v_j}a_k=c_{y_k}$, which is again a contradiction. If $i=0$ and $j=k$, then $g=a_1$, and so ${\lambda}=\tau_1$.
Suppose (c) holds, that is, ${\lambda}=c_{y_i}-g-\cdots-a_k-b_1-\cdots-b_k-e_1-c_s$, $0\leq i\leq k$, where $g$ is either $a_i$ or $a_{i+1}$ ($g=a_{i+1}$ if $i=0$). If $i>1$, then $c_{v_{i-1}}=c_{y_i}b_i=c_sb_i=c_{v_i}$, which is a contradiction. If $i=0$, then $c_{v_k}=c_{y_0}e_1=c_se_1=c_s$, which is a contradiction. If $i=1$ and $g=a_1$, then ${\lambda}$ is not minimal since $c_{y_1}-a_2$, so $a_1$ can be removed. Finally, if $i=1$ and $g=a_2$, then ${\lambda}=\tau_2$.
Suppose (d) holds, that is, ${\lambda}=c_{v_i}-g-\cdots-h-c_{v_j}$, $1\leq i<j\leq k$, where $g$ is either $b_i$ or $b_{i+1}$ and $h$ is either $b_j$ or $b_{j+1}$ (where $b_{k+1}=e_1$). In any case, ${\lambda}$ contains $b_j$, and so $c_{v_{j-1}}=c_{v_i}b_j=c_{v_j}b_j=c_{v_j}$, which is a contradiction.
Suppose (e) holds, that is, ${\lambda}=c_{v_i}-\cdots-e_1-c_s$, $1\leq i\leq k$. Then $c_{v_k}=c_{v_i}e_1=c_se_1=c_s$, which is a contradiction.
We have exhausted all possibilities and obtained that ${\lambda}$ must be equal to $\tau_1$ or $\tau_2$. The result follows.
\[lev5\] The path $\pi=a_1-\cdots-a_k-b_1-\cdots-b_k-e_1$ is a unique minimal $l$-path in ${\mathcal{G}}(S_0^k)$ with at least one endpoint that is not a constant.
We have that $\pi$ is an $l$-path by Lemmas \[lev2\] and \[lev2a\]. Suppose that ${\lambda}=e-\cdots-f$ is a minimal $l$-path in ${\mathcal{G}}(S_0^k)$ such that $e$ or $f$ is not a constant.
We claim that ${\lambda}$ does not contain any constant $c_z$. By Lemma \[lev3\], there is no constant $c_z$ such that ${\lambda}=e-\cdots-c_z-\cdots-f$ (since otherwise ${\lambda}$ would not be minimal). We may assume that $f$ is not a constant. But then $e$ is not a constant either since otherwise we would have that $ef$ is a constant and $ff=f$ is not a constant. But this is impossible since ${\lambda}$ is an $l$-path, and so $ef=ff$. The claim has been proved.
Thus all elements in ${\lambda}$ are generators from list (\[ecev2\]). We may assume that $e$ is to the left of $f$ (according to the ordering in (\[ecev2\])). Since ${\lambda}$ is an $l$-path, $e=ee=fe$. Hence, by Lemma \[lev2\], $e=a_p$ and $f=b_p$ (for some $p\in\{1,\ldots,k\}$) or $e=b_1$ and $f=e_1$ or $e=a_1$ and $f=e_1$.
Suppose that $e=a_p$ and $f=b_p$ for some $p$. Then, by Corollary \[cev2a\], ${\lambda}=a_p-\cdots-a_k-b_1-\cdots-b_p$. (Note that ${\lambda}=a_p-a_{p-1}-\cdots-a_1-e_1-b_k-\cdots-b_p$ is impossible since $a_1e_1\ne e_1a_1$.) If $p>1$ then, by Lemma \[lev2\], $c_{v_0}=a_pb_1=b_pb_1=c_{v_1}$, which is a contradiction. If $p=1$, then $c_{y_{k-1}}=a_1a_k=b_1b_k=c_{y_k}$, which is again a contradiction.
Suppose that $e=b_1$ and $f=e_1$. Then ${\lambda}=b_1-\cdots-b_k-e_1$, and so $c_{v_{k-1}}=b_1b_k=e_1b_k=c_{v_k}$, which is a contradiction.
Hence we must have $e=a_1$ and $f=e_1$. But then, by Corollary \[cev2a\], ${\lambda}=a_1-\cdots-a_k-b_1-\cdots-b_k-e_1=\pi$. The result follows.
\[teve\] For every even integer $n\geq2$, there is a band $S$ with knit degree $n$.
Let $n=2$. Consider the band $S=\{a,b,c,d\}$ defined by the following Cayley table: $$\begin{tabular}{c|cccc}
& $a$ & $b$ & $c$ & $d$ \\\hline
$a$ & $a$ & $b$ & $c$ & $d$ \\
$b$ & $b$ & $b$ & $b$ & $b$ \\
$c$ & $a$ & $b$ & $c$ & $d$ \\
$d$ & $d$ & $d$ & $d$ & $d$ \\
\end{tabular}$$ It is easy to see that the center of $S$ is empty and $a-b-c$ is a shortest $l$-path in ${\mathcal{G}}(S)$. Thus $\operatorname{kd}(S)=2$.
Let $n=2k$ where $k\geq2$. Consider the semigroup $S_0^k$ defined by (\[edco1\]). Then, by Corollary \[cev2\], $S_0^k$ is a band. The paths $\tau_1$, $\tau_2$, and $\pi$ from Lemmas \[lev4\] and \[lev5\] are the only minimal $l$-paths in ${\mathcal{G}}(S_0^k)$. Since $\tau_1$ has length $2k+1=n+1$, $\tau_2$ has length $2k+2=n+2$, and $\pi$ has length $2k=n$, it follows that $\operatorname{kd}(S_0^k)=n$.
The Odd Case {#ssodd}
------------
Suppose $n=2k+1\geq5$ is odd. We will obtain a band $S$ of knit degree $n$ by slightly modifying the construction of the band $S_0^k$ from Definition \[dco1\]. Recall that $S_0^k$ has knit degree $2k$ (see the proof of Theorem \[teve\]). We will obtain a band of knit degree $n=2k+1$ by simply removing transformations $e_1$ and $c_s$ from $S_0^k$.
\[dco2\] [Let $k\geq2$ be an integer. Consider the following subset of the semigroup $S^0_k$ from Definition \[dco1\]: $$\label{edco21}
S^1_k=S^0_k-\{e_1,c_s\}=\{a_1,\ldots,a_k,b_1,\ldots,b_k,c_{y_0},c_{y_1},\ldots,c_{y_k},c_{v_1},\ldots,c_{v_k}\}.$$ By Lemmas \[lev2\] and \[lev2a\], $S^1_k$ is a subsemigroup of $S^0_k$. ]{}
\[rdco2\] [Note that $r$ and $s$, which still occur in the domain (but not the image) of each element of $S_1^k$, are now superfluous. We can remove them from the domain of each element of $S^1_k$ and view $S^1_k$ as a semigroup of transformations on the set $$X=\{y_0,y_1,\ldots,y_k=v_0,v_1,\ldots,v_k,x_1,\ldots,x_k,u_1,\ldots,u_k\}.$$ ]{}
It is clear from the definition of $S_1^k$ that the multiplication table for $S_1^k$ is the multiplication table for $S_0^k$ (see Lemmas \[lev2\] and \[lev2a\]) with the rows and columns $e_1$ and $c_s$ removed. This new multiplication table is given by Lemmas \[lev2\] and \[lev2a\] if we ignore the multiplications involving $e_1$ or $c_s$. Therefore, the following lemma follows immediately from Corollary \[cev2\] and Lemmas \[lev4\] and \[lev5\].
\[levnew1\] Let $S_1^k$ be the semigroups defined by [(\[edco21\])]{}. Then $S_1^k$ is a band and $\tau=c_{y_0}-a_1-\cdots-a_k-b_1-\cdots-b_k-c_{v_k}$ is the only minimal $l$-path in ${\mathcal{G}}(S_1^k)$.
\[todd\] For every odd integer $n\geq5$, there is a band $S$ of knit degree $n$.
Let $n=2k+1$ where $k\geq2$. Consider the semigroup $S_1^k$ defined by (\[edco21\]). Then, by Lemma \[levnew1\], $S_1^k$ is a band and $\tau=c_{y_0}-a_1-\cdots-a_k-b_1-\cdots-b_k-c_{v_k}$ is the only minimal $l$-path in ${\mathcal{G}}(S_1^k)$. Since $\tau$ has length $2k+1=n$, it follows that $\operatorname{kd}(S_1^k)=n$.
The case $n=3$ remains unresolved. [**Open Question.**]{} Is there a semigroup of knit degree $3$?
Commuting Graphs with Arbitrary Diameters {#sald}
=========================================
In Section \[stx\], we showed that, except for some special cases, the commuting graph of any ideal of the semigroup $T(X)$ has diameter $5$. In this section, we use the constructions of Section \[smlp\] to show that there are semigroups whose commuting graphs have any prescribed diameter. We note that the situation is (might be) quite different in group theory: it has been conjectured that there is an upper bound for the diameters of the connected commuting graphs of finite non-abelian groups [@IrJa08 Conjecture 2.2].
\[tald\] For every $n\geq2$, there is a semigroup $S$ such that the diameter of ${\mathcal{G}}(S)$ is $n$.
Let $n\in\{2,3,4\}$. The commuting graph of the band $S$ defined by the Cayley table in the proof of Theorem \[teve\] is the cycle $a-b-c-d-a$. Thus the diameter of ${\mathcal{G}}(S)$ is $2$. Consider the semigroup $S$ defined by the following table: $$\begin{tabular}{r|rrrr}
& a & b & c & d\\ \hline
a & a & a & a & a \\
b & a & b & c & c \\
c & c & c & c & c \\
d & c & d & c & c
\end{tabular} \hspace{.5cm}$$ Note that $Z(S)=\emptyset$ and ${\mathcal{G}}(S)$ is the chain $a-b-c-d$. Thus the diameter of ${\mathcal{G}}(S)$ is $3$. The diameter of ${\mathcal{G}}(J_4)$ is $4$ (where $J_4$ is an ideal of $T(X)$ with $|X|=5$).
Let $n\geq5$. Suppose $n$ is even. Then $n=2k+2$ for some $k\geq2$. Consider the band $S_0^k$ from Definition \[dco1\]. Since $c_{y_0}$ and $a_1$ are the only elements of $S_0^k$ whose image contains $y_0$, they are the only elements of $S_0^k$ commuting with $c_{y_0}$ (see Lemma \[lcon\]). Similarly, $e_1$ and $c_s$ are the only elements commuting with $c_s$. Therefore, it follows from Corollary \[cev2a\] that $c_{y_0}-a_1-\cdots-a_k-b_1-\cdots-b_k-e_1-c_s$ is a shortest path in ${\mathcal{G}}(S_0^k)$ from $c_{y_0}$ to $c_s$, that is, the distance between $c_{y_0}$ and $c_s$ is $2k+2=n$. Since $a_1-\cdots-a_k-b_1-\cdots-b_k-e_1$ is a path in ${\mathcal{G}}(S_0^k)$, $c_{y_i}a_i=a_ic_{y_i}$ and $c_{v_i}b_i=b_ic_{v_i}$ ($1\leq i\leq k$), it follows that the distance between any two vertices of ${\mathcal{G}}(S_0^k)$ is at most $2k+2$. Hence the diameter of ${\mathcal{G}}(S_0^k)$ is $n$.
Suppose $n$ is odd. Then $n=2k+1$ for some $k\geq2$. Consider the band $S_1^k$ from Definition \[dco2\]. Then $c_{y_0}-a_1-\cdots-a_k-b_1-\cdots-b_k-c_{v_k}$ is a shortest path in ${\mathcal{G}}(S_1^k)$ from $c_{y_0}$ to $c_{v_k}$, that is, the distance between $c_{y_0}$ and $c_{v_k}$ is $2k+1=n$. As for $S_0^k$, we have $c_{y_i}a_i=a_ic_{y_i}$ and $c_{v_i}b_i=b_ic_{v_i}$ ($1\leq i\leq k$). Thus the distance between any two vertices of $S_1^k$ is at most $2k+1$, and so the diameter of ${\mathcal{G}}(S_1^k)$ is $n$.
Schein’s Conjecture {#ssch}
===================
The results obtained in Section \[smlp\] enable us to settle a conjecture formulated by B.M. Schein in 1978 [@Sc78 p. 12]. Schein stated his conjecture in the context of the attempts to characterize the $r$-semisimple bands.
A right congruence $\tau$ on a semigroup S is said to be modular if there exists an element $e\in S$ such that $(ex)\tau x$ for all $x\in S$. The radical $R_r$ on a band $S$ is the intersection of all maximal modular right congruences on $S$ [@Oe66]. A band $S$ is called *$r$-semisimple* if its radical $R_r$ is the identity relation on $S$.
In 1969, B.D. Arendt announced a characterization of $r$-semisimple bands [@Ar69 Theorem 18]. In 1978, B.M Schein pointed out that Arendt’s characterization is incorrect and proved [@Sc78 p. 2] that a band $S$ is $r$-semisimple if and only if it satisfies infinitely many quasi-identities: (1) and $(A_n)$ for all integers $n\geq1$, where $$\begin{aligned}
(1)\,\,\,\,\,&zx=zy{\Rightarrow}xy=yx,\notag\\
(A_n)\,\,\,\,\,&x_1x_2=x_2x_1\wedge x_2x_3=x_3x_2\wedge\ldots\wedge x_{n-1}x_n=x_nx_{n-1}\wedge\notag\\
&\wedge x_1x_1=x_nx_1\wedge x_1x_2=x_nx_2\wedge\ldots\wedge x_1x_n=x_nx_n {\Rightarrow}x_1=x_n.\notag\end{aligned}$$ Schein observed that $(A_1)$ and $(A_2)$ are true in every band, that $(A_3)$ easily follows from (1), and that Arendt’s characterization of $r$-semisimple bands is equivalent to (1). He used the last observation to show that Arendt’s characterization is incorrect by providing an example of a band $T$ for which (1) holds but $(A_4)$ does not. We note that Schein’s example is incorrect since the Cayley table in [@Sc78 p. 10], which is supposed to define $T$, does not define a semigroup because the operation is not associative: $(4*1)*1=10\neq8=4*(1*1)$. However, Schein was right that it is not true that condition (1) implies $(A_n)$ for all $n$. The semigroup $S_0^2$ (see Table 2) satisfies (1) but it does not satisfy $(A_5)$ since $a_1-a_2-b_1-b_2-e_1$ is an $l$-path (so the premise of $(A_5)$ holds) but $a_1\ne e_1$.
At the end of the paper, Schein formulates his conjecture [@Sc78 p. 12]: [**Schein’s Conjecture.**]{} For every $n>1$, $(A_n)$ does not imply $(A_{n+1})$. The reason that Section \[smlp\] enables us to settle Schein’s conjecture is the following lemma.
\[lscon\] Let $n\geq1$ and let $S$ be a band with no central elements. Then $S$ satisfies $(A_n)$ if and only if ${\mathcal{G}}(S)$ has no $l$-path of length $<n$.
First note that $(A_n)$ can be expressed as: for all $x_1,\ldots,x_n\in S$, $$\label{elscon}
x_1-\cdots-x_n\mbox{ and }x_1x_i=x_nx_i\mbox{ $(1\leq i\leq n)$}{\Rightarrow}x_1=x_n.$$ (Here, we allow $x-x$ and do not require that $x_1,\ldots,x_n$ be distinct.)
Assume $S$ satisfies $(A_n)$. Suppose to the contrary that ${\mathcal{G}}(S)$ has an $l$-path ${\lambda}=x_1-\cdots-x_k$ of length $<n$, that is, $k\leq n$. Then $x_1-\cdots-x_k-x_{k+1}-\cdots-x_n$, where $x_i=x_k$ for every $i\in\{k+1,\ldots,n\}$, and so $x_1=x_n=x_k$ by (\[elscon\]). This is a contradiction since ${\lambda}$ is a path.
Conversely, suppose that ${\mathcal{G}}(S)$ has no $l$-path of length $<n$. Let $x_1-\cdots-x_n$ and $x_1x_i=x_nx_i$ ($1\leq i\leq n$). Suppose to the contrary that $x_1\ne x_n$. If there are $i$ and $j$ such that $1\leq i<j\leq n$ and $x_i=x_j$, we can replace $x_1-\cdots-x_i-\cdots-x_j-\cdots-x_n$ with $x_1-\cdots-x_i-x_{j+1}-\cdots-x_n$. Therefore, we can assume that $x_1,\ldots,x_n$ are pairwise distinct. Recall that $S$ has no central elements, so all $x_i$ are vertices in ${\mathcal{G}}(S)$. Thus $x_1-\cdots-x_n$ is an $l$-path in ${\mathcal{G}}(S)$ of length $n-1$, which is a contradiction.
First, Schein’s conjecture is false for $n=3$.
\[pscon1\] $(A_3){\Rightarrow}(A_4)$.
Suppose a band $S$ satisfies $(A_3)$, that is, $$\label{essch1}
x_1x_2=x_2x_1 \wedge x_2x_3=x_3x_2 \wedge x_1x_1=x_3x_1 \wedge x_1x_2=x_3x_2 \wedge x_1x_3=x_3x_3 {\Rightarrow}x_1=x_3.$$ To prove that $S$ satisfies $(A_4)$, suppose that $$y_1y_2=y_2y_1 \wedge y_2y_3=y_3y_2 \wedge y_3y_4=y_4y_3 \wedge y_1y_1=y_4y_1 \wedge y_1y_2=y_4y_2 \wedge y_1y_3=y_4y_3 \wedge y_1y_4=y_4y_4.$$ Take $x_1=y_1$, $x_2=y_2y_3$, and $x_3=y_4$. Then $x_1,x_2,x_3$ satisfy the premise of (\[essch1\]): $$\begin{aligned}
x_1x_2&=y_1y_2y_3=y_1y_3y_2=y_4y_3y_2=y_3y_4y_2=y_3y_1y_2=y_3y_2y_1=y_2y_3y_1=x_2x_1,\notag\\
x_2x_3&=y_2y_3y_4=y_2y_4y_3=y_2y_1y_3=y_1y_2y_3=y_4y_2y_3=x_3x_2,\notag\\
x_1x_1&=y_1y_1=y_4y_1=x_3x_1,\,x_1x_2=y_1y_2y_3=y_4y_2y_3=x_3x_2,\,x_1x_3=y_1y_4=y_4y_4=x_3x_3.\notag\end{aligned}$$ Thus, by (\[essch1\]), $y_1=x_1=x_3=y_4$, and so $(A_4)$ holds.
Second, Schein’s conjecture is true for $n\ne3$.
\[pscon2\] If $n>1$ and $n\ne3$, then $(A_n)$ does not imply $(A_{n+1})$.
Consider the band $S=\{e,f,0\}$, where $0$ is the zero, $ef=f$, and $fe=e$. Then $e-0-f$, $ee=fe$, $e0=f0$, $ef=ff$, and $e\ne f$. Thus $S$ does not satisfy $(A_3)$. But $S$ satisfies $(A_2)$ since $(A_2)$ is true in every band. Hence $(A_2)$ does not imply $(A_3)$.
Let $n\geq4$. Then, by Theorems \[teve\] and \[todd\] and their proofs, the band $S$ constructed in Definition \[dco1\] (if $n$ is even) or Definition \[dco2\] (if $n$ is odd) has knit degree $n$. By Lemmas \[lev2\] and \[lev2a\], $S$ has no central elements. Since $\operatorname{kd}(S)=n$, there is an $l$-path in ${\mathcal{G}}(S)$ of length $n$ and there is no $l$-path in ${\mathcal{G}}(S)$ of length $<n$. Hence, by Lemma \[lscon\], $S$ satisfies $(A_n)$ and $S$ does not satisfy $(A_{n+1})$. Thus $(A_n)$ does not imply $(A_{n+1})$.
Problems {#spro}
========
We finish this paper with a list of some problems concerning commuting graphs of semigroups.
- Is there a semigroup with knit degree 3? Our guess is that such a semigroup does not exist.
- Classify the semigroups whose commuting graph is eulerian (proposed by M. Volkov). The same problem for hamiltonian and planar graphs.
- Classify the commuting graphs of semigroups.
- Is it true that for all natural numbers $n\geq 3$, there is a semigroup $S$ such that the clique number (girth, chromatic number) of ${\mathcal{G}}(S)$ is $n$?
- Classify the semigroups $S$ such that the clique and chromatic numbers of ${\mathcal{G}}(S)$ coincide.
- Calculate the clique and chromatic numbers of the commuting graphs of $T(X)$ and $\operatorname{End}(V)$, where $X$ is a finite set and $V$ is a finite-dimensional vector space over a finite field.
- Let ${\mathcal{G}}(S)$ be the commuting graph of a finite non-commutative semigroup $S$. An *${\mathit{rl}}$-path* is a path $a_1-\cdots-a_m$ in ${\mathcal{G}}(S)$ such that $a_1\ne a_m$ and $a_1a_ia_1=a_ma_ia_m$ for all $i=1,\ldots,m$. For ${\mathit{rl}}$-paths, prove the results analogous to the results for $l$-paths contained in this paper.
- Find classes of finite non-commutative semigroups such that if $S$ and $T$ are two semigroups in that class and ${\mathcal{G}}(S)\cong {\mathcal{G}}(T)$, then $S\cong T$.
Acknowledgments
===============
We are pleased to acknowledge the assistance of the automated deduction tool <span style="font-variant:small-caps;">Prover9</span> and the finite model builder <span style="font-variant:small-caps;">Mace4</span>, both developed by W. McCune [@McCune]. We also thank the developers of GAP [@Scel92], L.H. Soicher for GRAPE [@So06], and Aedan Pope and Kyle Pula for their suggestions after carefully reading the manuscript.
The first author was partially supported by FCT and FEDER, Project POCTI-ISFL-1-143 of Centro de Algebra da Universidade de Lisboa, by FCT and PIDDAC through the project PTDC/MAT/69514/2006, by PTDC/MAT/69514/2006 Semigroups and Languages, and byPTDC/MAT/101993/2008 Computations in groups and semigroups.
[99]{}
J. Araújo and J. Konieczny, Automorphism groups of centralizers of idempotents, *J. Algebra* [**269**]{} (2003), 227–239.
J. Araújo and J. Konieczny, Semigroups of transformations preserving an equivalence relation and a cross-section, *Comm. Algebra* [**32**]{} (2004), 1917–1935.
B.D. Arendt, Semisimple bands, *Trans. Amer. Math. Soc.* [**143**]{} (1969), 133–143.
C. Bates, D. Bundy, S. Perkins, and P. Rowley, Commuting involution graphs for symmetric groups, *J. Algebra* [**266**]{} (2003), 133–153.
E.A. Bertram, Some applications of graph theory to finite groups, *Discrete Math.* [**44**]{} (1983), 31–43.
D. Bundy, The connectivity of commuting graphs, *J. Combin. Theory Ser. A* [**113**]{} (2006), 995–1007.
A.H. Clifford and G.B. Preston, *The algebraic theory of semigroups*, Vol. I, Mathematical Surveys and Monographs, No. 7, American Mathematical Society, Providence, RI, 1964.
The GAP Group, *GAP – Groups, Algorithms, and Programming*, Version $4.4.12$, 2008,`http://www.gap-system.org`.
A. Iranmanesh and A. Jafarzadeh, On the commuting graph associated with the symmetric and alternating groups, *J. Algebra Appl.* [**7**]{} (2008), 129–146.
J. Konieczny, Semigroups of transformations commuting with idempotents, *Algebra Colloq.* [**9**]{} (2002), 121–134.
W. McCune, <span style="font-variant:small-caps;">Prover9</span> and <span style="font-variant:small-caps;">Mace4</span>, version LADR–2009–11A, <http://www.cs.unm.edu/~mccune/prover9/>
R.H. Oehmke, On maximal congruences and finite semisimple semigroups, *Trans. Amer. Math. Soc.* [**125**]{} (1966), 223–237.
A.S. Rapinchuk and Y. Segev, Valuation-like maps and the congruence subgroup property, *Invent. Math.* [**144**]{} (2001), 571–607.
A.S. Rapinchuk, Y. Segev, and G.M. Seitz, Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable, *J. Amer. Math. Soc.* [**15**]{} (2002), 929–978.
B.M. Schein, On semisimple bands, *Semigroup Forum* [**16**]{} (1978), 1–12.
Y. Segev, The commuting graph of minimal nonsolvable groups, *Geom. Dedicata* [**88**]{} (2001), 55–66.
L.H. Soicher, The GRAPE package for GAP, Version 4.3, 2006,`http://www.maths.qmul.ac.uk/~leonard/grape/`.
[^1]: Corresponding author.
|
---
abstract: 'High statistics Dalitz-plot distribution of $\eta \rightarrow \pi^+\pi^-\pi^0$ decay obtained recently by KLOE collaboration [@kloe] is fitted to the results of corresponding theoretical calculations in Chiral Perturbation Theory (ChPT) with unitarity corrections taken into account. The quark mass ratio $Q = \sqrt{(m^2_s - (m_d + m_u)^2/4)/(m^2_d - m^2_u)} $ can be otained from this analysis. We get $Q= 22.8\pm 0.4$ which differs from the value $Q_{DT} = 24.2$ that follows from Dashen’s theorem and agrees with recently calculated electromagnetic kaon mass difference.'
author:
- 'B.V. Martemyanov'
- 'V.S. Sopov'
title: 'Light quark mass ratio from Dalitz plot of $\eta \rightarrow \pi^+\pi^-\pi^0$ decay'
---
The possibility to extract light quark mass difference from $\eta \rightarrow \pi^+\pi^-\pi^0$ decay is known for a long time [@L]. In ChPT the decay width $\Gamma$ depends on quark mass ratios and theoretically calculable factor $\bar{\Gamma}$ [@GL]:
$$\Gamma = \left(\frac{Q_{DT}}{Q}\right)^4 {\bar \Gamma}~,
\label{gamma1}$$
where $$Q^{-2} = \frac{m^2_d - m^2_u}{m^2_s - {\hat m}^2}~,~~~{\hat m}=
\frac{m_d + m_u}{2}~,
\label{gamma2}$$ $m_u , m_d , m_s$ - are up, down and strange quark masses, $$Q^{-2}_{DT} = \frac{\left(\left( m^2_{K^0}-m^2_{K^+}\right)
- \left( m^2_{\pi^0}-m^2_{\pi^+}\right)\right) m^2_{\pi^0}}
{\left( m^2_K - m^2_{\pi^0}\right) m^2_K} = (24.2)^{-2}~,
\label{gamma3}$$ with $m^2_K =\left( m^2_{K^+}+m^2_{K^0}-m^2_{\pi^+}+m^2_{\pi^0}\right)/2$.
Note that $Q_{DT} = Q$ if electromagnetic mass differences of kaons and pions are equal to each other as Dashen’s theorem states [@Dashen]. Experimental (Particle Data Group) value $\Gamma = 291\pm 21 ~{\rm eV}~$ [@PDG] is far from one-loop ChPT value ${\bar \Gamma} = 167 \pm 50 ~{\rm eV}~$ [@GL] and from the values ${\bar \Gamma} = 209 \pm 20 ~{\rm eV}~$ [@Kambor] and ${\bar \Gamma} = 219 \pm 22 ~{\rm eV}~$ [@Anis] obtained with higher order corrections taken into account by dispersion method. In [@Kambor] the subtraction polinomial was taken from the decomposition of one-loop order amplitude and had therefore uncertainties connected to higher orders ChPT corrections. These uncertainties were further fixed [@MS] by fitting the experimental data [@gormley] on Dalitz-plot distribution in the decay considered: ${\bar \Gamma} = 213^{+3}_{-12} ~{\rm eV}~$. There was a conjecture in [@MS] that new experimental data on Dalitz-plot distribution will give slightly different value of ${\bar \Gamma}$. Now these new experimental data (contradicting to the old ones [@gormley]) are available [@kloe]. In what follows we will use them to get new value of ${\bar \Gamma}$ and new value of quark mass ratio $Q$, as a consequence.
We use the method of work [@MS] and remind it here for completeness. In order to simulate the experimental Dalitz-plot distribution we take it in a form $$1+ay+by^2+cy^3+dx^2$$ with $a=-1.075\pm 0.008$, $b=0.118\pm 0.009$, $c=0.13\pm 0.02$, $d=0.049\pm 0.008$ [@kloe] and $y,x$ defined in a standart way $$y=\frac{3T_0}{Q}-1~,~x=\frac{\sqrt{3}}{Q}\left(T_+-T_-\right)~,~
Q=T_++T_-+T_0~,$$ $T_+,T_-,T_0$ are the kinetic energies of pions in the rest frame $\eta \rightarrow \pi^+\pi^-\pi^0$ decay. We divide the Dalitz plot in $10\times 10$ bins ($x\times y$) that have equal number of events for the distribution considered. Then the number of events in each bin ($n$) is simulated by Gaussian distribution with variance equal to n. We used $n= 10 000$ to get the full statistics $N = 100 n =
1 000 000$ like in the experiment [@kloe]. From theoretical point of view the amplitude of $\eta \rightarrow \pi^+\pi^-\pi^0$ decay have an approximate solution from Eq.(5.28) of [@Kambor]. It contains the subtraction polinomial $$P(s) = \alpha +\beta s_a + \gamma s_a^2 + \delta (s_b-s_c)^2~,$$ where $s_a, s_b$ and $s_c$ are invariant masses squared of $\pi^+\pi^-,
\pi^+\pi^0$ and $\pi^-\pi^0$ pairs, respectively. For the values of parameters $\alpha, \beta, \gamma$ and $\delta$ within the regions $$\begin{aligned}
&\alpha = -1.28 \pm 0.14,~&\beta = 21.81 \pm 1.52 ~{\rm GeV}^{-2}\nonumber \\
&\gamma = 4.09 \pm 3.18 ~{\rm GeV}^{-4},~&\delta = 4.19 \pm 1.08~{\rm GeV}^{-4}
\label{allowed} \end{aligned}$$ (the case of zero subtraction points [@Kambor]) the “Minuit” fit of above simulated experimental Dalitz-plot distribution has terminated on the values $$\begin{aligned}
&\alpha_0 = -1.14,~&\beta_0 = 23.33 ~{\rm GeV}^{-2}\nonumber \\
&\gamma_0 = 1.03q
~{\rm GeV}^{-4},~&\delta_0 = 5.27 ~{\rm GeV}^{-4} \end{aligned}$$ with $\chi^2/N{d.o.f.} = 152/(100-4)$.
Three from four parameters are at the boundary of allowed region (\[allowed\]). This probably means that the guess [@Kambor] of the size of allowed region should be changed. Equally possible is the fit with the scaled values of parameters $\alpha, \beta, \gamma$ and $\delta$ because the normalization factor of the amplitude is not defined by the Dalitz-plot distribution. In our case the scaling of parameters $\alpha, \beta, \gamma$ and $\delta$ puts them outside the allowed region (\[allowed\]) and no freedom in the scaling (no error in ${\bar \Gamma}$) is possible. This way we get ${\bar \Gamma} = 229~ ~{\rm eV}$ what corresponds according to eqs. (\[gamma1\])-(\[gamma3\]) to the light quark mass ratio $Q = 22.8\pm 0.4$. The errors here are due to the errors in the experimental value of the width $\Gamma$. So, we conclude, high statistics Dalitz-plot distribution gives the value of light quark mass ratio $Q$ slightly lower than that from the assumption of equality of kaon and pion electromagnetic mass differences ($Q_{DT} = 24.2$). This is in agreement with calculations of electromagnetic mass differences for pions and kaons [@Donoghue; @Bijnens] which find large violations to Dashen’s theorem ($Q = 22.0\pm 0.6$). Our result agrees also very well with that of works [@Kambor] and [@Anis], where the values $Q=22.4\pm 0.9$ and $Q= 22.7\pm 0.8$ were obtained, correspondingly.
This work was partially supported by RFBR grant No.02-02-16957.
[99]{} B. Di Micco [*et al*]{} \[KLOE collaboration\], Light Meson Spectroscopy with the Kloe Experiment, hep-ex/0410072. H.Leutwyler, hep-ph/9609467. J.Gasser and H.Leutwyler, Nucl.Phys. [**B250**]{}, 539 (1985). R. Dashen, Phys. Rev.[**183**]{}, 1245 (1969) S. Eidelman [*et al.*]{}, Phys. Lett. [**B592**]{}, 1 (2004). J.Kambor, C.Wiesendanger and D.Wyler, Nucl.Phys. [**B465**]{}, 215 (1996). A.V.Anisovich and H.Leutwyler, Phys.Lett. [**B375**]{}, 335 (1996). B.V. Martemyanov and V.S. Sopov, Phys.Atom.Nucl.[**67**]{}, 424 (2004). M.Gormley et al., Phys.Rev. [**D2**]{}, 501 (1970). J. Donoghue and A. Perez, Phys. Rev. [**D55**]{}, 7075 (1997). J. Bijnens and J. Prades, Nucl. Phys. [**B490**]{},239 (1997).
|
---
abstract: 'When using an electron microscope for imaging of particles embedded in vitreous ice, the recorded image, or micrograph, is a significantly degraded version of the tomographic projection of the sample. Apart from noise, the image is affected by the optical configuration of the microscope. This transformation is typically modeled as a convolution with a point spread function. The Fourier transform of this function, known as the contrast transfer function (CTF), is oscillatory, attenuating and amplifying different frequency bands, and sometimes flipping their signs. High-resolution reconstruction requires this CTF to be accounted for, but as its form depends on experimental parameters, it must first be estimated from the micrograph. We present a new method for CTF estimation based on multitaper methods, which reduces bias and variance in the estimate. We also use known properties of the CTF and the background noise power spectrum to further reduce the variance through background subtraction and steerable basis projection. We show that the resulting power spectrum estimates better capture the zero-crossings of the CTF and yield accurate CTF estimates on several experimental micrographs.'
address:
- 'The Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ'
- 'Center for Computational Biology, Flatiron Institute, New York, NY'
- 'Department of Mathematics, Princeton University, Princeton, NJ'
author:
- Ayelet Heimowitz
- Joakim Andén
- Amit Singer
bibliography:
- 'ctf.bib'
title: 'Reducing Bias and Variance for CTF Estimation in Single Particle Cryo-EM'
---
contrast transfer function, cryo-electron microscopy, linear programming, multitaper estimator, spectral estimation, steerable basis expansion
Introduction
============
In recent years, single particle cryo-electron microscopy (cryo-EM) has emerged as a leading tool for resolving the 3D structure of macromolecules from multiple 2D projections of a specimen [@cheng2017cryo]. In this technique, multiple copies of a particle are embedded in vitreous ice and imaged in an electron microscope. This yields a set of micrographs, each containing several 2D particle projections.
The micrograph does not contain clean particle projections but is contaminated by several factors, including noise, ice aggregates and carbon film projection. The noise stems from an inherent limitation on the number of imaging electrons that can be applied to the specimen. The interference from carbon film and ice aggregates are due to the particular sample preparation techniques used.
The 2D projections in the micrograph are also distorted by convolution with a point spread function. This point spread function is due to the electron microscope configuration. It attenuates certain frequencies and flips the sign of certain frequency bands. A 3D density map reconstructed from distorted projections yields an unreliable representation of the particle [@frank2996book]. It is therefore important to estimate the point spread function and account for it during reconstruction.
To estimate these parameters, it is convenient to consider the Fourier transform of the point spread function, known as the *contrast transfer function* (CTF). This is due to two factors. First, the CTF has a simple expression in the polar coordinates of the spatial frequency. Second, its effect is directly visible in the frequency domain where the CTF acts as a pointwise multiplication rather than a convolution [@erickson1971ctf].
CTF estimation is one of the first steps in the single particle cryo-EM pipeline. Indeed, accounting for the CTF is needed in a variety of tasks, such as particle picking [@heimowitz2018apple], denoising [@bhamre2016denoising], class averaging [@scheres2012relion], ab initio reconstruction [@punjani2017cryosparc], refinement [@scheres2012relion; @punjani2017cryosparc; @tang2007eman2; @grant2018cistem] and heterogeneity analysis [@scheres2012relion].
The CTF is typically modeled as a sine function whose argument depends on the spatial frequency and several parameters of the objective lens of the microscope [@frank2996book]. The parameters we focus on in this paper are the defocus and astigmatism of the objective lens as these are unknown and must be estimated from the data. Additionally, the CTF is multiplied by a damping envelope, which suppresses the information in high frequencies [@sorzano2007damping].
When estimating the CTF parameters, it is common to first estimate the power spectrum of the micrograph. The observed micrograph image is typically modeled as a CTF-dependent term plus a noise term unaffected by CTF. The first term corresponds to a noiseless micrograph, that is, the tomographic projection of the sample filtered by the CTF @zhu19977gaussian. Modeling the unfiltered and filtered micrographs as 2D random fields, we find that their power spectra are closely related: the latter equals the former multiplied by the squared CTF. This multiplication induces concentric rings, known as Thon rings [@thon1971book], in the power spectrum of the filtered micrograph (see Fig. \[fig:astigmatism\]). Estimating the CTF therefore reduces to fitting the parameters of the CTF to the estimated power spectrum.
The vast majority of CTF estimation methods use a variant of the periodogram when estimating the power spectrum of the micrograph. This is due to its speed and simplicity. Unfortunately, the periodogram produces a biased and inconsistent estimate of the micrograph power spectrum.
Beyond these issues with the spectral estimators, fitting the CTF model to the estimated power spectrum is complicated by factors such as the high levels of noise present in the micrograph, [coincidence loss at the detector and more. The expected power spectrum equals the power spectrum of the clean, filtered micrograph plus a background term caused by the aforementioned complications. The background masks the true oscillations of the power spectrum of the particle projection. ]{} It is therefore important to estimate and remove the background from the estimated power spectrum [@zhu19977gaussian; @rohou2015ctffind4; @mindell2003ctffind3; @kai2016gctf].
Assuming that the micrograph power spectrum and the background were both estimated perfectly, the background-subtracted power spectrum equals the power spectrum of the filtered, clean micrograph. One way to estimate the CTF parameters is then to maximize the correlation of the background-subtracted power spectrum estimate and a squared CTF (or some monotonic function thereof) [@rohou2015ctffind4; @mindell2003ctffind3; @kai2016gctf; @tani1996correlation]. Optimizing the correlation then provides an estimate of the defocus and astigmatism. Another approach identifies a single ring in the estimated power spectrum and uses it to derive a closed-form solution of the CTF parameters @yan2017single_ring. In order to formulate this solution, all prior knowledge regarding the spherical aberration must be ignored.
In this paper, we present ASPIRE-CTF, which is a new method for CTF estimation, available as part of the ASPIRE package.[^1] We first estimate the power spectrum using a multitaper estimator [@babadi2014multitaper], further reducing the variance by averaging estimates from multiple regions of the micrograph. Using this estimated power spectrum, we estimate the background noise spectrum using linear programming (LP). Instead of using an approximate background model, our scheme ensures that the background-subtracted power spectrum estimate is non-negative and convex. We also show that the CTF is contained in the span of a small number of steerable basis functions. Thus, we further reduce the variance in our power spectrum estimate by projecting onto this span.
Given the power spectrum estimate, we provide two solutions for estimating the CTF parameters. Our first solution is similar to [@rohou2015ctffind4; @mindell2003ctffind3; @kai2016gctf; @tani1996correlation], where CTF parameters are estimated by maximizing the correlation of the square root of the power spectrum estimate with the absolute value of simulated CTFs. The second solution uses the spatial frequencies of several zero-crossings. Since we expect these zero-crossings to coincide with those of the squared CTF, we use them to define an overdetermined system of equations over the CTF parameters that we then solve. We note that, while our first solution is more robust, our second solution is faster to compute.
Our method is experimentally verified in Section \[sec:experiment\]. This is done via a comparison of the defocus estimates with that of @rohou2015ctffind4 [@kai2016gctf] on several datasets from the CTF challenge [@marabini2015challenge]. We show that our power spectrum estimation method is usually in agreement with one of the state-of-the-art methods @rohou2015ctffind4 [@kai2016gctf].
The main contribution of this paper, appearing in Sections \[subsubsec:variance1\]-\[sec:bk\], is our method for estimating the power spectrum of a micrograph. We reduce the variability of the power spectrum estimate, and are therefore the first to obtain an estimate where several zero-crossing rings of the CTF are easily recovered without additional assumptions.
We present the pipeline of our method in Fig. \[fig:pipeline\]. For each step of our suggested framework, we refer the reader to the appropriate section of the paper.
Notation {#notation .unnumbered}
--------
Given a 2D stationary random field ${\mathbf{x}}$ defined over ${\mathbb{Z}}^2$, we denote its autocovariance function by $R_{\mathbf{x}}$. The Fourier transform of $R_{\mathbf{x}}$ is known as the *power spectrum* of ${\mathbf{x}}$ and is given by $${S}_{\mathbf{x}}({\mathbf{g}}) = \sum_{n_1=-\infty}^\infty \sum_{n_2=-\infty}^\infty R_{\mathbf{x}}[m, n] \, {\mathrm{e}}^{{\mathrm{j}}2\pi (g_1 n_1 + g_2 n_2)},$$ for ${\mathbf{g}}= (g_1, g_2) \in [-1/2, 1/2]^2$ and ${\mathrm{j}}= \sqrt{-1}$. We denote magnitude of the spatial frequency vector ${\mathbf{g}}$ by ${{r}}$ and its counterclockwise angle with the positive x-axis by ${{\alpha}}$.
Materials and Methods
=====================
Problem formulation {#sec:formation}
-------------------
In the sample preparation stage of the single particle cryo-EM pipeline, many copies of a particle are embedded in vitreous ice. The imaging process uses an electron microscope to obtain a micrograph containing 2D projections of each instance. Under the weak-phase object approximation, we may describe this process by the linear model [@frank2996book; @thon1971book; @mindell2003ctffind3] $$\label{equ:image_formation}
{\mathbf{y}}= h_{\phi} * {\mathbf{x}}+ {\mathbf{e}},$$ where the clean tomographic projection ${\mathbf{x}}$ and the additive noise ${\mathbf{e}}$ are modeled as 2D stationary random fields [@frank2996book]. Since convolution preserves stationarity, the observed micrograph ${\mathbf{y}}$ is also a stationary random field. In this model, the clean projection ${\mathbf{x}}$ is convolved with the point spread function of the microscope $h_\phi$ which depends on a parameter vector $\phi$. We will at times denote this clean, but filtered, micrograph by ${\mathbf{z}}= h_\phi * {\mathbf{x}}$.
The CTF is the Fourier transform of the point spread function ${H_{\phi}}$ and may be modeled by [@rohou2015ctffind4] $${H_{\phi}}( {\mathbf{g}}) = - \sin( {\chi_{\phi}}( {\mathbf{g}}) ),
\label{equ:ctf_sine}$$ where ${\mathbf{g}}$ is the spatial frequency. Its phase is given by $${\chi_{\phi}}( {\mathbf{g}}) = \frac{1}{2 p^2} \pi \lambda {{r}}^2 {\Delta f}_\phi({{\alpha}}) - \frac{1}{2 p^4} \pi \lambda^3 {{r}}^4 C_s + w,$$ where $\lambda$ is the electron wavelength, $C_s$ is the spherical aberration, $w$ is the amplitude contrast, and $p$ is the pixel size. We also have the astigmatic defocus depth $${\Delta f}_\phi({{\alpha}}) = {\Delta f_1}+ {\Delta f_2}+ ( {\Delta f_1}- {\Delta f_2}) \cos ( 2 {{\alpha}}- 2 {\alpha_f}),
\label{equ:astigmatism}$$ where ${{\alpha}}$ is the polar angle of ${\mathbf{g}}$ and ${\Delta f_1}$, ${\Delta f_2}$, and ${\alpha_f}$ are the major and minor defocus depths and the defocus angle, respectively. These together form the defocus vector $\phi = ({\Delta f_1}, {\Delta f_2}, {\alpha_f})$, which parametrizes the CTF. The values ${\Delta f_1}$ and ${\Delta f_2}$ determine the amount of defocus along two perpendicular axes, while ${\alpha_f}$ specifies the counterclockwise angle between the major defocus axis and the positive x-axis. The difference ${\Delta f_1}- {\Delta f_2}$ measures the amount of *astigmatism* in the CTF. A visualization of the effect of astigmatism is provided in Fig. \[fig:astigmatism\].
The model allows us to discern several properties of the CTF. First, ${H_{\phi}}$ is real and oscillates between positive and negative values. As a result, it has several zero crossings. Second, the CTF is radially symmetric when ${\Delta f_1}= {\Delta f_2}$ (the non-astigmatic case). Third, with no spherical aberration (i.e., $C_s = 0$) the level sets of the CTF consist of ellipses centered at the origin. The spherical aberration $C_s$ thus accounts for small deviations from the elliptical shape.
While the parameters $\lambda$, $C_s$, and $w$ are typically known from the microscope configuration, the defocus parameters $\phi$ vary widely between experiments. We must therefore estimate them to obtain an accurate model of the CTF.
To estimate $\phi$, we turn to the power spectrum of the micrograph. The power spectra ${S}_{\mathbf{x}}$, ${S}_{\mathbf{y}}$, and ${S}_{\mathbf{e}}$ of ${\mathbf{x}}$, ${\mathbf{y}}$, and ${\mathbf{e}}$, respectively, are related by $$\label{equ:true_psd}
{S}_{\mathbf{y}}( {\mathbf{g}}) = \vert {H_{\phi}}( {\mathbf{g}}) \vert^2 \, {S}_{\mathbf{x}}( {\mathbf{g}}) + {S}_{\mathbf{e}}( {\mathbf{g}}).$$ This follows from and the fact that convolving a stationary random field with $h_\phi$ multiplies its power spectrum by the square Fourier transform magnitude $|{H_{\phi}}|^2$.
Equation suggests that estimates of the power spectra ${S}_{\mathbf{y}}$, ${S}_{\mathbf{x}}$, and ${S}_{\mathbf{e}}$ can be useful in resolving the CTF. We note that ${S}_{\mathbf{x}}$ and ${S}_{\mathbf{e}}$ are slowly decaying while $\vert {H_{\phi}}\vert^2$ oscillates rapidly in comparison. The background subtracted power spectrum is therefore approximately proportional to $|{H_{\phi}}|^2$. It follows that in order to estimate the defocus parameters $\phi$, we may estimate ${S}_{\mathbf{y}}-{S}_{\mathbf{e}}$ and maximize its correlation with $|{H_{\phi}}|^2$. This approach is used in [@rohou2015ctffind4; @mindell2003ctffind3; @kai2016gctf].
Another approach is to estimate $\phi$ from zero-crossings of ${S}_{\mathbf{y}}- {S}_{\mathbf{e}}$ [@yan2017single_ring; @tani1996correlation]. Specifically, for spatial frequencies where ${H_{\phi}}({\mathbf{g}}) = 0$, we have ${S}_{\mathbf{y}}( {\mathbf{g}}) - {S}_{\mathbf{e}}( {\mathbf{g}}) = 0$. Identifying these zero-crossings from estimates of ${S}_{\mathbf{y}}- {S}_{\mathbf{e}}$ thus constrains the zeros of ${H_{\phi}}$ and lets us estimate its defocus parameters $\phi$.
For both approaches, the first step is to estimate the background-subtracted power spectrum ${S}_{\mathbf{y}}- {S}_{\mathbf{e}}$. In the following, we propose an estimation method and show how the resulting estimate may be used to estimate $\phi$ by either maximizing correlation or matching zero-crossings.
As mentioned above, the CTF is also multiplied by an exponentially decreasing envelope function [@sorzano2007damping], which effectively acts as a low-pass filter on $h_{\phi} * {\mathbf{x}}$. In this paper, rather than include the envelope function in our analysis, we ignore high frequencies as they are strongly attenuated by the envelope. We also reduce the effect of the envelope function by estimating the CTF using the square root of our power spectrum estimate as in [@rohou2015ctffind4; @mindell2003ctffind3]. In this way, the effect of the envelope function on the two methods discussed is smaller.
Power spectrum estimation {#subsec:estimate}
-------------------------
In this section we present several methods for estimating the power spectrum of the micrograph. We first present the periodogram estimator and then show different methods for reducing its bias and variance.
### Periodogram estimator {#subsec:periodogram}
In an experimental setting, we only have access to an $N \times N$ sample of ${\mathbf{y}}$, given by the values ${\mathbf{y}}[k_1, k_2]$ for $(k_1, k_2) \in \{0, 1, \ldots, N-1\}^2$. Given these values, a common power spectrum estimator is provided by the periodogram [@oppenheim1989discrete] $$\label{equ:periodogram}
{{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}} ({\mathbf{g}}) = \frac{1}{N^2} \left\vert \sum_{k_1,k_2=0}^{N-1} {\mathbf{y}}\left[ k_1, k_2 \right] \, {\mathrm{e}}^{-{\mathrm{j}}2 \pi ( g_1 k_1 + g_2 k_2 ) } \right\vert^2,$$ for ${\mathbf{g}}\in [-1/2, 1/2]^2$. While ${{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}})$ may be calculated for any ${\mathbf{g}}$, it is typically calculated on the $N \times N$ grid $$M_N = \left\{ -\frac{1}{2}, -\frac{1}{2} + \frac{2}{N}, \ldots, \frac{1}{2} - \frac{2}{N} \right\}^2.$$ This enables the use of fast Fourier transforms (FFTs) for computing the periodogram with $O(N^2 \log N)$ computational complexity. Due to this and its ease of implementation, the periodogram is a popular spectral estimator in cryo-EM.
Since our goal is to estimate ${S}_{\mathbf{y}}$, let us consider how well it is estimated by the periodogram. The mean square error (MSE) of ${{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}$ at ${\mathbf{g}}$ is given by $${\operatorname{MSE}}({{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}}({\mathbf{g}})) = {\mathbb{E}}\left[\vert {{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}}) - {S}_{\mathbf{y}}({\mathbf{g}}) \vert^2\right].$$ To analyze the source of error, it is useful to define the *bias* and *variance* of the periodogram. The bias is defined as $${\operatorname{Bias}}({{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}}({\mathbf{g}})) = {\mathbb{E}}\left[{{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}}({\mathbf{g}})\right] - {S}_{{\mathbf{y}}}({\mathbf{g}})
\label{equ:bias}$$ and measures the deviation of the expectation from the true value, while the variance $${\operatorname{Var}}({{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}}({\mathbf{g}})) = {\mathbb{E}}\left[\left\vert {{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}}) - {\mathbb{E}}\left[{{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}})\right] \right\vert^2 \right]$$ measures the average deviation of the periodogram from its expectation. Both contribute to the MSE through the identity $${\operatorname{MSE}}({{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}})) = {\operatorname{Bias}}^2({{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}})) + {\operatorname{Var}}({{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}})).$$ A low MSE therefore requires low bias and low variance.
The periodogram, however, fails on both counts. First, while the periodogram is asymptotically unbiased [@percival1993multitapers; @thompson1982spectral], its bias remains large for small samples. Second, [the periodogram is an *inconsistent* estimator, that is, its variance does not decrease with an increase in sample size. Therefore, a periodogram that extends over the entire micrograph will have variance approximately equal to that of a periodogram that extends over some section of the micrograph.]{}
In the following sections, we will therefore consider different approaches to reducing both the bias and variance of the periodogram.
### Bartlett’s method {#subsubsec:variance1}
We first consider an approach for reducing variance called *Bartlett’s method* [@oppenheim1989discrete]. In this approach, the periodogram estimate is computed for several non-overlapping regions of the image. These estimates are then averaged, reducing the variance by a factor approximately equal to the number of regions used. It may therefore be tempting to drastically reduce the size of these regions. However, in experimental data, averaging over regions that are too small will increase the bias. Among other things, this would prevent us from properly estimating the low spatial frequencies.
We thus divide our image into $B$ non-overlapping blocks ${\mathbf{y}}_0, \ldots, {\mathbf{y}}_{B-1}$ of size $K \times K$. The averaged periodogram is $$\label{equ:periodogram_blocks}
{{\hat{S}}^{(\mathrm{b})}}_{\mathbf{y}}({\mathbf{g}}) = \frac{1}{B} \sum_{b=0}^{B-1} {{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}_b} ({\mathbf{g}}).$$ If each block ${\mathbf{y}}_b$ is independent of the others, we have ${\operatorname{Var}}({{\hat{S}}^{(\mathrm{b})}}_{\mathbf{y}}({\mathbf{g}})) = B^{-1} {\operatorname{Var}}({{\hat{S}}^{(\mathrm{p})}}_{\mathbf{y}}({\mathbf{g}}))$. Note that, since the block size is now $K \times K$, we sample ${\mathbf{g}}$ on $M_K$.
### Welch’s method {#subsec:bias}
The expected value of the periodogram estimator is known to be a convolution between the true power spectrum of the micrograph and a 2D Fejér kernel [@percival1993multitapers]. As the Fejér kernel has high sidelobes, this convolution leads to frequency leakage and therefore a high bias.
One method of lowering the bias of the periodogram estimation is tapering [@percival1993multitapers]. This multiplies the data ${\mathbf{y}}$ by a data taper ${\mathbf{w}}$ prior to computing the periodogram, resulting in a *modified periodogram*. While many options for data tapers exist, such as the Hann window [@vulovic2012taper], Babadi and Brown [@babadi2014multitaper] suggest the use of the zeroth-order discrete prolate spheroidal sequence (DPSS) [@slepian1978prolates]. The expected value of this modified periodogram is a convolution between the true power spectrum of the micrograph and a kernel with smaller sidelobes than those of the Fejér kernel [@percival1993multitapers]. This reduces the frequency leakage, and, therefore, the bias of the estimator.
While the taper may be applied to the entire micrograph, it is also possible to apply it to each block in Bartlett’s method . The resulting approach is known as *Welch’s method* [@welch1967wosa]. Welch also showed that further variance reduction is possible using overlapping (typically half-overlapping) blocks [@frenandez19977periodogram; @huang2003env; @frank2996book; @zhu19977gaussian]. This yields the *modified averaged periodogram*, $${{\hat{S}}^{(\mathrm{w})}}_{{\mathbf{y}}}({\mathbf{g}}) = \frac{1}{B} \sum_{b=1}^{B} {{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}_b \cdot {\mathbf{w}}}({\mathbf{g}})$$ where ${\mathbf{y}}_b \cdot {\mathbf{w}}$ is the pointwise multiplication of ${\mathbf{y}}_b$ and ${\mathbf{w}}$.
### Multitaper estimators {#subsec:multi}
As discussed in Section \[subsubsec:variance1\], one way to lower the variance in the periodogram is to average several estimates. For this reason, Thomson [@thompson1982spectral] suggested combining the estimates obtained from multiple tapers. Each taper yields a different estimate of the power spectrum, and averaging them significantly reduces the variance. A large number of tapers, however, results in significant smoothing of the power spectrum estimate, so the variance reduction needs to be balanced with an increase in bias for non-smooth power spectra. Thomson found that higher-order DPSSs were well-suited to this task and called the resulting power spectrum estimator the *multitaper estimator*. These estimators have recently demonstrated their usefulness for noise power spectrum estimation in cryo-EM [@anden2017factor; @anden2019multitaper]. For details of the DPSS data tapers we refer the reader to Appendix A.
Combining all the above methods for variance and bias reduction, we arrive at the multitaper estimator $$\label{equ:multitaper}
{{\hat{S}}^{(\mathrm{mt})}}_{{\mathbf{y}}}({\mathbf{g}}) = \frac{1}{L B} \sum_{b=0}^{B-1} \sum_{\ell=0}^{L-1} {{\hat{S}}^{(\mathrm{p})}}_{{{\mathbf{y}}}_b\cdot{\mathbf{w}}_\ell}({\mathbf{g}}),$$ where ${\mathbf{w}}_\ell$ is the $\ell$th out $L$ DPSSs for grids of size $K \times K$.
Figs. \[fig:compare\]-\[fig:compare2\] present a comparison between ${{\hat{S}}^{(\mathrm{p})}}_{{\mathbf{y}}}$, ${{\hat{S}}^{(\mathrm{b})}}_{\mathbf{y}}$, ${{\hat{S}}^{(\mathrm{w})}}_{\mathbf{y}}$, and ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$. The CTF oscillations are best resolved by the multitaper estimator ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$.
-------------------------------------------------- -------------------------------------------------- -------------------------------------------------- --------------------------------------------------
Periodogram Bartlett’s method Welch’s method Multitaper method
[{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{}
[{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{}
-------------------------------------------------- -------------------------------------------------- -------------------------------------------------- --------------------------------------------------
-------------------------------------------------- -------------------------------------------------- -------------------------------------------------- --------------------------------------------------
Periodogram Bartlett’s method Welch’s method Multitaper method
[{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{}
[{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{} [{width="0.22\linewidth"}]{}
-------------------------------------------------- -------------------------------------------------- -------------------------------------------------- --------------------------------------------------
Background subtraction {#sec:bk}
----------------------
In this section we present a method for removing the background spectrum and further reducing the variability of the estimator ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$. We do this by first estimating the radial profile of the background spectrum and removing this estimation from ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$. Further, we show that the set of squared CTFs is contained in the span of a steerable basis and project our estimate onto that basis.
### Estimating the background {#subsec:bksub1}
We saw in that the micrograph power spectrum can be expressed as the sum of two spectra: the clean, filtered power spectrum $\vert {H_{\phi}}({\mathbf{g}}) \vert^2\,{S}_{\mathbf{x}}({\mathbf{g}})$ and the background ${S}_{\mathbf{e}}$. The background-subtracted power spectrum is therefore $${S}_{{\mathbf{y}}} ({\mathbf{g}}) - {S}_{{\mathbf{e}}} ({\mathbf{g}}) = \vert {H_{\phi}}({\mathbf{g}}) \vert^2 \, {S}_{{\mathbf{x}}} ({\mathbf{g}}).$$ An estimate of ${S}_{\mathbf{y}}- {S}_{\mathbf{e}}$ is used by many methods to estimate the CTF parameters $\phi$ [@rohou2015ctffind4; @zhu19977gaussian]. Their success therefore depends on accurate estimation of the background ${S}_{\mathbf{e}}$.
The background is influenced by many factors and accurately modeling these factors is an open challenge. Many background estimation methods instead treat the background as a radially symmetric and slowly varying function [@frank2996book].
The background estimation problem can be formulated as a curve fitting problem [@frank2996book]. We note that the background should coincide with ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$ at the zero-crossings of ${H_{\phi}}$. Furthermore, the background should be strictly smaller than ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$ at spatial frequencies where ${H_{\phi}}$ does not have a zero-crossing (since ${S}_{\mathbf{x}}$ is strictly positive). We therefore estimate the background by minimizing the difference between ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$ and ${S}_{\mathbf{e}}$.
While the radial profile of the background is monotonically decreasing in most settings, this is not the case when a Gatan K2 direct detector is used in counting mode with a high dose rate. Rather, the background will be monotonically decreasing in the lower frequencies and monotonically increasing in higher frequencies [@li2013background]. Since a monotonically decreasing function, as well as a function that is at first monotonically decreasing and later monotonically increasing, must be convex, we model the background as the non-negative, convex function that is closest to, and no larger than, ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$.
We propose estimating the background ${S}_{\mathbf{e}}$ through LP. Specifically, we minimize the $\ell_1$ norm of the background-subtracted power spectrum estimate subject to several linear constraints. The first constraint ensures that the background-subtracted power spectrum estimate is non-negative, while the remaining constraints ensure that ${\hat{S}}_{\mathbf{e}}$ is a non-negative and convex.
Since we assume the background is radially symmetric, we consider its radial profile. To this end, we calculate the radial average of ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}({\mathbf{g}})$, which we denote, by a slight abuse of notation, ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}(r)$. The radial averaging is performed by projecting ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}({\mathbf{g}})$ on the circularly symmetric (i.e., purely radial) elements of a steerable basis (see Section \[subset:steerable\]).
The resulting linear program, whose result we denote by ${{\hat{S}}_{{\mathbf{e}}}^{(\mathrm{lp})}}$, is then $$\begin{array}{ll}
\displaystyle \operatorname*{minimize}_{{{\hat{S}}_{{\mathbf{e}}}}} & \underset{r=0,\frac{1}{K},\dots,\frac{m}{K}}{\sum} {{{\hat{S}}^{(\mathrm{mt})}}_{{\mathbf{y}}} \left( r \right)} - {{\hat{S}}_{{\mathbf{e}}}}\left( r \right)\\
\text{subject to} &
{{\hat{S}}_{{\mathbf{e}}}}( r ) \leq {{{\hat{S}}^{(\mathrm{mt})}}_{{\mathbf{y}}} \left( r \right)}, \quad r=0,\dots,\frac{m}{K}\\
& {{\hat{S}}_{{\mathbf{e}}}}\left( r + 1\right) + {{\hat{S}}_{{\mathbf{e}}}}\left( r -1\right) \ge 2 {{\hat{S}}_{{\mathbf{e}}}}\left( r \right), r=1,\dots,\frac{m}{K}\\
& {{\hat{S}}_{{\mathbf{e}}}}\left( r \right) \ge 0, \quad r=0,\dots,\frac{m}{K},
\end{array}$$ where ${{\hat{S}}_{{\mathbf{e}}}}= \begin{bmatrix} {\hat{S}}(0), \ldots, {\hat{S}}(m/K)) \end{bmatrix}^T$, and $0<m/K \le 0.5$ is the spatial frequency above which ${{\hat{S}}^{(\mathrm{mt})}}_{{\mathbf{y}}} \left( r \right)$ is typically dominated by noise. As its default, ASPIRE-CTF sets $m/K=3/8$.
We present the result of our linear program in Fig. \[fig:background\].
Expanding the 1D background spectrum to a 2D function, we again abuse notation slightly and set ${{\hat{S}}_{{\mathbf{e}}}^{(\mathrm{lp})}}({\mathbf{g}}) = {{\hat{S}}_{{\mathbf{e}}}^{(\mathrm{lp})}}(r)$ for all ${\mathbf{g}}\in [-1/2, 1/2]^2$. We denote the background-subtracted power spectrum estimate ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}({\mathbf{g}})-{{\hat{S}}_{{\mathbf{e}}}^{(\mathrm{lp})}}({\mathbf{g}})$ by ${{\hat{S}}^{(\mathrm{lp})}}_{\mathbf{z}}({\mathbf{g}})$ where ${\mathbf{z}}= h_{\phi} * {\mathbf{x}}$.
A different LP that can be used for background estimation was suggested in [@huang2003env]. However, contrary to our non-parametric approach which assumes convexity alone, [@huang2003env] suggests a LP based on parametric estimation.
### Expansion over a steerable basis {#subset:steerable}
In this section, we show that any function of the form - is contained in a low-dimensional subspace spanned by a set of steerable basis functions, such as a Fourier–Bessel basis [@zhao2013fourier; @Zhao2016fb] or prolate spheroidal wave functions (PSWFs) [@landa2017prolates; @landa2018prolates]. We will use this property to further reduce the variability of the power spectrum estimator by projecting the background-subtracted power spectrum estimate ${{\hat{S}}^{(\mathrm{lp})}}_{\mathbf{z}}({\mathbf{g}})$ onto this subspace.
A steerable basis consists of functions $f_{k,q}({{r}}) \, {\mathrm{e}}^{{\mathrm{j}}k {{\alpha}}}$, where $k \in {\mathbb{Z}}$ and $q = 0, \ldots, p_k-1$ for some $p_k \ge 0$. The radial part $f_{k,q}(r)$ depends on the specific choice of basis (e.g., in a Fourier–Bessel basis, it is a scaled Bessel function of order $q$) and does not enter explicitly into our analysis. We shall therefore leave it unspecified. A given function in polar coordinates may be decomposed in the basis as $$\label{equ:steerable}
x({{r}}, {{\alpha}}) = \sum_{k=-\infty}^\infty \sum_{q=0}^{\infty} a_{k,q} \, f_{k,q}({{r}}) \, {\mathrm{e}}^{{\mathrm{j}}k {{\alpha}}},$$ where $a_{k,q} \in {\mathbb{C}}$ is the coefficient corresponding to angular frequency $k$ and radial frequency $q$.
To determine the steerable basis expansion of the CTF , we consider its Taylor expansion around ${\Delta f_1}- {\Delta f_2}= 0$, $$\begin{gathered}
{H_{\phi}}( {\mathbf{g}}) = \sum_{\substack{n = 0 \\ n~\text{even}}}^{P} \frac{\left( -1 \right)^{\frac{n}{2}+1}}{n!} \sin ( {\chi_{\phi}}^0 ( {{r}})) C_{n,\phi}({\mathbf{g}}) \\+ \sum_{\substack{n=1 \\ n~\text{odd}}}^{P} \frac{\left( -1 \right)^{\frac{n+1}{2}}}{n!} \cos ( {\chi_{\phi}}^0 ( {{r}})) C_{n,\phi}({\mathbf{g}}) + R_P({\mathbf{g}}),\end{gathered}$$ where $R_P({\mathbf{g}})$ is the remainder term, $$C_{n,\phi}({\mathbf{g}}) = \left( \frac{1}{2}\pi \lambda \left( {\Delta f_1}- {\Delta f_2}\right) \cos \left( 2 ( {{\alpha}}- {\alpha_f}) \right) \frac{ {{r}}^2}{p^2} \right)^n,$$ and $${\chi_{\phi}}^0 ( {{r}}) = \frac{1}{2} \pi \lambda {{r}}^2 ({\Delta f_1}+{\Delta f_2}) - \frac{1}{2} \pi \lambda^3 {{r}}^4 C_s + w$$ is the non-astigmatic phase function.
The remainder term is bounded by a function of $$\left(\frac{{\Delta f_1}- {\Delta f_2}}{{\Delta f_1}+{\Delta f_2}}\right)^{(P+1)}$$ and is therefore small when astigmatism is small, which is the case for experimental cryo-EM data. We therefore conclude that $$\label{equ:ctf_approx}
{H_{\phi}}({\mathbf{g}}) \approx -\sin({\chi_{\phi}}^0 ({{r}})) - \cos({\chi_{\phi}}^0({{r}})) C_{1,\phi}({\mathbf{g}})$$ is a good approximation of the CTF.
Since $\cos(\alpha) = \frac{1}{2}({\mathrm{e}}^{-{\mathrm{j}}\alpha} + {\mathrm{e}}^{-{\mathrm{j}}\alpha})$, we rewrite as $$\begin{gathered}
\label{equ:taylor_ctf}
{H_{\phi}}({\mathbf{g}}) \approx -\sin({\chi_{\phi}}^0({{r}})) \\
-\frac{1}{4p^2} \cos({\chi_{\phi}}^0({{r}})) \pi \lambda ({\Delta f_1}-{\Delta f_2}) {\mathrm{e}}^{-{\mathrm{j}}2{\alpha_f}} r^2 \, {\mathrm{e}}^{{\mathrm{j}}2{{\alpha}}} \\
-\frac{1}{4p^2} \cos({\chi_{\phi}}^0({{r}})) \pi \lambda ({\Delta f_1}-{\Delta f_2}) {\mathrm{e}}^{{\mathrm{j}}2{\alpha_f}} r^2 \, {\mathrm{e}}^{-{\mathrm{j}}2{{\alpha}}}.\end{gathered}$$ Comparing and , we see that only terms corresponding to $k = -2$, $0$, and $2$ are present.
Concretely, we compute the coefficients $a_{k,q}$ of the expansion of $\sqrt{{{\hat{S}}^{(\mathrm{lp})}}_{\mathbf{z}}}$ over the steerable basis functions with radial frequencies to $k = 0, \pm 2$. The coefficients are computed through an inner product on a $K \times K$ grid: $$a_{k,q} = \frac{1}{K^2} \sum_{{\mathbf{g}}\in M_K} \sqrt{{{\hat{S}}^{(\mathrm{lp})}}_{\mathbf{z}}({\mathbf{g}})} f_{k, q}({{r}}) {\mathrm{e}}^{{\mathrm{j}}k {{\alpha}}},$$ where $r$ and ${{\alpha}}$ are the polar coordinates of ${\mathbf{g}}$. Evaluating for these $a_{k,q}$ and squaring the result then gives a new power spectrum estimate, which we denote as ${\hat{S}}_{\mathbf{z}}$.
In Fig. \[fig:PSD\_complete\], we present the results of our power spectrum estimation method on a micrograph from the EMPIAR-10028 dataset @wong2014empiar. This includes multitaper estimate ${{\hat{S}}^{(\mathrm{mt})}}_{\mathbf{y}}$ as well as the background-subtracted estimate ${{\hat{S}}^{(\mathrm{lp})}}_{\mathbf{z}}$ and the projection onto the steerable basis with $k = 0, \pm 2$. The result is smooth enough that many of the zero-crossings of the power spectrum can be easily resolved.
CTF Parameter Estimation
------------------------
In the previous sections we have introduced our method for estimating the background-subtracted power spectrum. In this section we will discuss two methods that use this estimate to recover the defocus and astigmatism of the micrograph.
### CTF estimation through correlation {#subsec:cross_corr}
The power spectrum ${S}_{\mathbf{x}}$ is a slowly-varying function of radial frequency. It follows that the oscillations in $|{H_{\phi}}({\mathbf{g}})|^2 \, {S}_{\mathbf{x}}({\mathbf{g}})$ are due to those of $|{H_{\phi}}({\mathbf{g}})|^2$. As a consequence, the square root of the background-subtracted power spectrum $|{H_{\phi}}({\mathbf{g}})| \, {S}_{\mathbf{x}}^{1/2}({\mathbf{g}})$ is proportional to the absolute value of the CTF (we note that using the square root instead of the actual power spectrum estimate reduces the influence of large values). It follows that the correlation test is a useful tool in estimating CTF parameters. Indeed, many CTF estimation methods therefore estimate the defocus parameters $\phi$ through correlation with a simulated CTF magnitude [@rohou2015ctffind4; @mindell2003ctffind3; @kai2016gctf].
The Pearson correlation of $|{H_{\phi}}({\mathbf{g}})|$ and ${\hat{S}}_{\mathbf{z}}^{1/2}({\mathbf{g}})$ is $$\label{equ:corr_def}
P_{cc}(\phi) = \frac{\sum_{{\mathbf{g}}\in R} |{H_{\phi}}({\mathbf{g}})| \, {\hat{S}}_{\mathbf{z}}^{1/2}({\mathbf{g}})}{\left( \sum_{{\mathbf{g}}\in R} |{H_{\phi}}({\mathbf{g}})|^2 \, \sum_{{\mathbf{g}}\in R} {\hat{S}}_{\mathbf{z}}({\mathbf{g}}) \right)^{1/2} },$$ where $R$ is the set of frequencies over which correlation is computed, and will be defined below. To optimize $P_{cc}(\phi)$, we first need an initial guess for the parameters $\phi$. For this, we follow @kai2016gctf and first consider non-astigmatic CTFs where ${\Delta f_1}= {\Delta f_2}$, which renders the value of ${\alpha_f}$ irrelevant. We thus calculate $P_{cc}(\phi)$ for $\phi = ({\Delta f}, {\Delta f}, 0)$ with ${\Delta f}$ on a 1D grid from ${\Delta f}_{\mathrm{min}}$ to ${\Delta f}_{\mathrm{max}}$ with a step of ${\Delta f}_{\mathrm{step}}$. Since ${H_{\phi}}$ is considered (at this stage) to be radially symmetric, we define the set of frequencies over which correlation is computed as $$R = \left\{ m_1, m_1+ \frac{2}{N}, \ldots, \frac{3}{8} \right\} \times \{0\},$$ where $m_1$ is the first maximum of the radial profile of ${{\hat{S}}^{(\mathrm{lp})}}_{\mathbf{z}}$. That is, we only consider frequencies ${\mathbf{g}}$ along a 1D radial profile and, furthermore, ignore the very low and very high frequencies (since the very low frequencies may dominate the cross-correlation result and the very high frequencies are strongly effected by the envelope function). The ${\Delta f}$ which maximizes $P_{cc}(\phi)$ on this grid is denoted ${\Delta f}_\star$.
To estimate the astigmatism of the CTF, we compute the principal directions of the second-order moments of ${\hat{S}}_{\mathbf{z}}^{1/2}$. Specifically, we form the $2 \times 2$ matrix ${\mathbf{M}}$ given by $$\begin{gathered}
\begin{aligned}
& M_{1,1} = {\textstyle \sum_{{\mathbf{g}}\in M_K}} \, g_1^2 \, {\hat{S}}_{\mathbf{z}}^{1/2}({\mathbf{g}}),\\
& M_{1,2} = M_{2,1} = {\textstyle \sum_{{\mathbf{g}}\in M_K}} \, g_{1} g_{2} \, {\hat{S}}_{\mathbf{z}}^{1/2}({\mathbf{g}}),\\
& M_{2,2} = {\textstyle \sum_{{\mathbf{g}}\in M_K}} g_2^2 \, {\hat{S}}_{\mathbf{z}}^{1/2}({\mathbf{g}}).
\end{aligned}\end{gathered}$$ The eigenvalues $\mu_1$ and $\mu_2$ of ${\mathbf{M}}$ estimate the size of the major and minor axes in ${\hat{S}}_{\mathbf{z}}^{1/2}$. We therefore expect the ratio $\mu_1/\mu_2$ to approximate ${\Delta f_1}/{\Delta f_2}$. Combining this with the estimated mean defocus ${\Delta f}_\star$, we get $$\begin{aligned}
\frac{1}{2}({\Delta f_1}+ {\Delta f_2}) &= {\Delta f}_\star \\
\frac{{\Delta f_1}}{{\Delta f_2}} &= \frac{\mu_1}{\mu_2},\end{aligned}$$ which has the solution $$\begin{aligned}
{\Delta f}_{1,\star} &= \frac{2\mu_1}{\mu_1+\mu_2} {\Delta f}_\star, \quad
{\Delta f}_{2,\star} &= \frac{2\mu_2}{\mu_1+\mu_2} {\Delta f}_\star.\end{aligned}$$
In order to improve our estimation of the defocus parameters, we run gradient descent on $P_{cc}(\phi)$. As we no longer approximate the image as non-astigmatic, we define the set of frequencies over which correlation is computed as $R = M_K$. We now initialize our gradient descent at $\phi = ({\Delta f}_{1,\star}, {\Delta f}_{2,\star}, {\alpha_f})$, where ${\alpha_f}$ is set as detailed in [@kai2016gctf] to an arbitrarily selected $0\le a < \pi/6$ (e.g. $a=\pi/12$) and $(a+\pi/6)$, $(a-\pi/6)$, $(a+\pi/3)$, $(a-\pi/3)$ or $(a-\pi/2)$. One run of gradient descent is performed for each value of ${\alpha_f}$ and the result with the highest value of $P_{cc}(\phi)$ is kept. The resulting $\phi$ is our defocus estimate for the micrograph.
We note that, as is done in [@rohou2015ctffind4; @mindell2003ctffind3; @kai2016gctf], we discard information in the lower and higher frequencies of $ {\hat{S}}_{\mathbf{z}}$. These frequencies can be determined by the user. As default values we use those suggested in [@rohou2015ctffind4].
### CTF estimation through zero-crossings {#subsec:zero_cross}
We have seen in the previous sections that the true background-subtracted power spectrum is $\vert {H_{\phi}}({\mathbf{g}}) \vert^2 {S}_{\mathbf{x}}({\mathbf{g}})$. Under the assumption that ${S}_{\mathbf{x}}({\mathbf{g}})$ is slowly-varying, it follows that at any frequency where the background-subtracted power spectrum reaches a minimal value of zero, the CTF must reach a zero-crossing.
Furthermore, even if the aforementioned assumption did not hold true, we could still infer the frequencies where the CTF reaches a zero-crossing. This is due to the fact that the zero-crossings of the CTF are known to create concentric, nearly elliptical rings, centered around the origin (see Section \[sec:formation\]). Therefore, this can be used a cue to differentiate between any minima of $\vert {H_{\phi}}({\mathbf{g}}) \vert^2 \, {S}_{\mathbf{x}}({\mathbf{g}})$ that stem from the zero-crossings of ${H_{\phi}}$ and any minima that stem from ${S}_{\mathbf{x}}$.
As we show in Fig. \[fig:PSD\_complete\](c), our estimation of the background-subtracted power spectrum enables easy detection of several elliptical rings where ${\hat{S}}_{\mathbf{z}}$ reaches its minima. To do this, we define any pixel with a value smaller than that of at least six out of its eight neighbors as a zero-crossing. Once the minima of ${\hat{S}}_{\mathbf{z}}$ are found, we discard any frequency that is not on a closed ring. Furthermore, we verify that the spatial frequencies of pixels residing on closed rings representing the minima of ${\hat{S}}_{\mathbf{z}}$ form ellipses approximately centered at the origin. We are then left with frequencies of several zero-crossings of the CTF.
Since we have ${H_{\phi}}({\mathbf{g}}) = - \sin({\chi_{\phi}}({\mathbf{g}}))$, we reach a zero-crossing of the CTF when ${\chi_{\phi}}({\mathbf{g}})$ is an integer multiple of $\pi$. Formally, the set of spatial frequencies on the $\ell$th ring of zero-crossings, denoted by $G_\ell$, satisfies $${\chi_{\phi}}({\mathbf{g}}_\ell) = \pi \ell.
\label{equ:zero_constraint}$$ Empirically, we are typically able to identify at least three rings, that is, three different values of $\ell$.
Combining for all ${\mathbf{g}}$ in $\hat{G}_\ell$ and combining these for different values of $\ell$, we obtain an overdetermined system of equations. Solving it yields an estimate for the defocus vector $\phi$. To solve the system, we use the method implemented in MATLAB (a variant of [@powel1970fsolve]).
We note that the estimated positions of CTF zeroes are extremely sensitive to the method of background subtraction. This method should therefore be used to obtain an initial estimate of the defocus. Refinement of the solution can be done as in Section \[subsec:cross\_corr\] using gradient-based optimization.
We note that typically both methods suggested in this section achieve similar results. However, while the zero-crossings-based method has lower computational complexity, the correlation-based method is more robust to noise. Therefore, for micrographs with very low SNR we recommend using the correlation-based method, while for cleaner micrographs we suggest using the zero-crossings method.
Results {#sec:experiment}
=======
We present experimental results for the ASPIRE-CTF framework presented in this paper. We apply our framework to datasets that are publicly available from the EMPIAR database [@iudin2016empiar] or the CTF challenge [@marabini2015challenge]. Unless otherwise stated, in the experiments below we use $L=4$ tapers and project the power spectrum onto the steerable PSWF basis.
Estimating CTF from movie frames
--------------------------------
The CTF may be estimated either from motion-corrected micrographs, from several frames averaged in real space or, alternatively, directly from the frames. This is done by estimating ${{\hat{S}}^{(\mathrm{mt})}}$ individually from each frame and averaging the estimates (see Fig. \[fig:pipeline\]). A benefit of estimating the CTF directly from the frames is that this practice enables us to correct for motion and estimate the CTF concurrently, thus speeding up the pipeline. Furthermore, any errors added by motion-correction will have no effect on the CTF estimation [@bartesaghi2014frames].
In this section, we compare the CTF estimates produced from motion-corrected micrographs to the estimation produced directly from the frames. We do this over several publicly available datasets, namely, EMPIAR-10002 [@bai2013empiar], EMPIAR-10028 [@wong2014empiar], EMPIAR-10242 [@zhang2019empiar], and EMPIAR-10249 [@herzik2019dataset]. A summary of these datasets appears in Table \[table:movies\].
While the EMPIAR-10028 and EMPIAR-10242 datasets contain both movies and motion-corrected micrographs, EMPIAR-10002 and EMPIAR-10249 contain movies alone. We therefore use MotionCor2 [@Zheng2017motion] to produce the motion-corrected micrograph for these two datasets. We present in Fig. \[fig:movies\] a comparison of the astigmatism (${\Delta f_1}-{\Delta f_2}$), average defocus (${\Delta f_1}/2+{\Delta f_2}/2$), and astigmatism angle ($\alpha_f$) as estimated from a motion-corrected micrograph with an estimate produced from the raw movie frames. We note that, as expected, the parameters estimated from each of these methods are nearly identical.
-------------- -------------------- ---------------- ------------ -------------- --------------- -----------------
Dataset Molecule Pixel size (Å) Spherical Voltage (kV) Microscope Detector
aberration
EMPIAR-10002 80S ribosome 1.77 2.0 300 Polara Falcon
EMPIAR-10028 80S ribosome 1.34 2.0 300 Polara Falcon
EMPIAR-10242 2N3R tau filaments 1.04 2.7 300 Titan Krios Gatan K2 Summit
EMPIAR-10249 HLA dehydrogenase 0.56 2.7 200 Talos Arctica Gatan K2 Summit
-------------- -------------------- ---------------- ------------ -------------- --------------- -----------------
CTF Challenge {#subsec:exp1}
-------------
The CTF challenge [@marabini2015challenge] consists of nearly $200$ micrographs of GroEL, 60S ribosome, apoferritin and TMV virus. These micrographs are taken from eight experimental datasets and one synthetic dataset, each referred to by a number ranging from $001$ to $009$. In the following, we restrict our attention to the experimental datasets, that is, sets $001$ through $008$.
The advantage of the CTF challenge is that each dataset is acquired using a different combination of microscope and camera, allowing for a qualitative comparison of CTF estimation methods for a variety of experimental setups. Notably, datasets $003$ and $004$ use a Gatan K2 direct detector in counting mode with a high electron dose, causing ${S}_{\mathbf{e}}$ to increase at high frequencies [@li2013background]. Additionally, dataset $008$ has an especially low signal-to-noise ratio (SNR), rendering CTF estimation difficult. A summary of these datasets is presented in [@marabini2015challenge].
The estimate of each micrograph’s power spectrum is computed as detailed in Section \[subsec:multi\]. Specifically, we divide the micrograph into half-overlapping blocks of size $K \times K$, where $K=512$, and use $L=4$ tapers in the estimation. We then estimate the background spectrum as detailed in Section \[subsec:bksub1\] and expand the background-subtracted power spectrum over the PSWF basis in order to reduce variability in the power spectrum estimate (Section \[subset:steerable\]). We use the correlation-based method (Section \[subsec:cross\_corr\]) to estimate defocus parameters, and denote the resulting vector of defocus parameters as $\phi_{a}^{(512)}$.
In some cases, a power spectrum of size $512 \times 512$ may not capture the oscillations of the power spectrum with sufficient accuracy [@rohou2015ctffind4]. We therefore compute a second estimate of the power spectrum using half-overlapping blocks of size $1024 \times 1024$. As this reduces the number of blocks, the variance of the estimator will grow. We therefore use $L=16$ data tapers in this case. We use this estimate of the power spectrum to estimate a vector of defocus parameters which we denote by $\phi_{a}^{(1024)}$.
We compare our results to the estimates produced by CTFFIND4 (version 4.1.13) and Gctf (version 1.06). We denote the vector of estimated defocus parameters produced by CTFFIND4 when using block of size $512 \times 512$ and $1024 \times 1024$ as $\phi_{c}^{(512)}$ and $\phi_{c}^{(1024)}$, respectively. We further denote the vector of estimated defocus parameters produced by Gctf when using block of size $512 \times 512$ and $1024 \times 1024$ as $\phi_{g}^{(512)}$ and $\phi_{g}^{(1024)}$, respectively. For each estimation method we select the vector of estimated defocus parameters that leads to highest correlation with the estimated power spectrum, that is $$\phi_j^* = \arg \underset{\phi_j \in \{\phi_j^{(512)}, \phi_j^{(1024)} \}}{\max} \left( \frac{1}{s_t} \sum_{m=1}^{s_t} P_{cc}^{m} (\phi_j ) \right),$$ where $j \in \{ a, c, g\}$, $s_t$ is the number of micrographs in the $t$th dataset and $P_{cc}^{m}$ is the correlation for the $m$th micrograph in the dataset, computed as in . We note that the correlation is computed with the power spectrum estimate suggested in [@mindell2003ctffind3]. That is, we compute the Pearson correlation coefficient between the background subtracted power spectrum computed as in [@mindell2003ctffind3] and using blocks of size $512 \times 512$ with $H_{\phi_j^{(512)}}$, and between the background subtracted power spectrum computed using blocks of size $1024 \times 1024$ with $H_{\phi_j^{(1024)}}$. In this manner, we choose the block size that best captures the oscillations of each dataset.
In order to compare the consistency of these $3$ methods, we present the differences between $\phi_a$, $\phi_c$ and $\phi_g$ in Tables \[tab:mean\]-\[tab:std\]. That is, for each micrograph $m$ we compute $$\begin{aligned}
& \epsilon_{j,k}({\Delta f_1}) = (\phi_j(1) - \phi_k(1)) / \phi_j(1)\\
& \epsilon_{j,k}({\Delta f_2}) = (\phi_j(2) - \phi_k(2)) / \phi_j(2)\\
& \epsilon_{j,k}(\alpha_f) = (\phi_j(3) - \phi_k(3)) / \phi_j(3)
\end{aligned}
\label{equ:dif_comparison}$$ where $j$ and $k$ are two estimation methods (ASPIRE-CTF, CTFFIND4 or Gctf). We report the mean and variance of $\epsilon$. We note that either $\epsilon_{a,g}$ or $\epsilon_{a,c}$ are often smaller than $\epsilon_{c,g}$, thus showing the ASPIRE-CTF estimate to be in the consensus of the three estimation vectors.
Fig. \[fig:vis\_compare\] contains a visual comparison between the power spectrum computed by our suggested framework and the power spectra computed by Gctf and CTFFIND4. We present the comparison over a micrograph from the eighth set of the CTF challenge as this set is known to be difficult. We note that the oscillations of the ASPIRE-CTF power spectrum are highly noticeable. In comparison, the variability of the power spectra computed by Gctf and CTFFIND4 make visual detection of oscillations challenging.
Runtime
-------
We compute runtime of ASPIRE-CTF and CTFFIND4 over dataset $001$ of the CTF challenge. For both methods, we partition the micrograph into blocks of size $512 \times 512$. When running ASPIRE-CTF we employ $L=4$ data tapers. Furthermore, we use the exhaustive search option for CTFFIND4, and perform an exhaustive 1D search in ASPIRE-CTF. While the CTF estimation results are comparable, there is a significant speedup when using ASPIRE-CTF. Runtime for ASPIRE-CTF is 22.5 seconds on average per micrograph, while runtime for CTFFIND4 is 541 seconds.
These experiments are run on a $2.6$ GHz Intel Core i7 CPU with four cores and $16$ GB of memory. We do not compare to the runtime of Gctf as it must be run on a GPU.
Consistency in low SNR
----------------------
To test consistency of results with changing SNR, we turn to the EMPIAR-10249 dataset @herzik2019dataset. This dataset consists of movies with $44$ frames per movie. Usually, all these frames, except for a few frames at the beginning and a few at the end, are motion-corrected and summed to create a micrograph. This is due to the fact that a micrograph created from as many motion-corrected frames as possible will have the best SNR.
We disregard the first frame and use MotionCor2 [@Zheng2017motion] to create $9$ motion-corrected micrographs. These consist of summing $5$, $8$, $13$, $18$, $23$, $28$, $33$, $38$, and $43$ motion-corrected frames, respectively. This gives us a sequence of micrographs with increasing SNR.
We estimated the CTF parameters independently from each micrograph in the manner detailed in Section \[subsec:exp1\]. Fig. \[fig:astigmatism\_vs\_defocus\] shows the astigmatism $|{\Delta f_1}-{\Delta f_2}|$ vs. mean defocus $({\Delta f_1}+ {\Delta f_2})/2$ of the CTF estimation for each method and over each micrograph. We see that while the average defocus values remain similar for all three methods, Gctf incurs a larger error in the astigmatism when 23 frames are used. On the other hand, our method and CTFFIND4 achieve consistent estimates regardless of the amount of frames averaged.
Conclusion
==========
In this paper we have presented a novel approach for power spectrum estimation of cryo-EM experimental data. Our approach uses the multitaper estimator, which often leads to reduced mean square error over Bartlett’s and Welch’s methods. Additionally, we presented a method for error reduction that is driven directly by the mathematical model of the contrast transfer function. We did this by projecting the power spectrum estimate onto a steerable basis and discarding any basis function where the CTF must be negligible. We showed that the combination of these two contributions leads to greatly reduced variability in our estimator.
We presented experimental results on twelve datasets, and showed that our method is well suited to both motion-corrected micrographs and raw movies data.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the Simons Foundation Math+X Investigator Award and the Moore Foundation Data-Driven Discovery Investigator Award. The authors thank B. Landa and I. Sason for help optimizing the PSWF code. The authors are also indebted to B. Landa, Y. Shkolnisky and A. Rohou for helpful comments and discussions. The Flatiron Institute is a division of the Simons Foundation.
Appendix A {#appendix-a .unnumbered}
==========
Zeroth-order discrete prolate spheroidal sequence (DPSS) [@slepian1978prolates] were proposed as data tapers in ([@babadi2014multitaper]; Section \[subsec:bias\]). Here we describe their generation.
The zeroth-order discrete prolate spheroidal sequence is a sequence of $d$ 1-D vectors, determined as the $d$ leading eigenvectors of the matrix $\mathbf{\mathcal{L}} \in \mathbb{R}^{K \times K}$, where $$\mathcal{L}[{k,m}] = \frac{\sin (\pi R (k-m)))}{\pi (k-m)},$$ and $R = \frac{2d}{N}$. We denote the resulting data tapers by $\mathbf{t}_1,\dots,\mathbf{t}_N$.
As the blocks $\mathbf{y}_b$ are 2D, that is $\mathbf{y}_b \in \mathbb{R}^{K \times K}$, the data tapers we use are defined as $$\mathbf{w}_{pq} = \mathbf{t}_p^T \mathbf{t}_q, \quad 0 \le p,q < d.$$ $$w_{dq+p} [k_1, k_2] = t_p[k_1] t_q[k_2].$$
Lastly, we note that $d$ is selected such that $$(d-1)^2 < L \le d^2.$$
References {#references .unnumbered}
==========
[^1]: <https://github.com/ComputationalCryoEM/ASPIRE-Python>
|
---
abstract: 'It has been demonstrated that excitable media with a tree structure performed better than other network topologies, it is natural to consider neural networks defined on Cayley trees. The investigation of a symbolic space called tree-shift of finite type is important when it comes to the discussion of the equilibrium solutions of neural networks on Cayley trees. Entropy is a frequently used invariant for measuring the complexity of a system, and constant entropy for an open set of coupling weights between neurons means that the specific network is stable. This paper gives a complete characterization of entropy spectrum of neural networks on Cayley trees and reveals whether the entropy bifurcates when the coupling weights change.'
address:
- 'Department of Applied Mathematics, National Dong Hwa University, Hualien 97401, Taiwan, ROC.'
- 'Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 81148, Taiwan, ROC.'
author:
- 'Jung-Chao Ban'
- 'Chih-Hung Chang\*'
- 'Nai-Zhu Huang'
bibliography:
- '../../grece.bib'
date: 'January 15, 2018'
title: Entropy bifurcation of neural networks on Cayley trees
---
[^1]
=1.5
Introduction
============
The human brain has recently been revealed as a system exhibiting traces of criticality; the corresponding spatiotemporal patterns are fractal-like. Gollo *et al.* [@GKC-SR2013] infer that criticality may arise from balanced dynamics within individual neurons. Neural networks have been developed to mimic brain behavior for the past few decades; they are widely applied in many disciplines such as signal propagation between neurons, deep learning, image processing, and information technology [@AA-N2003; @B-FTML2009; @CGM+-NNITo1992; @Fuk-NN2013]. Chernihovskyi *et al.* [@CMM+-JCN2005] implement cellular neural networks on simulating nonlinear excitable media and develop a relevant device to predict epileptic seizures. The overwhelming majority of neural network models adopts an $n$-dimensional lattice as the network’s topology. Gollo *et al.* [@GKC-PCB2009; @GKC-PRE2012; @GKC-SR2013] propose a neural network with a tree structure; excitable media with a tree structure performed better than other network topologies since it attains larger dynamic range (cf. [@AC-PRE2008; @KC-NP2006; @LSR-PRL2011]). It is of interest to ask the following problem.
**Problem 1.** How to measure the complexity of a tree structure neural network?
Alternatively, it is important to know how much information the neural network could store. On the other hand, it is of interest to know whether a neural network “avalanches", which means such a network is sensitive. More precisely, some small modification of parameters could lead to tremendously different dynamics such as the exponential decay of storage of information. One of the most frequently studied neural networks is the Hopfield neural network consisting of locally coupled neurons, in which the behavior of each neuron is represented by a differential equation. Beyond being essential for understanding the dynamics of differential equations, the investigation of equilibrium solutions is related to elucidating the long-term memory of brain. Whenever there are only finitely many equilibrium solutions, the investigation of equilibrium solutions is then equivalent to studying shift spaces in symbolic dynamical systems.
A one-dimensional shift space is a set consisting of right-infinite or bi-infinite words which avoid words in a so-called *forbidden set* $\mathcal{F}$ and is denoted by $\mathsf{X}_{\mathcal{F}}$. A shift space $\mathsf{X}_{\mathcal{F}}$ is called a shift of finite type (SFT) if $\mathcal{F}$ is a finite set. A significant invariant of shift spaces is the topological entropy, which reflects how much information a network can store. While there is an explicit formula for the entropy of $1$-d SFTs, there is no algorithm for the computation of the topological entropy of multidimensional SFTs so far (cf. [@LM-1995; @MP-ETDS2013; @MP-SJDM2013; @HM-AoM2010]).
Aubrun and Béal [@AB-TCS2012; @AB-TCS2013] introduce the notion of tree-shifts, which are shift spaces defined on Cayley trees, and then study the classification theory up to conjugacy, languages, and its application to automaton theory. It is noteworthy that such tree-shifts constitute an intermediate class in between one-sided and multidimensional shifts. Ban and Chang [@BC-2017; @BC-N2017] propose an algorithm for computing the entropy of a tree-shift of finite type (TSFT). The computation of the rigorous value of entropy is tricky due to the double exponential growth rate of the patterns for a TSFT (see Section 2 for more details).
For the case where TSFTs come from the equilibrium solutions of neural networks (on Cayley trees), the forbidden sets are constrained by the so-called *separation property*; this makes the entropy spectrum discrete (Theorem \[thm:CTNN-entropy-set\]). Elucidating the phenomenon of “neural avalanches” is related to the study of entropy bifurcation or entropy minimality problems. It is known that an irreducible $\mathbb{Z}^1$ SFT is entropy minimal; that is, any proper subshift $Y \subset X$ has smaller entropy than that of an irreducible SFT $X$. For $r \geq 2$, every $\mathbb{Z}^r$ SFT having the mixing property called *uniform filling property* is entropy minimal while there is a non-trivial *block gluing* $\mathbb{Z}^r$ SFT which is not entropy minimal. Readers are referred to [@BPS-TAMS2010; @LM-1995; @QS-ETDS2003] for more details. Proposition \[prop:entropy-region-W-equation\] gives an explicit formula for the coupling weights between neurons which make CTNNs entropy minimal, and the entropy bifurcation diagram is revealed (Figure \[fig:entropy-diagram-general-case\]). A remarkably novel phenomenon is that the entropy of a CTNN with the nearest neighborhood is either $0$ or $\ln d$, where $d$ is the number of children of each node.
The structure of this paper is as follows. Section 2 introduces the notion of tree-shifts and the algorithm for the computation of entropy of TSFTs. Section 3, aside from demonstrating how the investigation of the equilibrium solutions of CTNNs relates to the discussion of TSFTs, studies the learning problem of CTNNs; the necessary and sufficient condition of the forbidden sets of TSFTs corresponding to CTNNs is revealed. After demonstrating the discreteness of entropy spectrum of CTNNs, the entropy minimality problem is affirmatively solved in Section 3. Conclusion and discussion are given in Section 4.
Symbolic Dynamics on Cayley Trees {#sec:dynamics-cayley-tree}
=================================
This section recalls some definitions and results of symbolic dynamics on Cayley trees. A novel phenomenon about the entropy spectrum of tree-shifts of finite type is demonstrated herein.
Definitions and Notations
-------------------------
A Cayley tree, roughly speaking, is a graph without cycles. Two kinds of Cayley trees are mostly discussed: rooted Cayley trees and bi-rooted Cayley trees. A rooted $d$-ary Cayley tree (Figure \[fig:GMTS-binary-tree\]) can be seen as a directed graph such that the outdegree of each vertex is $d$ while a bi-rooted $d$-ary tree (also known as Bethe lattice, see [@Rozikov-2013] for more details) is an undirected graph such that the degree of each vertex is $d+1$. In this paper, we focus on the rooted Cayley tree for clarity, and the discussion can extend to the Bethe lattice. In the rest of this elaboration, we refer to rooted Cayley tree as Cayley tree unless otherwise stated.
Alternatively, a Cayley tree of order $d$ is a free semigroup $\Sigma^*$ generated by $\Sigma = \{g_1, g_2, \ldots, g_d\}$, where $d \in \mathbb{N}$. A *labeled tree* $t$ over a finite alphabet $\mathcal{A}$ is a function from $\Sigma^*$ to $\mathcal{A}$; a *node* of a labeled tree is an element of $\Sigma^*$, and the identity element relates to the *root* of the tree. Suppose $x = x_1 x_2 \ldots x_i, y = y_1 y_2 \ldots y_j \in \Sigma^*$ are nodes of a tree, we say that $x$ is a *prefix* of $y$ if and only if $i \leq j$ and $x_k = y_k$ for $1 \leq k \leq i$, and $xy = x_1 \cdots x_i y_1 \cdots y_j$ means the concatenation of $x$ and $y$. A subset $L \subset \Sigma^*$ is called *prefix-closed* if the prefix of every element of $L$ belongs to $L$. A *pattern* is a function $u: L \to \mathcal{A}$ with *support* $L$ and is called an *$(n+1)$-block* if its support $L = x \Delta_n$ for some $x \in \Sigma^*$, where $\Delta_n = \{y = y_1 y_2 \cdots y_n: y_i \in \Sigma \bigcup \{e\}\}$.
[0.45]{}
![A (rooted) binary Cayley tree is seen as a directed graph without cycles such that the outdegree of every vertex is $2$; herein we omit the arrow of the edge as seen in (A). A golden mean shift on binary Cayley tree is a tree-shift $\mathsf{X}_{\mathcal{F}}$ such that no consecutive $1$’s is allowed. Allowed and forbidden patterns are presented in (B).[]{data-label="fig:golden-mean-tree-shift"}](TreeNN-EntropyMinimal-20171215-pics)
[0.45]{}
![A (rooted) binary Cayley tree is seen as a directed graph without cycles such that the outdegree of every vertex is $2$; herein we omit the arrow of the edge as seen in (A). A golden mean shift on binary Cayley tree is a tree-shift $\mathsf{X}_{\mathcal{F}}$ such that no consecutive $1$’s is allowed. Allowed and forbidden patterns are presented in (B).[]{data-label="fig:golden-mean-tree-shift"}](TreeNN-EntropyMinimal-20171215-pics)
Suppose that $u$ is a pattern and $t$ is a labeled tree. Let $s(u)$ denote the support of $u$. We say that $u$ is accepted by $t$ if there exists $x \in \Sigma^*$ such that $u_y = t_{xy}$ for every node $y \in s(u)$. In this case, we say that $u$ is a pattern of $t$ rooted at the node $x$. A tree $t$ is said to *avoid* $u$ if $u$ is not accepted by $t$; otherwise, $u$ is called an *allowed pattern* of $t$ (see Figure \[fig:GMTS-forbidden\] for instance).
We denote by $\mathcal{T}$ (or $\mathcal{A}^{\Sigma^*}$) the set of all labeled trees on $\mathcal{A}$. The shift transformation $\sigma: \Sigma^* \times \mathcal{T} \to \mathcal{T}$ is defined by $(\sigma_w t)_x = t_{wx}$ for all $w, x \in \Sigma^*$. Given a collection of patterns $\mathcal{F}$, let $\mathsf{X}_{\mathcal{F}}$ denote the set of trees avoiding any element of $\mathcal{F}$. A subset $X \subseteq \mathcal{T}$ is called a *tree-shift* if $X = \mathsf{X}_{\mathcal{F}}$ for some $\mathcal{F}$. We say that $\mathcal{F}$ is *a set of forbidden patterns* (or *a forbidden set*) of $X$. A tree-shift $X = \mathsf{X}_{\mathcal{F}}$ is called a *tree-shift of finite type* (TSFT) if the forbidden set $\mathcal{F}$ is finite; we say that $\mathsf{X}_{\mathcal{F}}$ is a *Markov tree-shift* if $\mathcal{F}$ consists of two-blocks. Ban and Chang [@BC-TAMS2017] demonstrate that every TSFT can be treated as a Markov tree-shift after recoding, which extends a classical result in symbolic dynamical systems.
\[prop:TSFT-is-vertex-shift-is-Markov\] Every tree-shift of finite type is conjugated to a Markov tree-shift.
Proposition \[prop:TSFT-is-vertex-shift-is-Markov\] indicates that the investigation of Markov tree-shifts is essential for characterizing TSFTs. For the rest of this paper, a TSFT is referred to as a Markov tree-shift unless otherwise stated.
Entropy of tree-shifts
----------------------
An important invariant of shift spaces is topological entropy, which measures the growth rate of the number of the admissible patterns. Such an invariant reflects the complexity on its own right, we refer readers to [@ASY-1997] for more details. The *entropy* of tree-shifts is defined as $$\label{eq:entropy-tree-shift}
h(X)=\limsup_{n \rightarrow \infty} \frac{\ln^2 |B_{n}(X)|}{n},$$where $B_{n}(X)$ is the collection of $n$-blocks of $X$, $|B_{n}(X)|$ means the cardinality of $B_{n}(X)$, and $\ln^2 = \ln \circ \ln $. Ban and Chang indicate that the limit $h(X)=\lim_{n\rightarrow \infty} \ln^2 |B_{n}(X)|/n$ exists if $X$ is a TSFT and $h(X) \in \{0, \ln 2\}$ for each TSFT $X$ when $d=2$ [@BC-2017; @BC-N2017]; furthermore, a sufficient condition for positive entropy is revealed [@BC-JMP2017]. For the computation of entropy, Ban and Chang introduce the notion of *system of nonlinear recursive equations*.
\[def:SNRE\] Given $k \in \mathbb{N}$, we say that a sequence $\{\alpha_{1;n}, \alpha_{2;n}, \ldots, \alpha_{k;n}\}_{n \in \mathbb{N}}$ forms a *system of nonlinear recursive equations (SNRE)* of degree $(d, k)$ if $$\alpha_{i;n} = F_i(n) \quad \text{for} \quad n \geq 2, 1 \leq i \leq k,$$with initial condition $\alpha_{i;1} \in \mathbb{N}$ for $1 \leq i \leq k$, where $$F_i(n) = \sum_{c_1 + c_2 + \cdots + c_k = d} r_{i; c_1, \ldots, c_k} \alpha_{1; n-1}^{c_1} \alpha_{2; n-1}^{c_2} \cdots \alpha_{k; n-1}^{c_k}$$ with $r_{i; c_1, \ldots, c_k} \in \mathbb{Z}^+$.
Let $F = \{F_1, F_2, \ldots, F_k\}$ be defined in Definition \[def:SNRE\]. We also say that the sequence $\{\alpha_{1;n}, \alpha_{2;n}, \ldots, \alpha_{k;n}\}_{n \geq \mathbb{N}}$ is defined by $F$. For simplicity, $F$ is called the SNRE corresponding to $X$. Suppose that $F$ is given. For $1 \leq i \leq k$, we define the *indicator vector* $v_{F_i}$ of $F_i$ as $v_{F_i} = (r_{i; c_1, \ldots, c_k})$. Note that the indicator vector $v_{F_i}$ is unique up to permutation. For the convenience, we represent the indicator vector with respect to the lexicographic order. The matrix $I_F = \begin{pmatrix}
v_{F_1} \\
v_{F_2} \\
\vdots \\
v_{F_k}
\end{pmatrix}$ is called the *indicator matrix* of $F$. For example, suppose that the sequence $\{\alpha_{1;n}, \alpha_{2;n}\}_{n \geq \mathbb{N}}$ forms the SNRE $$\label{eq:example-SNRE}
\left\{
\begin{array}{l}
\alpha_{1;n} = F_1 = \alpha_{1;n-1}^{2} +\alpha_{2;n-1}^{2}, \\
\alpha_{2;n} = F_2 = 2 \alpha_{1;n-1} \alpha_{2;n-1}, \\
\alpha_{1;1} = \alpha_{2;1} = 1.%
\end{array}%
\right.$$ Then the corresponding indicator matrix is $$I_{F}= \begin{pmatrix}
1 & 0 & 1 \\
0 & 2 & 0
\end{pmatrix}.$$
Suppose $X$ is a TSFT over $\mathcal{A} = \{a_1, a_2, \ldots, a_k\}$. Let $$X_i = \{t \in X: t_{\epsilon} = a_i\} \quad \text{and} \quad \gamma_{i;n} = |B_n(X_i)|$$ for $1 \leq i \leq k$. It follows immediately that $\{\gamma_{1;n}, \gamma_{2;n}, \ldots, \gamma_{k;n}\}_{n \in \mathbb{N}}$ forms an SNRE. Furthermore, every SNRE of degree $(d, k)$ can be realized via a TSFT (cf. [@BC-N2017]). Let $F = \{F_1, \ldots, F_k\}$ be the representation of the SNRE of $X$. A subsystem called a *reduced system of nonlinear recursive equations* of $F$ is defined as follows.
\[def:RSNRE\] Suppose $X$ is a TSFT, and $F$ is the SNRE corresponding to $X$ with indicator matrix $I_F$. We call $E$ a *reduced system of nonlinear recurive equations* of $F$ if $E$ is an SNRE such that $I_{E}$ is a binary matrix satisfying the following conditions.
1. $I_{E} \leq I_{F}$;
2. there is exactly one nonzero entry in each row of $I_{E}$;
3. the initial condition of the sequence defined by $E$ is the same as the one defined by $F$.
Herein, two matrices $A, B \in \mathbb{Z}^{m \times n}$ with $A \leq B$ means that $A(i, j) \leq B(i, j)$ for $1 \leq i \leq m, 1 \leq j \leq n$.
Beyond defining the indicator matrix of an SNRE, a $k\times k$ nonnegative integral matrix $M_E$, called the *weighted adjacency matrix*, of a reduced SNRE $E$, is defined as $$\label{eq:weighted-matrix}
M_E (i, j) = \max \{m: \alpha_{j;n-1}^m|E_i\}, \quad 1 \leq i, j \leq k.$$ For example, consider the SNRE $F=\{F_i\}_{i=1}^{2}$ defined in with indicator matrix $$I_{F}= \begin{pmatrix}
1 & 0 & 1 \\
0 & 2 & 0
\end{pmatrix}.$$ Then a reduced SNRE $E$ of $F$ with indicator matrix $$I_{E} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix}$$ defines a sequence $\{\beta_{1;n}, \beta_{2;n}\}_{n \in \mathbb{N}}$ as follows. $$\left\{
\begin{array}{l}
\beta_{1;n} = E_1 = \beta_{1;n-1}^{2}, \\
\beta_{2;n} = E_2 = \beta_{1;n-1} \beta_{2;n-1}, \\
\beta_{1;1} = \beta_{2;1} = 1.
\end{array}%
\right.$$ Furthermore, the weighted adjacency matrix $M_E$ of $E$ is $$M_E = \begin{pmatrix}
2 & 0 \\
1 & 1
\end{pmatrix}.$$
A symbol $a_i \in \mathcal{A}$ is called *essential* if $\gamma_{i;n} \geq 2$ for some $n \in \mathbb{N}$; otherwise, we say that $a_i$ is *inessential*. Suppose that, for a TSFT $X$ over $\mathcal{A}$, each symbol in $\mathcal{A}$ is essential.
\[thm:algorithm-entropy\] Let $X$ be a TSFT and let $F$ be the representation of the SNRE of $X$. If every symbol is essential, then $$\label{eq:algorithm-entropy}
h(X) = \max \{\ln \rho _{M_E}: E \text{ is a reduced SNRE of } F\},$$ where $M_E$ is the weighted adjacency matrix of $E$ and $\rho _{M_E}$ is the spectral radius of $M_E$.
Suppose that, for a TSFT $X$, there are some inessential symbols, say, $a_{p_1}, \ldots, a_{p_j}$. Ban and Chang demonstrate that Theorem \[thm:algorithm-entropy\] still works provided, in , $M_E$ is replaced by $M_E'$, where $M_E'$ is the matrix obtained by deleting all the rows and columns indexed by those inessential symbols. Readers are referred to [@BC-2017] for more details.
Suppose that $d = 3, k = 4$. Let $X$ be a TSFT corresponds to the SNRE $$\left\{
\begin{array}{l}
\gamma_{1;n} = \gamma_{1;n-1} \gamma_{2;n-1} \gamma_{4;n-1} + \gamma_{4;n-1}^3, \\
\gamma_{2;n} = \gamma_{3;n-1} \gamma_{4;n-1}^2 + \gamma_{4;n-1}^3, \\
\gamma_{3;n} = \gamma_{1;n-1}^{2} \gamma_{2;n-1} + \gamma_{4;n}^3, \\
\gamma_{4;n} = \gamma_{4;n-1}^3, \\
\gamma_{i;1} = 1, 1 \leq i \leq 4.
\end{array}%
\right.$$ It is easily seen that $a_1, a_2, a_3$ are essential symbols and $a_4$ is inessential. The weighted adjacency matrix $M_E$ of the reduced SNRE $E$ which reaches the maximum in is $$M_E = \begin{pmatrix}
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 2 \\
2 & 1 & 0 & 0 \\
0 & 0 & 0 & 3
\end{pmatrix}.$$ Since $a_4$ is inessential, we replace $M_E$ with $$M_E' = \begin{pmatrix}
1 & 1 & 0 \\
0 & 0 & 1 \\
2 & 1 & 0
\end{pmatrix}.$$ Theorem \[thm:algorithm-entropy\] shows that the entropy of $X$ is $h(X) = \ln \rho_{M_E'} \approx \ln 1.839$, where $\rho_{M_E'}$ is the maximal root of $x^3 - x^2 - x - 1 = 0$.
\[prop:essential-symbols-ln-d\] Suppose $X$ is a tree-shift of finite type and let $F$ be the representation of the SNRE of $X$. If every symbol is essential, then $h(X) = \ln d$.
It suffices to show that there exists a reduced SNRE $E$ of $F$ such that $h(E) = \ln d$ since $h(X) \leq \ln d$ (cf. [@BC-N2017]). Let $E$ be a reduced SNRE of $F$. Then the weighted adjacency matrix $M_E$ satisfies $\sum\limits_{j=1}^k M_E(i, j) = d$ for $1 \leq i \leq d$. Since every symbol is essential, Theorem \[thm:algorithm-entropy\] infers that $h(X) \geq \ln \rho_{M_E}$, where $\rho_{M_E}$ is the spectral radius of $M_E$. This completes the proof since $\rho_{M_E} = d$.
Proposition \[prop:essential-symbols-ln-d\] infers the rigidity of entropy since it is a constant ($\ln d$) whenever there is no inessential symbol. Let $$\label{eq:matrix-row-sum-less-than-equal-to-d}
D = \{M \in \mathcal{M}_{\ell \times \ell}(\mathbb{Z}^+): \sum_{q=1}^{\ell} M(p, q) \leq d \text{ for } 1 \leq p \leq \ell, \ell \leq k\}$$ consist of nonnegative integral matrices whose dimension is less than or equal to $k$, and the summation of each row is less than or equal to $d$. Theorem \[thm:entropy-set-2symbol\] illustrates a complete characterization of the entropy of TSFTs.
\[thm:entropy-set-2symbol\] Let $H = \{h(X): X \text{ is a TSFT}\}$ be the entropy spectrum of TSFTs and $D$ is defined as in . Then $$\label{eq:entropy-spectrum-TSFT}
H = \{\ln \rho: \rho \text{ is the spectral radius of } M \in D\}.$$ More specifically, $H = \{\ln \lambda: 1 \leq \lambda \leq d\}$ if $k = 2$.
We start with demonstrating the case where $k = 2$ to clarify our idea. If $a_1$ and $a_2$ are both essential symbols, Proposition \[prop:essential-symbols-ln-d\] indicates that $h(X) = \ln d$. On the other hand, it is easily seen that $h(X) = 0$ provided $a_1$ and $a_2$ are both inessential. It remains to consider the case where exactly one symbol is essential.
Without loss of generality, we assume that $a_1$ is the essential symbol. It follows immediately from $a_2$ being inessential that, if $u \in \mathcal{F}$ such that $u_{\epsilon} = a_2$, $u_x = a_1$ for some $x = 1, 2, \ldots, d$. The SNRE of $X$ is then as follows. $$\left\{
\begin{aligned}
\gamma_{1;n} &= \sum_{c=0}^d \ell_c \gamma_{1;n-1}^c \gamma_{2;n-1}^{d-c}, \\
\gamma_{2;n} &= \gamma_{2;n-1}^d, n \geq 2, \\
\gamma_{i;1} &= 1, 1 \leq i \leq 2.
\end{aligned}
\right.$$ Since $a_1$ is essential, there exists $c < d$ such that $\ell_c > 0$. Let $\overline{c} = \max\{c: \ell_c > 0\}$ and let $E$ be the representation of the following reduced SNRE. $$\left\{
\begin{aligned}
\beta_{1;n} &= \ell_{\overline{c}} \beta_{1;n-1}^{\overline{c}} \beta_{2;n-1}^{d-\overline{c}}, \\
\beta_{2;n} &= \beta_{2;n-1}^d.
\end{aligned}
\right.$$ It follows from $M_E = \begin{pmatrix}
\overline{c} & d - \overline{c} \\
0 & d
\end{pmatrix}$ and $M_E' = (\overline{c})$ that $h(X) = \ln \overline{c}$. This shows that $H \subseteq \{\ln \lambda: 1 \leq \lambda \leq d\}$.
Conversely, for $1 \leq c \leq d$, let $X$ be a TSFT correspond to the SNRE $$\left\{
\begin{aligned}
\gamma_{1;n} &= \gamma_{1;n-1}^c \gamma_{2;n-1}^{d-c} + \gamma_{2;n-1}^d, \\
\gamma_{2;n} &= \gamma_{2;n-1}^d, \\
\gamma_{i;1} &= 1, 1 \leq i \leq 2.
\end{aligned}
\right.$$ Then $h(X) = \ln c$. The proof of $H = \{\ln \lambda: 1 \leq \lambda \leq d\}$ is thus complete.
Generally, the SNRE of $X$ is seen as $$\left\{
\begin{aligned}
&\gamma_{i;n} = \sum_{\mathbf{c}} \ell_{\mathbf{c}} \prod_{j=1}^k \gamma_{j;n-1}^{c_{i,j}}, \mathbf{c} = (c_{i,j}) \text{ satisfies } \sum_{j=1}^k c_{i, j} = d, n \geq 2, \\
&\gamma_{i;1} = 1, 1 \leq i \leq k.
\end{aligned}
\right.$$ It suffices to consider the case where there is inessential symbol. Without loss of generality, we may assume that $a_1, \ldots, a_{\ell}$ are essential and $a_{\ell + 1}, \ldots, a_k$ are inessential for some $1 \leq \ell \leq k-1$. For $1 \leq i \leq k$, let $\overline{\mathbf{c}}_i = (c_{i, 1}, \ldots, c_{i, k})$ satisfy $\sum_{j=1}^{\ell} c_{i,j} \geq \sum_{j=1}^{\ell} c_{i,j}'$ for all $\mathbf{c}' = (c_{i, 1}', \ldots, c_{i, k}')$, and let $E$ be the SNRE $$\left\{
\begin{aligned}
&\beta_{i;n} = \ell_{\mathbf{c}_i} \prod_{j=1}^k \gamma_{j;n-1}^{c_{i,j}}, n \geq 2, \\
&\beta_{i;1} = 1, 1 \leq i \leq k.
\end{aligned}
\right.$$ It is seen that $h(X) = \ln \rho$, where $\rho$ is the spectral radius of $M = (c_{i, j})_{1 \leq i, j \leq \ell}$. This elaborates $$H \subseteq \{\ln \rho: \rho \text{ is the spectral radius of } M \in D\}.$$ The demonstration of $H \supseteq \{\ln \rho: \rho \text{ is the spectral radius of } M \in D\}$ is similar to the discussion above, thus it is omitted. This completes the proof.
Suppose that $X = \mathsf{X}_{\mathcal{F}}$ is a TSFT over $\mathcal{A} = \{a_1, \ldots, a_k\}$. Recall that $\mathcal{F}$ consists of $2$-blocks. Let $\mathcal{B} = \mathcal{A}^{\Delta_1} \setminus \mathcal{F}$. For $\mathcal{A}' \subseteq \mathcal{A}$, set $$\mathcal{B}|_{\mathcal{A}'} = \{u \in \mathcal{B}: u_x \in \mathcal{A}' \text{ for } x \in \Delta_1\},$$ and let $X|_{\mathcal{A}'}$ denote the subshift generated by $\mathcal{B}|_{\mathcal{A}'}$. We say that $\mathcal{A}'$ is essential if every symbol $a \in \mathcal{A}'$ is essential. This section ends with Corollary \[cor:loop-imply-ln-d\], which comes immediately from the proof of Theorem \[thm:entropy-set-2symbol\] and is useful in the investigation of the entropy minimality problem of neural networks on Cayley trees.
\[cor:loop-imply-ln-d\] Suppose that $X = \mathsf{X}_{\mathcal{F}}$ is a TSFT over $\mathcal{A} = \{a_1, \ldots, a_k\}$. Then $h(X) = \ln d$ if and only if $X|_{\mathcal{A}'}$ is nontrivial for some essential set $\mathcal{A}' \subseteq \mathcal{A}$. More specifically, when $k=2$, $h(X) = \ln d$ if and only if $\mathcal{A}$ is essential or the $2$-block $u \in \mathcal{B}$ with $u_x = a$ for $x \in \Delta_1$ and $a$ is essential.
Obviously, $X|_{\mathcal{A}'}$ being nontrivial for some essential set $\mathcal{A}' \subseteq \mathcal{A}$ infers that $h(X) = \ln d$. For the converse direction, it suffices to show the case where $k = 2$, the general cases can be derived analogously. If $a_1$ and $a_2$ are inessential, then $h(X) = 0$, which is a contradiction. Without loss of generality, we may assume that $a_1$ is essential.
If $a_2$ is essential, Proposition \[prop:essential-symbols-ln-d\] demonstrates that $h(X) = \ln 2$. Otherwise, $h(X) = \ln d$ infers that the system $$\left\{
\begin{aligned}
\gamma_{1;n} &= \gamma_{1;n-1}^d, \\
\gamma_{2;n} &= \gamma_{2;n-1}^d,
\end{aligned}
\right.$$ must be a reduced SNRE of the original system. This derives the desired result.
Neural Networks on Cayley Trees
===============================
The overwhelming majority of models of neural networks are defined on $\mathbb{Z}^n$ lattice. While it is known that the characteristic shape of neurons is tree [@GKC-PCB2009], this section considers neural networks defined on Cayley trees. A *neural network on Cayley tree* (CTNN) is represented as $$\label{eq:cnn-tree}
\dfrac{d}{dt} x_w(t) = - x_w(t) + z + \sum_{v \in \mathcal{N}} a_v f(x_{wv}(t)), \quad w \in \Sigma^*,$$ for some finite set $\mathcal{N} \subset \Sigma^*$ known as the neighborhood, $v \in \mathcal{N}$, and $t\geq 0$. Herein, $x_w(t) \in \mathbb{R}$ represents the internal status of neuron at $w$; the map $f(s)$ is called the *output function* or *activation function*, and $z \in \mathbb{R}$ is called the *threshold*. The weighted parameters $A = (a_v)_{v \in \mathcal{N}}, a_v \in \mathbb{R},$ is called the *feedback template*, and Figure \[fig:TreeCNN\] shows the connection of a binary CTNN with the nearest neighborhood. Equation is derived by adopting Hopfield’s neural network ([@Hopfield-PNASU1982]) on the Cayley tree. Normatov and Rozikov [@NR-MN2006] show that harmonic functions on Cayley trees, which is a discrete time version of , are periodic with respect to normal subgroups of finite index. The present paper investigates the complexity of output solutions with respect to the output function $$\label{eq:piecewise-linear}
f(s)=\dfrac{1}{2}(|s+1|-|s-1|)$$ which is proposed by Chua and Yang [@CY-ITCS1988] and is widely applied to many disciplines such as signal propagation between neurons, pattern recognition, and self-organization.
![A neural network with the nearest neighborhood defined on binary trees. In this case, the neighborhood $\mathcal{N} = \{\epsilon, 0, 1\}$ and $a = a_{\epsilon}$.[]{data-label="fig:TreeCNN"}](TreeNN-EntropyMinimal-20171215-pics)
A *mosaic solution* $x = (x_w)_{w \in \Sigma^*}$ of is an equilibrium solution which satisfies $|x_w| > 1$ for all $w \in \Sigma^*$; its corresponding pattern $y = (y_w)_{w \in \Sigma^*} = (f(x_w))_{w \in \Sigma^*}$ is called a *mosaic output pattern*. Since the output function is piecewise linear with $f(s)=1$ (resp. $-1$) if $s \geq 1$ (resp. $s \leq -1$), the output of a mosaic solution $x = (x_w)_{w \in \Sigma^*}$ must be an element in $\left\{ -1,+1\right\}^{\Sigma^*}$, which is why we call it a *pattern*. Given a CTNN, we refer to $\mathbf{Y}$ as the output solution space; namely, $$\mathbf{Y} = \left\{ (y_w)_{w \in \Sigma^*}: y_w = f(x_w) \text{ and } (x_w)_{w \in \Sigma^*} \text{ is a mosaic solution of } \eqref{eq:cnn-tree} \right\} .$$
Learning problem of neural networks on Cayley trees
---------------------------------------------------
Learning problems (also called the inverse problems) are some of the most investigated topics in a variety of disciplines. From a mathematical point of view, determining whether a given collection of output patterns can be exhibited by a CTNN is essential for the study of learning problems. This section reveals the necessary and sufficient conditions for the capability of exhibiting the output patterns of CTNNs. The discussion is similar to the investigation in [@BC-NN2015; @BCLL-JDE2009; @Chang-ITNNLS2015], thus we only sketch the key procedures of the learning problems of CTNNs with the nearest neighborhood, namely, $\mathcal{N} = \Sigma \bigcup \{e\}$, for the compactness and self-containedness of this paper. A CTNN with the nearest neighborhood is realized as $$\label{eq:cnn-tree-nearest-nbd}
\dfrac{d}{dt} x_w(t) = - x_w(t) + z + a f(x_{w}(t)) + \sum_{i=1}^d a_i f(x_{wi}(t)),$$ where $a, a_1, \ldots, a_d \in \mathbb{R}$ and $w \in \Sigma^*$. Considering the mosaic solution $x = (x_w)_{w \in \Sigma^*}$, the necessary and sufficient conditions for $y_w = f(x_w) = 1$ is $$\label{eq:cnn-state+}
a - 1 + z > - \sum_{i=1}^d a_i y_{wi}.$$ Similarly, the necessary and sufficient conditions for $y_w = f(x_w) = -1$ is $$\label{eq:cnn-state-}
a - 1 - z > \sum_{i=1}^d a_i y_{wi}.$$
Let $$V^n = \{ v \in \mathbb{R}^n : v = (v_1, \ldots, v_n), \text{ and } |v_i| = 1, 1 \leq i \leq n \}.$$ Let $\alpha = (a_1, \ldots, a_d)$ represent the feedback template without the self-feedback parameter $a$. The basic set of admissible local patterns with the “$+$" state in the parent neuron is denoted as $$\widetilde{\mathcal{B}}_+( A, z) = \{v \in V^d: a - 1 + z > -\alpha \cdot v \},$$ where “$\cdot$" is the inner product in Euclidean space. Similarly, the basic set of admissible local patterns with the “$-$" state in the parent neuron is denoted as $$\widetilde{\mathcal{B}}_-( A, z) = \{v \in V^d: a - 1 - z > \alpha \cdot v \}.$$ Furthermore, the admissible local patterns induced by $(A, z)$ can be denoted by $$\mathcal{B}(A, z) = \mathcal{B}_+( A, z) \bigcup \mathcal{B}_-( A, z),$$ where $$\begin{aligned}
\mathcal{B}_+( A, z) &= \{v: v_{\epsilon} = 1 \text{ and } (v_1, \ldots, v_d) \in \widetilde{\mathcal{B}}_+( A, z)\}, \\
\mathcal{B}_-( A, z) &= \{v: v_{\epsilon} = -1 \text{ and } (v_1, \ldots, v_d) \in \widetilde{\mathcal{B}}_-( A, z)\}.\end{aligned}$$ Note that $\mathcal{B}(A, z)$ consists of two-blocks over $\mathcal{A} = \{1, -1\}$. For simplicity, we omit the parameters $(A, z)$ and refer to $\mathcal{B}$ as $\mathcal{B}(A, z)$.
Suppose $U$ is a subset of $V^n$, where $n \geq 2 \in \mathbb{N}$. Let $U^c = V^n \setminus U$. We say that $U$ satisfies the *linear separation property* if there exists a hyperplane $H$ that separates $U$ and $U^c$. More precisely, $U$ satisfies the separation property if and only if there exists a linear functional $g(z_1, z_2, \ldots, z_n) = c_1 z_1 + c_2 z_2 + \cdots + c_n z_n$ such that $$g(v) > 0 \quad \text{for} \quad v \in U \quad \text{and} \quad g(v) < 0 \quad \text{for} \quad v \in U^c.$$ Figure \[fig:separation\] interprets those $U \subset V^2$ satisfying the linear separation property.
![Suppose $U \subseteq V^2 = \{-1, 1\}^2$ presents a set of allowable local patterns. Linear separation property infers that $U$ and $V^2 \setminus U$, geometrically, must be separated by a straight line. Hence, there are only $12$ choices of $U$ when $d = 2$.[]{data-label="fig:separation"}](TreeNN-EntropyMinimal-20171215-pics)
Proposition \[prop:separation-property\] elucidates the necessary and sufficient condition for the learning problems of CTNNs; such a property holds for arbitrary neighborhood $\mathcal{N}$ provided $\mathcal{N}$ is prefix-closed. The proof of Proposition \[prop:separation-property\] is similar to the discussion in [@BC-NN2015], thus it is omitted.
\[prop:separation-property\] A collection of patterns $\mathcal{B} = \mathcal{B}_+ \bigcup \mathcal{B}_-$ can be realized in if and only if either of the following conditions is satisfied:
1. $-\mathcal{B}_+ \subseteq \mathcal{B}_-$ and $\mathcal{B}_-$ satisfies linear separation property;
2. $-\mathcal{B}_- \subseteq \mathcal{B}_+$ and $\mathcal{B}_+$ satisfies linear separation property.
Let $$\label{eq:parameter-space}
\mathbb{R}^{d+2} = \{ (A, z) |\ A \in \mathbb{R}^{d+1}, z \in \mathbb{R} \}$$ denote the parameter space. Theorem \[thm:partition\] demonstrates that $\mathbb{R}^{d+2}$ can be partitioned into finitely equivalent sub-regions such that two sets of parameters induce identical basic sets of admissible local patterns if and only if they belong to the same partition in the parameter space. We skip the proof of Theorem \[thm:partition\] for the compactness of this paper since the demonstration is similar to ths discussion in [@HJLL-IJBCASE2000].
\[thm:partition\] There exists a positive integer $K = K(d)$ and a unique collection of open subsets $\{P_i\}_{i=1}^K$ of the parameter space satisfying
1. $\mathbb{R}^{d+2} = \bigcup\limits_{i=1}^K \overline{P}_k$;
2. $P_i \bigcap P_j = \varnothing$ for all $i \neq j$;
3. $\mathcal{B}(A, z) = \mathcal{B}(A', z')$ if and only if $(A, z), (A', z') \in P_i$ for some $1 \leq i \leq K$.
Herein, $\overline{P}$ indicates the closure of $P$ in $\mathbb{R}^{d+2}$.
A straightforward examination asserts that, whenever a set of parameters $(A, z)$ is given, the output solution space $\mathbf{Y}$ is a Markov tree-shift since $\mathbf{Y} = \mathsf{X}_{\mathcal{F}}$, where $\mathcal{F} = \{-1, 1\}^{\Delta_1} \setminus \mathcal{B}(A, z)$.
We consider the case where $d = 2$ as an example. Note that, whenever the parameters $a_1$ and $a_2$ are determined, and partition the $a$-$z$ plane into $25$ regions; the “order" (i.e., the relative position) of lines $a - 1 +(-1)^{\ell} z = (-1)^{\ell} (a_1 y_{w1} + a_2 y_{w2})$, $\ell = 1, 2$, can be uniquely determined by the following procedures:
1) The signs of $a_1, a_2$ (i.e., the parameters are positive or negative).
2) The magnitude of $a_1, a_2$ (i.e., $|a_1| > |a_2|$ or $|a_1| < |a_2|$).
This partitions $a$-$z$ plane into $8 \times 25 = 200$ sub-regions. According to Theorem \[thm:partition\], the parameter space $\mathcal{P}^4$ is partitioned into less than $200$ equivalent sub-regions.
Entropy bifurcation of neural networks on Cayley trees
------------------------------------------------------
Suppose that, for each neuron of a neural network on Cayley tree, we substitute its output pattern $1$ (resp. $-1$) with $+$ (resp. $-$); then the output solution space $\mathbf{Y}$ of a CTNN is a Markov tree-shift over $\mathcal{A} = \{+, -\}$. We denote the TSFT $\mathbf{Y}$ by $\mathbf{Y}_{\mathcal{B}}$ when we want to emphasize the basic set of admissible patterns $\mathcal{B}$. This subsection investigates the entropy and the entropy bifurcation diagram of $\mathbf{Y}$.
We start with the following lemma, for which the proof can be done via straightforward elucidation, thus it is omitted.
\[lem:zero-entropy-trivial case\] Suppose that $\mathbf{Y}_{\mathcal{B}}$ is an output solution space such that $u_{\epsilon} = v_{\epsilon}$ for all $u, v \in \mathcal{B}$. Then $h(\mathbf{Y}_{\mathcal{B}}) = 0$.
Based on Lemma \[lem:zero-entropy-trivial case\], we may assume that, for each basic set of admissible local patterns $\mathcal{B}$, there exist $u, v \in \mathcal{B}$ such that $u_{\epsilon} \neq v_{\epsilon}$. We call such a set of local patterns $\mathcal{B}$ *nontrivial*; an output solution space $\mathbf{Y}_{\mathcal{B}}$ is called *nontrivial* if its corresponding set of local patterns $\mathcal{B}$ is nontrivial.
\[thm:CTNN-entropy-set\] Suppose that $\mathbf{Y}$ is an output solution spaces of . Then $h(\mathbf{Y}) = 0$ or $\ln d$.
Lemma \[lem:zero-entropy-trivial case\] suggests that we only need to consider nontrivial output solution spaces; that is, there exist $u, v \in \mathcal{B}$ such that $u_{\epsilon} = +$ and $v_{\epsilon} = -$. Proposition \[prop:essential-symbols-ln-d\] demonstrates that $h(\mathbf{Y}) = \ln d$ if both symbols $+$ and $-$ are essential. It remains to consider the case where exactly one symbol is inessential.
Without loss of generality, we may assume that $-$ is inessential. In other words, if $u \in \mathcal{B}$ satisfies $u_{\epsilon} = -$, then $u_i = -$ for $1 \leq i \leq d$. Proposition \[prop:separation-property\] shows that there exists $v \in \mathcal{B}$ such that $v_{\epsilon} = v_1 = \cdots = v_d = +$; Corollary \[cor:loop-imply-ln-d\] indicates that $h(\mathbf{Y}_{\mathcal{B}}) = \ln d$. This completes the proof.
The well-known entropy minimality problem investigates when the entropy of any proper subspace is strictly smaller than the entropy of the original shift space. For the case of CTNNs, the entropy minimality problem is equivalent to investigating under what condition $h(\mathbf{Y}_{\mathcal{B}'}) < h(\mathbf{Y}_{\mathcal{B}})$, where $\mathcal{B}'$ is obtained by deleting a pattern in $\mathcal{B}$. Furthermore, it follows from Theorem \[thm:CTNN-entropy-set\] that the change of entropy is from $\ln d$ to $0$; in other words, simply removing a pattern from the basic set of admissible local patterns $\mathcal{B}$ makes significant influence to the original space. Equation characterizes those parameters which make such a tremendous influence.
The discussion in the previous subsection shows that, once the parameters $a_1, \ldots, a_d$ are fixed, and partition the $a$-$z$ plane into $(2^d+1)^2$ regions. We encode these regions by $[p, q]$ for $0 \leq p, q \leq 2^d$ and denote the corresponding basic set of admissible local patterns as $\mathcal{B}_{[p, q]}$. More specifically, $\mathcal{B}_{[p, q]} = \mathcal{B}_{[p, q]; +} \bigcup \mathcal{B}_{[p, q]; -}$ which satisfies $|\mathcal{B}_{[p, q]; +}| = p$ and $|\mathcal{B}_{[p, q]; -}| = q$. For simplicity, we denote $\mathbf{Y}_{\mathcal{B}_{[p, q]}}$ by $\mathbf{Y}_{[p, q]}$. The following proposition comes immediately.
Suppose that the parameters $a_1, \ldots, a_d$ are given. Then $\mathbf{Y}_{[p, q]} \cong \mathbf{Y}_{[q, p]}$ for $0 \leq p, q \leq 2^d$.
Since $\mathcal{B}_{[p, q]; +} = - \mathcal{B}_{[q, p]; -}$ and $\mathcal{B}_{[p, q]; -} = - \mathcal{B}_{[q, p]; +}$, the desired results is then derived.
Suppose that the parameters $a_1, \ldots, a_d$ are given. A pair of parameters $(a, z)$ is called *critical* if, for each $r > 0$, there exists $(a', z'), (a'', z'') \in B_r (a, z)$ such that $h(\mathbf{Y}_{\mathcal{B}'}) = \ln d$ and $h(\mathbf{Y}_{\mathcal{B}''}) = 0$, where $B_r (a, z)$ is the $r$-ball centered at $(a, z)$ and $\mathcal{B}' = \mathcal{B}(A', z'), \mathcal{B}'' = \mathcal{B}(A'', z''), A' = (a', a_1, \ldots, a_d)$, and $A'' = (a'', a_1, \ldots, a_d)$.
\[prop:entropy-region-W-equation\] Suppose that the parameters $a_1, \ldots, a_d$ are given. Then $h(\mathbf{Y}_{[p, q]}) = 0$ if and only if $$\min\{p, q\} = 0 \text{ or } \max\{p, q\} = 1,$$ where $0 \leq p, q \leq 2^d$. Furthermore, let $\ell$ be the index such that $|a_{\ell}| = \min \{|a_i|: 1 \leq i \leq d\}$. Then $(a, z)$ is critical if and only if $$\label{eq:W-shape}
a - 1 = \big||z| - |a_{\ell}|\big| - \sum_{i \neq \ell} |a_i|.$$
Observe that the proof of Theorem \[thm:CTNN-entropy-set\] demonstrates that $h(\mathbf{Y}_{[p, q]}) = 0$ if and only if $\min\{p, q\} = 0$ or $\max\{p, q\} = 1$. It remains to show that $(a, z)$ is critical if and only if $(a, z)$ satisfies .
Let $C = \{\sum_{i=1}^d \ell_i a_i: \ell_i \in \{-1, 1\} \text{ for all } i\}$, and let $$K_1 = \max C \quad \text{and} \quad K_2 = \max C \setminus \{K_1\}$$ be the largest and the second largest elements in $C$, respectively. A careful but straightforward verification asserts that $(a, z)$ is critical if and only if $$a - 1 = \left| |z| -\dfrac{K_1 - K_2}{2}\right| - \dfrac{K_1 + K_2}{2}.$$ (See Figure \[fig:entropy-diagram-general-case\] for more information.) The desired result follows from the fact that $$K_1 = \sum_{i=1}^d |a_i| \quad \text{and} \quad K_2 = \sum_{i \neq \ell} |a_i| - |a_{\ell}|.$$
![Entropy bifurcation diagram of neural networks on Cayley trees. Whenever all the parameters are fixed except $a$ and $z$, the $a$-$z$ plane is partitioned into $(2^d + 1)^2$ regions. A CTNN has entropy $\ln d$ if and only if the parameter $(a, z)$ is above the red line.[]{data-label="fig:entropy-diagram-general-case"}](TreeNN-EntropyMinimal-20171215-pics)
As the end of this section, we give the following example to clarify the investigation of entropy bifurcation diagrams of neural networks on the binary Cayley tree.
A neural network on the binary Cayley tree is represented as $$\dfrac{d}{dt} x_w(t) = - x_w(t) + z + a f(x_{w}(t)) + a_1 f(x_{w1}(t)) + a_2 f(x_{w2}(t)),$$ where $w \in \Sigma^*$ and $\Sigma = \{1, 2\}$. The necessary and sufficient conditions for $y_w = 1$ and $y_w = -1$ are $$a - 1 + z > - (a_1 y_{w1} + a_2 y_{w2})
\quad \text{and} \quad
a - 1 - z > a_1 y_{w1} + a_2 y_{w2},$$ respectively. Suppose that $a_1, a_2$ satisfy $0 < -a_1 < a_2$. It follows from $a_1 - a_2 < -a_1 - a_2 < a_1 + a_2 < -a_1 + a_2$ that, whenever $a$ and $z$ are fixed, the “ordered” basic set of admissible local patterns $\mathcal{B} = \mathcal{B}_+ \bigcup \mathcal{B}_-$ must obey $$\mathcal{B}_+ \subseteq \{(+, -, +), (+, +, +), (+, -, -), (+, +, -)\}$$ and $$\mathcal{B}_- \subseteq \{(-, +, -), (-, -, -), (-, +, +), (-, -, +)\}.$$
![Entropy bifurcation diagram of neural networks on the binary Cayley tree. The $a$-$z$ plane is partitioned into $25$ sub-regions.[]{data-label="fig:entropy-diagram-binary-tree"}](TreeNN-EntropyMinimal-20171215-pics)
If the parameters $a$ and $z$ locate in the region $[3, 2]$(cf. Figure \[fig:entropy-diagram-binary-tree\]), then the basic set is $$\mathcal{B}_{[3, 2]} = \{(+, -, +), (+, +, +), (+, -, -), (-, +, -), (-, -, -)\}.$$ Theorem \[thm:CTNN-entropy-set\] and Proposition \[prop:entropy-region-W-equation\] conclude that $h(\mathbf{Y}_{[3, 2]}) = \ln 2$ and $(a, z)$ is critical if and only if $a - 1 = \big| |z| + a_1 \big| - a_2$.
Conclusion and Discussion
=========================
In this paper, motivated by Gollo *et al.*’s works (cf. [@GKC-PCB2009; @GKC-PRE2012; @GKC-SR2013]), we study the dynamical behavior which tree structure neural networks are capable of. More specifically, we focus on equilibrium solutions known as mosaic solutions since they are related to the long-term memory of the brain and are applied in a wide range of disciplines. Entropy, a frequently used invariant, reveals the growth rate of the amount information stored in a (tree structure) neural network. Alternatively, positive entropy reflects that adding one more neuron stores exponential times of memory relative to the original system. A small modification of coupling weights resulting in different entropy means the neural network is sensitive or in some critical status. We elaborate the criticality of a neural network by whether or not the neural network is entropy minimal.
After demonstrating the entropy spectrum of tree structure neural networks is discrete, we illustrate the necessary and sufficient condition for determining if a neural network is sensitive. Furthermore, the formula for coupling weights of critical neural networks is indicated.
Since the activation function considered in this article is piecewise linear transformation $f(s) = \dfrac{1}{2} (|s+1| - |s-1|)$, the output patterns of mosaic solutions are binary patterns. That is, the coloring set $\mathcal{A}$ consists of only two symbols. It is of interest that what conclusion we can derive when $\mathcal{A}$ consists of $k$ symbols for some integer $k \geq 3$. Furthermore, we focus on the rooted Cayley tree as the network’s topology in the whole discussion; it is also of interest whether or not our results remain true for Bethe lattice. Related work is in preparation.
[^1]: \*Author to whom any correspondence should be addressed.
|
---
abstract: |
The assumption that the universe is homogeneous and isotropic on large scales is one of the fundamental postulates of cosmology. We have tested the large scale homogeneity of the galaxy distribution in the Sloan Digital Sky Survey Data Release One (SDSS-DR1) using volume limited subsamples extracted from the two equatorial strips which are nearly two dimensional (2D). The galaxy distribution was projected on the equatorial plane and we carried out a 2D multi-fractal analysis by counting the number of galaxies inside circles of different radii $r$ in the range $5 \, h^{-1} {\rm Mpc}$ to $150 \,
h^{-1} {\rm Mpc}$ centred on galaxies. Different moments of the count-in-cells were analysed to identify a range of length-scales ($60-70 \, h^{-1} {\rm Mpc}$ to $150 h^{-1} {\rm Mpc}$ ) where the moments show a power law scaling behaviour and to determine the scaling exponent which gives the spectrum of generalised dimension $D_q$. If the galaxy distribution is homogeneous, $D_q$ does not vary with $q$ and is equal to the Euclidean dimension which in our case is 2. We find that $D_q$ varies in the range $1.7$ to $2.2$. We also constructed mock data from random, homogeneous point distributions and from N-body simulations with bias $b=1, 1.6$ and $2$, and analysed these in exactly the same way. The values of $D_q$ in the random distribution and the unbiased simulations show much smaller variations and these are not consistent with the actual data. The biased simulations, however, show larger variations in $D_q$ and these are consistent with both the random and the actual data. Interpreting the actual data as a realisation of a biased universe, we conclude that the galaxy distribution is homogeneous on scales larger than $60-70 \, h^{-1} {\rm Mpc}$.
author:
- |
Jaswant Yadav$^1$[^1], Somnath Bharadwaj$^2$[^2], Biswajit Pandey$^2$[^3] and T.R.Seshadri$^1$[^4]\
$^1$ Department of Physics & Astrophysics, University of Delhi, Delhi 110007,India.\
$^2$ Department of Physics and Metereology and Centre for Theoretical Studies, IIT Kharagpur, 721 302, India.
title: Testing homogeneity on large scales in the Sloan Digital Sky Survey Data Release One
---
methods: numerical - galaxies: statistics - cosmology: theory - cosmology: large scale structure of universe
Introduction
============
The primary aim and objective of all galaxy redshift surveys is to determine the large scale structures in the universe. Though the galaxy distribution exhibits a large variety of structures starting from groups and clusters, extending to superclusters and an interconnected network of filaments which appears to extend across the whole universe, we expect the galaxy distribution to be homogeneous on large scales. The assumption that the universe is homogeneous and isotropic on large scales is known as the Cosmological Principle and this is one of the fundamental pillars of cosmology. In addition to determining the large scale structures, galaxy redshift surveys can also be used to verify that the galaxy distribution does indeed become homogeneous on large scales and thereby validate the Cosmological Principal. Further, these can be used to investigate the scales at which this transition to homogeneity takes place. In this paper we test whether the galaxy distribution in the SDSS-DR1 (@abaz) is [*actually*]{} homogeneous on large scales.
A large variety of methods have been developed and used to quantify the galaxy distribution in redshift surveys, prominent among these being the two-point correlation function $\xi(r)$ (@pee) and its Fourier transform the power spectrum $P(k)$. There now exist very precise estimates of $\xi(r)$ (eg. SDSS, @zevi; 2dFGRS, @haw) and the power spectrum $P(k)$ (eg. 2dFGRS, @perci; SDSS @teg2) determined from different large redshift surveys. On small scales the two point correlation function is found to be well described by the form $$\xi(r)=(\frac{r}{r_0})^{\g}
\label{eq:1}$$ where $\g =1.75\pm 0.03 $ and $ r_0=6.1\pm 0.2\, h^{-1} {\rm
Mpc}$ for the SDSS (@zevi) and $\g =1.67\pm 0.03
$ and $ r_0=5.05\pm 0.26\, h^{-1} {\rm Mpc}$ for the 2dFGRS (@haw). The power law behaviour of $\xi(r)$ suggests a scale invariant clustering pattern which would violate homogeneity if this power-law behaviour were to extend to arbitrarily large length-scales. Reassuringly, the power law form for $\xi(r)$ does not hold on large scales and it breaks down at $r> 16 h^{-1} {\rm Mpc} $ for SDSS and at $r> 20 h^{-1} {\rm Mpc} $ for 2dFGRS. The fact that the values of $\xi(r)$ fall sufficiently with increasing $r$ is consistent with the galaxy distribution being homogeneous on large scales. A point to note is that though the $\xi(r)$ determined from redshift surveys is consistent with the universe being homogeneous at large scales it does not actually test this. This is because the way in which $\xi(r)$ is defined and determined from observations refers to the mean number density of galaxies and therefore it presupposes that the galaxy distribution is homogeneous on large scales. Further, the mean density which we compute is only that on the scale of the survey. It will be equal to the mean density in the universe only if the transition to homogeneity occurs well within the survey region. To verify the large scale homogeneity of the galaxy distribution it is necessary to consider a statistical test which does not presuppose the premise which is being tested. Here we consider one such test, the “multi-fractal dimension” and apply it to the SDSS-DR1.
A fractal is a geometric object such that each part of it is a reduced version of the whole i.e. it has the same appearance on all scales. Fractals have been invoked to describe many physical phenomena which exhibit self-similarity. A multi-fractal is an extension of the concept of a fractal. It incorporates the possibility that the particle distribution in different density environments may exhibit a different scaling or self-similar behaviour. The fact that the galaxy clustering is scale-invariant over a range of length-scales led @peter to propose that the galaxies had a fractal distribution. The later analysis of @coleman seemed to bear out such a proposition whereas @bor claimed that the fractal description was valid only on small scales and the galaxy distribution was consistent with homogeneity on large scales. A purely fractal distribution would not be homogeneous on any length-scale and this would violate the Cosmological Principle. Further, the mean density would decrease if it were to be evaluated for progressively larger volumes and this would manifest itself as an increase in the correlation length $r_0$ (eq. \[eq:1\]) with the size of the sample. However, this simple prediction of the fractal interpretation is not supported by data, instead $r_0$ remains constant for volume limited samples of CfA2 redshift survey with increasing depth [@martinez].
The analysis of the ESO slice project [@gujo] confirms large scale homogeneity whereas the analysis of volume limited samples of SSRS2 [@cappi] is consistent with both the scenarios of fractality and homogeneity. A similar analysis [@haton] carried out on APM-Stromlo survey exhibits a fractal behaviour with a fractal dimension of $ 2.1\pm 0.1$ on scales up to $40\, h^{-1}{\rm
Mpc}$. Coming to the fractal analysis of the LCRS, @amen find a fractal behaviour on scales less than $\sim$ $30 h^{-1}\,{\rm Mpc}$ but are inconclusive about the transition to homogeneity. A multi-fractal analysis by @bharad1 shows that the LCRS exhibits homogeneity on the scales $80$ to $200 \,
h^{-1}\, {\rm Mpc}$. The analysis of @kur shows this to occur at a length-scale of $\sim 30 \, h^{-1}\,{\rm Mpc}$, whereas @best fails to find a transition to homogeneity even on the largest scale analysed. The fractal analysis of the PSCz [@pan] shows a transition to homogeneity on scales of $30\, h^{-1} {\rm Mpc}$. Recently @baris have performed a fractal analysis of SDSS EDR and find that a fractal distribution continues to length-scales of $200\,h^{-1}{\rm Mpc}$ whereas @hog analyse the SDSS LRG to find a convergence to homogeneity at a scale of around $70\, h^{-1} {\rm
Mpc}$.
In this paper we use the multi-fractal analysis to study the scaling properties of the galaxy distribution in the SDSS-DR1 and test if it is consistent with homogeneity on large scales. The SDSS is the largest galaxy survey available at present. For the current analysis we have used volume limited subsamples extracted from the two equatorial strips of the SDSS-DR1. This reduces the number of galaxies but offers several advantages. The variation in the number density in these samples are independent of the details of the luminosity function and is caused by the clustering only. The larger area and depth of these samples provide us the scope to investigate the scale of homogeneity in greater detail.
The model with $\Omega_{m0}=0.3$, $\Omega_{\Lambda0}=0.7$, $h=0.7$ and a featureless, adiabatic, scale invariant primordial power spectrum is currently believed to be the minimal model which is consistent with most cosmological data (@efst; @perci1; @teg1). Estimates of the two point correlation function $\xi(r)$ (LCRS, @tuck; SDSS, @zevi; 2dFGRS, @haw) and the power spectrum $P(k)$ (LCRS, @lin; 2dFGRS, @perci; SDSS, @teg2) are all consistent with this model. In this paper we use N-body simulations to determine the length-scale where the transition to homogeneity occurs in the model and test if the actual data is consistent with this.
There are various other probes which test the cosmological principle. The fact that the Cosmic Microwave Background Radiation (CMBR) is nearly isotropic $(\Delta T/T \sim 10^{-5})$ can be used to infer that our space-time is locally very well described by the Friedmann-Robertson-Walker metric (@ehl). Further, the CMBR anisotropy at large angular scales $(\sim 10^{o})$ constrains the [*rms*]{} density fluctuations to $\delta\rho/\rho
\sim 10^{-4} $ on length-scales of $1000\,h^{-1}{\rm Mpc}$ (e.g. @wu). The analysis of deep radio surveys (e.g. FIRST, @bale) suggests the distribution to be nearly isotropic on large scales. By comparing the predicted multipoles of the X-ray Background to those observed by HEAO1 (@sch) the fluctuations in amplitude are found to be consistent with the homogeneous universe (@lah). The absence of big voids in the distribution of Lyman-$\alpha$ absorbers is inconsistent with a fractal model (@nus).
A brief outline of the paper follows. In Section 2 we describe the data and the method of analysis, and Section 3 contains results and conclusions.
Data and method of analysis
===========================
SDSS and the N-body data
------------------------
SDSS is the largest redshift survey at present and our analysis is based on the publicly available SDSS-DR1 data (@abaz). Our analysis is limited to the two equatorial strips which are centred along the celestial equator ($\delta=0^{\circ}$), one in the Northern Galactic Cap (NGP) spanning $91^{\circ}$ in [*r.a.*]{} and the other Southern Galactic Cap (SGP) spanning $65^{\circ}$ in [*r.a.*]{}, their thickness varying within $\mid \delta \mid \le 2.5^{\circ}$ in [*dec.*]{} We constructed volume limited subsamples extending from $z=0.08$ to $0.2$ in redshift ([*i.e.*]{} $235\,h^{-1}
{\rm Mpc}$ to $571\,h^{-1}{\rm Mpc}$ comoving in the radial direction) by restricting the absolute magnitude range to $-22.6\leq M_r \leq -21.6$. The resulting subsamples are two thin wedges of varying thickness aligned with the equatorial plane. Our analysis is restricted to slices of uniform thickness $\pm 4.1 \,h^{-1}{\rm Mpc}$ along the equatorial plane extracted out of the wedge shaped regions. These slices are nearly 2D with the radial extent and the extent along [*r.a.*]{} being much larger than the thickness. We have projected the galaxy distribution on the equatorial plane and analysed the resulting 2D distribution (Figure \[fig:1\]). The SDSS-DR1 subsamples that we analyse here contains a total of 3032 galaxies and the subsamples are exactly same as those analysed in @pandey.
We have used a Particle-Mesh (PM) N-body code to simulate the dark matter distribution at the mean redshift $z=0.14$ of our subsample. A comoving volume of $[645 h^{-1} {\rm Mpc}]^3$ is simulated using $256^3$ particles on a $512^3$ mesh with grid spacing $1.26 h^{-1} {\rm Mpc}$. The set of values $(\Omega_{m0},\Omega_{\Lambda0},h)=(0.3,0.7,0.7)$ were used for the cosmological parameters, and we used a power spectrum characterised by a spectral index $n_s=1$ at large-scales and with a value $\Gamma=0.2$ for the shape parameter.The power spectrum was normalised to $\sigma_8=0.84$ (WMAP, @sperg) . Theoretical considerations and simulations suggest that galaxies may be biased tracer of the underlying dark matter distribution (e.g., @kais; @mo; @dekel; @taru and @yoshi). A “sharp cutoff” biasing scheme (@cole) was used to generate particle distributions. This is a local biasing scheme where the probability of a particle being selected as a galaxy is a function of local density only. In this scheme the final dark-matter distribution generated by the N-body simulation was first smoothed with a Gaussian of width $5
h^{-1} {\rm Mpc}$. Only the particles which lie in regions where the density contrast exceeds a critical value were selected as galaxy. The values of the critical density contrast were chosen so as to produce particle distributions with a low bias $b=1.2$ and a high bias $b=1.6$. An observer is placed at a suitable location inside the N-body simulation cube and we use the peculiar velocities to determine the particle positions in redshift space. Exactly the same number of particles distributed over the same volume as the actual data was extracted from the simulations to produce simulated NGP and SGP slices. The simulated slices were analysed in exactly the same way as the actual data.
Methods of Analysis
-------------------
A fractal point distribution is usually characterised in terms of its fractal dimension. There are different ways to calculate this, and the correlation dimension is one of the methods which is of particular relevance to the analysis of galaxy distributions. The formal definition of the correlation dimension involves a limit which is meaningful only when the number of particles is infinite and hence this cannot be applied to galaxy surveys with a limited number of galaxies. To overcome this we adopt a “working definition” which can be applied to a finite distribution of $N$ galaxies. It should be noted that our galaxy distribution is effectively two dimensional, and we have largely restricted our discussion to this situation.
Labelling the galaxies from $1$ to $N$, and using $\x_i$ and $\x_j$ to denote the comoving coordinates of the $i$ th and the $j$ th galaxies respectively, the number of galaxies within a circle of comoving radius $r$ centred on the $i$ th galaxy is $$n_i(r)=\sum_{j=1}^{N}\th(r-\mid \x_i-\x_j \mid)
\label{eq:2}$$ where $\th(x)$ is the Heaviside function defined such that $\th(x)=0$ for $x<0$ and $\th(x)=1$ for $x\ge0$. Averaging $n_i(r)$ by choosing $M$ different galaxies as centres and dividing by the total number of galaxies gives us $$C_2(r)=\frac{1}{MN}\sum_{i=1}^{M}n_i(r)$$ which may be interpreted as the probability of finding a galaxy within a circle of radius $r$ centred on another galaxy. If $C_2(r)$ exhibits a power law scaling relation $C_2(r)\propto
r^{D_2}$, the exponent $D_2$ is defined to be the correlation dimension. Typically, a power law scaling relation will hold only over a limited range of length-scales $r_1 \le r \le r_2$, and it may so happen that the galaxy distribution has different correlation dimensions over different ranges of length-scales.
It is clear that $C_2(r)$ is closely related to the volume integral of the two point correlation function $\xi(r)$. In a situation where this has a power law behaviour $\xi(r)=(\frac{r}{r_0})^{\g}$, the correlation dimension is $D_2=2-\g$ on scales $r<r_0$. Further, we expect $D_2=2$ on large scales where the galaxy distribution is expected to be homogeneous and isotropic.
In the usual analysis the two point correlation does not fully characterise all the statistical properties of the galaxy distribution, and it is necessary to also consider the higher order correlations [*eg.*]{} the three point and higher correlations. Similarly, the full statistical quantification of a fractal distribution also requires a hierarchy of scaling indices. The multi-fractal analysis used here does exactly this. It provides a continuous spectrum of generalised dimension $D_q$, the Minkowski-Bouligand dimension, which is defined for a range of $q$.
The definition of the generalised dimension $D_q$ closely follows the definition of the correlation dimension $D_2$, the only difference being that we use the $(q-1)$th moment of $n_i(r)$. The quantity $C_2(r)$ is now generalised to $$C_q(r)=\frac{1}{MN}\sum_{i=1}^{M}[n_i(<r)]^{q-1}
\label{eq:3}$$ which is used to define the Minkowski-Bouligand dimension $$D_q=\frac{1}{q-1}\frac{d\log{C_q(r)}}{d\log{r}}
\label{eq:4}$$ Typically $C_q(r)$ will not exhibit the same scaling behaviour over the entire range of length-scales, and it is possible that the spectrum of generalised dimension will be different in different ranges of length-scales. The correlation dimension corresponds to the generalised dimension at $q=2$, whereas $D_1$ corresponds to the box counting dimension. The other integer values of $q$ are related to the scaling of higher order correlation functions. A mono-fractal is characterised by a single scaling exponent [*ie.*]{} $D_q$ is a constant independent of $q$, whereas the full spectrum of generalised dimensions is needed to characterise a multi-fractal. The positive values of $q$ give more weightage to the regions with high number density whereas the negative values of $q$ give more weightage to the underdense regions. Thus we may interpret $D_q$ for $q > 0$ as characterising the scaling behaviour of the galaxy distribution in the high density regions like clusters whereas $q<0$ characterises the scaling inside voids. In the situation where the galaxy distribution is homogeneous and isotropic on large scales, we expect $D_q=2$ independent of the value of $q$.
There are a variety of different algorithms which can be used to calculate the generalised dimension, the Nearest Neighbour Interaction(@badi) and the Minimal Spanning Tree (@suth) being some of them. We have used the correlation integral method which we present below.
The two subsamples, NGP and SGP contain 1936 and 1096 galaxies respectively and they were analysed separately. For each galaxy in the subsample we considered a circle of radius $r$ centred on the galaxy and counted the number of other galaxies within the circle to determine $n_i(r)$ (eq. \[eq:2\]). The radius $r$ was increased starting from $5\, h^{-1} \, {\rm Mpc}$ to the largest value where the circle lies entirely within the subsample boundaries. The values of $n_i(r)$ determined using different galaxies as centres were then averaged to determine $C_q(r)$ (eq. \[eq:4\]). It should be noted that the number of centres falls with increasing $r$, and for the NGP there are $\sim 800$ centres for $r=80h^{-1} \, {\rm Mpc}$ with the value falling to $\sim 100$ for a radius of $r=150h^{-1} \, {\rm
Mpc}$. The large scale behaviour of $C_q(r)$ was analysed to determine the range of length-scales where it exhibits a scaling behaviour and to identify the scaling exponent $D_q$ as a function of $q$.
In addition to the actual data, we have also constructed and analysed random distributions of points. The random data contains exactly the same number points as there are galaxies in the actual data distributed over exactly the same region as the actual NGP and SGP slices. The random data are homogeneous and isotropic by construction, and the results of the multi-fractal analysis of this data gives definite predictions for the results expected if the galaxy distribution were homogeneous and isotropic. The random data and the simulated slices extracted from the N-body simulations were all analysed in exactly the same way as the actual data. We have used 18 independent realisations of the random and simulated slices to estimate the mean and the $1-\sigma$ error-bars of the spectrum of generalised dimensions $D_q$.
Results and Discussions
=======================
Figures \[fig:f1\] and \[fig:f2\] show $C_q(r)$ at $q=-2$ and $2$, respectively, for the actual data, for one realisation of the random slices and for one realisation of the simulated slices for each value of the bias. The behaviour of $C_q(r)$ at other values of $q$ is similar to the ones shown here. Our analysis is restricted to $-4 \le q \le 4$. We find that $C_q(r)$ does not exhibit a scaling behaviour at small scales $(5\, h^{-1} \, {\rm Mpc} \le r \le 40 \,
h^{-1} \, {\rm Mpc})$. Further, the small-scale behaviour of $C_q(r)$ in the actual data is different from that of the random slices and is roughly consistent with the simulated slices for $b=1.6$. We find that $C_q(r)$ shows a scaling behaviour on length-scales of from somewhere around $60-70 \, h^{-1} \, {\rm Mpc}$ to $150 \,
h^{-1} \, {\rm Mpc}$. Although the actual data, the random and simulated slices all appear to converge over this range of length-scales indicating that they are all roughly consistent with homogeneity, there are small differences in the slopes. We have used a least-square fit to determine the scaling exponent or generalised dimension $D_q$ shown in Figure \[fig:f3\].
Ideally we would expect $D_q=2$ for a two dimensional homogeneous and isotropic distribution. We find that for the actual data $D_q$ varies in the range $1.7$ to $2.2$ in the NGP and $1.8$ to $2.1$ in the SGP on large-scales. In both the slices the value of $D_q$ decreases with increasing $q$, and it crosses $D_q=2$ somewhere around $q=-1$. The variation of $D_q$ with $q$ shows a similar behaviour in the random slices, but the range of variation is much smaller $(1.9 \le D_q \le 2.1)$. Comparing the actual data with the random data we find that the actual data lies outside the $1-\sigma$ error-bars of the random data (not shown here) for most of the range of $q$ except around $q=-1$ where $D_q=2$ for both the actual and random data. Accepting this at face value would imply that the actual data is not homogeneous at large scales.
Considering the simulated data, we find that the variation in $D_q$ depends on the value of the bias $b$. For the unbiased simulations $D_q$ shows very small variations $(1.9 \le D_q \le 2.1)$ and the results are very close to those of the random data. We find that increasing the bias causes the variations in $D_q$ to increase. In all cases $D_q$ decreases with increasing $q$ and it crosses $D_q=2$ around $q=-1$. Increasing the bias has another effect in that it results in larger $1-\sigma$ error-bars.
Comparing the simulated data with the random data and the actual data we find that the unbiased simulations are consistent with the random data but not the actual data. The actual data lies outside the $1-\sigma$ error-bars of the unbiased model. This implies that the unbiased model has a transition to homogeneity at $60-70 \, h^{-1} \, {\rm
Mpc}$. The spectrum of generalised dimensions as determined from the unbiased simulations on length-scales $60-70 \, h^{-1} \, {\rm Mpc}$ to $150 \, h^{-1} \, {\rm
Mpc}$ is different from that of the actual data [*ie.*]{} the unbiased model fails to reproduce the large scale properties of the galaxy distribution in our volume limited subsamples of the SDSS-DR1.
The simulations with bias $b=1.6$ and $b=2$ have larger $1-\sigma$ error-bars and these are consistent with both the random and the actual data. Interpreting the actual data as being a realisation of a biased universe, we conclude that it has a transition to homogeneity at $60-70 \, h^{-1} \, {\rm Mpc}$ and the galaxy distribution is homogeneous on scales larger than this.
The galaxy subsample analysed here contains the most luminous galaxies in the SDSS-DR1. Various investigations have shown the bias to increase with luminosity (@nor; @zevi) and the subsample analysed here is expected to be biased with respect to the underlying dark matter distribution. @sel have used the halo model in conjunction with weak lensing to determine the bias for a number of subsamples with different absolute magnitude ranges. The brightest sample which they have analysed has galaxies with absolute magnitudes in the range $-23 \le
M_r \le -22$ for which they find a bias $b=1.94 \pm 0.2$. Our results are consistent with these findings.
A point to note is that the $1-\sigma$ error-bars of the spectrum of generalised dimension $D_q$ increases with the bias. This can be understood in terms of the fact that $C_q(r)$ is related to volume integrals of the correlation functions which receives contribution from all length-scales. The fluctuations in $C_q(r)$ can also be related to volume integrals of the correlation functions. Increasing the bias increases the correlations on small scales $(\le 40-50 \, h^{-1} {\rm
Mpc})$ which contributes to the fluctuations in $C_q(r)$ at large scales and causes the fluctuations in $D_q$ to increase.
The galaxies in nearly all redshift surveys appear to be distributed along filaments. These filaments appear to be interconnected and they form a complicated network often referred to as the “cosmic web”. These filaments are possibly the largest coherent structures in galaxy redshift surveys. Recent analysis of volume limited subsamples of the LCRS [@bharad2] and the same SDSS-DR1 subsamples analysed here [@pandey] shows the filaments to be statistically significant features of the galaxy distribution on length-scales $\le
70-80 \, h^{-1} {\rm Mpc}$ and not beyond. Larger filaments present in the galaxy distribution are not statistically significant and are the result of chance alignments. Our finding that the galaxy distribution is homogeneous on scales larger than $60-70 \, h^{-1}
{\rm Mpc}$ is consistent with the size of the largest statistically significant coherent structures namely the filaments.
Acknowledgements {#acknowledgements .unnumbered}
================
SB would like to acknowledge financial support from the Govt. of India, Department of Science and Technology (SP/S2/K-05/2001). JY and BP are supported by fellowships of the Council of Scientific and Industrial Research (CSIR), India. JY and TRS would like to thank IUCAA for the facilities at IUCAA Reference Centre at Delhi University. TRS thanks IUCAA for the support provided through the Associateship Program.
The SDSS-DR1 data was downloaded from the SDSS skyserver http://skyserver.sdss.org/dr1/en/. Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.
[99]{}
Abazajian K. [et al.]{}, 2003, , 126, 2081
Amendola L., Palladino E., 1999, , 514, L1
Badii R., Politi A., 1984, Physical Review Letters, 52, 1661
Baleisis A., Lahav O., Loan A.J., Wall J.V., 1998, , 297,545
Best J. S., 2000, , 541, 519
Bharadwaj S., Gupta A. K., Seshadri T. R., 1999, , 351, 405
Bharadwaj S., Bhavsar S. P., Sheth J. V., 2004, , 606, 25
Baryshev Y. V., Bukhmastova Y. L., 2004, Astronomy Letters, 30, 444
Borgani S., 1995, , 251, 1
Cappi A., Benoist C., da Costa L. N., Maurogordato S., 1998, , 335, 779
Cole S., Hatton S., Weinberg D.H., Frenk C.S., 1998, , 300, 945
Coleman P. H., Pietronero L., 1992, , 213, 311
Dekel A., Lahav O., 1999, , 520, 24
Efstathiou G. [et al.]{}, 2002, , 330, L29
Ehlers J., Green P., Sachs R.K., 1968, J Math Phys, 9, 1344
Guzzo L., 1997, New Astronomy, 2, 517
Hawkins E. [et al.]{}, 2003, ,346,78
Hatton S., 1999, , 310, 1128
Hogg D. W., Eistenstein D. J., Blanton M. R., Bahcall N. A., Brinkmann J., Gunn J. E., Schneider D. P., 2004, Submitted to ,(astro-ph/0411197)
Kaiser N., 1984, , 284, L9
Kurokawa T., Morikawa M., Mouri H., 2001, , 370, 358
Lahav O., 2002, Classical and Quantum Gravity, 19, 3517
Lin H., Kirshner R. P., Shectman S. A., Landy S. D., Oemler A., Tucker D. L., Schechter P. L., 1996, , 471, 617
Mart[í]{}nez V. J., L[' o]{}pez-Mart[í]{} B., Pons-Border[í]{}a M.,2001, , 554, L5
Mo H.J., White S.D.M., 1996, , 282, 347
Nusser A., Lahav O., 2000, , 313, L39
Norberg P. [et al.]{}, 2001, , 328, 64
Pandey B., Bharadwaj S., 2005, , 357, 1068
Pan J., Coles P., 2000, , 318, L51
Peebles P.J.E., 1980, The Large-Scale Structures of the Universe, Princeton University Press, Princeton, NJ
Percival W. J. [et al.]{}, 2001, , 327, 1297
Percival W. J. [et al.]{}, 2002, , 337, 1068
Pietronero L., 1987, Physica A, 144, 257
Scharf C. A., Jahoda K., Treyer M., Lahav O., Boldt E., Piran T., 2000, , 544, 49
Spergel D. N. [et al.]{}, 2003, , 148, 175
Sutherland W., Efstathiou G., 1991, , 248, 159
Taruya A., Suto Y., 2001,, 542, 559
Tegmark M. [et al.]{}, 2004, PRD, 69, 103501
Tegmark M. [et al.]{}, 2004, , 606, 702
Tucker D. L. [et al.]{}, 1997, MNRAS, 285, 5
Seljak U. [et al.]{}, 2005, PRD, 71, 043511
Wu K., Lahav O., Rees M., 1999, , 397, 225
Yoshikawa K., Taruya A., Jing Y.P., Suto Y., 2001, , 558, 520
Zehavi I. [et al.]{}, 2002, ,571,172
[^1]: Email: jaswant@physics.du.ac.in
[^2]: Email: somnathb@iitkgp.ac.in
[^3]: Email: pandey@cts.iitkgp.ernet.in
[^4]: Email: trs@physics.du.ac.in
|
---
abstract: 'The prototype compound for the *neutral-ionic* phase transition, namely TTF-CA, is theoretically investigated by first-principles density functional theory calculations. The study is based on three neutron diffraction structures collected at 40, 90 and 300 K (Le Cointe et al., Phys. Rev. B **51**, 3374 (1995)). By means of a topological analysis of the total charge densities, we provide a very precise picture of intra and inter-chain interactions. Moreover, our calculations reveal that the thermal lattice contraction reduces the indirect band gap of this organic semi-conductor in the neutral phase, and nearly closes it in the vicinity of the transition temperature. A possible mechanism of the *neutral-ionic* phase transition is discussed. The charge transfer from TTF to CA is also derived by using three different techniques.'
author:
- 'V. Oison'
- 'C. Katan'
- 'P. Rabiller'
- 'M. Souhassou'
- 'C. Koenig'
bibliography:
- 'pr-v7.bib'
title: 'Neutral-ionic phase transition : a thorough ab-initio study of TTF-CA\'
---
Introduction:
=============
Charge transfer salts presenting mixed stacks with alternating donor (D) and acceptor (A) molecules have been extensively studied over the last 20 years for their original neutral-ionic phase transitions (NIT).[@mayerle; @torrance] Very recently, this class of materials has gained renewed interest as it has been demonstrated experimentally that in some cases the conversion from the neutral state (N) to the ionic state (I) or from I to N can also be induced by photo-irradiation.[@koshihara] Despite intensive theoretical work, the mechanism of the phase transition and photo-conversion has not yet been clarified. In these systems, unlike in other classes of compounds, no dominant interaction has been evidenced and the nature of the NIT must be related to a subtle interplay of different type of interactions.
Depending on the choice of D and A molecules as well as on molecular packing, different type of NIT have been observed mainly under pressure and sometimes also under temperature variation. Continuous and discontinuous NIT have been reported presenting different amplitudes of charge transfer (CT) variations from D to A, often with a *dimerization* in DA pairs along the stacking axis. The TTF-CA complex made from tetrathiafulvalene (D = TTF) and p-chloranil (A = CA) molecules is considered as the prototype compound for NIT. At atmospheric pressure, it undergoes a first order NIT at a critical temperature of about 80 K. This symmetry breaking phase transition leads to a ferroelectric low temperature phase where the initially planar D and A molecules are both deformed and displaced to form DA pairs along the stacking chains.[@lecointe] According to vibrational spectroscopy [@girlando2] and CT absorption spectra [@jacobsen] the CT has been estimated to be about 0.2 e$^-$ in the neutral high temperature phase and 0.7 e$^-$ in the ionic low temperature phase. TTF-CA is also a typical example showing N-I and I-N conversion under photo-irradiation.[@koshihara] Recently, ultrafast (ps) optical switching from I to N has been observed by femtosecond reflection spectroscopy.[@iwai]
Numerous theoretical approaches have been developed to study the NIT. For many years, the balance between the energy needed to ionize molecules and the gain in the Madelung energy has been at the heart of the concept of NIT.[@torrance; @iwai] More advanced models mainly concern extensions of the one-dimensional (1D) modified Hubbard model which includes on-site Coulomb repulsion, transfer integral and on-site energy. Anusooya-Pati et al.[@anusooya] have recently shown that such a model has a continuous NIT between a diamagnetic band insulator and a paramagnetic Mott insulator. Therefore, the ground state at the NIT may be metallic but unconditionally unstable to dimerization. Such a metallic behavior near the N-I borderline has been indeed reported.[@saito] Depending on the strength of other interaction such as long range Coulomb interaction, electron-phonon (Peierls) or electron-molecular-vibration (Holstein) couplings, continuous or discontinuous ionicity changes can be generated and the dimerization instability is more or less affected.[@anusooya]
Despite increasing computer power, first-principles calculations are still very rare in the field of molecular crystals. These calculations offer a unique tool for analyzing, at a microscopic level, CT salts without any beforehand assumption concerning the relative strengths of electronic interactions. They have already contributed to evidence the coupling between CT variation and anisotropic three-dimensional (3D) lattice contractions in the presence of hydrogen bonds [@oison] and to analyze the quantum intra and inter-chain interactions in the mixed stack CT crystal of TTF-2,5Cl$_2$BQ.[@katanjpcm] This compound has twice as less atoms per unit cell as TTF-CA which makes the computational effort considerably lower, but only few experimental work is available and structural data are limited to the ambient condition phase. From an artificial structure obtained by relaxing all atomic positions within the experimental unit cell, it has been shown that the weak molecular distortion due to symmetry breaking induces a non-negligible contribution to the total CT variation at the NIT.[@katanjpcm]
Starting from first-principles Density Functional Theory (DFT) calculations, we present here a thorough study of TTF-CA. Our main purpose is not to describe the phase transition itself but to reach a precise understanding of the electronic ground states on both sides and far from the transition point. This work gives a new and complementary insight for this family of CT complexes. After a brief section concerning computational details, we present, in Section \[sec:tb\], a complete analysis of the valence and conduction bands with the help of a tight-binding model fitted to the ab-initio results. This will lead us to discuss a possible mechanism of the phase transition itself. In Section \[sec:topo\], the 3D total electron density at 300 and 40 K will be analyzed in details using Bader’s topological approach.[@bader] Section \[sec:CT\] will be devoted to the determination of the CT in the high and low temperature phases from our tigh-binding model, from Bader’s approach and from a simple model based on isolated molecules calculations. The last method will only be briefly mentionned as we defer detailed description to a forthcoming publication.
\[sec:co\]Computational details
===============================
Our ab-initio ground state electronic structure calculations (frozen lattice) are based on the experimental structures obtained by neutron-scattering experiments at 300, 90 and 40 K.[@lecointe] The symmetry is monoclinic with two equivalent chains with alternation of TTF and CA molecules along the stacking axis $\mathbf{a}$. This leads to 52 atoms per unit cell. Above 81 K, the space group is $P12_1/n1$ and TTF and CA are located on inversion centers. At 300 K : ${a}= 7.40$, ${b}= 7.62$, ${c}=
14.59$ Åand $\beta=99,1^\circ$ and at 90 K : ${a}= 7.22$, ${b}= 7.59$, ${c}= 14.49$ Å and $\beta=99,1^\circ$. At 40 K, the space group is $P1n1$ with ${a}= 7.19$, ${b}= 7.54$, ${c}= 14.44$ Å and $\beta=98,6^\circ$. TTF and CA are slightly distorded and displaced and form DA pairs in a ferroelectric arrangement in the **a** direction.
We performed electronic structure calculations within the framework of DFT using the local density approximation (LDA) parametrization by Perdew and Zunger [@pz], Becke’s gradient correction to the exchange energy [@becke] and Perdew’s gradient correction to the correlation energy.[@perdew] The main difficulty in treating the molecular compounds of this family is the presence of non negligible contributions of dynamical electronic interactions of van der Waals type. They are known to be poorly described within traditional DFT approximations, more advanced treatments being far too time consuming for such complex systems. Nevertheless, we believe that the presence of a significant CT between D and A makes our ab-initio results reliable for the determination of the occupied electronic states, especially in the I phase where electrostatic interactions largely dominate.
We used the projector augmented wave (PAW) method [@blo] which uses augmented plane waves to describe the full wave functions and densities without shape approximation. The core electrons are described within the frozen core approximation. The version of the CP-PAW code used for all our calculations considers only $\mathbf{k}$ points where the wave functions are real. In order to increase the number of $\mathbf{k}$ points along $\mathbf{a}^\ast$, we had to double and treble the unit cell along the $\mathbf{a}$ direction and our calculations showed that with 3 $\mathbf{k}$ points along $\Gamma (0,0,0)\rightarrow
\mathrm{X} (1,0,0)$ results are well converged as for TTF-2,5Cl$_2$BQ.[@katanjpcm] The band structures calculations have been carried out with a plane wave cutoff of 30 Ry for the wave functions and 120 Ry for the densities, using 8 $\mathbf{k}$ points: $\Gamma (0,0,0)$, $\mathrm{X}/2 (1/2,0,0)$, $\mathrm{X} (1,0,0)$, $\mathrm{Y} (0,1,0)$, $\mathrm{W}/2 (1,1/2,0)$, $\mathrm{W} (1,1,0)$, $\mathrm{Z} (0,0,1)$, and $\mathrm{S} (1,0,1)$ in units of $(\mathbf{a}^\ast/2,
\mathbf{b}^\ast/2,\mathbf{c}^\ast/2)$.
As the topological analysis requires very accurate electron densities, we have first verified that results do not deteriorate when using only 3 $\mathbf{k}$ points along $\Gamma \rightarrow \mathrm{X}$ instead of 8 and then increased the plane wave cutoff to 50 Ry for the wave functions. The subsequent topological analysis was performed using the new InteGriTy software package [@integrity] which achieves topological analysis following Bader’s approach [@bader] on electron densities given on 3D grids. In order to obtain very accurate results, a grid spacing close to 0.10 a.u has been choosen.
\[sec:tb\]Ab-initio band structure
==================================
Our ab-initio DFT calculations in the crystal provide not only the total electron density $\mathrm{n(\mathbf{r})}$ but also for each band the energies and wave functions at all $\mathbf{k}$ points given in section \[sec:co\]. The present section is devoted to the sole valence bands (VB) and conduction bands (CB) which are directly related to the strong anisotropy observed experimentally in transport and excitation properties.[@tokura; @mitani; @okamoto] The detailed analysis of $\mathrm{n(\mathbf{r})}$ will be given in section \[sec:topo\].
Quasi 1D shape of valence and conduction bands
----------------------------------------------
The dispersion curves for the VB and CB are given in Fig. \[disp\] for the experimental structures at 300 K (high temperature phase) and 40 K (low temperature phase). In both cases, they are separated by more than 1 eV from the other occupied and unoccupied bands forming thus a nearly isolated four-band system. The presence of two equivalent chains in the unit cell related by a gliding plane makes the VB and CB twofold, splittings in some particular directions being due to small interactions between the symmetry related chains. The dispersion is maximum along $\Gamma \rightarrow \mathrm{X}$ which is the direction of reciprocal space corresponding to the chain axis $\mathbf{a}$.[@katanssc]
300 K 40 K
The isodensity representation of the VB states are given in Fig. \[isoVB\]. At 300 K for $\mathbf{k}=\Gamma$ (Fig. \[isoVB\]a), the density is located on TTF with a shape very close to that of the highest occupied molecular orbital (HOMO) of TTF.[@katanTTF] At $\mathbf{k}=\mathrm{X}$, (Fig. \[isoVB\]b), hybridization occurs and the density centered on the acceptor molecule is similar to the one of the lowest unoccupied molecular orbital (LUMO) of CA.[@katanCA] The same happens in TTF-2,5Cl$_2$BQ [@katanssc] and is a direct consequence of the presence of an inversion center in the crystal combined to different symmetries of the molecular orbitals which are involved. As the inversion center is lost at 40 K, hybridization is no more symmetry forbidden at $\mathbf{k}=\Gamma$ and the pairing of TTF and CA is clearly evidenced in Fig. \[isoVB\]c (DA pair on the bottom). On these figures one can identify the regions of wave function overlap responsible for the intra-chain hopping integrals.
$\Gamma$ at 300 K $\mathrm{X}$ at 300 K $\Gamma$ at 40 K
The dispersion is much less important along $\Gamma \rightarrow \mathrm{Y}$, nearly vanishing in the third direction and very similar for both structures. As for TTF-2,5Cl$_2$BQ,[@katanjpcm] the band gap is most probably indirect: $E_g = 0.05~\mbox{eV}$ at 300 K and $0.15~\mbox{eV}$ at 40 K. It is well known that the LDA used in our calculations underestimate the gap of semiconductors; these values are thus lower estimates of the true band gap. However, because of missing data a direct comparison to experiments is impossible. Optical absorption measurements [@torrance] showed a first broad peak around 0.7 eV for the N phase which is attributed to the ionization of one DA pair. It may correspond either to band to band transitions or to an exciton peak. The optical spectra of TTF-2,5Cl$_2$BQ [@torrance] shows a shift to higher energies which is coherent with a larger calculated band gap for TTF-2,5Cl$_2$BQ: 0.14 eV in the high temperature structure.[@katanjpcm]
\[subsec:tb\]Tight-binding model
--------------------------------
As shown in Fig. \[isoVB\], the ab-initio VB and CB can be interpreted in the frame of a tight-binding model as linear combinations of the HOMO of TTF and the LUMO of CA. One should emphasize that in such a model, hybridizations with lower bands are completely neglected. The interaction with deeper occupied states is nevertheless indirectly taken into account via the values of the parameters, which are fitted to the VB and CB given by an ab-initio calculation of all states. We will follow a similar procedure to the one used for TTF-2,5Cl$_2$BQ,[@katanjpcm] but here the task is more involved as we have to handle four bands instead of two. Let us write $\mathrm{\vert{\mathbf{r},D}\rangle}$ and $\mathrm{\vert{\mathbf{r'},A}\rangle}$ respectively the HOMO of TTF located at position $\mathbf{r}$ and the LUMO of CA at $\mathbf{r'}$. The VB Bloch functions are then defined by: $$\begin{aligned}
\mathrm{\vert\Psi^{lk}\rangle\!\!\!\!}&&
\mathrm{= \sum_{n}e^{i\mathbf{k}\mathbf{R_n}}\biggl(
C^{lk}_D\vert{\mathbf{R_n},D}\rangle + C^{lk}_{D'}
\vert{\mathbf{R_n}+\mathbf{\nu},D'}\rangle}
\nonumber\\
&&
\mathrm{+ \; C^{lk}_A \vert{\mathbf{R_n}+\mathbf{\tau},A}\rangle
+ C^{lk}_{A'} \vert{\mathbf{R_n}+\mathbf{\tau}+\mathbf{\nu},A'}
\rangle\biggr)}.
\label{eqpsi}\end{aligned}$$ Here $\mathrm{l}=1,2$ is the band index; $\mathbf{R_n}$ corresponds to a primitive translation; $\mathbf{\tau} = \mathrm{\mathbf{a}/2}$ represents a translation between A and D along the chain axis and $\mathbf{\nu} = \mathrm{\mathbf{a}/2+\mathbf{b}/2+\mathbf{c}/2}$ a translation between the two symmetry equivalent chains, the molecules from the second chain being denoted by $\mathrm{D'}$ and $\mathrm{A'}$. The molecular orbitals $\mathrm{\vert{D}\rangle}$, $\mathrm{\vert{A}\rangle}$, $\mathrm{\vert{D'}\rangle}$ and $\mathrm{\vert{A'}\rangle}$ are calculated using ab-initio simulations for isolated molecules in their experimental crystalline conformations. For each $\mathbf{k}$ point, the weight $\vert\mathrm{C^{lk}_j}\vert^2$ ($\mathrm{j=}$ A or D) is obtained by mean square minimization in the unit cell of the difference between the ab-initio value of $\mathrm{\vert\Psi^{lk}(\mathbf{r})\vert^2}$ calculated in the crystal and the one deduced from expression (\[eqpsi\]). Fig. \[Cak2\] displays these weights at the 6 corners of the reduced Brillouin zone used in these calculations for both the high (left) and low (right) temperature phases. In the former, the valence band at the $\Gamma$ is a pure TTF state for symmetry reasons: $\mathrm{C_A^{l\Gamma}}=0$. The band gap separates nearly pure D occupied states from nearly pure A empty states in the whole $\mathrm{Z\!-\!U\!-\!Y\!-}\Gamma$ plane, whereas electronic states are of mixed D-A nature in the $\mathrm{X\!-\!W\!-\!T\!-\!S}$ plane. In the latter, a significant mixing of D-A states takes place in the whole Brillouin zone, for the VB as well as for the CB.
300 K 40 K
The charge transfer $\mathrm{\rho_{VB}}$ from TTF to CA is given by the mean value of the weights of the acceptors in the VB: $$\begin{aligned}
\mathrm{\rho_{VB}}
&&
\mathrm{= \frac{1}{2}\sum_{l}\biggl(
\langle\vert C^{lk}_A \vert^2 \rangle +
\langle\vert C^{lk}_{A'} \vert^2 \rangle \biggl)}
\nonumber\\
&&
\mathrm{= \frac{V}{8\pi^3} \int_{ZB}\sum_l |C_A^{lk}|^2d^3\mathbf{k}}.
\label{eqrhovb}\end{aligned}$$ Fig. \[Cak2\] shows clearly that the increase of $\mathrm{\rho_{VB}}$ in the low temperature phase results mainly from additionnal hybridization, which is no longer symmetry forbidden, in the whole $\mathrm{\Gamma\!-\!Y\!-\!U\!-\!Z}$ plane.
In order to model VB and CB in terms of a tight-binding scheme, we define the on-site energies by $\mathrm{\langle D \vert H \vert D \rangle}
\mathrm{= \langle D' \vert H \vert D' \rangle}
\mathrm{= E_0^D}$ and $\mathrm{\langle A \vert H \vert A \rangle}
\mathrm{= \langle A' \vert H \vert A' \rangle}
\mathrm{= E_0^A.}$ Two families of intermolecular interactions have been defined, the first corresponding to interactions between molecules of the same species shown on Fig. \[parajj\] and the second to interaction between TTF and CA shown on Fig. \[paraDA-ab\] and Fig. \[paraDA-pipi\]. Due to the different symmetries of the HOMO of TTF and LUMO of CA, the sign of the hopping integrals between TTF and CA is alternately plus and minus. For example, along the stacking chains one has $\langle \mathbf{0},\mathrm{D \vert H} \vert \mathbf{a}/2,\mathrm{A}
\rangle = \mathrm{t}$ and $\langle \mathbf{a}/2,\mathrm{A \vert H} \vert
\mathbf{a}, \mathrm{D} \rangle = - \mathrm{t}$. The symmetry breaking in the low temperature phase leads to additionnal terms so that the latter hopping integrals become respectively $\mathrm{t} + \varepsilon$ and $-\mathrm{t} + \varepsilon$ whereas $\mathrm{\pm t_{i}}$, $\mathrm{\pm \theta_{i}}$ and $\mathrm{\delta_i^j}$ ($\mathrm{i = 1}$ or $2$ and $\mathrm{j= D}$ or $\mathrm{A}$) become $\mathrm{\pm t_{i} + \varepsilon_i}$, $\mathrm{\pm \theta_{i} + \lambda_i}$ and $\mathrm{\delta_i^j \pm
\upsilon_i^j}$ respectively.
Using the ab-initio energies and the weights $\mathrm{\vert C^{lk}_j \vert^2}$ at the 6 corners of the reduced Brillouin zone used in these calculations, we determine the parameters given on Table \[tab:jj\] to \[tab:def\] and the curves plotted in Fig. \[disp\] and Fig. \[Cak2\]. Some parameters like $\mathrm{\beta_1^j}$ and $\mathrm{\beta_2^j}$ (Fig. \[parajj\]) cannot be determined separately because of the small number of $\mathbf{k}$ points available, but all these contributions are found to be negligible. In Fig. \[disp\] the degeneracy is lifted at $\Gamma$ due to the small inter-chain interactions $\mathrm{\delta_i^j}$ (Fig. \[parajj\]), at $\mathrm{S}$ due to $\mathrm{\theta_i}$ (Fig. \[paraDA-pipi\]) and, in the low temperature phase at $\mathrm{Y}$ due to $\mathrm{\upsilon_i^j}$. The $\mathrm{\theta_i}$ are also responsible for the small HOMO-LUMO hybridization at $\mathrm{Y}$ and $\mathrm{Z}$ (Fig. \[Cak2\]).
The transfer integral $\pm\mathrm{t}$ along the chain axis (Fig. \[paraDA-ab\]) dominates and its estimate is in agreement with previous ones.[@jacobsen; @girlando1; @tokura; @katanssc] In the low temperature phase, $\mathrm{t}$ increases significantly (Table \[tab:DA\]) partly due to the lattice contraction and partly to the symmetry breaking.[@katanjpcm] As for TTF-2,5Cl$_2$BQ, the main effect of the latter is expressed by the additional parameter $\varepsilon$ which is as large as a third of $\mathrm{t}$ (Table \[tab:def\]) and which leads to a large HOMO-LUMO hybridization in the $\mathrm{Z\!-\!U\!-\!Y\!-}\Gamma$ plane. The other additional parameters $\mathrm{\varepsilon_i}$, $\mathrm{\lambda_i}$ and $\mathrm{\upsilon_i^j}$ have negligible values.
------------------------------------------------------------------------------------------------------
300 K 40 K 300 K 40 K
---------------------------------- ---------- ---------- ----------------------- ---------- ----------
$\mathrm{E_0^D}$ 2.894 3.191 $\mathrm{E_0^A}$ 3.048 3.348
$\mathrm{\beta^D_a}$ 0.002 0.001 $\mathrm{\beta^A_a}$ $-$0.017 $-$0.024
$\mathrm{\beta^D_b}$ $-$0.022 $-$0.005 $\mathrm{\beta^A_b}$ $-$0.023 $-$0.003
$\mathrm{\beta^D_1 + \beta^D_2}$ 0.003 0.004 $\mathrm{\beta^A_1 0.002 0.003
+ \beta^A_2}$
$\mathrm{\delta^D_1}$ $-$0.002 $-$0.003 $\mathrm{\delta^A_1}$ $-$0.004 $-$0.004
$\mathrm{\delta^D_2}$ $-$0.004 $-$0.005 $\mathrm{\delta^A_2}$ $-$0.002 $-$0.003
------------------------------------------------------------------------------------------------------
: \[tab:jj\] On site energies and interaction parameters (eV) between two molecules of the same species as defined in Fig. \[parajj\].
300 K 40 K
--------------------- ------- -------
$\mathrm{t}$ 0.167 0.206
$\mathrm{t_1 -t_2}$ 0.003 0.005
$\mathrm{\theta_1}$ 0.016 0.012
$\mathrm{\theta_2}$ 0.005 0.010
: \[tab:DA\] Interaction parameters (eV) between TTF and CA as defined in Fig. \[paraDA-ab\] and Fig. \[paraDA-pipi\].
---------------------------------------------------------------------------------------------------------------------------------------------------------
$\mathrm{\varepsilon}$ $\mathrm{\varepsilon_1 + $\mathrm{\lambda_1 + \lambda_2}$ $\mathrm{\upsilon^D_1 + \upsilon^D_2}$ $\mathrm{\upsilon^A_1 +
\varepsilon_2}$ \upsilon^A_2}$
------------------------ -------------------------- ---------------------------------- ---------------------------------------- -------------------------
0.061 0.001 -0.002 -0.003 -0.004
---------------------------------------------------------------------------------------------------------------------------------------------------------
: \[tab:def\] Additionnal deformation parameters (eV) due to the symmetry breaking in the low temperature phase (40 K) as defined in Section \[subsec:tb\].
Towards a metallic state?
-------------------------
Our calculations at 300 and 40 K present the electronic structure far from the transition. In order to understand how the band structure is changed by thermal contraction, we performed also a calculation just above the NIT for the crystal structure known at 90 K. The corresponding dispersion curves are given on Fig. \[disp90\]. They display a closed indirect band gap between $\Gamma$ and $\mathrm{Y}$, with nearly pure D occupied states at $\mathrm{Y}$ at a higher energy than nearly pure A empty states at $\Gamma$, as our method imposes a fully occupied VB. This is of course unphysical and is due to the DFT-LDA error on the band gap value. It leads however to interesting conclusions: (i) the thermal lattice contraction reduces the indirect band gap in the neutral phase, going toward a metallic state and (ii) as this band gap closes, an electron transfer occurs necessarily from D to A in the vicinity of the transition temperature, leading to an increase of the ionicity of the molecules, even in a frozen lattice.
Within our tight-binding model, the band gap decrease results from (i) the decrease of the difference $\mathrm{E_0^A-E_0^D}$ between the on site energies, (ii) the increase of the interaction between two TTF molecules along $\mathbf{b}$ ($\mathrm{\vert\beta_b^D\vert}$) which increases the energy of the valence band at $\mathrm{Y}$ and (iii) the increase of the interaction between the two CA molecules along $\mathbf{a}$ ($\mathrm{\vert\beta_a^A\vert}$) which decreases the energy of the CB at $\Gamma$ (Table \[tab:90\]). The cell contraction increases also $\mathrm{t}$ resulting in a slight increase of the weights of the acceptors in the VB $\mathrm{\vert C_A^{lk}\vert^2}$ in the $\mathrm{X\!-\!W\!-\!T\!-\!S}$ plane from 0.36 to 0.38 and a very small variation of $\mathrm{\rho_{VB}}$ as observed experimentally [@jacobsen] before the transition. One can notice that $\mathrm{\vert C_A^{lk}\vert^2}$ in the $\mathrm{X\!-\!W\!-\!T\!-\!S}$ plane has already reached at 90 K the value that we obtained at 40 K.
Of course, in the actual crystal, the charge transfer is largely facilitated by dynamical effects in the vicinity of the transition temperature, when the gap becomes narrow. Thermal 1D excitations from D to A have already been observed.[@mhl] They are precursors of the global changes in the crystal which certainly involve intra and intermolecular vibrations. Calculations on a A-D-A model complex [@oison] have recently shown that the CT variation occuring in the VB (along the mixed stack $\pi$-chains) induces an intra-molecular electronic redistribution, affecting deep molecular states as far as 10 eV below the frontier orbitals. These deep states are also concerned in the hydrogen bonds (along OH-chains resulting from coplanar alternating D and A molecules in the $\mathbf{b} \pm
\mathbf{a}/2$ direction) so that their modifications are directly related to the lateral lattice contractions. In the low temperature phase, the gap opens again between mixed D-A states both in the VB and CB, in the whole Brillouin zone. This suggest an electronic mechanism for the NIT where the electronic redistribution associated to the structural phase transition allows the crystal to avoid the metallic state.[@anusooya]
300 K 90 K
-------------------------- ---------- ----------
$\mathrm{E_0^A - E_0^D}$ 0.154 0.135
$\mathrm{\beta^D_b}$ $-$0.022 $-$0.028
$\mathrm{\beta^A_a}$ $-$0.017 $-$0.022
$\mathrm{t}$ 0.167 0.189
: \[tab:90\] Parameters of the tight-binding model (eV) mostly affected by the structural changes occuring between 300 and 90 K.
\[sec:topo\]Intermolecular interactions
=======================================
The previous section showed that VB and CB are clearly 1D. The aim of the present section is to explore thoroughly the total electron density $\mathrm{n(\mathbf{r})}$ given by our first-principle calculations. This is essential to elucidate all features, not only the 1D aspects, presented by this NIT.
We could plot isodensity representations of $\mathrm{n(\mathbf{r})}$ in some selected planes to evidence the most important intermolecular couplings. But, the presence of two chains per unit cell related by a gliding plane makes it an endless task and Bader’s approach [@bader] is a clever manner to circumvent this difficulty.
Bader’s theory
--------------
Within the Quantum Theory of Atoms in Molecules [@bader] $\mathrm{n(\mathbf{r})}$ can be analysed in details by means of its topological properties. The topological features of $\mathrm{n(\mathbf{r})}$ are characterized by analysing its gradient vector field $\mathbf{\nabla}\mathrm{n}(\mathbf{r})$. Here we will focus on bond critical points (CP) where $\mathbf{\nabla}\mathrm{n}(\mathbf{r})$ vanishes. They are characterized by the density at the CP $\mathrm{n(\mathbf{r}_{CP})}$, a positive curvature ($\lambda_3$) parallel and two negative curvatures ($\lambda_1,\lambda_2$) perpendicular to the bond path. The ellipticity of a bond, defined as $\epsilon = \lambda_1/\lambda_2 - 1$, describes the deviation from cylindrically symmetric bonds. For closed shell interactions, further information is obtained by using Abramov’s [@abramov] expression for the kinetic energy density $\mathrm{G(\mathbf{r}_{CP})=
3/10 (3\pi^2)^{2/3} n(\mathbf{r}_{CP})^{5/3}
+ 1/6 \nabla^2 n(\mathbf{r}_{CP})}$, and, combined with the local form of the Virial theorem [@bader], the local contribution to the potential energy density $\mathrm{V(\mathbf{r}_{CP})= 1/4 \times \nabla^2 n(\mathbf{r}_{CP})
- 2 G(\mathbf{r}_{CP})}$ which have already been used to study hydrogen bond strengths.[@espinosa] In the present work, $\mathrm{V(\mathbf{r}_{CP})}$ will be used to quantify the strength (intensity) of the intermolecular interactions. It allows to quantitatively compare interactions between two or more atoms. This procedure is less speculative than the comparison of interatomic distances to the sum of van der Waals radii [@lecointe], or related criteria [@zefirov; @collet], as van der Waals radii are not defined with sufficient accuracy [@pauling; @bondi; @baur] and fail to take into account different relative orientations.
\[sub:intra\]Intra-chain interactions
-------------------------------------
In the high temperature phase (300 K), the strongest contact within the chains is the one connecting $\mathrm{S_4}$ to $\mathrm{C_{13}}$-$\mathrm{C_{15}}$ (Fig. \[cpintra\] and Table \[tab:cpintra\]). It corresponds to the one already visible in the VB (Fig. \[isoVB\]) and has thus a significant contribution resulting from HOMO-LUMO overlap. It has a quite large ellipticity which is related to the fact that interaction do not occur between two atoms but rather between $\mathrm{S_4}$ and the bridge of $\mathrm{C_{13}}$-$\mathrm{C_{15}}$. Such a 2D attractor is also found for the interaction $\mathrm{C_{16}}\cdots\mathrm{C_{1}}$-$\mathrm{C_{2}}$ which is 40 $\%$ less intense than the former and somewhat less than $\mathrm{S_1}\cdots\mathrm{Cl_{2}}$, $\mathrm{S_1}\cdots\mathrm{Cl_{4}}$, $\mathrm{Cl_{3}}\cdots
\mathrm{C_{4}}$ and $\mathrm{Cl_{1}}\cdots\mathrm{C_{6}}$. These four latter atom-atom interactions show that $\mathrm{Cl}$ plays an important role for the electronic coupling along the chains. The acceptor in the middle (left) of Fig. \[cpintra\] is coupled to TTF on bottom (top) by $\mathrm{Cl_{1}},\mathrm{Cl_{3}}\cdots
\mathrm{C_{6}},\mathrm{C_{4}}$ ($\mathrm{S_2}$) on the left side and $\mathrm{Cl_{2}},\mathrm{Cl_{4}}\cdots\mathrm{S_1}$ ($\mathrm{C_{5}},\mathrm{C_{3}}$) on the right side.
In the low temperature phase (40 K), the pairing of the molecules in DA pairs is clearly visible on these local contacts (bottom of the figure on the right part of Fig. \[cpintra\] and Table \[tab:cpintra\]). The $\mathrm{S_4}\cdots\mathrm{C_{13}}$-$\mathrm{C_{15}}$ contact remains the dominant one with a 50 $\%$ increase of its strength. Due to molecular reorientation, the contact $\mathrm{C_{16}}\cdots\mathrm{C_{1}}$-$\mathrm{C_{2}}$ has changed into a more atom-atom like contact between $\mathrm{C_{2}}$ and $\mathrm{C_{14}}$. Its intensity has more than doubled and with $\mathrm{S_4}\cdots\mathrm{C_{13}}$-$\mathrm{C_{15}}$ they dominate all other local contacts. The latter have also increased by about 35 $\%$ and additionnal strong contacts $\mathrm{O_{2}}\cdots\mathrm{S_{2}}$ and $\mathrm{S_3}\cdots\mathrm{Cl_2}$ have appeared. These contacts confirm [@lecointe] that inside the pair, the CA molecule is essentially connected to the left part of the TTF molecule below. Between the DA pairs (top of the right part of Fig. \[cpintra\]), we observe a small decrease and a more atom-atom like contact for $\mathrm{S_3}\cdots\mathrm{C_{14}}$-$\mathrm{C_{16}}$ whereas the $\mathrm{S}\cdots\mathrm{Cl}$ and $\mathrm{Cl}\cdots\mathrm{C}$ contacts moderately increase, $\mathrm{Cl_4}\cdots\mathrm{C_3}$ being the strongest one.
300 K 40 K
-------------------------------------------------------- ------------ ------------------------------- ------------ ------------------------------- ------------ -------------------------------
$\epsilon$ $\mathrm{V(\mathbf{r}_{CP})}$ $\epsilon$ $\mathrm{V(\mathbf{r}_{CP})}$ $\epsilon$ $\mathrm{V(\mathbf{r}_{CP})}$
$\mathrm{S_4}\cdots\mathrm{C_{13}}$-$\mathrm{C_{15}}$ 7.4 -9.3 5.8 -16.5
$\mathrm{S_3}\cdots\mathrm{C_{14}}$-$\mathrm{C_{16}}$ 7.4 -9.3 4.4 -7.8
$\mathrm{S_1}\cdots\mathrm{Cl_2}$ 1.3 -7.0 1.6 -10.6
$\mathrm{S_2}\cdots\mathrm{Cl_1}$ 1.3 -7.0 1.3 -7.8
$\mathrm{Cl_3}\cdots\mathrm{C_4}$ 1.3 -6.7 1.3 -8.9
$\mathrm{Cl_4}\cdots\mathrm{C_3}$ 1.3 -6.7 1.2 -8.7
$\mathrm{O_2}\cdots\mathrm{S_2}$ 2.1 -8.9
$\mathrm{Cl_1}\cdots\mathrm{C_6}$ 2.2 -6.0 2.5 -8.1
$\mathrm{Cl_2}\cdots\mathrm{C_5}$ 2.2 -6.0 1.9 -6.8
$\mathrm{S_3}\cdots\mathrm{Cl_2}$ 2.2 -8.1
$\mathrm{S_1}\cdots\mathrm{Cl_4}$ 1.3 -6.0 1.6 -7.8
$\mathrm{S_2}\cdots\mathrm{Cl_3}$ 1.3 -6.0 1.1 -7.1
$\mathrm{C_{16}}\cdots\mathrm{C_{1}}$-$\mathrm{C_{2}}$ 6.0 -5.6
$\mathrm{C_{14}}\cdots\mathrm{C_{2}}$ 4.5 -11.7
-------------------------------------------------------- ------------ ------------------------------- ------------ ------------------------------- ------------ -------------------------------
: \[tab:cpintra\] Strongest intra-chain contacts for high (300 K) and low temperature (40 K) phases. $\epsilon$ and $\mathrm{V(\mathbf{r}_{CP})}$ give respectively the ellipticity and potential energy density (kJ. mol$^{-1}$) at each bond critical point. At 40 K, intra and inter stend for interactions inside a DA pair and between two DA pairs respectively.
\[sub:inter\]Inter-chain interactions
-------------------------------------
The inter-chain interactions can be split in two groups, the first corresponding to interactions between two chains separated by a primitive translation along $\mathbf{b}$ and the second to the interactions between two chains related by the gliding plane. In the $\mathbf{c}$ direction the chains are too far from each other to allow direct atom-atom interactions. In Fig. \[cpinter\] and Table \[tab:cpinter\] we have reported the properties of the strongest inter-chain contacts. For all of them except the two $\mathrm{Cl}\cdots\mathrm{H}$ contacts, $\epsilon$ is quite small and the interactions are diatomic. On Fig. \[cpinter\], it is clear that $\mathrm{Cl_4}$ interacts with the $\mathrm{C}-\mathrm{H_2}$ bond. In TTF-CA, the importance of hydrogen bonds has already been underlined. [@batail; @lecointe; @oison] Our results show clearly that they dominate all other intra and inter-chain interactions in both high and low temperature phases. $\mathrm{O_1}\cdots\mathrm{H_3}$ in the OH chains is stronger than $\mathrm{O_1}\cdots\mathrm{H_2}$ as it is shorter and more straight, the CA and TTF molecular planes being aligned in the $\mathbf{b} \pm \mathbf{a}/2$ direction. The other strong interactions reported in Table \[tab:cpinter\] have comparable $\mathrm{V(\mathbf{r}_{CP})}$’s to the intra-chain ones given in Table \[tab:cpintra\] but there are fewer of them in the same energy range.
In the low temperature phase, the loss of inversion symmetry leads to two inequivalent distances for each contact. As already discussed by Le Cointe et al. [@lecointe], some of the distance reductions (Table \[tab:cpinter\]) are larger than those expected from the thermal unit cell contraction which is about 1 $\%$ along $\mathbf{b}$ and $\mathbf{c}$ and 3 $\%$ along $\mathbf{a}$ directions. The hydrogen bonds are the most affected and become 80 $\%$ and 50 $\%$ stronger for the short and long bonds respectively. All other interactions except the long $\mathrm{S}\cdots\mathrm{H_2}$ bond, undergo a smaller strengthening.
------------------------------------------------ ------------------------------- ------ ------------------------------- ----------- ------------------------------- -----------
$\mathrm{V(\mathbf{r}_{CP})}$ d $\mathrm{V(\mathbf{r}_{CP})}$ $\Delta$d $\mathrm{V(\mathbf{r}_{CP})}$ $\Delta$d
$\mathrm{O_1}\cdots\mathrm{H_3}$ -16.3 2.35 -26.7 -7 $\%$ -24.3 -5 $\%$
$\mathrm{S_3}\cdots\mathrm{Cl_1}$ -6.8 3.57 -10.3 -5 $\%$ -8.7 -3 $\%$
$\mathrm{Cl_1}\cdots\mathrm{H_3}$ -6.7 3.03 -10.8 -6 $\%$ -8.0 -1 $\%$
$\mathrm{O_1}\cdots\mathrm{H_2}$ -12.0 2.50 -21.4 -8 $\%$ -18.3 -6 $\%$
$\mathrm{Cl_4}\cdots\mathrm{C_4}-\mathrm{H_2}$ -8.8 3.53 -11.6 -3 $\%$ -11.0 -2 $\%$
$\mathrm{S_1}\cdots\mathrm{H_2}$ -7.8 2.93 -10.4 -4 $\%$ -7.1 +1 $\%$
$\mathrm{Cl_2}\cdots\mathrm{Cl_3}$ -7.5 3.53 -9.1 -2 $\%$ -8.1 -1 $\%$
------------------------------------------------ ------------------------------- ------ ------------------------------- ----------- ------------------------------- -----------
: \[tab:cpinter\] Strongest inter-chain contacts for high (300 K) and low temperature (40 K) phases. $\mathrm{V(\mathbf{r}_{CP})}$ corresponds to the potential energy density (kJ. mol$^{-1}$) at the bond critical point. d is the distance in Å between the two atoms of the first column. $\Delta$d gives the relative variation of this distance with respect to its value at 300 K.
Discussion
----------
Most of the contacts discussed in the Section \[sub:intra\] and \[sub:inter\] have already been pointed out on the basis of a detailed comparison between interactomic distances and van der Waals radii. [@lecointe] Our results demonstrate that a topological analysis of the total charge density allows to go a step further. It leads to a very precise picture of the atoms involved in the different contacts (bond path) and provides a quantative way to compare their shape ($\epsilon$) and intensity ($\mathrm{V(\mathbf{r}_{CP})}$). From Table \[tab:cpinter\] for the inter-chain interactions and Table \[tab:cpintra\] and Ref. for the intra-chain ones, we find the following relation between the relative variations of the potential energy density $\Delta\mathrm{V(\mathbf{r}_{CP})}= \mathrm{(V_{40K} - V_{300K})/
V_{300K}}$ and the length of the bonds $\Delta\mathrm{d = (d_{40K} - d_{300K})/d_{300K}}$: $$\begin{aligned}
\Delta\mathrm{V(\mathbf{r}_{CP})}\simeq -10\times\Delta\mathrm{d}.\end{aligned}$$ Thus, the strengthening (diminution) of a given contact defined by $\mathrm{V(\mathbf{r}_{CP})}$ is directly correlated to the reduction (augmentation) in distance.
All our results on TTF-CA show that HOMO-LUMO overlap and hydrogen bonds dominate the intermolecular interactions in both high and low temperature structures and drive the molecular deformation and reorientation occuring at the NIT. The other inter-chain contacts do not show any peculiar behavior and are dragged along with the former. In addition, many strong interactions inside and between the chains involve Cl atoms which must largely contribute to stabilize these crystal structures.
\[sec:CT\]Charge Transfer
=========================
CT is a key factor of the NIT but it is is well known that it is not a well defined quantity and there are numerous ways to extract it from experimental or theoretical results. In TTF-CA it has already been estimated from the intensity of the CT absorption band [@jacobsen], from the dependence on molecular ionicity of either bond lengths [@lecointe] or vibrational frequencies. [@girlando2] In this section, we will use three different techniques to estimate the CT between TTF and CA. The first two are based on our ab-initio calculations in the crystal while the third is obtained from a model build on isolated molecules calculations.
From the tight-binding model
----------------------------
Within our tight-binding model, the CT in the VB is obtained by integrating the analytic expression of $\vert \mathrm{C_A^{lk}}\vert^2$ (Eq. (\[eqrhovb\])). We have first estimated $\mathrm{\rho_{VB}}$ by using the parameters determined in Section \[subsec:tb\], leaving out all those which are smaller than 0.005 eV in both phases. A second estimate has been obtained by keeping only the parameters related to the dispersion along $\Gamma\rightarrow\mathrm{X}$. In that case, $\vert \mathrm{C_A^{lk}}\vert^2$ is given by $$\begin{aligned}
\mathrm{
|C_A^{lk}|^2} &=& \frac{1}{2} \label{eqcak2}\\
&-&\mathrm{\frac{(E^A - E^D)}{2\sqrt{(E^A - E^D)^2 +
4t^2 sin^2\frac{\mathbf{k}\mathbf{a}}{2} +
4\varepsilon^2 cos^2\frac{\mathbf{k}\mathbf{a}}{2}}}}.\nonumber\end{aligned}$$ Both approaches lead to the same values of $\mathrm{\rho_{VB}}$ indicated in Table \[tab:CT\] where the corresponding experimental values have also been reported in the last column. Even though experiments lead to coarse estimates of the CT as its determination is always indirect, our high temperature value seems too large. Two reasons can be put forward to explain this discrepancy: (i) the tight-binding model is based on a linear combination of molecular orbitals of isolated molecules which disregards their deformation due to intermolecular interactions in the crystal and (ii) the underestimate of the dynamical part of the electron-electron interactions due to the LDA approximation. Among other things, the later will affect the band gap which is directly included via $\mathrm{E^D - E^A}$ in Eq. \[eqcak2\]. In order to get a feeling for the influence of the band gap on $\mathrm{\rho_{VB}}$, we increased it by 0.5 eV in both phases to locate it approximately near the first absorption peak.[@torrance] The corresponding values have also been reported in Table \[tab:CT\]. In the high temperature phase, it produces a significant decrease of 0.16 e$^-$ whereas it is only of 0.03 e$^-$ in the low temperature phase. Thus $\mathrm{\rho_{VB}}$ as well as its variation between 300 and 40 K becomes in much better agreement with the experimental results.
------- ---------------- -------------------------------- ------------- ----------- -------------
Tight Tight Integration Isolated exp.[^1]
Binding Binding Atomic molecules
$\mathrm{E_g}$ $\mathrm{E_g}+0.5~\mathrm{eV}$ Basin Model
300 K 0.59 0.43 0.48 0.47 0.2$\pm$0.1
40 K 0.72 0.69 0.64 0.60 0.7$\pm$0.1
------- ---------------- -------------------------------- ------------- ----------- -------------
: \[tab:CT\] Estimates of the charge transfer (CT) from TTF to CA.
From Bader’s theory
-------------------
Within Bader’s approach, a basin can be uniquely associated to each atom. It is defined as the region containing all gradient paths terminating at the atom. The boundaries of this basin are never crossed by any gradient vector trajectory and atomic moments are obtained by integration over the whole basin.[@bader; @integrity] The CT between TTF and CA is obtained by summing the atomic charges belonging to each molecule. For this purpose very high precision is needed as we are looking for a few tenths of an electron out of a total of 116 valence electrons. The corresponding CT are reported in Table \[tab:CT\]. These estimates are no more directly affected by the band gap error due to the LDA as they are based on the sole occupied states. Accordingly, they are both at 300 and 40 K smaller than those obtained by the tight-binding model and thus closer to the experimental results. Nevertheless, in the high temperature phase, the CT seems still somewhat overestimated.
From a molecular model
----------------------
We obtained a third estimate of the CT by means of a simple model based on isolated molecules calculations which will only be briefly reported in the present paper. Here, we are looking for the charge transfer $\rho$ which minimizes the total energy in a fixed structure, i.e. either in the high or low temperature structure. The first contribution to the variation of the total energy is the molecular one (Carloni et al. [@carloni]): $$\begin{aligned}
\mathrm{
\Delta E_{mol}(\rho) = \rho(\epsilon_A^0-\epsilon_D^0)
+ \frac{1}{2} \rho^2 (U_D+U_A)},
\label{eq503}\end{aligned}$$ where the reference state is the one without any CT, $\epsilon_j^0$ is the energy of the relevant orbital in the reference state and $\mathrm{U_j}$ the Coulomb repulsion of an electron in that orbital. The second contribution is the electrostatic one limited to the Madelung contribution $ q_i q_j / r_{ij}$. The atomic charges $q_i$ and $q_j$ are deduced from PAW calculations on isolated molecules by using a model density reproducing the multipole moments of the true molecular charge density.[@blo2] The remaining contributions to the total energy are usually accounted for by a van der Waals term which is constant in a given structure and which can thus be disregarded. The CT values given by this simple model are also given in Table \[tab:CT\].\
Discussion
----------
It is satisfying that all our three estimates are in close agreement with each other in both low and high temperature phases. At 40 K, they are also in good agreement with the experimental value. At 300 K, we observe a systematic overestimate of about 0.2 e$^-$ with respect to the values obtained by Jacobsen et al. [@jacobsen], our result being closer to the one deduced from intermolecular bond lengths (0.4 e$^-$).[@lecointe]\
This discrepancy may have different origins. First of all, the CT is not a uniquely defined quantity as its evaluation relies on different (and not necessarily equivalent) models based either on theoretical or on experimental results. Starting from experimental data, the models can only be very simple and most of the results published in the literature are based on the behavior of isolated molecules upon ionization: bond-lengths, intramolecular vibration frequencies, optical spectra..., neglecting thus all crystalline effects apart from the charge transfer on these molecular properties.\
On the other hand, our theoretical results are based on the determination of the charge density. They are all affected by the error due to the LDA approximation to the exchange-correlation potential. It is well known that this approximation, even in its gradient-corrected version, underestimates the dynamical part of the electron-electron interactions. In weakly interacting systems where these dynamical effects (or so-called van der Waals interactions) are dominant, this leads to an overestimate of the electron delocalization and an overbinding effect, together with an underestimate of the band-gap. An overestimate of the intermolecular CT is thus expected. However, it has already been shown [@oison] that in the ionic phase the multipolar static effects are considerably stronger than in the neutral phase, so that the LDA error on the CT should also be considerably reduced in the low temperature phase.
In our tight-binding model we have clearly evidenced the effect of the underestimate of the band gap. Within Bader’s approach, the CT determination only relies on the occupied states, but leads nevertheless to an overestimate in the neutral phase. Regarding our model based on isolated molecules calculations, it has been shown that the ionization energy of TTF [@katanTTF] and the electron affinity of CA [@katanCA] are correctly obtained within the gradient-corrected LDA approximation used in the present study. But our model takes into account neither the atomic multipolar contributions nor the polarization due to the crystal field.
Thus, our three estimates are coherent and lead probably to a correct determination of the CT in the ionic phase, and a too large one in the neutral phase. At this stage however, the discrepancy with the available experimental values has not to be attributed to the sole LDA problem, but also to the definition itself of this CT. A good way to check the rather high value obtained in this work at 300 K is to perform an experimental charge density acquisition by X-ray diffraction and use Bader’s approach to extract an experimental CT value coherent with our theoretical estimate.
Conclusion
==========
This is the first extensive ab-initio study of the electronic ground states of TTF-CA, prototype compound of the NIT. We should remind here that temperature is included only via the experimental structures, all calculations being performed in a frozen lattice.
The valence and conduction bands are dominantly of 1D character along the stacking chains and are essentially a linear combination of the HOMO of D and the LUMO of A as in TTF-2,5Cl$_2$BQ.[@katanjpcm] The weak molecular distorsion due to the symmetry breaking part of the NIT is responsible for the main contribution to the charge transfer variation. We found also that the thermal lattice contraction leads to a decrease of the band gap in the neutral phase, going toward a metallic state. This is consistent with the theoretical work of Anusooya-Pati et al. [@anusooya] who found a metallic ground state at the transition which is unstable to dimerization. It has also to be related with the metallic behavior observed by Saito et al. [@saito] in an other DA alternating CT complex.
A detailed analysis of all intermolecular bond critical points shows that molecular deformation and reorientation occuring at the NIT are driven by both the HOMO-LUMO overlap and the hydrogen bonds. Quite many intermolecular contacts involving chlorine atoms have also been evidenced. Moreover, we have found a direct relationship between the relative variations of the potential energy density at the critical point and the distance between the atoms involved in the intermolecular bond.
Three estimates of the charge transfer have been obtained from our tight-binding model, from integration over the atomic basins defined within Bader’s approach and from a simple model based on isolated molecules calculations. All three estimates are in close agreement with each other and give a value of about 0.45 e$^-$ at 300 K and 0.65 e$^-$ at 40 K. Experimental charge density collection on TTF-CA is under progress in order to check the rather high value obtained at 300 K.\
All these results provide a possible picture for the mechanism of the NIT. The energy dispersions in VB and CB are mainly along $\mathbf{a}^\ast$ and $\mathbf{b}^\ast$ with an indirect gap between $\Gamma$ and $\mathrm{Y}$. Above the transition temperature, the VB has dominant D character at $\mathrm{Y}$ and the CB a pure A character at $\Gamma$. When temperature is lowered, the slow decrease of the cell parameters induces a slow variation of all transfer integrals and the band gap progressively vanishes. There is a point where the indirect band gap closes. Pockets of holes at $\mathrm{Y}$ and electrons at $\Gamma$ appear which activate a new mechanism assisted by intra [@katanTTF; @venuti] and intermolecular [@moreac; @okimoto] vibrations creating DA pairs (and longer $\ldots$DADA$\ldots$ strings) for which the inversion symmetry is locally broken. This picture is consistent with the observed pretransitionnal effects [@mhl] described as ionic segments in the neutral phase [@okimoto; @nagaosa; @koshihara; @luty] sometimes called CT exciton strings.[@kuwata] The rapid increase of CT along the chains may then activate the hydrogen bonds via intra-molecular electron-electron interactions between the valence $\pi$ and deep $\sigma$ molecular orbitals.[@oison] This enables the whole crystal to topple into the ionic state at the transition point and is consistent with the sudden decrease of the cell parameter $\mathbf{b}$.[@lecointe] Once the whole crystal has switched the band gap opens and both VB and CB have strong D-A character leading to a macroscopic change of the CT.
This work has benefited from collaborations within (1) the $\Psi_k$-ESF Research Program and (2) the Training and Mobility of Researchers Program “Electronic Structure” (Contract: FMRX-CT98-0178) of the European Union and (3) the International Joint Research Grant “Development of charge transfert materials with nanostructures” (Contract: 00MB4). Parts of the calculations have been supported by the “Centre Informatique National de l’Enseignement Supérieur” (CINES—France). We would like to thank P.E. Blöchl for his PAW code and A. Girlando for useful discussions.
[^1]: From Ref.
|
=22.5cm -1.5cm -0.3cm 0.3cm ‘@=11 addtoreset[equation]{}[section]{} ‘@=12
\#1[(\[\#1\])]{} \#1[Fig \[\#1\]]{} ø[[O]{}]{} §[[S]{}]{} [H]{} \#1[[**\#1**]{}]{} c\#1 =1000000
KEK/TH/756\
April 2001\
\
Avinash Dhar $^{1*}$ and Yoshihisa Kitazawa $^{\dagger}$\
\
[*Laboratory for Particle and Nuclear Physics,*]{}\
[*High Energy Accelerator Research Organization (KEK),*]{}\
[*Tsukuba, Ibaraki 305-0801, JAPAN.*]{}\
=1000000 **ABSTRACT\
**
We investigate Schwinger-Dyson equations for correlators of Wilson line operators in non-commutative gauge theories. We point out that, unlike what happens for closed Wilson loops, the joining term survives in the planar equations. This fact may be used to relate the correlator of an arbitrary number of Wilson lines eventually to a set of [*closed*]{} Wilson loops, obtained by joining the individual Wilson lines together by a series of well-defined cutting and joining manipulations. For closed loops, we find that the non-planar contributions do not have a smooth limit in the limit of vanishing non-commutativity and hence the equations do not reduce to their commutative counterparts. We use the Schwinger-Dyson equations to derive loop equations for the correlators of Wilson observables. In the planar limit, this gives us a [*new*]{} loop equation which relates the correlators of Wilson lines to the expectation values of closed Wilson loops. We discuss perturbative verification of the loop equation for the $2$-point function in some detail. We also suggest a possible connection between Wilson line based on an arbitrary contour and the string field of closed string.
------------------------------------------------------------------------
Introduction
============
Non-commutative gauge theories are realized on branes in the zero slope limit in the presence of a large NS-NS B-field [@CDS; @DH; @AAJ; @CK; @C2; @CH; @VS; @SW1]. Recently these theories have attracted a lot of attention. Various aspects of these theories have been studied in [@MR; @HI; @NS; @AIIKKT; @L; @C1; @I; @IIKK1; @SW2; @DMWY; @DRT; @BM; @AMNS; @MRS; @AD; @IIKK2; @DG; @GW; @F; @JMW; @HKLM; @DMR; @W; @GMS; @S; @KRS; @MW; @DK1].
In ordinary gauge theories, a generic gauge-invariant observable is provided by an arbitrary closed Wilson loop. Non-commutative gauge theories have more general gauge-invariant observables, defined on [*open*]{} contours. Such gauge-invariant observables in non-commutative gauge theories were constructed in [@IIKK1]. Different aspects of these were studied in [@AD; @IIKK2; @RU; @DR; @GHI; @DW; @DK2; @HL; @DT; @OO; @LM; @O]. Roughly speaking, these gauge-invariant observables can be written as Fourier transforms of open Wilson lines. In the operator formalism they are given by the following expression W\_C\[y\]=[Tr]{}(Pexp{i\_C d \_y\_() A\_(x+y())} e\^[ik.x]{} ), \[oneone\] where =i\_, \[onetwo\] and the trace in (\[oneone\]) is over both the gauge group, taken here to be $U(N)$, as well as the operator Hilbert space. These open Wilson lines are gauge-invariant in non-commutative gauge theories, unlike in ordinary gauge theories, provided the momentum $k^\mu$ associated with the Wilson line is fixed in terms of the straight line joining the end points of the path $C$, given by $y^\mu(\sigma)$ where $0 \leq \sigma \leq 1$, by the relation y\_(1)-y\_(0)=\_k\_. \[onethree\] The path $C$ is otherwise completely arbitrary. When $k$ vanishes, the two ends of the path $C$ must meet and we have a closed Wilson loop.
In an earlier work [@DK2], based on a perturbative analysis of correlation functions of straight Wilson lines with generic momenta, we had suggested that at large momenta the Wilson lines are bound into the set of closed Wilson loops that can be formed by joining the Wilson lines together in all possible different ways. In the present work we will establish a more general connection between correlators of Wilson lines and expectation values of Wilson loops in a non-perturbative setting, for arbitrary Wilson lines. In this generic case, however, the closed Wilson loops to which the Wilson lines are related are not formed by simply joining the Wilson lines together, but by more complicated cutting and joining manipulations. Also, the statement is valid for arbitrary momenta, not necessarily large, carried by the Wilson lines. We will use the framework of Schwinger-Dyson equations and the closely related loop equations in the planar limit. In the context of non-commutative gauge theories similar equations have been studied earlier in [@FKKT; @DW2; @AZD].
This paper is organized as follows. In the next section we summarize some aspects of operator formulation of non-commutative gauge theories, which is used throughout this paper, and in particular list some useful identities. In Sec.3 we derive Schwinger-Dyson equations for the correlators of open and closed Wilson observables and discuss these at finite $N$ as well as in the planar limit. As in commutative gauge theories, the splitting term disappears from the planar equations. However, unlike in the case of closed Wilson loops, the joining term survives in the planar equations for open Wilson lines. This has the consequence of eventually relating them to closed Wilson loops. We also find that at finite $N$, the splitting term does not reduce to the ordinary gauge theory result in the limit in which the non-commutativity is removed. We trace this result to the UV-IR mixing in non-commutative gauge theories. In Sec.4 we use the results of Sec.3 to write down loop equations for the correlators of open and closed Wilson observables. We consider the loop equation for the $2$-point function of the open Wilson lines in the planar limit and discuss the verification of this equation in ’t Hooft perturbation theory in some detail. Sec.5 contains a discussion of a possible connection between a Wilson line based on an arbitrary contour and string field for closed string and the non-perturbative meaning of the new loop equations derived here. In the Appendix, we give details of the perturbative calculations.
Non-commutative gauge theories - operator formulation
=====================================================
We will be working in $4$-dimensional Euclidean space with a generic non-commutativity parameter in (\[onetwo\]). The non-commutative gauge theory action that we will consider is S=[1 4g\^2]{}[Tr]{}(F\_(x))\^2 + \[twoone\] where $$F_{\mu\nu}(\hat x)=\hat\del_\mu A_\nu(\hat x)-\hat\del_\nu
A_\mu(\hat x) +i[A_\mu(\hat x),A_\nu(\hat x)].$$ The dots stand for possible bosonic and fermionic matter coupled to the gauge field and the trace ${\rm Tr} = {\rm tr}_{U(N)} \ {\rm tr}_{\cal H}$ is over the gauge group $U(N)$ as well as the operator Hilbert space $\cal H$. To define this latter trace more precisely, let us assume that $\theta_{\mu\nu}$ has the canonical form, \_[01]{}=-\_[10]{}=\_a, \_[23]{}=-\_[32]{}=\_b, \[twoonea\] with all other components vanishing, and let us define the operators a=[[x]{}\_0 +[x]{}\_1 ]{}, b=[[x]{}\_2 +[x]{}\_3 ]{}, \[twooneb\] which satisfy the standard harmonic oscillator algebra, =1, \[b,b\^\]=1. \[twoonec\] The operator Hilbert space trace is then defined by \_[H]{} O(x)=(2)\^2 \_a \_b \_[n\_a,n\_b]{} <n\_a,n\_b|O(x)|n\_a,n\_b>. \[twooned\] Note that with this definition of the operator Hilbert space trace, the coupling constant $g$ appearing in the action (\[twoone\]) is dimensionless.
We use the standard Weyl operator ordering, O(x)= d\^4y O(y) \^[(4)]{}(x - y) \[twotwo\] where the [*operator*]{} delta-function is defined in terms of the Heisenberg group elements by \^[(4)]{}(x - y)= e\^[-ik.y]{} e\^[ik.x]{}. \[twothree\] The use of this operator delta-function simplifies many calculation because it shares some properties of the usual delta-function. For example, (\[twotwo\]) and \_[H]{}\^[(4)]{}(x - y)=1. \[twofour\] There are, of course, differences as in the following identity which encodes the star product: \^[(4)]{}(x - y) \^[(4)]{}(x - z) =e\^[-[i 2]{} \_+ \_-]{} \^[(4)]{}(x-y\_+) \^[(4)]{}(y\_-). \[twofive\] where $y_+={y+z \over 2}$ and $y_-=y-z$. Below we give two identities involving these operator delta-functions which will be used in deriving the Schwinger-Dyson equations in the next section. The first one “joins” together two operators which appear inside two different traces, d\^4z \_a [Tr]{}\[O\_1(x) t\^a \^[(4)]{}(x - z)\] [Tr]{}\[O\_2(x) t\^a \^[(4)]{}(x - z)\] = [Tr]{}\[O\_1(x) O\_2(x)\], \[twosix\] and the second one “splits” two operators which are inside the same trace, d\^4z \_a [Tr]{}\[O\_1(x) t\^a \^[(4)]{}(x - z) O\_2(x) t\^a \^[(4)]{}(x - z)\] = [1 (2)\^4 [det]{}]{} [Tr]{}\[O\_1(x)\] [Tr]{}\[O\_2(x)\].\
\[twoseven\] Here the $t^a$’s are the generators for the gauge group, which we have taken to be U(N), with the normalization dictated by the completeness condition \_a t\^a\_[ij]{}t\^a\_[kl]{}=\_[il]{}\_[jk]{} \[twosevena\]
Wilson observables and cyclic symmetry
--------------------------------------
The generic gauge-invariant Wilson observable is given in (\[oneone\]). We will also need the Wilson operator W\_C\[y\]\_[0s]{}=Pexp{i\_C d \_y\_() A\_(x+y()-y(0))} e\^[ik\_s.x]{}. \[twoeight\] Here the subscripts ‘$0s$’ indicate that the path-ordered phase factor runs from $\sigma=0$ to $\sigma=s$ along the curve $C$, and $y(s)-y(0)=\theta k_s$. The Wilson operator $\hat W_C[y]_{s1}$, which runs from $\sigma=s$ to $\sigma=1$, is defined similarly: W\_C\[y\]\_[s1]{}=Pexp{i\_C d \_y\_() A\_(x+y()-y(s))} e\^[ik\_s.x]{} \[twonine\] where $y(1)-y(s)=\theta \tilde k_s$. These operators are related to the Wilson observable as follows: (W\_C\[y\]\_[01]{})=W\_C\[y\] e\^[ik.y(0)]{}. \[twoten\]
The Wilson observable $W_C[y]$ possesses a “cyclic symmetry” because of the trace over both the gauge group and the operator Hilbert space. To arrive at a mathematical expression of this symmetry, we note that W\_C\[y\]\_[01]{}=e\^[[i 2]{}k\_s k]{} W\_C\[y\]\_[0s]{} W\_C\[y\]\_[s1]{}, \[twoeleven\] and, [^1] therefore, W\_C\[y\] &=& e\^[-ik.y(0)]{} e\^[[i 2]{}k\_s k]{} [Tr]{}(W\_C\[y\]\_[0s]{} W\_C\[y\]\_[s1]{})\
&=& e\^[-ik.y(s)]{} [Tr]{}(W\_[C\_s]{}\[y\_s\])\
&& W\_[C\_s]{}\[y\_s\], \[twotwelve\] where in the second step we have used the cyclic property of the trace and recombined the two operators in the opposite order. The contour $C_s$ is given by y\_s() &=& y(+ s), 0 (1-s)\
&=& y(-1+s)+y(1)-y(0), (1-s)1. \[twothirteen\] It is obtained from the curve $C$ by cutting it at a point $\sigma=s$ and rejoining the two pieces in the opposite order, as shown in Fig. 1.
(400,150)(0,0) (150,75)(50,90,270) (300,125)(50,180,270) (300,25)(50,90,180) (160,75)(230,75) (158,75)\[r\][$C$]{} (150,129)\[lb\][$\sigma=0$]{} (98,75)\[rt\][$s$]{} (150,25)\[lt\][$\sigma=1$]{} (235,75)\[l\][$C_s$]{} (250,129)\[lb\][$\sigma=0$]{} (302,75)\[l\][$(1-s)$]{} (250,25)\[lt\][$\sigma=1$]{}
\
[*Fig. 1: Cyclic symmetry of Wilson line*]{}
If the original curve is a straight line, then the transformed curve is also a straight line shifted by an amount $s(\theta k)$. It is easy to see more directly that such shifts are a symmetry of the straight Wilson line. This symmetry was used very effectively in [@DK2] to simplify perturbative calculations. More generally, the cyclic symmetry relates Wilson observables defined on contours that are nontrivially different. As we shall see later, the quantity V\^[(k)]{}\_[C]{}\[y\]=i\_C dy\_() e\^[-ik.y()]{} \[twofourteen\] frequently appears in perturbative calculations of multipoint Wilson line correlation functions. It is easy to see that in fact this quantity is invariant under the cyclic symmetry (\[twothirteen\]), and that may be reason for its appearance. It is also noteworthy that the above quantity is very similar to the vector vertex operator of open string theory. It would be interesting to have a better understanding of these connections and the implications of the cyclic symmetry.
Schwinger-Dyson equation
========================
In this section we will first derive the Schwinger-Dyson equation for multipoint correlators of Wilson observables and then analyse it at finite $N$ as well as in the planar limit. The Schwinger-Dyson equation follows from the standard functional integral identity 0=d\^4z \_a [ A\_\^a(z)]{} . \[threeone\] Using W\_C\[y\]=i \_C dy\_(s) e\^[[i 2]{}k\_s k]{} W\_C\[y\]\_[0s]{}(t\^a\^[(4)]{}(x-z)) W\_C\[y\]\_[s1]{} \[threetwo\] and the joining and splitting identities, (\[twosix\]) and (\[twoseven\]), we get &&[1 g\^2]{} <[Tr]{}(W\_[C\_1]{}\[y\_1\]D\_F\_(x)) [Tr]{}(W\_[C\_2]{}\[y\_2\]) (W\_[C\_n]{}\[y\_n\])>\
&=& i\^n\_[l=2]{}\_[C\_l]{}dy\_[l]{}(s) e\^[[i 2]{}k\_[ls]{} k\_l]{} <[Tr]{}(W\_[C\_2]{}\[y\_2\]) (W\_[C\_l]{}\[y\_l\]\_[0s]{}W\_[C\_1]{}\[y\_1\] W\_[C\_l]{}\[y\_l\]\_[s1]{}) (W\_[C\_n]{}\[y\_n\])>\
&+& [i (2)\^4 [det]{}]{}\_[C\_1]{}dy\_[1]{}(s) e\^[[i 2]{}k\_[1s]{} k\_1]{} <[Tr]{}(W\_[C\_1]{}\[y\_1\]\_[0s]{}) [Tr]{}(W\_[C\_1]{}\[y\_1\]\_[s1]{})[Tr]{}(W\_[C\_2]{}\[y\_2\]) (W\_[C\_n]{}\[y\_n\])>.\
\[threethree\] In this equation $y_l(s)-y_l(0)=\theta k_{ls}$ and each of the contours $C_1, C_2, \cdots C_n$ may be open or closed.
Closed Wilson loop
------------------
Let us first consider a single closed Wilson loop. In this case equation (\[threethree\]) reduces to <[Tr]{}(W\_C\[y\]D\_F\_(x))> =[i (2)\^4 [det]{}]{}\_C dy\_(s) <[Tr]{}(W\_C\[y\]\_[0s]{}) [Tr]{}(W\_C\[y\]\_[s1]{})> \[threefour\] Here $C$ is a closed curve. The right hand side of (\[threefour\]) has a disconnected piece. However, because of momentum conservation, the disconnected piece contributes only when $y(s)=y(0)$. In fact, both $<{\rm Tr}(\hat W_C[y]_{0s})>$ and $<{\rm Tr}(\hat W_C[y]_{s1})>$ are proportional to $(2\pi)^4 \delta^{(4)}(k_s)$. One of these gives rise to the total space-time volume $V$, while the other factor can be rewritten as $(2\pi)^4 {\rm det}\theta \ \delta^{(4)}(y(s)-y(0))$. Thus, we may rewrite (\[threefour\]) as &&[1 g\^2]{} <[Tr]{}(W\_C\[y\]D\_F\_(x))>\
&=& [i V]{} \_C dy\_(s) \^[(4)]{}(y(s)-y(0)) <[Tr]{}(W\_C\[y\]\_[0s]{})> <[Tr]{}(W\_C\[y\]\_[s1]{})>\
&& + [i (2)\^4 [det]{}]{}\_C dy\_(s) <[Tr]{}(W\_C\[y\]\_[0s]{}) [Tr]{}(W\_C\[y\]\_[s1]{})>\_[conn.]{} \[threefive\]
The second term on the right hand side of (\[threefive\]) contains the connected part of the $2$-point function of open Wilson lines. At finite $N$, it is easy to see that this term is down by a factor of $1/N^2$ relative to the other terms in the equation. In the planar limit, therefore, this term drops out and the planar equation looks formally like the correponding equation in commutative gauge theory. This is consistent with the perturbative result that, except for in an overall phase, the dependence on the non-commutative parameter $\theta$ drops out of planar diagrams. However, there are new gauge-invariant observables in non-commutative gauge theory, the open Wilson lines, and so there are new equations. As we shall see shortly, these new equations have a non-trivial planar limit. One might then say that it is these new equations that reflect the new physics of non-commutative gauge theory.
At finite $N$ the second term on the right hand side of (\[threefive\]) contributes. One might wonder whether this term reduces to its commutative counterpart in the limit of small non-commutative parameter. An argument has been presented in [@AZD] suggesting that this is the case. However, we find that, in fact, the small $\theta$ limit of this term is not smooth, at least in perturbation theory, as we will now show.
At the lowest order in perturbation theory, the second term on the right hand side of (\[threefive\]) evaluates to \_C dy\_(s) [1 k\_s\^2]{} . \[threesix\] For simplicity, let us specialize to a rectangular contour of sides $L_1$ and $L_2$. We will also take $\theta$ to be of the form in (\[twoonea\]) with $\theta_a=\theta_b=\theta_0$. In this case the diagrams that contribute to (\[threesix\]) are shown in Fig. 2.
(450,125)(0,0) (50,25)(50,100) (50,100)(200,100) (200,100)(200,25) (100,25)(50,25) (197,25)(103,25) (250,25)(250,100) (250,100)(400,100) (400,100)(400,25) (300,25)(250,25) (397,25)(303,25) (100,25)(30,0,180)[3]{}[8]{} (325,100)(325,25)[3]{}[8]{} (102,28)\[b\][$s$]{} (47,60)\[rb\][$L_2$]{} (150,22)\[lt\][$L_1$]{} (197,22)\[lt\][$\sigma=0$]{} (203,28)\[lb\][$\sigma=1$]{} (302,28)\[b\][$s$]{} (247,60)\[rb\][$L_2$]{} (350,22)\[lt\][$L_1$]{} (397,22)\[lt\][$\sigma=0$]{} (403,28)\[lb\][$\sigma=1$]{}
\
[*Fig. 2: Lowest order diagrams contributing to the non-planar term*]{}
We can easily evaluate (\[threesix\]) in this case. The result is i(L\_[1]{}-L\_[2]{}) \[threeseven\] where $\phi=L_1\theta^{-1}L_2$ is the magnetic flux passing through the rectangular contour and f()= (1-e\^[-is]{}). \[threeeight\] In the limit of small non-commutativity, $\phi$ is large, and then $f(\phi) \sim {\rm ln}\phi$. In this case the right hand side of (\[threefive\]) is divergent with the leading term going as $\sim
{\rm ln}\phi/\theta_0^2$. So we see that if we take the limit of small non-commutativity first, keeping $N$ finite, we do not recover the commutative result. It is easy to see in perturbation theory that the origin of this problem lies in UV-IR mixing. It has been argued in [@MRS] that this phenomenon renders loop diagrams finite in non-commutative field theory. Now, the diagrams in Fig. 2 that contribute to the right hand side of (\[threefive\]) at order $1/N^2$ in the lowest order in ’t Hooft perturbation theory actually come from non-planar one-loop diagrams on the left hand side of this equation, as shown in Fig. 3.
(450,150)(0,0) (50,50)(50,125) (50,125)(200,125) (200,125)(200,50) (200,50)(50,50) (250,50)(250,125) (250,125)(400,125) (400,125)(400,50) (400,50)(250,50) (100,50)(30,0,180)[3]{}[8]{} (150,100)(70,225,315)[3]{}[12]{} (350,125)(350,50)[3]{}[8]{} (350,100)(70,225,315)[3]{}[12]{} (100,53)\[b\][$s$]{} (47,85)\[rb\][$L_2$]{} (150,47)\[lt\][$L_1$]{} (300,53)\[b\][$s$]{} (247,85)\[rb\][$L_2$]{} (350,47)\[lt\][$L_1$]{}
\
[*Fig. 3: Non-planar one-loop diagrams giving rise to diagrams in Fig. 2.*]{}
In fact, the relevant amplitude is g\^4N dy\_1dy\_3 dy\_2dy\_4 e\^[ipq]{} \[threeeighta\] We can estimate the above momentum integral as follows && e\^[ipq]{}\
&=& {
[ll]{} [14\^2]{}[1(y\_1-y\_3)\^2]{} [14\^2]{}[1(y\_2-y\_4)\^2]{} &\
[12(2)\^4]{} [1\_0\^2]{}[ln]{}([|y\_1-y\_3||y\_2-y\_4|]{}) &
. In commutative gauge theory these diagrams have short distance singularities which are linearly divergent. In the non-commutative theory they get regularized at the non-commutativity scale, as can be seen from the above expression. The singularities of the commutative theory reappear in the limit of small non-commutativity. This is what is reflected in the singular behaviour of the right hand side of (\[threefive\]) for small non-commutativity.
Open Wilson lines
-----------------
The generic equation satisfied by the $n$-point function of Wilson lines is (\[threethree\]). The second term on the right hand side of this equation has a disconnected part which is given by &&<[Tr]{}(W\_[C\_1]{}\[y\_1\]\_[0s]{})> <[Tr]{}(W\_[C\_1]{}\[y\_1\]\_[s1]{})[Tr]{}(W\_[C\_2]{}\[y\_2\]) (W\_[C\_n]{}\[y\_n\])>\
&+& <[Tr]{}(W\_[C\_1]{}\[y\_1\]\_[s1]{})><[Tr]{}(W\_[C\_1]{}\[y\_1\]\_[0s]{})[Tr]{}(W\_[C\_2]{}\[y\_2\]) (W\_[C\_n]{}\[y\_n\])>. \[threenine\] Because of momentum conservation, the first term contributes only for $y(s)=y(0)$, while the second term contributes only for $y(s)=y(1)$. In either term we get back the original $n$-point function. This is just like for the closed Wilson loop discussed above. The connected part of the second term on the right hand side can be easily seen to be down by a factor of $1/N^2$ compared to the other terms in the equation. In the planar limit, therefore, it drops out, leaving only the “joining” term (the first term on the right hand side), apart from the disconnected term mentioned above. We then have the result that the planar Schwinger-Dyson equation for Wilson lines expresses any $n$-point function entirely in terms of $(n-1)$-point functions. By iterating this procedure $(n-1)$ times we may, in principle, express any $n$-point function entirely in terms of [*closed*]{} Wilson loops.
The simplest example of the above phenomenon is provided by the $2$-point function. In this case, the planar Schwinger-Dyson equation reads &&[1 g\^2]{} <[Tr]{}(W\_[C\_1]{}\[y\_1\]D\_F\_(x)) [Tr]{}(W\_[C\_2]{}\[y\_2\])>\
&=& i\_[C\_2]{}dy\_[2]{}(s) e\^[[i 2]{}k\_[2s]{} k\_2]{} <[Tr]{}(W\_[C\_2]{}\[y\_2\]\_[0s]{}W\_[C\_1]{}\[y\_1\] W\_[C\_2]{}\[y\_2\]\_[s1]{})>\
&+& [i (2)\^4 [det]{}]{}\_[C\_1]{}dy\_[1]{}(s) e\^[[i 2]{}k\_[1s]{} k\_1]{} . \[threeten\] We see that the right hand side involves closed Wilson loops, apart from the the $2$-point function itself. The closed curves involved are obtained by first traversing the curve $C_1$ given by $y_1(\sigma), \ 0
\leq \sigma \leq 1$ and then the curve given by y() &=& y\_2(+s)-y\_2(s)+y\_1(1), 0 (1-s)\
&=& y\_2(-1+s)-y\_2(s)+y\_1(0), (1-s) 1 \[threeeleven\] for different values of $s, \ 0 \leq s \leq 1$. Note that the closed curves obtained in this way are continuous because of momentum conservation.
Similarly, the $3$-point function involves two different $2$-point functions, &&<[Tr]{}(W\_[C\_2]{}\[y\_2\]\_[0s]{}W\_[C\_1]{}\[y\_1\] W\_[C\_2]{}\[y\_2\]\_[s1]{})[Tr]{}(W\_[C\_3]{}\[y\_3\])>,\
&&<[Tr]{}(W\_[C\_3]{}\[y\_3\]\_[0s]{}W\_[C\_1]{}\[y\_1\] W\_[C\_3]{}\[y\_3\]\_[s1]{})[Tr]{}(W\_[C\_2]{}\[y\_2\])>, \[threetwelve\] depending on whether $C_1$ and $C_2$ or $C_1$ and $C_3$ combine into a single curve. These $2$-point functions are themselves related to closed loops, as discussed above. Thus, the $3$-point function can eventually be related to closed Wilson loops, there being two distinct sets of structures for the closed contours involved. These closed contours can be obtained explicitly, as we have done above for the case of the $2$-point function. For $n$-point function one eventually gets $(n-1)!$ distinct structures for the closed loops.
The above discussion establishes a general link between the correlators of Wilson lines and the expectation value of closed Wilson loops. In a previous work [@DK2] we have presented a perturbative proof for long straight Wilson lines to be bound together into closed loops. The connection we have found here is more general in that the Wilson lines are based on arbitrary contours and the momenta need not be large. The closed loops we now find are also more general. We believe that the planar Schwinger-Dyson equation indeed supports our previous claim for the high energy behaviour of the Wilson lines. This is because, firstly, it can be argued that the planar approximation is always valid in the high energy limit (or large $\theta$ limit) since the splitting term is supressed by $1/{\rm det}\theta$. Secondly, we need to repeatedly insert the equation of the motion operator into the Wilson line correlator in order to eventually relate it to Wilson loops. Such an operation picks up contact terms in the sense that it makes two Wilson lines touch each other. In high energy limit, we expect to rediscover such contact terms, although the real singularities are expected to be regulated by the noncommutativity. Finally, the set of relevant closed loops may collapse to that found in [@DK2], which is the set of extreme configurations of Wilson loops obtained by simply joining the Wilson lines end-to-end.
Loop equation for Wilson lines
==============================
In this section we will first derive the loop equation for multipoint correlators of Wilson lines. We will then consider the case of the $2$-point function in detail and verify the planar loop equation in this case upto second order in ’t Hooft perturbation theory.
The loop equation is basically the Schwinger-Dyson equation (\[threethree\]) with the insertion of the equation of motion operator replaced by a geometric variation of the contour. This is done with the help of the identity &=& e\^[[i 2]{}k\_[’]{} k\_]{} W\_C\[y\]\_[0]{}(i\_y\_() F\_(x)) W\_C\[y\]\_[’]{}(i\_[’]{} y\_(’) F\_(x))W\_C\[y\]\_[’1]{}\
&& -(-’)e\^[[i 2]{}k\_k]{} W\_C\[y\]\_[0]{}(i\_y\_() D\_F\_(x))W\_C\[y\]\_[1]{} \[fourone\] where, as before, $\theta k_\tau=y(\tau)-y(0)$ and $\tilde k_{\tau}=k-k_\tau$. Note that this identity is valid at interior points of the Wilson line. At the boundaries of the Wilson line one has to be more careful. However, if we vary [*both*]{} the ends keeping $k$ fixed, and assume that the tangents to the contour at the ends are identical, [^2] then a very similar identity is valid at the ends also.
We need to separate out the equation of motion piece on the right hand side of (\[fourone\]). This may formally be done by defining the “loop laplacian” [@POLY] \_[0]{} \^\_[-]{}dt [\^2 y\_(+t/2) y\_(-t/2)]{}. \[fourtwo\] In principle, if the quantum theory is regularized then the first term in (\[fourone\]) does not have any singularities as $\tau
\rightarrow \tau'$ and so the loop laplacian picks up only the delta-function term on the right hand side of this equation. [^3] We then get -[\^2 W\_C\[y\] y\^2()]{}=e\^[[i 2]{}k\_k]{} W\_C\[y\]\_[0]{}(i\_y\_() D\_F\_(x))W\_C\[y\]\_[1]{}. \[fourthree\] Using this in (\[threethree\]) we get the loop equation for Wilson line correlators && -[1 g\^2]{}[\^2 y\_1\^2()]{}<W\_[C\_1]{}\[y\_1\]W\_[C\_2]{}\[y\_2\] W\_[C\_n]{}\[y\_n\]>\
&=& \^n\_[l=2]{} \_[C\_l]{} ds (i\_y\_1().i\_s y\_l(s)) e\^[-ik\_1.y\_1()-ik\_l.y\_l(s)]{}\
&&\
&& + [1 (2)\^4 [det]{}]{} \_[C\_[1]{}]{} ds (i\_y\_1().i\_s y\_1(s)) e\^[-ik\_1.y\_1() +[i 2]{}k\_[1s]{} k\_1]{}\
&& , \[fourfour\] where the contours $C_{1\tau}$ and $C_{1s}$ are as defined in (\[twothirteen\]). Also, the operator $\hat W_C[y]$ has been defined in (\[twoeight\]) and $W_C[y]$ is the gauge-invariant observable defined in (\[oneone\]).
Two-point function
------------------
We will now discuss the case of the $2$-point function in some detail. For the $2$-point function, the loop equation reduces to && -[1 g\^2]{}[\^2 y\_1\^2()]{}<W\_[C\_1]{}\[y\_1\]W\_[C\_2]{}\[y\_2\]>\
&=& \_[C\_2]{} ds (i\_y\_1().i\_s y\_2(s)) e\^[-ik\_1.y\_1()-ik\_2.y\_2(s)]{} <[Tr]{}(W\_[C\_[1]{}]{}\[y\_[1]{}\] W\_[C\_[2s]{}]{}\[y\_[2s]{}\])>\
&& + [1 (2)\^4 [det]{}]{} \_[C\_[1]{}]{} ds (i\_y\_1().i\_s y\_[1]{}(s)) e\^[-ik\_1.y\_1() +[i 2]{}k\_[1s]{} k\_1]{}\
&& <[Tr]{}(W\_[C\_[1]{}]{}\[y\_[1]{}\]\_[0s]{}) [Tr]{}(W\_[C\_[1]{}]{}\[y\_[1]{}\]\_[s1]{}) W\_[C\_2]{}\[y\_2\]>. \[fourfive\]
We are interested in the planar limit of this equation. In this limit only the disconnected part of the $3$-point function, appearing on the right hand side of (\[fourfive\]), survives. This disconnected part is $$<{\rm Tr}(\hat W_{C_{1\tau}}[y_{1\tau}]_{0s})>
<{\rm Tr}(\hat W_{C_{1\tau}}[y_{1\tau}]_{s1}) W_{C_2}[y_2]>+<{\rm Tr}(\hat
W_{C_{1\tau}}[y_{1\tau}]_{s1})>< {\rm Tr}(\hat
W_{C_{1\tau}}[y_{1\tau}]_{0s}) W_{C_2}[y_2]>.$$ Because of momentum conservation, the first term survives only when $y_{1\tau}(s)=y_{1\tau}(0)$, while the second term survives only when $y_{1\tau}(s)=y_{1\tau}(1)$. Assuming that the contour $C_{1\tau}$ has no self-intersection, the first condition is satisfied only for $s=0$, while the second condition is satisfied only for $s=1$. Thus the disconnected part of the $3$-point function takes the form $$(<{\rm Tr}(\hat W_{C_{1\tau}}[y_{1\tau}]_{0s})>+<{\rm Tr}(\hat
W_{C_{1\tau}}[y_{1\tau}]_{s1})>)<{\rm Tr}(\hat W_{C_{1\tau}}[y_{1\tau}])
W_{C_2}[y_2]>,$$ which, using (\[twotwelve\]), is equivalent to $$e^{ik_1.y_1(\tau)} \
(<{\rm Tr}(\hat W_{C_{1\tau}}[y_{1\tau}]_{0s})>+<{\rm Tr}(\hat
W_{C_{1\tau}}[y_{1\tau}]_{s1})>)<W_{C_1}[y_1] W_{C_2}[y_2]>.$$ Using this in (\[fourfive\]), together with the fact that $\del_\tau
y_\mu(\tau)$ gives tangents at the two ends of the contour $C_{1\tau}$, which are equal by construction, we get && -[1 g\^2]{}[\^2 y\_1\^2()]{}<W\_[C\_1]{}\[y\_1\]W\_[C\_2]{}\[y\_2\]>\
&=& -[(\_y\_1())\^2 (2)\^4 [det]{}]{} \_[C\_[1]{}]{} ds (<[Tr]{}(W\_[C\_[1]{}]{}\[y\_[1]{}\]\_[0s]{})>+ <[Tr]{}(W\_[C\_[1]{}]{}\[y\_[1]{}\]\_[s1]{})>)<W\_[C\_1]{}\[y\_1\] W\_[C\_2]{}\[y\_2\]>\
&& +\_[C\_2]{} ds (i\_y\_1().i\_s y\_2(s)) e\^[-ik\_1.y\_1()-ik\_2.y\_2(s)]{} <[Tr]{}(W\_[C\_[1]{}]{}\[y\_[1]{}\] W\_[C\_[2s]{}]{}\[y\_[2s]{}\])>. \[foursix\] This is the final form of the planar loop equation for the $2$-point function. Notice that the first term on the right hand side of this equation is proportional to $(\del_\tau y_1(\tau))^2$ and also it involves the original $2$-point function. Taken together with the left hand side, the two terms have the form of string hamiltonian acting on the $2$-point function. However, it is not clear that this term is really physically meaningful. In fact, a corresponding term in the loop equation for the commuting gauge theory is often ignored in the regularized theory. In the present case also, an evaluation of the coefficient of $(\del_\tau y_1(\tau))^2$ cannot be done unambiguously. This is because such a calculation involves computation of amplitudes for splitting off of tiny bits at the two ends of the Wilson line defined on the contour $C_{1\tau}$. The computation of this is delicate and needs a regulator. So the physical significance of this term remains unclear.
We should mention here that equations (\[fourfour\]) and (\[foursix\]) are new type of loop equations since there is no analogue of these in commutative gauge theory. Also, it is clear that the planar equation (\[foursix\]) relates the $2$-point function of open Wilson lines to the expectation value of a closed Wilson loop. The latter may be obtained by solving the planar loop equation for a closed Wilson loop. Thus one needs equations for both types of Wilson observables to form a closed system of equations.
Perturbative verification
-------------------------
Perhaps the most interesting aspect of the loop equation (\[foursix\]) is its stringy interpretation. Investigating this aspect of the loop equation is bound to be inherently non-perturbative. In fact, recently such an exercise has been successfully carried out in [@PR; @DGO] for the loop equation in commutative gauge theory, using the AdS/CFT correspondence. A similar exercise for the present non-commutative case seems to require a better understanding of the connection between non-commutative gauge theory and its conjectured gravity dual [@MR; @HI], and is beyond the scope of the present work. Here we will restrict ourselves to a perturbative verification of (\[foursix\]). We will do the computations upto the second order in the ’t Hooft coupling.
Separating out the momentum conserving delta-function, we may parametrize the $2$-point function as =(2)\^4 \^[(4)]{}(k\_1+k\_2) G\_[C\_1C\_2]{}\[y\_1, y\_2\], \[fourseven\] where, in perturbation theory, G\_[C\_1C\_2]{}\[y\_1, y\_2\]=G\^[(1)]{}\_[C\_1C\_2]{}\[y\_1, y\_2\]+\^2 G\^[(2)]{}\_[C\_1C\_2]{}\[y\_1, y\_2\]+ . \[foureight\] Here $\lambda=g^2N$ is the ’t Hooft coupling constant. As indicated in (\[foureight\]), in perturbation theory the function $G_{C_1C_2}[y_1, y_2]$ starts at first order in the ’t Hooft coupling and is order one in $N$. The left hand side of (\[foursix\]) is, therefore, of order $N$, the same as the right hand side.
The lowest order diagram contributing to the $2$-point function is shown in Fig. 4.
(400,100)(0,0) (200,50)(50,120,240) (200,50)(50,300,60) (157,75)(243,25)[3]{}[16]{} (147,50)\[r\][$C_1$]{} (255,50)\[l\][$C_2$]{}
\
[*Fig. 4: Lowest order diagram contributing to the $2$-point function.*]{}
A simple calculation gives the result G\^[(1)]{}\_[C\_1C\_2]{}\[y\_1, y\_2\]=[1 k\_1\^2]{}V\_[C\_1]{}\^[(k\_1)]{}\[y\_1\]. V\_[C\_2]{}\^[(k\_2)]{}\[y\_2\]. \[fournine\] where $V_C^{(k)}[y]$ has been defined in (\[twofourteen\]). Operating the loop laplacian on (\[fournine\]), we get the lowest order expression for the left hand side of (\[foursix\]) [^4] (i\_y\_[1]{}() e\^[-ik\_1.y\_1()]{} ) V\_[C\_2]{}\^[(k\_2)]{}\[y\_2\]. \[fourten\] In arriving at this expression we have used that $k.V_C^{(k)}[y]=0$, which is true because of the identity $k.y(1)=k.y(0)$ which follows from the definition of $k$, (\[onethree\]).
On the right hand side of (\[foursix\]), at the lowest order in $\lambda$, the first term does not contribute. The relevant contribution comes from the second term by setting the gauge field to zero in each of the two Wilson line operators involved in making the closed Wilson loop. Omitting the momentum conserving delta-function, we get precisely the expression in (\[fourten\]).
At the next order in $\lambda$, there are several different types of diagrams that contribute to the $2$-point function. Fig. 5 shows a representative example from each type.
(450,125)(0,0) (90,75)(50,120,240) (60,75)(50,300,60) (77,12)\[b\][(a)]{} (40,75)\[r\][$C_1$]{} (112,75)\[l\][$C_2$]{} (47,100)(15,53,247)[3]{}[5]{} (50,45)(100,45)[3]{}[5]{} (190,75)(50,120,240) (160,75)(50,300,60) (177,12)\[b\][(b)]{} (150,45)(200,45)[3]{}[5]{} (150,105)(200,105)[3]{}[5]{} (290,75)(50,120,240) (260,75)(50,300,60) (277,12)\[b\][(c)]{} (250,105)(275,75)[3]{}[4]{} (250,45)(275,75)[3]{}[4]{} (275,75)(310,80)[3]{}[3]{} (275,75)[2]{} (390,75)(50,120,240) (360,75)(50,300,60) (377,12)\[b\][(d)]{} (390,80)(410,80)[3]{}[2]{} (340,80)(360,80)[3]{}[2]{} (375,80)(15,0,180)[3]{}[4]{} (375,80)(15,180,360)[3]{}[4]{} (390,80)[2]{} (360,80)[2]{}
\
[*Fig. 5: Examples of diagrams contributing to the $2$-point function at second order in perturbation theory.*]{}
We have done a calulation of the quantity $G^{(2)}_{C_1C_2}[y_1,
y_2]$, which gives the second order contribution to the $2$-point function. Some details of this calculation and the result are given in the Appendix.
At the second order in $\lambda$, both terms on the right hand side of (\[foursix\]) contribute. The contribution of the first term comes from the lowest order calculation of the $2$-point function, while that of the second term comes from a one gauge boson exchange. In the Appendix we have discussed in detail how each of these contributions arises as a result of operating the loop laplacian on $G^{(2)}_{C_1C_2}[y_1, y_2]$. Here we only mention that some of the terms from different sets of diagrams that appear in the calculation of the left hand side of (\[foursix\]) have a structure that does not occur on the right hand side. However, there are non-trivial cancellations between the contributions of different sets of diagrams. We have checked that many such terms disappear from the overall result for the left hand side as well, but we have not attempted a complete verification of this.
It would be interesting to extend the present perturbative analysis to all orders in $\lambda$. To verify the new loop equation (\[fourfour\]) non-perturbatively, one needs to understand what the multipoint correlators of Wilson lines based on arbitrary contours map on to in the string/supergravity dual. A better understanding of the non-commutative gauge theory/string theory duality than we have at present appears to be necessary for this.
Discussion
==========
In this paper we have investigated Schwinger-Dyson and loop equations in non-commutative gauge theory. A major difference from the commutative case is the existence of gauge-invariant Wilson line observables based on open contours, in addition to those on closed contours. The Schwinger-Dyson and loop equations in non-commutative gauge theories, therefore, involve both types of gauge-invariant Wilson observables. In the planar limit, the equations for a closed Wilson loop simplify and, like in their commutative counterparts, involve only closed loops. There are, however, [*new*]{} equations, those for open Wilson lines. These involve closed Wilson loops as well, so both types of Wilson observables are needed for a closed set of equations in the planar limit. In fact, as we have seen, these latter equations determine correlators of open Wilson lines entirely in terms of closed Wilson loops.
Recently in several works it has been argued [@GHI; @HL; @DT; @OO; @LM; @O] that local operators in non-commutative gauge theory with straight Wilson lines attached to them are dual to bulk supergravity modes. In this context it is relevant to ask what bulk observables are dual to Wilson lines based on arbitrary open contours. This question is also important for a non-perturbative study of the new loop equations derived here. Our proposal is to identify a Wilson line based on an arbitrary open contour with the operator dual to bulk [*closed*]{} string. This proposal is based on the following reasoning.
The momentum variable appearing in a Wilson line satisfies the condition (\[onethree\]). This is a constraint on the contour enforced by gauge invariance. The contour is otherwise arbitrary. This condition may be regarded as a boundary condition on the curves involved. A generic curve with this boundary condition may be parametrized as $y(\sigma)=\sigma(\theta k)+y'(\sigma)$, where $0 \leq
\sigma \leq 1$ and $y'(\sigma)$ satisfies [*periodic*]{} boundary conditions. Thus the freedom contained in a generic Wilson line is exactly the one needed to describe a closed string!
Actually, we can take this line of reasoning further. Let us confine our attention to smooth curves, with the additional condition that the tangents to the curve at the two ends are equal. In this case we may parametrize the curves as y() &=& y\_0()+y(),\
y\_0() &=& y\_0(0)+(k),\
y() &=& \_[n=1]{}\^(\_n e\^[-2i n ]{}+ \_n e\^[2i n ]{}). \[fiveone\] As the above parametrization suggests, what we are going to do is to assume that deviations of the given curve from a straight line are small and expand the Wilson line, based on the given curve, around the corresponding straight Wilson line. This gives W\_C\[y\] &=& W\_[C\_0]{}\[y\_0\]+\_0\^1 d y\_() ([W\_C\[y\] y\_()]{} )\_[y=y\_0]{}\
&& +[1 2!]{} \_0\^1 d\_0\^1 d’ y\_() y\_(’) ([\^2 W\_C\[y\] y\_() y\_(’)]{} )\_[y=y\_0]{}+\[fivetwo\] Here $C_0$ refers to the straight line contour.
The first term in the above equation is known to be the non-commutative gauge theory operator dual to the bulk closed string tachyon. The second term vanishes, since $\bigg({\delta W_C[y] \over
\delta y_\mu(\sigma)} \bigg)_{y=y_0}$ is independent of $\sigma$, which can be easily verified using the cyclic symmetry of a straight Wilson line, and since $\delta y(\sigma)$ has no zero mode. The first non-trivial contribution comes from the third term. Using a generalization of the identity in (\[fourone\]), we may rewrite this term as && \_0\^1 d\_0\^1 d’ y\_() y\_(’) \
&& + \_0\^1 d y\_() y\_() [Tr]{} (\_[C\_0]{}(0, ) (il\_ D\_F\_ (x+y\_0())) \_[C\_0]{}(, 1) e\^[ik.x]{} )\
&& + \_0\^1 d y\_() i\_y\_() [Tr]{} (\_[C\_0]{}(0, ) F\_(x+y\_0()) \_[C\_0]{}(, 1) e\^[ik.x]{} ),\
\[fivethree\] where $\hat {\cal U}_{C_0}(\sigma_1, \sigma_2)$ is the path-ordered phase factor, running along the straight line contour $C_0$, from the point $\sigma_1$ to $\sigma_2$. Note that in this notation $W_C[y]=
{\rm Tr}(\hat {\cal U}_{C_0}(0, 1) \ e^{ik.\hat x})$. Substituiting for $\delta y(\sigma)$ from (\[fiveone\]) in this expression and extracting the part of the $n=1$ term symmetric in the indices $\mu,
\nu$, we get precisely the operator that has been identified in [@OO] as being dual to the bulk graviton (in the bosonic string), polarized along the brane directions, modulo factors that connect the open string metric with the closed string metric and terms involving the scalar fields. Note that the last term in (\[fivethree\]) is purely antisymmetric in the indices $\mu, \nu$ and hence contributes only to the operator dual to the bulk antisymmetric tensor field.
It seems quite likely that the above procedure gives us [*all*]{} the gauge theory operators dual to bulk string modes. It is, therefore, tempting to identify the Wilson line based on generic curves of the type described by (\[fiveone\]) as dual to the bulk [*closed*]{} string. An expansion of the Wilson line around the corresponding straight line contour would then be like the expansion of the [ *closed string field*]{} in terms of the various string modes carrying a definite momentum. If this is true, then multipoint correlators of Wilson lines should be identified with closed string scattering amplitudes. In particular, the $2$-point function would then have the interpretation of closed string propagator and (\[foursix\]) would be the equation of motion satisfied by the propagator. Such a non-perturbative interpretation of (\[foursix\]), or more generally (\[fourfour\]), should further enhance our understanding of gauge theory/string theory duality. The above discussion applies to the bosonic string. It would be interesting to extend these ideas to the case of the superstring.
Appendix
========
In this appendix we will give some details of the calculation of $G^{(2)}_{C_1C_2}[y_1, y_2]$. We will also describe how the action of the loop laplacian on it reproduces the right hand side of the loop equation (\[foursix\]).
The diagrams that contribute to $G^{(2)}_{C_1C_2}[y_1, y_2]$ can be collected into four different types of groups. A representative from each of these has been shown in Fig. 5. There are six self-energy type of diagrams, Fig. 5(a). Their total contribution to $G^{(2)}_{C_1C_2}[y_1, y_2]$ is && [1 k\_1\^2]{} \_[\_1]{}\_[\_2>[\_2]{}’>[\_2”]{}]{}\
&& +1 2 \[aone\] where a dot on $y$ stands for a derivative with respect to the argument and the last contribution is obtained by the $1
\leftrightarrow 2$ interchange of the subscripts on $k$, $y$ and $\sigma$.
There are two diagrams in the second set represented by Fig. 5(b). Their total contribution to $G^{(2)}_{C_1C_2}[y_1, y_2]$ is && d\^4z e\^[ik\_1.z]{} \_[\_1>[\_1]{}’]{} \_[\_2>[\_2]{}’]{} . \[atwo\]
In the third set, represented by Fig. 5(c), also there are two diagrams. Their total contribution to $G^{(2)}_{C_1C_2}[y_1, y_2]$ is && -[1 k\_1\^2]{} d\^4z \_[\_1]{}\_[\_2>[\_2]{}’]{} [e\^[ik\_1.(z+y\_2(\_2)-y\_1(\_1))]{} 4 \^2 z\^2]{}\
&& ([1 4 \^2 |z+y\_2(\_2)-y\_2([\_2]{}’)+ k\_1|\^2]{}-[1 4 \^2 |z+y\_2(\_2)-y\_2([\_2]{}’)|\^2]{} )\
&& +1 2 \[athree\]
Finally, we have the gauge boson self-energy diagrams like Fig. 5(d), including those with ghosts. Their total contribution to $G^{(2)}_{C_1C_2}[y_1, y_2]$ is && -[1 \^2]{} d\^4z \_[\_1]{}\_[\_2]{} e\^[ik\_1.(z-y\_1(\_1)+y\_2(\_2))]{} [y\_[1]{}(\_1)y\_[2]{}(\_2) 4 \^2 z\^2]{}\
&& ( \_(-[\_z]{}\^2 +2i\_k\_1.\_z-5k\_1\^2)+8[\_z]{}\_\_) ( [1 4 \^2 |z+k\_1|\^2]{} -[1 4 \^2 z\^2]{} ). \[afour\]
Let us now evaluate the action of the loop laplacian, $-\del^2/ \del
y_1^2(\tau)$, on the expression for the second order contribution to the $2$-point function given above. In the first term in (\[aone\]), the only dependence on $y_1$ is in the form of $V_{C_1\mu}^{(k_1)}[y_1]$, which has been defined in (\[twofourteen\]). Applying the loop laplacian on it gives the result -[\^2 y\_1\^2()]{}V\_[C\_1]{}\^[(k\_1)]{}\[y\_1\]= (k\_1\^2\_-k\_[1]{}k\_[1]{}) y\_[1]{}() e\^[-ik\_1.y\_1()]{}. \[afive\] The $k_1^2$ in the first term above cancels the factor of $1/k_1^2$ in front of the first term in (\[aone\]). The rest of this factor can be seen to precisely reproduce that contribution of the second term on the right hand side of the loop equation (\[foursix\]) in which a self-energy insertion is present on the contour $C_{2s}$. The three terms correspond to the three possibilities that the marked point $s$ on the contour $C_{2s}$ is entirely above, entirely below or in-between the points where the self-energy insertion takes place. The second term in (\[afive\]) gives rise to the following contribution from the first term in (\[aone\]): && -[k\_1.y\_1() k\_1\^2]{}e\^[-ik\_1.y\_1()]{} \_[\_2>[\_2]{}’>[\_2”]{}]{} .\
\[asix\] This can be simplified to && [ik\_1.y\_1() k\_1\^2]{}e\^[-ik\_1.y\_1()]{} \_[\_2>[\_2]{}’]{}(y\_2(\_2).y\_2([\_2]{}’)) (e\^[ik\_1.y\_2(\_2)]{}-e\^[ik\_1.y\_2([\_2]{}’)]{})\
&& ( [1 4 \^2 |y\_2(\_2)-y\_2([\_2]{}’)+k\_1|\^2]{} -[1 4 \^2 |y\_2(\_2)-y\_2([\_2]{}’)|\^2]{} ) \[aseven\] Now, it is easy to see that on the right hand side of (\[foursix\]) there are no terms at this order having the above structure. Therefore, for consistency of the loop equation, (\[aseven\]) must get cancelled by another term in the $2$-point function. In fact, this does happen and the required term comes from the first term of (\[athree\]). In this term also $y_1$ is in the form of $V_{C_1\mu}^{(k_1)}[y_1]$, so applying the loop laplacian results in two terms because of (\[afive\]). Let us look at the second term. It is && [k\_1.y\_1() k\_1\^2]{}e\^[-ik\_1.y\_1()]{} d\^4z \_[\_2>[\_2]{}’]{} [e\^[ik\_1.(z+y\_2(\_2))]{} 4 \^2 z\^2]{}\
&& ([1 4 \^2 |z+y\_2(\_2)-y\_2([\_2]{}’)+ k\_1|\^2]{}-[1 4 \^2 |z+y\_2(\_2)-y\_2([\_2]{}’)|\^2]{} ) \[aeight\] In the last term in the square brackets above, let us rewrite $2k_1.\del_z$ as $-i\{(k_1+i\del_z)^2+\del_z^2-k_1^2\}$ and then use $(k_1+i\del_z)^2(e^{ik_1.z}/z^2)=e^{ik_1.z}\del_z^2(1/z^2)$ and $\del_z^2(1/z^2)=-4\pi^2\delta^{(4)}(z)$. As a result of this simplification, one of the terms we get is precisely (\[aseven\]), but with opposite sign. So this unwanted term cancels in a rather nontrivial way, since the cancellation involves terms which come from two entirely different diagrams.
Going back to (\[aone\]), let us now look at the second term, which is obtained from the first by $1 \leftrightarrow 2$ interchange: && [1 k\_1\^2]{} \_[\_1]{}\_[\_1>[\_1]{}’>[\_1”]{}]{} \[anine\] The $y_1$ structure of this term is much more complicated than that of the first term in (\[aone\]). So the result of applying the loop laplacian on it is also more complicated. For example, let us consider the first term in the above expression. If the loop laplacian acts on $y_1({\sigma_1}'')e^{-ik_1.y_1({\sigma_1}'')}$, the result is simple and, in fact, just reproduces that contribution of the second term on the right hand side of the loop equation (\[foursix\]) in which a self-energy insertion is present on the contour $C_{1\tau}$ entirely above the marked point $\tau$. A similar operation of the loop laplacian on the other two terms in (\[anine\]) reproduces the other two contributions in which the self-energy insertion is either entirely below the marked point $\tau$ or across it. On the other hand, if the loop laplacian acts on the propagator $1/4 \pi^2
|y_1(\sigma_1)-y_1({\sigma_1}')|^2$, the result is a delta-function type of contribution. Together with a similar contribution from the last term in (\[anine\]) (the middle term has no contribution of this type since the delta-function does not click), this precisely reproduces the entire contribution of the first term on the right hand side of the loop equation in this order.
Let us now go to the next term, (\[atwo\]). Here, the loop laplacian may act on any of the four propagators, resulting in a delta-function. This gives four different terms and these precisely reproduce that contribution of the second term on the right hand side of the loop equation (\[foursix\]) in which a gauge boson is exchanged between the two contours $C_{1\tau}$ and $C_{2s}$. For this it is essential to remember that the loop on the right hand side, ${\rm Tr}(\hat
W_{C_{1\tau}}[y_{1\tau}] \hat W_{C_{2s}}[y_{2s}])$, involves the hatted operators. As defined in (\[twotwelve\]), these differ from the unhatted ones in that the argument of the gauge field is shifted by the starting point of the contour. For the contours $C_{1\tau}$ and $C_{2s}$, the starting points are respectively $y_{1\tau}(0)=y_1(\tau)$ and $y_{2s}(0)=y_2(s)$. The four terms mentioned above correspond to the four differnt possibilities of the two ends of the gauge field propagator landing above or below the marked points on the two contours.
Thus, we see that the action of the loop laplacian on the second order contribution to the $2$-point function reproduces all the terms expected on the right hand side of the loop equation (\[foursix\]). There are also other terms produced in the process of applying the loop laplacian on the contribution of individual diagrams to the $2$-point function. We have seen above an explicit example of one such term which, however, eventually disappeared because of a nontrivial cancellation with another term. We expect that all other similar terms (which are produced in the process of applying the loop laplacian on the contributions of different diagrams to the $2$-point function, but are not present on the right hand side of the loop equation) will eventually disappear through similar cancellations, but we have not attempted a complete verification of this.
[999]{} A. Connes, M. Douglas and A. Schwarz, “Non-commutative Geometry and Matrix Theory: Compactification on Tori”, JHEP [**9802**]{} (1998) 002, hep-th/9711162. M.R. Douglas and C. Hull, “D-branes and the non-commutative torus,” JHEP [**9802**]{} (1998) 008, hep-th/9711165. F. Ardalan, H. Arfaei and M.M. Sheikh-Jabbari, “Mixed branes and matrix theory on non-commutative torus,” hep-th/9803067; “Non-commutative geometry from strings and branes”, JHEP [**9902**]{} (1999) 016, hep-th/9810072. Y. Cheung and M. Krogh, “Non-commutative geometry from $D_0$ branes in a background B-field”, Nucl. Phys. [**B528**]{} 185 (1999), hep-th/9803031. H. Garcia-Compean, “On the deformation quantization description of matrix compactifications”, Nucl. Phys. [**B541**]{} 651 (1999), hep-th/9804188. C. Chu and P. Ho, “Non-commutative Open String and D-brane”, Nucl. Phys. [**B550**]{} 151 (1999), hep-th/9812219. V. Schomerus, “D-branes and deformation quantization”, JHEP [**9906**]{} (1999) 030, hep-th/9903205. N. Seiberg and E. Witten, “String theory and non-commutative geometry”, JHEP [**9909**]{} (1999) 032, hep-th/9908142. N. Nekrasov and A.S. Schwarz, “Instantons on non-commutative $R^4$ and $(2, 0)$ superconformal six dimensional theory”, Comm. Math. Phys. [**198**]{} (1998) 689, hep-th/9802068. J. Maldacena and J. Russo, “Large N limit of non-commutative gauge theories”, hep-th/9908134. A. Hashimoto and N. Itzhaki, “Non-commutative Yang-Mills and the AdS/CFT correspondence”, Phys. Lett. [**B465**]{} (1999) 142, hep-th/9907166. H. Aoki, N. Ishibashi, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, “Non-commutative Yang-Mills in IIB matrix model”, Nucl. Phys. [**565**]{} (2000) 176, hep-th/9908141. M. Li, “Strings from IIB matrices”, Nucl. Phys. [**499**]{} (1997) 149, hep-th/9612222. L. Cornalba, “D-brane physics and noncommutative Yang-Mills theory”, hep-th/9909081. N. Ishibashi, “A relation between commutative and non-commutative descriptions D-branes”, hep-th/9909176. N. Ishibashi, S. Iso, H. Kawai and Y. Kitazawa, “Wilson loops in non-commutative Yang-Mills”, Nucl. Phys. [**B573**]{} (2000) 573, hep-th/9910004. N. Seiberg and E. Witten, “D1/D5 system and singular CFT”, JHEP [**9904**]{} (1999) 017, hep-th/9903224. A. Dhar, G. Mandal, S.R. Wadia and K.P. Yogendran, “D1/D5 system with B field, non-commutative geometry and the CFT of the Higgs branch”, Nucl. Phys. [**B575**]{} (2000) 177, hep-th/9910194. S.R. Das, S. Kalyana Rama and S. Trivedi, “Supergravity with self-dual B fields and instantons in non-commutative gauge theory”, JHEP [**0003**]{} (2000) 004, hep-th/9911137. I. Bars and D. Minic, “Non-commutative geometry on a discrete periodic lattice and gauge theory”, Phys. Rev. [**D62**]{} (2000) 1050, hep-th/9910091. J. Ambjorn, Y. Makeenko, J. Nishimura and R. Szabo, “Finite N matrix models of non-commutative gauge theory”, JHEP [**9911**]{} (1999) 029, hep-th/9911041; “Non-perturbative dynamics of non-commutative gauge theory”, Phys. Lett. [**B480**]{} 399, hep-th/0002158; “Lattice gauge fields and discrete non-commutative Yang-Mills theory”, JHEP [**0007**]{} (2000) 013, hep-th/0003208. S. Minwalla, M.V. Raamsdonk and N. Seiberg, “Non-commutative perturbative dynamics”, JHEP [**0007**]{} (2000) 013, hep-th/0003160. O. Andreev and H. Dorn, “On open string sigma-model and non-commutative gauge fields”, Phys. Lett. [**B476**]{} (2000) 402, hep-th/9912070. N. Ishibashi, S. Iso, H. Kawai and Y. Kitazawa, “String scale in non-commutative Yang-Mills”, Nucl. Phys. [**B583**]{} (2000) 159, hep-th/0004038. S.R. Das and B. Ghosh, “A note on supergravity duals of non-commutative Yang-Mills theory”, JHEP [**0006**]{} (2000) 043, hep-th/0005007. L. Alvarez-Gaume and S.R. Wadia, “Gauge theory on a quantum phase space”, hep-th/0006219. A.H. Fatollahi, “Gauge symmetry as symmetry of matrix coordinates”, hep-th/0007023. D. Jatkar, G. Mandal and S.R. Wadia, “Nielsen-Olesen vortices in noncommutative abelian higgs model”, JHEP [**0009**]{} (2000) 018, hep-th/0007078. J.A. Harvey, P. Kraus, F. Larsen and E. Martinec, “D-branes and strings as non-commutative solitons”, JHEP [**0007**]{} (2000) 042, hep-th/0005031. K. Dasgupta, S. Mukhi and G. Rajesh, “Non-commutative tachyons”, JHEP [**0006**]{} (2000) 022, hep-th/0005006. E. Witten, “Non-commutative tachyons and string field theory”, hep-th/0006071. R. Gopakumar, S. Minwalla and A. Strominger, “Symmetry restoration and tachyon condensation in open string theory”, hep-th/0007226; “Non-commutative solitons”, JHEP [**0005**]{} (2000) 020, hep-th/0003160. N. Seiberg, “A note on background independence in non-commutative gauge theories, matrix model and tachyon condensation”, hep-th/0008013. P. Kraus, A. Rajaraman and S. Shenker, “Tachyon condensation in non-commutative gauge theory”, hep-th/0010016. G. Mandal and S.R. Wadia, “Matrix model, noncommutative gauge theory and the tachyon potential”, hep-th/0011094. A. Dhar and Y. Kitazawa, “Wilson loops in strongly coupled noncommutative gauge theories’, hep-th/0010256, to appear in Phys. Rev. [**D**]{}. Soo-Jong Rey and R. von Unge, “S-duality, non-critical open string and non-commutative gauge theory”, Phys.Lett. [**B499**]{} (2001) 215, hep-th/0007089. S.R. Das and Soo-Jong Rey, “Open Wilson lines in non-commutative gauge theory and tomography of holographic dual supergravity”, Nucl.Phys. [**B590**]{} (2000) 453, hep-th/0008042. D.J. Gross, A. Hashimoto and N. Itzhaki, “Observables of non-commutative gauge theories”, hep-th/0008075. A. Dhar and S.R. Wadia, “A note on gauge invariant operators in non-commutative gauge theories and the matrix model”, Phys. Lett. [**B495**]{} (2000) 413, hep-th/0008144. A. Dhar and Y. Kitazawa, “High energy behaviour of Wilson lines”, JHEP [**0102**]{} (2001) 004, hep-th/0012170. H. Liu, “\*-Trek II”, hep-th/0011125. S.R. Das ans S. Trivedi, “Supergravity couplings to noncommutative branes, open Wilson line and generalized star products”, hep-th/0011131. Y. Okawa and H. Ooguri, “How noncommutative gauge theories couple to gravity”, hep-th/0012218; “Energy momentum tensors in matrix theory and in noncommutative gauge theories”, hep-the/0103124. H. Liu and J. Michelson, “Supergravity couplings of noncommutative D-branes”, hep-th/0101016. K. Okuyama, “Comments on open Wilson lines and generalized star products”, hep-th/0101177. M. Fukuma, H. Kawai, Y. Kitazawa and A. Tsuchiya, “String field theory from IIB matrix model”, Nucl. Phys. [**B510**]{} (1998)158, hep-th/9705128. A. Dhar and S.R. Wadia, unpublished. M. Abou-Zeid and H. Dorn, “Dynamics of Wilson observables in non-commutative gauge theory”, hep-th/0009231. A.M. Polyakov, “Gauge fields and strings”, Harwood Academic Publishers (1997), pp. 119. A.M. Polyakov and V.S. Rychkov, “Gauge fields-strings duality and the loop equation”, hep-th/0002106; “Loop dynamics and AdS/CFT correspondence”, hep-th/0005173. N. Drukker, D.J. Gross and H. Ooguri, “Wilson loops and minimal surfaces”, Phys. Rev. [**D60**]{} (1999) 125006, hep-th/9904191.
[^1]: In the following $\hat W_C[y]_{01}$, which goes over the full parametric range, will be denoted by $\hat W_C[y]$ for notational convenience.
[^2]: Under these conditions the variation of contour at the end points is effectively like at an interior point.
[^3]: In practice the separation of the two terms on the right hand side of (\[fourone\]) is a nontrivial issue. For a discussion in the case of commutative gauge theory, see, for example, [@POLY; @PR; @DGO].
[^4]: Here and in the following we have omitted the momentum conserving delta-function factor $(2\pi)^4 \delta^{(4)}(k_1+k_2)$ which is present on both sides of (\[foursix\]).
|
---
abstract: 'We give necessary and sufficient conditions for existence of solutions to a general system of complex Monge-Ampère equations on Fano horosymmetric manifolds. In particular, we get necessary and sufficient conditions for existence of coupled Kähler-Ricci solitons, Mabuchi metrics and twisted Kähler-Einstein metrics in terms of combinatorial data of the manifold.'
address:
- |
Thibaut Delcroix\
Institut de Recherche Mathématique Avancée\
UMR 7501, Université de Strasbourg et CNRS\
7 rue René Descartes\
67000 Strasbourg, France
- |
Jakob Hultgren\
Matematisk institutt\
Universitetet i Oslo\
Niels Henrik Abels hus\
Moltke Moes vei 35\
0851 Oslo, Norway
author:
- Thibaut Delcroix
- Jakob Hultgren
bibliography:
- 'HoroCMab.bib'
title: 'Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds'
---
Introduction
============
The background for the present paper is the study of Kähler-Einstein metrics, i.e. Kähler metrics $\omega$ that satisfy the Einstein condition $$\operatorname{Ric}\omega = \lambda \omega$$ where $\operatorname{Ric}\omega$ is the Ricci curvature form of $\omega$ and $\lambda\in {\mathbb{R}}$. This condition has been especially fruitful to study in the Kähler setting, since here it reduces from a tensor equation to a second order scalar differential equation (a complex Monge-Ampère equation). Moreover, the problem of determining when a given Kähler manifold admits a Kähler-Einstein metric has turned out to be connected to certain subtle properties in algebraic geometry. Indeed, by the Yau-Tian-Donaldson conjecture which was recently confirmed in [@CDS15a; @CDS15b; @CDS15c] (see also [@Tia15]), a compact Kähler manifold admits a Kähler-Einstein metric if and only if it is K-stable, an algebraic condition due to Tian and Donaldson which is modeled on Geometric Invariant Theory. Over the previous decades, several generalizations of Kähler-Einstein metrics have been introduced. We will recall a few of them here. Perhaps most notable are Kähler-Ricci solitons. These are defined by the property that they evolve under Ricci flow by holomorphic automorphisms of the manifold. As such, they appear as the large time limit of the Ricci flow on many Kähler manifolds. Another important generalization is given by twisted Kähler-Einstein metrics, satisfying the Einstein condition up to a fixed tensor on the manifold. Among other things, these provide a generalization of Kähler-Einstein metrics to the setting of log Fano manifolds. Moreover, in the more general case when the fixed tensor is allowed to be singular, they played an important role in the proof of the Yau-Tian-Donaldson conjecture. A third generalization of Kähler-Einstein metrics was introduced by Mabuchi in [@Mab01]. These metrics, which often goes under the name Mabuchi metrics, were recently given an interpretation in terms of a certain moment map on the space of Kähler metrics in the canonical class of the manifold, motivating further study (see [@Don17; @Yao]). Finally, a fourth generalization is given by coupled Kähler-Einstein metrics, introduced by Witt Nyström and the second named author in [@HWN17]. These are tuples of Kähler metrics satisfying a certain coupled version of the Einstein condition. Here the Monge-Ampère equation of Kähler-Einstein metrics is replaced by a system of Monge-Ampère equations.
Twisted Kähler-Einstein metrics, Kähler-Ricci solitons and coupled Kähler-Einstein metrics are all associated to different generalizations of the Yau-Tian-Donaldson conjecture where the condition of K-stability has been modified to take the given generalization into account. The first two of these generalized Yau-Tian-Donaldson conjectures was settled in [@DS16]. It should be said, however, that even in the light of [@CDS15a; @CDS15b; @CDS15c] and [@DS16] determining if a given manifold admits a (twisted) Kähler-Einstrein metric or a Kähler-Ricci soliton is not a straight forward task. The condition of K-stability is not readily checkable. Consequently, along a line parallel to the Yau-Tian-Donaldson conjecture, much work has been done to find simple, explicit conditions for existence of Kähler-Einstein metrics on special classes of manifolds. A starting point for one branch of this is the result of Wang and Zhu in 2004 [@WZ04], saying that a toric Fano manifold admits a Kähler-Einstein metric if and only if the barycenter of the associated polytope vanishes. In addition, Wang and Zhu proved that any toric Fano manifold admit a Kähler-Ricci soliton. These results have since been extended to more general settings: (see for example [@BB13; @DelKE; @Yao; @Hul]). In particular, in [@DelKE] the first named author gives explicit conditions for existence of Kähler-Einstein metrics on group compactifications. In this paper we study a general system of Monge-Ampère equations which incorporates the four generalizations of Kähler-Einstein metrics mentioned above as well as any combination of them (see Equation \[eqn\_general\] below). Our main theorem is a necessary and sufficient condition for existence of solutions to this equation on horosymmetric manifolds, a class of manifolds introduced by the first named author in [@DelHoro]. This class is strictly included in the class of spherical manifolds and strictly includes the class of group compactifications. We deduce this theorem as a consequence of two technical results. The first of these is that Yau’s higher order estimates along the continuity method for Monge-Ampère equations, as well as local solvability, apply to a suitable continuity path containing Equation \[eqn\_general\] (see Theorem \[thm:ReductionToC0\] below). The second of these is that the $C^0$-estimates on toric manifolds proved by Wang and Zhu extends to the current setting of Equation \[eqn\_general\] on horosymmetric manifolds. This is done by proving $C^0$-estimates along a continuity path of real Monge-Ampère equations on convex polyhedral cones (see Theorem \[thm:C0Estimates\] below). As a byproduct we get necessary conditions for existence of solutions to these system of real Monge-Ampère equations. Moreover, these conditions are also sufficient in the cases when the data defining the cones is induced by horosymmetric manifolds.
The paper is organized as follows: in the remaining of this introductory section, we state our main results. Section \[sec:reductionC0\] is devoted to the proof of Theorem \[thm:ReductionToC0\]. The proof of Theorem \[thm:C0Estimates\] is given in Section \[sec\_C0\]. We then explain in Section \[sec:horosym\] how these two results apply to the case of horosymmetric manifolds, after recalling the definitions and tools to deal with such manifolds. Finally, we include in Section \[sec:examples\] an illustration of our results on low dimensional horosymmetric manifolds.
Setup {#sec:setup}
-----
We will now formulate the system of Monge-Ampère equations we will consider. Let $X$ be a Fano manifold of dimension $n$ and $K$ a compact subgroup of the automorphism group $\mathrm{Aut}(X)$ of $X$. For a positive integer $k$, fix $K$-invariant Kähler forms $\theta_1,\ldots,\theta_k$ and a $K$-invariant semi-positive (1,1)-form $\gamma$ such that $$[\gamma] + \sum_{i=1}^{k}[\theta_i]=c_1(X).$$ Let $\theta_0$ denote a Kähler form of unit mass such that $$\gamma + \sum_{i=1}^{k}\theta_i = \operatorname{Ric}(\theta_0).$$ Fix a $k$-tuple of (real) holomorphic vector fields $(V_i)$, commuting with the action of $K$ such that the subgroup generated by $JV_i$ lies in $K$ for each $i$ (where $J$ denotes the complex structure on $X$). Let $f_i$ denote the $K$-invariant (real valued) ${\sqrt{-1}\partial\bar\partial}$-potentials for the Lie derivatives $L_{V_i}\theta_i$, normalized so that $\int_Xf_i\theta_i^n=0$. Let $(h_i)$ denote a $k$-tuple of real valued smooth concave functions on the real line which satisfy $\int_Xe^{h_i\circ f_i}\theta_i^n=1$.
We consider in this paper systems of equations whose solutions are $k$-tuples $(\phi_i) = (\phi_1,\ldots,\phi_k)$ of $K$-invariant functions on $X$ such that for all $i\in \{1,\ldots,k\}$, $\theta_i+{\sqrt{-1}\partial\bar\partial}\phi_i$ is Kähler and $$\label{eqn_general}
e^{h_i(f_i+V_i(\phi_i))}(\theta_i+{\sqrt{-1}\partial\bar\partial}\phi_i)^n=
e^{-\sum_{m=1}^k\phi_m}\theta_0^n.$$
When $\gamma=0$ and $k=1$, this equation reduces to an equation considered by Mabuchi in [@Mab03] and whose solutions define what he calls multiplier hermitian structures. Different choices of $h_1$ recover the definitions of Kähler-Einstein metrics (if $h_1$ is constant), Kähler-Ricci solitons (if $h_1$ is affine) and Mabuchi metrics (if $e^{h_1}$ is affine).
When $\gamma=0$, $k>1$ and $h_i$ is affine we get the system of equations defining coupled Kähler-Einstein metrics and coupled Kähler-Ricci solitons introduced in [@HWN17] and [@Hul]. Moreover, putting $\gamma=0$, $k>1$ and choosing $e^{h_i}$ affine we get a natural definition for *coupled Mabuchi metrics*, generalizing both Mabuchi metrics and coupled Kähler-Einstein metrics.
Finally, choosing a non-zero $\gamma$ allows to consider twisted versions of the above equations.
In terms of the associated Kähler forms $\omega_i=\theta_i+{\sqrt{-1}\partial\bar\partial}\phi_i$, is equivalent to $$\operatorname{Ric}\omega_i - \sqrt{-1} \partial \bar \partial h_i(f_i+V_i(\phi_i)) = \gamma + \sum_{m=1}^k \omega_m$$ for $i\in \{1,\ldots,k\}$. If the functions $h_1,\ldots,h_k$ are invertible a third formulation is given by considering the coupled Ricci potentials, i.e. functions $F_1,\ldots,F_k$ such that $${\sqrt{-1}\partial\bar\partial}F_i = \operatorname{Ric}\omega_i - \sum_m \omega_m.$$ Then $(\phi_i)$ solves if and only if $h_i^{-1}\circ F_i$ is a ${\sqrt{-1}\partial\bar\partial}$-potential of $L_{V_i}(\omega_i)$ for each $i$. Note that in the case of Mabuchi metrics, it is implicitly part of our assumptions that the affine function $e^{h_1\circ f_1}$ is positive. This condition was identified by Mabuchi as a necessary condition for existence of Mabuchi metrics and it is actually of the same nature as the assumption that $X$ is Fano that we make to consider positive Kähler-Einstein metrics or Kähler-Ricci solitons.
A continuity path and reduction to $C^0$-estimates {#sec:intro_red}
--------------------------------------------------
We will consider the following continuity path for : $$e^{h_i(f_i+V_i(\phi_i))}(\theta_i+{\sqrt{-1}\partial\bar\partial}\phi_i)^n=
e^{-t\sum_{m=1}^k\phi_m}\theta_0^n.
\label{eq:ComplexContPath}$$ where $0\leq t \leq 1$. Note that given a solution $(\phi_i)$ and constants $C_1,\ldots, C_k$ such that $\sum_k C_k=0$, we get a new solution by considering the tuple $(\phi_i+C_i)$. Fixing a point $x_0\in X$ we will assume any solution $(\phi_1,\ldots,\phi_k)$ of to be normalized to satisfy $$\label{eq:Normalization}
\phi_1(x_0) = \ldots = \phi_k(x_0).$$
Given a continuity path of the type , we will say that $C_0$-estimates hold on an interval $[t_0,t]\subset [0,1]$ if there exist a constant such that for any solution $(\phi_i)$ at $t'\in [t_0,t]$ $$\label{eq:C0Assumption}
\max_i\sup_X |\phi_i| < C.$$ Using well-known techniques of Yau and Aubin we will reduce existence of solutions of to a priori $C^0$-estimates for .
\[thm:ReductionToC0\] Let $t\in (0,1]$ and assume that for each $t_0\in (0,t)$, $C^0$-estimates hold for on $[t_0,t]$. Then has a solution for any $t'\in [0,t]$. In particular, if $t=1$ then has a solution.
Canonical metrics of Kähler-Einstein type on horosymmetric manifolds {#sec:intro_horo}
--------------------------------------------------------------------
On a manifold $X$ equipped with an automorphism group of large dimension, it often turned out that complex Monge-Ampère equations such as the Kähler-Einstein equation could be translated into a real Monge-Ampère equation with data encoded by combinatorial information on the action of $\mathrm{Aut}(X)$. The major example is that of toric manifolds [@WZ04] but several bigger such classes of manifolds were studied over the years, notably homogeneous toric bundles [@PS10] and group compactifications [@DelKE]. Horosymmetric manifolds were introduced by the first author in [@DelHoro] as a generalization of the above classes, and the tools to translate complex Monge-Ampère equations into real Monge-Ampère equations were developed in the same paper.
In the present article, we will study our general system of complex Monge-Ampère equations on horosymmetric manifolds. It yields a complete combinatorial caracterization of existence of solutions (under a few additional mild assumptions such as invariance of the data under a compact subgroup). In particular we recover and generalize a wide array of recent results on existence of canonical metrics.
A horosymmetric manifold is a manifold $X$ equipped with an action of a connected linear reductive complex group $G$ such that the action admits an open orbit $G\cdot x$ which is a homogeneous bundle over a generalized flag manifold with fiber a complex symmetric space. Several combinatorial data were associated to such manifolds and their line bundles in [@DelHoro]. For this introduction we just need the following. We choose (wisely) a maximal torus $T$ and Borel subgroup $T\subset B$ of $G$. The lattice $\mathcal{M}=\mathfrak{X}(T/T_x) \subset \mathfrak{X}(T)$ is called the spherical lattice. The root system of $G$ splits into three (possibly empty) parts $\Phi^+=\Phi_{Q^u}\cup \Phi_s^+\cup (\Phi_L^+)^{\sigma}$ (see Section \[sec:horosym\] for details). Define $\bar{C}^+$ as the cone of elements $p\in\mathcal{M}\otimes {\mathbb{R}}$ such that $\kappa(p,\alpha)\geq 0$ for all $\alpha\in\Phi_s^+$, where $\kappa$ denotes the Killing form.
To a nef class $[\gamma]$ on the horosymmetric manifold $X$ we associate, by elaborating on [@DelHoro], a *toric polytope* $\Delta^{{\mathrm{tor}}}_{\gamma}\subset \mathcal{M}\otimes {\mathbb{R}}$ (well defined up to translation by an element of $\mathfrak{X}(T/[G,G])\otimes {\mathbb{R}}$) and an isotropy character $\chi_i\in \mathfrak{X}(T)$. Furthermore, for the anticanonical line bundle, there is a canonical choice of representative $\Delta^{{\mathrm{tor}}}_{ac}$ among the possible toric polytopes.
We now consider the setup of Section \[sec:setup\] on a horosymmetric Fano manifold, with the additional assumption that the $\theta_i$ and $\gamma$ are invariant under a fixed maximal compact subgroup $K$ of $G$, that the vector fields $V_i$ commute with the action of $G$, and that the classes $[\theta_i]$ and $[\gamma]$ are in the subspace generated by semiample line bundles whose restriction to the fiber of the open orbit are trivial. Note that horosymmetric manifolds are Mori Dream Spaces and as such their nef cone is generated by classes of line bundles. The subspace defined above always contain the anticanonical class and is equal to the full nef cone as long as the symmetric fiber does not have any Hermitian factor. The assumption on the vector fields allows to associate to $V_i$ an affine function $\ell_i$ on $\mathcal{M}\otimes {\mathbb{R}}$ (see Section \[sec:HamiltHoro\]).
Using the assumption $[\gamma]+\sum_i[\theta_i]=c_1(X)$ we may and do choose the toric polytopes of each class so that the Minkowski sum $\Delta^{{\mathrm{tor}}}_{\gamma}+\sum_i\Delta_{\theta_i}^{{\mathrm{tor}}}$ is equal to the canonical $\Delta^{{\mathrm{tor}}}_{ac}$. For the statement of the main theorem, we introduce the following notations. The Duistermaat-Heckman polynomial $P_{DH}$ is defined on $\mathfrak{X}(T)\otimes{\mathbb{R}}$ by $$P_{DH}(q):=\prod_{\alpha\in\Phi_{Q^u}\cup \Phi_s^+}\kappa(\alpha,q).$$ Let $2\rho_H=\sum_{\alpha\in\Phi_{Q^u}\cup \Phi_s^+}\alpha$ considered as an element of $\mathcal{M}\times {\mathbb{R}}$ (*i.e.* composed with $\mathcal{P}$ in the notations of Section \[sec:horosym\]). Define the $\ell_i$-modified Duistermaat-Heckman barycenters by $$\label{eq:defn_bar}
\mathbf{bar}^{DH}_i= \int_{\Delta^{{\mathrm{tor}}}_{\theta_i}\cap \bar{C}^+} p e^{h_i\circ \ell_i (p)}P_{DH}(\xi_i+p)dp$$ where $dp$ is the Lebesgue measure normalized so that $\int_{\Delta^{{\mathrm{tor}}}_{\theta_i}\cap \bar{C}^+} e^{h_i\circ \ell_i (p)}P_{DH}(\xi_i+p)dp=1$.
\[thm:cpld\_can\_horo\] On a Fano horosymmetric manifold $X$, there exists a solution to the system if and only if $$0 \in \mathrm{Relint}\left( t\sum_i\mathbf{bar}^{DH}_i+(1-t)\sum_i\Delta^{{\mathrm{tor}}}_{\theta_i} +\Delta^{{\mathrm{tor}}}_{\gamma}-2\rho_H+-(\bar{C}^+)^{\vee} \right)$$
As a consequence, we obtain numerous generalizations or alternate proofs of recent results, and we shall illustrate this with a few corollaries. Let us first consider the non-coupled and non-twisted case (so $\Delta_{\theta_1}^{{\mathrm{tor}}}=\Delta_{ac}^{{\mathrm{tor}}}$), and first the case when $h_1$ is constant. We obtain the following generalization of [@WZ04; @PS10; @Li11; @DelKE; @Yao17] as well as an alternate proof to a particular case of [@DelKSSV].
A horosymmetric Fano manifold $X$ is Kähler-Einstein if and only if $\mathbf{bar}^{DH}_1-2\rho_H\in \mathrm{Relint}((\bar{C}^+)^{\vee})$. The greatest Ricci lower bound $R(X)$ of a horosymmetric Fano manifold $X$ is equal to $$\sup\left\{t\in ]0,1[~;~ 2\rho_H+\frac{t}{1-t}(2\rho_h-\mathbf{bar}^{DH}_1) \in \mathrm{Relint}(\Delta_{ac}^{{\mathrm{tor}}}-(\bar{C}^+)^{\vee}) \right\}$$
The variant for Kähler-Ricci solitons and greatest Bakry-Emery-Ricci lower bounds follow from allowing $h_1$ to be an affine function. Note that the possible affine function is fully determined up to constant by the conditions ensuring that the translated polytope lies in the linear span of $(\bar{C}^+)^{\vee}$.
The case of Mabuchi metrics is solved by allowing $h_1$ to be the logarithm of an affine function, and we obtain a generalization and alternate proof of [@Yao; @LZ17]. One can actually formulate the problem purely in terms of the polytope:
The horosymmetric Fano manifold $X$ admits a Mabuchi metric if and only there exists $\xi\in (\mathfrak{X}(T/[G,G])\otimes {\mathbb{R}})^*$ such that replacing $h_1\circ\ell_1$ by $\xi+C$ in the definition of the barycenter (for an appropriate normalizing contant $C$), both of the following conditions are satisfied:
- $\xi(p)+C>0$ on $\delta_{ac}^{{\mathrm{tor}}}\cap \bar{C}^+$, and
- $\mathbf{bar}^{DH}_1-2\rho_H\in \mathrm{Relint}((\bar{C}^+)^{\vee})$.
Finally, consider the case of coupled Kähler-Ricci solitons on horosymmetric manifolds. We obtain the following generalization of the second author’s existence result for toric manifolds [@Hul]. We place ourselves in the general coupled but not twisted setting, and assume that all the $h_i$ are affine. Again we may forget the $h_i$ and $\ell_i$ to formulate the problem only in terms of the polytope and the data of elements $\xi_i\in (\mathfrak{X}(T/[G,G])\otimes {\mathbb{R}})^*$ and normalizing constants $C_i$.
The decomposition $\sum_i\theta_i=c_1(X)$ admits coupled Kähler-Ricci solitons if and only if there exists $\xi_i\in (\mathfrak{X}(T/[G,G])\otimes {\mathbb{R}})^*$ such that $\sum_i \mathbf{bar}_i^{DH}-2\rho_H\in \mathrm{Relint}((\bar{C}^+)^{\vee})$ where $h_i\circ\ell_i$ is replaced by $\xi_i+C_i$ in the definition of the barycenters with appropriate normalizing constants $C_i$.
Finally we note that there are new results in the toric case. The obvious new results are for the coupled canonical metrics we have defined in the present paper of course, but it seems also noticeable that our theorem, in the simpler case of twisted Kähler-Einstein metrics, was never observed and proved before.
Let $X$ be toric Fano manifold. Let $\gamma$ be a $K$-invariant semipositive $(1,1)$-form and $\theta$ a Kähler form such that $\theta+\gamma\in c_1(X)$. Let $\Delta_{\gamma}$ be the polytope associated to $\gamma$ and let $\mathbf{bar}$ be the barycenter of the moment polytope of $[\theta]$. Then there exists a solution $\omega\in [\theta]$ to the twisted Kähler-Einstein equation $\operatorname{Ric}(\omega)=\omega+\gamma$ if and only if $0\in \mathrm{Relint}(\mathbf{bar}+\Delta_{\gamma})$.
In the last section of the article we will illustrate these corollaries on some low dimensional examples of horosymmetric manifolds. We take advantage of this occasion to determine the best horosymmetric structure on Fano threefolds. Most are toric, but admit a lower rank horospherical structure, and some are symmetric but not horospherical. We obtain that way new examples of Fano threefolds with Mabuchi metrics or without Mabuchi metrics.
We investigate on one rank one horospherical example the existence of a different type of canonical metric: instead of taking $h_1$ to be the logarithm of an affine function as in the case of Mabuchi metrics, we allow an integral multiple of such. In other words, we consider the case when $e^{h_1}$ is a power of an affine function. This provides an interpolation between the case of Mabuchi metrics and the case of Kähler-Ricci solitons. We illustrate on the example the fact that there may be no Mabuchi metrics, but a canonical metric for $e^{h_1}$ the square of an affine function.
Finally, we consider two rank one horospherical Fano fourfolds which admit coupled Kähler-Einstein pairs but no single Kähler-Einstein metric. For one, it is just the horospherical point of view on the second author’s example [@Hul], and the other is a natural variant.
Real Monge-Ampère equations on polyhedral cones
-----------------------------------------------
As mentioned in the previous section, on horosymmetric manifolds reduces to a real Monge-Ampère equation. One main ingredient in the proof of Theorem \[thm:cpld\_can\_horo\] is an apriori $C^0$-estimate for these real Monge-Ampère equations. The convex geometric setting we will use to state these $C^0$-estimates is a slight generalization of the one proposed by horosymmetric manifolds.
Let ${\mathfrak{a}}$ be a (real) vector space, $r=\dim {\mathfrak{a}}$ and ${\mathfrak{a}}^+$ be a convex polyhedral cone in ${\mathfrak{a}}$. We will implicitly fix a basis of ${\mathfrak{a}}$ and corresponding norms $|\cdot|$ and Lebesgue measures on ${\mathfrak{a}}$ and its dual $\mathfrak{a}^*$. Moreover, let $J$ be a continuous function on ${\mathfrak{a}}^+$, positive on $\mathring{{\mathfrak{a}}}^+$ and vanishing on $\partial {\mathfrak{a}}$. Assume also the function defined on $\mathring{{\mathfrak{a}}}^+$ by $j=-\ln J$ is smooth and convex. Finally, let $\Delta_1,\ldots,\Delta_k$ be convex bodies in $\mathfrak{a}^*$ and, for each $i\in \{1,\ldots,k\}$, let $G_i$ be a continuous function on $\Delta_i$, smooth and positive on $\mathring{\Delta}_i$, and such that $ \int_{\Delta_i} G_i = 1. $
The statement of the theorem will involve a few technical conditions on $j$ and $g_1,\ldots,g_k$. To state them we first need some terminology. Given a convex function $f: \mathring{{\mathfrak{a}}}^+\rightarrow {\mathbb{R}}$, we define its *asymptotic function* $$f_{\infty}:{\mathfrak{a}}^+\rightarrow {\mathbb{R}}\cup \{\infty\}
\qquad \xi \mapsto \lim_{t\rightarrow \infty} f(x+t\xi)/t$$ for any choice of $x\in \mathring{{\mathfrak{a}}}^+$ (it is standard that for a convex function, $f_{\infty}$ does not depend on $x$). Moreover, for any convex body $\Delta\subset \mathfrak{a}^*$, let $\mathcal{P}(\Delta)$ denote the space of all smooth, strictly convex functions $u:\mathfrak{a}\rightarrow {\mathbb{R}}$ such that $\overline{du({\mathfrak{a}}^+)} = \Delta$ and $$\label{eq:PDelta}
\sup_{{\mathfrak{a}}^+} |u-v_{\Delta}| < \infty,$$ where $v_\Delta$ is the support function of $\Delta$. We note that $f_{\infty}=v_{\Delta}|_{{\mathfrak{a}}^+}$ for any $f\in \mathcal{P}(\Delta)$.
From now on, we will let $\Delta$ denote the Minkowski sum $\Delta = \sum \Delta_i$. For every positive $M\in {\mathbb{R}}$ we define $K_M$ as the following subset of ${\mathfrak{a}}^+$ $$K_M = \left\{x\in a^+;\; \exists u\in \mathcal{P}(\Delta),\, j(x) + u(x) - \inf_{a^+} (j+u) \leq M \right\}.$$
\[Jassumption\] We assume the following on $j$, $\Delta$ and $g_1,\ldots,g_k$:
1. \[Jass\_proper\] $j_{\infty}+v_{\Delta}\geq \epsilon |x|$ for some $\epsilon>0$,
2. \[Jass\_translate\] for any $M>0$, $j-j_{\infty}$ is bounded from below on $K_M$
3. $j_\infty$ is continuous and finite on $a^+$
4. \[gass\_integrable\] $G_i^{-\epsilon}$ is integrable for some $\epsilon>0$.
Fix, for all $i\in\{1,\ldots,k\}$, a reference function ${{u}^{{\mathrm{ref}}}_{i}} \in \mathcal{P}(\Delta_i)$. We consider the continuity path of system of equations of the form $$\label{eq:RealContPath}
{\det(d^2{{u}^{t}_{i}})}G_i(d{{u}^{t}_{i}}) = J \prod_m e^{-t{{u}^{t}_{m}}-(1-t){{u}^{{\mathrm{ref}}}_{m}}}$$ where $0\leq t\leq 1$, and we are only interested in solutions $({{u}^{t}_{i}})\in \prod_i \mathcal{P}(\Delta_i)$. Similarly as in Section \[sec:intro\_red\], we assume any solution to be normalized to satisfy $${{u}^{t}_{1}}(0)={{u}^{t}_{2}}(0)=\cdots={{u}^{t}_{k}}(0).$$
Let ${\mathbf{bar}_{i}}\in \Delta_i$ denote the barycenter of $\Delta_i$ with respect to the measure with potential $G_i$ $${\mathbf{bar}_{i}} = \int_{\Delta_i} p G_i(p)$$ and let $F_t$ denote the function on ${\mathfrak{a}}^+$ given by $$F_t = t\sum_i{\mathbf{bar}_{i}} +(1-t)v_{\Delta} +j_{\infty}.$$ For $t\in [0,1]$ we will use $(\ddagger_t)$ to denote the condition that $F_t\geq 0$ on $\mathfrak{a}^+$ with equality only if $t=1$, $-\xi\in \mathfrak{a}^+$ and $j_\infty(-\xi)=-j_\infty (\xi)$.
\[thm:C0Estimates\] Let $t_0>0$ and $t\in (t_0,1]$. Assume $(\ddagger_t)$ is true. Then there are $C^0$-estimates for on $[t_0, t]$. Moreover, if $(\ddagger_t)$ is not true, then has no solution at $t$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
It is a pleasure for the authors to thank the organizers of the Spring School *Flows and Limits in Kähler Geometry* in Nantes, part of a thematic semester of the Centre Henri Lebesgue, where the joint work leading to this article was initiated. The first author is partially supported by the ANR Project FIBALGA ANR-18-CE40-0003-01.
Reduction to $C^0$ estimates {#sec:reductionC0}
============================
Let $s\in [0,1]$ and $t\in [0,1]$. To prove Theorem \[thm:ReductionToC0\] we will consider the following more general version of : $$\label{eq:GenComplexContPath}
e^{sh_i(f_i+V_i(\phi_i))}\omega_i^n = e^{-t\sum_{m=1}^k \phi_i}\omega_0^n.$$
We will begin by proving that the set of $[t,s]\in [0,1)\times [0,1]$ such that is solvable is open. Then we will establish a priori higher order estimates on solutions to assuming a priori $C_0$-estimates. Finally we will prove a priori $C_0$-estimates for solutions to when $(t,s)\in \{0\}\times [0,1]$. Using the Calabi-Yau Theorem to produce a solution at $(t,s)=(0,0)$, this will be enough to conclude the proof of Theorem \[thm:ReductionToC0\].
Openness
--------
We define the following Banach spaces $$A= \left\{(\phi_i)\in \left(C^{4,\alpha}(X)\right)^k: \phi_i \textnormal{ is } K-\textnormal{invariant for all } i \right\}$$ and $$B = \left\{(v_i)\in \left(C^{2,\alpha}(X)\right)^k: v_i \textnormal{ is } K-\textnormal{invariant for all } i \right\}.$$ Moreover, let $A_{(\theta_i)}$ be the set of $k$-tuples $(\phi_i)\in A$ such that $\omega_i:=\theta_i+{\sqrt{-1}\partial\bar\partial}\phi_i>0$ for all $i$. Moreover, for each $i$, let $g_i$ denote the function $$g_i = h_i(f_i+V_i(\phi_i)).$$ Let $$F:[0,1]\times [0,1]\times A_{(\theta_i)} \rightarrow B$$ be defined by $$F(t,s,(\phi_i)) =
\begin{pmatrix}
\log\frac{\omega_1^n}{\theta_1^n} + g_1+{V_1}(\phi_1) + t\sum \phi_i - \hat \phi_1 \\
\vdots \\
\log \frac{\omega_k}{\theta_k^n} +g_k+{V_k}(\phi_k) + t\sum \phi_i - \hat \phi_k
\end{pmatrix}$$ where $\hat\phi_i$ is the average of $\phi_i$ with respect to $\theta_i$ $$\hat\phi_i = \frac{\int_X \phi_i \theta_i^n}{\int_X \theta_i^n}.$$ Up to normalization, it follows that $F(t,s,(\phi_i))=0$ if and only if $(\phi_i)$ is a solution to at $(t,s)$. Moreover, in this case the measure $$\mu := e^{sg_i}\omega_i^n$$ is independent of $i$.
To any Kähler metric $\omega=\sqrt{-1}\sum \omega_{j,m} \omega_{j,m} dz\wedge d\bar z$ and function $g$ we may associate a complex (1,0)-vector field $$\frac{1}{\sqrt{-1}}\sum_{j,m} \frac{\partial g}{\partial \bar z_m} dz_j.$$ The following lemma characterizes the variation of $g_i$ with respect ot $\phi_i$.
\[lemma:VariationOfExtraFactor\] Let $(v_i)\in A$. Then $$\left.\frac{dh_i(f_i+V_i(\phi_i+tv_i))}{dt} \right|_{t=0} = \left\langle\operatorname{grad}^\mathbb C_{\omega_i} h_i(f_i+V_i(\phi_i)),v_i\right\rangle.
\label{eq:gVariation}$$
By the chain rule, it suffices to prove this in the case $h_i=id$. In this case it is well known. However, for completeness we include a proof of it. By straight forward differentiation, the left hand side of is $V_i(v_i)$. Moreover, by the definition of $f_i$ $$\begin{aligned}
{\sqrt{-1}\partial\bar\partial}\left( f_i + V_i(\phi_i)\right) & = & L_{V_i}(\theta_i) + L_{V_i}({\sqrt{-1}\partial\bar\partial}\phi_i) \nonumber \\
& = & L_{V_i}(\omega_i) \nonumber \\
& = & d(V_i\rfloor \omega_i). \nonumber\end{aligned}$$ It follows that $$\bar \partial \left( f_i + V_i(\phi_i)\right) - V_i\rfloor \omega_i \label{eq:holo10form}$$ is a $\partial$-closed (0,1)-form. Conjugating gives a holomorphic $(1,0)$-form which, since $X$ is Fano, must vanish. Hence $\eqref{eq:holo10form}$ vanishes. Choosing coordinates $(z_1,\ldots,z_n)$ such that $ \omega_i = \sqrt{-1}\sum_j dz_j\wedge d z_j $ and writing $V_i = \sum_j V_i^j \partial/\partial \bar z_j $ we get $$\begin{aligned}
\sum_j \frac{\partial\left( f_i + V_i(\phi_i)\right)}{\bar z_j} d\bar z_j & = & \bar \partial \left( f_i + V_i(\phi_i)\right) \nonumber \\
& = & V_i\rfloor \omega_i \nonumber \\
& = &
\left.\left(\sum_jV_i^j\frac{\partial}{\partial z_j}\right)\right\rfloor \left( \sqrt{-1}\sum_j dz_j\wedge d\bar z_j\right) \nonumber \\
& = & \sqrt{-1}\sum_j V_i^j d\bar z_j, \nonumber\end{aligned}$$ hence $\partial ( f_i + V_i(\phi_i))/d\bar z_j = \sqrt{-1}V_i^j$ for each $j$ and $$\begin{aligned}
\operatorname{grad}^\mathbb C_\omega\left( f_i + V_i(\phi_i)\right) & = & \frac{1}{\sqrt{-1}}\sum_j \frac{\partial\left( f_i + V_i(\phi_i)\right)}{d \bar z_j} \frac{\partial}{\partial z_j} \nonumber \\
& = & \sum_j V_i^j \frac{\partial}{\partial z_j} \nonumber \\
& = & V_i. \nonumber\end{aligned}$$ This proves the lemma.
Now, let $\Delta_{\omega_i,sg_i}$ be the $sg_i$-weighted Laplacian $$\Delta_{\omega_i,sg_i} = \Delta_{\omega_i} + \operatorname{grad}^\mathbb C_{\omega_i} sg_i$$ where $\Delta_{\omega_i}$ is the usual Laplace-Beltrami operator of the metric $\omega_i$.
\[lemma:Linearization\] The linearization of $F$ at $(t,\phi)$ with respect to the second argument is given by $H:A\rightarrow B$ defined by $$\label{eq:Linearization}
H(v_1,\ldots,v_k) = \begin{pmatrix}
-\Delta_{\omega_1,sg_1} v_1 + t\sum v_i - \hat v_1\\
\vdots \\
-\Delta_{\omega_k,sg_k} v_k + t\sum v_i - \hat v_k\\
\end{pmatrix}.$$ where $\hat v_i = \int \hat v_i \theta_i^n$. Moreover, $H$ is elliptic. Finally, assume $F(t,\phi)=0$ and let $\langle\cdot,\cdot\rangle$ be the inner product on $(C^{2,\alpha}(X))^k$ given by $$\langle(u_i),(v_i)\rangle = \sum_i \int_X u_iv_i d\mu$$ Then $\langle H(u_1,\ldots,u_k) , (v_i)\rangle = \langle (u_i),H(v_1,\ldots,v_k) \rangle$ for any $(u_i),(v_i)\in (C^{2,\alpha})^k$.
By Lemma \[lemma:VariationOfExtraFactor\], the linearization of $g_i$ with respect to $\phi$ is $\operatorname{grad}^\mathbb C_{\omega_i} g( u_i )$. Together with a standard computation (see for example the proof of Lemma 3 in [@Hul]), this proves the first part of the lemma.
The second part of the lemma, i.e. that $H$ is elliptic, follows exactly as in the proof of Lemma 3 in [@Hul].
The third part of the lemma is a consequence of standard properties for $\Delta_{\omega_i,sg_i}$, namely $$\label{eq:WeightedLaplacianPI}
\int_X (\Delta_{\omega_i,sg_i} u)v e^{sg_i}\omega_i^n = \int \langle du,dv \rangle_{\omega_i} e^{sg_i}\omega_i^n = \int_X u(\Delta_{\omega_i,sg_i} v) e^{sg_i}\omega_i^n.$$ From this, the third part of the lemma follows as in the proof of Lemma 3 in [@Hul].
\[lemma:HInjective\] Assume $(t,s)\in [0,1)\times [0,1]$ and $(v_i)\in A$ satisfies for all $i$ $$\Delta_{\omega_i,sg_i} v_i = \lambda\sum_{m=1}^k v_m
\label{eq:LaplaceEigenfunction}$$ for a $k$-tuple $\omega_1,\ldots,\omega_k$ satisfying for all $i$ $$\operatorname{Ric}\omega_i - \sqrt{-1} \partial \bar \partial sg_i = \gamma + t\sum_{m=1}^k \omega_m + (1-t)\sum_{m=1}^k \theta_m.
\label{eq:RicContinuityMethod}$$ Then either $\lambda > t$ or $v_i$ is constant for all $i$.
The proof of this lemma follows closely the proof of Theorem 1.3 in [@Pin18] and the proof of Lemma 4 in [@Hul]. The crucial point that is new in this setting is the following Weitzenböck identity: $$\label{eq:WeitzenbockId}
\int_X \left\langle d\left(\Delta_{\omega_i,sg_i} u\right), du\right\rangle_{\omega_i} e^{g_i}\omega_i^n \geq \int_X \left(\operatorname{Ric}_{\omega_i} - \sqrt{-1} \partial \bar \partial g_i\right) \left(\operatorname{grad}^\mathbb C_{\omega_i} u,\overline{\operatorname{grad}^\mathbb C_{\omega_i} u}\right) e^{g_i}\omega_i^n$$ Using and following the argument in [@Hul] or [@Pin18] proves the lemma. We will now prove . In the following computation we will suppress the index on $\omega_i$ and $g_i$. Instead, lower index $i,\bar i,j,\bar j,p$ and $ \bar p$ will denote covariant differentiation. $$\begin{aligned}
\int_X \sum_i (\Delta_\omega u)_i u_{\bar i} e^{g}\omega^n & = & \int_X (\Delta_\omega u)_i u e^{g}\omega^n \nonumber \\
& = & \int_X \sum_{ij} u_{j\bar j i}u_{\bar i}e^{g}\omega^n \nonumber \\
& = & \int_X \sum_{ij} \left(u_{ij\bar j}-R^p_{ji\bar j}\right)u_{\bar i}e^{g}\omega^n \nonumber \\
& = & \int_X \operatorname{Ric}_{i\bar p}u_pu_{\bar i}e^{g}\omega^n + \int_X u_{ij}u_{\bar i\bar j}e^{g}\omega^n - \int_X u_{ij}u_{\bar i}g_{\bar j}e^{g}\omega^n. \nonumber
\end{aligned}$$ Moreover, $$\begin{aligned}
\int_X \left(\operatorname{grad}^\mathbb C_{\omega_i} g (u)\right)_iu_{\bar i} e^{g}\omega^n
& = & \int_X \left(u_jg_{\bar j}\right)_iu_{\bar i} e^{g}\omega^n \nonumber \\
& = & \int_X u_{ji}g_{\bar j}u_{\bar i} e^{g}\omega^n
+ \int_X u_j g_{\bar ji} u_{\bar i} e^{g}\omega^n \nonumber
\end{aligned}$$ We get $$\begin{aligned}
\int_X \sum_i (\Delta_\omega u + \operatorname{grad}^\mathbb C_{\omega} g (u))_i u_{\bar i} e^{g}\omega^n & = & \int_X \left(\operatorname{Ric}_{i\bar p}u_pu_{\bar i}-u_j g_{\bar ji} u_{\bar i}\right)e^{g}\omega^n \nonumber \\
& & + \int_X u_{ij}u_{\bar i\bar j}e^{g}\omega^n, \nonumber
\end{aligned}$$ and follows.
Estimate on $\Delta_{\theta_i}\phi_i$
-------------------------------------
\[lemma:LaplacianEstimate\] Assume $(\phi_i)$ satisfies $\eqref{eq:GenComplexContPath}$ for some $(t,s)\in [0,1]\times [0,1]$. Then $$\max_i \sup_X |\Delta_{\theta_i}\phi_i| \leq C$$ where $C$ depends only on $\max_i ||\phi_i||_{C^0(X)}$.
The proof follows the standard method of Yau. In particular, see [@Pin18] and [@Hul] where the cases of coupled Kähler-Einstein metrics and coupled Kähler-Ricci solitons is treated. The method is based on expanding the quantity $$\Delta_{\omega_i} \left(e^{-C_1\phi_i}(n+\Delta_{\theta_i}\phi_i)\right)$$ at a point where $e^{-C_1\phi_i}(n+\Delta_{\theta_i}\phi_i)$ attains its maximum. To prove the lemma, some extra care need to be taken when estimating $$\begin{aligned}
\Delta_{\theta_i}g_i & = & \Delta_{\theta_i}\left(h_i(f_i+V_i(\phi_i))\right) \nonumber \\
& = & h_i'\Delta_{\theta_i} \left(f_i+V_i(\phi_i)\right) + h_i''\left|d(f_i+V_i(\phi_i))\right|^2_{\theta_i}. \label{eq:LaplaceOfgi}\end{aligned}$$ where $h_i'$ and $h_i''$ are the first and second derivatives of $h$ at the point given by $f_i+V_i(\phi_i)$. Now, by Corollary 5.3, page 768 in [@Zhu00], $|f_i+V_i(\phi_i)|$ can be bounded by a constant independent of $\omega_i$. It follows that $h'$ is bounded. Moreover, $|\Delta_{\theta_i}(f_i+V_i(\phi_i)) |$ can be bounded as in [@Zhu00], page 769. Finally, by concavity of $h$, the second term is bounded from above by 0. Apart from this, the proof follows the arguments in [@Pin18] and [@Hul].
$C^0$-estimates for $t=0$
-------------------------
\[lemma:C0Estimatet=0\] There is a constant $C$ such that any solution $(\phi_i)$ to at $t=0$ and $s\in [0,1]$ satisfies $$\label{eq:C0Estimatet=0}
\sup_X \left|\phi_i-\sup_X\phi_i\right|<C$$ for all $i$.
Let $$F^i_s = -s g_i + \log \frac{\omega_0^n}{\theta_i^n}+c_s$$ where $c_s$ is a constant such that $$\int_X e^{F^i_s}\theta_i^n = \int_X \theta_i^n.$$ This means each $\phi_i$ satisfies the equation $$\omega_i^n = e^{F^i_s}\theta_i^n.$$
As above, we note that by Corollary 5.3, page 768 in [@Zhu00], $ |g_i| = |h_i(f_i+V_i(\phi_i))| $ can be bounded by a constant independent of $\phi_i$. Moreover, since $\int \omega_i^n=\int \theta_i^n$ $$|c_s| = \left|\log\int_X e^{-s g_i}\omega_i^n - \log\int_X \theta_i^n\right|\leq s\sup_X |g_i|.$$ It follows that $|F_i|$ can be bounded by a constant independent of $\phi_i$. By invariance of at $t=0$ we may assume $\sup \phi_i = -1$ for all $i$. Applying the argument in [@Tia96], page 157-159, we get a uniform constant $C$ satisfying .
Proof of Theorem \[thm:ReductionToC0\]
--------------------------------------
First of all, by Lemma \[lemma:Linearization\], Lemma \[lemma:HInjective\] and a standard application of the Implicit Function Theorem, the set of $(t,s)$ such that is solvable is open in $[0,1)\times [0,1]$. Moreover, by the Calabi-Yau Theorem we may find $\phi_1,\ldots,\phi_k$ solving at $(t,s)=(0,0)$. Since $\omega_0$ is $K$-invariant, $\phi_i$ will be $K$-invariant for each $i$. We will now consider the path $$(t,s)\in A:=\left(\{0\}\times [0,1]\right) \cup \left([0,1]\times \{1\}\right) \subset [0,1]\times [0,1].$$ In other words, we consider $(s,t)$ such that either $t=0$ or $s=1$. By the assumed $C^0$ estimates in , Lemma \[lemma:LaplacianEstimate\] and Lemma \[lemma:C0Estimatet=0\] any solution to has bounded $C^0$ norm and bounded Laplacian. It then follows from Theorem 1 in [@Wan12] that the $C^{2,\alpha}$-norm $\sup_i ||\phi_i||_{C^{2,\alpha}}$ of any solution to along this path is uniformly bounded. We concluded that the set of $(t,s)\in A$ such that is solvable is open and closed in $A$, hence is solvable at $s=1$ for any $t\in [0,1]$. This proves the theorem.
$C^0$-estimates for coupled real Monge-Ampère equations on polyhedral cones {#sec_C0}
===========================================================================
In this section we will prove Theorem \[thm:C0Estimates\]. The proof essentially follows the argument in [@DelKE] with adaptations for the coupled case from [@Hul]. Both of these are in turn generalizations of the argument in [@WZ04].
Set $$\nu_t:=t\sum_m{{u}^{t}_{m}}+(1-t)\sum_m{{u}^{{\mathrm{ref}}}_{m}}+j$$ on $\mathring{\mathfrak{a}}^+$. Note that $\nu_t-j\in{\mathcal{P}(\Delta_{})}$, hence by Assumption \[Jassumption\](\[Jass\_proper\]), $\nu_t$ is smooth, strictly convex and proper. As a consequence, the following real number $m_t\in {\mathbb{R}}$, point $x_t\in\mathring{{\mathfrak{a}}}^+$ and bounded convex set $A_t\subset \mathring{{\mathfrak{a}}}^+$ are well-defined: $$m_t=\min \nu_t = \nu_t(x_t) \qquad A_t :=
\{x\in \mathring{{\mathfrak{a}}}^+ ~;~ m_t\leq \nu_t(x) \leq m_t+1 \}.$$ For a function $g$ in ${\mathfrak{a}}^+$ and $x,\xi\in {\mathfrak{a}}^+$, we will use the notation $d_x g(\xi)$ to denote the directional derivative of $g$ at the point $x$ in the direction $\xi$. As in [@WZ04], the first step is to reduce $C^0$-estimates on to estimates on $\nu_t$.
\[lem:RedToEstOnNu\] Assume $t\geq t_0>0$ and the following hold: $$\begin{aligned}
|m_t| & \leq & \C, \label{eq:mt_bound} \\
\nu_t(x) & \geq & \C|x-x_t|-\C \label{eq:lin_growth} \\
|x_t| & \leq & \C. \label{eq:xt_bound}\end{aligned}$$ Then $$\max_i \sup_{{\mathfrak{a}}^+} |{{u}^{t}_{i}}-{{u}^{{\mathrm{ref}}}_{i}}| \leq \C.$$
We follow the argument in [@Hul]. There is only one added difficulty which is solved by using Assumption \[Jassumption\](\[gass\_integrable\]). To prove the lemma, first recall that the Legendre transform $u\mapsto \mathcal{L}(u)$ is such that, for $u, v \in \mathcal{P}(\Delta)$, $\sup_{{\mathfrak{a}}^+} |u-v| = \sup_{\Delta} |\mathcal{L}(u)-\mathcal{L}(v)|$. Furthermore, if $u\in {\mathcal{P}(\Delta_{})}$ then $\mathcal{L}(u)$ is bounded on $\Delta$, hence to get the conclusion it is enough to prove estimates on $\mathcal{L}({{u}^{t}_{i}})$ for each $i$.
For each $i$, let $w_i^t=\mathcal{L}({{u}^{t}_{i}})$. We may first obtain a bound on $w_i^t$ at one given point. Note that $w_i^t(d{{u}^{t}_{i}}(0))=-{{u}^{t}_{i}}(0)$. Furthermore, ${{u}^{t}_{i}}(0)= \frac{1}{N}\sum_l{{u}^{t}_{l}}(0)$ by the normalization we chose. Since ${{u}^{t}_{l}}\in{\mathcal{P}(\Delta_{l})}$ for all $l$, and $|x_t|$ is controlled, $|{{u}^{t}_{i}}(0)-\frac{1}{N}\sum_l{{u}^{t}_{l}}(x_t)|$ is also controlled. Moreover, we claim that $|j(x_t)|$ is uniformly bounded. To see this, let $\gamma\in \mathring{{\mathfrak{a}}}$ and note that since $d_{x_t}j \in \Delta$ we have that $x_t$ is in the compact set $$\label{eq:xtset}
B(0,C_4)\cap \{dj(\gamma)\geq -v_\Delta(\gamma)\}.$$ Since $j\rightarrow +\infty$ as we approach the boundary of ${\mathfrak{a}}^+$ we get that is included in the interior of $a^+$. This means $j$ is bounded on , proving the claim. Moreover, since $|j(x_t)|$ is bounded we get that $|{{u}^{t}_{i}}(0)-m_t|$, and hence $|{{u}^{t}_{i}}(0)|$ is bounded. Let now $\hat{w}_i^t$ denote the average of $w_i^t$ on $\Delta_i$. The conclusion of the Lemma will follow from using $$\begin{aligned}
\sup_{\Delta_i} |w_i^t(p)-w_i^t(0)| &
\leq \C|w_i^t-\hat{w}_i^t|_{1-r/s} \\
& \leq \C|dw_i^t|_{L^s(\Delta_i)}\end{aligned}$$ for $s>r$ (recall that $r$ is the dimension of $\mathfrak{a}$), where the second inequality holds by Sobolev inequalities. Using Assumption \[Jassumption\](\[gass\_integrable\]), we choose $p>1$ such that $g_i^{-1/(p-1)}$ is integrable. We have $$\begin{aligned}
\int_{\Delta_i} |dw_i^t|^s & =
\int_{\Delta_i} (|dw_i^t|^sG_i^{1/p})G_i^{-1/p} \\
& \leq \left(\int_{\Delta_i} |dw_i^t|^{ps}G_i\right)^{1/p} \left(\int_{\Delta_i} G_i^{-1/(p-1)}\right)^{(p-1)/p} \\
& \leq \C\left(\int_{\Delta_i} |dw_i^t|^{ps}G_i\right)^{1/p}\end{aligned}$$ To finally bound $|dw_i^t|_{L^s(\Delta_i)}$, we Legendre transform back to use the equation: $$\begin{aligned}
\int_{\Delta_i} |dw_i^t|^{ps}g_i & =
\int_{{\mathfrak{a}}^+} |x|^{ps}G_i(d{{u}^{t}_{i}}){\det(d^2{{u}^{t}_{i}})} \\
& = \int_{{\mathfrak{a}}^+} |x|^{ps}e^{-\nu_t(x)} \\
& \leq C\end{aligned}$$ thanks to the uniform linear growth estimate on $\nu_t$.
The first step in proving the estimates , and on $\nu_t$ is to get bounds on $\mathrm{Vol}(A_t)$.
There exists a constant $\Cl{lowvol}>0$ independent of $t$ such that $$\Cr{lowvol}\leq \mathrm{Vol}(A_t).$$
We have $d\nu(\xi) < j_\infty(\xi) + v_\Delta(\xi)$ which is bounded on $a^+$ by continuity of $j_\infty$. This means $\nu_t<m_t+C|x-x_t|$ on $(x_t+a^+)$ for some constant C>0, hence the set $(x_t+a^+)\cap B(x_t,1/C)$ is included in $A_t$ for some uniform $r>0$. This gives the desired volume bound.
On the other hand, there is an upper bound in terms of $m_t$:
There exists a constant $\Cl{upvol}$ such that, for $t\geq t_0$, $$\mathrm{Vol}(A_t)\leq \Cr{upvol}e^{m_t/2}.$$
The proof follows exactly the lines of that of Wang and Zhu [@WZ04] (see also Donaldson [@Don08]). The only requirement for the proof to work is to prove an inequality of the form $${\det(d^2\nu_t)}\geq \Cl{comparison}e^{-m_t}$$ for some constant $\Cr{comparison}$ which does not depend on $t\geq t_0$. This is easily obtained starting from one of the equations by noting that $g_i$ is continuous positive on $\mathring{\Delta}_i$, hence $1/g_i\geq c>0$ for some constant $c$.
From this it follows, following the proof of Wang and Zhu or Donaldson with only notational changes:
\[lem\_mlinear\]
- There exists a constant $C$ independent of $t$, such that $|m_t|\leq C$.
- There exist a constant $\kappa >0$ and a constant $C$, both independent of $t$, such that for $x\in \mathfrak{a}^+$, $$\nu_t(x) \geq \kappa |x-x_t|-C.$$
Recall that $(\ddagger_t)$ denotes the condition that $$F_t(\xi) = t\sum_i{\mathbf{bar}_{i}}(\xi) +(1-t)v_{\Delta}(\xi) +j_{\infty}(\xi) \geq 0$$ with equality only if $t=1$, $-\xi\in \mathfrak{a}^+$ and $j_\infty(-\xi)=-j_\infty (\xi)$. By Lemma \[lem:RedToEstOnNu\] and Lemma \[lem\_mlinear\], the $C^0$-estimates of Theorem \[thm:C0Estimates\] are reduced to proving a bound on $|x_t|$. To prove the first part of Theorem \[thm:C0Estimates\] it thus suffices to rule out the case $|x_t|\rightarrow \infty$ whenever $(\ddagger_t)$ holds. Most of the rest of this section is dedicated to this. However, before we start we prove the second point of Theorem \[thm:C0Estimates\], namely that if there is a solution to at $t$ then $(\ddagger_t)$ holds.
\[lem\_obstruction\] Assume there exists a solution in $\prod_i{\mathcal{P}(\Delta_{i})}$ to at time $t$. Then $(\ddagger)_t$ does not hold.
The proof is based on the vanishing of $$\label{eq:stokes}
0 = \int_{\mathfrak{a}^+} d\nu_t(\xi) e^{-\nu_t},$$ for any $\xi \in \mathfrak{a}$, which follows from integration by parts, Assumption \[Jassumption\](\[Jass\_proper\]) and the the fact that $J=0$ on $\partial {\mathfrak{a}}^+$. (see *e.g.* [@DelKE Section 5] for details). By definition of $\nu_t$, the left hand side of splits as a sum $$\label{eq:stokesexpanded}
0 = t\sum_m\int_{\mathfrak{a}^+} d{{u}^{t}_{m}}(\xi)e^{-\nu_t}
+(1-t)\sum_m\int_{\mathfrak{a}^+} d{{u}^{{\mathrm{ref}}}_{m}}(\xi)e^{-\nu_t}
+ \int_{\mathfrak{a}^+} d j(\xi)e^{-\nu_t}.$$ Note that, by definition of ${\mathbf{bar}_{k}}$, the fact that $G_i$ has unit mass on $\Delta_i$ and the equation, we have $$\label{eq:bar}
\int_{\mathfrak{a}^+} d{{u}^{t}_{m}}(\xi)e^{-\nu_t}={\mathbf{bar}_{m}}(\xi).$$ Restricting to elements $\xi$ of $\mathfrak{a}^+$, we have by (strict) convexity $d{{u}^{{\mathrm{ref}}}_{i}}(\xi)< v_{\Delta_i}(\xi)$. Moreover, $d j(\xi)\leq j_{\infty}(\xi)$ with strict inequality for some $\xi\in {\mathfrak{a}}^+$ whenever $-\xi\notin \mathfrak{a}^+$ or $j_{\infty}(\xi)\neq j_{\infty}(-\xi)$. Since the total mass of $e^{-\nu_t}$ is one, this ends the proof of the lemma.
To conclude the proof of Theorem \[thm:C0Estimates\], we assume that the $C^0$-estimates fail on some interval $[t_0,t']\subset [0,1]$. By Lemma \[lem:RedToEstOnNu\] and Lemma \[lem\_mlinear\] it follows that we may pick a converging sequence $t_i$ of elements in $[t_0,t']$, whose limit we denote by $t_{\infty}$, such that $\lim |x_{t_i}|=\infty$ and $\xi_i:=x_{t_i}/|x_{t_i}|$ admits a limit $\xi_{\infty}\in \mathfrak{a}^+$. We will prove that $(\ddagger_{t_\infty})$ does not hold and hence, by monotonicity of $F_t$ with respect to $t$, that $(\ddagger_{t'})$ does not hold. This is essentially the content of the following lemma:
\[lem\_limits\] Under the assumptions in the preceding paragraph $$F_{t_\infty}(\xi_\infty) = \big(t_{\infty}\sum_m \mathrm{bar}_m +(1-t_{\infty})v_{\Delta}+j_{\infty}\big)(\xi_{\infty}) = 0.$$ Moreover, if $t_\infty = 1$ then $-\xi_\infty\notin {\mathfrak{a}}^+$ or $j_\infty(-\xi_\infty)\not=-j_\infty(\xi_\infty)$.
We abbreviate the indices $t_i$ by $i$ from now on. The proof again relies on the vanishing , with $\xi=\xi_i$. Using this, it will follow from and the following lemma:
\[lem:conv\] $$\begin{aligned}
\int_{a^+} d j(\xi_i) e^{-\nu} & \rightarrow & j_\infty(\xi_\infty) \\
\int_{a^+} d {{u}^{{\mathrm{ref}}}_{k}}(\xi_i) e^{-\nu} & \rightarrow & v_\Delta(\xi_\infty)\end{aligned}$$
To prove Lemma \[lem:conv\] we will first identify suitable sets on which $e^{-\nu_i}$ concentrate. Let $U_{R,M} = B(x_i,R)\cap K_M$.
\[lem:consmass\] $$\int_{a^+\setminus U_{R,M}} e^{-\nu} \leq e^{-m_t-\min\{M,\epsilon R\}}R^n|B(0,1)|$$ and $$\int_{\partial U_{R,M}} e^{-\nu} \leq e^{-m_t-\min\{M,\epsilon R\}}R^{n-1}|\partial B(0,1)|.$$
First of all, note that the complement of $K_M$ in $\mathring{{\mathfrak{a}}}^+$ is given by $$\mathring{{\mathfrak{a}}}^+\setminus K_M = \{x \in {\mathfrak{a}}^+; \; \forall u\in \mathcal{P},\, j(x)+u(x) -\inf_{{\mathfrak{a}}^+} (j+u) > M\}.$$ In particular, since $$\sum_m tu_m+(1-t)\sum_m{{u}^{{\mathrm{ref}}}_{m}}\in \mathcal{P}(\Delta)$$ we get that $$\nu = j+\sum_m tu_m+(1-t)\sum_m{{u}^{{\mathrm{ref}}}_{m}} > \inf \nu + M$$ outside $K_M$. The lemma follows from this and the fact that, by Lemma \[lem\_mlinear\], $\nu\geq\epsilon R+m_t$ on $a^+\setminus B(x_t,R)$.
We also get the following bound:
\[lem:boundgrad\] For any $\xi\in \mathfrak{a}$, $$\left|\int_{U_{M,R}} d j(\xi) e^{-\nu}\right| < C|\xi|$$ where $C$ is independent of $u$.
By Stokes’ theorem $$\begin{aligned}
\left|\int_{U_{M,R}} d j(\xi) e^{-\nu}\right| & = & \left|\int_{U_{M,R}} d \nu(\xi) e^{-\nu} - \int_{U_{M,R}} d \left(\sum_m tu_m+(1-t)\sum_m{{u}^{{\mathrm{ref}}}_{m}}\right)(\xi) e^{-\nu}\right| \nonumber \\
& = & \left|\int_{\partial U_{M,R}} n(\xi) e^{-\nu}\right| + \left|\int_{U_{M,R}} d \left(\sum_m tu_m+(1-t)\sum_m{{u}^{{\mathrm{ref}}}_{m}}\right)(\xi) e^{-\nu}\right| \nonumber \\
& \leq & |\xi|\int_{\partial U_{M,R}} e^{-\nu} + v_\Delta(\xi) \int_{U_{M,R}} e^{-\nu} \label{eq:StokesEq}\end{aligned}$$ where $n$ is a unit length inward pointing normal on $\partial U_{M,R}$. By Lemma \[lem:consmass\], is bounded.
We are now ready to prove Lemma \[lem:conv\].
As in [@DelKE], let $\gamma$ be a point in $a^+$ such that the closure of $ B(\gamma,R) $ is contained in the interior of $a^+$. By assumption, $j \geq j_\infty + C$ on $U_{M,R}$ for some $C\in {\mathbb{R}}$. Let $\xi_i'=(x_i-\gamma)/|x_i|$ and $\xi_i'' = \gamma/|x_i|$. It follows by convexity of $j$ that for any $y \in B(0,R)$ such that $x_i+y\in U_{M,R}$ $$\begin{aligned}
d_{x_i+y}j(\xi_i') & \geq & \frac{j(x_i+y)-j(\gamma+y)}{|x_i-\gamma|} \nonumber \\
& \geq & \frac{j_\infty(x_i+y)+C-j(\gamma+y)}{|x_i-\gamma|} \nonumber \\
& \geq & j_\infty\left(\xi_i+\frac{y}{|x_i|}\right)\frac{|x_i|}{|x_i-\gamma|} -\frac{C+\sup_{B(\gamma,R)} |j|}{|x_i-\gamma|} \nonumber \\
& \rightarrow & j_\infty(\xi_\infty) \label{eq:gradconv}\end{aligned}$$ uniformly in $y$ as $|x_i|\rightarrow \infty$ since $j_\infty$ is continuous on $a^+$ and $\xi_i+y/|x_i|\rightarrow \xi$.
By Lemma \[lem:boundgrad\] $$\left|\int_{U_{M,R}} d j(\xi_\infty) e^{-\nu} dx - \int_{U_{M,R}} d j(\xi_i') e^{-\nu} dx \right| \leq C|\xi_i-\xi_\infty|$$ which vanishes as $i\rightarrow \infty$. We conclude that $$\int_{U_{M,R}} d j(\xi_\infty) e^{-\nu} dx \geq j_\infty(\xi_\infty)\int_{U_{M,R}} e^{-\nu} dx - o(1)$$ where $o(1)\rightarrow 0$ as $i\rightarrow \infty$. Moreover, by standard convexity properties, $d j(\xi_\infty)\leq j_\infty(\xi)$ everywhere on ${\mathfrak{a}}^+$. This proves the first point in the lemma. The second point in the lemma follows by a similar argument.
To prove the first part we assume the $C^0$-estimates does not hold. As explained in the paragraph before Lemma \[lem\_limits\] this means we may pick a sequence $t_i$ converging to $t_{\infty}$, such that $\lim |x_{t_i}|=\infty$ and $\xi_i := \lim x_{t_i}/|x_{t_i}|$ admits a limit $\xi_{\infty}\in \mathfrak{a}^+$. By Lemma \[lem\_limits\] this means $$F_{t_\infty}(\xi_\infty) = \lim F_{t_i}(\xi_i) = \lim F_{t_i}(\xi_i) - \int_{{\mathfrak{a}}^+} d\nu_i(\xi_i) e^{-\nu_i} = 0.$$ Thus, to conclude the first part of the theorem we just need to rule out the case that $t_\infty = 1$, $-\xi_\infty\in {\mathfrak{a}}^+$ and $j_\infty(-\xi_\infty) = - j_\infty(\xi_\infty)$. By the latter of these conditions we have that $dj(\xi_\infty)$ is constant, hence $$\int_X dj(\xi_\infty) e^{-\nu_i} = j_\infty(\xi_\infty)$$ for all $i$. Since $$F_1(\xi_\infty) = \big(\sum_m \mathrm{bar}_m +j_{\infty}\big)(\xi_{\infty}) = 0$$ we may plug in $\xi=\xi_\infty$ and $t=t_i$ into to get $$0 = (1-t_i)\left(j_\infty(\xi_\infty) + \int_{{\mathfrak{a}}^+} \sum_m d{{u}^{{\mathrm{ref}}}_{m}}(\xi_\infty)e^{-\nu_i}\right).$$ This means $$\int_{{\mathfrak{a}}^+} \sum_m d{{u}^{{\mathrm{ref}}}_{m}}(\xi_\infty)e^{-\nu_i} = -j_\infty(\xi_\infty)$$ for all $i$. However, since the right hand side of this is independent of $i$ and the left hand side is strictly less than $v_\Delta(\xi_\infty)$ and converging towards $v_\Delta(\xi_\infty)$ we get a contradiction. This proves the first part of the theorem. The second part of the theorem is given by Lemma \[lem\_obstruction\].
Case of Horosymmetric manifolds {#sec:horosym}
===============================
In this section we will recall the definition of horosymmetric manifolds, and the tools developed in [@DelHoro] to study their Kähler geometry. We will then use these tools to translate in the horosymmetric setting to a real Monge-Ampère equation (see Theorem \[thm:translation\_horo\] below). Combining with Theorem \[thm:ReductionToC0\] and Theorem \[thm:C0Estimates\], we obtain the proof of Theorem \[thm:cpld\_can\_horo\].
Setting
-------
Let $G$ denote a connected complex linear Lie group, and assume that the Fano manifold $X$ is equipped with an action of $G$. We assume that there exist a point $x$ in $G$, a parabolic subgroup $P$ of $G$, a Levi subgroup $L$ of $P$ and a complex Lie algebra involution $\sigma$ of the Lie algebra of $L$ such that $G\cdot x$ is open and dense in $X$, and the Lie algebra of the isotropy group $G_x$ of $x$ is the direct sum of the Lie algebra of the unipotent radical of $P$ and of the Lie subalgebra of the Lie algebra of $L$ fixed by $\sigma$. In other words, the manifold $X$ is *horosymmetric* as introduced in [@DelHoro]. We fix such a point $x$ and denote by $H$ the group $G_x$.
Let us fix $T_s$ a torus in $L$, maximal for the property that $\sigma$ acts on $T_s$ as the inverse. Let $T$ be a $\sigma$-stable maximal torus of $L$ containing $T_s$, $\Phi$ the root system of $G$ with respect to $T$ and $\Phi_L$ that of $L$. We let $Q$ denote the Borel subgroup opposite to $P$, $Q^u$ denote its unipotent radical, and $\Phi_{Q^u}$ the roots of $Q^u$. Let $B$ denote a Borel subgroup of $G$, such that $T\subset B\subset Q$, with corresponding positive roots $\Phi^+$, and such that for any $\beta\in \Phi_L^+=\Phi_L\cap \Phi^+$, either $\sigma(\beta)=\beta$ or $-\sigma(\beta)\in \Phi_L^+$. We further introduce the notation $\Phi_s=\Phi_L\setminus \Phi_L^{\sigma}$.
Given $\beta\in \Phi_s$, we define the associated *restricted root* $\bar{\beta}$ by $\bar{\beta}=\beta-\sigma(\beta)$, and let $\bar{\Phi}$ denote the set of all restricted roots, called the *restricted root system*. To any restricted root $\alpha$ is associated an integer $m_{\alpha}$, its *multiplicity*, equal to the number of $\beta\in \Phi_s$ such that $\bar{\beta}=\alpha$.
We fix $\theta$ a Cartan involution of $G$ commuting with $\sigma$ and $K=G^{\theta}$ the corresponding maximal compact subgroup. Let ${\mathfrak{a}}_s$ denote the real vector space $i{\mathfrak{k}}\cap {\mathfrak{t}}_s$. It may naturally be identified with $\mathfrak{Y}(T_s)\otimes {\mathbb{R}}$, where $\mathfrak{Y}(T_s)$ denotes the set of one parameter subgroups of $T_s$, or also with $\mathfrak{Y}(T/(T\cap H))\otimes {\mathbb{R}}$, since $T_s\rightarrow T/(T\cap H)$ is an isogeny.
The restricted root system $\bar{R}\subset \mathfrak{X}(T/(T\cap H))$ defines restricted Weyl chambers in $\mathfrak{a}_s$, and recall that such a Weyl chamber is a fundamental domain for the action of $K$ on $G/H$.
We define $\mathcal{H}$ and $\mathcal{P}$ on $\mathfrak{t}$ by $\mathcal{H}(t)=(t+\sigma(t))/2$ and $\mathcal{P}(t)=(t-\sigma(t))/2$. It provides an isomorphism $$(\mathcal{H},\mathcal{P}):\mathfrak{X}(T)\otimes {\mathbb{R}}\rightarrow \mathfrak{X}(T/T_s)\otimes {\mathbb{R}}\oplus \mathfrak{X}(T/(T\cap H))\otimes {\mathbb{R}}.$$
Data associated to real line bundles and Kähler forms
-----------------------------------------------------
The Picard group of a general spherical variety is described in [@Bri89]. First, the Picard group of horosymmetric Fano manifold $X$ is generated by the $B$-stable prime divisors (which are Cartier). These divisors are exactly closures of codimension one $B$-orbits in $X$ and are of two types: the $G$-stable prime divisors, which are also closures of codimension one $G$-orbits, and the closures in $X$ of codimension one $B$-orbits contained in the open $G$-orbit $G/H$, called *colors*.
The linear relations between these divisors are fully encoded by the *spherical lattice* $\mathcal{M}=\mathfrak{X}(T/(T\cap H))$, which consists of the weights of $B$-eigenvalues in the field of rational functions on $G/H$, and the map $\rho$ which sends a $B$-stable divisor to the element of the dual $\mathcal{M}_{\mathbb{Q}}^*$ induced by restriction of the associated divisorial valuation to the $B$-eigenvalues in $\mathbb{C}(G/H)$. The relations in the Picard group are exactly the $\sum_{D}\rho(D)(m)=0$ for $m\in \mathcal{M}$, where the sum runs over all $B$-stable prime divisors of $X$.
### Case of linearized line bundles
Let $\mathcal{L}$ denote a $G$-linearized line bundle on $X$. Then as in [@DelHoro] we may associate to it
- a Lie algebra character $\chi_{\mathcal{L}}:{\mathfrak{t}}\rightarrow {\mathbb{C}}$ defined by $\exp(t)\cdot \xi=e^{\chi(\mathcal{H}(t))}\xi$ for $\xi\in \mathcal{L}_{eH}$, called the *isotropy character*,
- a *special divisor* $D_{\mathcal{L}}$, defined as the ${\mathbb{R}}$-divisor equal to $1/k$ times the divisor of a $B$-semi-invariant section of some tensor power $\mathcal{L}^k$ whose $B$-weight vanishes on $T_s$.
Note that we changed a bit the definitions with respect to [@DelHoro], and consider the isotropy character directly as the Lie algebra character of $T$ induced by the more natural isotropy subgroup character. By definition, it is an element of $\mathfrak{X}(T/T_s)\otimes {\mathbb{R}}$. The special divisor $D_{\mathcal{L}}$ is $B$-stable and thus decomposes as $D_{\mathcal{L}}=\sum_{D}n_DD$ where $D$ runs over the $B$-stable prime divisors of $X$. Then we associate to $\mathcal{L}$ its *special polytope*, defined as $$\Delta_{\mathcal{L}}=\{m\in\mathcal{M}\otimes{\mathbb{R}};\rho(D)(m)+n_D\geq 0, \forall D\}$$ Furthermore, to a $K$-invariant Hermitian metric $q$ on $\mathcal{L}$, we may associate two functions:
- the *quasipotential* $\phi_q:G\rightarrow {\mathbb{R}}$ defined by $\phi(g)=-2\ln |g\cdot \xi|_{q}$, for some fixed $\xi\in \mathcal{L}_{eH}$,
- and the *toric potential* $u_q:\mathfrak{a}_s\rightarrow {\mathbb{R}}$ defined by $u(a)=\phi(\exp(a))$.
### Forgetting the linearization
Note that for any line bundle $\mathcal{L}$ on $X$, there exists a tensor power $\mathcal{L}^k$ which admits a $G$-linearization, and that two linearizations of the same line bundle differ by a character of $G$ [@KKLV89; @KKLV89]. Modifying the $G$-linearizations of $L$ by a character $\beta$ of $G$ and multiplying the reference element $\xi$ by $s\in{\mathbb{C}}^*$ has the following effects.
The Lie algebra isotropy character does not change, provided we assume that the center $Z(G)$ of $G$ does not act trivially on $X$, which is easily obtained by quotienting $G$ by the fixator of $X$ and we will assume this from now on. Indeed, in this situation, the intersection of $H$ and $Z(G)$ is trivial. The special divisor is changed to the linearly equivalent divisor obtained by adding the relation associated to the element of $\mathcal{M}\otimes {\mathbb{R}}$ induced by $\beta$. The special polytope is translated accordingly. The quasipotential $\phi$ transforms to the function $\phi-2\ln|s\beta(g)|$ and $u$ changes accordingly.
As a result, the isotropy character is a well defined data associated to a line bundle, while the special divisor and special polytope are well defined modulo transformation by a character of $G$. Similarly, the quasipotential and toric potential are well defined up to addition of a constant and a function associated to a character of $G$, and actually depend only on the curvature of $q$, if one allows for *real* characters of $G$, that is, elements of $\mathfrak{X}(T/(T\cap [G,G]))\otimes {\mathbb{R}}$.
### Case of arbitrary real $(1,1)$-classes
Since $X$ is spherical, any real $(1,1)$-class $\Omega$ on $X$ is the class of a real line bundle, that is $\Omega=tc_1(L_1)/k_1+(1-t)c_1(L_2)/k_2$ for some real number $t$, integers $k_1$, $k_2$ and line bundles $L_1$ and $L_2$. It is then straightforward to extend the definitions: the isotropy character of $\Omega$ is $\chi_{\Omega}=t\chi_{L_1}/k_1+(1-t)\chi_{L_2}/k_2$, the special divisor is $D_{\Omega}=tD_{L_1}/k_1+(1-t)D_{L_2}/k_2$ and the special polytope is defined from the special divisor as before. Note again that the isotropy character is well defined, while the last two are defined only modulo the action of a *real* character of $G$ (an element of $\mathfrak{X}(T/(T\cap [G,G]))\otimes {\mathbb{R}}$). Similarly, given a $K$-invariant $(1,1)$-form $\omega\in \Omega$, we can associate to it a quasipotential and a toric potential, well defined up to a constant and the action of a real character of $G$.
In particular, the above constructions apply to Kähler classes and $K$-invariant Kähler forms. The Kähler assumption on the other hand will impose additional conditions such as the convexity of the toric potential $u$.
It is possible to fix the real character by requiring that the derivative of $u$ at the origin is zero in directions coming from the action of the center of $G$. We will fix a normalization of the toric potentials when relevant.
Special polytope, toric polytope and moment polytope {#sec:corresp_pol}
----------------------------------------------------
Let us recall the relations between different polytopes associated to a real line bundle on a horosymmetric manifold (see [@DelHoro] for details).
Assume $\Omega$ is the class of a real line bundle. The polytope $\Delta_{\Omega}^+:=\chi_{\Omega}+\Delta_{\Omega}$ is called the moment polytope of $\Omega$. It coincides with Kirwan’s moment polytope for $\Omega$, or equivalently Brion’s moment polytope for $\Omega$ if $\Omega$ is the class of an ample ${\mathbb{Q}}$-line bundle. Conversely, the special divisor is the image of the moment polytope under the natural projection $\mathfrak{X}(T)\otimes {\mathbb{R}}\rightarrow
\mathfrak{X}(T_s)\otimes{\mathbb{R}}$ and the identification of the latter with $\mathfrak{X}(T/(T\cap H))\otimes {\mathbb{R}}$.
Let $\omega$ be a $K$-invariant Kähler form in $\Omega$. Let us now explain the relation between the toric potential $u$ of $\omega$ and the special polytope $\Delta_{\Omega}$. There is another natural polytope associated to it, the *toric polytope* $\Delta^{{\mathrm{tor}}}_{\Omega}$, defined as the $\bar{W}$-invariant polytope which is the convex hull of the images by the restricted Weyl group $\bar{W}$ of the special polytope. Again, the toric polytope is well defined only up to translation by a real character of $G$. Now the relation between the toric potential of a Kähler form in $\Omega$ and the toric polytope is $$\{d_au;a\in\mathfrak{a}_s\}=\mathrm{Int}(-2\Delta^{{\mathrm{tor}}})$$ up to translation by an element of $\mathfrak{X}(T/(T\cap [G,G]))\otimes {\mathbb{R}}$.
Finally recall the equivalence between the conditions $\Delta^+_{\Omega}\cap \mathfrak{X}(T/[L,L])\otimes{\mathbb{R}}\neq 0$ and $\Delta^+_{\Omega}-\chi_{\Omega}=\Delta_{\Omega}=\Delta^{{\mathrm{tor}}}_{\Omega}\cap \bar{C}^+$.
The connected group of equivariant automorphisms, and Hamiltonian functions {#sec:HamiltHoro}
---------------------------------------------------------------------------
Consider the action by multiplication on the right of the normalizer $N_G(H)$ of $H$ on $G/H$, that is $n\cdot gH=gn^{-1}H$ for $g\in G$, $n\in N_G(H)$. This action obviously factorizes through $N_G(H)/H$ and the action of the neutral component of this group extends to $X$. It may further be identified with the action of the neutral component of the group of $G$-equivariant automorphisms of $X$ (see [@DelKSSV]). From [@BP87], we know that $N_G(H)/H$ is diagonalizable and more precisely, its Lie algebra is identified with $\mathfrak{Y}((T\cap N_G(H))/(T\cap H))\otimes {\mathbb{C}}$.
The manifold $X$ is thus actually equipped with an action of the group $\tilde{G}=G\times N_G(H)/H$, and we let $\tilde{K}$ denote the maximal compact subgroup of $\tilde{G}$ obtained as a product of $K$ and the maximal compact subgroup of $N_G(H)/H$.
Let $\omega$ denote a $\tilde{K}$-invariant Kähler form on $X$. Let $\chi$, $\phi$ and $u$ denote the associated data. Let $V$ denote a holomorphic vector field on $X$ commuting with the action of $G$, and generating a purely non-compact subgroup of automorphisms. Let $f_{V,\omega}$ denote the function on $X$ defined, up to normalization, by $L_V\omega=i\partial\bar{\partial}f_{V,\omega}$. We naturally identify $V$ with an element of $\mathfrak{Y}((T_s \cap N_G(H)))\otimes {\mathbb{R}}\subset \mathfrak{a}_s$.
\[prop:HamiltonianHoro\] The function $f_{V,\omega}$ is up to a normalizing additive constant equal to the $\tilde{K}$-invariant function defined by $f_{V,\omega}(\exp(a)\cdot x)= -d_a u(V)$ for $a\in \mathfrak{a}_s$.
Let $\pi:G\rightarrow G/H$ denote the quotient map. Note that the quasipotential $\phi$ is an $i\partial\bar{\partial}$-potential for $\pi^*\omega$ on $G$. Furthermore, $\pi$ is equivariant for the action of $T_s\cap N_G(H)$ on the right on $G/H$ and $G$, so that $$i\partial\bar{\partial}L_V\phi= L_V\pi^*\omega
=\pi^*L_V\omega= i\partial\bar{\partial}(f_{V,\omega}\circ\pi)$$ It is straightforward, using the definition of the quasipotential, to compute that $(L_V\phi)(ke^ah)=-d_au(V)$ for any $a\in {\mathfrak{a}}_s$, $k\in K$, $h\in H$. As a consequence, the difference between $-d_au(V)$ and $f_{V,\omega}\circ\pi$ is a bounded $K\times H$-invariant pluriharmonic function on $G$. In other words, this difference is a constant.
How to determine special polytopes and isotropy character
---------------------------------------------------------
We will in the following and for our main result only be interested in classes of real line bundles satisfying the following assumption.
\[assumption\_isotropy\] We assume that the isotropy character of the class is trivial on $[\mathfrak{l},\mathfrak{l}]\cap \mathfrak{h}$.
By linearity, it is enough to determine the special polytopes on some real generators of the subspace of $(1,1)$-classes satisfying the above assumption. The $G$-stable prime divisors are the special divisors of the line bundles they determine, since those admit unique $G$-invariant sections (up to a character of $G$). For the same reasons, their isotropy characters are trivial.
From the description of colors of horosymmetric manifolds in [@DelHoro], it is then enough to consider in addition those colors which are the inverse image of the single color $D$ by some $f:G/H\rightarrow G/P_{\alpha}$ where $P_{\alpha}$ is the maximal proper parabolic subgroup of $G$ containing $B^-$ with $\alpha$ as unique root of $P^u$. In this case, there is an obvious $B$-semi-invariant section whose divisor is $f^{-1}(D)$: the pull-back of the unique $B$-semi-invariant section of $\mathcal{O}_{G/P_{\alpha}}(D)$. The latter has $B$-weight equal to the fundamental weight $\varpi$ of $\alpha$. Hence the special divisor associated to $\mathcal{O}_X(\overline{f^{-1}(D)}$ is $f^{-1}(D)+\mathrm{div}(\varpi\circ\mathcal{P})$, and the isotropy character coincides with $\varpi\circ\mathcal{H}$.
The anticanonical line bundle {#sec:anticanonical}
-----------------------------
For the anticanonical line bundle, which is of utmost importance here, the special divisor was obtained from Brion’s anticanonical divisor [@Bri97] in [@DelHoro]: it is the divisor with coefficient zero for colors coming from the symmetric fiber, and $m_D-\sum_{\alpha\in \Phi_{Q^u}\cup \Phi_s^+} \alpha\circ\mathcal{P}(\rho(D))$ for the others, where $m_D=1$ if the divisor is $G$-stable and $m_D=\alpha^{\vee}(\sum_{\beta\in \Phi_s^+\cup \Phi_{Q^u}}\beta)$ if $D$ comes from the simple root $\alpha\in \Phi_{Q^u}$. Its isotropy character is $\sum_{\alpha\in \Phi_{Q^u}}\alpha\circ\mathcal{H}$. Unlike general line bundles, the anticanonical line bundle on $X$ admits a canonical $G$ or $\tilde{G}$-linearization, induced by the natural linearization of the action of the full automorphism group on the tangent bundle. We denote by $\Delta_{ac}$, $\Delta^{{\mathrm{tor}}}_{ac}$, etc the canonical choice of representatives (in the classes of equivalence under translation by a character) of data associated to the anticanonical line bundle, induced by this particular linearization. Similarly, for a metric on the anticanonical line bundle there is a canonical choice of toric potential up to an additive constant. More precisely, given a metric $h$ and a non-zero element $\xi$ in the fiber of $K_X^{-1}$ over $x$, the toric potential of the metric is, up to an additive constant, the function defined by $$u(a)=-\ln \lvert \exp(a)\cdot \xi\rvert^2_h$$ where the group action is the (anti)canonical one.
Recall also the correspondence between Hermitian metrics on the anticanonical line bundle and volume forms. Let $s$ denote a local trivialization of the anticanonical line bundle, and $s^{-1}$ the induced local trivialization of the canonical line bundle. Then $i^{n^2}s^{-1}\wedge \overline{s^{-1}}$ defines a local reference volume form. A Hermitian metric $h$ on $K_X^{-1}$ is associated with the volume form $\Omega_h$ defined locally by $$\Omega_h = \lvert s_1 \rvert_h^2 i^{n^2}s^{-1}\wedge \overline{s^{-1}}$$ and the converse correspondence is obvious by the same formula. In the case of the volume form $\omega^n$ induced by a Kähler form $\omega$, it should be further noticed that the curvature form of the metric associated with $\omega^n$ is none other than the Ricci form $\operatorname{Ric}(\omega)$ of $\omega$.
Monge-Ampère operator
---------------------
We describe here the Monge-Ampère operator on Kähler forms provided the Kähler classes satisfy Assumption \[assumption\_isotropy\].
The differential $d_au $ of a function defined on ${\mathfrak{a}}_s$ is identified with an element of $\mathfrak{X}(T)\otimes{\mathbb{R}}$ via the projection $\mathcal{P}$. Given a fixed choice of basis for ${\mathfrak{a}}_s$, the real Monge-Ampère operator $u\mapsto \det(d_a^2u)$ is well defined as a real valued function for any $a\in {\mathfrak{a}}_s$. Let us denote by $I$ the function defined on ${\mathfrak{a}}_s$ by $$I(a)=\prod_{\alpha\in \Phi_{Q^u}}e^{-2\alpha(a)}
\prod_{\beta\in \Phi_s^+} \sinh(-2\beta(a)).$$ Let $P_{DH}$ denote the Duistermaat-Heckman polynomial defined on $\mathfrak{X}(T)\otimes {\mathbb{R}}$ by $$P_{DH}(p)=\prod_{\alpha\in \Phi_{Q^u}\cup \Phi_s^+}\kappa\left(\alpha,p\right),$$ where $\kappa$ denote the Killing form. Let ${\mathfrak{a}}_s^-$ denote the negative restricted Weyl chamber, defined as the set of all $a\in {\mathfrak{a}}_s$ such that $\beta(a)\leq 0$ for all $\beta\in \Phi_s^+$.
[[@DelHoro]]{} \[thm:MAHoro\] There is a choice of non zero element $\xi$ in the fiber over $x$ of the anticanonical bundle of $G/H$ such that for any $a\in \mathrm{Int}({\mathfrak{a}}_s^-)$, for any $K$-invariant Kähler form satisfying Assumption \[assumption\_isotropy\], we have $$I(a) (\omega^n)_{e^a\cdot x} =
\det(d^2_au) P_{DH}(2\chi-d_au)
i^{n^2}((e^a\cdot\xi)^{-1}\wedge \overline{(e^a\cdot\xi)^{-1}})$$
Note that while [@DelHoro] proves the result only for rational classes, it is immediate to extend the result to arbitrary Kähler forms using our definition of isotropy character and toric potential.
Translating the equations
-------------------------
We have now gathered enough from [@DelHoro] to translate the system of equations in terms of the toric potentials. We place ourselves in the setting of Section \[sec:intro\_horo\].
In particular, each vector field $V_i$ commutes with the action of $G$, and as such is identified with an element of $\mathfrak{Y}(T_s\cap N_G(H))\otimes {\mathbb{R}}$. We denote by $\chi_i$ the isotropy character of $\theta_i$. Let $w$ denote a fixed toric potential for $\gamma$. Let $u_i$ denote a toric potential for $\omega_i$, and assume, without loss of generality, that the function $w+\sum_i u_i$ is the canonical (up to constant) choice of toric potential for a metric on the anticanonical line bundle. We further assume that the toric potentials are defined consistently within a fixed class, i.e. that the differences $ u_i^{\heartsuit}-u_i^{\diamondsuit} $ are bounded functions. In particular, the function on $\mathfrak{a}_s$ corresponding to the difference $\phi_i^t$ is $u_i^t-u_i^{{\mathrm{ref}}}$.
Let $C_i$ be the constants defined by $ C_i=\int_{-(2\Delta_i^{{\mathrm{tor}}}\cap \bar{C}^+)} p(V_i)P_{DH}(2\chi_i-p)dp.$ Let $G_i$ denote the function defined by $ G_i(p)= Ce^{h_i(-p(V_i)+C_i)}P_{DH}(2\chi_i-p)$, where $C$ is the constant which ensures $\int_{-(2\Delta_{\omega_i}^{{\mathrm{tor}}}\cap \bar{C}^+)}G_i(p)dp=1$. The constant $C$ is independent of $i$ by assumption on the functions $h_i$. We set $J(a):=e^{-w(a)}I(a)$. Finally, fix the normalization in the toric potentials of the form $\sum_i u_i^{\heartsuit}$ by the condition $$\int_{\mathfrak{a}_s^-}e^{-\sum_i u_i^{\heartsuit}(a)}J(a)da = 1.$$ Using the results on horosymmetric manifolds recalled earlier in the section and these notations, we obtain the precise relation between the system of complex Monge-Ampère equations and the system of real Monge-Ampère equations .
\[thm:translation\_horo\] Assume the functions $\phi_i^t$ are solutions to Equation \[eq:ComplexContPath\], then the functions $u_i^t$ are solutions, on $\mathrm{Int}({\mathfrak{a}}_s^-)$, to the system of equations $$G_i(d_au_i^t)\det(d^2_au_i^t)
= e^{-\sum_{l=1}^{k}(tu_l^t+(1-t)u_l^{{\mathrm{ref}}})(a)}
J(a).$$
Let $\xi$ be the element of $(K_X^{-1})_x$ produced by Theorem \[thm:MAHoro\], then by the same result we have for any $i$, $$I(a)((\omega_i^t)^n)_{e^a\cdot x}=
\det(d^2_au_i^t) P_{DH}(2\chi_i-d_au_i^t)i^{n^2}((e^a\cdot\xi)^{-1}\wedge \overline{(e^a\cdot\xi)^{-1}}).$$ By Proposition \[prop:HamiltonianHoro\], and the choice of normalization of Hamiltonian we have $f_{V_i,\omega_i^t}(e^a\cdot x)=-d_au_i^t(V_i)+C_i$ for the constant $C_i$ independent of $t$ defined before the statement of the theorem.
We now turn to the right hand side of Equation \[eq:ComplexContPath\]. By definition of $\theta_0$, by the assumption on $w+\sum_i u_i^{\heartsuit}$, and by the correspondences recalled in Section \[sec:anticanonical\], $$(\theta_0^n)_{e^a\cdot x} =
e^{-w(a)+\sum_j u_j^{{\mathrm{ref}}}(a)}i^{n^2}((e^a\cdot\xi)^{-1}\wedge \overline{(e^a\cdot\xi)^{-1}}).$$ By definition of $\phi_i^t$, we then have $$(e^{-t\sum \phi_j}\theta_0^n)_{e^a\cdot x} =
e^{-w(a)}e^{t\sum_j u_j^t(a)+(1-t)\sum_j u_j^{{\mathrm{ref}}}(a)}i^{n^2}((e^a\cdot\xi)^{-1}\wedge \overline{(e^a\cdot\xi)^{-1}})$$ and it concludes the proof.
We are now able to carry out the proof of Theorem \[thm:cpld\_can\_horo\].
It follows from the successive application of Theorem \[thm:ReductionToC0\], of Theorem \[thm:translation\_horo\], and of Theorem \[thm:C0Estimates\], then a few manipulations on the conditions. We thus only need to check that the data provided by Theorem \[thm:translation\_horo\] verify the assumptions of Theorem \[thm:C0Estimates\].
For this we identify the remaining data: we are working on the vector space ${\mathfrak{a}}={\mathfrak{a}}_s$, the cone is ${\mathfrak{a}}^+={\mathfrak{a}}_s^-$, and the polytopes $\Delta_i$ are the $\Delta_i=-2(\Delta^{{\mathrm{tor}}}_{\theta_i}\cap \bar{C}^+)$. The functions $g_k$ is a product of a polynomial with the exponential of an affine function hence satisfy Assumption \[Jassumption\] (\[gass\_integrable\]). For the function $J$, note that $$j_{\infty}=v_{-2(\Delta_{\gamma}^{{\mathrm{tor}}}\cap \bar{C}^+)}+2\sum_{\alpha\in \Phi_{Q^u}\cup \Phi_s^+}\alpha.$$ It is thus continuous and finite on ${\mathfrak{a}}^+$, $j-j_{\infty}$ is bounded from below on the whole cone ${\mathfrak{a}}^+$ since $\ln\sinh(x)\leq x$. Finally, Assumption \[Jassumption\] (\[Jass\_proper\]) is satisfied since we always have, by the description of the anticanonical divisor, $2\sum_{\alpha\in \Phi_{Q^u}\cup \Phi_s^+}\alpha\circ\mathcal{P} \in \mathrm{Int}(2\Delta_{ac}^{{\mathrm{tor}}})=\mathrm{Int}(2\Delta_{\gamma}^{{\mathrm{tor}}}+2\sum_i\Delta_{\theta_i}^{{\mathrm{tor}}})$.
Examples {#sec:examples}
========
We will illustrate in this section some of the consequences of our main result, but this also serves us as a pretext to present the natural horosymmetric structure on several Fano threefolds. This is even easier now that the connected automorphism groups of Fano threefolds have been completely determined in [@CPS18].
Recall that there are five homogeneous Fano threefolds: the quadric $Q$, products of projective spaces ${\mathbb{P}}^3$, ${\mathbb{P}}^2\times {\mathbb{P}}^1$, $({\mathbb{P}}^1)^3$ and the full flag threefold, usually denoted by $W$. It turns out from examination of the remaining threefolds with infinite connected automorphism groups that the non-toric, possibly spherical threefold, necessarily have a connected *reductive* automorphism group isogenous to $\operatorname{SL}_2\times {\mathbb{C}}^*$. Any horosymmetric threefold under the action of $\operatorname{SL}_2\times {\mathbb{C}}^*$ is either symmetric or rank two horospherical. It is not hard to check that any horospherical *threefold* has to be toric.
On the other hand, toric manifolds in general are horospherical under the action of a maximal connected *reductive* automorphism group, and considering this more precise horospherical structure reduces the rank of the action unless the latter group is a torus. For toric Fano threefolds which are not homogeneous, the possible connected reductive automorphism groups are in the list: $\operatorname{SL}_3\times {\mathbb{C}}^*$, $\operatorname{SL}_2^2\times {\mathbb{C}}^*$, $\operatorname{SL}_2\times ({\mathbb{C}}^*)^2$.
In this section we will determine the symmetric Fano threefolds, as well as the best horospherical structures on toric Fano threefolds. For the latter, we will focus on the Fano threefolds equipped with a rank one horospherical structure, and only mention that all the others either appear in the list of rank $2$ smooth and Fano embeddings of $\operatorname{SL}_2/U\times {\mathbb{C}}^*$ in Pasquier’s thesis [@Pas06 Chapitre 7], or are products of ${\mathbb{P}}^1$ with a toric Del Pezzo surface. We will then study some rank one Fano $\operatorname{SL}_2\times \operatorname{SL}_3$-horospherical *fourfolds* including the toric example of Fano manifold with no Kähler-Einstein metric but a pair of coupled Kähler-Einstein metrics obtained by the second author in [@Hul].
In this section, we will often identify Fano threefolds by their identifier as used for example in [@CPS18]. We will use for this the notation $F^3_{I}$ for the threefold with identifier $I$. For example, $F^3_{2.33}$ denotes the blowup of the projective space ${\mathbb{P}}^3$ along a line. Note that the number before the period in the identifier is the Picard rank of the threefold.
In this section, the group $G$ will be a product of special linear groups and of one-dimensional tori. We fix as maximal torus $T$ the subgroup of diagonal matrices, and as Borel subgroup $B$ the subgroup of upper-diagonal matrices. The positive root system $\Phi^+\subset \mathfrak{X}(T)$ will be the one associated to these choices.
A horospherical homogeneous space $G/H$ is fully determined by the data of the parabolic $P=N_G(H)$ and of the spherical lattice $\mathcal{M}\subset \mathfrak{X}(P)$. Furthermore, Pasquier obtained very explicit criterions to determine if a horospherical embedding is Fano and smooth. These allow to determine the horospherical structures on smooth Fano threefolds easily. For symmetric threefolds, we will use the classification of Fano symmetric varieties with low rank obtained by Ruzzi [@Ruz12].
Finally, we will work in this section rather with moment polytope than toric polytopes, and refer to Section \[sec:corresp\_pol\] for the correspondence with the toric polytope which allows to apply the results as stated in Section \[sec:intro\_horo\].
Warm up: Hirzebruch surfaces and their blow-down
------------------------------------------------
Before considering threefolds, let us first consider the non-homogeneous horospherical surfaces. The only possibility for the group action is $G=\operatorname{SL}_2$. It is of rank one, type $A_1$, we use the notations $\Phi^+=\{\alpha\}$, $\mathfrak{X}(T)={\mathbb{Z}}\frac{\alpha}{2}$ and have (up to a constant) $P_{DH}(x\frac{\alpha}{2})=x$. The normalizer $P$ of $H$ has to be the opposite Borel subgroup (the lower triangular matrices), but there remains a choice of the generator $\mu$ of the lattice $\mathcal{M}\subset \mathfrak{X}(P)=\mathfrak{X}(T)$. We use the notation $\mu=k\alpha/2$ for some positive integer $k$.
There are always two choices of $G$-equivariant embeddings of $G/H$ in this situation, for each $k$: one toroidal embedding $\mathbb{F}_k$ and one colored embedding ${\mathbb{P}}(1,1,k)$. The toroidal ones are none other than the Hirzebruch surfaces. In particular, $\mathbb{F}_1$ is the blow up of ${\mathbb{P}}^2$ at one point, and for higher $k$, $\mathbb{F}_k$ is not Fano. On the other hand the colored are, as the notation suggests, weighted projective planes. They are Fano, but singular as soon as $k\geq 2$. The anticanonical moment polytope of ${\mathbb{P}}(1,1,k)$ is $\Delta=[0,(1+k/2)\alpha]\subset \mathfrak{X}(T)\otimes {\mathbb{R}}.$
Set $M:=1+k/2$, $V(a,b)=\int_0^Mt(at+b) dt$, and $X(a,b)=\int_0^M t^2(at+b)dt$ for $a,b\in {\mathbb{R}}$. The necessary condition for existence of Mabuchi metrics is satisfied on these weighted projective planes if for the unique pair $(a,b)$ such that $X(a,b)=V(a,b)=V(0,1)$, the affine function $t\mapsto at+b$ is positive on $[0,M]$.
A straightforward computation using $V(a,b)=aM^3/3+bM^2/2$, $X(a,b)=aM^4/4+bM^3/3$ shows that $b=3(3M-4)/M$, $a=6(3-2M)/M^2$. We recover that $a=0$ in the case $k=1$ which corresponds to ${\mathbb{P}}^2$, and in this situation $b=1>0$ so that unsurprisingly there exists a Kähler-Einstein metric. In the cases $k>1$, one has $0<-b/a<M$, so that the corresponding affine function is not positive on the polytope, that is there cannot be any reasonable singular Mabuchi metric on ${\mathbb{P}}(1,1,k)$.
Fano rank one $\operatorname{SL}_2\times \operatorname{SL}_2$-horospherical threefolds
--------------------------------------------------------------------------------------
The group $\operatorname{SL}_2\times \operatorname{SL}_2$ is of rank two, type $A_1\times A_1$. We have $\Phi^+=\{\alpha_1,\alpha_2\}$ and $\mathfrak{X}(T)={\mathbb{Z}}\frac{\alpha_1}{2}+{\mathbb{Z}}\frac{\alpha_2}{2}$. The only generalized flag manifold for the action of $\operatorname{SL}_2\times \operatorname{SL}_2$ is ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The parabolic $P$ is the Borel subgroup consisting of pairs of lower triangular matrices (opposite to $B$), thus $\mathfrak{X}(P)=\mathfrak{X}(T)$. For any horospherical subgroup $H$ with normalizer equal to $P$, we have have the same Duistermaat-Heckman, and its expression is up to a multiplicative constant $P_{DH}(x\frac{\alpha_1}{2}+y \frac{\alpha_2}{2})=xy$. From the work of Pasquier [@Pas06; @Pas08], we deduce the list of choices of $\mathcal{M}$ that allow smooth and Fano embeddings of $G/H$ and list these. Up to obvious symmetries, we have the following possibilities.
The first possibility is the product case, if $\mathcal{M}={\mathbb{Z}}\alpha_1/2$. In this case, $G/H$ is the product ${\mathbb{C}}^2\setminus \{0\}\times {\mathbb{P}}^1$ and the smooth and Fano embeddings are the products $F^3_{3.28}=\mathbb{F}_1\times {\mathbb{P}}^1$ and $F^3_{2.34}={\mathbb{P}}^2\times {\mathbb{P}}^1$. Their moment polytopes are $\{t\alpha_1/2+\alpha_2; t\in [t_-,3]\}$ where $t_-=0$ for $F^3_{2.34}$ and $t_-=1$ for $F^3_{3.28}$.
The second possibility is $\mathcal{M}={\mathbb{Z}}\frac{\alpha_1+\alpha_2}{2}$. The corresponding horospherical homogeneous space admits a unique smooth and Fano embedding: the Fano threefold $F^3_{3.31}$ which is the $\operatorname{SL}_2\times \operatorname{SL}_2$-equivariant ${\mathbb{P}}^1$-bundle ${\mathbb{P}}(\mathcal{O}\oplus\mathcal{O}(1,1))$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The moment polytope is $\{t(\alpha_1+\alpha_2)/2;t\in [1,3]\}$.
The last possibility for $\mathcal{M}$ is ${\mathbb{Z}}\frac{\alpha_1-\alpha_2}{2}$. There are then three (up to symmetry) smooth and Fano embeddings: the fully colored one, which is the projective space ${\mathbb{P}}^3$, the single colored $F^3_{2.33}$ and the toroidal $F^3_{3.25}$. The geometrical description of the action is straighforward: consider ${\mathbb{C}}^4$ as the sum ${\mathbb{C}}^2\oplus {\mathbb{C}}^2$ and the corresponding componentwise action of $\operatorname{SL}_2\times \operatorname{SL}_2$. The induced action on ${\mathbb{P}}^3$ is the rank one horospherical structure, with two colored closed orbits which are two disjoint lines: the natural inclusions of the projectivization of both ${\mathbb{C}}^2$ summands. Then $F^3_{2.33}$ is the blow-up of ${\mathbb{P}}^3$ along one of these lines, while $F^3_{3.25}$ is the blow up along both lines. The moment polytopes are $\{(4-t)\alpha_1/2+t\alpha_2/2; t\in [t_-,t_+]\}$ where $(t_-,t_+)$ is $(0,4)$ for ${\mathbb{P}}^3$, $(0,3)$ for $F^3_{2.33}$, and $(1,3)$ for $F^3_{3.25}$.
Let us now see how the conditions for existence of Kähler-Einstein metrics or Mabuchi metrics on these manifolds reduce to simple computations in one variable in this setting. Note that the results here are not new (see [@NSY17]). The manifold $F^3_{3.25}$ is known to be Kähler-Einstein. This follows from the obvious symmetry of its moment polytope and of $P_{DH}$ under the exchange of $\alpha_1$ and $\alpha_2$.
For Mabuchi metrics on $F^3_{3.31}$, we consider an arbitrary affine function on the polytope, which we may write $t\mapsto at+b$ under the parametrization used before. Then the barycenter involved in the search for Mabuchi metric is $$\mathbf{bar}(a,b):=\frac{\int_1^3 (at+b)t^3dt}{\int_1^3 (at+b)t^2dt}\frac{\alpha_1+\alpha_2}{2}=\frac{363a+150b}{300a+130b}(\alpha_1+\alpha_2)$$ It follows first that it is different from $\alpha_1+\alpha_2$ if $a=0$ *i.e.* there are no Kähler-Einstein metrics. More precisely, the barycenter is $\alpha_1+\alpha_2$ if and only if $63a+20b=0$, and under this condition the affine function $t\mapsto at+b$ is positive on $[1,3]$, so there are Mabuchi metrics on $F^3_{3.31}$.
Similarly, for $F^3_{2.33}$ define a barycenter depending on a choice of affine function $t\mapsto at+b$ by $$\mathbf{bar}(a,b):=2\alpha_1+\frac{\int_0^3 (at+b)t^2(4-t)dt}{\int_0^3 (at+b)t(4-t)dt}\frac{\alpha_2-\alpha_1}{2} = 2\alpha_1+\frac{72a+35b}{10(7a+4b)}(\alpha_2-\alpha_1)$$ The barycenter coincides with $\alpha_1+\alpha_2$ if and only if $2a-5b=0$, and under this condition the affine function $t\mapsto at+b$ is positive on $[0,3]$, so there are Mabuchi metrics on $F^3_{2.33}$ as well.
Fano rank one $\operatorname{SL}_3$-horospherical threefolds
------------------------------------------------------------
The group $\operatorname{SL}_3$ is of rank two, type $A_2$. We have $\Phi^+=\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}$ and $\mathfrak{X}(T)={\mathbb{Z}}\frac{\alpha_1+2\alpha_2}{3}+{\mathbb{Z}}\frac{2\alpha_1+\alpha_2}{3}.$
The only flag manifold of dimension $2$ under $\operatorname{SL}_3$ is the projective space ${\mathbb{P}}^2$. We may thus assume that $P$ is conjugate to the stabilizer of a plane in ${\mathbb{C}}^3$. We assume that $-\alpha_1$ is a root of $P^u$. The characters of $P$ are those characters in the weight lattice of $\operatorname{SL}_3$ which are orthogonal to $\alpha_2$ with respect to the Killing form. In other words, this is the one dimensional ${\mathbb{Z}}(2\alpha_1+\alpha_2)/3$. The Duistermaat-Heckman polynomial in this section will be $P_{DH}(x\frac{\alpha_1+2\alpha_2}{3}+y \frac{2\alpha_1+\alpha_2}{3})=xy(x+y) $. Using again the work of Pasquier, we deduce the possible horospherical homogeneous spaces with smooth and Fano embeddings, as well as these embeddings.
The first case is $\mathcal{M}={\mathbb{Z}}(2\alpha_1+\alpha_2)/3$. There are two smooth and Fano embeddings of the corresponding horospherical homogeneous space. The colored one is isomorphic to ${\mathbb{P}}^3$, with moment polytope $\{t(2\alpha_1+\alpha_2)/3; t\in[0,4]\}$. The toroidal one is the ${\mathbb{P}}^1$-bundle $F^3_{2.35}={\mathbb{P}}(\mathcal{O}\oplus\mathcal{O}(1))$ over ${\mathbb{P}}^2$, with moment polytope $\{t(2\alpha_1+\alpha_2)/3; t\in[2,4]\}$.
The second case is $\mathcal{M}={\mathbb{Z}}(4\alpha_1+2\alpha_2)/3$. In this case, the toroidal embedding $F^3_{2.36}={\mathbb{P}}(\mathcal{O}\oplus\mathcal{O}(2))$ over ${\mathbb{P}}^2$, with moment polytope $\{t(2\alpha_1+\alpha_2)/3; t\in[1,5]\}$ is the only smooth and Fano embedding. It is easy to check that $F^3_{2.36}$ does not admit any Mabuchi metric. We consider for this the barycenter, depending on the variables $a$ and $b$ defining the affine function $t\mapsto at+b$, defined by $$\mathbf{bar}(a,b):=\frac{\int_1^5(at+b)t^3dt}{\int_1^5(at+b)t^2dt}\frac{2\alpha_1+\alpha_2}{3}=\frac{781a+195b}{585a+155b}(2\alpha_1+\alpha_2)$$ The above barycenter is equal to $2\alpha_1+\alpha_2$ if and only if $196a+40b=0$, but under this condition, the affine function $t\mapsto at+b$ is not positive on the segment $[1,5]$.
However if we consider the variant of Mabuchi metrics given by considering powers of affine functions, we get the existence of a canonical metric of this type on $F^3_{2.36}$ as soon as we consider the squares of affine functions. To see this, consider $$\mathbf{bar}^{(2)}(a,b):=\frac{\int_1^5(at+b)^2t^3dt}{\int_1^5(at+b)^2t^2dt}\frac{2\alpha_1+\alpha_2}{3}=\frac{3255a^2+1562ab+195b^2}{2343a^2+1170ab+155b^2}(2\alpha_1+\alpha_2)$$ The above barycenter is equal to $2\alpha_1+\alpha_2$ if and only if the second order equation $912a^2+398ab+40b^2=0$ is satisfied. Choosing (without loss of generality) $b=912$, there are two solutions $a_{\pm}=-199\pm\sqrt{3121}$. The affine function $a_-t+b$ changes sign on $[1,5]$, but not $a_+t+b$ which remains positive.
Symmetric Fano threefolds under the action of $\operatorname{SL}_2\times {\mathbb{C}}^*$
----------------------------------------------------------------------------------------
The group $G=\operatorname{SL}_2\times {\mathbb{C}}^*$ is of rank two, type $A_1$. We have $\Phi^+=\{\alpha\}$ and $\mathfrak{X}(T)={\mathbb{Z}}\frac{\alpha}{2}\oplus {\mathbb{Z}}f$ (where $f$ is a generator of $\mathfrak{X}(T/[G,G])$).
There is only one possible rank two involution on $G$ (up to conjugation): $\sigma(g,s)=((g^t)^{-1},s^{-1})$ for $(g,s)\in \operatorname{SL}_2\times {\mathbb{C}}^*$. Up to quotienting the ${\mathbb{C}}^*$ factor so that it acts effectively, there are three possible symmetric subspaces: $H_A:=\operatorname{SO}_2\times \{1\}$, $H_B:=N(\operatorname{SO}_2)\times \{1\}$, and the intermediate case $H_C:=\{(g,g\sigma(g)^{-1}/I_2),g\in N(\operatorname{SO}_2)\}$. For the last subgroup, $I_2$ denotes the identity matrix in $\operatorname{SL}_2$, and the quotient is well defined since for any $g\in N(\operatorname{SO}_2)$, $g\sigma(g)^{-1}= \pm I_2 \in Z(\operatorname{SL}_2)$. In any case, the Duistermaat-Heckman polynomial is given (as always up to a multiplicative constant) by $P_{DH}(x\frac{\alpha}{2}+yf)=x$.
From Ruzzi [@Ruz12] we know the smooth and Fano $G$-equivariant embeddings of these symmetric spaces. This provides 12 Fano threefolds. We draw in Figures \[fig:RA\], \[fig:RB\] and \[fig:RC\] the moment polytopes, and include the data of the spherical lattice $\mathcal{M}$ as the dotted grid, while the dashed line represents the single wall of the positive restricted Weyl chamber in $\mathcal{M}\otimes {\mathbb{R}}$. We denote by RA1, RA2, RA3 the Fano embeddings of $G/H_A$, by RB1, RB2, RB3 the Fano embeddings of $G/H_B$, and by RC1 to RC6 the Fano embeddings of $G/H_C$ as in the figures.
(-.5,-.5) node[RA1]{}; (-1,-2) grid\[xstep=1,ystep=1\] (3,2);
(0,0) node[$\bullet$]{}; (0,0) – (1,0) node\[above right\][$\alpha$]{};
(0,-2) – (0,2);
(0,-1) – (2,-1) – (2,1) – (0,1) – cycle;
(-.5,-.5) node[RA2]{}; (-1,-2) grid\[xstep=1,ystep=1\] (3,2);
(0,0) node[$\bullet$]{}; (0,0) – (1,0) node\[above right\][$\alpha$]{};
(0,-2) – (0,2);
(0,-1) – (3,-1) – (1,1) – (0,1) – cycle;
(-.5,-.5) node[RA3]{}; (-1,-2) grid\[xstep=1,ystep=1\] (3,2);
(0,0) node[$\bullet$]{}; (0,0) – (1,0) node\[above right\][$\alpha$]{};
(0,-2) – (0,2);
(0,-1) – (2,-1) – (2,0) – (1,1) – (0,1) – cycle;
(-.5,-.5) node[RB1]{}; (-1\*[1]{},-2\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (3\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,0) node\[above right\][$\alpha$]{};
(0,-2\*[1]{}) – (0,2\*[1]{});
(0,-[1]{}) – (3\*[1]{}/2,-[1]{}) – (3\*[1]{}/2,[1]{}) – (0,[1]{}) – cycle;
(-.5,-.5) node[RB2]{}; (-1\*[1]{},-2\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (3\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,0) node\[above right\][$\alpha$]{};
(0,-2\*[1]{}) – (0,2\*[1]{});
(0,-[1]{}) – (5\*[1]{}/2,-[1]{}) – ([1]{}/2,[1]{}) – (0,[1]{}) – cycle;
(-.5,-.5) node[RB3]{}; (-1\*[1]{},-2\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (3\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,0) node\[above right\][$\alpha$]{};
(0,-2\*[1]{}) – (0,2\*[1]{});
(0,-[1]{}) – (3\*[1]{}/2,-[1]{}) – (3\*[1]{}/2,0) – ([1]{}/2,[1]{}) – (0,[1]{}) – cycle;
(-.5,.5) node[RC1]{}; (-2\*[1]{},-3\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (2\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,-[1]{}/2) node\[above right\][$\alpha$]{};
(-2\*[1]{},-2\*[1]{}) – (2\*[1]{},2\*[1]{});
(-3\*[1]{}/2,-3\*[1]{}/2) – (3\*[1]{}/2,-3\*[1]{}/2) – (3\*[1]{}/2,3\*[1]{}/2) – cycle;
(-.5,.5) node[RC2]{}; (-2\*[1]{},-3\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (2\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,-[1]{}/2) node\[above right\][$\alpha$]{};
(-2\*[1]{},-2\*[1]{}) – (2\*[1]{},2\*[1]{});
(-[1]{}/2,-[1]{}/2) – (3\*[1]{}/2,-5\*[1]{}/2) – (3\*[1]{}/2,3\*[1]{}/2) – cycle;
(-.5,.5) node[RC3]{}; (-2\*[1]{},-3\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (2\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,-[1]{}/2) node\[above right\][$\alpha$]{};
(-2\*[1]{},-2\*[1]{}) – (2\*[1]{},2\*[1]{});
(-[1]{}/2,-[1]{}/2) – (3\*[1]{}/2,-5\*[1]{}/2) – (3\*[1]{}/2,-[1]{}/2) – ([1]{}/2,[1]{}/2) – cycle;
(-.5,.5) node[RC4]{}; (-2\*[1]{},-2\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (2\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,-[1]{}/2) node\[above right\][$\alpha$]{};
(-2\*[1]{},-2\*[1]{}) – (2\*[1]{},2\*[1]{});
(-3\*[1]{}/2,-3\*[1]{}/2) – ([1]{}/2,-3\*[1]{}/2) – (3\*[1]{}/2,-[1]{}/2) – (3\*[1]{}/2,3\*[1]{}/2) – cycle;
(-.5,.5) node[RC5]{}; (-2\*[1]{},-2\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (2\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,-[1]{}/2) node\[above right\][$\alpha$]{};
(-2\*[1]{},-2\*[1]{}) – (2\*[1]{},2\*[1]{});
(-[1]{}/2,-[1]{}/2) – ([1]{}/2,-3\*[1]{}/2) – (3\*[1]{}/2,-3\*[1]{}/2) – (3\*[1]{}/2,3\*[1]{}/2) – cycle;
(-.5,.5) node[RC6]{}; (-2\*[1]{},-2\*[1]{}) grid\[xstep=[1]{},ystep=[1]{}\] (2\*[1]{},2\*[1]{});
(0,0) node[$\bullet$]{}; (0,0) – ([1]{}/2,-[1]{}/2) node\[above right\][$\alpha$]{};
(-2\*[1]{},-2\*[1]{}) – (2\*[1]{},2\*[1]{});
(-[1]{}/2,-[1]{}/2) – ([1]{}/2,-3\*[1]{}/2) – (3\*[1]{}/2,-3\*[1]{}/2) – (3\*[1]{}/2,-[1]{}/2) – ([1]{}/2,[1]{}/2) – cycle;
Given their moment polytopes, it is straightforward to compute (using [@Bri89]) their Picard rank and anticanonical degree. It is often enough to identify the Fano threefolds. Namely, this allows to immediately identify RA2, RA3, RB2, RB3, RC1, RC2, and RC3. On the other hand, it is clear from the combinatorial data that RA1 and RB1 are the products of ${\mathbb{P}}^1$ with the wonderful compactification of their two dimensional symmetric factor (respectively $\operatorname{SL}_2/\operatorname{SO}_2\subset {\mathbb{P}}^1\times {\mathbb{P}}^1$ and $\operatorname{SL}_2/N_{\operatorname{SL}_2}(\operatorname{SO}_2)\subset {\mathbb{P}}^2$). For RC4, it follows from [@CPS18] that $F^3_{2.29}$ is the only threefold with appropriate Picard rank, anticanonical degree, and that admits an *almost effective* action of $\operatorname{SL}_2\times {\mathbb{C}}^*$. For the last two, let us provide a geometrical description which allows to identify them.
Consider the standard quadratic form $z_1^2+z_2^2+z_3^2+z_4^2+z_5^2$ on ${\mathbb{C}}^5$. Write ${\mathbb{C}}^5={\mathbb{C}}^3\times {\mathbb{C}}^2$ and note that the above quadratic form is the sum of the standard quadratic forms on both factors. Consider the induced action of $P(O_3\times O_2)\simeq \operatorname{GL}_2$ on the quadric in ${\mathbb{P}}^4$. It admits five orbits: the closed orbit given by inclusion of the $1$-dimensional quadric from the first factor, the two fixed points given by the two components of the $0$-dimensional quadric from the second factor, the codimension $1$ orbit consisting of elements which are the image of products of non-zero elements of both affine quadrics, and finally the open orbit, isomorphic to $G/H_C$. This action coincides with the structure of $G/H_C$-embedding of RC1. From the caracterization of equivariant morphisms between $G/H_C$-embeddings [@Kno91 Theorem 4.1], we deduce that RC5 is the blowup of RC1 at one of the fixed point, while RC6 is the blowup of RC1 at both fixed points. Note also that RC4 is the blowup of RC1 along the $1$-dimensional quadric.
We summarize the correspondence in the following table:\
RA1 RA2 RA3 RB1 RB2 RB3 RC1 RC2 RC3 RC4 RC5 RC6
-------------- -------------- ------------- -------------- -------------- -------------- ------- ------------------ -------------- -------------- -------------- --------------
$F^3_{3.27}$ $F^3_{3.31}$ $F^3_{4.8}$ $F^3_{2.34}$ $F^3_{2.36}$ $F^3_{3.22}$ $Q^3$ ${\mathbb{P}}^3$ $F^3_{2.35}$ $F^3_{2.29}$ $F^3_{2.30}$ $F^3_{3.19}$
Note in particular that RA3, RB3, RC4, RC5, RC6 are neither homogeneous nor toric, and that $\operatorname{SL}_2\times {\mathbb{C}}^*$ is up to isogeny a maximal connected reductive automorphism group of each.
To illustrate our results on these examples, let us consider the existence of Mabuchi metrics on those threefolds. It is known that RC4 and RC6 admit Kähler-Einstein metrics. For the other three, they are the lowest dimensional examples where our results give new existence or non-existence for Mabuchi metrics.
For RC5, we write points in the moment polytope as $xf+y\alpha$ where $x$ and $y$ are parameters, $f$ is orthogonal to $\alpha$ and the vertices of the moment polytope are given by their coordinates $(x,y)$ as the elements of the set $\{(-3,0),(0,3),(1,2),(1,0) \}$. Searching for a Mabuchi metric amounts to searching for an affine function $x\mapsto ax+b$ (note the dependence only in $x$), positive on $[-3,1]$, such that $$\left(\int_{-3}^0\int_0^{3+x}+\int_0^1\int_0^{3-x}\right) \left((xf+(y-1)\alpha)(ax+b)y\right) dy dx=x_bf+y_b\alpha$$ satisfies $x_b=0$ and $y_b>0$. The condition on $x_b$ imposes $49a-20b=0$, and under this condition, the affine function vanishes at $-49/20\in [-3,1]$. As a consequence, RC5 does not admit any Mabuchi metric.
Consider now RA3, we again fix coordinates $(x,y)$ with respect to a basis $(f,\alpha)$ where $f$ is orthogonal to $\alpha$ and such that the coordinates of the vertices of the moment polytope are $(-1,0)$, $(-1,2)$, $(0,2)$, $(1,1)$ and $(1,0)$. We search for an affine function $x\mapsto ax+b$, positive on $[-1,1]$, such that $$\left(\int_{-1}^0\int_0^{2}+\int_0^1\int_0^{2-x}\right)\left((xf+(y-1)\alpha)(ax+b)y\right) dy dx=x_bf+y_b\alpha$$ satisfies $x_b=0$ and $y_b>0$. The condition $x_b=0$ implies $112a-65b=0$ and the other conditions are then satisfied, so RA3 admits a Mabuchi metric.
Similar verifications show that RB3 also admits a Mabuchi metric.
Rank one $\operatorname{SL}_2\times \operatorname{SL}_3$-horospherical Fano fourfolds and Futaki’s example
----------------------------------------------------------------------------------------------------------
The group $\operatorname{SL}_2\times \operatorname{SL}_3$ is of rank three and of type $A_1\times A_2$. We have $\Phi^+=\{\alpha_1,\alpha_2,\alpha_3,\alpha_2+\alpha_3\}$ and $\mathfrak{X}(T)={\mathbb{Z}}\varpi_1\oplus {\mathbb{Z}}\varpi_3+{\mathbb{Z}}\varpi_2$ where $\varpi_1=\frac{\alpha_1}{2}$, $\varpi_2=\frac{2\alpha_2+\alpha_3}{3}$ and $\varpi_3=\frac{\alpha_2+2\alpha_3}{3}$.
We are interested in rank one, four dimensional horospherical homogeneous spaces that do not split as products. As a consequence, we assume that $P$ is the parabolic with $\Phi_{P^u}=\{-\alpha_1,-\alpha_2\}$, and that $\mathcal{M}={\mathbb{Z}}e \subset \mathfrak{X}(P)={\mathbb{Z}}\varpi_1+{\mathbb{Z}}\varpi_2$ where $e=k_1\varpi_1+k_2\varpi_2$ for some non-zero relative integers $k_1$ and $k_2$. From Pasquier’s work, there are smooth and Fano embeddings exactly for $(k_1,k_2)\in\{\pm(1,1),\pm(-1,1),\pm(1,2),\pm(-1,2)\}$, including, for each, the toroidal ${\mathbb{P}}^1$-bundle ${\mathbb{P}}(\mathcal{O}\oplus\mathcal{O}(k_1,k_2))$ over ${\mathbb{P}}^1\times {\mathbb{P}}^2$.
Let us concentrate first on the case $(k_1,k_2)=(-1,1)$, whose toroidal embedding corresponds to Hultgren’s example [@Hul]. The toroidal embedding $X={\mathbb{P}}(\mathcal{O}\oplus\mathcal{O}(1,-1))$ has Picard rank three. By the description of the Picard group for horospherical varieties, and the criterion for ampleness, we can choose parameters $(s_1,s_2,s_3)$ for $0\leq s_1\leq s_2\leq s_3$ parametrizing semi-positive real line bundles $L(s_1,s_2,s_3)$ (with associated class of $B$-stable divisor represented by $(s_3-s_1)D_+ +(s_2-s_3)D_- + s_3D_{\alpha_1}$). The moment polytope of $L(s_1,s_2,s_3)$ is then $$\Delta(s_1,s_2,s_3)=\{s_3\varpi_2+t(\varpi_1-\varpi_2); s_1\leq t \leq s_2 \}.$$ In particular, the anticanonical moment polytope is $\Delta(1,3,5)$.
(-3,-1) grid\[xstep=1,ystep=1\] (6,6); (0,0) node[$\bullet$]{}; (0,0) node\[below left\][$0$]{}; (1,4) – (3,2) node\[above right\][$\Delta(s_1,s_2,s_3)$]{}; (0,5) – (5,0); (1,0) – (1,4); (3,0) – (3,2); (0,5) node[$\bullet$]{}; (0,5) node\[above left\][$s_3\varpi_2$]{}; (5,0) node[$\bullet$]{}; (5,0) node\[below\][$s_3\varpi_1$]{}; (1,0) node[$\bullet$]{}; (1,0) node\[below\][$s_1\varpi_1$]{}; (3,0) node[$\bullet$]{}; (3,0) node\[below\][$s_2\varpi_1$]{}; (5,5) node\[above right\][$\mathfrak{X}(P)$]{}; (1,-1) – (-3,3) node\[above right\][$\mathcal{M}\otimes {\mathbb{R}}$]{};
The Duistermaat-Heckamn polynomial on $\mathfrak{X}(P)\otimes{\mathbb{R}}$ is, up to a multiplicative constant, given by $P_{DH}(x_1\varpi_1+x_2\varpi_2)=x_1x_2^2$. The barycenter of $\Delta(s_1,s_2,s_3)$ with respect to the Duistermaat-Heckman polynomial is $$\mathbf{bar}(s_1,s_2,s_3)=s_3\varpi_2+\frac{2}{5}\frac{6(s_2^5-s_1^5)-15s_3(s_2^4-s_1^4)+10s_3^2(s_2^3-s_1^3)}{3(s_2^4-s_1^4)-8s_3(s_2^3-s_1^3)+6s_3^2(s_2^2-s_1^2)}(\varpi_1-\varpi_2)$$ In particular, for the anticanonical line bundle, $\mathbf{bar}(1,3,5)=5\varpi_2+\frac{244}{125}(\varpi_1-\varpi_2)\neq 2\varpi_1+3\varpi_2= 2\rho_H$, so we recover that $X$ is not Kähler-Einstein.
For a decomposition of the anticanonical line bundle in a sum of two ample real line bundles $L(1,3,5)=L(s_1,s_2,s_3)+L(1-s_1,3-s_2,5-s_3)$ (valid if $0<s_1<1$, $s_1<s_2<2+s_1$ and $s_2<s_3<2+s_2$) we see that the equation $\mathbf{bar}(s_1,s_2,s_3)+\mathbf{bar}(1-s_1,3-s_2,5-s_3)=2\rho_H$ boils down to a quartic equation in $s_3$ provided $s_1$ and $s_2$ are fixed, so that we can precisely determine if there is a solution as well as its exact value. For the obvious choice $s_1=1/2$ and $s_2=3/2$ we do not find a solution, so we arbitrarily consider $s_1=1/4$ and $s_2=3/2$, to obtain the quartic equation $$30720s_3^4-184000s_3^3+386272s_3^2-348246s_3+115587=0.$$ It admits two real roots (with complicated expression), one of which is approximately $2,6831$ and satisfies the Kähler condition for the decomposition. As a conseuquence, there exist pairs of coupled Kähler-Einstein metrics on this Fano fourfold.
To conclude on this homogeneous space $G/H$, let us describe all four smooth and Fano embeddings whose existence is ensured by [@Pas08]. The elementary description is as follows. Consider ${\mathbb{C}}^5$ as the product ${\mathbb{C}}^2\times {\mathbb{C}}^3$, and the induced action of $\operatorname{SL}_2\times \operatorname{SL}_3$ on ${\mathbb{P}}^4$. There are three orbits under this action: the line ${\mathbb{P}}({\mathbb{C}}^2\times\{0\})$, the plane ${\mathbb{P}}(\{0\}\times{\mathbb{C}}^3)$, and an open orbit isomorphic to $G/H$. The smooth and Fano embeddings of $G/H$ are the projective space ${\mathbb{P}}^4$ and its blowups along the line, the plane, and both. This fourth possibility coincides with the toroidal embedding considered above.
Let us now turn to the case $(k_1,k_2)=(-1,2)$ which corresponds to the embedding $X={\mathbb{P}}(\mathcal{O}\oplus\mathcal{O}(-1,2))$. In this case, we may again describe the semipositive real line bundles $L_{s_1,s_2,s_3}$ using three real parameters $0\leq s_1\leq s_2 \leq s_3$. The anticanonical line bundle is $L(1,3,7)$. The moment polytope is $$\Delta(s_1,s_2,s_3)=\{s_3\varpi_2+t(\varpi_1-2\varpi_2); s_1\leq t \leq s_2\}.$$ Let $\mathbf{bar}(s_1,s_2,s_3)$ denote again the barycenter of $\Delta(s_1,s_2,s_3)$ with respect to the Duistermaat-Heckman polynomial.
Let us consider the decompositions of the anticanonical of the form $L(1,3,7)=L(1/2,3/2,z)+L(1/2,3/2,7-z)$, which is a decomposition into Kähler classes for $3/2<z<11/2$. The equation $\mathbf{bar}(1/2,3/2,z)+\mathbf{bar}(1/2,3/2,7-z)=2\varpi_1+3\varpi_2$ translates to the quartic equation $$10(z^2-7z)^2+261(z^2-7z)+1631=0$$ which admits two real solutions $z_{\pm}$ $$\frac{3}{2}<z_-=\frac{35-\sqrt{5(\sqrt{2881}-16)}}{10}<z_+=\frac{35-\sqrt{5(\sqrt{2881}-16)}}{10}<\frac{11}{2}.$$ We thus obtain two examples of decompositions with coupled Kähler-Einstein metrics. Note that $z=7/2$ is not a solution of the above equation, so there are no Kähler-Einstein metrics on $X$.
|
---
author:
- Qiang Tu
- Chuanxi Wu
- Xueting Qiu
- 'Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China [^1]'
title: 'The distributional hyper-Jacobian determinants in fractional Sobolev spaces'
---
[**Abstract:**]{} In this paper we give a positive answer to a question raised by Baer-Jerison in connection with hyper-Jacobian determinants and associated minors in fractional Sobolev spaces. Inspired by recent works of Brezis-Nguyen and Baer-Jerison on the Jacobian and Hessian determinants, we show that the distributional $m$th-Jacobian minors of degree $r$ are weak continuous in fractional Sobolev spaces $W^{m-\frac{m}{r},r}$, and the result is optimal, satisfying the necessary conditions, in the frame work of fractional Sobolev spaces. In particular, the conditions can be removed in case $m=1,2$, i.e., the $m$th-Jacobian minors of degree $r$ are well defined in $W^{s,p}$ if and only if $W^{s,p} \subseteq W^{m-\frac{m}{r},m}$ in case $m=1,2$.
[**Key words:**]{} Hyper-Jacobian, Higher dimensional determinants, Fractional Sobolev spaces, Distributions.
[**2010 MR Subject Classification:**]{} 46E35, 46F10, 42B35.
Introduction and main results
=============================
Fix integer $m\geq 1$ and consider the class of non-smooth functions $u$ from $\Omega$, a smooth bounded open subset of $\mathbb{R}^N$, into $\mathbb{R}^n$( $N\geq 2$). The aim of this article is to identify when the hyper($m$th)-Jacobian determinants and associated minors of $u$, which were introduced by Olver in [@O], make sense as a distribution.
In the case $N=n$ and $m=1$, starting with seminal work of Morrey[@MC], Reshetnyak[@RY] and Ball[@BJ] on variational problems of non-linear elasticity, it is well known that the distributional ($1$th-)Jacobian determinant $\mbox{Det}(Du)$ of a map $u\in W^{1,\frac{N^2}{N+1}}(\Omega,\mathbb{R}^N)$ (or $u\in L^{q}\cap W^{1,p}(\Omega,\mathbb{R}^N)$ with $\frac{N-1}{p}+\frac{1}{q}=1$ and $N-1< p\leq \infty$) is defined by $$\mbox{Det}(Du):=\sum_{j} \partial_j(u^i(\mbox{adj} Du)^i_j),$$ where $\mbox{adj}Du$ means the adjoint matrix of $Du$. Furthermore, Brezis-Nguyen [@BN] extended the range of the map $u\mapsto \mbox{Det} (Du)$ in the framework of fractional Sobolev spaces. They showed that the distributional Jacobian determinant $\mbox{Det}(Du)$ for any $u\in W^{1-\frac{1}{N},N}(\Omega,\mathbb{R}^N)$ can be defined as $$\langle\mbox{Det}(Du), \psi\rangle:=\lim_{k\rightarrow \infty}\int_{\Omega}\det(Du_k)\psi dx~~~\forall \psi \in C_{c}^{1}(\Omega, \mathbb{R}),$$ where $u_k\in C^1(\overline{\Omega}, \mathbb{R}^N)$ such that $u_k\rightarrow u$ in $ W^{1-\frac{1}{N},N}$. They pointed out that the result recovers all the definitions of distributional Jacobian determinants mentioned above, except $N=2$, and the distributional Jacobian determinants are well-defined in $W^{s,p}$ if and only if $W^{s,p}\subseteq W^{1-\frac{1}{N},N}$ for $1<p<\infty$ and $0<s<1$.
In the case $n=1$ and $m=2$, similar to the results in [@BN], the distributional Hessian(2th-Jacobian) determinants are well-defined and continuous on $W^{2-\frac{2}{N},N}(\mathbb{R}^N)$ (see [@IT; @BJ]). Baer-Jersion [@BJ] pointed out that the continuous results of Hessian determinant in $W^{2-\frac{2}{N},N}(\mathbb{R}^N)$ with $N\geq 3$ implies the known continuity results in space $W^{1,p}(\mathbb{R}^N)\cap W^{2,q}(\mathbb{R}^N)$ with $1<p,r<\infty$, $\frac{2}{p}+\frac{N-2}{q}=1$, $N\geq 3$ (see [@DGG; @DM; @FM]). Furthermore they showed that the distributional Hessian determinants are well-defined in $W^{s,p}$ if and only if $W^{s,p}\subseteq W^{2-\frac{2}{N},N}$ for $1<p<\infty$ and $1<s<2$.
For $m>2$, $m$th-Jacobian, as a generalization of ordinary Jacobian, was first introduced by Escherich [@EG] and Gegenbauer [@GL]. In fact, the general formula for hyper-Jacobian can be expressed by using Cayley’s theory of higher dimensional determinants. All these earlier investigations were limited to polynomial functions until Olver [@O] turn his attention to some non-smooth functions. Especially he showed that the $m$th-Jacboian determinants (minors) of degree $r$ can be defined as a distribution provided $$u\in W^{m-[\frac{m}{r}],\gamma}(\Omega, \mathbb{R}^n)\cap W^{m-[\frac{m}{r}]-1,\delta} (\Omega, \mathbb{R}^n) ~\mbox{with}~\frac{r-t}{\gamma}+\frac{t}{\delta}\leq 1, t:=m ~\mbox{mod}~r$$ or $$u\in W^{m-[\frac{m}{r}], \gamma}(\Omega, \mathbb{R}^n)~\mbox{with}~\gamma\geq \max\{\frac{ Nr}{N+t}\}.$$ Bare-Jersion [@BJ] raised an interesting question: whether do there exist fractional versions of this result? I.e., is the $m$th-Jacboian determinant of degree $r$ continuous from space $W^{m-\frac{m}{r},r}$ into the space of distributions? Our first results give a positive answer to the question. We refer to Sec. 2 below for the following notation.
\[hm-thm-1\] Let $q,n,N$ be integers with $2\leq q\leq \underline{n}:=\min\{n,N\}$, for any integer $1\leq r\leq q$, multi-indices $\beta\in I(r,n)$ and $\bm{\alpha}= (\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$ ($j=1,\cdots,m$), the $mth$-Jacobian $(\beta, \bm{\alpha})$-minor operator $u \longmapsto M_{\bm{\alpha}}^{\beta}(D^mu) (\mbox{see}~ (\ref{hm-pre-for-1})):C^m(\Omega,\mathbb{R}^n)\rightarrow \mathcal{D}'(\Omega)$ can be extended uniquely as a continuous mapping $u \longmapsto \mbox{Div}_{\bm{\alpha}}^{\beta}(D^mu):W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)\rightarrow \mathcal{D}'(\Omega)$. Moreover for all $u,v\in W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)$, $\psi \in C^{\infty}_c(\Omega,\mathbb{R})$, we have $$\begin{split}
&\left|\langle\mbox{Div}_{\bm{\alpha}}^{\beta}(D^mu)-\mbox{Div}_{\bm{\alpha}}^{\beta}(D^mv),\psi\rangle\right|\\
&\leq C_{r,q,n,N,\Omega}\|u-v\|_{W^{m-\frac{m}{q},q}}\left(\|u\|_{W^{m-\frac{m}{q},q}}^{r-1} +\|v\|_{W^{m-\frac{m}{q},q}}^{r-1}\right)\|D^m\psi\|_{L^{\infty}}.
\end{split}$$
We recall that for $0<s<\infty$ and $1\leq p<\infty$, the fractional Sobolev space $W^{s,p}(\Omega)$ is defined as follows: when $s<1$ $$W^{s,p}(\Omega):=\left\{u\in L^p(\Omega)\mid \left(\int_{\Omega}\int_{\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy\right)^{\frac{1}{p}}<\infty\right\},$$ and the norm $$\|u\|_{W^{s,p}}:=\|u\|_{L^p}+\left(\int_{\Omega}\int_{\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy\right)^{\frac{1}{p}}.$$ When $s>1$ with non-integer, $$W^{s,p}(\Omega):=\{u\in W^{[s],p}(\Omega)\mid D^{[s]} u\in W^{s-[s],p}(\Omega)\},$$ the norm $$\|u\|_{W^{s,p}}:=\|u\|_{W^{[s],p}}+\left(\int_{\Omega}\int_{\Omega} \frac{|D^{[s]}u(x)-D^{[s]}u(y)|^p}{|x-y|^{N+(s-[s])p}}dxdy\right)^{\frac{1}{p}}.$$
It is worth pointing out that we may use the same method to get a similar result, see Corollary \[hm-cor-3-1\], for $u\in W^{m-\frac{m}{q},q}(\Omega)$ with $m\geq 2$. Theorem \[hm-thm-1\] and Corollary \[hm-cor-3-1\] recover not only all the definitions of Jacobian and Hessian determinants mentioned above, but also the definitions of $m$-th Jacobian in [@O] since the following facts
1. $W^{m-[\frac{m}{r}],\gamma}(\Omega, \mathbb{R}^n)\cap W^{m-[\frac{m}{r}]-1,\delta} (\Omega, \mathbb{R}^n)\subset W^{m-\frac{m}{r},r}(\Omega, \mathbb{R}^n)$ with continuous embedding if $\frac{r-t}{\gamma}+\frac{t}{\delta}\leq 1$ $(1<\delta<\infty, 1<r\leq N)$, where $t:=m ~\mbox{mod}~r$.
2. $W^{m-[\frac{m}{r}], \gamma}(\Omega, \mathbb{R}^n)\subset W^{m-\frac{m}{r},r}(\Omega, \mathbb{R}^n)$ $(1<r\leq N)$ with continuous embedding if $\gamma\geq \max\{\frac{ Nr}{N+t}\}$.
Similar to the optimal results for the ordinary distributional Jacobian and Hessian determinants in [@BN; @BJ], an natural question is that wether the results in Theorem \[hm-thm-1\] is optimal in the framework of the space $W^{s,p}$? I.e., is the distributional $m$-th Jacobian minors of degree $r$ well-defined in $W^{s,p}(\Omega, \mathbb{R}^n)$ if and only if $W^{s,p}(\Omega, \mathbb{R}^n)\subset W^{m-\frac{r}{m},r}(\Omega, \mathbb{R}^n)$? Such a question is connected with the construction of counter-examples in some special fractional Sobolev spaces. Indeed, the above conjecture is obviously correct in case $r=1$. Our next results give a partial positive answer in case $r>1$.
\[hm-thm-2\] Let $m, r$ be integers with $1< r\leq \underline{n}$, $1<p<\infty$ and $0<s<\infty$ be such that $W^{s,p}(\Omega, \mathbb{R}^n) \nsubseteq W^{m-\frac{m}{r},r}(\Omega, \mathbb{R}^n)$. If the condition $$\label{hm-thm-2-for-1}
1<r<p, s=m-m/r ~\mbox{non-integer}$$ fails, then there exist a sequence $\{u_k\}_{k=1}^{\infty} \subset C^{m}(\overline{\Omega}, \mathbb{R}^n)$, multi-indices $\beta\in I(r,n)$, $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j\in I(r,N)$ and a function $\psi\in C_c^{\infty}(\Omega)$ such that $$\label{hm-thm-2-for-2}
\lim_{k\rightarrow \infty} \|u_k\|_{s,p} =0, ~~~~\lim_{k\rightarrow \infty} \int_{\Omega} M^{\beta}_{\bm{\alpha}}(D^mu) \psi dx=\infty,$$
one still unanswered question is whether the above optimal results hold in case (\[hm-thm-2-for-1\]). We give some discuss in Sec. 4 and give positive answers in case $m=1$ and $2$. Indeed
\[hm-thm-3\] Let $m=1~\mbox{or}~2$ and $r,s,p$ be as in Theorem \[hm-thm-2\]. Then there exist a sequence $\{u_k\}_{k=1}^{\infty} \subset C^{m}(\overline{\Omega}, \mathbb{R}^n)$, multi-indices $\beta\in I(r,n)$, $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j\in I(r,N)$ and a function $\psi\in C_c^{\infty}(\Omega)$ such that (\[hm-thm-2-for-2\]) holds.
Furthermore, we give reinforced versions of optimal results, see Theorem \[hm-thm-4-14\], for $u\in W^{2-\frac{2}{r},r}(\Omega)$ with $1<r\leq N$. we expect that there are reinforced versions of optimal results for $W^{m-\frac{m}{r},r}(\Omega)$($m>2$), for instance there exist a sequence $\{u_k\}_{k=1}^{\infty} \subset C^{m}(\overline{\Omega})$ and a function $\psi\in C_c^{\infty}(\Omega)$ such that $$\lim_{k\rightarrow \infty} \|u_k\|_{s,p} =0, ~~~~\lim_{k\rightarrow \infty} \int_{\Omega} M_{\bm{\alpha}}(D^mu) \psi dx=\infty$$ for any $s,p$ with $W^{s,p}(\Omega) \nsubseteq W^{m-\frac{m}{r},r}(\Omega)$.
This paper is organized as follows. Some facts and notion about higher dimensional determinant and hyper-Jacobian are given in Section 2. In Section 3 we establish the weak continuity results and definitions for distributional hyper-Jacobian minors in fractional Sobolev space. Then we turn to the question about optimality and get some positive results in Section 4.
Higher dimensional determinants
===============================
In this section we collect some notation and preliminary results for hyper-Jacobian determinants and minors. Fist we recall some notation and facts about about ordinary determinants and minors, whereas further details can be found in [@GMS].
Fix $0\leq k\leq n$, we shall use the standard notation for ordered multi-indices $$\label{subnotation01}
I(k,n):=\{\alpha=(\alpha_1,\cdot\cdot\cdot,\alpha_k) \mid \alpha_i ~\mbox{integers}, 1\leq \alpha_1 <\cdot\cdot\cdot< \alpha_k\leq n\},$$ where $n \geq 2$. Set $I(0,n)=\{0\}$ and $|\alpha|=k$ if $\alpha \in I(k,n)$. For $\alpha\in I(k,n)$,
1. $\overline{\alpha}$ is the element in $I(n-k,n)$ which complements $\alpha$ in $\{1,2,\cdot\cdot\cdot,n\}$ in the natural increasing order.
2. $\alpha-i$ means the multi-index of length $k-1$ obtained by removing $i$ from $\alpha$ for any $i \in \alpha$.
3. $\alpha+j$ means the multi-index of length $k+1$ obtained by adding j to $\alpha$ for any $j\notin \alpha$, .
4. $\sigma(\alpha,\beta)$ is the sign of the permutation which reorders $(\alpha,\beta)$ in the natural increasing order for any multi-index $\beta$ with $\alpha\cap \beta=\emptyset$. In particular set $\sigma(\overline{0},0):=1$.
Let $n,N \geq 2$ and $A=(a_{ij})_{n \times N}$ be an $n \times N$ matrix. Given two ordered multi-indices $\alpha\in I (k,N)$ and $\beta \in I(k,n)$, then $A_{\alpha}^{\beta}$ denotes the $k \times k $-submatrix of $A$ obtained by selecting the rows and columns by $\beta$ and $\alpha$, respectively. Its determinant will be denoted by $$M_{\alpha}^{\beta}(A):=\det A_{\alpha}^{\beta},$$ and we set $M_{0}^{0}(A):=1$. The adjoint of $A_{\alpha}^{\beta}$ is defined by the formula $$(\mbox{adj}~ A_{\alpha}^{\beta})_j^i:= \sigma(i,\beta-i) \sigma(j,\alpha-j) \det A_{\alpha-j}^{\beta-i},~~~~ i \in \beta, j\in \alpha.$$ So Laplace formulas can be written as $$M_{\alpha}^{\beta}(A)= \sum_{j \in \alpha} a_{ij} (\mbox{adj}~ A_{\alpha}^{\beta})_j^i,~~~~ i\in\beta.$$
Next we pay attention to the higher dimensional matrix and determinant.
An $m$-dimensional matrix $\bm{A}$ of order $N^m$ is a hypercubical array of $N^m$ as $$\bm{A}=(a_{l_1l_2\cdot\cdot\cdot l_m})_{N\times\cdot\cdot\cdot\times N},$$ where the index $l_i\in \{1,\cdot\cdot\cdot N\}$ for any $1\leq i\leq m$.
\[hm-def-2-1\] Let $\bm{A}$ be an $m$-dimensional matrix, then the (full signed) determinant of $\bm{A}$ is the number $$\det \bm{A}=\sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_N} \Pi_{s=2}^m\sigma(\tau_s) a_{1\tau_2(1)\cdot\cdot\cdot\tau_m(1)} a_{2 \tau_2(2)\cdot\cdot\cdot\tau_m(2)}\cdot\cdot\cdot a_{N\tau_2(N)\cdot\cdot\cdot\tau_m(N)},$$ where $S_N$ is the permutation group of $\{1,2,\cdot\cdot\cdot,N\}$ and $\sigma(\cdot)$ is the sign of $\cdot$.
For any $1\leq i\leq m$ and $1\leq j\leq N$, the $j$-th $i$-layer of $\bm{A}$, the $(m-1)$-dimensional matrix denoted by $\bm{A}|_{l_i=j}$, which generalizing the notion of row and column for ordinary matrices, is defined by $$\bm{A}|_{l_i=j}:=(a_{l_1l_2\cdot\cdot l_{i-1}jl_{i+1}\cdot\cdot\cdot l_m)})_{N\times\cdot\cdot\cdot\times N}.$$ According to Definition \[hm-def-2-1\], we can easily obtain that
\[hm-lem-2-11\] Let $\bm{A}$ be an $m$-dimensional matrix and $1\leq i\leq m$. $\bm{A}'$ is a matrix such that a pair of $i$-layers in $\bm{A}$ is interchanged, then $$\det \bm{A}'=
\begin{cases}
(-1)^{m-1}\det \bm{A}~~~~~~~~i=1,\\
-\det \bm{A}~~~~~~~~ i\geq 2.
\end{cases}$$
For any $\bm{A}$ and $1\leq i<j\leq m$, the $(i,j)$-transposition of $\bm{A}$, denoting by $\bm{A}^{T(i,j)}$, is a $m$-dimensional matrix defined by $$a'_{l_1,\cdot\cdot\cdot,l_i,\cdot\cdot\cdot,l_j,\cdot\cdot\cdot,l_m}=a_{l_1,\cdot\cdot\cdot,l_j,\cdot\cdot\cdot,l_i,\cdot\cdot\cdot,l_m}$$ for any $l_1,\cdot\cdot\cdot,l_m=1,\cdot\cdot\cdot,N$. where $$\bm{A}^{T(i,j)}:=(a'_{l_1l_2\cdot\cdot\cdot\cdot\cdot l_m)})_{N\times\cdot\cdot\cdot\times N}.$$ Then we have
Let $\bm{A}$ be an $m$-dimensional matrix and $1\leq i<j\leq m$, if $m$ is odd and $1<i<j\leq m$ or $m$ is even, then $$\det \bm{A}^{T(i,j)}=\det \bm{A}.$$
According to the definition of the $m$-dimensional determinant, we only to show the claim in case $m$ is even ,$i=1$ and $j=2$. $$\begin{split}
\det \bm{A}&=\sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_N} \Pi_{s=2}^m\sigma(\tau_s) a_{1\tau_2(1)\cdot\cdot\cdot\tau_m(1)} a_{2 \tau_2(2)\cdot\cdot\cdot\tau_m(2)}\cdot\cdot\cdot a_{N\tau_2(N)\cdot\cdot\cdot\tau_m(N)}\\
&=\sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_N} \Pi_{s=2}^m\sigma(\tau_s) a_{\tau_2^{-1}(1)1\tau_3\circ \tau^{-1}_2(1)\cdot\cdot\cdot\tau_m\circ\tau_2^{-1}(1)} a_{\tau_2^{-1}(2) 2\tau_3\circ\tau_2^{-1}(2)\cdot\cdot\cdot\tau_m\circ\tau_2^{-1}(2)}\cdot\cdot\cdot a_{\tau_2^{-1}(N) N \tau_3\circ\tau_2^{-1}(N)\cdot\cdot\cdot\tau_m\circ \tau_2^{-1}(N)}\\
&=\sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_N} (\sigma(\tau_2))^{m-2} \sigma(\tau^{-1}_2)\sigma(\tau_3\circ\tau^{-1}_2)\cdot\cdot\cdot \sigma(\tau_m\circ\tau^{-1}_2)\\
&\cdot a'_{1\tau_2^{-1}(1)\tau_3\circ \tau^{-1}_2(1)\cdot\cdot\cdot\tau_m\circ\tau_2^{-1}(1)} a'_{2\tau_2^{-1}(2) \tau_3\circ\tau_2^{-1}(2)\cdot\cdot\cdot\tau_m\circ\tau_2^{-1}(2)}\cdot\cdot\cdot a'_{N\tau_2^{-1}(N) \tau_3\circ\tau_2^{-1}(N)\cdot\cdot\cdot\tau_m\circ \tau_2^{-1}(N)}\\
&=\sum_{\tau'_2,\cdot\cdot\cdot,\tau'_m\in S_N} \Pi_{s=2}^m\sigma(\tau'_s) a'_{1\tau'_2\tau'_3(1)\cdot\cdot\cdot\tau'_m(1)} a'_{2\tau'_2(2) \tau'_3(2)\cdot\cdot\cdot\tau'_m(2)}\cdot\cdot\cdot a'_{N\tau'_2(N) \tau'_3(N)\cdot\cdot\cdot\tau'_m(N)}.
\end{split}$$
More generally, suppose $\bm{A}$ be an $m$-dimensional matrix of order $N_1\times \cdot\cdot\cdot\times N_m$, $1\leq r\leq \min\{N_1,\cdot\cdot\cdot,N_m\}$, and an type of multi-index $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ where $\alpha^j:=(\alpha^j_1,\cdot\cdot,\cdot, \alpha^j_r)$, $\alpha^j_i \in \{1,2,\cdot\cdot\cdot,N_j\}$ and $\alpha^j_{i_1}\neq \alpha^j_{i_2}$ for $i_1\neq i_2$. Define the $\bm{\alpha}$-minor of $\bm{A}$, denoted by $\bm{A}_{\bm{\alpha}}$, to be the $m$-dimensional matrix of order $r^m$ as $$\bm{A}_{\bm{\alpha}}=(b_{l_1l_2\cdot\cdot\cdot l_m})_{r\times\cdot\cdot\cdot\times r},$$ where $b_{l_1l_2\cdot\cdot\cdot l_m}:=a_{\alpha^1_{l_1}\alpha^2_{l_2}\cdot\cdot\cdot\alpha^m_{l_m}}$. Its determinant will be denoted by $$M_{\bm{\alpha}}(\bm{A}):=\det \bm{A}_{\bm{\alpha}}.$$ If $\alpha^j$ is not increasing, let $\widetilde{\alpha^j}$ be the increasing multi-indices generated by $\alpha^j$ and $\widetilde{\bm{\alpha}}:=(\widetilde{\alpha^1},\cdot\cdot\cdot,\widetilde{\alpha^m})$, then Lemma \[hm-lem-2-11\] implies that $M_{\bm{\alpha}}(\bm{A})$ and $M_{\widetilde{\bm{\alpha}}}(\bm{A})$ differ only by a sign. Without loss of generality, we can assume $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j\in I(r,N_j)$. Moreover we set $M_{\bm{0}}(\bm{A}):=1$.
Next we pay attention to hyper-Jacobian determinants and minors for a map $u\in C^{m}(\Omega, \mathbb{R}^n)$. We will denote by $D^mu$ the hyper-Jacobian matrix of $u$, more precisely, $D^m u$ is a $(m+1)$-dimensional matrix with order $n\times N\times\cdot\cdot\cdot\times N$ given by $$D^mu:=(a_{l_1l_2\cdot\cdot\cdot l_{m+1}})_{n\times N\times\cdot\cdot\cdot\times N}$$ where $$a_{l_1l_2\cdot\cdot\cdot l_{m+1}}=\partial_{l_2}\partial_{l_3}\cdot\cdot\cdot\partial_{l_{m+1}} u^{l_1}.$$ Then for any $\beta\in I(r,n)$, $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j\in I(r,N)$ and $1\leq r\leq \min\{n,N\}$, the $m$th-Jacobian $(\beta, \bm{\alpha})$-minor of $u$, denoted by $M^{\beta}_{\bm{\alpha}}(D^mu)$, is the determinant of the $(\beta, \bm{\alpha})$- minor of $D^mu$, i.e., $$\label{hm-pre-for-1}
M^{\beta}_{\bm{\alpha}}(D^mu):=M_{(\beta,\bm{\alpha})}(D^m u).$$ In particular if $N=n$ and $\beta=\alpha^1=\cdot\cdot\cdot=\alpha^m=\{1,2,\cdot\cdot\cdot,N\}$, $\det (D^mu)$ is called the $m$-th Jacobian determinant of $u$. Similarly, the hyper-Jacobian matrix $D^m u$ of $u\in C^{m}(\Omega)$ is a $m$-dimensional matrix with order $N\times\cdot\cdot\cdot\times N$ and the $m$th-Jacobian $\bm{\alpha}$-minor of $u$ is defined by $M_{\bm{\alpha}}(D^mu)$.
In order to prove the main results, some lemmas, which can be easily manipulated by the definition of hyper-Jacobian minors, are introduced as follows.
\[hm-lem-2-2\] Let $u=(v,\cdots,v)\in C^{m}(\Omega, \mathbb{R}^n)$ with $v\in C^{m}(\Omega)$. For any $\beta\in I(r,n)$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j\in I(r,N)$, $1\leq r\leq \underline{n}$ $$M^{\beta}_{\bm{\alpha}}(D^mu)=
\begin{cases}
r! M_{\bm{\alpha}}(D^mv)~~~~m~\mbox{is even},\\
0~~~~~~~~~~~~~~~~~m~\mbox{is odd}.
\end{cases}$$
\[hm-lem-2-1\] Let $u\in C^{m}(\Omega, \mathbb{R}^n)$, $\beta\in I(r,n)$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j\in I(r,N)$, $1\leq r\leq \underline{n}$. Then for any $1\leq i\leq m$ $$M^{\beta}_{\bm{\alpha}}(D^mu)=\sum_{\tau_1,\cdot\cdot\cdot,\tau_{i-1},\tau_{i+1},\cdot\cdot\cdot,\tau_m\in S_r} \Pi_{s\in \overline{i}}\sigma(\tau_s) M^{\overline{0}}_{\alpha^i} (Dv(i)),$$ where $M^{\overline{0}}_{\alpha^i}(\cdot)$ is the ordinary minors and $v(i)\in C^1(\Omega, \mathbb{R}^r)$ can be written as $$v^j(i)=\partial_{\alpha^1_{\tau_1(j)}}\cdot\cdot\cdot \partial_{\alpha^{i-1}_{\tau_{i-1}(j)}} \partial_{\alpha^{i+1}_{\tau_{i+1}(j)}} \cdot\cdot\cdot \partial_{\alpha^{m}_{\tau_{m}(j)}}u^{\beta_j},~~~~~~j=1,\cdots,r.$$
Hyper-jacobians in fractional Sobolev spaces
============================================
In this section we establish the weak continuity results for the Hyper-jacobian minors in the fractional Sobolev spaces $W^{m-\frac{m}{q},q}(\Omega, \mathbb{R}^n)$.
Let $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$, we set $$\bm{\widetilde{\alpha}}=(\alpha^1+(N+1),\cdot\cdot\cdot,\alpha^m+(N+m)), R(\bm{\widetilde{\alpha}}):=\{(i_1,\cdot\cdot\cdot,i_m)\mid i_j\in \alpha^j+(N+j)\}.$$ For any $I=(i_1,\cdot\cdot\cdot,i_m)\in R(\bm{\widetilde{\alpha}})$, $$\widetilde{\bm{\alpha}}-I:=(\alpha^1+(N+1)-i_1,\cdot\cdot\cdot,\alpha^m+(N+m)-i_m);$$ $$\sigma(\widetilde{\bm{\alpha}}-I,I):=\Pi_{s=1}^m \sigma(\alpha^s+(N+s)-i_s,i_s);$$ $$\partial_I:=\partial_{x_{i_1}}\cdot\cdot\cdot\partial_{x_{i_m}};~~~~ \widetilde{x}:=(x_1,\cdots,x_N,x_{N+1},\cdots,x_{N+m}).$$ We begin with the following simple lemma:
\[hm-lem-3-1\] Let $u \in C^m(\Omega, \mathbb{R}^n)$, $\psi\in C_c^m(\Omega)$, $0\leq r\leq \underline{n}:=\min\{n,N\}$, $\beta\in I(r,n)$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$ ($1\leq j\leq m$), then $$\label{hm-lem-3-1}
\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^mu)\psi dx=\sum_{I\in R(\bm{\widetilde{\alpha}})}(-1)^m\sigma(\widetilde{\bm{\alpha}}-I,I)\int_{\Omega\times [0,1)^m} M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mU) \partial_{I} \Psi d\widetilde{x},$$ for any extensions $U\in C^m(\Omega\times[0,1)^m,\mathbb{R}^n)\cap C^{m+1}(\Omega\times (0,1)^m,\mathbb{R}^n)$ and $\Psi\in C^m_c(\Omega\times[0,1)^m,\mathbb{R})$ of $u$ and $\psi$, respectively.
It is easy to show the results in case $r=0,1$ or $\underline{n}=1$. So we give the proof only for the case $2\leq r\leq \underline{n}$. Denote $$U_i:=\begin{cases}
U|_{x_{N+i+1}=\cdot\cdot\cdot=x_{N+m}=0},~~~~1\leq i\leq m-1,\\
U,~~~~i=m.
\end{cases}
\Psi_i:=\begin{cases}
\Psi|_{x_{N+i+1}=\cdot\cdot\cdot=x_{N+m}=0},~~~~1\leq i\leq m-1,\\
\Psi,~~~~i=m.
\end{cases}$$ $$\Omega_i:=\Omega\times [0,1)_{x_{N+1}}\times \cdot\cdot\cdot\times [0,1)_{x_{N+i}};~~~~\widetilde{x_i}:=(x, x_{N+1},\cdot\cdot\cdot x_{N+i}).$$ Applying the fundamental theorem of calculus and the definition of $M_{\bm{\alpha}}^{\beta}(D^mu)$, we have $$\label{hm-lem-3-for-1}
\begin{split}
\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^mu)\psi dx&=-\int_{\Omega_1} \partial_{N+1} \left(M_{\bm{\alpha}}^{\beta}(D^mU_1)\Psi_1\right) d\widetilde{x_1}\\
&=-\int_{\Omega_1} \partial_{N+1}M_{\bm{\alpha}}^{\beta}(D^mU_1)\Psi_1 d\widetilde{x_1}-\int_{\Omega_1}M_{\bm{\alpha}}^{\beta}(D^mU_1)\partial_{N+1}\Psi_1 d\widetilde{x_1}.
\end{split}$$ According to the Lemma \[hm-lem-2-1\], $M_{\bm{\alpha}}^{\beta}(D^mU_1)$ can be written as $$M_{\bm{\alpha}}^{\beta}(D^mU_1)= \sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_r} \Pi_{s=2}^m\sigma(\tau_s) M^{\overline{0}}_{\alpha^1} (DV_1),$$ where $\overline{0}:=\{1,2,\cdot\cdot\cdot,r\}$ and $$V_1(\widetilde{x_1}):=(V_1^1(\widetilde{x_1}),\cdot\cdot\cdot,V_1^r(\widetilde{x_1})),~~~~V^j_1=\partial_{\alpha^2_{\tau_2(j)}}\cdot\cdot\cdot\partial_{\alpha^{m}_{\tau_{m}(j)}}u^{\beta_j}.$$ Then $$\label{hm-lem-3-for-12}
\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^mu)\psi dx=\sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_r} \Pi_{s=2}^m\sigma(\tau_s) \left\{-\int_{\Omega_1} \partial_{N+1}M^{\overline{0}}_{\alpha^1} (DV_1)\Psi_1 d\widetilde{x_1}-\int_{\Omega_1}M^{\overline{0}}_{\alpha^1} (DV_1)\partial_{N+1}\Psi_1 d\widetilde{x_1} \right\}.$$ We denote the first part integral on the right-hand side by $I$, Laplace formulas of the $2$-dimensional minors imply that $$\begin{split}
I&=-\sum_{i\in \alpha^1} \sum_{j=1}^r \int_{\Omega_1} \sigma(i, \alpha^1-i) \sigma(j, \overline{0}-j) \partial_{N+1}\partial_i V_1^j M^{\overline{0}-j}_{\alpha^1-i} (DV_1)\Psi_1 d\widetilde{x_1}\\
&=\sum_{i\in \alpha^1} \sum_{j=1}^r \int_{\Omega_1} \sigma(i, \alpha^1-i) \sigma(j, \overline{0}-j) \partial_{N+1} V_1^j \left(\partial_i M^{\overline{0}-j}_{\alpha^1-i} (DV_1)\Psi_1+ M^{\overline{0}-j}_{\alpha^1-i} (DV_1)\partial_i\Psi_1\right) d\widetilde{x_1}.\\
\end{split}$$ Since $$\sum_{i\in \alpha^1} \sigma(i, \alpha^1-i) \sigma(j, \overline{0}-j)\partial_i M^{\overline{0}-j}_{\alpha^1-i} (DV_1)=0$$ for any $j$, it follows that $$\begin{split}
I&=\sum_{i\in \alpha^1} \sum_{j=1}^r \int_{\Omega_1} \sigma(i, \alpha^1-i) \sigma(j, \overline{0}-j) \partial_{N+1} V_1^j M^{\overline{0}-j}_{\alpha^1-i} (DV_1)\partial_i\Psi_1 d\widetilde{x_1}\\
&=\sum_{i\in \alpha^1} \int_{\Omega_1} \sigma(i, \alpha^1-i) \sigma(N+1, \alpha^1-i) M^{\overline{0}}_{\alpha^1+(N+1)-i} (DV_1)\partial_i\Psi_1 d\widetilde{x_1}\\
&=-\sum_{i\in \alpha^1} \int_{\Omega_1} \sigma(\alpha^1+(N+1)-i,i) M^{\overline{0}}_{\alpha^1+(N+1)-i} (DV_1)\partial_i\Psi_1 d\widetilde{x_1}.
\end{split}$$ Combing with (\[hm-lem-3-for-12\]), we obtain that $$\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^mu)\psi dx= -\sum_{i_1\in \alpha^1+(N+1)} \sigma(\alpha^1+(N+1)-i_1,i_1) \sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_r} \Pi_{s=2}^m\sigma(\tau_s) \int_{\Omega_1} M^{\overline{0}}_{\alpha^1+(N+1)-i_1} (DV_1)\partial_{i_1}\Psi_1 d\widetilde{x_1}.$$ For any $i_1\in \alpha^1+(N+1)$, we denote $\gamma:=\alpha^1+(N+1)-i_1$, then $$\label{hm-lem-3-for-2}
\begin{split}
&\sum_{\tau_2,\cdot\cdot\cdot,\tau_m\in S_r} \Pi_{s=2}^m\sigma(\tau_s) M^{\overline{0}}_{\alpha^1+(N+1)-i_1} (DV_1)=\sum_{\tau_1,\tau_2,\cdot\cdot\cdot,\tau_m\in S_r} \Pi_{s=1}^m\sigma(\tau_s) \partial_{\gamma_{\tau_1(1)}}V_1^1 \cdot\cdot\cdot \partial_{\gamma_{\tau_1(r)}}V_1^r\\
&=\sum_{\tau_1,\tau_2,\cdot\cdot\cdot,\tau_m\in S_r} \Pi_{s=1}^m\sigma(\tau_s) \left(\partial_{\gamma_{\tau_1(1)}}\partial_{\alpha^2_{\tau_2(1)}} \cdot\cdot\cdot \partial_{\alpha^m_{\tau_m(1)}} U_1^{\beta_1}\right)\cdot\cdot\cdot \left( \partial_{\gamma_{\tau_1(r)}} \partial_{\alpha^2_{\tau_2(r)}} \cdot\cdot\cdot \partial_{\alpha^m_{\tau_m(r)}} U_1^{\beta_r}\right)\\
&= M^{\beta}_{\bm{\alpha}(i_1)} (D^m U_1),
\end{split}$$ where $\bm{\alpha}(i_1):=(\alpha^1+(N+1)-i_1,\alpha^2,\cdot\cdot\cdot,\alpha^m )$. Hence $$\begin{split}
\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^mu)\psi dx&= -\sum_{i_1\in \alpha^1+(N+1)} \sigma(\alpha^1+(N+1)-i_1,i_1) \int_{\Omega_1} M^{\beta}_{\bm{\alpha}(i_1)} (D^m U_1)\partial_{i_1}\Psi_1 d\widetilde{x_1}\\
&=\sum_{i_1\in \alpha^1+(N+1)} \sigma(\alpha^1+(N+1)-i_1,i_1) \int_{\Omega_2} \partial_{N+2} \left( M^{\beta}_{\bm{\alpha}(i_1)} (D^m U_2)\partial_{i_1}\Psi_2\right) d\widetilde{x_2}.
\end{split}$$ An easy induction and the argument similar to the one used in (\[hm-lem-3-for-1\])-(\[hm-lem-3-for-2\]) shows that $$\begin{split}
\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^mu)\psi dx=(-1)^j\sum_{s=1}^j\sum_{i_s\in \alpha^s+(N+s)} \Pi_{s=1}^j \sigma(\alpha^s+(N+s)-i_s,i_s) \int_{\Omega_j} M_{\bm{\alpha}(i_1i_2\cdot\cdot\cdot i_j)}^{\beta}(D^mU_j)\partial_{i_1i_2\cdot\cdot\cdot i_j}\Psi_jd\widetilde{x_j}
\end{split}$$ for any $1\leq j\leq m$, where $$\bm{\alpha}(i_1i_2\cdot\cdot\cdot i_j):=(\alpha^{1}+(N+1)-i_1,\cdot\cdot\cdot,\alpha^{j}+(N+j)-{i_j},\alpha^{j+1},\cdot\cdot\cdot,\alpha^{m}).$$
\[hm-lem-3-2\] Let $u,v \in C^m(\Omega, \mathbb{R}^n)$ and $\psi\in C_c^m(\Omega)$ and $2\leq q\leq \underline{n}$. Then for any $1\leq r\leq q$, $\beta\in I(r,n)$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$, $$\left|\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^m u) \psi dx- \int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^m v) \psi dx\right|\leq C\|u-v\|_{W^{m-\frac{m}{q},q}}(\|u\|^{r-1}_{W^{m-\frac{m}{q},q}}+ \|v\|^{r-1}_{W^{m-\frac{m}{q},q}}) \|D^m \psi\|_{L^{\infty}},$$ the constant $C$ depending only on $q,r,m,n,N$ and $\Omega$.
Let $\widetilde{u}$ and $\widetilde{v}$ be extensions of $u$ and $v$ to $\mathbb{R}^N$ such that $$\|\widetilde{u}\|_{W^{m-\frac{m}{q},q}(\mathbb{R}^N,\mathbb{R}^n)}\leq C \|u\|_{W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)},~~~~\|\widetilde{v}\|_{W^{m-\frac{m}{q},q}(\mathbb{R}^N,\mathbb{R}^n)}\leq C\|v\|_{W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)}$$ and $$\|\widetilde{u}-\widetilde{v}\|_{W^{m-\frac{m}{q},q}(\mathbb{R}^N,\mathbb{R}^n)}\leq C \|u-v\|_{W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)},$$ where $C$ depending only on $q,m,n,N$ and $\Omega$.
According to a well known trace theorem of Stein in [@STE1; @STE2], where $W^{m-\frac{m}{q},q}(\mathbb{R}^N)$ is identified as the space of traces of $W^{m,q}(\mathbb{R}^N\times(0,+\infty)^m)$, there is a bounded linear extension operator $$E:W^{m-\frac{m}{q},q}(\mathbb{R}^N,\mathbb{R}^n)\rightarrow W^{m,q}(\mathbb{R}^N\times (0,+\infty)^m,\mathbb{R}^n).$$
Let $U$ and $V$ be extensions of $\widetilde{u}$ and $\widetilde{v}$ to $\mathbb{R}^N\times (0,+\infty)^m$, respectively, i.e., $$U=E\widetilde{u},~~V=E\widetilde{v}.$$ We then have $$\|D^mU\|_{L^{q}(\Omega \times (0,1)^m)}\leq C\|u\|_{W^{m-\frac{q}{m},q}(\Omega,\mathbb{R}^n)},~~~~\|D^mV\|_{L^{q}(\Omega \times (0,1)^m)}\leq C \|v\|_{W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)}$$ and $$\|D^mU-D^mV\|_{L^{q}(\Omega \times (0,1)^m)}\leq C \|u-v\|_{W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)}.$$ Let $\Psi \in C^m_c(\Omega\times [0,1)^m)$ be an extension of $\psi$ such that $$\|D^m\Psi\|_{L^{\infty}(\Omega\times [0,1)^m)}\leq C\|D^m\psi\|_{L^{\infty}(\Omega)}.$$ According to Lemma \[hm-lem-3-1\], we have $$\label{hm-lem-for-3-31}
\begin{split}
&\left|\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^m u) \psi dx- \int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^m v) \psi dx\right|\leq \sum_{I\in R(\bm{\widetilde{\alpha}})}\int_{\Omega\times [0,1)^m} \left|M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mU)-M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mV)\right| |\partial_{I} \Psi| d\widetilde{x}\\
&\leq \| D^m \Psi\|_{L^{\infty}(\Omega\times [0,1)^m)} \sum_{I\in R(\bm{\widetilde{\alpha}})}\int_{\Omega\times [0,1)^m} \left|M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mU)-M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mV)\right| d\widetilde{x}.
\end{split}$$ Note that for any $I\in R(\bm{\widetilde{\alpha}})$ $$\begin{split}
&\left|M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mU)-M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mV)\right|\\
&\leq \sum_{\tau_1,\cdot\cdot\cdot, \tau_m\in S_r} |\partial_{\tau_1(1)\cdot\cdot\cdot\tau_m(1)}U^{\beta_1} \cdot\cdot\cdot \partial_{\tau_1(r)\cdot\cdot\cdot\tau_m(r)}U^{\beta_r}- \partial_{\tau_1(1)\cdot\cdot\cdot\tau_m(1)}V^{\beta_1}\cdot\cdot\cdot \partial_{\tau_1(r)\cdot\cdot\cdot\tau_m(r)}V^{\beta_r}| \\
&\leq \sum_{\tau_1,\cdot\cdot\cdot, \tau_m\in S_r} \sum_{s=1}^{r} |D^mU|^{s-1}|D^mU-D^mV||D^mV|^{r-s}\\
&\leq C|D^mU-D^mV|(|D^mU|^{r-1}+|D^mV|^{r-1}).
\end{split}$$ Combining with (\[hm-lem-for-3-31\]), we can easily obtain $$\begin{split}
&\left|\int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^m u) \psi dx- \int_{\Omega} M_{\bm{\alpha}}^{\beta}(D^m v) \psi dx\right|\\
&\leq C \int_{\Omega\times [0,1)^m} |D^mU-D^mV|(|D^mU|^{r-1}+|D^mV|^{r-1}) d\widetilde{x} \|D^m\Psi\|_{L^{\infty}(\Omega\times [0,1)^m)}\\
&\leq C \|u-v\|_{W^{m-\frac{m}{q},q}}(\|u\|^{r-1}_{W^{m-\frac{m}{q},q}}+ \|v\|^{r-1}_{W^{m-\frac{m}{q},q}}) \|D^m \psi\|_{L^{\infty}}.
\end{split}$$
According to the above lemma, we can give the definitions of distributional $m$th-Jacobian minors of $u$ with degree less that $q$ when $u\in W^{m-\frac{m}{q},q}(\Omega, \mathbb{R}^n)$ ($2\leq q\leq \underline{n}$).
\[hm-def-3-1\] Let $u\in W^{m-\frac{m}{q},q}(\Omega, \mathbb{R}^n)$ with $2\leq q \leq \underline{n}$. For any $0\leq r\leq q$, $\beta\in I(r,n)$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$, the distributional $m$th-Jacobian $(\beta, \bm{\alpha})$-minors of $u$, denoted by $\mbox{Div}_{\bm{\alpha}}^{\beta}(D^mu)$, is defined by $$\langle \mbox{Div}_{\bm{\alpha}}^{\beta}(D^mu), \psi \rangle:=
\begin{cases}
\int_{\Omega} \psi(x)dx,~~~~~~~~~~~r=0; \\
\lim_{k\rightarrow \infty} \int_{\Omega} M^{\beta}_{\bm{\alpha}}(D^mu_k)\psi dx,~~~~ 1\leq r\leq q\\
\end{cases}$$ for any $\psi\in C^m_c(\Omega)$ and any sequence $\{u_k\}_{k=1}^{\infty}\subset C^m(\overline{\Omega},\mathbb{R}^n)$ such that $u_k\rightarrow u$ in $W^{m-\frac{m}{q},q}(\Omega,\mathbb{R}^n)$.
Obviously this quantity is well-defined since Lemma \[hm-lem-3-2\] and the fact that $C^m(\overline{\Omega}, \mathbb{R}^n)$ is dense in $W^{m-\frac{m}{q},q}(\Omega, \mathbb{R}^n)$.
It is clear that Theorem \[hm-thm-1\] is a consequence of Lemma \[hm-lem-3-2\] and Definition \[hm-def-3-1\].
According to the trace theory and the approximate theorem, we obtain a fundamental representation of the distributional m-th Jacobian minors in $W^{m-\frac{m}{q},q}$.
\[hm-pro-3-1\] Let $u\in W^{m-\frac{m}{q},q}(\Omega, \mathbb{R}^n)$ with $2\leq q \leq \underline{n}$. For any $0\leq r\leq q$, $\beta\in I(r,n)$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$ , $$\int_{\Omega} \mbox{Div}_{\bm{\alpha}}^{\beta}(D^mu)\psi dx=\sum_{I\in R(\bm{\widetilde{\alpha}})}(-1)^m\sigma(\widetilde{\bm{\alpha}}-I,I)\int_{\Omega\times [0,1)^m} M_{\widetilde{\bm{\alpha}}-I}^{\beta}(D^mU) \partial_{I} \Psi d\widetilde{x}$$ for any extensions $U\in W^{m,q}(\Omega\times[0,1)^m,\mathbb{R}^n)$ and $\Psi\in C^m_c(\Omega\times[0,1)^m)$ of $u$ and $\psi$, respectively.
Note that the $m$-dimensional matrix $D^m u$ is symmetric if $u\in C^m(\Omega)$, i.e., $(D^mu)^{T(i,j)}=D^mu$ for any $1\leq i<j\leq m$. An argument similar to the one used in Lemma \[hm-lem-3-1\] and \[hm-lem-3-2\] show that
\[hm-cor-3-1\] Let $u\in W^{m-\frac{m}{q},q}(\Omega)$ with $2\leq q \leq N$ and $m\geq 2$. For any $0\leq r\leq q$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$, Then the $m$-th Jacobian $\bm{\alpha}$-minor operator $u \longmapsto M_{\bm{\alpha}}(D^mu):C^m(\Omega)\rightarrow \mathcal{D}'(\Omega)$ can be extended uniquely as a continuous mapping $u \longmapsto \mbox{Div}_{\bm{\alpha}}(D^mu):W^{m-\frac{m}{q},q}(\Omega)\rightarrow \mathcal{D}'(\Omega)$. Moreover for all $u,v\in W^{m-\frac{m}{q},q}(\Omega)$, $\psi \in C^{\infty}_c(\Omega,\mathbb{R})$ and $1\leq r\leq q$, we have $$\begin{split}\left|\langle\mbox{Div}_{\bm{\alpha}}(D^mu)-\mbox{Div}_{\bm{\alpha}}(D^mv),\psi\rangle\right|\leq C_{r,q,N,\Omega}\|u-v\|_{W^{m-\frac{m}{q},q}}\left(\|u\|_{W^{m-\frac{m}{q},q}}^{r-1} +\|v\|_{W^{m-\frac{m}{q},q}}^{r-1}\right)\|D^m\psi\|_{L^{\infty}},
\end{split}$$ where the constant depending only on $r, q, N$ and $ \Omega$. In particular, the distributional minor $\mbox{Div}_{\bm{\alpha}}(D^mu)$ can be expressed as $$\int_{\Omega} \mbox{Div}_{\bm{\alpha}}(D^mu)\psi dx=\sum_{I\in R(\bm{\widetilde{\alpha}})}(-1)^m\sigma(\widetilde{\bm{\alpha}}-I,I)\int_{\Omega\times [0,1)^m} M_{\widetilde{\bm{\alpha}}-I}(D^mU) \partial_{I} \Psi d\widetilde{x}$$ for any extensions $U\in W^{m,q}(\Omega\times[0,1)^m)$ and $\Psi\in C^m_c(\Omega\times[0,1)^m)$ of $u$ and $\psi$, respectively.
The optimality results in fractional Sobolev spaces
===================================================
In this section we establish the optimality results of Theorem 1 in the framework of spaces $W^{s,p}$. Before proving the main results, we state some interesting consequences (see [@BM2 Theorem 1 and Proposition 5.3]):
\[hm-lem-4\] For $0\leq s_1<s_2<\infty$, $1\leq p_1, p_2,p\leq \infty$, $s=\theta s_1+(1-\theta) s_2$, $\frac{1}{p}=\frac{\theta}{p_1}+\frac{1-\theta}{p_2}$ and $0<\theta<1$, the inequality $$\|f\|_{W^{s,p}(\Omega)}\leq C \|f\|_{W^{s_1,p_1}(\Omega)}^{\theta} \|f\|_{W^{s_2,p_2}(\Omega)}^{1-\theta}.$$ holds if and only if the following condition fails $$s_2\geq 1~\mbox{is an integer}, ~p_2=1~\mbox{and}~s_2-s_1\leq 1-\frac{1}{p_1}.$$
\[hm-pro-1\] The following equalities of spaces holds:
1. $W^{s,p}(\Omega)=F^s_{p,p}(\Omega)$ if $s>0$ is a non-integer and $1\leq p\leq \infty$.
2. $W^{s,p}(\Omega)=F^s_{p,2}(\Omega)$ if $s\geq 0$ is an integer and $1<p<\infty$.
The definition of Triebel-Lizorkin spaces $F^s_{p,q}$ can be seen in [@BM2; @TH].
\[hm-rem-41\] If $1<r\leq N$, according to the embedding properties of the Triebel-Lizorkin spaces $F^s_{p,q}$, see e.g. [@TH page 196], and Proposition \[hm-pro-1\], we consider all possible cases:
1. $s-m+\frac{m}{r}>\max\{0,\frac{N}{p}-\frac{N}{r}\}$, then the embedding $W^{s,p}(\Omega)\subset W^{m-\frac{m}{r},r}(\Omega)$ holds;
2. $s-m+\frac{m}{r}<\max\{0,\frac{N}{p}-\frac{N}{r}\}$, the embedding fails;
3. $s-m+\frac{m}{r}=\max\{0,\frac{N}{p}-\frac{N}{r}\}$, there are three sub-cases:
1. if $p\leq r$, then the embedding $W^{s,p}(\Omega)\subset W^{m-\frac{m}{r},r}(\Omega)$ holds;
2. if $p>r$ and $m-\frac{m}{r}$ integer, the embedding $W^{s,p}(\Omega)\subset W^{m-\frac{m}{r},r}(\Omega)$ holds;
3. if $p>r$ and $m-\frac{m}{r}$ non-integer, the embedding fails.
In order to solve the optimality results, we just consider three cases: $$\begin{split}
&(1) 1<p\leq r, s+\frac{m}{r}<m+\frac{N}{p}-\frac{N}{r};\\
&(2)1< r<p, 0<s<m-\frac{m}{r};\\
&(3) 1<r<p, s=m-m/r ~\mbox{non-integer}.
\end{split}$$
Without loss of generality, one may assume that $n=N$, $(-8,8)^N\subset \Omega$, and $\bm{\alpha'}=(\alpha',\cdots,\alpha')$ with $\alpha'=(1,2,\cdot\cdot\cdot,r)$. First we establish the optimality results in case $1< r<p, 0<s<m-\frac{m}{r}$.
\[hm-pro-41\] Let $m, r$ be integers with $1< r\leq \underline{n}$, $p>r$ and $0<s<m-\frac{m}{r}$. Then there exist a sequence $\{u_k\}_{k=1}^{\infty} \subset C^{m}(\overline{\Omega}, \mathbb{R}^N)$ and a function $\psi\in C_c^{\infty}(\Omega)$ such that $$\label{hm-thm-for-1}
\lim_{k\rightarrow \infty} \|u_k\|_{s,p} =0, ~~~~\lim_{k\rightarrow \infty} \int_{\Omega} M^{\alpha'}_{\bm{\alpha'}}(D^mu_k) \psi dx=\infty.$$
For any integer $k$, we define $u_k: \Omega \rightarrow \mathbb{R}^N$ as $$u_k^i(x)= k^{-\rho} \sin (k x_i),~~1\leq i\leq r-1;~~~~
u_k^i(x)= 0,~~r< i\leq N$$ and $$u_k^r(x)= k^{-\rho} (x_r)^m \prod_{j=1}^{r-1} \sin (\frac{m\pi}{2}+k x_j).$$ Where $\rho$ is a constant such that $s<\rho<m-\frac{m}{r}$. Since $\|D^{[s]+1}u_k\|_{L^{\infty}}\leq C k^{[s]+1-\rho}$ and $\|u_k\|_{L^{\infty}}\leq C k^{-\rho}$, it follows that $$\|u_k\|_{s,p} \leq C\|u_k\|^{1-\theta}_{L^p}\|u_k\|^{\theta}_{[s]+1,p}\leq C k^{s-\rho}.$$ Where $\theta=\frac{s}{[s]+1}$. Let $\psi\in C^{\infty}_c(\Omega)$ be such that $$\label{hm-th2-for-2}
\psi(x)=\prod_{i=1}^N \psi'(x_i), ~\mbox{with}~\psi'\in C^1_c((0,\pi)), \psi'\geq 0 ~\mbox{and}~\psi'=1~ \mbox{in}~(\frac{1}{4}\pi,\frac{3}{4}\pi).$$ Then $$\int_{\Omega} M^{\alpha'}_{\bm{\alpha'}} (D^m u_k) \psi dx\geq m!\int_{(\frac{1}{4}\pi, \frac{3}{4}\pi)^N} k^{mr-\rho r-m} \prod_{j=1}^{r-1} \sin^2 (\frac{m\pi}{2}+kx_j) dx=Ck^{mr-\rho r-m}.$$ Hence the conclusion (\[hm-thm-for-1\]) holds.
Next we establishing the optimality results in case $1<r<p, s=m-m/r ~\mbox{non-integer}$ by constructing a lacunary sum of atoms, which is inspired by the work of Brezis and Nguyen [@BN].
\[hm-pro-42\] Let $m, r$ be integers with $1< r\leq \underline{n}$, $p>r$ and $s=m-m/r$ non-integer. Then there exist a sequence $\{u_k\}_{k=1}^{\infty} \subset C^{m}(\overline{\Omega}, \mathbb{R}^N)$ and a function $\psi\in C_c^{\infty}(\Omega)$ satisfying the conditions (\[hm-thm-for-1\]).
Fix $k>>1$. Define $v_k=(v_k^1,\cdots,v_k^N):\Omega\rightarrow \mathbb{R}^N$ as follows $$v_k^i=\begin{cases} \sum_{l=1}^k \frac{1}{n_l^{s}(l+1)^{\frac{1}{r}}} \sin( n_l x_i), ~~~~1\leq i\leq r-1;\\
(x_r)^m\sum_{l=1}^k \frac{1}{n_l^{s}(l+1)^{\frac{1}{r}}} \prod_{j=1}^{r-1} \sin(\frac{m\pi}{2}+n_l x_j),~~~~i=r;\\
0,~~~~~~~~r+1\leq i\leq N.
\end{cases}$$ Where $n_l=k^{\frac{r^2}{m}} 8^l$ for $1\leq l\leq k$. Let $\psi\in C^{\infty}_c(\Omega)$ be defined as (\[hm-th2-for-2\]). We claim that $$\label{hm-thm-for-2}
\|v_k\|_{s,p} \leq C,~~~~ \int_{\Omega} M^{\alpha'}_{\bm{\alpha'}}(D^mv_k) \psi dx \geq C \ln k,$$ where the constant $C$ is independent of $k$.
Assuming the claim holds, we deduce $u_k= (\ln k)^{-\frac{1}{2r}} v_k$ and $\psi$ satisfies the conditions (\[hm-thm-for-1\]). Hence it remains to prove (\[hm-thm-for-2\]).
On the one hand $$\begin{split}
M_{\bm{\alpha'}}^{\alpha'} (D^m v_k) &=\left\{ \prod_{i=1}^{r-1} \left( \sum_{l_i=1}^k \frac{n_{l_i}^{\frac{m}{r}}}{(l_i+1)^{\frac{1}{r}}} \sin(\frac{m\pi}{2}+n_{l_i} x_i)\right)\right\}\times \left(m! \sum_{l_r=1}^k \frac{1}{n_{l_r}^{s}(l_r+1)^{\frac{1}{r}}} \prod_{j=1}^{r-1} \sin(\frac{m\pi}{2}+n_{l_r} x_j) \right) \\
&=m! \sum_{(l_1,\cdot\cdot\cdot,l_r)\in G} \frac{1}{n_{l_r}^{s}(l_r+1)^{\frac{1}{r}}} \prod_{i=1}^{r-1} \left(\frac{n_{l_i}^{\frac{m}{r}}}{(l_i+1)^{\frac{1}{r}}} \sin(\frac{m\pi}{2}+n_{l_i} x_i) \sin(\frac{m\pi}{2}+n_{l_r} x_i) \right)\\
&+m! \sum_{l=1}^k \frac{1}{l+1} \prod_{i=1}^{r-1} \sin^2(\frac{m\pi}{2}+n_l x_i),
\end{split}$$ where $$G:=\{(l_1,\cdot\cdot\cdot,l_r)\mid (l_1,\cdot\cdot\cdot,l_r)\neq (l,\cdot\cdot\cdot,l) ~\mbox{for}~l,l_1,\cdot\cdot\cdot,l_r=1,\cdot\cdot\cdot,k\}.$$
Hence $$\label{hm-thm-for-6}
\int_{\Omega} M^{\beta}_{\bm{\alpha'}}(D^mv_k) \psi dx\geq C \sum_{l=1}^k \frac{1}{l+1} \int_{(\frac{1}{4}\pi, \frac{3}{4}\pi)^N} \prod_{i=1}^{r-1} \sin^2(\frac{m\pi}{2}+n_l x_i) dx-CI,$$ where $$I:= \left| \int_{\Omega} \psi(x) \sum_{(l_1,\cdot\cdot\cdot,l_r)\in G} \frac{1}{n_{l_r}^{s}(l_r+1)^{\frac{1}{r}}} \prod_{i=1}^{r-1} \left(\frac{n_{l_i}^{\frac{m}{r}}}{(l_i+1)^{\frac{1}{r}}} \sin(\frac{m\pi}{2}+n_{l_i} x_i) \sin(\frac{m\pi}{2}+n_{l_r} x_i) \right) dx\right|.$$ Since $n_l=k^{\frac{r^2}{m}} 8^l$, it follows that $$\label{hm-thm-for-3}
\frac{n_{l_i}}{n_{l_j}}\leq |n_{l_i}-n_{l_j}|~\mbox{for any}~l_i,l_j=1,\cdot\cdot\cdot,k~\mbox{with}~l_i\neq l_j,$$ $$\label{hm-thm-for-4}
\min_{i\neq j} |n_{l_i}-n_{l_j}|\geq k^{\frac{r^2}{m(r-1)}}$$ and $$\label{hm-thm-for-5}
\{n_l\mid l=1,\cdot\cdot\cdot,k\}\cap \{z\in \mathbb{R}\mid 2^{n-1}\leq |z|< 2^n\}~\mbox{has at most one element for any }~n\in \mathbb{N}.$$ For any $(l_1,\cdot\cdot\cdot,l_r)\in G$, there exists $1\leq i_0\leq r-1$ such that $l_{i_0}\neq l_r$, it follows from (\[hm-th2-for-2\]), (\[hm-thm-for-3\]) and (\[hm-thm-for-4\]) that $$\begin{split}
&\left|\frac{1}{n_{l_r}^{s}(l_r+1)^{\frac{1}{r}}} \int_{\Omega} \psi(x) \prod_{i=1}^{r-1} \left(\frac{n_{l_i}^{\frac{m}{r}}}{(l_i+1)^{\frac{1}{r}}} \sin(\frac{m\pi}{2}+n_{l_i} x_i) \sin(\frac{m\pi}{2}+n_{l_r} x_i) \right) dx\right|\\
&\leq \frac{C}{n_{l_r}^{s}(l_r+1)^{\frac{1}{r}}} \prod_{i=1}^{r-1} \frac{n_{l_i}^{\frac{m}{r}}}{(l_i+1)^{\frac{1}{r}}} \left|\int_0^{\pi} \psi'(x_i) \sin(\frac{m\pi}{2}+n_{l_i} x_i) \sin(\frac{m\pi}{2}+n_{l_r} x_i) dx_i \right|\\
&\leq C \prod_{i=1}^{r-1} \left(\frac{n_{l_i}}{n_{l_r}}\right)^{\frac{m}{r}} \min \{\frac{1}{|n_{l_i}-n_{l_r}|^m},1\} \|D^m\psi\|_{L^{\infty}}\\
&\leq \frac{C}{|n_{l_{i_0}}-n_{l_r}|^{m-\frac{m}{r}}}\\
&\leq C k^{-r}.
\end{split}$$ Combine with (\[hm-thm-for-6\]), we find $$\int_{\Omega} M^{\alpha'}_{\bm{\alpha'}}(D^mv_k) \psi dx\geq C \sum_{l=1}^k \frac{1}{l+1} -C,$$ which implies the second inequality of (\[hm-thm-for-2\]). On the other hand, in order to prove the first inequality of (\[hm-thm-for-2\]), it is enough to show that $$\label{hm-thm-for-10}
\|v'_k\|_{s,p} \leq C,$$ where $v'_k:=(v^1_k,v^2_k,\cdot\cdot\cdot,v^{r-1}_k,\frac{v^r_k}{(x_r)^m})$. In fact, the Littlewood-Paley characterization of the Besov space $B^s_{p,p}([0,2\pi]^N)$ (e.g. [@TH]) implies that $$\label{hm-thm-for-11}
\|v'_k\|_{s,p}\leq C \left(\|v'_k\|^p_{L^p([0,2\pi]^N)}+\sum_{j=1}^{\infty} 2^{sjp}\|T_j(v'_k)\|^p_{L^p([0,2\pi]^N)}\right)^{\frac{1}{p}}.$$ Here the bounded operators $T_j:L^p\rightarrow L^p$ are defined by $$T_j\left(\sum a_n e^{in\cdot x}\right)=\sum_{2^j\leq |n|< 2^{j+1}} \left( \rho(\frac{|n|}{2^{j+1}})- \rho(\frac{|n|}{2^{j}})\right)a_ne^{in\cdot x},$$ where $\rho\in C_c^{\infty}(\mathbb{R})$ is a suitably chosen bump function. Then we have $$\label{hm-thm-for-12}
\|T_j(v'_k)\|^p_{L^p([0,2\pi]^N)}\leq C_p \sum_{l=1}^k \frac{1}{n_l^{sp}(l+1)^{\frac{p}{r}}} \|T_j(g_{l,k})\|^p_{L^p([0,2\pi]^N)},$$ where $g_{l,k}=(\sin(n_l x_1),\cdot\cdot\cdot,\sin(n_l x_{r-1}),\prod_{j=1}^{r-1} \sin(\frac{m\pi}{2}+n_l x_j))$. Indeed, since $\sin(n_l x_i)=\frac{1}{2i}(e^{in_lx_i}-e^{-in_lx_i})$, $g_{l,k}$ can be written as $$g_{l,k}(x)= \sum_{\varepsilon\in \{-1,0,1\}^{r-1}} a_{\varepsilon} e^{n_l i \varepsilon\cdot \widehat{x}},$$ where $\widehat{x}=(x_1,\cdot\cdot\cdot,x_{r-1})$, $|a_{\varepsilon}|\leq 1$ for any $\varepsilon$. Set $$S(j,l)=\{\varepsilon\in \{-1,0,1\}^{r-1}\mid 2^{j-1}\leq n_l |\varepsilon|<2^{j+2}\}$$ and $$\chi(j,l)=\begin{cases}
1~~~~~S(j,l)\neq \emptyset\\
0~~~~~S(j,l)= \emptyset
\end{cases}.$$ Hence $$\label{hm-thm-for-13}
\|T_j(g_{l,k})\|^p_{L^p([0,2\pi]^N)}\leq C_{r,N} \chi(j,l).$$ For any $j$, if $S(j,l)\neq \emptyset$, then $\frac{2^{j-1}}{\sqrt{r-1}}\leq n_l< 2^{j+2}$, which implies that $\sum_{l=1}^{k}\chi(j,l)<[\frac{\log_2(r-1)}{6}]+1$. Thus, applying (\[hm-thm-for-11\]), (\[hm-thm-for-12\]) and (\[hm-thm-for-13\]), we have $$\begin{split}
\|v'_k\|_{s,p}^p&\leq C_{p,s,N,r}\left( \|v'_k\|^p_{L^p([0,2\pi]^N)}+\sum_{j=1}^{\infty} \sum_{l=1}^k\frac{2^{sjp}}{n_l^{sp}(l+1)^{\frac{p}{r}}}\chi(j,l)\right)\\
&\leq C_{p,s,N,r}\left( \|v'_k\|^p_{L^p([0,2\pi]^N)}+ \sum_{l=1}^k\frac{1}{(l+1)^{\frac{p}{r}}} \left(\sum_{j=1}^{\infty}\chi(j,l)\right)\right).
\end{split}$$ which implies (\[hm-thm-for-10\]) since $\sum_{j=1}^{\infty}\chi(j,l)\leq [\frac{\log_2(r-1)}{2}]+4$ for any $l$.
Clearly Theorem \[hm-thm-2\] is a consequence of Proposition \[hm-pro-41\] and \[hm-pro-42\] as explained in Remark \[hm-rem-41\].
Next we pay attention to the optimality results in case $1<p\leq r, s+\frac{m}{r}<m+\frac{N}{p}-\frac{N}{r}$.
\[hm-pro-43\] Let $m, r$ be integers with $1<p\leq r\leq \underline{n}$ and $s+\frac{m}{r}<m+\frac{N}{p}-\frac{N}{r}$. If there exist a function $g\in C_c^{\infty}(B(0,1), \mathbb{R}^n)$, $\beta\in I(r,n)$ and $\bm{\alpha}=(\alpha^1,\alpha^2,\cdot\cdot\cdot,\alpha^m)$ with $\alpha^j \in I(r,N)$ such that $$\label{hm-thm-for-15}
\int_{B(0,1)} M_{\bm{\alpha}}^{\beta} (D^m g(x)) |x|^m dx \neq 0.$$ Then there exist a sequence $\{u_k\}_{k=1}^{\infty} \subset C^{m}(\overline{\Omega}, \mathbb{R}^N)$ and a function $\psi\in C_c^{\infty}(\Omega)$ satisfying the conclusions (\[hm-thm-2\]).
For any $0<\varepsilon<<1$ we set $$u_{\varepsilon}=\varepsilon^\rho g(\frac{x}{\varepsilon}),$$ where $\rho$ is a constant such that $s-\frac{N}{p}<\rho<m-\frac{N}{r}-\frac{m}{r}$.
On the one hand, Lemma \[hm-lem-4\] implies that $$\|u_{\varepsilon}\|_{s,p}\leq C\|u_{\varepsilon}\|^{\theta}_{L^p}\|u_{\varepsilon}\|^{1-\theta}_{[s]+1,p}\leq C\varepsilon^{\rho+\frac{N}{p}-s}\|g\|^{\theta}_{L^p}\|D^{[s]+1}g\|_{L^p}^{1-\theta},$$ where $\theta=\frac{[s]+1-s}{[s]+1}$. On the other hand, let $\psi\in C^{\infty}_c(\Omega)$ be such that $\psi(x)= |x|^m+ O(|x|^{m+1})$ as $x\rightarrow 0$. Then $$\begin{split}
&\int_{\Omega} M_{\bm{\alpha}}^{\beta} (D^m u_{\varepsilon}) \psi dx=\varepsilon^{\rho r-rm+N} \int_{B(0,1)} M_{\bm{\alpha}}^{\beta}(D^m g(x)) \psi(\varepsilon x) dx\\
&=\varepsilon^{\rho r -rm+N+m} \int_{B(0,1)} M_{\alpha}^{\beta}(D^mg(x)) |x|^m dx +O( \varepsilon^{\rho r-rm+N+m+1}).
\end{split}$$ Take $\varepsilon = \frac{1}{k}$ and hence the conclusion is proved.
In order to establishing the optimality results in case $1<p\leq r, s+\frac{m}{r}<m+\frac{N}{p}-\frac{N}{r}$, a natural problem is raised whether there exists $g\in C_c^{\infty}(B(0,1), \mathbb{R}^N)$ such that the conclusion (\[hm-thm-for-15\]) holds. We have positive answers to the problem in case $m=1$ or $2$, see Theorem \[hm-thm-3\], according the following Lemma:
\[hm-lem-3\] Let $g\in C_c^{\infty}(B(0,1))$ be given as $$\label{hm-lem-for-1}
g(x)=\int_0^{|x|} h(\rho) d\rho$$ for any $x\in \mathbb{R}^N$, where $h\in C_c^{\infty}((0,1))$ and satisfies $$\int_0^1 h(\rho)d\rho=0,~~~~\int_0^1h^{r}(\rho)\rho^{-r+N+s-1} d\rho\neq 0.$$ Here $r\geq 2,s\geq 1$ are integers. Then for any $\alpha \in I(r,N)$, we have $$\label{hm-lem-for-2}
\int_{B(0,1)} M_{\alpha}^{\alpha}( D^2 g(x)) |x|^s dx\neq 0.$$
It is easy to see that $$D^2 g=\frac{1}{|x|^3}(A+B),$$ where $A=(a_{ij})_{N\times N}$ and $B=(b_{ij})_{N\times N}$ are $N\times N$ matrices such that $$a_{ij}=h(|x|)|x|^2\delta_{i}^{j},~~b_{ij}=\left(h'(|x|)|x|-h(|x|)\right)x_ix_j,~~~~i,j=1,\ldots,N.$$ Using Binet formula and the fact $\mbox{rank}(B)=1$, one has $$\begin{aligned}
M_{\alpha}^{\alpha}(A+B)&=M_{\alpha}^{\alpha}(A)+\sum_{i\in \alpha}\sum_{j\in\alpha}\sigma(i,\alpha-i)\sigma(j,\alpha-j)b_{ij}M_{\alpha-i}^{\alpha-j}(A)\\
&=h^r(|x|)|x|^{2r} - h^r(|x|)|x|^{2r-2}\sum_{i\in\alpha} x_i^2+h^{r-1}(|x|)h'(|x|)|x|^{2r-1} \sum_{i\in\alpha} x_i^2,\end{aligned}$$ Hence $$\begin{aligned}
\int_{B(0,1)}M_{\alpha}^{\alpha}( D^2 g)|x|^s dx=\int_{B(0,1)} |x|^{-3r+s}M_{\alpha}^{\alpha}(A+B) dx={\setcounter{RomanNumber}{1}\Roman{RomanNumber}}-{\setcounter{RomanNumber}{2}\Roman{RomanNumber}}+{\setcounter{RomanNumber}{3}\Roman{RomanNumber}},\end{aligned}$$ where $${\setcounter{RomanNumber}{1}\Roman{RomanNumber}}:=\int_{B(0,1)} h^r(|x|) |x|^{-r+s} dx,$$ $${\setcounter{RomanNumber}{2}\Roman{RomanNumber}}:=\int_{B(0,1)} h^r(|x|) |x|^{-r-2+s} \sum_{i\in \alpha} x_i^2dx,$$ and $${\setcounter{RomanNumber}{3}\Roman{RomanNumber}}:=\int_{B(0,1)} h^{r-1}(|x|)h'(|x|)|x|^{-r-1+s} \sum_{i\in \alpha} x_i^2dx.$$ Then integration in polar coordinates gives $${\setcounter{RomanNumber}{3}\Roman{RomanNumber}}=\frac{r-N-s}{N} 2\pi\prod_{i=1}^{N-2}I(i) \int_0^1 h^r(\rho) \rho^{-r+N+s-1} d\rho,$$ where $I(i)=\int_0^{\pi} \sin^i \theta d\theta$. Similarly, $${\setcounter{RomanNumber}{2}\Roman{RomanNumber}}=\frac{r}{N} 2\pi\prod_{i=1}^{N-2}I(i) \int_0^1 h^r(\rho) r^{-r+N+s-1} d\rho,$$ and $${\setcounter{RomanNumber}{1}\Roman{RomanNumber}}= 2\pi\prod_{i=1}^{N-2}I(i) \int_0^1 h^r(\rho) \rho^{-r+N+s-1} d\rho,$$ which implies (\[hm-lem-for-2\]), and then the proof is complete.
Note that if $m=2$ and $g= (g',\cdots,g')$ with $g'\in C^{2}(\Omega)$, then Lemma \[hm-lem-2-2\] implies $$M_{\bm{\alpha}}^{\alpha}(D^2g)=r! M_{\alpha^1}^{\alpha^2}(D^2g')$$ for any $\bm{\alpha}=(\alpha^1,\alpha^2)$, $\alpha\in I(r,N)$. Hence Theorem \[hm-thm-3\] is the consequence of Proposition \[hm-pro-41\], \[hm-pro-42\], \[hm-pro-43\] and Lemma \[hm-lem-3\].
In particular, we can give a reinforced versions of optimal results in case $m=2$.
\[hm-thm-4-14\] Let $1< r\leq N$, $1<p<\infty$ and $0<s<\infty$ be such that $W^{s,p}(\Omega) \nsubseteq W^{2-\frac{2}{r},r}(\Omega)$. Then there exist a sequence $\{u_k\}_{k=1}^{\infty} \subset C^{m}(\overline{\Omega})$ and a function $\psi\in C_c^{\infty}(\Omega)$ such that $$\lim_{k\rightarrow \infty} \|u_k\|_{s,p} =0, ~~~~\lim_{k\rightarrow \infty} \int_{\Omega} M^{\alpha'}_{\alpha'}(D^2u_k) \psi dx=\infty.$$
We divide our proof in three case:
**Case 1:** $1<p\leq r$ and $s+\frac{2}{r}<2+\frac{N}{p}-\frac{N}{r}$
Apply Lemma \[hm-lem-3\] and the argument similar to one used in Proposition \[hm-pro-43\].
**Case 2:** $r<p$ and $0<s<2-\frac{2}{r}$
For $k>>1$, we set $$u_k:=k^{-\rho} x_{r}\Pi_{i=1}^{r-1} \sin^2(kx_i),$$ where $\rho$ is a constant with $s<\rho<2-\frac{2}{r}$. According to the facts that $\|u_k\|_{L^{\infty}}\leq C k^{-\rho}$ and $\|D^2u_k\|_{L^{\infty}}\leq C k^{2-\rho}$, it follows that $$\|u_k\|_{s,p}\leq C \|u_k\|_{L^p}^{1-\frac{s}{2}} \|u_k\|_{2,p}^{\frac{s}{2}}\leq C k^{s-\rho}.$$ On the other hand, Let $\psi\in C^{\infty}_c(\Omega)$ be defined as (\[hm-th2-for-2\]), the (4.1) in [@BJ Proposition 4.1] implies that $$\begin{split}
&\left |\int_{\Omega} M_{\alpha'}^{\alpha'} (D^2 u_k) \psi dx\right|\geq \left| \int_{(\frac{1}{4}\pi, \frac{3}{4}\pi)^N} M_{\alpha'}^{\alpha'}(D^2u_{k})dx\right|\\
&\geq k^{2r-2-r\rho} 2^r \int_{(\frac{1}{4}\pi, \frac{3}{4}\pi)^N} x_r^{r-2} \left(\prod_{i=1}^{r-1} \sin(kx_i)\right)^{2r-2} \left(\sum_{j=1}^{r-1} \cos^2(kx_j)\right)dx\\
&=Ck^{2r-2-r\rho}.
\end{split}$$
**Case 3:** $2<r<p$ and $s=2-\frac{2}{r}$
For any $k\in \mathbb{N}$ with $k\geq 2$, define $u_k$ with $$u_k(x)=\frac{1}{(\ln k)^{\frac{1}{2r}}} x_r \sum_{l=1}^k \frac{1}{n_l^{2-\frac{2}{r}} l^{\frac{1}{r}}} \prod_{i=1}^{r-1} \sin^2 (n_l x_i)~~~~x\in \mathbb{R}^N,$$ where $n_l=k^{r^{3l}}$. Let $\psi\in C^{\infty}_c(\Omega)$ be defined as (\[hm-th2-for-2\]). The argument similar to the one used in [@BJ Proposition 5.1] shows that $$\|u_k\|_{W^{s,p}(\Omega)}\leq C \|u_k\|_{W^{s,p}((0,2\pi)^N)}\leq C \frac{1}{(\ln k)^{\frac{1}{2r}}}$$ and $$\left| \int_{\Omega} M_{\alpha'}^{\alpha'}(D^2u_{k}) \psi dx\right|=C \left| \int_{(0,2\pi)^{r}} M_{\alpha'}^{\alpha'}(D^2u_{k}) \prod_{i=1}^r \psi'(x_i) dx_1\cdot\cdot\cdot dx_r\right|
\geq C (\ln k)^{\frac{1}{2}}.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by NSF grant of China ( No. 11131005, No. 11301400) and Hubei Key Laboratory of Applied Mathematics (Hubei University).
[1]{}
E. Baer and D. Jerison, *Optimal function spaces for continuity of the Hessian determinant as a distribution*, J. Funct. Anal., **269** (2015), 1482-1514.
J. Ball, *Convexity conditions and existence theorems in nonlinear elasticity*, Arch. Ration. Mech. Anal., **63** (1977), 337-403.
H. Brezis and P. Mironescu, *Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces,* J. Evol. Equ., **4** (2001), 387-404.
H. Brezis and P. Mironescu, *Gagliardo-Nirenberg inequalities and non-inequalities: The full story,* Ann. Inst. H. Poincaré Anal. Non Lin$\acute{e}$aire, **35** (2018), 1355-1376.
H. Brezis and H. Nguyen, *The Jacobian determinant revisited,* Invent. Math., **185** (2011), 17-54.
L. D’Onofrio, F. Giannetti and L. Greco, *On weak Hessian determinants*, Rend. Mat. Acc. Lincei (9), **16** (2005), 159-169.
B. Dacorogna and F. Murat, *On the optimality of certain Sobolev exponents for the weak continuity of determinants*, J. Funct. Anal., **105** (1992), 42-62.
G. Escherich, *Die Determinanten hoheren Ranges und ihre Verwendung zur Bildung von Invarianten*, Denkshr. Kais. Akad. Wiss., **43** (1882), 1-12.
I. Fonseca and J. Mal$\acute{y}$, *From Jacobian to Hessian: distributinal form and relaxation*, Riv. Mat. Univ. Parma, **7** (2005), 45-74.
E. Gagliardo, *Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili*, Rend. Semin. Mat. Univ. Padova, **27** (1957), 284-305.
L. Gegenbauer, *$\ddot{U}$ber Determinanten hoheren Ranges*, Denkshr. Kais. Akad. Wiss., **43** (1882), 17-32.
M. Giaquinta, G. Modica and J. Souček, *Cartesian currents in the calculus of variations, I, II,* Springer-Verlag, Berlin, 1998.
T. Iwaniec, *On the concept of the weak Jacobian and Hessian*, Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., **83** (2001) 181-205, Univ. Jyväskylä, Jyväskyla.
C. Morrey, *Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften*, vol. 130, Springer, New York(1966). Y. Reshetnyak, *The weak convergence of completely additive vector-valued set functions*, Sib. Mat. Zh., **9** (1968), 1386-1394.
P. Olver, *Hyper-Jacobians, determinantal ideals and weak solutions to variational problems*, Proc. Roy. Soc. Edinburgh Sect. A, **3-4** (1983), 317-340.
E. Stein, *The characterization of functions arising as potentials I*, Bull. Amer. Math. Soc., **67** (1961), 102-104.
E. Stein, *The characterization of functions arising as potentials II*, Bull. Amer. Math. Soc., **68** (1962), 577-582.
H. Treibel, *Theory of Functions Spaces*, Monogr. Math., **78**, Birkhäuser Verlag, Basel, 1983.
[^1]: *Email addresses*: qiangtu@whu.edu.cn(Qiang Tu), cxwu@hubu.edu.cn (Chuanxi Wu), qiuxueting1996@163.com (Xueting Qiu).
|
---
abstract: 'We introduce the light-weight carbon fiber and aluminum gondola designed for the [<span style="font-variant:small-caps;">Spider</span>]{} balloon-borne telescope. [<span style="font-variant:small-caps;">Spider</span>]{} is designed to measure the polarization of the Cosmic Microwave Background radiation with unprecedented sensitivity and control of systematics in search of the imprint of inflation: a period of exponential expansion in the early Universe. The requirements of this balloon-borne instrument put tight constrains on the mass budget of the payload. The [<span style="font-variant:small-caps;">Spider</span>]{} gondola is designed to house the experiment and guarantee its operational and structural integrity during its balloon-borne flight, while using less than 10% of the total mass of the payload. We present a construction method for the gondola based on carbon fiber reinforced polymer tubes with aluminum inserts and aluminum multi-tube joints. We describe the validation of the model through Finite Element Analysis and mechanical tests.'
author:
- |
J.$\,$D. Soler, P. A. R. Ade, M. Amiri, S. J. Benton, J. J. Bock, J. R. Bond, S. A. Bryan, C. Chiang, C. C. Contaldi, B. P. Crill, O. P. Doré, M. Farhang, J. P. Filippini, L. M. Fissel, A. A. Fraisse, A. E. Gambrel, N. N. Gandilo, S. Golwala, J. E. Gudmundsson, M. Halpern, M. Hasselfield, G. C. Hilton, W. A. Holmes, V. V. Hristov, K. D. Irwin, W. C. Jones, Z. D. Kermish, C.-L. Kuo, C. J. MacTavish, P. V. Mason, K. G. Megerian, L. Moncelsi, T. Morford, J. M. Nagy, C. B. Netterfield, R. O’Brient, A. S. Rahlin, C. D. Reintsema, J. E. Ruhl, M. C. Runyan, J. A. Shariff, A. Trangsrud, C. Tucker, R. S. Tucker, A. D. Turner, A. C. Weber, D. V. Wiebe, E. Y. Young Institute d’Astrophysique Spatiale. CNRS. France;\
Department of Astronomy and Astrophysics. University of Toronto. Canada;\
School of Physics & Astronomy. Cardiff University. UK;\
Department of Physics & Astronomy. University of British Columbia. Canada;\
Department of Physics. University of Toronto. Canada;\
Department of Physics. California Institute of Technology. USA;\
CITA. University of Toronto. Canada;\
Canadian Institute for Advanced Research. Canada;\
Jet Propulsion Laboratory. USA;\
Department of Physics. Case Western Reserve University. USA;\
Department of Physics. Princeton University. USA;\
Department of Physics. Stanford University. USA;\
School of Mathematics, Statistics & Computer Science. University of KwaZulu-Natal. South Africa;\
Theoretical Physics Blackett Laboratory. Imperial College. UK;\
National Institute of Standards and Technology. USA;
bibliography:
- 'InstrumentationRefs.bib'
- 'jdslib.bib'
- 'CMBrefs.bib'
- 'ISMrefs.bib'
title: 'Design and construction of a carbon fiber gondola for the [<span style="font-variant:small-caps;">**Spider**</span>]{} balloon-borne telescope'
---
INTRODUCTION {#sec:intro}
============
Balloon-borne experiments are limited in mass by the buoyancy of the helium balloons used to lift them. The mass available for scientific equipment on board the payload is constrained by the mass of the structural elements that guarantee the integrity of the experiment during the balloon-borne flight. The structure of an ideal payload must therefore combine durability and high strength per unit of mass. The optimization for high strength and low weight is possible using composite materials and aggressive light-weighting techniques derived from the results of detailed Finite Element Analysis (FEA).
This document describes the design of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola: a pointed platform, which combines composite materials with light-weighted aluminum elements to provide support for the [<span style="font-variant:small-caps;">Spider</span>]{} instrument [@filippini2010; @fraisse2013; @rahlin2014]. The design of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola was motivated by the large volume cryogenic vessel necessary to achieve the goals of the experiment [@fraisse2013; @odea2011]. The structure was custom made to accommodate the cryostat, which houses a set of six telescopes. This cryostat is larger and heavier than any previous balloon-borne experiment cryostats[@crill2003; @maciasperez2007; @fissel2010; @reichborn2010]. The [<span style="font-variant:small-caps;">Spider</span>]{} gondola also allows the motion of the cryostat in azimuth and elevation and protects its structural integrity during flight and after termination. Some of the products of the [<span style="font-variant:small-caps;">Spider</span>]{} design such as the carbon fiber and aluminum sunshields as well as the pivot have successfully flown in the BLASTPol Long Duration Balloon (LDB) flight from Antarctica in December 2010 (BLASTPol10) and 2012 (BLASTPol12)[@fissel2010; @pascale2008].
The technique used for the design of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola is modular and can be extended to future balloon-borne experiments. Many of the parts, such as the outer frame multi-tube joints, are custom made but can be easily adapted to other payload geometries. Others, such as the sunshield hubs and inserts, can be used to produce structures with multiple geometries. The motor assemblies and elevation drive sets can be scaled to accommodate the necessities of other balloon-borne instruments. The specific requirements of the gondola change from experiment to experiment and even from campaign to campaign. However, the state of the art Computer-Assisted Design (CAD) tools and novel composite materials used in the [<span style="font-variant:small-caps;">Spider</span>]{} design can be applied to a broad range of experiments.
This proceeding is organized as follows: Section \[mecha:intro\] describes the structural requirements for the balloon-borne platform and presents the critical scenarios considered for its design. Section \[mecha:suspensionelements\] introduces the suspension elements, which link the payload to the flight train. Section \[mecha:outerframe\] describes the design and construction of the carbon fiber and aluminum structure composing the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame. Finally, Section \[mecha:sunshields\] presents the design and construction techniques used for the sunshield frames.
![Rendering of the [<span style="font-variant:small-caps;">Spider</span>]{} LDB cryostat. The corresponding dimensions are in millimeters.[]{data-label="mecha:spidercryostat"}](SpiderLDBCryostatViews.eps){height="0.25\textheight"}
Design Benchmarks {#mecha:intro}
=================
The size of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola is directly related to the scientific goals of the experiment. The size of the [<span style="font-variant:small-caps;">Spider</span>]{} refracting telescopes is determined by the angular resolution adequate for the observation of the imprint of primordial gravitational waves in the polarization of the Cosmic Microwave Background (CMB) polarization[@seljak1997]. The size of the cryostat is determined by the diameter and length of the telescope inserts plus the volume necessary to hold enough cryogens to achieve the necessary integration times. In sum, the size of the gondola is defined by the dimensions of the cryostat, illustrated in Figure \[mecha:spidercryostat\].
Further geometrical limits are set by the space in the high bay where the gondola is assembled in Antarctica and the limits of the deployment vehicle. The launch procedure requires that no part of the gondola intersects a plane 20 from the vertical axis passing through the attachment point between the launch vehicle and the gondola as shown in Figure \[LaunchVehicle\]. Additionally, in order to transport the gondola to the integration and launch location, the separate components of the gondola must fit in standard shipping containers with interior dimensions 2.38 m in height by 2.34$\,$m in width, and can be 5.71$\,$m or 12.03$\,$m deep according to the International Organization for Standardization (ISO) standard 6346.
![Geometrical limits for the gondola design set by the Columbia Scientific Balloon Facility (CSBF) deployment vehicle in Antarctica [@CSBF2011].[]{data-label="LaunchVehicle"}](LaunchVehicleRequirementsAntarctica.eps){width="0.85\linewidth"}
The balloons and the facilities necessary to fly [<span style="font-variant:small-caps;">Spider</span>]{} are provided by the Columbia Scientific Balloon Facility (CSBF), a division of the National Aeronautics and Space Administration (NASA) dedicated to balloon flights. The maximum mass of the scientific payload is approximately 2268$\,$kg (5000$\,$lb), which corresponds to the maximum gross lift of the balloon (8000$\,$lb) minus the mass of the balloon itself, the flight train, the Support Instrumentation Package (SIP), the solar arrays, the ballast hopper and enough ballast to maintain the altitude during the duration of the flight. The dry mass of the [<span style="font-variant:small-caps;">Spider</span>]{} cryostat is roughly $\sim$1600$\,$kg (3500$\,$lb) when fully integrated and loaded with cryogens. The remaining 668$\,$kg (1500$\,$lb) are distributed between the motorized systems, which allow pointing in azimuth and elevation, the Attitude Control System (ACS), the flight computers, the serial hub, batteries, the solar array, pointing sensors, sunshields, miscellaneous electronic boxes, and the frame, which supports the whole assembly.
The [<span style="font-variant:small-caps;">Spider</span>]{} Gondola
--------------------------------------------------------------------
The [<span style="font-variant:small-caps;">Spider</span>]{} payload, shown fully assembled during compatibility test in Figure \[FullGondola\], is composed of three main parts: the outer frame or gondola, made out of aluminum and Carbon Fiber Reinforced Polymer (CFRP) tubes; the cryostat, which is effectively an inner frame trunnion-mounted onto the outer frame at two points along a horizontal axis; and a set of sunshields, which attaches to the outer frame.
![Left: Schematic drawing of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola including a 1$\,$m-tall penguin for scale. Right: photograph of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola fully assembled during CSBF compatibility test in 2013.[]{data-label="FullGondola"}](SpiderFrontPenguinandLabels.eps "fig:"){height="0.42\textheight"} ![Left: Schematic drawing of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola including a 1$\,$m-tall penguin for scale. Right: photograph of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola fully assembled during CSBF compatibility test in 2013.[]{data-label="FullGondola"}](FullGondola.eps "fig:"){height="0.42\textheight"}
The payload is designed to work in two configurations. In the *laboratory configuration*, the gondola sits on top of a customized aluminum cart with pneumatic tires, which allow moving the gondola without the need of a hoist. In the *flight configuration*, the pivot hangs from a hoist and the gondola is suspended from the cables. The blocks attaching the outer frame to the cart are removed and the ballast hooper and the solar arrays are attached to the square frame beneath the SIP.
The outer frame is a customized truss structure composed of CFRP tubes with aluminum inserts, which are fastened together by multi-tube aluminum joints. It is suspended from the pivot by three cables as illustrated in Figure \[mecha:suspensionangles\]. All the beams in the structure are distributed in triangles in order to maintain axial forces and minimize moment on truss elements. All angles and distances have been chosen in order to locate the payload elements and optimize the force distribution. The technique used to construct the outer frame is described in Section \[mecha:outerframe\].
The outer frame geometry allows 360 rotation of the cryostat around the trunnions. This feature facilitates the integration of the gondola and the installation of the telescope inserts in the cryostat. During the flight, the elevation limits are set by the elevation drive. The telescope can point between 20 and 50 in elevation. The minimum elevation corresponds to the orientation where the telescope observes the ground. The maximum elevation is the orientation where the telescope sees the balloon. Both values are obtained by modeling the [<span style="font-variant:small-caps;">Spider</span>]{} beams as a set of truncated cones with a 10 aperture and coincident with the telescope apertures.
The entire gondola can rotate to any azimuthal angle. The center-of-mass of the system is on the rotational axis so that translation of the gondola does not generate torques that re-orient the telescope. The limits of observation in azimuth are constrained by the exposure of the telescope to direct sunlight. Therefore, the geometry of the sunshields defines the area of the sky, which can be observed by the experiment [@soler2014therm].
![Position of suspension cables and hanging angles indicated on a front (shown in the left hand side) and a right view of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola.[]{data-label="mecha:suspensionangles"}](SpiderGondolaCables.eps){height="0.4\textheight"}
The [<span style="font-variant:small-caps;">Spider</span>]{} cryostat was designed and manufactured by RedStone Aerospace. The structural support of the interior components of the cryostat relies on G10/aluminum flexures symmetrically placed on the cylinder sides, as described in previous SPIE proceedings[@gudmundsson2010; @runyan2010]. The analysis of the structural integrity of the interior of the cryostat is beyond the scope of this document but it is described on the Redstone Aerospace design reports[@redstone2008].
The [<span style="font-variant:small-caps;">Spider</span>]{} sunshield is also a truss structure composed of CFRP tubes with aluminum inserts, which are fastened together by multi-tube aluminum joints. The sunshields have less demanding structural requirements than the outer frame; thus the technique used for their construction is modular and less restrictive as discussed in Section \[mecha:sunshields\].
Critical Design Scenarios {#DesignScenarios}
-------------------------
The geometry of the outer frame was defined through a series of beam mesh simulations made with the SolidWorks[^^]{} Simulation package. In these simulations, the frame is modeled by a series of truss elements, which have specified cross sections and material properties. The trusses are connected with ideal nodes, which constrain rotation or translation depending on the design scenario. Gravity and accelerations can be defined in the model and loads can be applied to the nodes and trusses. The result of the beam mesh simulation is a load table, which presents the axial forces, bending forces, and stresses on multiple segments of each truss. The beam mesh model is run and tested in a set of critical design scenarios defined by large accelerations and shocks suffered by the structure during the balloon-borne flight.
Initially, the beam mesh synthetic model is used for optimizing the distribution of the truss elements. The resulting load tables are used to test and select the materials of the frame. The selection of the CFRP tubes in the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame is made using a series of beam mesh simulations using tubes of different diameters and wall thicknesses. Once the tubes are selected, the load tables from each study are used as an input for the simulation of individual joints and inserts. The following are the critical scenarios considered in the design of [<span style="font-variant:small-caps;">Spider</span>]{}.
### Chute Shock {#DS:ParachuteShock}
All structural components of the gondola must survive a vertical acceleration of 97.8$\,$m$\,$s$^{-2}$ (10$\,$g). Such acceleration results from the parachute opening and breaking the free-fall of the gondola after balloon termination. The pull produced by the parachute is named chute-shock, and it is described as “*a load 10 times the weight of the payload applied vertically at the suspension point*” according to the CSBF structural requirements for balloon gondolas [@CSBF2011]. This is the dominant scenario in the design considerations of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola as it is the one, which translates into largest forces on the structural elements.
The simulation of this scenario was made using two redundant models. The first was made by applying the forces corresponding to the tension on the suspension cables to the corresponding nodes and constraining the nodes and trusses where the main loads are located. The second was made by assigning the mass of the different components to their corresponding support trusses and nodes and then applying a vertical acceleration of 97.8 m s$^{-2}$. Both methods overestimate the stresses on the frame, so their results are adequate for validating the design. The results of the first simulation produce slightly larger values of stress in the truss elements and those are the values used for validating the model.
The results of the simulation give a minimum Factor of Safety (FOS) of 4.14 for gondola structural elements, when comparing the tensile strength of the CFRP tubes (1,896.05$\,$MPa) to the maximum stress. This minimum FOS corresponds to the tensile stress on both of the tubes directly below the front suspension cables.
The effect of the chute shock on the cryostat was included in the design specifications used by RedStone Aerospace for the design of the [<span style="font-variant:small-caps;">Spider</span>]{} cryostat. Each trunnion mount has a FOS of 1.24. The minimum FOS in the suspension cables and the spreader bar are 2.85 and 3.79 respectively.
### Pin release (Uneven loading)
At launch, the gondola will be suspended from a pin located on the end of the arm of the deployment vehicle. The vehicle moves in order to align the payload beneath the balloon before releasing. However, the wind can displace the balloon to form an angle of a few degrees with the vertical axis. In a gondola suspended by cables, such as [<span style="font-variant:small-caps;">Spider</span>]{}, the misalignment results in the momentary concentration of the load in one or one pair of the suspension cables and subsequently an abrupt pull produced by the flight train catching the payload after it is released from the pin. The result of this violent motion is a large angular acceleration of the cryostat. This angular acceleration was estimated to be around 600 s$^{-2}$ in the launch of the BLAST LDB flight from Antarctica in December 2006 (BLAST06). This sudden rotation damaged the lock pin in the BLAST LDB flight from Kiruna, Sweden in 2005 (BLAST05), and destroyed the elevation drive in the 2009 flight of the E and B Experiment (EBEX)[@reichborn2010].
The prescription given by CSBF for treating this scenario is: “*a load five times the weight of the payload applied at the suspension point and 45 to the vertical ... if flexible cable suspension systems are used, they must be able to withstand uneven loading caused by cable buckling*”. The worst scenario in case of cable buckling would be having the whole payload hanging momentarily from a single cable, a situation in which the minimum FOS is 1.10 for the [<span style="font-variant:small-caps;">Spider</span>]{} suspension cables.
For the [<span style="font-variant:small-caps;">Spider</span>]{} gondola, the effect of a 600 s$^{-2}$ angular acceleration is mainly received by three elements: the cryostat assembly, the elevation drive, which locks the cryostat in a fixed elevation position, and the outer frame, which ultimately receives the impact of the accelerating cryostat. The cryostat assembly is supported by G10 flexures designed to support the internal components of the cryostat during the chute shock. The flexures located in the extremes of the cylinder are located at 1$\,$m from the rotational axis and have enough mechanical advantage to support the internal components.
The elevation drive-locking pin of [<span style="font-variant:small-caps;">Spider</span>]{} is located 1 m from the rotation axis of the cryostat. This distance was chosen to increase the mechanical advantage of the pin and minimize the effect of the shock on the gondola. This locking strategy produces minimum FOS of 5.14, 2.16, and 1.48 for the cryostat, the trunnion and the locking arm respectively [@shariff2014].
The effect of the deployment vehicle pin release on the outer frame is modeled by applying the force of the elevation drive-locking pin on the corresponding node. This force is calculated using the torque resulting from the cryostat moment of inertia and the angular acceleration. The simulation of this load on the outer frame gives a minimum safety factor of 2.45 on the truss elements. Additionally, we calculated the effect of a lateral acceleration of 5g on the gondola elements. The model of the outer frame of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola has a minimum safety factor of 4.1 for the truss elements in this case. The FOS considerations on the cryostat are redundant with the results of the 10g analysis.
### Landing: Resting on the base at 5g {#DS:Landing}
The surface winds in Antarctica make an upright landing very unlikely, even for a gondola with legs like BLASTPol [@fissel2010]. Consequently, to save weight, no attempt was made in the design of [<span style="font-variant:small-caps;">Spider</span>]{} to prevent the payload from rolling over onto one of its sides after landing. Mechanically, the most demanding instant during such a landing is the first contact with the ground. In this stage the most exposed elements of the gondola are the lower frame and the supporting elements of the SIP. In order to simulate these landing conditions, the gondola was required to withstand a load equivalent to its mass at 5g while resting on its lower plane. The results of the simulation give a FOS of 3.77 on the truss elements.
### On cart at 1g {#DS:OnCart}
During integration, the outer frame is fastened to the cart on the nodes in the corners of the upper square frame. The loads produced by the cryostat, the boxes and the SIP have been tested at 10g on the outer frame in the parachute shock scenario. Given that the support points are different, a new simulation was made to guarantee the structural integrity of the payload during integration. This case is not as critical as the chute shock but it is very important given that scientists and non-flight equipment are going to be around the structure during the integration. With an additional load of 200$\,$kg (441lb) on the deck, the FOS of the truss elements is larger than 10.
### Frequency Analysis
As a consequence of the pointing requirements, the minimum feedback rate of the control system is 10$\,$Hz. To accommodate this, the outer frame is designed to be rigid with resonant frequencies over that value. The minimum resonant frequency of the outer frame is located at 29.45$\,$Hz and the next harmonic is at 81.50$\,$Hz. These resonant modes are show in Figure \[mecha:resonantmodes\] and they are calculated from a model of vibration response made with a static beam mesh model in SolidWorks[^^]{}.
The [<span style="font-variant:small-caps;">Spider</span>]{} outer frame profits from the excellent vibration damping provided by the fiber winding in the CFRP. Similar to the behavior of handle structural tubes in mountain and road bikes, the [<span style="font-variant:small-caps;">Spider</span>]{} tube mesh absorbs the vibrations of the structure. Nevertheless, particular attention has been devoted to maintaining all mechanical tolerances to minimize backlash from motors and to servicing the bearing units with low-temperature grease to avoid further vibrations.
![Displacement diagram corresponding to the first two resonance modes of the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame at 29.45 (left) and 81.50$\,$Hz (right).[]{data-label="mecha:resonantmodes"}](SpiderGondolaDesignFrequencyDisplacementX.eps){height="0.25\textheight"}
Suspension elements {#mecha:suspensionelements}
===================
The [<span style="font-variant:small-caps;">Spider</span>]{} gondola is attached to a steel wire ladder that composes the flight train using a customized universal joint. The universal joint is attached at one end to the ladder and to the pivot shaft at the other end. The shaft rotates inside a steel casing supported by a thrust bearing. Each of the three suspension cables is attached to steel tabs welded to a steel casing. The suspension cables end in thimbles, which are fastened to the tabs by round pin anchor shackles. The two front cables are attached to a carbon fiber and aluminum spreader bar that avoids the effect of lateral forces on the outer frame. A single cable directly attaches the pivot to the back of the gondola as illustrated in Figure \[mecha:suspensionangles\].
Universal Joint
---------------
The universal joint used on [<span style="font-variant:small-caps;">Spider</span>]{}, shown in Figure \[mecha:universaljoint\], is a reproduction of the design successfully flown in the BLAST test flight from Fort Sumner, NM in 2003 (BLAST03), BLAST05, BLASTPol10, and BLASTPol12 and based on the original design by AMEC Dynamic Structures. It is composed of a monolithic steel core with four cylindrical shafts where two bored aluminum yokes are free to rotate. A steel snap ring at the end of each shaft retains the aluminum yoke. One of the aluminum yokes connect to the flight train while the other connects to the pivot shaft. The two-rotation-axis design of the universal joint allows free rotation up to 135 in each direction. The larger thermal expansion coefficient of aluminum compared to steel guarantees that the side elements are free to rotate even when exposed to the temperature changes experienced during the LDB flight. An initial design of the [<span style="font-variant:small-caps;">Spider</span>]{} universal joint included oil-lubricated phosphor bronze flanged sleeve bearings to reduce the friction between the steel shaft and the aluminum bore. However, given the demanding pointing control requirements, the latest version of the universal joint uses a set of needle bearings[@shariff2014]. At chute shock, the steel core and aluminum plates of the universal joint have FOSs of 7.48 and 3.22 respectively.
![Left: Exploded view of the [<span style="font-variant:small-caps;">Spider</span>]{} Universal Joint. Part of the pivot steel casing is visible at the bottom. The upper segment of the universal joint is fastened to a coupling block provide by CSBF. Right: Spreader Bar.[]{data-label="mecha:universaljoint"}](UniversalJointX.eps){height="0.3\textheight"}
Suspension Cables
-----------------
The [<span style="font-variant:small-caps;">Spider</span>]{} gondola is suspended by cables made with braided Technora[^^]{}, a high modulus polyamide known for its high strength, heat resistance, low stretch, and better resistance to flex-fatigue than other high modulus aramid fibers [@syntheticropes]. The Technora[^^]{} cables are considerably lighter than steel ropes of equivalent strength and make integration operations more manageable. The degradation of the synthetic fibers by exposure to ultraviolet (UV) light is prevented using an aluminized mylar sleeve. This sleeve also prevents overheating, which would produce thermal degradation of the fibers.
The [<span style="font-variant:small-caps;">Spider</span>]{} suspension ropes are custom made by Helinets, a supplier of helicopter cargo lines. [<span style="font-variant:small-caps;">Spider</span>]{} uses five 5/8 inch diameter segments for the back and front suspension cables. Each rope is hand braided and pre-stretched. This process results in length tolerance of around 0.30$\,$m (1$\,$ft). For this reason, the synthetic cables have to be combined with turnbuckles that allow adjusting the pin-to-pin distances.
The 5/8 and 3/4$\,$inch ropes are rated for maximum axial loads of 38,700$\,$lbf and 58,500$\,$lbf. The minimum FOS of the suspension ropes is determined by CSBF: “*Each cable, cable termination and cable attachment must have an ultimate strength greater than five times the weight of the payload divided by the sine of the angle that the cable makes with horizontal, which should be larger than 30, in a normal flight configuration.*”. The FOS and nominal length of each cable are summarized in Table \[mecha:suspensionropesTable\].
-------------- ------------ -------- ------
Rope Pin-to-pin Load FOS
length (m) at 5g
Back 3.997 26,416 1.47
Front top 1.810 31,063 1.25
Front bottom 2.094 25,511 1.50
-------------- ------------ -------- ------
: Minimum safety factors on the [<span style="font-variant:small-caps;">Spider</span>]{} gondola suspension cables. The pin-to-pin distance refers to the nominal distance between the coupling points and thus it corresponds to the length of the rope plus the turnbuckle plus the shackle.[]{data-label="mecha:suspensionropesTable"}
Spreader Bar
------------
A spreader bar cancels the horizontal components of the tension on the suspension cables to minimize the lateral forces acting on the outer frame structure. The [<span style="font-variant:small-caps;">Spider</span>]{} spreader bar consists of a CFRP tube and two aluminum inserts that attach to the closed spelter socket end of the suspension cables as shown in Figure \[mecha:universaljoint\].
The CFRP tubing was a straightforward solution for the spreader bar given the analysis made for the design of the outer frame, which will be described in Section \[mecha:outerframe\]. Given the role of the spreader bar, the design goal of the inserts is to maintain the force produced by the suspension cables acting on the axis of the tube, and this was achieved by constructing the part around the node where the axis of the cables and the axis of the tube intersect. The minimum safety factor obtained from simulations of the inserts is 1.28.
Since the design of the inserts guarantees on-axis force, the main failure mode for the CFRP tube is Euler buckling. Euler buckling, also known as elastic instability, is characterized by failure of a structural member subjected to high compressive stress. The criterion derived by Euler for determining the critical force in columns with no consideration for lateral forces is given by Equation \[EulerBuclking\]. In the case of the CFRP tube the flexural rigidity (a product of the moment of inertia of the cross sectional area, $I$, and the modulus of elasticity, $E$) is particularly difficult to calculate due to the multiple orientations of the fibers on the tube and the characteristics of the epoxy. The CFRP tube’s flexural rigidity is guaranteed by the manufacturer[^1] to be 43.61$\,$kN$\,$m$^{2}$. Considering the worst case scenario, which is the beam being fixed at both ends $\kappa=$0.50, the critical force for this beam is 325.45$\,$kN (73,162$\,$lbf) and corresponds to a safety factor of 3.79. $$\label{EulerBuclking}
F = \frac{\pi^2(EI)}{(\kappa L)^2},$$ where L is the length of the tube.
---------------- ----- ------------ ---------- ----------- ---------------- --------
Component Qty Material Mass Design Manufacturer FOS
(g) at 10g
Insert Joint 2 Al 7075-T6 2,301.93 JDS Quickparts 1.28
Beam 1 CFRP 2,268.29 AA70430A CST composites 3.79
Shoulder bolts 4 Steel 735.44 91259A313 McMaster Carr
---------------- ----- ------------ ---------- ----------- ---------------- --------
: Properties of the components of the [<span style="font-variant:small-caps;">Spider</span>]{} spreader bar.[]{data-label="mecha:spiderspreaderbartable"}
Outer Frame {#mecha:outerframe}
===========
The [<span style="font-variant:small-caps;">Spider</span>]{} outer frame, shown in Figure \[OuterFrameJointFigure\], is made up of two kinds of components: CFRP tubes with aluminum inserts on their ends and multitube aluminum joints. The fastening of the inserts and the CFRP tube is made with an epoxy adhesive. The inserts are bolted into the faces of the aluminum joints that constrain the orientation angle of each tube. The final mass of the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame is 193.76$\,$kg (427.17$\,$lbs), accounting for only 9.4% of the total mass of the experiment. The conceptual design of the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame is inspired by truss structures made with composite materials and it is largely influenced by the design of carbon fiber bicycles. The original evaluation of the CFRP tubing design[@martin_thesis2008] pointed out the difficulty of fastening multiple composite tubes and included several structures constructed with aluminum beams, such as the SIP cage and the reaction wheel support. The final version of the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame is entirely made of CFRP tubes held together with monolithic, custom-made multitube aluminum joints and a common aluminum insert glued to each end of the CFRP tubes.
![The [<span style="font-variant:small-caps;">Spider</span>]{} gondola frame. Left: schematic showing the labeling of the multitube joints. Right: photograph of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola without the cryostat.[]{data-label="OuterFrameJointFigure"}](SpiderSchematicNodesGondola.eps "fig:"){height="0.30\textheight"} ![The [<span style="font-variant:small-caps;">Spider</span>]{} gondola frame. Left: schematic showing the labeling of the multitube joints. Right: photograph of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola without the cryostat.[]{data-label="OuterFrameJointFigure"}](SpiderGondolaBarePhoto.eps "fig:"){height="0.30\textheight"}
The geometry of the outer frame is set by the scientific requirements of the experiment. The outer frame encloses the lower portion of the cryostat allowing its rotation. It also provides two square frames: one for mounting the deck with the attitude control electronics and one for supporting the SIP.
Material Selection and Construction Technique {#mecha:technique}
---------------------------------------------
### Carbon-Fiber-Reinforced Polymer (CFRP) Tubing
CFRP is a very strong and light-weight material. Although it can be relatively expensive compared to aluminum and glass fiber, its high strength-to-weight ratio and good rigidity make it the perfect candidate for a light-weight structure such as the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame.
CFRP tubes are constructed by pultrusion, a continuous process for manufacture of composite materials with constant cross-section. The reinforced fibers are pulled through a resin, followed by a separate preforming system, and then into a heated die where the resin undergoes polymerization. The resin used for the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame tubes is epoxy. CFRP tubes are not weldable and drilling causes damage to the winding of the fibers, therefore they require metallic inserts to join them [@adams_adhesives1997].
The CFRP tubes of the [<span style="font-variant:small-caps;">Spider</span>]{} frame are provided by CST Composites, an Australian company with experience in manufacturing high quality filament wound tubing for industrial and marine applications. The selected product is the carbon/epoxy tubing AA70430A with 70.4$\,$mm inner diameter (ID) and a 3$\,$mm wall thickness. The linear density of this product is 1.056$\,$kg$\,$m$^{-1}$. It has a flexural rigidity (IE) of 43.61$\,$kN$\,$m$^{2}$ and maximum tensile strength of over 1,896.05$\,$MPa (275,000$\,$psi).
The selection of this product was made using a parametric static beam mesh study in Solidworks[^^]{}. The parachute shock and landing scenarios (\[DesignScenarios\]) were iteratively run on the model while changing the diameter and wall thickness of the tube until an optimal compromise between strength and estimated mass was achieved. According to FEA, the largest axial load on the final design of the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame is a 117.2$\,$kN (26,350$\,$lbf), resulting from the vertical pull of the front suspension cables at chute shock. Comparison of this axial load with the mechanical test performed on prototypes of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola tubes to test the adhesive joint results in minimum FOS of 1.72 at room temperature and 1.37 when pulled at lower temperatures in a bath of dry ice.
### Adhesive Joint
The adhesive fastener selected for the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame is the 3M[^^]{}Scotch-Weld[^^]{}Epoxy Adhesive 2216 B/A Gray. This is a flexible, two-part, room-temperature-curing epoxy with high peel and shear strength. This product is recommended for bonding metals and plastics with good retention of strength after environmental aging and resistance to extreme shock, vibration and flexing. The Scotch-Weld[^^]{}2216 is widely used in aircraft and aerospace applications and it meets the DOD-A-82720 Military Specification for *adhesive, modified-epoxies, flexible, and two-parts*.
The main difficulty in the construction of bonded joints is the assembly process. One of the main concerns in the case of the [<span style="font-variant:small-caps;">Spider</span>]{} CFRP tube was galvanic corrosion. This can occur between aluminum and CFRP due to their large contact potential [@adams_adhesives1997]. Selecting materials with lower contact potentials like titanium and CFRP can minimize this effect. However, the higher specific strength and corrosion preventing features of titanium do not justify the greater cost, which is 5 to 10 times higher than aluminum [@callister_materials2005]. Instead the [<span style="font-variant:small-caps;">Spider</span>]{} assembly avoids galvanic corrosion by ensuring electrical insulation between the two materials. This can be achieved by ensuring that the adhesive layer completely covers each overlapping surface thus allowing the use of aluminum for the inserts. The use of fishing lines as spacers was tested and dismissed because of the additional complexity in the assembly process and poor results in pull tests.
The final prescription for the surface preparation of the adhesive joint between the CFRP tubes and the aluminum joints is:
1. Hand-sand the bonding surfaces of the aluminum insert to remove the corrosion layer in the metal and impurities on the surface.
2. Hand-sand the inside of the CFRP tube without breaking the outer layer of epoxy, which keeps the carbon fibers together.
3. Cover the edge of the outer face of the CFRP tube with masking tape to facilitate the cleaning of the tube. Clean both surfaces using acetone.
4. Activate the glue by combining the two elements of the formula. 18 g of glue is enough for one outer frame insert. 3 g of glue is enough for one sunshield insert.
5. Remove the bubbles trapped in the glue by putting the mix in a vacuum desiccator for two minutes.
6. Cover both surfaces distributing the glue in uniform thin layers.
7. Bring together both pieces, slightly rotating the insert.
8. Install the tube in a clocking jig (in the case of the outer frame tubes) or in the assembly (in the case of the sunshield tubes).
The working time of the Scotch-Weld[^^]{}2216 is 90 minutes. However, the glue thickens after 60 minutes of activation and it becomes harder to work with it. It is recommended to work in small batches to facilitate the gluing process. The curing of the outer frame tubes is made keeping the tube in a vertical position to avoid contact between the aluminum and the CFRP before the glue solidifies. The glue cures after 8 to 12 hours depending on the temperature conditions.
Because it is difficult to predict the strength and durability of adhesive joints, we experimentally pull-tested the assembled tubes. Quality assessment of each batch of glue is made by preparing a test assembly where the [<span style="font-variant:small-caps;">Spider</span>]{} inserts are replaced by pull test inserts. The pull test of the first batch of tubes was performed at Sling-Choker MFG Ltd. where the adhesive joint was pulled to destruction when the load reached 201.1 kN (45,200 lbf) at room temperature and 161.2$\,$kN (36,230 lbf) when the tube was immersed in dry ice. This corresponds to a minimum FOS of 1.37 when compared to the largest axial load on the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame. The gluing of all the [<span style="font-variant:small-caps;">Spider</span>]{} outer frame tubes was done between September and October 2009. Pull test of the control tubes is scheduled at different times before the deployment of [<span style="font-variant:small-caps;">Spider</span>]{} to assess the adhesive joint after aging, stress on shipping, and exposure to sunlight.
Inserts
-------
The inserts, shown in Figure \[Insert\], are monolithic pieces of aluminum. Depending on their location on the outer frame, the inserts are made with either 6061-T6 or 7075-T6 aluminum. Both are tempered and precipitation hardened aluminum alloys. 6061-T6 is one of the most commonly used alloys of aluminum and it is often used in structure components of balloon-borne experiments. 7075-T6 has a yield strength comparable to many steels but it is more difficult to machine and considerably more expensive than 6061-T6. The selection of each type of aluminum depends on the particular force in each joint.
![[<span style="font-variant:small-caps;">Spider</span>]{} outer frame insert. Left: render of the insert. Right: preparation of the adhesive joint between the CFRP tube and the aluminum insert.[]{data-label="Insert"}](Insert.eps "fig:"){height="0.17\textheight"} ![[<span style="font-variant:small-caps;">Spider</span>]{} outer frame insert. Left: render of the insert. Right: preparation of the adhesive joint between the CFRP tube and the aluminum insert.[]{data-label="Insert"}](Insert2.eps "fig:"){height="0.17\textheight"}
The inserts have two main sections: an adhesive joint section and a bolt fastening section. The adhesive joint is made between the cylindrical section of the insert and the CFRP tube. A fillet inside of the cavity reinforces the cylindrical section. The neck of the piece is machined to permit the location of the mounting bolts. All the inserts have a 3.175$\,$mm diameter canal that connects the interior of the cylindrical section with the mounting face and allows the air to circulate from the inside of the glued tube into the exterior. The FEA analysis of the inserts shows a FOS of 1.37 and 1.43 for the 6061-T6 and 7075-T6 aluminum inserts. The flange of the insert has been pull tested to destruction at 183.4$\,$kN (41,230$\,$lbf), which corresponds to a FOS of 1.79 with respect to the greatest axial stress during chute shock.
Joints
------
The CFRP are fastened in multi-tube joints in the nodes of the structure. The geometry of each multi-tube joint is determined by the orientation of the CFRP tubes set during the frame design. The joints are polyhedrons with faces perpendicular to the axes of the tubes. Each face fits the base of the insert and the eight-bolt pattern, thus the dimensions of the insert flange ultimately determine the overall size of the joint.
The diversity of angles in the outer frame required customized design for at least half of the tube connecting joints. In total, 13 different joints were designed for the outer frame, some of them with a mirrored version located on the other side of the gondola. The locations and names of the joints are shown in Figure \[OuterFrameJointFigure\]. There are three joints connected to the suspension cables (A and D), two joints supporting the block bearings where the cryostat sits (C), four joints supporting the flywheel (Z), two joints supporting the elevation drive (B), four joints forming the frame for the deck and providing attachment points for the cart and the sunshields (F and G), and four joints forming the frame where the SIP sits.
All the aluminum joints include light-weighting pockets determined by FEAs. Each pocket reduces stress concentration in the joint while minimizing its mass. The final geometry of each pocket aims to simplify the machining process. However, the tubes orientations determine result in non-trivial joint geometries and their machining required a six-axis Computer Numerical Control (CNC) mill.
Floor
-----
The floor of the [<span style="font-variant:small-caps;">Spider</span>]{} gondola is the deck which supports the Attitude Control System (ACS) rack, the flight computers, the star cameras, the batteries and other electronics [@benton2014; @gandilo2014]. The floor is made with two Teklam Corp. A510C 48 in$\times$96$\,$in and 1.0$\,$inch thick aluminum/aluminum honeycomb panels. These panels are manufactured with Teklam process specification TPS-A-A-500, which guarantees their “*use for primary and secondary aerospace applications*”. The density of the honeycomb panel is 4.54$\,$kg$\,$m$^{-2}$ (0.93 lb ft$^{-2}$), which corresponds to a total mass of approximately 13.5$\,$kg (29.7$\,$lb) per section.
Each panel rests on eight aluminum supports. The lower ends of the aluminum supports fit around the CFRP tubes and are secured using a complementary bracket as shown in Figure \[mecha:spiderfloor\]. The bottom bracket has a hole pattern to accommodate the line of sight transmitters.
![Rendering of [<span style="font-variant:small-caps;">Spider</span>]{}’s aluminum honeycomb deck and aluminum supports.[]{data-label="mecha:spiderfloor"}](SpiderFloor.eps){height="0.25\textheight"}
Sunshields {#mecha:sunshields}
==========
The [<span style="font-variant:small-caps;">Spider</span>]{} sunshield frame is mounted to the base of the outer frame using four aluminum supports. The frame is covered with two layers of aluminized mylar to protect the experiment from exposure to direct sunlight[@soler2014therm]. The [<span style="font-variant:small-caps;">Spider</span>]{} sunshield frame is built using a combination of CFRP tubing and aluminum joints in a simplified version of the design used for the outer frame. This design was originally developed for the balloon-borne telescope BLASTPol and tested during the Antarctic flights of that experiment in 2010 and 2012[@fissel2010; @galitzki2014].
The observation of the BLASTPol astronomical targets required shielding the telescope from direct sunlight at 40 from the sun position in azimuth. This was achieved with the construction of a baffle located around the 1.8$\,$m-diameter primary mirror and extending 7$\,$m in the bore-sight direction. The baffle frame is a truncated cylindrical structure with a hexagonal base made of CFRP tubes with aluminum inserts connected to aluminum hubs as shown in Figure \[BLASTpolBaffle\].
The CFRP tubes selected for thie BLASTPol baffle were the CST composites AA20020A: a 20.0$\,$mm ID and 2.0$\,$mm wall thickness CFRP tube. The selection of this product was made following a series of beam mesh simulations as described in Section \[mecha:outerframe\]. Although the design constraints and loads are not as demanding as for the outer frame, the location of the baffle structure directly around the observation field of the telescope demanded special attention to the rigidity and structural integrity after launch. The structure required aggressive light-weighting to avoid increasing the moment of inertia of the cryostat over the capacity of the elevation drive. Additionally, the resonant frequencies of the structure had to be kept over $15\,$Hz to avoid resonance with the natural frequencies of the experiment.
The structure of the BLASTPol baffle starts with a hexagonal ring whose dimensions are determined by the clearance space around the primary mirror. Three more hexagonal rings are set at equal distances between the first ring and the minimum distance necessary to keep the sunlight off the primary and secondary mirrors at 40 in azimuth from the position of the sun. Each hexagonal ring is offset by 30 from the orientation of the previous ring to produce a triangular truss structure between each ring.
![BLASTPol baffle assembly. The detail shows the multi-tube coupling to the aluminum hub.[]{data-label="BLASTpolBaffle"}](BLASTpolBaffleHubDetail.eps){height="0.28\textheight"}
The joints connecting the tubes are aluminum octahedrons with two parallel faces (top and bottom) and six faces perpendicular to the base forming a hexagon. They are centered on the node where the axes of the tubes converge, making the faces of the hexagon perpendicular to the tube axis. The inserts are made with aluminum plates, which mount to the hexagonal faces of the hub with two 1/4-20 bolts. The plates are welded to an aluminum cylinder oriented in the direction of the axis of the tube. This cylinder serves as the coupling surface for the adhesive joint with the CFRP tube as shown in the zoomed-in detail of Figure \[BLASTpolBaffle\].
The large area covered by the mylar makes the wind force an important aspect of the design of the sunshield frame. The magnitude of the wind force can be calculated from: $$\label{airforce}
F_{air} = \frac{1}{2}\rho A C v^{2}.$$ Where $A$ is the effective area facing the wind, $v$ is the velocity of the wind, and $C$ is the drag coefficient, which depends on the geometry of the shields[@batchelor2000introduction]. In order to validate the model, a worst-case scenario was considered where $A$ corresponds to the largest effective area of the shields and $C = 1$ corresponding to a flat surface perpendicular to the airflow. The BLASTPol structure has a FOS above 2.0 under the load of 20$\,$knot (10.3 m$\cdot$s$^{-1}$) wind.
The [<span style="font-variant:small-caps;">Spider</span>]{} sunshields follow the same construction technique and design concept as the BLASTPol baffle: the structure is a cylinder with a hexagonal base composed of CFRP tubes fastened by aluminum inserts and joints. The main modification with respect to the design of the BLASTPol baffle is the diameter of the CFRP tubing, which has been increased to support larger loads. The chosen product is the CST Composites AA32018, a 32.0$\,$mm ID and 1.8$\,$mm wall thickness CFRP tube.
![[<span style="font-variant:small-caps;">Spider</span>]{} sunshield assembly. The detail shows the multitube coupling at the aluminum hub.[]{data-label="SpiderSunshields"}](SpiderSunshieldsHubDetailX.eps){height="0.35\textheight"}
The [<span style="font-variant:small-caps;">Spider</span>]{} sunshields also implement a new feature in the construction technique: the combination of custom-angle fixed aluminum inserts with universal-joint-type insert, which can be oriented in two different angles as shown in Figure \[SpiderSunshields\]. A universal-joint-type insert follows the tube incidence angle in two directions: one defined by the tube insert and yoke block hinge; the other is defined by the rotation of the yoke over the plate fastened to the aluminum hub. This versatile type of insert constitutes a general solution for the assembly of structures with moderate loads such as baffles and sunshields.
The gluing of this structure is made in place after *dry* assembling the structure. The distance between the joints is constrained by cutting the tubes to length with $\pm$1$\,$mm tolerance. The assembly is temporarily held together using ropes around the tubes. In the absence of an extended flat surface to use as a reference for the assembly, the ropes constrain rotation of the hubs and ensure that the elements are held in the right configuration. The gluing is made triangle by triangle in several stages.
As for the BLASTPol baffle, one of the main considerations in the design of the [<span style="font-variant:small-caps;">Spider</span>]{} sunshields is the lateral and vertical wind force. The sunshield frame is designed to hold the pivot during parachute shock and maintain structural integrity when covered with mylar and exposed to $20\,$knot ($10.3\,$m$\cdot$s$^{-1}$) wind. Its lowest resonance frequency is located over 15$\,$Hz, guaranteeing that the vibrational modes will not affect the pointing of the gondola.
Conclusions
===========
A light-weight gondola is one of the characteristics that make [<span style="font-variant:small-caps;">Spider</span>]{} unique as a suborbital experiment. Combining the use of composite materials and detailed FEA, the [<span style="font-variant:small-caps;">Spider</span>]{} gondola pushes the limits fixed by the maximum gross lift of stratospheric balloons. Its initial Antarctic flight will measure or powerfully constrain the B-mode signal from primordial gravitational waves and provide a landmark in the design of balloon-borne telescopes.
The [<span style="font-variant:small-caps;">Spider</span>]{} collaboration gratefully acknowledges the support of NASA (award numbers NNX07AL64G and NNX12AE95G), the Lucille and David Packard Foundation, the Gordon and Betty Moore Foundation, the Natural Sciences and Engineering Research Council (NSERC), the Canadian Space Agency (CSA), and the Canada Foundation for Innovation. We thank the JPL Research and Technology Development Fund for advancing detector focal plane technology. W. C. Jones acknowledges the support of the Alfred P. Sloan Foundation. A. S. Rahlin is partially supported through NASAs NESSF Program (12-ASTRO12R-004). J. D. Soler acknowledges the support of the European Research Council under the European Union’s Seventh Framework Programme FP7/2007-2013/ERC grant agreement number 267934. J. D. Soler thanks Taylor G. Martin and Marco P. Viero for their valuable comments on computer-aided design and carbon fiber gluing techniques. Logistical support for this project in Antarctica is provided by the U.S. National Science Foundation through the U.S. Antarctic Program. We would also like to thank the Columbia Scientific Balloon Facility (CSBF) staff for their continued outstanding work.
[^1]: CST composites
|
---
abstract: 'The recombination model is applied to the production of and at all in central Au+Au collisions. Since no light quarks are involved in the hadronization, those hidden-strange particles present a clean slate for the study of the role of strange quarks in large-physics. We find that shower $s$ quarks have negligible effect for $p_T<6$ GeV/c, in which range the thermal $s$ quarks make the dominant contributions to the formation of and $\Omega^-$. We show that the same effective temperature of the $s$ quarks is responsible for the shape of the spectra of both $\phi$ and $\Omega^-$. We predict that the ratio of to will show a peak at $p_T\approx 6$ GeV/c due to the effect of the hard partons. We also give reasons on the basis of the dependence that cannot be formed by means of $K^+K^-$ coalescence.'
author:
- 'Rudolph C. Hwa$^1$ and C. B. Yang$^{1,2}$'
title: 'Production of $\phi$ and $\Omega^-$ at RHIC in the Recombination Model'
---
Introduction
============
The production of strange particles has always been a subject of great interest in heavy-ion collisions because of their relevance to possible signatures of deconfinement and flavor equilibration [@1; @2]. Strangeness enhancement that has been observed at various colliding energies is a phenomenon associated with soft particles in the bulk matter [@3; @4]. At high transverse momentum ($p_T$), on the other hand, the production of jets does not favor strange particles, whose fragmentation functions are suppressed compared to those for non-strange particles. At intermediate range between the two extremes the distribution depends sensitively on both the strangeness content and the production mechanism. It has been shown that in that range the spectra of $\pi, K$, and $p$ can be well described by parton recombination [@5]. In this paper we study the production of and $\Omega^-$, both of which consist of only strange quarks. Without the participation of the non-strange quarks, they present a clean platform for the examination of the transverse momentum spectra of the $s$ and $\bar s$ quarks. Our aim is to learn about the transition from the enhanced thermal quarks to the suppressed shower quarks in the strange sector.
Since the hidden-strange particles ($s\bar s, sss$) are expected to have small hadronic cross sections due to the OZI rule [@6; @7; @8], and are less likely to be affected by final-state interaction with co-movers in heavy-ion collisions, compared to kaons and hyperons. Their spectra should therefore reveal more directly their formation mechanism. We shall apply the recombination model, as in [@5], and predict the shapes of their spectra beyond the intermediate range where data do not yet exist. We can also calculate the $\Omega/\phi$ ratio and show how different it is from the $p/\pi$ ratio that has been the definitive signature of parton [@9; @10; @11]. If the prediction of the peaking of the $\Omega/\phi$ ratio at $p_T\approx 6$ GeV/c is verified by experiments, it will be another piece of evidence in support of at high $p_T$.
It should be emphasized that the focus of our work is on the hadronization of partons and not on the hydrodynamical evolution of the initial dense system. Thus we make no assertions on the nature of the thermal partons, but take their spectra from low-data phenomenologically in the framework of the model. The shower parton distributions are known from previous study [@12], so our calculation of the hidden-strange hadronic spectra involves the of various combinations of thermal and shower strange quarks. The enhanced thermal $s$ and the suppressed shower $s$ lead to an interesting departure from the more familiar scenario in the non-strange sector.
Formulation of the Problem
==========================
We shall assume that all hadrons produced in heavy-ion collisions are formed by of quarks and/or antiquarks, the original formulation of which is given in [@13; @14]. Later improvements are described in [@15] for hadronic collisions, in [@16] for $pA$ collisions and in [@5] for $AA$ collisions. For any colliding system the invariant inclusive distribution of a produced meson with momentum $p$ in a 1D description of the process is $$\begin{aligned}
p^0{dN_M \over dp} = \int {dp_1 \over p_1}{dp_2
\over p_2}F_{q\bar{q}'} (p_1, p_2) R_M(p_1, p_2, p) ,
\label{1}\end{aligned}$$ and for a produced baryon $$\begin{aligned}
p^0{dN_B\over dp}=\int {dp_1\over p_1}{dp_2\over p_2}{dp_3\over p_3}\,
F_{qq'q''}(p_1,p_2,p_3)\,R_B(p_1,p_2,p_3,p) .
\label{2}\end{aligned}$$ The properties of the medium created by the collisions are imbedded in the joint quark distributions $F_{q\bar q'}$ and $F_{qq'q''}$. The functions (RF) $R_M$ and $R_B$ depend on the hadron structure of the particle produced, but not on the medium out of which quarks hadronize. In the valon model description of hadron structure [@14; @17], the RFs are $$\begin{aligned}
R_M(p_1, p_2, p) =g_{M}\,y_1y_2\, G_M (y_1 y_2 ) , \label{3}\\
R_B(p_1,p_2,p_3,p)=g_{B}\,y_1y_2y_3\,G_B(y_1,y_2,y_3) , \label{4}\end{aligned}$$ where $y_i=p_i/p$, and $g_M$ and $g_B$ are statistical factors. $G_M$ abd $G_B$ are the non-invariant probability densities of finding the valons with momentum fractions $y_i$ in a meson and a baryon, respectively.
Equations (\[1\]) and (\[2\]) apply for the produced hadron having any momentum $\vec p$. We consider $\vec p$ only in the transverse plane, and write $p_T$ as $p$ so that $dN/p_Tdp_T$ becomes $(pp^0)^{-1}$ times the right-hand sides of Eqs. (\[1\]) and (\[2\]). For and production, $F_{q\bar q'}$ is $F_{s\bar s}$, and $F_{qq'q''}$ is $F_{sss}$. The RFs are very narrow in momentum space, since both and are loosely bound systems of the constituent $s$ quark. We shall approximate $G_\phi$ and $G_\Omega$ by $\delta$-functions: $$\begin{aligned}
G_\phi(y_1,y_2)&=&\delta(y_1-1/2)\,\delta(y_2-1/2) , \label{5} \\
G_\Omega(y_1,y_2,y_3)&=&\delta(y_1-1/3)\,\delta(y_2-1/3)\,\delta(y_3-1/3) , \label{6}\end{aligned}$$ and set $g_\phi=3/4$ and $g_\Omega=1/2$ for $J=1$ and $J=3/2$, respectively. Using these in Eqs. (\[3\]) and (\[4\]), and then in (\[1\]) and (\[2\]), we obtain the simple algebraic expressions $$\begin{aligned}
{dN_\phi\over pdp}&=&{3\over 4pp_0}\,F_{s\bar s}(p/2, p/2) , \label{7}\\
{dN_\Omega\over pdp}&=&{1\over 2pp_0}\,F_{sss}(p/3, p/3, p/3) , \label{8}\end{aligned}$$ where $p_0=(m^2+p^2)^{1/2}$, $m$ being the mass of or $\Omega^-$, as the case may be.
In [@5] we have described how the joint parton distributions can receive contributions from the thermal ($\cal T$) and shower ($\cal S$) sources. In a schematic way they can be expressed as $$\begin{aligned}
F_{s\bar s}&=&{\cal TT+TS+SS}, \label{9}\\
F_{sss}&=&\kappa\,({\cal TTT+TTS+TSS+SSS}), \label{10}\end{aligned}$$ where showers from more than 1 jet are neglected. In Eq. (\[10\]) the multiplicative factor $\kappa$ is added to the $sss$ distribution to allow for the possible constraint arising from the competition among various channels of hadronization that can limit the number of $s$ quarks available for the formation of $\Omega^-$.
We parameterize the invariant $s$ quark in the thermal source as $$\begin{aligned}
{\cal T}(p_1) = p_1{dN_s^{\rm th}\over dp_1} = C_s\, p_1\exp (-p_1/T_s), \label{11}\end{aligned}$$ where $C_s$ and $T_s$ are two parameters to be determined by fitting the low-data; they are expected to be different from those in the non-strange sector. The distribution of shower $s$ quark in central Au+Au collisions is, as in [@5], $$\begin{aligned}
{\cal S}(p_2) = \xi \sum_i \int^{\infty}_{k_0}dk\, k\,
f_i(k)\, S^s_i (p_2/k) ,
\label{12}\end{aligned}$$ where $S_i^s$ is the shower parton distribution (SPD) for an $s$ quark in a shower initiated by a hard parton $i$, $f_i(k)$ is the transverse-momentum ($k$) of hard parton $i$ at midrapidity in central Au+Au collisions, and $\xi$ is the average fraction of hard partons that can emerge from the dense medium to hadronize. As in [@5], $f_i(k)$ is taken from Ref. [@18], $k_0$ is set at 3 GeV/c, and $\xi$ is found to be 0.07.
Equations (\[7\])-(\[12\]) completely specify the problem. There are 3 parameters to vary to fit the spectra of and $\Omega^-$: they are $C_s, T_s$ and $\kappa$. All aspects of the semi-hard shower partons have been fixed by previous studies [@5]. The three parameters are all related to the soft thermal partons, the properties of which, we have stated from the outset, are to be determined phenomenologically.
Results
=======
We first vary $C_s$ and $T_s$ to fit the data on the of in central Au+Au collisions at $\sqrt s=200$ GeV [@19]. For up to 3 GeV/c, which is the extent to which data exist, the entire can be accounted for by the of the thermal partons only. The thermal-thermal component is shown by the dashed line in Fig. 1. The values of $C_s$ and $T_s$ used for the fit are $$\begin{aligned}
C_s=8.64\ {\rm (GeV/c)^{-1}}, \qquad\qquad T_s=0.385\ {\rm GeV/c}. \label{13}\end{aligned}$$ The contributions from thermal-shower (dash-dot line) and shower-shower (line with crosses) do not become important until $p_T\approx 7$ GeV/c. The solid line in Fig. 1 indicates the sum of all contributions. The dominance of the thermal component for $p_T<6$ GeV/c is due to the fact that the production of $s$ quarks in a shower is highly suppressed [@12]. For that reason the role of the shower partons in the formation of in the intermediate region is insignificant compared to that for the production of pions [@5]. Shower-shower recombination that can be related to fragmentation can become important, but not until $p_T>8$ GeV/c.
![ Transverse momentum distribution of in central Au+Au collisions. Data are from [@19]. The solid line is the sum of the three contributions: $\cal TT$ (dashed line); $\cal TS$ (dash-dot line); $\cal SS$ (line with crosses). ](fig1.eps){width="45.00000%"}
Next, for the production of there is only one parameter $\kappa$ to vary. Again, the of thermal partons only dominates throughout the whole region shown in Fig. 2. All other contributions that involve the participation of at least one shower parton are increasingly negligible with increasing number of shower $s$ quark. The of $\cal SSS$ can be identified with the fragmentation of hard partons to in the same sense that the of only the shower $uud$ in a jet gives the proton by fragmentation [@5], even though the fragmentation function for does not exist. Evidently, the contribution to the spectrum from fragmentation is negligible until is extremely large.
![ Transverse momentum distribution of in central Au+Au collisions. Data are from [@20]. The heavy solid line is the sum of the four contributions: $\cal TTT$ (dashed line); $\cal
TTS$ (dash-dot line); $\cal TSS$ (light solid line); $\cal SSS$ (line with crosses).](fig2.eps){width="45.00000%"}
The value of $\kappa$ used in the fit of the data [@20] is $$\kappa=0.037 . \label{14}$$ It affects only the overall normalization of the spectrum, not the relative magnitudes of the different components. Thus our predictive power is in the shape of spectrum, not in the normalization. The agreement with the data in Fig. 2 is clearly excellent. Apart from the different powers of $p_1$ in Eq.(\[11\]) that appear in (\[9\]) and (\[10\]), the shape is controlled by $T_s$, which is common for both and production. The low level of production is a phenomenological fact that is embodied in the smallness of $\kappa$ in Eq.(\[14\]). The underlying physics is most likely the demand for $s$ quarks in the formation of lower-mass particles, such as kaons, hyperons and meson, so that their availability for forming the higher-mass is significantly reduced. To consider all channels of hadronization simultaneously would require a study similar to those carried out in [@21; @22; @23], which is beyond the scope of this work, where our emphasis is on the dependences of and $\Omega^-$.
{width="45.00000%"}
Having obtained and spectra, let us present their ratio $R_{\Omega/\phi}(p_T)$, shown in Fig. 3. The dashed line is the ratio of only the thermal contributions. It is linearly rising because, on account of Eqs. (\[9\]) and (\[10\]), it is proportional to $p$ at large $p$ after the common exponential terms of the two spectra are cancelled. The solid line shows the effect when the thermal-shower is taken into account. Although the effect of shower partons on is small, the effect on above $p_T=5$ GeV/c is substantial enough to turn the dashed line in Fig. 3 into a peak at around 6 GeV/c. Confirmation of this peak by future experimental data would be a definitive validation of the theoretical approach taken here and, in particular, the importance of thermal-shower recombination.
Discussion
==========
The strange sector differs from the non-strange sector in several ways. Firstly, the values of $C_s$ and $T_s$ given in Eq.(\[13\]) are different from $$C=23.2\ {\rm (GeV/c)^{-1}} , \qquad\qquad T=0.317\ {\rm GeV/c} , \label{15}$$ for the non-strange quarks [@5]. Strangeness enhancement refers to the excess of $s$ quarks in $AA$ collisions compared to the scaled $pp$ collisions, but the total number of $s$ quarks is still much less than that of light quarks in nuclear collisions. That difference is reflected in $C_s$ being much less than $C$. The inverse slope $T_s$ is, however, higher than $T$, since hydrodynamical flow gives the more massive $s$ quark a larger mean transverse momentum than it does the light quarks. The combined effect of larger $T_s$ and weaker shower $s$ quarks results in the thermal partons becoming a dominant contributor to the formation of and over a much wider range of than in the case of non-strange quarks. For pion production the thermal-shower becomes important for $p_T>3$ GeV/c, whereas for it is not until $p_T>7$ GeV/c.
The above comment cannot be checked directly by experiments. However, the predicted ratio of to can be tested. The rising portion of that ratio in Fig. 3 is much broader than that for the $p/\pi$ ratio [@24], and is an indication of the dominance of thermal-thermal recombination. The overall shape of $R_{\Omega/\phi}$ is very different from that of $R_{p/\pi}$. The latter peaks at $p_T\approx 3$ GeV/c with an experimental value exceeding 1, whereas the former is predicted to peak at $p_T\approx 6$ GeV/c with a value less than 0.4. Neither $R_{p/\pi}$ nor $R_{\Omega/\phi}$ exhibits properties that can be associated with fragmentation. The difference between them reflects the differences between $u, d$ and $s$ quarks on the one hand, and between non-strange and hidden-strange hadrons on the other.
Within the framework of it is possible for us to examine the reality of formation through $K^+K^-$ coalescence, which is a mechanism that has been advocated in certain models [@25; @26]. If $H_h(p_T)$ denotes the invariant inclusive distribution, $p^0dN_h/dp_T$, of hadron $h$ at $y=0$ in heavy-ion collisions, then the coalescence process of $K^++K^-\rightarrow \phi$ implies by use of Eq. (1) that $$\begin{aligned}
H_\phi^{[KK]}(p_T) \propto H_K^2(p_T/2) \label{16}\end{aligned}$$ apart from a multiplicative constant associated with the RF. On the other hand, if is produced by $s\bar s$ as we have done here, then the same procedure yields $$\begin{aligned}
H_\phi^{[ss]}(p_T) \propto {\cal T}_s^2(p_T/2) , \label{17}\end{aligned}$$ where the dominance of the thermal parton is used. Thus it is a matter of comparing $H_K(p)$ with ${\cal T}_s(p)$, which are the invariant distributions of the entities that recombine. Since a kaon is formed by $\bar sq$ recombination, where $q$ denotes either $u$ or $d$, whose thermal ${\cal T}(p)$ is characterized by $C$ and $T$ shown in Eq. (\[15\]), the exponential part of $H_K(p)$ must have an inverse slope $T'$ that is between $T$ and $T_s$, i.e., $~e^{-p/T'}$ with $2/T'=T^{-1}+T_s^{-1}$. That is to be compared to the exponential part of ${\cal T}_s^2(p/2)={\cal T}_s(p)$, which is $e^{-p/T_s}$ \[cf. Eq. (\[11\])\]. In view of our good fit of the data in Fig. 1, we conclude that an alternative fit using $H_K(p)$ characterized by $T'$ would fail. It therefore follows from the consideration of the dependence alone that cannot be formed by $K^+K^-$ coalescence. This conclusion is consistent with that of [@19] based on the centrality independence of the $\phi/K^-$ ratio.
In summary, we have shown that both and are formed mainly by the of thermal $s$ quarks, have reproduced the shape of the of from a fit of that of $\phi$, and have made a prediction of the ratio of to that peaks at $p_T\approx 6$ GeV/c. Thermal-shower does not become important until is at around that peak, and parton fragmentation does not dominate until $p_T$ is much higher. Finally, we have shown how the dependence disfavors the formation of by $K^+K^-$ coalescence.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Nu Xu for his encouragement and communication in the pursuit of this problem. This work was supported, in part, by the U. S. Department of Energy under Grant No. DE-FG02-96ER40972 and by the Ministry of Education of China under Grant No. 03113.
[99]{}
J. Rafelski and B. Müller, Phys. Rev. Lett. [**48**]{}, 1066 (1982).
P. Koch, B. Müller and J. Rafelski, Phys. Rep. [**142**]{}, 167 (1986).
P. Braun-Munzinger, K. Redlich and J. Stachel, in [*Quark Gluon Plasma 3*]{}, edited by R. C. Hwa and X.-N. Wang (World Scientific, Singapore, 2004), p. 491. F. Antinori, nucl-ex/0404032, plenary talk at [*Quark Matter 2004*]{}, 17th Int’l Conf. on Ultra Relativistic Nucleus-Nucleus Collisions (Oakland, CA, January 2004), edited by H. G. Ritter and X. N. Wang.
R. C. Hwa and C. B. Yang, nucl-th/0401001, Phys. Rev. C (to be published).
S. Okubo, Phys. Lett. [**5**]{}, 165 (1963).
G. Zweig, CERN Report No. 8417/TH412 (1964).
J. Iizuka [*et al.*]{}, Prog. Theor. Phys. [**35**]{}, 1061 (1966).
R. C. Hwa, and C. B. Yang, Phys. Rev. C [**67**]{}, 034902 (2003).
V. Greco, C. M. Ko, and P. Lévai, Phys. Rev.Lett. [**90**]{}, 202302 (2003); Phys. Rev. C [**68**]{}, 034904 (2003).
R. J. Fries, B. Müller, C. Nonaka and S. A.Bass, Phys. Rev. Lett. [**90**]{}, 202303 (2003); Phys. Rev. C [**68**]{}, 044902 (2003).
R. C. Hwa, and C. B. Yang, hep-ph/0312271.
K. P. Das and R. C. Hwa, Phys. Lett. B [**68**]{}, 459 (1977).
R. C. Hwa, Phys. Rev. D [**22**]{}, 1593 (1980).
R. C. Hwa, and C. B. Yang, Phys. Rev. C [**66**]{}, 025205 (2002).
R. C. Hwa, and C. B. Yang, Phys. Rev. C [**65**]{}, 034905 (2002).
R. C. Hwa, and C. B. Yang, Phys. Rev. C [**66**]{}, 025204 (2002).
D. K. Srivastava, C. Gale, and R. J. Fries, Phys.Rev. C [**67**]{}, 034903 (2003).
J. Adams [*et al.*]{} (STAR Collaboration), nucl-ex/0406003.
C. Suire (for the STAR Collaboration), Nucl. Phys. A [**715**]{}, 470c (2003).
A. Białas, Phys. Lett. B [**442**]{}, 449 (1998).
J. Zimányi, T. S. Biró, T. Csörgő and P. Lévai, Phys. Lett. B [**472**]{}, 243 (2000).
R. C. Hwa, and C. B. Yang, Phys. Rev. C [**66**]{}, 064903 (2002). S. S. Adler [*et al.*]{} (STAR Collaboration), Phys. Rev. C [**69**]{}, 034909 (2004).
H. Sorge, Phys. Rev. C [**52**]{}, 3291 (1995).
M. Bleicher [*et al.*]{}, J. Phys. G: Nucl. Part. Phys. [**25**]{}, 1859 (1999); S. A. Bass [*et al.*]{}, Phys. Rev. [**C 60**]{}, 021902 (1999).
|
---
abstract: 'The similarities between linearized gravity and electromagnetism are known since the early days of General Relativity. Using an exact approach based on tidal tensors, we show that such analogy holds only on very special conditions and depends crucially on the reference frame. This places restrictions on the validity of the “gravito-electromagnetic” equations commonly found in the literature.'
---
Gravito-electromagnetic analogy based on tidal tensors
======================================================
The topic of the gravito-electromagnetic analogies has a long story, with different analogies being unveiled throughout the years. Some are purely formal analogies, like the splitting of the Weyl tensor in electric and magnetic parts, e.g. [@Maartens]; but others (e.g [@DSX; @CostaHerdeiro2008; @Jantzen; @Natario; @Ruggiero:2002]) stem from certain physical similarities between the gravitational and electromagnetic interactions. The linearized Einstein equations (see e.g. [@DSX; @Ruggiero:2002; @Gravitation; @and; @Inertia]), in the harmonic gauge $\bar{h}_{\alpha\beta}^{\,\,\,\,\,\,,\beta}=0$, take the form $\square\bar{h}^{\alpha\beta}=-16\pi T^{\alpha\beta}/c^{4}$, similar to Maxwell equations in the Lorentz gauge: $\square A^{\beta}=-4\pi j^{\beta}/c$. That suggests an analogy between the trace reversed time components of the metric tensor $\bar{h}_{0\alpha}$ and the electromagnetic 4-potential $A_{\alpha}$. Defining the 3-vectors usually dubbed gravito-electromagnetic fields, the time components of these equations may be cast in a Maxwell-like form, e.g. eqs (16)-(22) of [@Ruggiero:2002]. Furthermore (on certain special conditions, see section \[sec:Linearized-Gravity\]) geodesics, precession and forces on gyroscopes are described in terms of these fields in a form similar to their electromagnetic counterparts, e.g. [@Ruggiero:2002], [@Gravitation; @and; @Inertia]. Such analogy may actually be cast in an exact form using the 3+1 splitting of spacetime (see [@Jantzen; @Natario]).
These are analogies comparing physical quantities (electromagnetic forces) from one theory with inertial gravitational forces (i.e. fictitious forces, that can be gauged away by moving to a freely falling frame, due to the equivalence principle); it is clear that (non-spinning) test particles in a gravitational field move with zero acceleration $DU^{\alpha}/d\tau=0$; and that the spin 4-vector of a gyroscope undergoes Fermi-Walker transport $DS^{\alpha}/d\tau=S_{\sigma}U^{\alpha}DU^{\sigma}/d\tau$, with no real torques applied on it. In this sense the gravito-electromagnetic fields are pure coordinate artifacts, attached to the observer’s frame.
However, these approaches describe also (not through the “gravito-electromagnetic” fields themselves, but through their derivatives; and, again, under very special conditions) tidal effects, like the force applied on a gyroscope. And these are covariant effects, implying physical gravitational forces.
Herein we will discuss under which precise conditions a similarity between gravity and electromagnetism occurs (that is, under which conditions the physical analogy $\bar{h}_{0\mu}\leftrightarrow A_{\mu}$ holds, and Eqs. like (16)-(22) of [@Ruggiero:2002] have a physical content). For that we will make use of the tidal tensor formalism introduced in [@CostaHerdeiro2008]. The advantage of this formalism is that, by contrast with the approaches mentioned above, it is based on quantities which can be covariantly defined in both theories — tidal forces (the only physical forces present in gravity) — which allows for a more transparent comparison between the electromagnetic (EM) and gravitational (GR) interactions.
[>p[39.6ex]{}c>p[41.9ex]{}c]{} & Worldline deviation: & & Geodesic deviation: & & & & Force on magnetic dipole: & & Force on gyroscope: & & & & Maxwell Equations: & & Eqs. Grav. Tidal Tensors: & $E_{\,\,\,\alpha}^{\alpha}=4\pi\rho_{c}$ & (3a) & $\mathbb{E}_{\,\,\,\alpha}^{\alpha}=4\pi\left(2\rho_{m}+T_{\,\,\alpha}^{\alpha}\right)$ & (3b)$E_{[\alpha\beta]}=\frac{1}{2}F_{\alpha\beta;\gamma}U^{\gamma}$ & (4a) & $\mathbb{E}_{[\alpha\beta]}=0$ & (4b)$B_{\,\,\,\alpha}^{\alpha}=0$ & (5a) & $\mathbb{H}_{\,\,\,\alpha}^{\alpha}=0$ & (5b) & & &
\
[$\rho_{c}=-j^{\alpha}U_{\alpha}$ and $j^{\alpha}$ are, respectively, the charge density and current 4-vector; $\rho_{m}=T_{\alpha\beta}U^{\alpha}U^{\beta}$ and $J^{\alpha}=-T_{\,\beta}^{\alpha}U^{\beta}$ are the mass/energy density and current (quantities measured by the observer of 4-velocity $U^{\alpha}$); $T_{\alpha\beta}\equiv$ energy-momentum tensor; $S^{\alpha}\equiv$ spin 4-vector; $\star\equiv$ Hodge dual. We use $\tilde{e}_{0123}=-1$.]{}
The tidal tensor formalism unveils a new gravito-electromagnetic analogy, summarized in Table \[analogy\], based on exact and covariant equations. These equations make clear key differences, and under which conditions a similarity between the two interactions may occur.
Eqs. (1) are the worldline deviation equations yielding the relative acceleration of two neighboring particles (connected by the infinitesimal vector $\delta x^{\alpha}$) with the *same* 4-velocity $U^{\alpha}$ (and the same $q/m$ ratio, in the electromagnetic case). These equations manifest the physical analogy between electric tidal tensors: $\mathbb{E}_{\alpha\beta}\leftrightarrow E_{\alpha\beta}$.
Eq. (2a) yields the electromagnetic force exerted on a magnetic dipole moving with 4-velocity $U^{\alpha}$, and is the covariant generalization of the usual 3-D expression $\mathbf{F_{EM}}=\nabla(\mathbf{S}.\mathbf{B})q/2m$ (valid only in the dipole’s proper frame); Eq. (2b) is exactly the Papapetrou-Pirani equation for the gravitational force exerted on a spinning test particle. In both (2a) and (2b), Pirani’s supplementary condition $S_{\mu\nu}U^{\nu}=0$ is assumed (c.f. [@CostaHerdeiro2009]). These equations manifest the physical analogy between magnetic tidal tensors: $B_{\alpha\beta}\leftrightarrow\mathbb{H}_{\alpha\beta}$.
Taking the traces and antisymmetric parts of the EM tidal tensors, one obtains Eqs. (3a)-(6a), which are explicitly covariant forms for each of Maxwell equations. Eqs. (3a) and (6a) are, respectively, the time and space projections of Maxwell equations $F_{\ \ \ ;\beta}^{\alpha\beta}=4\pi j^{\alpha}$; i.e., they are, respectively, covariant forms of $\nabla\cdot\mathbf{E}=4\pi\rho_{c}$ and $\nabla\times\mathbf{B}=\partial\mathbf{E}/\partial t+4\pi\mathbf{j}$; Eqs. (4a) and (5a) are the space and time projections of the electromagnetic Bianchi identity $\star F_{\ \ \ ;\beta}^{\alpha\beta}=0$; i.e., they are covariant forms for $\nabla\times\mathbf{E}=-\partial\mathbf{B}/\partial t$ and $\nabla\cdot\mathbf{B}=0$. These equations involve only tidal tensors and sources, which can be seen substituting the following decomposition (or its Hodge dual) in (4a) and (6a): $$F_{\alpha\beta;\gamma}=2U_{[\alpha}E_{\beta]\gamma}+\epsilon_{\alpha\beta\mu\sigma}B_{\,\,\,\gamma}^{\mu}U^{\sigma}\
.\label{Fdecomp}$$ It is then straightforward to obtain the *physical* gravitational analogues of Maxwell equations: one just has to apply the same procedure to the gravitational tidal tensors, i.e., write the equations for their traces and antisymmetric parts (that is more easily done decomposing the Riemann tensor in terms of the Weyl tensor and source terms, see [@CostaHerdeiro2007] sec. 2), which leads to Eqs. (3b) - (6b). Underlining the analogy with the situation in electromagnetism, Eqs. (3b) and (6b) turn out to be the time-time and and time-space projections of Einstein equations $R_{\mu\nu}=8\pi(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T_{\,\,\,\alpha}^{\alpha})$, and Eqs. (4b) and (5b) the time-space and time-time projections of the algebraic Bianchi identities $\star R_{\ \ \ \gamma\beta}^{\gamma\alpha}=0$.
Gravity vs Electromagnetism
---------------------------
*Charges* — the gravitational analogue of $\rho_{c}$ is $2\rho_{m}+T_{\,\,\alpha}^{\alpha}$ ($\rho_{m}+3p$ for a perfect fluid) $\Rightarrow$ in gravity, pressure and all material stresses contribute as sources.
*Ampere law* — in stationary (in the observer’s rest frame) setups, $\star F_{\alpha\beta;\gamma}U^{\gamma}$ vanishes and equations (6a) and (6b) match up to a factor of 2 $\Rightarrow$ currents of mass/energy source gravitomagnetism like currents of charge source magnetism.
*Symmetries of Tidal Tensors* — The GR and EM tidal tensors do not generically exhibit the same symmetries, signaling fundamental differences between the two interactions. In the general case of fields that are time dependent in the observer’s rest frame (that is the case of an intrinsically non-stationary field, or an observer moving in a stationary field), the electric tidal tensor $E_{\alpha\beta}$ possesses an antisymmetric part, which is the covariant derivative of the Maxwell tensor along the observer’s worldline; there is also an antisymmetric contribution $\star F_{\alpha\beta;\gamma}U^{\gamma}$ to $B_{\alpha\beta}$. These terms consist of time projections of EM tidal tensors (cf. decomposition \[Fdecomp\]), and contain the laws of electromagnetic induction. The gravitational tidal tensors, by contrast, are symmetric (in vacuum, in the magnetic case) and spatial, manifesting the absence of analogous effects in gravity.
*Gyroscope vs. magnetic dipole* — According to Eqs. (2), both in the case of the magnetic dipole and in the case of the gyroscope, it is the magnetic tidal tensor, *as seen by the test particle* ($U^{\alpha}$ in Eqs. (2) is the gyroscope/dipole 4-velocity), that determines the force exerted upon it. Hence, from Eqs. (6), we see that the forces can be similar only if the fields are stationary (besides weak) in the gyroscope/dipole frame, i.e., when it is at “rest” in a stationary field. Eqs. (2) also tell us that in gravity the angular momentum $S$ plays the role of the magnetic moment $\mu=S(q/2m)$; the relative minus sign manifests that masses/charges of the same sign attract/repel one another in gravity/electromagnetism, as do charge/mass currents with parallel velocity.
Linearized Gravity\[sec:Linearized-Gravity\]
============================================
If the fields are stationary in the observer’s rest frame, the GR and EM tidal tensors have the same symmetries, which by itself does not mean a close similarity between the two interactions (note that despite the analogy in Table 1, EM tidal tensors are linear, whereas the GR ones are not). But in two special cases a matching between tidal tensors occurs: ultrastationary spacetimes (where the gravito-magnetic tidal tensor is linear, see [@CostaHerdeiro2008] Sec. IV) and linearized gravitational perturbations, which is the case of interest for astronomical applications.
Consider an arbitrary electromagnetic field $A^{\alpha}=(\phi,\mathbf{A})$ and arbitrary perturbations around Minkowski spacetime in the form[^1]
$$ds^{2}=-c^{2}\left(1-2\frac{\Phi}{c^{2}}\right)dt^{2}-\frac{4}{c}\mathcal{A}_{j}dtdx^{j}+\left[\delta_{ij}+2\frac{\Theta_{ij}}{c^{2}}\right]dx^{i}dx^{j}\ .\label{Linear pert}$$
**Tidal effects.** — The GR and EM tidal tensors from these setups will be in general very different, as is clear from equations (3-6), and as one may check from the explicit expressions in [@CostaHerdeiro2008].
But if one considers time independent fields, and a static observer of 4-velocity $U^{\mu}=c\delta_{0}^{\mu}$, then the *linearized* gravitational tidal tensors match their electromagnetic counterparts identifying $(\phi,A^{i})\leftrightarrow(\Phi,\mathcal{A}^{i})$ (in expressions below colon represents partial derivatives; $\epsilon_{ijk}\equiv$ Levi Civita symbol): $$\mathbb{E}_{ij}\simeq-\Phi_{,ij}\stackrel{\Phi\leftrightarrow\phi}{=}E_{ij},\quad\mathbb{H}_{ij}\simeq\epsilon_{i}^{\,\,\, lk}\mathcal{A}_{k,lj}\stackrel{\mathcal{A}\leftrightarrow A}{=}B_{ij}\ .\label{MatchingLinear}$$ This suggests the physical analogy $(\phi,A^{i})\leftrightarrow(\Phi,\mathcal{A}^{i})$, and defining the “gravito-electro-magnetic fields” $\mathbf{E_{G}}=-\nabla\Phi$ and $\mathbf{B_{G}}=\nabla\times\bm{\mathcal{A}}$, in analogy with the electromagnetic fields $\mathbf{E}=-\nabla\phi,$ $\mathbf{B}=\nabla\times\mathbf{A}$. In terms of these fields we have $\mathbb{E}_{ij}\simeq(E_{G})_{i,j}$ and $\mathbb{H}_{ij}\simeq(B_{G})_{i,j}$, in analogy with the electromagnetic tidal tensors $E_{ij}=E_{i,j}$ and $B_{ij}=B_{i,j}$.
The matching (\[MatchingLinear\]) means that a gyroscope at rest (relative to the static observer) will feel a force $F_{G}^{\alpha}$ similar to the electromagnetic force $F_{EM}^{\alpha}$ on a magnetic dipole, which in this case take the very simple forms (time components are zero): $$\mathbf{F_{EM}}=\frac{q}{2mc}\nabla(\mathbf{B}.\mathbf{S});\ \ \ \ \ F_{G}^{j}=-\frac{1}{c}\mathbb{H}^{ij}S_{i}\approx-\frac{1}{c}(B_{G})^{i,j}S_{i}\ \Leftrightarrow\ \mathbf{F_{G}}=-\frac{1}{c}\nabla(\mathbf{B_{G}}.\mathbf{S})\ .\label{FG_Stationary}$$ Had we considered gyroscopes/dipoles with different 4-velocities, not only the expressions for the forces would be more complicated, but also the gravitational force would significantly differ from the electromagnetic one, as one may check comparing Eqs. (12) with (17)-(20) of [@CostaHerdeiro2008]. This will be exemplified in section \[Translational-vs.-Rotational\].
The matching (\[MatchingLinear\]) also means, by similar arguments, that the relative acceleration between two neighboring masses $D^{2}\delta x^{i}/d\tau^{2}=-\mathbb{E}^{ij}\delta x_{j}$ is similar to the relative acceleration between two charges (with the same $q/m$): $D^{2}\delta x^{i}/d\tau^{2}=E^{ij}\delta x_{j}(q/m)$, *at the instant* when the test particles have 4-velocity $U^{\alpha}=c\delta_{0}^{\alpha}$ (i.e., are *at rest* relative to the static observer $\mathcal{O}$).
**Gyroscope precession.** — The evolution of the spin vector of the gyroscope is given by the Fermi-Walker transport law, which, for a gyroscope at rest reads $DS^{i}/d\tau=0$; hence, we have, in the coordinate basis, Eq. (\[PrecessGen\]a). The last term of Eq. (\[PrecessGen\]a) vanishes if we express $\mathbf{S}$ in the local orthonormal tetrad $e^{\hat{\alpha}}$: $S^{i}=S^{\hat{i}}e_{\,\hat{i}}^{i}$, where to linear order $e_{\,\hat{i}}^{i}=\delta_{\ \hat{i}}^{i}-\Theta_{\ \ \hat{i}}^{i}/c^{2}$; in this fashion we obtain Eq. (\[PrecessGen\]b), which is similar to the precession of a magnetic dipole in a magnetic field $d\mathbf{S}/dt=q\mathbf{S}\times\mathbf{B}/2mc$: $$\frac{dS^{i}}{dt}=-c\Gamma_{0j}^{i}S^{j}=-\frac{1}{c}\left[(\mathbf{S}\times\mathbf{B_{G}})^{i}+\frac{1}{c}\frac{\partial\Theta_{\ }^{ij}}{\partial t}S_{j}\right]\ \ (a);\ \ \ \ \ \ \frac{dS^{\hat{i}}}{dt}=-\frac{1}{c}(\mathbf{S}\times\mathbf{B_{G}})^{\hat{i}}\ \ (b).\label{PrecessGen}$$ Thus, in the special case of gyroscope precession, the linear gravito-electromagnetic analogy holds even if the fields vary with time.
**Geodesics.** — The space part of the equation of geodesics $U_{\ ,\beta}^{\alpha}U^{\beta}=-\Gamma_{\beta\gamma}^{\alpha}U^{\beta}U^{\gamma}$ is given, to first order in the perturbations and in test particle’s velocity, by ($a^{i}\equiv d^{2}x^{i}/dt^{2})$:$$\begin{aligned}
\mathbf{a} & = & \nabla\Phi+\frac{2}{c}\frac{\partial\bm{\mathcal{A}}}{\partial t}-\frac{2}{c}\mathbf{v}\times(\nabla\times\bm{\mathcal{A}})-\frac{1}{c^{2}}\left[\frac{\partial\Phi}{\partial t}\mathbf{v}+2\frac{\partial\Theta_{\ j}^{i}}{\partial t}v^{j}\mathbf{e_{i}}\right]\ .\label{geoGeneral}\end{aligned}$$ Comparing with the electromagnetic Lorentz force:$$\mathbf{a}=\frac{q}{m}\left[-\nabla\phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}+\frac{\mathbf{v}}{c}\times(\nabla\times\mathbf{A})\right]=\frac{q}{m}\left[\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right]\,,\label{Lorentz}$$ these equations do not manifest, in general, a close analogy. Note that the last term of (\[geoGeneral\]), which has no electromagnetic analogue, is, for the problem at hand (see next section), of the same order of magnitude as the second and third terms. But when one considers stationary fields, then (\[geoGeneral\]) takes the form $\mathbf{a}=-\mathbf{E_{G}}-2\mathbf{v}\times\mathbf{B_{G}}/c$ analogous to (\[Lorentz\]).
Note the difference between this analogy and the one from the tidal effects considered above: in the case of the latter, the similarity occurs only when the *test particle* sees time independent *fields* (fields $\equiv$ derivatives of potentials/of metric perturbations); for geodesics, it is when *the observer* (not the test particle!) sees a time independent *potential* ($\phi$)*/metric perturbations*($\Phi,\Theta_{ij}$).
\[Translational-vs.-Rotational\]Translational vs. Rotational Mass Currents
--------------------------------------------------------------------------
The existence of a similarity between gravity and electromagnetism thus relies on the time dependence of the mass currents: if the currents are (nearly) stationary, for instance from a spinning celestial body, the gravitational field generated is analogous to a magnetic field; an example is the gravitomagnetic field due to the rotation of the Earth, detected on LAGEOS data by [@Ciufolini; @Lageos] (and which is also the subject of experimental scrutiny by the Gravity Probe B and the upcoming LARES missions). But when the currents seen by the observer vary with time — e.g. the ones resulting from translation of the celestial body, considered in [@SoffelKlioner] — then the dynamics differ significantly.
**Rotational Currents.** — We will start by the well known analogy between the electromagnetic field of a spinning charge (charge $Q$, magnetic moment $\mu$) and the gravitational field (in the far region $r\rightarrow\infty$) of a rotating celestial body (mass $m$, angular momentum $J$), see Fig.\[fig1\]
{width="123.2mm"}
\[fig1\]
The electromagnetic field of the spinning charge is described by the 4-potential $A^{\alpha}=(\phi,\mathbf{A})$, given by (\[GravPot\]a). The spacetime around the spinning mass is asymptotically described by the linearized Kerr solution, obtained by putting in (\[Linear pert\]) the perturbations (\[GravPot\]b) :$$\phi=\frac{Q}{r}\ ,\ \ \ \mathbf{A}=\frac{1}{c}\frac{\bm{\mathbf{\mu}}\times\mathbf{r}}{r^{3}}\ \ \ (a);\ \ \ \ \ \ \Phi=\frac{M}{r}\ ,\ \ \ \bm{\mathcal{A}}=\frac{1}{c}\frac{\mathbf{J}\times\mathbf{r}}{r^{3}}\ ,\ \ \Theta_{ij}=\Phi\delta_{ij}\ \ \ (b).\ \ \ \label{GravPot}$$ For the observer at rest $\mathcal{O}$ the gravitational tidal tensors asymptotically match the electromagnetic ones, identifying the appropriate parameters:[$$\mathbb{E}_{ij}\simeq\frac{M}{r^{3}}\delta_{ij}-\frac{3Mr_{i}r_{j}}{r^{5}}\stackrel{M\leftrightarrow Q}{=}E_{ij};\ \ \ \ \mathbb{H}_{ij}\simeq\frac{3}{c}\left[\frac{(\mathbf{r}.\mathbf{J})}{r^{5}}\delta_{ij}+2\frac{r_{(i}J_{j)}}{r^{5}}-5\frac{(\mathbf{r}.\mathbf{J})r_{i}r_{j}}{r^{7}}\right]\stackrel{J\leftrightarrow\mu}{=}B_{ij}$$ ]{}(all the time components are zero for this observer). This me[ans that]{} $\mathcal{O}$ will find a similarity between *physical* (i.e., tidal) gravitational forces and their electromagnetic counterparts: the gravitational force $F_{G}^{i}=-\mathbb{H}^{ji}S_{j}/c$ exerted on a gyroscope carried by $\mathcal{O}$ is similar to the force $F_{EM}^{i}=qB^{ji}S_{j}/2mc$ on a magnetic dipole; and the worldline deviation $D^{2}\delta x^{i}/d\tau^{2}=-\mathbb{E}^{ij}\delta x_{i}$ of two masses dropped from rest is similar to the deviation between two charged particles with the same $q/m$.
Moreover, observer $\mathcal{O}$ will see test particles moving on geodesics described by equations analogous to the electromagnetic Lorentz force (see Fig. \[fig1\]).
**Translational Currents.** — For the observer $\bar{\mathcal{O}}$ moving with velocity $\mathbf{w}$ relative to the mass/charge of Fig. \[fig1\], however, the electromagnetic and gravitational interactions will look significantly different. For simplicity we will specialize here to the case where $\mathbf{J}=\bm{\mu}=0$, so that the mass/charge currents seen by $\bar{\mathcal{O}}$ arise solely from translation. To obtain the electromagnetic 4-potential $A^{\bar{\alpha}}$ in the frame $\bar{\mathcal{O}}$, we apply the boost $A^{\bar{\alpha}}=\Lambda_{\ \alpha}^{\bar{\alpha}}A^{\alpha}=(\bar{\phi},\bar{\mathbf{A}})$, where $\Lambda_{\ \alpha}^{\bar{\alpha}}\equiv\partial\bar{x}^{\bar{\alpha}}/\partial x^{\alpha}$, using the expansion of Lorentz transformation (as done in e.g. [@WillNordvedt1972]): $$t=\bar{t}\left(1+\frac{w^{2}}{2c^{2}}+\frac{3w^{4}}{8c^{4}}\right)+\left(1+\frac{w^{2}}{2c^{2}}\right)\frac{\mathbf{\bar{x}}.\mathbf{w}}{c^{2}};\ \ \ \ \mathbf{x}=\mathbf{\bar{x}}+\frac{1}{2c^{2}}(\bar{\mathbf{x}}.\mathbf{w})\mathbf{w}+\left(1+\frac{w^{2}}{2c^{2}}\right)\mathbf{w}\bar{t}\ ,\label{Boost}$$ yielding, to order $c^{-2}$, $A^{\bar{\alpha}}=(\bar{\phi},\mathbf{\bar{A}})$, with $\bar{\phi}=Q(1+w^{2}/2c^{2})/r$ and $\mathbf{\bar{A}}=-Q\mathbf{w}/rc$. To obtain $A^{\bar{\alpha}}$ in the coordinates ($\bar{x}^{i},\bar{t}$) of $\bar{\mathcal{O}}$, we must also express $r$ (which denotes the distance between the source and the point of observation, in the frame $\mathcal{O}$) in terms of $R\equiv|\mathbf{\bar{r}}+\mathbf{w}\bar{t}|$, i.e., the distance between the source and the point of observation in the frame $\bar{\mathcal{O}}$. Using transformation (\[Boost\]), we obtain: $r^{-1}=R^{-1}[1-(\mathbf{w}.\mathbf{R})^{2}/(2R^{2}c^{2})]$, and finally the electromagnetic potentials seen by $\bar{\mathcal{O}}$:$$\bar{\phi}=\frac{Q}{R}\left(1+\frac{w^{2}}{2c^{2}}-\frac{(\mathbf{w}.\mathbf{R})^{2}}{4R^{2}c^{2}}\right);\ \ \ \ \ \ \mathbf{\bar{A}}=-\frac{1}{c}\frac{Q}{R}\mathbf{w}\ .\label{EMF Obar}$$ The metric of the spacetime around a point mass, in the coordinates of $\bar{\mathcal{O}}$, is also obtained using transformation (\[Boost\]), which is accurate to Post Newtonian order, by an analogous procedure. First we apply the Lorentz boost $g_{\bar{\alpha}\bar{\beta}}=\Lambda_{\ \bar{\alpha}}^{\alpha}\Lambda_{\ \bar{\beta}}^{\beta}g_{\alpha\beta}$ to the metric (\[GravPot\]) (with $\mathcal{A}=0$); then, expressing $r$ in terms of $R$, we finally obtain (note that, although we are not putting the bars therein, indices $\alpha=0,i$ in the following expressions refer to the coordinates of $\bar{\mathcal{O}}$): $$\begin{aligned}
g_{00} & = & -1+2\frac{M}{Rc^{2}}+\frac{4Mw^{2}}{Rc^{4}}-\frac{M(\mathbf{w}.\mathbf{R})^{2}}{c^{4}R^{3}}\equiv-1+\frac{2\bar{\Phi}}{c^{2}};\ \ \nonumber \\
\ g_{0i} & = & \frac{4Mw_{i}}{Rc^{3}}\equiv-\frac{2\bar{\mathcal{A}}_{i}}{c^{2}};\ \ \ \ \ \ \ \ \ \ g_{ij}=\left[1+2\frac{M}{Rc^{2}}\right]\delta_{ij}\equiv\left[1+2\frac{\bar{\Theta}}{c^{2}}\right]\delta_{ij}\ ,\label{GObar}\end{aligned}$$ where we retained terms up to $c^{-4}$ in $g_{00}$, up to $c^{-3}$ in $g_{i0}$ and $c^{-2}$ in $g_{ij}$, as usual in Post-Newtonian approximation. This matches, to linear order in $M$, Eqs. (5) of [@SoffelKlioner] for the case of one single source; or e.g. Eqs. (11) of [@Nordvedt1988] (in the case of the latter, an additional gauge choice, Eq. (19) of [@WillNordvedt1972], was made). The metric (\[GObar\]), like the electromagnetic potential (\[EMF Obar\]), is now time dependent, since $\mathbf{R}(\bar{t})=\mathbf{\bar{r}}+\mathbf{w}\bar{t}$.
The gravitational tidal tensors seen by $\bar{\mathcal{O}}$ are ([$\mathbb{E}_{\alpha0}=\mathbb{E}_{0\alpha}=\mathbb{H}_{\alpha0}=\mathbb{H}_{0\alpha}=0$]{}):[$$\begin{aligned}
\mathbb{E}_{ij} & = & -\bar{\Phi}_{,ij}-\frac{2}{c}\frac{\partial}{\partial\bar{t}}\bar{\mathcal{A}}_{(i,j)}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial\bar{t}^{2}}\bar{\Theta}\delta_{ij}\nonumber \\
& = & \frac{M\delta_{ij}}{R^{3}}\left[1+\frac{3w^{2}}{c^{2}}-\frac{9}{2}\frac{(\mathbf{R}.\mathbf{w})^{2}}{c^{2}R^{2}}\right]-\frac{3MR_{i}R_{j}}{R^{5}}\left[1+\frac{2w^{2}}{c^{2}}-\frac{5(\mathbf{R}.\mathbf{w})^{2}}{2c^{2}R^{2}}\right]\nonumber \\
& & -\frac{3Mw_{i}w_{j}}{c^{2}R^{3}}+\frac{6Mw_{(i}R_{j)}(\mathbf{R}.\mathbf{w})}{c^{2}R^{5}};\label{Egij}\\
\mathbb{H}_{ij} & = & \epsilon_{i}^{\,\,\, lk}\bar{\mathcal{A}}_{k,lj}-\frac{1}{c}\epsilon_{ij}^{\ \ l}\frac{\partial\bar{\Theta}_{,l}}{\partial\bar{t}}\ =\frac{M}{cR^{3}}\left[3\epsilon_{ij}^{\ \ k}w_{k}-\frac{3}{R^{2}}(\mathbf{R}.\mathbf{w})\epsilon_{ij}^{\ \ k}R_{k}-\frac{6}{R^{2}}(\mathbf{R}\times\mathbf{w})_{i}R_{j}\right],\label{Hij}\end{aligned}$$ ]{}which significantly differ from the electromagnetic ones ([$E_{0\alpha}=B_{0\alpha}=0$]{}):[$$\begin{aligned}
E_{ij} & = & -\bar{\phi}_{,ij}-\frac{1}{c}\frac{\partial}{\partial\bar{t}}\bar{A}_{i;j}\ =\ E_{i,j}\nonumber \\
& = & \frac{Q\delta_{ij}}{R^{3}}\left[1+\frac{w^{2}}{2c^{2}}-\frac{3}{4}\frac{(\mathbf{R}.\mathbf{w})^{2}}{c^{2}R^{2}}\right]-\frac{3QR_{i}R_{j}}{R^{5}}\left[1+\frac{w^{2}}{2c^{2}}-\frac{5(\mathbf{R}.\mathbf{w})^{2}}{4c^{2}R^{2}}\right]\nonumber \\
& & -\frac{Qw_{i}w_{j}}{2c^{2}R^{3}}+\frac{3Qw_{[i}R_{j]}(\mathbf{R}.\mathbf{w})}{c^{2}R^{5}};\label{Eij}\\
E_{i0} & = & -\frac{1}{c}\frac{\partial}{\partial\bar{t}}\bar{\phi}_{;i}-\frac{1}{c^{2}}\frac{\partial^{2}\bar{A}_{i}}{\partial\bar{t}^{2}}\ \equiv\frac{1}{c}\ \frac{\partial E_{i}}{\partial\bar{t}}\ =\ \frac{Q}{cR^{3}}\left[w_{i}-\frac{3(\mathbf{R}.\mathbf{w})R_{i}}{R^{2}}\right];\label{Ei0}\\
B_{ij} & = & \epsilon_{i}^{\ lm}\bar{A}_{m;lj}\ \equiv\ B_{i,j}=\frac{Q}{cR^{3}}\left[\epsilon_{ij}^{\ \ k}w_{k}-\frac{3}{R^{2}}(\mathbf{R}\times\mathbf{w})_{i}R_{j}\right];\label{Bij}\\
B_{i0}\ & = & \frac{1}{c}\frac{\partial B_{i}}{\partial\bar{t}}=-\ \frac{3Q}{c^{2}R^{5}}(\mathbf{R}.\mathbf{w})(\mathbf{R}\times\mathbf{w})_{i}\ .\label{Bi0}\end{aligned}$$ ]{}Note in particular that, unlike their gravitational counterparts, $E_{\alpha\beta}$ and $B_{\alpha\beta}$ are not symmetric, and have non-zero time components. The antisymmetric parts $E_{[ij]}=E_{[i,j]}$ and $B_{[ij]}=B_{[i,j]}$ above are (vacuum) Maxwell equations $\nabla\times\mathbf{E}=-(1/c)\partial\mathbf{B}/\partial t$ and $\nabla\times\mathbf{B}=(1/c)\partial\mathbf{E}/\partial t$, implying that a time varying electric/magnetic field endows the magnetic/electric tidal tensor with an antisymmetric part. For instance, a time varying electric field will always induce a force on a magnetic dipole. The fact that $\mathbb{E}_{\alpha\beta}$ and $\mathbb{H}_{\alpha\beta}$ are symmetric reflects the absence of analogous gravitational effects. The time component $B_{i0}$ means that the force on a magnetic dipole (magnetic moment $\mu=q/2m$) will have a time component $(F_{EM})_{0}=(1/c)\bm{\mu}.\partial\mathbf{B}/\partial t$, which (see [@CostaHerdeiro2009] sec. 1.2) is minus the power transferred to the dipole by Faraday’s law of induction (and is reflected in the variation of the dipole’s proper mass $m=-P^{\alpha}U_{\alpha}/c^{2}$). Again, this is an effect which has no gravitational counterpart: $\mathbb{H}_{\alpha0}=\mathbb{H}_{0\alpha}=0$, thus $(F_{G})_{0}=0$, and the proper mass of the gyroscope is a constant of the motion.
The space part of the geodesic equation for a test particle of velocity $\mathbf{v}$ is:[$$\begin{aligned}
\mathbf{a} & = & \nabla\bar{\Phi}+\frac{2}{c}\frac{\partial\bm{\bar{\mathcal{A}}}}{\partial\bar{t}}-2\mathbf{v}\times(\nabla\times\bm{\bar{\mathcal{A}}})-\frac{3}{c^{2}}\frac{\partial}{\partial\bar{t}}\left(\frac{M}{R}\right)\mathbf{v}\label{GeoTrans}\\
& = & -\frac{M}{R^{3}}\left[1+\frac{2w^{2}}{c^{2}}-\frac{3(\mathbf{R}.\mathbf{w})^{2}}{2c^{2}R^{2}}\right]\mathbf{R}+\frac{3M(\mathbf{R}.\mathbf{w})}{c^{2}R^{3}}\mathbf{w}-\frac{4M}{c^{2}R^{3}}\mathbf{v}\times(\mathbf{R}\times\mathbf{w})+\frac{3}{c^{2}}\frac{M}{R^{3}}(\mathbf{R}.\mathbf{w})\mathbf{v}\ ,\nonumber \end{aligned}$$ ]{}which matches equation (10) of [@SoffelKlioner], or (7) of [@Nordvedt1973], again, in the special case of only one source, and keeping therein only linear terms in the perturbations and test particle’s velocity $\mathbf{v}$.
Comparing with its electromagnetic counterpart[$$\left(\frac{m}{q}\right)\mathbf{a}=\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}=\frac{Q}{R^{3}}\left[1+\frac{w^{2}}{2c^{2}}-\frac{3(\mathbf{R}.\mathbf{w})^{2}}{4c^{2}R^{2}}\right]\mathbf{R}-\frac{1}{2}\frac{Q(\mathbf{R}.\mathbf{w})}{c^{2}R^{3}}\mathbf{w}+\frac{Q}{c^{2}R^{3}}\mathbf{v}\times(\mathbf{R}\times\mathbf{w})$$ ]{}we find them similar to a certain degree (up to some factors), except for the last term of (\[GeoTrans\]). That term signals a difference between the two interactions, because it means that there is a velocity dependent acceleration which is parallel to the velocity; that is in contrast with the situation in electromagnetism, where the velocity dependent accelerations arise from magnetic forces, and are thus always perpendicular to $\mathbf{v}$.
As expected from Eqs. (\[PrecessGen\]) (and by contrast with the other effects), the precession of a gyroscope carried by $\bar{\mathcal{O}}$, Eq. (\[PrecessTrans\]b) takes a form analogous to the precession of a magnetic dipole, Eq. (\[PrecessTrans\]a), if we express $\mathbf{S}$ in the local orthonormal tetrad $e^{\hat{i}}$, non rotating relative to the inertial observer at infinity, such that $S^{i}=(1-M/R)S^{\hat{i}}$: $$\frac{d\mathbf{S}}{d\bar{t}}=\frac{q}{2m}\frac{Q}{c^{2}R^{3}}\left[\mathbf{S}\times(\mathbf{R}\times\mathbf{w})\right]\ \ (a);\ \ \ \ \ \ \frac{dS^{\hat{i}}}{d\bar{t}}=\frac{2M}{c^{2}R^{3}}\left[(\mathbf{R}\times\mathbf{w})\times\mathbf{S}\right]^{\hat{i}}\ \ (b)\ .\label{PrecessTrans}$$ If instead of the gyroscope comoving with observer $\bar{\mathcal{O}}$ (with constant velocity $\mathbf{w}$), we had considered a gyroscope moving in a circular orbit, then an additional term would arise in analogy with Thomas precession for the magnetic dipole; for a circular geodesic that term amounts to $-1/4$ of expression (\[PrecessTrans\]b), and we would obtain the well known equation for geodetic precession (e.g. [@Gravitation; @and; @Inertia]).
Conclusion
==========
We conclude our paper by discussing some of the implications of our conclusions in the approaches usually found in literature. In the framework of linearized theory, e.g. [@Ruggiero:2002; @Gravitation; @and; @Inertia], Einstein equations are often written in a Maxwell-like form; likewise, geodesics, precession and gravitational force on a spinning test particle are cast (in terms of 3-vectors defined in analogy with the electromagnetic fields $\mathbf{E}$ and $\mathbf{B}$) in a form similar to, respectively, the Lorentz force on a charged particle, the precession and the force on a magnetic dipole.
We have concluded that the actual physical similarities between gravity and electromagnetism (on which the physical content of such approaches relies) occur only on very special conditions. For tidal effects, like the forces on a gyroscopes/dipoles, the analogy manifest in Eqs. (\[FG\_Stationary\]) holds only when the *test particle* sees time independent *fields*. In the example of analogous systems considered in section \[Translational-vs.-Rotational\], this means that the center of mass of the gyroscope/dipole must not move relative to the central body. In the case of the analogy between the equation of geodesics and the Lorentz force law (see Fig. \[fig1\]), as manifest in equation (\[geoGeneral\]), it is in the *potentials/metric perturbations*, as seen by *the observer* (not the test particle!), that the time independence is required. The latter condition is not as restrictive as the one of the tidal effects: consider for instance observers moving in circular orbits around a static mass/charge; such observers see an unchanging spacetime, and unchanging electromagnetic potentials, so, for them, the equation of geodesics and Lorentz force take similar forms (such analogy may actually be cast in an exact form, see [@Natario; @Jantzen]). However, those observers see a time-varying electric field $\mathbf{E}$ (constant in magnitude, but varying in direction), which, by means of equations (4) and (6), implies that the tidal tensors are not similar to the gravitational ones[^2].
Finally, as a consequence of this analysis, a distinction, from the point of view of the analogy with electrodynamics, between effects related to (stationary) rotational mass currents, and those arising from translational mass currents, becomes clear: albeit in the literature both are dubbed “gravitomagnetism”, one must note that, while the former are clearly analogous to magnetism, in the case of the latter the analogy is not so close.
We thank the anonymous referee for very useful comments and suggestions.
[Ruggiero-Tartaglia 2002]{} 1998, *Class. Q. Grav.*, 15, 705 [\[]{}arXiv:gr-qc/9704059\]
1991, *Phys. Rev. D.*, 43, 3273
2002, *Nuovo Cim.*, 117B 743 [\[]{}arXiv:gr-qc/0207065\].
2008, *Phys. Rev. D*, 78, 024021
2007 [\[]{}arXiv:gr-qc/0612140\]
2009, *Int. J. Mod. Phys. A*, 24, 1695
2004, *Nature* 431, 958; Ciufolini I., Pavlis E., Peron R., *New Astron.* 11, 527-550 (2006). The measurement of the Lense-Thirring effect is expected to be improved to an accuracy of a few percent by the upcoming LARES mission: http://lares.diaa.uniroma1.it/
1995, *Gravitation and Inertia* **(Princeton Univ. Press)**
2008, *Phys. Rev. D,* 78, 024033
1992, *Ann. Phys.,* 215, 1 [\[]{}arXiv:gr-qc/0106043\].
2007, *Gen. Rel. Grav.,* 39, 1477 [\[]{}arXiv:gr-qc/0701067\].
1973, *Phys. Rev. D*, **7,** 2347
1972, *APJ*, 177, 757
1988, *Int. J. Theoretical Physics,* **27,** 2347
[^1]: In the previous sections we were putting $c=1$. In this section we re-introduce the speed of light in order to facilitate comparison with relevant literature.
[^2]: The electromagnetic field $F^{\alpha\beta}$ is not constant along the worldline of an observer moving in a circular orbit (radius $R$, angular velocity $\bm{\Omega}$, velocity $\mathbf{w}=\bm{\Omega}\times\mathbf{R}$) around a point charge. Its variation endows the magnetic tidal tensor with an antisymmetric part, and the electric tidal tensor with a time component: $dF^{0i}/d\tau=Qw^{i}/cR^{3}=-2E^{[i0]}=-\epsilon^{ijk}B_{[jk]}$. This means that they significantly differ from the GR tidal tensors seen by an observer in circular motion around a point mass.
Note that both the GR and the EM tidal tensors for these analogous problems can be obtained from, respectively, Eqs. (\[Egij\])-(\[Hij\]) and (\[Eij\])-(\[Bi0\]), making therein $\mathbf{R}.\mathbf{w}=0$ (corresponding to circular motion), despite the fact that these expressions were originally derived for an observer with constant velocity. This is because, as can be seen from their definitions in Table \[analogy\], it is the 4-velocity $U^{\alpha}$ (regardless of the way it varies), at the given point, that determines the tidal tensors.
|
---
abstract: |
A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree $2p^2$, we determine exactly the possible sets of Hopf Galois structure types.
[**Keywords:**]{} Galois theory, Hopf algebra, Hopf action, computational algebra system Magma.
---
[**Computation of Hopf Galois structures on separable extensions and classification of those for degree twice an odd prime power**]{}
Teresa Crespo and Marta Salguero
Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain, e-mail: teresa.crespo@ub.edu, msalguga11@alumnes.ub.edu
Introduction
============
A Hopf Galois structure on a finite extension of fields $L/K$ is a pair $(H,\mu)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ is a Hopf action of $H$ on $L$, i.e a $K$-linear map $\mu: H \to
{\operatorname{End}}_K(L)$ giving $L$ a left $H$-module algebra structure and inducing a $K$-vector space isomorphism $L\otimes_K H\to{\operatorname{End}}_K(L)$. Hopf Galois structures were introduced by Chase and Sweedler in [@C-S]. For separable field extensions, Greither and Pareigis [@G-P] give the following group-theoretic equivalent condition to the existence of a Hopf Galois structure.
\[G-P\] Let $L/K$ be a separable field extension of degree $g$, ${{\widetilde{L}}}$ its Galois closure, $G={\mathrm{Gal}}({{\widetilde{L}}}/K), G'={\mathrm{Gal}}({{\widetilde{L}}}/L)$. Then there is a bijective correspondence between the set of Hopf Galois structures on $L/K$ and the set of regular subgroups $N$ of the symmetric group $S_g$ normalized by $\lambda (G)$, where $\lambda:G \hookrightarrow S_g$ is the monomorphism given by the action of $G$ on the left cosets $G/G'$.
For a given Hopf Galois structure on a separable field extension $L/K$ of degree $g$, we will refer to the isomorphism class of the corresponding group $N$ as the type of the Hopf Galois structure. The Hopf algebra $H$ corresponding to a regular subgroup $N$ of $S_g$ normalized by $\lambda (G)$ is the Hopf subalgebra ${{\widetilde{L}}}[N]^G$ of the group algebra ${{\widetilde{L}}}[N]$ fixed under the action of $G$, where $G$ acts on ${{\widetilde{L}}}$ by $K$-automorphisms and on $N$ by conjugation through $\lambda$. The Hopf action is induced by $n \mapsto n^{-1}(\overline{1})$, for $n \in N$, where we identify $S_g$ with the group of permutations of $G/G'$ and $\overline{1}$ denotes the class of $1_G$ in $G/G'$. It is known that the Hopf subalgebras of ${{\widetilde{L}}}[N]^G$ are in 1-to-1 correspondence with the subgroups of $N$ stable under the action of $G$ (see e.g. [@CRV] Proposition 2.2) and that, given two regular subgroups $N_1, N_2$ of $S_g$ normalized by $\lambda (G)$, the Hopf algebras ${{\widetilde{L}}}[N_1]^G$ and ${{\widetilde{L}}}[N_2]^G$ are isomorphic if and only if the groups $N_1$ and $N_2$ are $G$-isomorphic.
Childs [@Ch1] gives an equivalent condition to the existence of a Hopf Galois structure introducing the holomorph of the regular subgroup $N$ of $S_g$. We state the more precise formulation of this result due to Byott [@B] (see also [@Ch2] Theorem 7.3).
\[theoB\] Let $G$ be a finite group, $G'\subset G$ a subgroup and $\lambda:G\to {\operatorname{Sym}}(G/G')$ the morphism given by the action of $G$ on the left cosets $G/G'$. Let $N$ be a group of order $[G:G']$ with identity element $e_N$. Then there is a bijection between $${\cal N}=\{\alpha:N\hookrightarrow {\operatorname{Sym}}(G/G') \mbox{ such that
}\alpha (N)\mbox{ is regular}\}$$ and $${\cal G}=\{\beta:G\hookrightarrow {\operatorname{Sym}}(N) \mbox{ such that }\beta
(G')\mbox{ is the stabilizer of } e_N\}$$ Under this bijection, if $\alpha\in {\cal N}$ corresponds to $\beta\in {\cal G}$, then $\alpha(N)$ is normalized by $\lambda(G)$ if and only if $\beta(G)$ is contained in the holomorph ${\mathrm{Hol}}(N)$ of $N$.
[**Notation.**]{} In the sequel, $L/K$ will denote a finite separable field extension, ${{\widetilde{L}}}$ the normal closure of $L/K$, $G$ the Galois group ${\mathrm{Gal}}({{\widetilde{L}}}/K)$, $G'$ the Galois group ${\mathrm{Gal}}({{\widetilde{L}}}/L)$.
In [@CS1], we presented a program written in the computational algebra system Magma which determines all Hopf Galois structures on a separable field extension of a given degree $g$. It was built on Theorem \[G-P\] and was effective up to degree 11. In this paper we present a program based on Theorem \[theoB\] which determines all Hopf Galois structures on a separable field extension up to degree 31 and uses the Magma database of transitive groups which derives from the classification given in [@Hu]. We note that the bound on the degree is imposed by the fact that transitive groups are not classified beyond degree 31 and not by the effectiveness of our program. As for the one in [@CS1], the program presented here distinguishes almost classically Galois structures, decides for the remaining ones if the Galois correspondence is bijective and classifies the Hopf Galois structures in Hopf algebra isomorphism classes. In Section \[algorithm\] we give a description of the program and in Section \[table\] we show a table which summarizes the obtained results. The Magma code and the output of the program for each degree can be found in [@CSw].
In [@CS2] we studied Hopf Galois structures on a separable field extension of degree $p^n$, for $p$ an odd prime, $n\geq 2$. In this paper we consider separable field extensions of degree $2p^n$. In Section \[2pn\], we consider the general case $n\geq 2$. Theorems \[2pn1\] and \[2pn2\] concerns Hopf Galois structure types on a separable field extension of degree $2p^n$. The first one proves that the occurrence of cyclic type implies the occurrence of dihedral type and the second one that the occurrence of dihedral type excludes the occurrence of any type of exponent smaller that $2p^n$.
In Section \[section2p2\] we study in more detail the case $n=2$. We describe the five groups of order $2p^2$ and determine the corresponding automorphism groups. In Theorem \[2p2\] we obtain that if a separable field extension of degree $2p^2$ has a Hopf Galois structure whose type is one of the two groups of exponent $2p$ containing an element of order $2p$, then it has a Galois structure of type the only group of exponent $2p$ not containing an element of order $2p$. Finally Corollary \[sets\] gives all possible sets of Hopf Galois structure types for separable field extensions of degree $2p^2$.
We note that the results obtained have been intuited by performing an analysis of the outputs of our program.
Description of the computation procedure {#algorithm}
========================================
Given a separable field extension $L/K$ of degree $g$, ${{\widetilde{L}}}$ its Galois closure, $G={\mathrm{Gal}}({{\widetilde{L}}}/K),$ $G'={\mathrm{Gal}}({{\widetilde{L}}}/L)$, the action of $G$ on the left cosets $G/G'$ is transitive, hence the morphism $\lambda:G \rightarrow S_g$ identifies $G$ with a transitive subgroup of $S_g$, which is determined up to conjugacy. Moreover, if we enumerate the left cosets $G/G'$ starting with the one containing $1_G$, $\lambda(G')$ is equal to the stabilizer of $1$ in $\lambda(G)$. Therefore considering all separable field extensions $L/K$ of degree $g$ is equivalent to considering all transitive groups $G$ of degree $g$, up to conjugation. We note that these groups have been classified in [@Hu] up to $g=31$ and are included in the data base of the program Magma. We shall denote by $gTk$ the group given by Magma as `TransitiveGroup(g,k)`.
The program consists in an auxiliary function Automorphisms, a Main Function which determines the Hopf Galois structures and the additional functions giving the properties of these structures. We describe in more detail the two first ones since the others have already been described in [@CS1].
[**The function Automorphisms**]{}
Given a pair of integers $(g,k)$ this function returns the group ${\operatorname{Aut}}(G)$ of automorphisms of the group $G=gTk$ and the group ${\operatorname{Aut}}(G,G')$ of automorphisms of $G$ sending $G'$ to itself. In order to obtain the latter, the function uses the permutation representation of ${\operatorname{Aut}}(G)$ to obtain a group $P$ of permutations isomorphic to ${\operatorname{Aut}}(G)$. It then computes the set `stabims` of images of $G'$ under ${\operatorname{Aut}}(G)$ and the action of the generators of ${\operatorname{Aut}}(G)$ on this set. This gives the embedding `act` of $P$ into ${\operatorname{Sym}}(\verb;stabims;)$ and then the preimage of the stabilizer of 1 in ${\operatorname{Sym}}(\verb;stabims;)$ is a subgroup $Q$ of $P$ corresponding to ${\operatorname{Aut}}(G,G')$ by the permutation representation.
[**The main function**]{}
1. For each regular subgroup $N$ of $S_g$, we order it so that $n_j(1)=j$, for $1\leq j \leq g$, and compute its embedding $\lambda(N)$ in $S_g$ induced by the action on itself by left translation.
2. Given a transitive group $G$ of degree $g$ and a type of regular subgroups $N$ of $S_g$, we determine the subgroups $G^*$ of $H:={\mathrm{Hol}}(N)$ which are isomorphic to $G$ and transitive and such that the stabilizer ${\mathrm{Stab}}(G^*,1)$ of $1$ in $G^*$ is isomorphic to the stabilizer $G'$ of $1$ in $G$.
3. For each $G^*$ obtained in Step 1, we look for an isomorphism from $G^*$ to $G$ sending ${\mathrm{Stab}}(G^*,1)$ to $G'$. Let $f$ be the isomorphism from $G^*$ to $G$ provided by Magma. If $|G|=g$, then $G'$ is trivial and $f$ will do. If this is not the case, we compare $f({\mathrm{Stab}}(G^*,1))$ to the images of $G'$ by all automorphisms of $G$. If, for some $a \in {\operatorname{Aut}}G$, we have $f({\mathrm{Stab}}(G^*,1))=a(G')$, then $h:=f \circ a^{-1}$ is the wanted isomorphism. Then $\beta=h^{-1}$ is the embedding $\beta$ as in Theorem \[theoB\].
4. We order $T:=G/G'$ so that $t_j(1)=j$, for $1\leq j \leq g$.
5. For each pair $(G^*,h)$ obtained in Step 2, we compute the whole set of isomorphisms $G^*$ to $G$ sending ${\mathrm{Stab}}(G^*,1)$ to $G'$ by composing $h$ with each element in ${\operatorname{Aut}}(G,G')$. We obtain then all $\beta$’s from $G$ to ${\mathrm{Hol}}(N)$ as in Theorem \[theoB\]. For each such $\beta$ we determine the corresponding $\alpha(N)$ as in the proof of Theorem \[theoB\]. We obtain then all regular subgroups of $S_g$ isomorphic to $N$ and normalized by $\lambda(G)$.
We further determine those Hopf Galois structures for which the Galois correspondence is bijective and partition the set of Hopf Galois structures of a given type in Hopf algebra isomorphism classes with an optimized version of the algorithm presented in [@CS1] (see also [@CSw]). Previously we compute the embedding $\lambda(G)$ in $S_g$ induced by the action of $G$ by left translation on the set $T$ of cosets of $G$ modulo $G'$ accordingly with the ordering of $T$ in Step 3. Taking into account [@CS1] Proposition 6, we know that an almost classically Galois structure lies alone in its isomorphism class. Hence, we put these apart when performing the partition in Hopf algebra isomorphism classes. Furthermore we determine the Hopf Galois structures for which the Galois correspondence is bijective in a more effective way.
Extensions of degree $2p^n$ {#2pn}
===========================
If $N$ is a group of order $2p^n$, with $p$ an odd prime, its unique $p$-Sylow subgroup is either a cyclic group of order $p^n$ or a group of order $p^n$ and exponent $<p^n$. In the first case, $N$ is either a cyclic group or a dihedral group. In the second case, $N$ has exponent $<2p^n$.
\[2pn1\] Let $L/K$ be a separable extension of degree $2p^n$, $p$ an odd prime, $n\geq 1$. If $L/K$ has a Hopf Galois structure of cyclic type, then it has a Hopf Galois structure of dihedral type.
We shall see that ${\mathrm{Hol}}(D_{2p^n})$ contains a subgroup isomorphic to $C_{2p^n}$, acting regularly on $D_{2p^n}$ and such that its normalizer in ${\operatorname{Sym}}(D_{2p^n})$ is contained in ${\mathrm{Hol}}(D_{2p^n})$. The result will then follow from Theorem \[theoB\]. Let us write $D_{2p^n}=\langle r,s \mid r^{p^n}=s^2=1, srs=r^{-1} \rangle$. The automorphism group of $D_{2p^n}$ is generated by the automorphisms $\varphi$ and $\psi$ defined as follows.
$$\begin{array}{cccl} \varphi:&s & \mapsto & rs \\ &r & \mapsto & r \end{array} \quad \begin{array}{cccl} \psi:&s & \mapsto & s \\ &r & \mapsto & r^i \end{array}$$
where $i$ has order $p^{n-1}(p-1)$ modulo $p^n$ and $1 \leq i \leq p^n$. The automorphisms $\varphi$ and $\psi$ have orders $p^n$ and $p^{n-1}(p-1)$, respectively and satisfy $\psi\varphi\psi^{-1}=\varphi^i$. We have then ${\mathrm{Hol}}(D_{2p^n})=\langle r,s,\varphi,\psi\rangle$, with $r,\varphi$ of order $p^n$ and commuting with each other; $s$ of order $2$ and $\psi$ of order $p^{n-1}(p-1)$ commuting with each other and with the further relations
$$srs=r^{-1}, \varphi s \varphi^{-1}=rs, \psi r \psi^{-1}=r^i, \psi\varphi\psi^{-1}=\varphi^i.$$
Now, the powers of the element $s\varphi$ satisfy
$$(s\varphi)^k = \left\{ \begin{array}{ll} r^{-k/2} \varphi^k & \text{ if \ } k \text{\ is even} \\sr^{-(k-1)/2} \varphi^k & \text{ if \ } k \text{\ is odd} \end{array} \right.$$
Hence $s\varphi$ has order $2p^n$ and $\langle s\varphi \rangle$ acts regularly on $D_{2p^n}$. Indeed, we have
$$(s\varphi)^k (Id) = \left\{ \begin{array}{ll} r^{-k/2} & \text{ if \ } k \text{\ is even} \\sr^{-(k-1)/2} & \text{ if \ } k \text{\ is odd} \end{array} \right.$$
We want to see now that the normalizer of $\langle s\varphi \rangle$ in ${\operatorname{Sym}}(D_{2p^n})$ is included in ${\mathrm{Hol}}(D_{2p^n})$. We consider the element $h=\varphi^k \psi \in {\mathrm{Hol}}(D_{2p^n})$, where $k= -(i-1)/2$, if $i$ is odd, and $k=(i+p^n-1)/2$ if $i$ is even. We may check that $h$ has order $p^{n-1}(p-1)$ and satisfy $h(s\varphi)h^{-1}= r^k s \varphi^i \in \langle s\varphi \rangle$. Hence the normalizer of $\langle s\varphi \rangle$ in ${\mathrm{Hol}}(D_{2p^n})$ has order $2p^{2n-1} (p-1)$ and is then the whole normalizer of $\langle s\varphi \rangle$ in ${\operatorname{Sym}}(D_{2p^n})$.
To prove the next theorem we need the following technical lemma.
\[le\] Let $n\geq 1$ be an integer number and $p$ be an odd prime number. Then $p+1$ has order $p^{n-1}$ modulo $p^n$. More precisely
$$(p+1)^{p^{n-2}} \equiv 1+p^{n-1} \pmod{p^n}, \quad (p+1)^{p^{n-1}} \equiv 1 \pmod{p^n}.$$
We have $(p+1)^{p^{n-2}}=1+ p^{n-1} +\sum_{k=2}^{p^{n-2}} {p^{n-2} \choose k} p^k$. Now $v_p \left( {p^{n-2} \choose k}\right)=n-2-v_p(k)$ (see [@R]), hence $v_p \left( {p^{n-2} \choose k}p^k \right)=n-2+k-v_p(k) \geq n$ for $k\geq 2$. Now $(p+1)^{p^{n-1}} \equiv (1+p^{n-1})^p \equiv 1 \pmod{p^n}$.
\[2pn2\] Let $L/K$ be a separable extension of degree $2p^n$, $p$ an odd prime, $n\geq 1$. If $L/K$ has a Hopf Galois structure of dihedral type, then it has no Hopf Galois structure of type $N$, for any $N$ of exponent $<2p^n$.
If $L/K$ has a Hopf Galois structure of dihedral type, then $G={\mathrm{Gal}}({{\widetilde{L}}}/K)$ embeds into the holomorph ${\mathrm{Hol}}(D_{2p^n})$ as a transitive subgroup. We shall prove that every transitive subgroup of ${\mathrm{Hol}}(D_{2p^n})$ contains an element of order $p^n$. Let us write $D_{2p^n}=\langle r,s \rangle$ and ${\mathrm{Hol}}(D_{2p^n})=\langle r,s,\varphi,\psi\rangle$, as in the proof of Theorem \[2pn1\]. We have $|{\operatorname{Aut}}D_{2p^n}|=p^{2n-1}(p-1), |{\mathrm{Hol}}(D_{2p^n}|=2p^{3n-1}(p-1)$. We write $H:={\mathrm{Hol}}(D_{2p^n})$. A subgroup $G$ of $H$ is transitive if and only if $[G:{\mathrm{Stab}}_H(1)\cap G]=2p^n$. For a transitive subgroup $G$ of $H$, we have then $|G|=2p^ld$, with $n\leq l \leq 3n-1$, $d\mid p-1$. Let ${\operatorname{Syl}}(G)$ be a $p$-Sylow subgroup of $G$ (it is unique except for $p=3$, $d=2$). We have the equalities between indices
$$\begin{array}{ll} & [G:{\mathrm{Stab}}_H(1) \cap {\operatorname{Syl}}(G)]=[G:{\operatorname{Syl}}(G)][{\operatorname{Syl}}(G):{\mathrm{Stab}}_H(1)\cap {\operatorname{Syl}}(G)], \\
& [G:{\mathrm{Stab}}_H(1) \cap {\operatorname{Syl}}(G)]=[G:{\mathrm{Stab}}_H(1)\cap G][{\mathrm{Stab}}_H(1)\cap G:{\mathrm{Stab}}_H(1)\cap {\operatorname{Syl}}(G)].\end{array}$$
Taking into account that $[G:{\operatorname{Syl}}(G)]$ and $[{\mathrm{Stab}}_H(1)\cap G:{\mathrm{Stab}}_H(1)\cap {\operatorname{Syl}}(G)]$ are prime to $p$ and $[{\operatorname{Syl}}(G):{\mathrm{Stab}}_H(1)\cap {\operatorname{Syl}}(G)]$ is a $p$-power, we obtain
$$G \text{\ transitive \ } \Rightarrow [{\operatorname{Syl}}(G):{\mathrm{Stab}}_H(1)\cap {\operatorname{Syl}}(G)]=p^n.$$
Let us determine now ${\operatorname{Syl}}(H)$. We have $|{\operatorname{Syl}}(H)|=p^{3n-1}$. The elements $r, \varphi, \psi^{p-1}$ belong to ${\operatorname{Syl}}(H)$. We write $\chi:=\psi^{p-1}$. Now $\langle \varphi, \chi \rangle = \langle \varphi \rangle \rtimes \langle \chi \rangle$ has order $p^{2n-1}$ and $\langle r, \varphi, \chi \rangle = \langle r \rangle \rtimes \langle \varphi ,\chi \rangle$ has order $p^{3n-1}$, hence ${\operatorname{Syl}}(H)=\langle r, \varphi, \chi \rangle$. Therefore ${\operatorname{Syl}}(H)\cap {\mathrm{Stab}}_H(1)=\langle \varphi, \chi \rangle$ has order $p^{2n-1}$. By Lemma \[le\], $p+1$ has order $p^{n-1}$ modulo $p^n$, hence we may assume $\chi \varphi \chi^{-1}=\varphi^{p+1}$. We will now characterize the elements of order $p^n$ in $H$. We determine first the elements of order $p^n$ in ${\operatorname{Syl}}({\operatorname{Aut}}D_{2p^n})=\langle \varphi,\chi\rangle$. We shall prove
$$\label{eq1}
\varphi^i \chi^j, 0\leq i \leq p^{n}-1, 0\leq j \leq p^{n-1}-1 \text{\ has order \ }p^n \Leftrightarrow \varphi^i \text{ \ has order \ }p^n$$
By induction, we prove $(\varphi^i \chi^j)^k=\varphi^{i(\sum_{l=0}^{k-1} (p+1)^{lj})}\chi^{kj}$. Since $\chi$ has order $p^{n-1}$, we have in particular $(\varphi^i \chi^j)^{p^{n-1}}=\varphi^{i(\sum_{l=0}^{p^{n-1}-1} (p+1)^{lj})}$. Letting $j=p^a j_0$, with $p\nmid j_0$, we have
$$\sum_{l=0}^{p^{n-1}-1} (p+1)^{lj}=\dfrac{1-(p+1)^{p^{n-1}j}}{1-(p+1)^j}=\dfrac{1-(p+1)^{p^{n+a-1}j_0}}{1-(p+1)^{p^aj_0}}.$$
By Lemma \[le\], we have $(p+1)^{p^{n+a-1}}=1+p^{n+a}+\lambda p^{n+a+1}$, for some integer $\lambda$ and $(p+1)^{p^a}=1+p^{a+1}+\mu p^{a+2}$, for some integer $\mu$, which implies $(p+1)^{p^{n+a-1}j_0}=1+j_0 p^{n+a}+\lambda' p^{n+a+1}$, for some integer $\lambda'$ and $(p+1)^{p^aj_0}=1+j_0p^{a+1}+\mu' p^{a+2}$, for some integer $\mu'$. We obtain then
$$\label{eq2}
\sum_{l=0}^{p^{n-1}-1} (p+1)^{lj} \equiv p^{n-1} \pmod{p^n},$$
hence
$$\varphi^{i(\sum_{l=0}^{p^{n-1}-1} (p+1)^{lj})}= 1 \Leftrightarrow p\mid i \Leftrightarrow \varphi^i \text{ \ has order \ } \leq p^n.$$
We have then proved (\[eq1\]). Now, for $f \in {\operatorname{Aut}}(D_{2p^n})$, we have
$$(r^m,f)^k=(r^mf(r^m)\dots f^{k-1}(r^m),f^k),$$
hence, if $f$ has order $p^n$, then $(r,f)$ has order $p^n$. Now, if $f=\varphi^{pi} \chi^j$, then $(r^m,f)^{p^{n-1}}=(r^{m(\sum_{l=0}^{p^{n-1}-1} (p+1)^{lj})},{\operatorname{Id}})$. By (\[eq2\]), we obtain that $(r^m,\varphi^{pi} \chi^j)$ has order $p^n$ if and only if $r^m$ has order $p^n$ if and only if $p\nmid m$. Together we have obtained ${\mathrm{ord}}(r^m,\varphi^i \chi^j)=p^n$ if and only if $p\nmid m$ or $p\nmid i$. From this characterization of the elements in $H$ of order $p^n$, we obtain that the subset $F$ of elements of order $<p^n$ in $H$ is a subgroup of $H$ and has order $p^{3n-3}$. Moreover $|F\cap {\mathrm{Stab}}_H(1)|=p^{2n-2}$, hence $[F:F\cap {\mathrm{Stab}}_H(1)]=p^{n-1}$.
Let now $G$ be a subgroup of $H$ with no element of order $p^n$. We have then ${\operatorname{Syl}}(G) \subset F$ and $[{\operatorname{Syl}}(G):{\mathrm{Stab}}_H(1)\cap {\operatorname{Syl}}(G)]\leq [F:F\cap {\mathrm{Stab}}_H(1)]=p^{n-1}$, hence $G$ is not transitive. We have then proved that every transitive subgroup of ${\mathrm{Hol}}(D_{2p^n})$ contains an element of order $p^n$.
Let now $N$ be a group of order $2p^n$ and exponent $<2p^n$. We shall prove that ${\mathrm{Hol}}(N)$ has no elements of order $p^n$. Let $P$ be the unique $p$-Sylow subgroup of the group $N$. We have then $N=P\rtimes C_2$ and $P$ is a non-cyclic group of order $p^n$. If $\varphi \in {\operatorname{Aut}}N$, then $\varphi(P)=P$. By [@Ko], Corollary 4.3, $\varphi_{|P}^{p^{n-1}}={\operatorname{Id}}_P$. For $a$ an element of order 2 in $N$, we have $\varphi(a)=ax$, for some $x \in P$ satisfying $axa=x^{-1}$. In the particular case where $N$ is the direct product of $P$ and $C_2$, the only possibility is $\varphi(a)=a$. Now $\varphi(a)=ax$ implies
$$\varphi^{p^{n-1}}(a)=ax\varphi(x)\varphi^2(x) \dots\varphi^{p^{n-1}-1}(x)$$
and by [@Ko], Theorem 4.4, we have $x\varphi(x)\varphi^2(x) \dots\varphi^{p^{n-1}-1}(x)=e$, hence $\varphi^{p^{n-1}}(a)=a$. Since an element $\varphi$ in ${\operatorname{Aut}}(N)$ is determined by $\varphi_{|P}$ and $\varphi(a)$, we have obtained $\varphi^{p^{n-1}}={\operatorname{Id}}_N$ for all $\varphi \in {\operatorname{Aut}}(N)$. Now, since $|{\operatorname{Syl}}_p({\mathrm{Hol}}(N))|=|{\operatorname{Syl}}_p(N)||{\operatorname{Syl}}_p({\operatorname{Aut}}(N))|$, we have ${\operatorname{Syl}}_p({\mathrm{Hol}}(N))=P\rtimes {\operatorname{Aut}}(N)$. For $(y,\varphi) \in P\rtimes {\operatorname{Aut}}(N)$, we have
$$(y,\varphi)^{p^{n-1}}=(y\varphi(y)\varphi^2(y) \dots\varphi^{p^{n-1}-1}(y),\varphi^{p^{n-1}}).$$
By [@Ko], Theorem 4.4, we have $y\varphi(y)\varphi^2(y) \dots\varphi^{p^{n-1}-1}(y)=e$ and we have proved above that $\varphi^{p^{n-1}}={\operatorname{Id}}_N$. We have then that ${\mathrm{Hol}}(N)$ has no elements of order $p^n$.
In conclusion, if $L/K$ has a Hopf Galois structure of dihedral type, we have proved that $G={\mathrm{Gal}}({{\widetilde{L}}}/K)$ contains an element of order $p^n$. Since for $N$ a group of order $2p^n$ and exponent $<2p^n$, ${\mathrm{Hol}}(N)$ has no elements of order $p^n$, $G$ cannot be a subgroup of ${\mathrm{Hol}}(N)$, hence $L/K$ has no Hopf Galois structure of type $N$.
Extensions of degree $2p^2$ {#section2p2}
===========================
Let $p$ denote an odd prime. If $G$ is a group of order $2p^2$, the $p$-Sylow subgroup $Syl_p$ of $G$ is normal in $G$. Hence $G$ is the direct or semidirect product of $Syl_p$ and $C_2$. The abelian groups of order $2p^2$ are $C_{p^2} \times C_2 \simeq C_{2p^2}$ and $C_p \times C_p \times C_2 \simeq C_p \times C_{2p}$.
1) If $Syl_p=C_{p^2}$ and $G$ is not abelian, since ${\operatorname{Aut}}C_{p^2} \simeq ({\mathbf Z}/p^2 {\mathbf Z})^*$ has a unique element of order 2, there is a unique semidirect product and $G$ is isomorphic to the dihedral group $D_{2p^2}$.
2) If $Syl_p=C_p\times C_p=\langle a \rangle \times \langle b \rangle$ and $G$ is not abelian, since ${\operatorname{Aut}}(C_p\times C_p) \simeq {\mathrm{GL}}(2,p)$ has three elements of order 2, up to conjugation, namely $\left(\begin{smallmatrix} -1&0\\0&1 \end{smallmatrix} \right), \left(\begin{smallmatrix} 1&0\\0&-1 \end{smallmatrix} \right), \left(\begin{smallmatrix} -1&0\\0&-1 \end{smallmatrix} \right)$, there are three possible actions of $C_2$ on $Syl_p$:
$$\begin{array}{ccc} a & \mapsto & a^{-1} \\ b & \mapsto & b \end{array} , \quad \begin{array}{ccc} a & \mapsto & a \\ b & \mapsto & b^{-1} \end{array}, \quad \begin{array}{ccc} a & \mapsto & a^{-1} \\ b & \mapsto & b^{-1} \end{array}.$$
The two first ones give both clearly groups isomorphic to $D_{2p} \times C_p$. The third one gives a group which we denote by $(C_p\times C_p) \rtimes C_2$. This group is not isomorphic to $D_{2p} \times C_p$ since the center of $D_{2p} \times C_p$ has order $p$ whereas the center of $(C_p\times C_p) \rtimes C_2$ is trivial.
There are then 5 groups of order $2p^2$, up to isomorphism. We determine now the automorphism group for each of them.
1) ${\operatorname{Aut}}C_{2p^2} \simeq ({\mathbf Z}/2p^2 {\mathbf Z})^*$ is cyclic of order $p(p-1)$.
2) For $G=C_p\times C_p \times C_2=\langle a \rangle \times \langle b \rangle \times \langle c \rangle$, $c$ is the unique element of order 2. An element in ${\operatorname{Aut}}G$ is then given by $a \mapsto a^i b^j, b\mapsto a^k b^l, c \mapsto c$, with $0\leq i,j,k,l \leq p-1, p\nmid il-jk$. We have then ${\operatorname{Aut}}G \simeq {\mathrm{GL}}(2,p)$ and $|{\operatorname{Aut}}G|=(p^2-1)(p^2-p)=p(p+1)(p-1)^2$.
3) For $G=D_{2p} \times C_p$, we write $D_{2p}=\langle r,s|r^p=s^2=1,srs=r^{-1} \rangle$ and denote by $c$ a generator of $C_p$. An automorphism of $G$ is given by $r \mapsto r^i, s \mapsto r^j s, c \mapsto c^k$, with $1\leq i,k\leq p-1, 0\leq j \leq p-1$. We have then $|{\operatorname{Aut}}G|=p(p-1)^2$.
4) For $G=(C_p\times C_p) \rtimes C_2$, we write $C_p\times C_p=\langle a \rangle \times \langle b \rangle$ and $C_2=\langle c \rangle$. An automorphism of $G$ is given by $a \mapsto a^ib^j, b \mapsto a^k b^l, c \mapsto a^mb^n c$, with $0\leq i,j,k,l,m,n\leq p-1, p\nmid il-jk$. We have then $|{\operatorname{Aut}}G|=(p^2-1)(p^2-p)p^2=p^3(p+1)(p-1)^2$.
5) For $G=D_{2p^2}=\langle r,s|r^{p^2}=s^2=1,srs=r^{-1} \rangle$, an automorphism is given by $r\mapsto r^i, s \mapsto r^j s$, with $0\leq i,j\leq p^2-1, p\nmid i$. We have then $|{\operatorname{Aut}}G|=(p^2-p)p^2=p^3(p-1)$.
\[2p2\] Let $L/K$ be a separable extension of degree $2p^2$, $p$ an odd prime. If $L/K$ has a Hopf Galois structure of type $C_p\times C_{2p}$ or $C_p \times D_{2p}$, then it has a Hopf Galois structure of type $(C_p\times C_p)\rtimes C_2$.
Let us assume that $L/K$ has a Hopf Galois structure of type $C_p\times C_{2p}$. Then $G={\mathrm{Gal}}(\widetilde{L}/K)$ is isomorphic to a transitive subgroup of ${\mathrm{Hol}}(C_p\times C_{2p})$. We shall see that ${\mathrm{Hol}}(C_p\times C_p)\rtimes C_2)$ contains a subgroup isomorphic to $C_p\times C_{2p}$, acting regularly on $(C_p\times C_p)\rtimes C_2$ and such that its normalizer in ${\operatorname{Sym}}((C_p\times C_p)\rtimes C_2)$ is contained in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$. Let us write $(C_p\times C_p)\rtimes C_2=(\langle a \rangle \times \langle b \rangle) \rtimes \langle c \rangle$. An automorphism $\varphi$ of $(C_p\times C_p)\rtimes C_2$ is given by
$$\begin{array}{cccl} \varphi:& a & \mapsto & a^i b^j \\& b & \mapsto & a^k b^l \\& c & \mapsto & a^m b^n c \end{array} \quad p \nmid il-jk.$$
Let us consider the elements $(a,{\operatorname{Id}}), (b,{\operatorname{Id}}), (c,\varphi_1)$ in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$, where $\varphi_1$ is defined by $\varphi_1(a)=a^{-1},
\varphi_1(b)=b^{-1}, \varphi_1(c)=c$. We may check that $(a,{\operatorname{Id}})$ and $(b,{\operatorname{Id}})$ have order $p$, $(c,\varphi_1)$ has order 2 and all three elements commute with each other, hence generate a subgroup $F_1$ isomorphic to $C_p\times C_{2p}$. We may see that the orbit of $1$ under the action of $F_1$ contains $(C_p\times C_p)\rtimes C_2$. Indeed
$$1 \stackrel{(a,{\operatorname{Id}})^i}{\longmapsto}a^i \stackrel{(b,{\operatorname{Id}})^j}{\longmapsto}a^i b^j \stackrel{(c,\varphi_1)}{\longmapsto}a^{-i} b^{-j} c.$$
By computation we obtain that the elements in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$ which normalize $F_1$ are precisely those of the form $(x,\psi_1)$, where $x$ is any element in $(C_p\times C_p)\rtimes C_2$ and $\psi_1$ is defined by
$$\begin{array}{cccl} \psi_1:& a & \mapsto & a^i b^j \\& b & \mapsto & a^k b^l \\& c & \mapsto & c \end{array} \quad p \nmid il-jk.$$
We obtain then than the normalizer of $F_1$ in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$ has order $2p^2(p^2-1)(p^2-p)$ and is then the whole normalizer of $F_1$ in ${\operatorname{Sym}}((C_p\times C_p)\rtimes C_2)$. Theorem \[theoB\] gives then the wanted result.
Let us assume now that $L/K$ has a Hopf Galois structure of type $C_p\times D_{2p}$. Then $G={\mathrm{Gal}}(\widetilde{L}/K)$ is isomorphic to a transitive subgroup of ${\mathrm{Hol}}(C_p\times D_{2p})$. We shall see that ${\mathrm{Hol}}(C_p\times C_p)\rtimes C_2)$ contains a subgroup isomorphic to $C_p\times D_{2p}$, acting regularly on $(C_p\times C_p)\rtimes C_2$ and such that its normalizer in ${\operatorname{Sym}}((C_p\times C_p)\rtimes C_2)$ is contained in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$.
Let us consider the elements $(a,{\operatorname{Id}}), (b,{\operatorname{Id}}), (c,\varphi_2)$ in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$, where $\varphi_2$ is defined by $\varphi_2(a)=a,
\varphi_2(b)=b^{-1}, \varphi_2(c)=c$. We may check that $(c,\varphi_2)$ has order 2, commutes with $(b,{\operatorname{Id}})$ and satisfies $(c,\varphi_2)(a,{\operatorname{Id}})(c,\varphi_2)=(a,{\operatorname{Id}})^{-1}$. The three elements generate then a subgroup $F_2$ isomorphic to $C_p\times D_{2p}$. We may see that the orbit of $1$ under the action of $F_2$ contains $(C_p\times C_p)\rtimes C_2)$. Indeed
$$1 \stackrel{(a,{\operatorname{Id}})^i}{\longmapsto}a^i \stackrel{(b,{\operatorname{Id}})^j}{\longmapsto}a^i b^j \stackrel{(c,\varphi_2)}{\longmapsto}a^{-i} b^{j} c.$$
By computation we obtain that the elements in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$ which normalize $F_2$ are precisely those of the form $(x,\psi_2)$, where $x$ is any element in $(C_p\times C_p)\rtimes C_2$ and $\psi_2$ is defined by
$$\begin{array}{cccl} \psi_2:& a & \mapsto & a^i \\& b & \mapsto & b^l \\& c & \mapsto & a^mc \end{array} \quad p \nmid il.$$
We obtain then than the normalizer of $F_2$ in ${\mathrm{Hol}}((C_p\times C_p)\rtimes C_2)$ has order $2p^3(p-1)^2$ and is then the whole normalizer of $F_2$ in ${\operatorname{Sym}}((C_p\times C_p)\rtimes C_2)$. Again Theorem \[theoB\] gives the wanted result.
\[sets\] Let $L/K$ be a separable extension of degree $2p^2$, $p$ an odd prime. Then either $L/K$ has no Hopf Galois structures or the set of types of Hopf Galois structures on $L/K$ is one of the following:
$$\begin{array}{c} \{ D_{2p^2} \}, \quad \{ (C_p\times C_p) \rtimes C_2 \}, \quad \{ D_{2p^2}, C_{2p^2} \}, \quad \{ (C_p\times C_p) \rtimes C_2, C_p\times C_{2p} \}, \\ \{ (C_p\times C_p) \rtimes C_2,C_p \times D_{2p} \}, \quad \{ (C_p\times C_p) \rtimes C_2, C_p\times C_{2p}, C_p \times D_{2p} \}.\end{array}$$
By Theorems \[2pn1\] and \[2p2\] the possibilities not considered in the corollary do not occur. Let us see now that all cases listed do occur by exhibiting an example. We denote again by ${{\widetilde{L}}}$ a Galois closure of $L/K$ and $G={\mathrm{Gal}}({{\widetilde{L}}}/K)$. For $G=S_{2p^2}$, the whole symmetric group in $2p^2$ letters, $L/K$ has no Hopf Galois structures, by [@G-P] Corollary 4.8. If for some group $N$ of order $2p^2$, we have either $G=N$ or $G={\mathrm{Hol}}(N)$, then $L/K$ has a Hopf Galois structure of type $N$. If for some group $N$ of order $2p^2$, we have $|G|>|{\mathrm{Hol}}(N)|$, then $L/K$ has no Hopf Galois structures of type $N$ by Theorem \[theoB\]. These facts, together with Theorems \[2pn1\] and \[2p2\], allows us to obtain the results listed in the following table giving the types $N$ occurring for each $G$ in the first column, except those marked with a question mark.
Galois group $\diagdown$ types $C_{2p^2}$ $D_{2p^2}$ $C_p\times C_{2p}$ $C_p \times D_{2p}$ $(C_p\times C_p) \rtimes C_2$
------------------------------------------------- ------------ ------------ -------------------- --------------------- -------------------------------
$G={\mathrm{Hol}}(D_{2p^2})$ No Yes No No No
$G={\mathrm{Hol}}((C_p\times C_p) \rtimes C_2)$ No No No No Yes
$G=C_{2p^2}$ Yes Yes No No No
$G={\mathrm{Hol}}(C_p\times C_{2p})$ No No Yes No Yes
$G={\mathrm{Hol}}(C_p\times D_{2p})$ No No No ? Yes Yes
$G=C_p\times C_{2p}$ No No Yes Yes ? Yes
We now resolve the question marks.
If $G={\mathrm{Hol}}(C_p\times D_{2p})$, let us see that it does not have structures of type $N:=C_p\times C_{2p}$. Equivalently, we shall prove that $G$ do not embed into ${\mathrm{Hol}}(N)$. To this end, we observe that the $p$-Sylow groups ${\operatorname{Syl}}(G)$ and ${\operatorname{Syl}}({\mathrm{Hol}}(N))$ have both order $p^3$. We shall prove that they are not isomorphic. Let us write $N=C_p\times C_p \times C_2=\langle a \rangle \times \langle b \rangle \times \langle c \rangle$, as above. We consider the following elements in ${\mathrm{Hol}}(N)$: $A=(a,{\operatorname{Id}}), B=(b^{-1},{\operatorname{Id}}), C=(1,\varphi)$, where $\varphi$ is defined by $\varphi(a)=ab, \varphi(b)=b, \varphi(c)=c$. These three elements have order $p$ and satisfy $AB=BA, BC=CB, AC=BCA$, hence generate a subgroup of ${\mathrm{Hol}}(N)$ isomorphic to the Heisenberg group $H_p$ (see [@CS2] 3.2). We have then ${\operatorname{Syl}}({\mathrm{Hol}}(N)) \simeq H_p$. Let us write now $D_{2p}\times C_p=\langle r,s,c \rangle$, as above. We consider the elements $(r,{\operatorname{Id}}), (c,{\operatorname{Id}}), (1,\psi)$ in $G$, where $\psi$ is defined by $\psi(r)=r,\psi(s)=rs, \psi(c)=c$. These elements have order $p$ and commute with each other, hence generate a subgroup of $G$ isomorphic to $C_p\times C_p \times C_p$. We have then ${\operatorname{Syl}}(G) \simeq C_p\times C_p \times C_p$.
Finally if $G=C_p\times C_{2p}$, we shall see that $L/K$ has Hopf Galois structures of type $C_p\times D_{2p}$. To this end, we prove that $C_p\times C_{2p}$ embeds regularly in ${\mathrm{Hol}}(C_p\times D_{2p})$. We write $D_{2p}=\langle r,s|r^p=s^2=1,srs=r^{-1} \rangle$ and denote by $c$ a generator of $C_p$. The elements $(r,{\operatorname{Id}})$ and $(c,{\operatorname{Id}})$ have order $p$ and commute with each other. The element $(s,\varphi_0)$, where $\varphi_0$ is defined by $\varphi_0(r)=r^{-1}, \varphi_0(s)=s, \varphi_0(c)=c$ has order 2 and commutes with $(r,{\operatorname{Id}})$ and $(c,{\operatorname{Id}})$, hence the three elements together generate a subgroup isomorphic to $C_p\times C_{2p}$. Moreover, this subgroup is regular since the orbit of $1$ under its action is the whole $(C_p\times D_{2p})$. Indeed, we have
$$1 \stackrel{(r,{\operatorname{Id}})^i}{\longmapsto}r^i \stackrel{(c,{\operatorname{Id}})^j}{\longmapsto}r^i c^j \stackrel{(s,\varphi_0)}{\longmapsto}sr^{-i} c^{j}=r^i s c^j.$$
Summary of the program output {#table}
=============================
Table \[fig\] is a compendium of the computation results. In it we give for every degree $g$ the total number of transitive groups of degree $g$ and the number Max of transitive groups of degree $g$ whose order does not exceed the order of the holomorphs of all the groups of order $g$; the number of possible types of Hopf Galois structures; the total number of Hopf Galois structures and the number of the almost classically Galois ones; the number of Hopf Galois structures with bijective Galois correspondence and the number of those which are not almost classically Galois; the number of Hopf algebra isomorphism classes in which the Hopf Galois structures are partitioned (which correspond to $G$-isomorphism classes of the corresponding regular groups $N$) and the number of those for Galois extensions (i.e. when $G'={\mathrm{Gal}}({{\widetilde{L}}}/L)$ is trivial); and finally the execution times in seconds and the memory used in megabytes (except for $g=16$ which could not be computed at once).
\[fig\]
---- ------- ------ ---- ------- ------ ------ ------ ------- ------ ---------------- ----------------
12 301 129 5 249 56 81 25 165 48 $\approx$ 18 $\approx$ 31
13 9 6 1 6 6 6 0 6 1 $\approx$ 1 $\approx$ 18
14 63 25 2 32 14 19 5 26 6 $\approx$ 2 $\approx$ 22
15 104 11 1 8 8 8 0 8 1 $\approx$ 1 $\approx$ 22
16 1954 1906 14 49913 2636 9331 6695 26769 6717 – –
17 10 5 1 5 5 5 0 5 1 $\approx$ 1 $\approx$ 30
18 983 528 5 881 123 253 130 525 79 $\approx$ 206 $\approx$ 113
19 8 6 1 6 6 6 0 6 1 $\approx$ 1 $\approx$ 26
20 1117 170 5 434 79 156 77 266 55 $\approx$ 57 $\approx$ 48
21 164 26 2 78 22 46 24 42 8 $\approx$ 5 $\approx$ 26
22 59 18 2 36 14 19 5 26 6 $\approx$ 5 $\approx$ 26
23 7 4 1 4 4 4 0 4 1 $\approx$ 1 $\approx$ 18
24 25000 9738 15 14908 844 2682 1838 8353 1896 $\approx$ 9730 $\approx$ 1327
25 211 90 2 106 70 74 4 82 12 $\approx$ 32 $\approx$ 175
26 96 37 2 58 22 35 13 46 6 $\approx$ 12 $\approx$ 27
27 2392 1547 5 6699 766 1100 334 2030 547 $\approx$ 5375 $\approx$ 500
28 1854 214 4 388 84 143 59 256 40 $\approx$ 63 $\approx$ 33
29 8 6 1 6 6 6 0 6 1 $\approx$ 1 $\approx$ 22
30 5712 483 4 479 99 197 98 373 36 $\approx$ 113 $\approx$ 40
31 12 8 1 8 8 8 0 8 1 $\approx$ 1 $\approx$ 22
---- ------- ------ ---- ------- ------ ------ ------ ------- ------ ---------------- ----------------
: Summary of results[]{data-label="fig"}
Acknowledgments. {#acknowledgments. .unnumbered}
================
We are very thankful to Professor Derek Holt for kindly sending to us the Magma code of the function Automorphisms.
[1]{} N.P. Byott, *Uniqueness of Hopf Galois structure for separable field extensions*. Comm. Algebra 24 (1996), 3217-3228. Corrigendum, ibid., 3705. S.U. Chase, M. Sweedler, *Hopf Algebras and Galois Theory*. Lecture Notes in Mathematics, Vol. 97, Springer Verlag, 1969. L. N. Childs, *On the Hopf Galois theory for separable field extensions*. Comm. Algebra 17 (1989), 809-825. L. N. Childs, *Taming wild extensions: Hopf algebras and local Galois module theory*, AMS 2000. T. Crespo, A. Rio, M. Vela, *On the Galois correspondence theorem in separable Hopf Galois theory*, Publ. Mat. 60 (2016), 221-234. T. Crespo, M. Salguero, *Computation of Hopf Galois structures on low degree separable extensions and classification of those for degrees $p^2$ and $2p$*, Publ. Mat., to appear; arXiv:1802.09948. T. Crespo, M. Salguero, *Hopf Galois structures on separable field extensions of odd prime power degree*, J. Algebra 519 (2019), 424-439. T. Crespo, M. Salguero, *Algorithmic Hopf Galois theory: Magma code, tables and results*, https://sites.google.com/view/algorithmichg/ C. Greither, B. Pareigis, *Hopf Galois theory for separable field extensions*. J. Algebra 106 (1987), 239-258. A. Hulpke, *Constructing transitive permutation groups*, Journal of Symbolic Computation 39 (2005), 1-30. T. Kohl, *Classification of the Hopf Galois Structures on Prime Power Radical Extensions*, J. Algebra, 207 (1998), 525-546. M. Romagny, *$p$-adic formulas* https://perso.univ-rennes1.fr/matthieu.romagny/notes/p\_adic\_formulas.pdf.
|
---
abstract: 'Consider an $N\times N$ hermitian random matrix with independent entries, not necessarily Gaussian, a so called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as $N\to\infty$, the same as that of hermitian random matrices from GUE. We prove this conjecure for a certain subclass of hermitian Wigner matrices.'
address: ' Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden'
author:
- Kurt Johansson
title: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices
---
Introduction and main results
=============================
Consider a probability measure $\mathbb P_N$ on the space of all $N\times
N$ hermitian matrices. We will be interested in the statistical properties of the spectrum as $N$ becomes large, in particular in features that are insensitive to the details of the particular sequence of probability measures we are considering. It is believed, on the basis of numerical simulations, that for many types of hermitian random matrix ensembles, i.e. choices of $\mathbb P_N$, the local statistical properties of the eigenvalues are the same as for the Gaussian Unitary Ensemble (GUE), where $d\mathbb P_N(M)=Z_N^{-1}
\exp(-\frac N2{\text{Tr\,}}M^2) dM$. Here $dM$ is Lebesgue measure on the space $\mathcal{H}_N\sim \mathbb R^{N^2}$ of all $N\times N$ hermitian matrices. The asymptotic eigenvalue density as $N\to\infty$ (density of states) is given by the Wigner semicircle law $\rho(t)=\frac
1{2\pi}\sqrt{(4-t^2)_+}$. Let $\rho_N(x_1,\dots,x_N)$ be the induced probability density on the eigenvalues. The semicircle law is the limit of the one-dimensional marginal density as $N\to\infty$. The [*m - point correlation function*]{} $$\label{1.1}
R_m^{(N)}(x_1,\dots,x_m)=\frac{N!}{(N-m)!}\int_{\mathbb
R^{N-m}}\rho_N(x)dx_{m+1}\dots dx_N,$$ is given by, [@Me] ch. 5, [@TW], $$\label{1.2}
R_m^{(N)}(x_1,\dots,x_m)=\det (K_N(x_i,x_j))_{i,j=1}^m,$$ where the kernel $K_N(x,y)$ is given by $$\label{1.3}
K_N(x,y)=\frac{\kappa_{N-1}}{\kappa_N}\frac{p_N(x)p_{N-1}(y) -
p_{N-1}(x)p_N(y)}{x-y}e^{-N(x^2+y^2)/4}.$$ Here $p_N(x)=\kappa_Nx^N+\dots$ are the normalized orthogonal polynomials with respect to the weight function $\exp(-Nx^2/2)$ on $\mathbb R$ (rescaled Hermite polynomials). From these formulas, and Plancherel-Rotach asymptotics for the Hermite polynomials it follows that $$\label{1.4}
\lim_{N\to\infty}\frac{1}{(N\rho(u))^m}R_m^{(N)}(u+\frac{t_1}{N\rho(u)},\dots,
u+\frac{t_m}{N\rho(u)})=\det\left(\frac{\sin\pi(t_i-t_j)}{\pi(t_i-t_j)}
\right) _{i,j=1}^m$$ if $\rho(u)>0$. It has been proved, [@PS], [@DKMVZ], [@BI], that this is also true in other invariant ensembles of the form $d\mathbb P_N(M)=Z_N^{-1}\exp(-N{\text{Tr\,}}V(M))dM$. The orthogonal polynomials in (1.3) are then replaced by polynomials orthogonal with respect to $\exp(-NV(x))$ on $\mathbb R$. That the ensemble is invariant means that the probability measure is invariant under the conjugation $M\to U^{-1}MU$, with a unitary matrix $U$. Sufficient control of the limit (\[1.4\]) for all $m\ge 1$, makes it possible to determine the asymptotic spacing distribution, i.e. distances between nearest neighbour eigenvalues, see [@DKMVZ]. More precisely, let $\{t_N\}$ be a sequence such that $t_N\to\infty$ but $t_N/N\to 0$ as $N\to\infty$ and define, [@KS], [@DKMVZ], $S_N(s,x)$, $s\ge 0$, $x\in\mathbb R^N$, to be the symmetric function, which for $x_1<\dots<x_N$ is defined by $$\label{1.5}
S_N(s,x)=\frac 1{2t_N}\#\{1\le j\le
N-1\,;\,x_{j+1}-x_j\le\frac{s}{N\rho(u)},
|x_j-u|\le\frac{t_N}{N\rho(u)}\}.$$ Given an hermitian matrix $M$ let $x_1(M)<\dots x_N(M)$ be its eigenvalues; we write $x(M)=(x_1(M),\dots, x_N(M))$. Then it is proved in [@DKMVZ] that $$\label{1.6}
\lim_{N\to\infty}\mathbb E_N[S_N(s, x(M))]=\int_0^s p(\sigma)d\sigma,$$ for a large class of invariant ensembles. Here $p(\sigma)$ is the density of the $\beta=2$ local spacing distribution, the [*Gaudin distribution*]{}, given by the probability density $$\label{1.7}
p(s)=\frac{d^2}{ds^2}\det(I-K)_{L^2(0,s)},$$ where $K$ is the operator on $L^2(0,s)$ with kernel $K(t,s)=\sin\pi(t-s)/\pi(t-s)$, the [*sine kernel*]{}, see [@Me].
The aim of the present paper is to extend (\[1.4\]) and (\[1.6\]) to other, non-invariant ensembles. It is conjectured, see [@Me] p.9, that (\[1.4\]) and (\[1.6\]) should hold also for so called [*Wigner matrices*]{} where the elements are independent but not necessarily Gaussian variables. In this case the probability measure is not invariant under conjugation by unitary matrices. For other results on Wigner matrices see for example [@Ba], [@Kh], [@KKP], [@Po], [@SS1] and [@SS2]. In particular, in [@So] the universality of the fluctuations of the largest eigenvalue is established. To be more precise, consider the complex random variables $w_{jk}$, $1\le j\le k$ with independent laws $P_{jk}=P_{jk}^R\otimes P_{jk}^I$, where $P_{jj}^I=\delta_0$. Let $\mathcal{W}^p$, a class of Wigner ensembles, denote the class of all $\{P_{jk}\}_{1\le j\le k}$ which satisfy $$\label{1.8}
\int zdP_{jk}(z)=0\quad,\quad\int |z|^2dP_{jk}(z)=\sigma^2$$ for all $1\le j\le k$, and furthermore $$\label{1.9}
\sup_{j,k}\int|z|^pdp_{jk}(z)<\infty.$$ If $w_{kj}=\bar w_{jk}$, $W=(w_{jk})_{j,k=1}^N$ is an $N\times N$ [*hermitian Wigner matrix* ]{}.
Fix $a>0$ and let $\phi_a(t)=(\pi a^2)^{-1/2}\exp(-t^2/a^2)$ be a Gaussian density function. Define $Q_{jk}^{R,I}=\phi_a\ast
P_{jk}^{R,I}$, $1\le j<k$, $Q_{jj}^R=\phi_{a\sqrt{2}}\ast P_{jj}^R$, $j\ge 1$ and $Q_{jj}^I=\delta_0$. Then $Q$ is also a Wigner ensemble and we let $\mathcal{W}^p_a$ denote the subclass of $\mathcal{W}^p$ obtained in this way. Note that although $\mathcal{W}^p_a$ does not contain all Wigner ensembles it does contain cases where the distribution of the matrix elements have very different shapes, so in this sense it is rather broad, and proving universality in $\mathcal{W}^p_a$ clearly shows that the universality is not restricted to the invariant ensembles. Another way to describe this ensemble of random matrices is as follows. Let $V$ be a GUE-matrix with the probability measure $Z_N^{-1}\exp(-\frac 12{\text{Tr\,}}V^2)dV$ and let $W$ be an $N\times N$ Wigner matrix with distribution $P\in\mathcal{W}^p$, i.e. the law of $w_{jk}$ is $P_{jk}$. We will assume that the variance $\sigma^2=1/4$, which can always be achieved by rescaling. Then $W+aV$ has the distribution $Q$, and we write $$M=\frac{1}{\sqrt{N}}(W+aV)\notag.$$ We can think of this in terms of Dyson’s Brownian motion model, [@Dy], $W+aV$ is obtained from $W$ by letting the matrix elements execute a Brownian motion for a time $a^2$, see sect. 2. If $P\in\mathcal{W}^p$ and $W$ is is an $N\times N$ hermitian matrix we let $P^{(N)}$ denote the distribution of $H=W/\sqrt{N}=(h_{jk})$, i.e. $$dP^{(N)}(H)=\prod_{1\le j\le k\le N} dP_{jk}(\sqrt {N}h_{jk})\notag.$$ The matrix $M$ has the distribution $Q^{(N)}$, which is given by $$\label{1.10}
dQ^{(N)}(M)=2^{-N/2}\left(\frac{N}{\pi
a^2}\right)^{N^2/2}\left(\int_{\mathcal{H}_N}e^{-\frac{N}{2a^2}{\text{Tr\,}}(M-H)^2}dP^{(N)}(H)\right)dM,$$ and this is the measure we will study. The asymptotic distribution of the eigenvalues $x_1,\dots,x_N$ of $M$ is the semicircle law $$\label{1.11}
\rho(u)=\frac{2}{\pi(1+4a^2)}\sqrt{(1+4a^2-u^2)_+}.$$ The following proposition will be proved in sect. 2 using an argument from [@BH1], [@BH2].
\[eigenmeas\] The symmetrized eigenvalue measure on $\mathbb R^N$ induced by $Q^{(N)}$ has a density $$\label{1.12}
\rho_N(x)=\int_{\mathcal{H}_N}\rho_N(x;y(H))dP^{(N)}(H)$$ where $$\label{1.12'}
\rho_N(x;y)=\left(\frac{N}{2\pi
a^2}\right)^{N/2}\frac{\Delta_N(x)}{\Delta_N(y)}
\det(e^{-\frac{N}{2a^2}(x_j-y_k)^2})_{j,k=1}^N$$ and $\Delta_N(x)=\prod_{1\le i<j\le N}(x_i-x_j)$ is the Vandermonde determinant.
The main result of the present paper is that for Wigner ensembles from $\mathcal{W}^p_a$ we can prove (\[1.4\]) and (\[1.6\]), and thus extend the universality to a rather broad class of Wigner matrices.
\[corr\] Fix $a>0$ and assume that $|u|\le \sqrt{1/2+2a^2}$. Let $R_M^{(N)}(x_1,\dots,x_m)$ be the correlation functions, defined by (\[1.1\]), of the eigenvalue measure $\rho_N$, (\[1.12\]), for $Q^{(N)}$, (\[1.10\]). Let $f\in L_c^\infty(\mathbb R^m)$, the set of all $L^\infty$ functions on $\mathbb R^m$ with compact support, and set for $x\in\mathbb R^N$ $$(Sf)(x)=\sum_{i_1,\dots,i_m} ' f(x_{i_1},\dots,x_{i_m})\notag,$$ where the sum is over all distinct indices from $\{1,\dots, N\}$. If $Q\in\mathcal{W}_a^p$ with $p>2(m+2)$, then $$\begin{aligned}
\label{1.12''}
&\lim_{N\to\infty}\int_{\mathcal{H}_N}(Sf)(N\rho(u)(x_1(M)-u),\dots
N\rho(u)(x_N(M)-u))dQ^{(N)}(M)\\
&\lim_{N\to\infty}\int_{\mathbb R^m}f(t_1,\dots,t_m)\frac 1{(N\rho(u))^m}
R_m^{(N)}(u+\frac {t_1}{N\rho(u)},\dots,u+\frac {t_m}{N\rho(u)})d^mt\notag\\
&=\int_{\mathbb
R^m}f(t_1,\dots,t_m)\det(\frac{\sin\pi(t_i-t_j)}{\pi(t_i-t_j)})_{i,j=1}^m
d^mt.\notag\end{aligned}$$
The condition on $u$ is made just to simplify the saddle-point argument in sect. 3; the result should hold for any $u$ with $\rho(u)>0$.
We can also prove that the spacing distribution is the same as for GUE.
\[spac\] Fix any $a>0$ and assume that $Q\in\mathcal{W}_a^{6+\epsilon}$, $\epsilon>0$. Let $S_N(s,x)$ be defined by (\[1.5\]). Then, for any $s\ge 0$, $$\label{1.13}
\lim_{N\to\infty}\int_{\mathcal{H}_N}
S_N(s,x(M))dQ^{(N)}(M)=\int_0^sp(\sigma)d\sigma,$$ where $p(s)$ is given by (\[1.7\]).
The theorems will be proved in sect. 4 after the preparatory work in sect. 2 and 3.
The correlation functions
=========================
We will start by proving Proposition 1.1 using the Harish-Chandra/Itzykson-Zuber formula following [@BH1], [@BH2]. After that we will give a formula for the correlation functions of $\rho_N(x;y)$, which is very close to the formula in [@BH3], but our derivation will be different. A central role will be played by non-intersecting one-dimensional Brownian motions and we will use the formulas of Karlin and McGregor. Also we will discuss the relation to Dyson’s Brownian motion model. This connection can be found in [@Gr] and we will only give an outline.
Let $F(x)$ be a continuous symmetric function on $\mathbb R^N$. By Fubini’s theorem $$\label{2.1}
\int_{\mathcal{H}_N} F(x(M))dQ^{(N)}(M)=c_N^{(1)}\int_{\mathcal{H}_N}\left(
\int_{\mathcal{H}_N}F(x(M))e^{-\frac N{2a^2}{\text{Tr\,}}(M-H)^2}dM\right)dP^{(N)}(H)$$ with $c_N^{(1)}=2^{-N/2}(N/\pi a^2)^{N^2/2}$. In the right hand side of (\[2.1\]) we make the substitution $M=U^{-1}RU$, with $U\in U(N)$ and $R\in\mathcal{H}_N$, and then integrate over $U(N)$. If we use Fubini’s theorem again, we obtain $$c_N^{(1)}\int_{\mathcal{H}_N}\left(
\int_{\mathcal{H}_N}F(x(R))\left(\int_{U(N)}e^{-\frac N{2a^2}{\text{Tr\,}}(U^{-1}RU-H)^2}dU\right)dR\right)dP^{(N)}(H).\notag$$ Here we have also used the fact that $dM=dR$. The integral over $U(N)$ can now be evaluated using the Harish-Chandra/Itzykson-Zuber formula, [@Ha],[@IZ], see also [@Me] A.5. We obtain the integral $$c_N^{(1)}c_N^{(2)}\int_{\mathcal{H}_N}\left(
\int_{\mathcal{H}_N}F(x)\frac
1{\Delta_N(x)\Delta_N(y)}\det(e^{-N(x_j-y_k)^2/2a^2})_{j,k=1}^NdR\right)
dP^{(N)}(H),\notag$$ where $y_1,\dots,y_N$ are the eigenvalues of $H$ and $c_N^{(2)}=(a^2/N)^{N(N-1)/2} \prod_{j=1}^Nj!$. The integrand in the middle integral depends only on the eigenvalues $x$ of $R$ and hence we can integrate out the other degrees of freedom in the standard way, [@Me] ch. 3, and obtain, after using Fubini’s theorem, $$\begin{aligned}
\label{2.2}
&\int_{\mathcal{H}_N} F(x(M))dQ^{(N)}(M)\notag\\&=
c_N^{(1)}c_N^{(2)}c_N^{(3)}\int_{\mathcal{H}_N}\left(
\int_{\mathbb R^N}F(x)
\frac
{\Delta_N(x)}{\Delta_N(y)}\det(e^{-N(x_j-y_k)^2/2a^2})_{j,k=1}^Nd^Nx\right)
dP^{(N)}(H)\end{aligned}$$ with $c_N^{(3)}=\pi^{N(N-1)/2}\prod_{j=1}^N(j!)^{-1}$. We see that $c_N^{(1)}c_N^{(2)}c_N^{(3)}=(N/2\pi a^2)^{N/2}$ and since (\[2.2\]) holds for arbitrary bounded, continuous, symmetric $F(x)$ we have proved that the symmetrized eigenvalue measure is given by \[1.12\]. This proves proposition \[eigenmeas\].
Let $p_t(x,y)$ be the transition probability of a Markov process $X(t)$ on $\mathbb R$ with continuous paths. Consider $N$ independent copies of the process $(X_1(t),\dots,X_N(t))$ and assume that this is a strong Markov process in $\mathbb R^N$. Suppose that the particles start at positions $y_1<\dots<y_N$ at time 0. The probability density that they are at positions $x_1<\dots<x_N$ at time $S$ given that their paths have not intersected anytime during the time interval $[0,S]$ is, by a theorem of Karlin and McGregor, [@KM], $$\det(P_S(y_j,x_k))_{j,k=1}^N.\notag$$ Hence, the conditional probability density that the particles are at positions $y_1<\dots<y_N$ at time 0, at positions $x_1<\dots<x_N$ at time $S$, at positions $z_1<\dots<z_N$ at time $S+T$, given that their paths have not intersected in the time interval $[0,S+T]$ is $$\label{2.3}
q_{S,T}(x;y;z)\doteq \frac 1{\mathcal{Z}_N}\det(P_S(y_j,x_k))_{j,k=1}^N
\det(P_T(x_j,z_k))_{j,k=1}^N,$$ where $$\mathcal{Z}_N=\int_{x_1<\dots<x_N}\det(P_S(y_j,x_k))_{j,k=1}^N
\det(P_T(x_j,z_k))_{j,k=1}^Nd^Nx;\notag$$ we assume that $\mathcal{Z}_N>0$. Note that the expression (\[2.3\]) is a symmetric function of $x_1,\dots,x_N$, so we can regard it as a probability measure on $\mathbb R^N$. Our next lemma shows that we can obtain $\rho_N(x;y)$ defined by (\[1.12’\]) as a limit of the measure in (\[2.3\]).
\[qst\] Let $z_j=j-1$, $1\le j\le N$ and let $p_t(x,y)=(2\pi
t)^{-1/2}\exp((x-y)^2/2t)$ be the transition probability for Brownian motion. Then, for any $x\in\mathbb R^N$ and $y_1<\dots ,y_N$, $$\label{2.4}
\lim_{T\to\infty}q_{S,T}(x;y;z)=\frac 1{(2\pi
S)^{N/2}}\frac{\Delta_N(x)}{\Delta_N(y)}
\det(e^{-(x_j-y_k)^2/2S})_{j,k=1}^N\doteq q_S(x;y).$$
Note that $\rho_N(x;y)=q_{a^2/N}(x;y)$.
Write $$\begin{aligned}
\label{2.5}
&\det(P_S(y_j,x_k))_{j,k=1}^N
\det(P_T(x_j,z_k))_{j,k=1}^N\notag\\&=
\frac 1{(2\pi)^N(TS)^{N/2}}\det(e^{-(x_j-y_k)^2/2S})_{j,k=1}^N
\prod_{j=1}^Ne^{-\frac{x_j^2+z_j^2}{2T}}\det(e^{x_jz_k/2T})_{j,k=1}^N.\end{aligned}$$ Note that $\mathcal{Z}_N$ is the conditional probability density of going from $y_1<\dots ,y_N$ to $z_1<\dots ,z_N$ without collosions, i.e. $$\begin{aligned}
\label{2.6}
\mathcal{Z}_N&=\det(p_{S+T}(y_j,z_k))_{j,k=1}^N\notag\\
&=\frac 1{(2\pi)^{N/2}(S+T)^{N/2}}\prod_{j=1}^Ne^{-\frac{y_j^2+z_j^2}
{2(S+T)}} \det(e^{y_jz_k/2(S+T)})_{j,k=1}^N.\end{aligned}$$ Now, since $z_j=j-1$, we have two Vandermonde determinants in (\[2.5\]) and (\[2.6\]). If we evaluate these, take the quotient between (\[2.5\]) and (\[2.6\]) and then take the limit $T\to\infty$, we obtain the right hand side of (\[2.4\]).
Proposition 1.1 and lemma 2.1 establish a link between the eigenvalue distribution of $M=(W+aV)/\sqrt{N}$ and the non-intersecting Brownian paths. If we set $S=a^2/N$, then the right hand side of (\[2.4\]) and (\[1.12’\]) are identical; $y_1<\dots<y_N$ are the eigenvalues of $H=W/\sqrt{N}$. This relation can also be seen in another way, which we will now outline. Let $X(t)= (x_{jk}(t))_{j,k=1}^N$ be an $N\times N$ Hermitian matrix, where ${\text{Re\,}}x_{jk}(t)$, ${\text{Im\,}}x_{jk}(t)$, $j\le k$ are independent Brownian motions with variance $(1+\delta_{jk})/2$. Assume that $X(0)=H$ is distributed according to $P^{(N)}$. Then the distribution of $X(a^2/N)$ is the same as that of $M=(W+aV)/\sqrt{N}$. Following Dyson, [@Dy], see also [@PR], it is possible to derive a stochastic differential equation for the eigenvalues $\lambda_1(t),\dots,\lambda_N(t)$ of $X(t)$, $$\label{2.7}
d\lambda_i=dB_i+\sum_{k\neq i}\frac 1{\lambda_i-\lambda_k} dt,$$ where $B_i$ are independent standard Brownian motions on $\mathbb R$, and with the intial conditions $\lambda_i(0)=y_i$, $1\le i\le N$. We can also consider the problem of non-intersecting Brownian motions in a different way than that of Karlin and McGregor. Namely, let $K=\{x\in\mathbb R^N\,;\,x_1<\dots<x_N\}$ and consider Brownian motion in $\mathbb R^N$ starting at $y\in K$ and conditioned to remain in $K$ forever. As proved in [@Gr], see also [@HW], [@Pi], if $\lambda_i$ are the components of the $N$-dimensional conditioned Brownian motion they satisfy the stochastic differential equation (\[2.7\]) with the same initial conditions. This gives another way to obtain (\[1.12’\]) without using the Harish-Chandra, Itzykson/Zuber formula. Actually, we can turn the argument around and give a proof of this formula.
We turn now to the computation of the correlation functions of the right hand side of (\[2.4\]), but we start more generally with (\[2.3\]). This can be analyzed using the techniques of [@TW], compare the analysis of the Schur measure, [@Ok], in [@Jo1], and see also [@Jo2]. For completeness, let us outline the result we need from [@TW]. Let $(\Omega,\mu)$ be a measure space. Assume that $\phi_j,\psi_j\in L^2(\Omega,\mu)$, $1\le j\le N$, and $f\in
L^\infty(\Omega,\mu)$. Set $$Z_N[f]=\frac 1{N!}\int_{\Omega_N}\det(\phi_j(x_k))_{j,k=1}^N
\det(\psi_j(x_k))_{j,k=1}^N\prod_{j=1}^Nf(x_j)d\mu(x_j)\notag$$ and $$A=\left(\int_\Omega\phi_j(x)\psi_j(x)d\mu(x)\right)_{j,k=1}^N.\notag$$
\[TW\] ([@TW]) Assume that $Z_N[1]\neq 0$. Then $A$ is invertible and we can define $$\label{2.8}
K_N(t,s)=\sum_{j,k=1}^N\psi_k(t)(A^{-1})_{kj}\phi_j(s).$$ Then, for any $g\in L^\infty(\Omega,\mu)$, $$\label{2.9}
\frac{Z_N[1+g]}{Z_N[1]}=\det(I+K_Ng)_{L^2(\Omega)}.$$ If we define a density on $\Omega^N$ by $$\label{2.10}
u_N(x)=\frac 1{N!Z_N[1]}\det(\phi_j(x_k))_{j,k=1}^N
\det(\psi_j(x_k))_{j,k=1}^N,$$ then it has the correlation functions $$\label{2.11}
\frac{N!}{(N-M)!}\int_{\Omega^{N-m}}u_N(x)dx_{m+1}\dots
dx_N=\det(K_N(x_i,x_j))_{i,j=1}^N.$$
We will indicate the main steps in the proof of (\[2.9\]) of which (\[2.11\]) is a consequence, see [@TW]. Set $$B=\left(\int_\Omega\phi_j(x)\psi_j(x)g(x)d\mu(x)\right)_{j,k=1}^N.\notag$$ Then, by the formula $$Z_N[f]=\det\left(\int_\Omega\phi_j(x)\psi_j(x)f(x)d\mu(x)\right)_{j,k=1}^N,
\notag$$ which goes all the way back to [@An], and which is not difficult to prove by expanding the determinants, we see that $\det A=Z_N[1]\neq 0$, so $A$ is invertible and $$\label{2.12}
\frac{Z_N[1+g]}{Z_N[1]}=\frac{\det(A+B)}{\det A}=\det(I+A^{-1}B).$$ Now, $$(A^{-1}B)_{jk}=\int_\Omega\psi_k(x)
\left(\sum_{\ell=1}^N(A^{-1})_{j\ell}\phi_\ell(x)g(x)\right)d\mu(x),\notag$$ and we define $T:\mathbb C^N\to L^2(\Omega,d\mu)$ and $S:L^2(\Omega,d\mu)\to
\mathbb C^N$ by the kernels $T(x,k)=\psi_k(x)$ and $S(j,x)=\sum_{\ell=1}^N (A^{-1})_{j\ell}\phi_\ell(x_g(x)$. Then, by (\[2.12\]) and a determinant identity, $$\begin{aligned}
\frac{Z_N[1+g]}{Z_N[1]}&=\det(I+ST)_{\mathbb
C^N}=\det(I+TS)_{L^2(\Omega,d\mu)}\notag\\
&=\det(I+K_Ng)_{L^2(\Omega,d\mu)}\notag,\end{aligned}$$ with $K_N$ given by (\[2.8\]). Note that $K_Ng$, which means first multiplication by $g$ and then application of the operator on $L^2(\Omega,d\mu)$ with kernel $K_N$, is a finite rank operator.
Observe now that if we take $\Omega=\mathbb R$, $d\mu(x)=dx$, $\phi_j(x)=P_T(x,z_j)$ and $\psi_j(x)=p_S(y_j,x)$, then (\[2.3\]) is a probability density of the form (\[2.10\]) and we can apply the proposition. Note that $$(A)_{jk}=\int_{\mathbb R}p_T(x,z_j)p_S(y_k,x)dx=p_{S+T}(y_k,z_j).\notag$$ The kernel which gives the correlation functions is $$K_N^{S,T}(u,v)=\sum_{k=1}^Np_S(y_k,v)\left(\sum_{j=1}^N
(A^{-1})_{jk}p_T(u,z_j)\right).\notag$$ Let $A_k(v)$ be the matrix we obtain from $A$ by replacing column $k$ by $(p_T(v,z_1)\dots p_T(v,z_N))^T$. Then, by Kramers’ rule, $$\label{2.13}
K_N^{S,T}(u,v)=\sum_{k=1}^Np_S(y_k,v)\frac{\det A_k(v)}{\det A}.$$ This formula and proposition \[TW\] is the basis for the next proposition. The result is closely related to the result derived in [@BH3] by different methods.
\[prop2.4\] The correlation functions for $q_S(x;y)$ defined by (\[2.4\]) are given by $$\begin{aligned}
\label{2.14}
R_m^{N}(x_1,\dots,x_m;y)&\doteq\frac{N!}{(N-m)!}\int_{\mathbb R^{N-m}}
q_S(x;y)dx_{m+1}\dots dx_N\\
&=\det(K_N^S(x_i,x_j;y))_{i,j=1}^m,\notag\end{aligned}$$ where $$\begin{aligned}
\label{2.15}
&K_N^S(u,v;y)=\frac{e^{(v^2-u^2)/2S}}{(v-u)S(2\pi
i)^2}\int_{\gamma}dz\int_{\Gamma}dw(1-e^{(v-u)z/S})\\
&\times\frac 1z\left(w+z-v-S\sum_j\frac{y_j}{(w-y_j)(z-y_j)}\right)
e^{(w^2-2vw-z^2+2uz)/2S}.\notag\end{aligned}$$ Here $\gamma$ is the union of the curves $t\to -t+i\omega$, $t\in\mathbb
R$ and $t\to t-i\omega$, $t\in\mathbb R$ with a fixed $\omega>0$, and $\Gamma:\mathbb R\ni t\to it$.
We have to show that with $p_t(u,v)=(2\pi t)^{-1/2}\exp(-(u-v)^2/2t)$ and $z_j=j-1$ the limit of the right hand side of (\[2.13\]) as $T\to\infty$ can be written as (\[2.15\]). The result then follows from lemma \[qst\], proposition \[TW\] and the dominated convergence theorem. We see that $$\label{2.16}
\det A=\frac
1{(2\pi(S+T))^{N/2}}\prod_{j=1}^Ne^{-\frac{z_j^2+y_j^2}{2(S+T)}}
\prod_{1\le i<j\le N}(e^{\frac{y_j}{S+T}}-e^{\frac{y_i}{S+T}})$$ by the formula for a Vandermonde determinant. Let $\Gamma^\ast_M$ be the curve $t\to t+iM$, $t\in \mathbb R$, $M$ fixed. Then $$p_T(z_j,v)=\frac 1{\sqrt{2\pi
T}}e^{-\frac{z_j^2}{2(S+T)}-\frac{v^2}{2T}}
\frac 1{\sqrt{2\pi}}\int_{\Gamma^\ast_M}e^{-\frac{\tau^2}2+z_j(\frac vT
+i\tau\sqrt{\frac{S}{2T(S+T)}})}d\tau\notag$$ Hence, $$\begin{aligned}
\det A_k(v)&=\frac 1{2\pi\sqrt{T}}\frac 1{(2\pi (S+T))^{(N-1)/2}}
\left(\prod_{j=1}^Ne^{-\frac{z_j^2}{2(S+T)}}\right)\left(
\prod_{j\neq k}e^{-\frac{y_j^2}{2(S+T)}}\right)\notag\\
&\times e^{-\frac{v^2}{2T}}\int_{\Gamma^\ast_M}
e^{-\frac{\tau^2}2}\det\tilde{A}_k(v)d\tau,\notag\end{aligned}$$ where $\tilde{A}_k(v)$ is the matrix we get from $(\exp(\frac{z_jy_k}{S+T}))_{j,k=1}^N$ by replacing column $k$ by $(\exp(z_j(\frac vT+i\tau\sqrt{\frac{S}{2T(S+T)}})))_{j=1}^N$. Since $z_{j}=j-1$ we have a Vandermonde determinant and we obtain $$\begin{aligned}
\label{2.17}
\det A_k(v)&=\sqrt{\frac{S+T}T}\frac 1{(2\pi (S+T))^{N/2}}
\left(\prod_{j=1}^Ne^{-\frac{z_j^2}{2(S+T)}}\right)\left(
\prod_{j\neq k}e^{-\frac{y_j^2}{2(S+T)}}\right)\notag\\
&\times e^{-\frac{v^2}{2T}}\frac 1{\sqrt{2\i}}\int_{\Gamma^\ast_M}
e^{-\frac{\tau^2}2}
\prod_{1\le i<j\le N}(e^{\frac{y_j}{S+T}}-e^{\frac{y_i}{S+T}})d\tau,\end{aligned}$$ where $y_k$ should be replaced by $(S+T)(\frac
vT+i\tau\sqrt{\frac{S}{2T(S+T)}}) $. Take the quotient of (\[2.16\]) and (\[2.17\]) and let $T\to\infty$. This gives $$\lim_{T\to\infty}\frac{\det A_k(v)}{\det A}=\frac 1{\sqrt{2\pi}}
\int_{\Gamma^\ast_M}
e^{-\frac{\tau^2}2}\prod_{j\neq k}\left(\frac{v+i\sqrt{S}\tau-y_j}
{y_k-y_j}\right)d\tau.\notag$$ Choose $M$ so that $v-\sqrt{S}M=L$, where $L$ is given, and make the change of variables $w=v+i\sqrt{S}\tau$. Then $$\lim_{T\to\infty}\frac{\det A_k(v)}{\det A}=
\frac 1{i\sqrt{2\pi S}}
\int_{\Gamma_L}e^{\frac{(w-v)^2}{2S}}\prod_{j\neq k}\left(\frac{
w-y_j}{y_k-y_j}\right)dw,\notag$$ where $\Gamma_L:t\to L+it$, $t\in\mathbb R$. Thus, using (\[2.13\]), $$K_N^S(u,v;y)=\frac 1{2\pi iS}\sum_{k=1}^Ne^{-(y_k-u)^2/2S}
\int_{\Gamma_L}e^{\frac{(w-v)^2}{2S}}\prod_{j\neq k}\left(\frac{
w-y_j}{y_k-y_j}\right)dw.\notag$$ Let $\gamma$ be a curve surrounding $y_1,\dots,y_N$ and choose $L$ so large that $\gamma$ and $\Gamma$ do not intersect. The residue theorem gives $$\frac 1{2\pi i}\int_\gamma\frac{e^{-(z-u)^2/2S}}{w-z}\prod_{j=1}^N
\frac{w-y_j}{z-y_j}dz =
\sum_{k=1}^Ne^{-(y_k-u)^2/2S}\prod_{j\neq k}\left(\frac{
w-y_j}{y_k-y_j}\right)\notag$$ for all $w\in\Gamma_L$. Thus, $$\label{2.18}
K_N^S(u,v;y)=\frac {e^{\frac{v^2-u^2}{2S}}}{(2\pi i)^2S}\int_\gamma
dz\int_{\Gamma_L} dwe^{\frac 1{2S}(w^2-2vw-z^2+2uz)}\frac 1{w-z}
\prod_{j=1}^N\frac{w-y_j}{z-y_j}.$$ In (\[2.18\]) we make the change of variables $z\to bz$, $w\to bw$ with $b\in\mathbb R$ close to 1. This will modify the contours but we can use Cauchy’s theorem to deform back to $\gamma$ and $\Gamma_L$. Now, take the derivative with respect to $b$ and then put $b=1$. This gives the equation $$\begin{aligned}
0&=K_N^S(u,v;y)+\frac {e^{\frac{v^2-u^2}{2S}}}{(2\pi i)^2S^2}\int_\gamma
dz\int_{\Gamma_L} dw\frac 1{w-z}e^{\frac
1{2S}(w^2-2vw-z^2+2uz)}\notag\\
&\times\left[w^2-z^2+uz-vw+S\sum_{j=1}^N\left(\frac{w}{w-y_j}-
\frac{z}{z-y_j}\right)\right]\prod_{j=1}^N\frac{w-y_j}{z-y_j}\notag.\end{aligned}$$ This can be written $$\begin{aligned}
&\frac{\partial}{\partial u}((u-v)K_N^S(u,v;y))=
-\frac {e^{\frac{v^2-u^2}{2S}}}{(2\pi i)^2S^2}\int_\gamma
dz\int_{\Gamma_L} dw\notag\\&\left[w+z-v-S\sum_{j=1}^N
\frac{y_j}{(w-y_j)(z-y_j)}\right]
e^{(w^2-2vw-z^2)/2S}e^{uz/S}\prod_{j=1}^N\frac{w-y_j}{z-y_j},\notag\end{aligned}$$ and integration of this formula gives (\[2.15\]). In this last formula we can choose $L$ arbitrarily and take $\gamma$ to be the curve in the proposition by using Cauchy’s formula, This completes the proof.
We now take $S=a^2/N$ and set $$\label{2.19}
\mathcal{K}_N(u,v;y)=e^{\frac{N(u^2-v^2)}{2a^2}+\omega(u-v)}K_N^{a^2/N}
(u,v;y),$$ where $\omega$ is a constant that will be specified later. Note that we can replace $K_N^{a^2/N}$ with $\mathcal{K}_N$ in (\[2.14\]) without changing the correlation functions, so we can just as well work with $\mathcal{K}_N$. Set $$\begin{aligned}
&f_N(z)=\frac 1{2a^2}(z^2-2uz)+\frac 1N\sum_{j=1}^N\log(z-y_j)\notag\\
&g_N(z,w)=\frac 1{a^2z}\left(w+z-u-\frac{a^2}{N}
\sum_{j=1}^N\frac{y_j}{(w-y_j)(z-y_j)}\right)\notag\\
&h(z,w)=\frac{e^{\omega(u-v)}}{N\rho(u)(v-u)}e^{\frac N{a^2}(u-v)w}(
e^{N(u-v)w/a^2}-e^{N(u-v)(w-z)/a^2})\notag,\end{aligned}$$ so that $$\label{2.20}
\mathcal{K}_N(u,v;y)=N\rho(u)\int_\gamma\frac{dz}{2\pi
i}\int_\Gamma\frac{dw}{2\pi i} h(z,w)g_N(z,w)e^{N(f_N(w)-f_N(z))}.$$ These are the formulas we will use in the asymptotic analysis. A straightforward computation shows that $$\label{2.21}
g_N(z,w)=\frac 1zf_N'(z)+\frac{f_N'(z)-f_N'(w)}{z-w}.$$
Asymptotics
===========
The eigenvalues $y_1,\dots,y_N$ of the Wigner matrix $H$ converge to the semicircle law $$\label{3.1}
\sigma(t)=\frac 2{\pi}\sqrt{1-t^2},\quad |t|\le 1.$$ In order to be able to perform the saddle point analysis of (\[2.20\]) we need uniform control of the convergence of $f_N(z)$ to its limit $$\label{3.0}
f(z)=\frac 1{2a^2}(z^2-2uz)+\int_{-1}^1\log (z-t)\sigma(t)dt.$$ In order to show this we must start with some probability estimates. Write $\Omega_{R,\eta}=\{z\in\mathbb C\,;\,|{\text{Re\,}}z|\le R,
\eta\le |{\text{Im\,}}z|\le R\}$.
\[lem3.1\] Let $F\in L^\infty(\mathbb R^N)$ be symmetric and let $\eta>0$ and $R>0$ be given. Assume that $P\in\mathcal{W}^p$, $p>4$ and $0<\xi<\min(\frac 12-\frac 2p,\frac 1{16})$. Then, there is a probability measure $\tilde{P}^{(N)}$ on $\mathcal{H}_N$ such that $$\label{3.1'}
\left|\int_{\mathcal{H}_N}F(x(H))dP^{(N)}(H)-
\int_{\mathcal{H}_N}F(x(H))d\tilde{P}^{(N)}(H)\right|\le N^{2-p(\frac
12-\xi)}||F||_\infty ,$$ and $$\label{3.1''}
\sup_{z\in\Omega_{R,\eta}}\left|\frac
1N{\text{Tr\,}}\log(z-H)-\int_{-1}^1\log(z-t)\sigma(t)dt\right|\le CN^{-\xi}$$ a.s. with respect to $\tilde{P}^{(N)}$.
Given $P\in\mathcal{W}^p$ we introduce a cut-off $L>0$ and define a new probability measure $P_L\in\mathcal{W}^p$ by $$dP_{L,jk}^{R,I}(t)=\frac 1{d_{L,jk}}\chi_{[-L,L]}(t)dP_{jk}^{R,I}(t)
\quad, 1\le j\le k\notag$$ where $d_{L,jk}$ is a normalization constant. Note that $P_{L,jk}$ is supported in $K=[-L,L]^2$. Set $d^{(N)}_L=\prod_{1\le j\le k\le
N}d_{L,jk}$. Then, $$\begin{aligned}
\label{3.2}
\left|\int_{\mathcal{H}_N}F(x(H))dP^{(N)}(H)-
\int_{\mathcal{H}_N}F(x(H))dP_L^{(N)}(H)\right|&\le
||F||_\infty(1-d^{(N)}_L)(1+\frac
1{d^{(N)}_L})\notag\\
&\le\frac{CN^2}{L^p}||F||_\infty\end{aligned}$$ for some constant $C$. The last estimate follows from $$\label{3.2''}
1-d^{(N)}_L=P[\text{some\,} |W_{jk}|\ge L]\le N^2\sup_{1\le j\le
k}\frac{E[|W_{jk}|^p]}{L^p}\le \frac{CN^2}{L^p}$$ by (\[1.9\]). Set $D_N=\Omega_{R,\eta}\cap\frac 1N\mathbb Z^2$ and note that $\# D_N\le
CN^2$ for some constant $C$ that only depend on $R,\eta$. For a given function $f$ set $$A_N(f;\delta)=\{H\in\mathcal{H}_N\,;\,|\frac 1N{\text{Tr\,}}(f(H))-\int_{-1}^1f(t)d \sigma(t)|\le\delta\},\notag$$ where $\sigma(t)$ is the semicircle law (\[3.2\]). Set $$\label{3.2'}
A_N(\delta)=\bigcap_{z\in D_N}A_N(f_z,\delta),$$ where $f_z(t)=\log(z-t)$ (principal branch). To estimate the probability of $A_N(\delta)$ under $P^{(N)}$ we will use a result of Guionnet and Zeitouni, [@GZ]. Let $$|f|_{\mathcal{L}}=\sup_{t,s\in\mathbb R}\frac{|f(x)-f(y)|}{|x-y|},\notag$$ and $||f||_{\mathcal{L}}=||f||_\infty+|f|_{\mathcal{L}}$. Then, by [@GZ], corollary 1.6a), and the discussion before this corollary, given $\epsilon>0$, there are positive constants $C_0(\epsilon)$, $C_1$ and $C_2$ such that if we write $$\label{3.3}
\delta_1(N)=C_1L^2|f|_{\mathcal{L}}N^{-1}+C_2(\epsilon)||f||_{\mathcal{L}}
N^{-1/4+\epsilon},$$ then $$\label{3.4}
P_L^{(N)}\left[|\frac 1N{\text{Tr\,}}f(H)-\int_{-1}^1f(t)\sigma(t)dt|\ge\delta\right]\le
4\exp\left[-\frac{C_2N^2}{L^4 |f|_{\mathcal{L}}}(\delta-\delta_1(N))^2\right]$$ for any $\delta>\delta_1(N)$. Since under $P_L^{(N)}$ all $|H_{jk}|\le\sqrt{2}(L/\sqrt{N})$, the spectral radius is $\le 2L$. Thus, the left hand side of (\[3.4\]) is unchanged if we replace $f=f_z$ with $f=f_z^L(t)$, where $f_z^L(t)=\log(z-t)$ if $|t|\le 2L$, $f_z^L(t)=\log(z-2L)$ if $t>2L$ and $f_z^L(t)=\log(z+2L)$ if $t<-2L$. Now, $f_z^L(t)$ is Lipschitz and there is a constant $C_3$, independent of $L$, such that $|f_z^L(t)|_{\mathcal{L}}\le C_3 $ and $||f_z^L(t)||_{\mathcal{L}}\le C_3(1+\log L)$ for all $z\in\Omega_{z,\eta}$. Take $L=L_N=N^{1/2-\xi}$ and $\epsilon=1/6$ in (\[3.3\]). Then $\delta_1(N)\le CN^{-2\xi}$ and if we choose $\delta=N^\xi$ in (\[3.4\]) we obtain $$\label{3.5}
P_L^{(N)}\left[|\frac 1N{\text{Tr\,}}f_z(H)-\int_{-1}^1f_z(t)\sigma(t)dt|\ge N^{-\xi}\right]\le
c_1\exp(-c_2N^{2\xi})$$ for some positive constants $c_1, c_2$. If we use (\[3.5\]) we see that the probability of the complement of the event in (\[3.2’\]) can be estimated as $$\label{3.6}
P_{L_N}^{(N)}[A_N(N^{-\xi})^c]\le CN^2e^{-c_2N^{2\xi}}.$$ Set $$d\tilde{P}^{(N)}(H)=(P_{L_N}^{(N)}[A_N(N^{-\xi})])^{-1}\chi_{A_N(N^{-\xi})}
(H) dP_{L_N}^{(N)}.\notag$$ Note that $N^2/L_N^p=N^{2-p(1/2-\xi)}$, so combining (\[3.2\]), (\[3.2’\]) and (\[3.6\]) we obtain the estimate (\[3.1’\]). From the definition of $A_N(\delta)$ we see that (\[3.1”\]) holds for $z\in D_N$, but then a straightforward approximation argument extends it to all $z\in\Omega_{R,\eta}$. This completes the proof of lemma 3.1.
We now come to the central asymptotic result.
\[lem3.2\] Let $\Omega_{R,\eta}$ be as above, let $\xi\in(0,1/2]$ and let $K$ be a compact subset of $\mathbb R$. Also let $u_N$ be a sequence such that $u_N\to u$ as $N\to\infty$. Furthermore, let $Y_{R,\eta}$ be the set of all $y\in\mathbb R^N$ such that $$\label{3.7}
\sup_{z\in\Omega_{R,\eta}}\left|\frac 1N\sum_{j=1}^N\log(z-y_j)-
\int_{-1}^1\log(z-t)\sigma(t)dt\right|\le CN^{-\xi}$$ for some constant $C$ and all $N\ge 1$, where $\sigma(t)$ is given by (\[3.1\]). Then, we can find $R_0>0$, $\eta_0>0$ and a constant $C$ such that for all $y\in Y_{R_0,\eta_0}$, $\tau\in K$, $|u|\le\sqrt{1/2+2a^2}$ and $N\ge 1$, $$\left|\frac{1}{N\rho(u)}\mathcal{K}_N(u_N,u_N+\frac{\tau}{N\rho(u)};y)-
\frac{\sin\pi \tau}{\pi\tau}\right|\le C(|u-u_N|+N^{-\xi}),$$ where $\rho(u)$ is given by (\[1.11\]).
It follows from the formula (\[2.20\]) that $$\label{3.9}
\frac{1}{N\rho(u)}\mathcal{K}_N(u_N,u_N+\frac{\tau}{N\rho(u)};y)
=N\int_\gamma\frac{dz}{2\pi i}\int_\Gamma\frac{dw}{2\pi i}
h(z,w)g_N(z,w)e^{N(f_N(w)-f_N(z))} ,$$ where $g_N(z,w)$ is given by (\[2.21\]), $$f_N(z)=\frac 1{2a^2}(z^2-2u_Nz)+\frac 1N\sum_{j=1}^N\log(z-y_j)
\notag$$ and $$h(z,w)=\frac{e^{\omega_0\tau}}{\tau}\left(e^{-\tau w/a^2\rho(u)}-
e^{-\tau(w-z)/a^2\rho(u)}\right)\notag$$ We have taken $\omega=\omega_0/N\rho(u)$, where $\omega_0$ is given by (\[3.16\]) below. The integral in (\[3.9\]) will be analyzed using a saddle point argument. It follows from (\[3.7\]) and Cauchy’s integral formula that there is a constant $C$ such that for all $N\ge
1$, $\tau\in K$, $y\in Y_{R/2,2\eta}$ and $|u|\le\sqrt{1/2+2a^2}$, $$\begin{aligned}
\label{3.10}
&|f_N'(z)-f'(z)|\le C(N^{-\xi}+|u-u_N|)\\
&|f_N''(z)-f''(z)|\le CN^{-\xi}.\notag\end{aligned}$$ A computation shows that, $$f'(z)=\frac 1{a^2}(z-u)+2(z-\sqrt{z^2-1}).\notag$$ Set $S(w)=(w+1/w)/2$ with inverse $S^{-1}(z)=z+\sqrt{z^2-1}$, where $\sqrt{z^2-1}=\sqrt{z-1}\sqrt{z+1}$ (principal argument). The function $S$ maps $\{|w|>1\}$ to $\mathbb C\setminus [-1,1]$ and $|w|=1$ is mapped to $[-1,1]$. Note that $$f'(S(w))=\frac w{2a^2}+(2+\frac 1{2a^2})\frac 1w-\frac u{a^2}.\notag$$ Write $u=\sqrt{1+4a^2}\cos\theta_c$, where $\theta_c\in [0,\pi]$. Our assumption on $u$ means that $|\cos\theta_c|\le 1/2$. Note that $f'(S(w))=0$ has the solutions $w_c^{\pm}=\sqrt{1+4a^2}\exp(\pm
i\theta_c)$. Hence the critical points for $f$ are $z_c^{\pm}=S(w_c^{\pm})$.
We will now define some contours that we will use. Pick $\delta>0$ (small), see below. Set, for some $\epsilon>0$ (small), $\gamma_1^+(t)=S(\sqrt{1+4a^2}e^{i\delta}-t)$, $-\infty<t\le 0$, $\gamma_2^+(t)=S(\sqrt{1+4a^2}e^{it})$, $\delta\le t\le\theta_c-\epsilon$, $\gamma_3^+(t)=S(\sqrt{1+4a^2}e^{it})$, $\theta_c-\epsilon\le t\le\theta_c+\epsilon$, $\gamma_4^+(t)=S(\sqrt{1+4a^2}e^{it})$, $\theta_c+\epsilon\le t\le\pi-\delta$ and $\gamma_5^+(t)=S(\sqrt{1+4a^2}e^{i(\pi-\delta)}-t)$, $0\le t<\infty$. Also, set $\gamma_j^-(t)=\overline{\gamma_j^+(t)}$, $1\le j\le
5$. Then, we can take $\gamma=\sum_{j=1}^5(\gamma_j^+-\gamma_j^-)
=\gamma^+-\gamma^-$ in (\[3.9\]). Let $t_0\in (1/\sqrt{1+4a^2}, 1)$ be such that ${\text{Im\,}}S(t_0w_c^+)=\eta$, and write $\alpha={\text{Re\,}}S(t_0w_c^+)$. Set, for some $\epsilon>0$ (small), $\Gamma_1^+(t)=\alpha+it$, $0\le
t\le\eta$, $\Gamma_2^+(t)=S(tw_c^+)$, $t_0\le t\le 1-\epsilon$, $\Gamma_3^+(t)=S(tw_c^+)$, $1-\epsilon\le t\le 1+\epsilon$ and $\Gamma_4^+(t)=S(tw_c^+)$, $1+\epsilon\le t$. Also, set $\Gamma_j^-(t)= \overline{\Gamma_j^+(t)}$, $1\le j\le 4$. We can then take $\Gamma=\sum_{j=1}^4(\Gamma_j^+-\Gamma_j^-)=\Gamma^+-\Gamma^-$ in (\[3.9\]). Set $$\label{3.10'}
L_N^{bd}(\tau;y)=N\int_{\gamma^b_3}\frac{dz}{2\pi i}
\int_{\Gamma^d_3}\frac{dw}{2\pi i}h(z,w)g_N(z,w)e^{N(f_N(w)-f_N(z))},$$ where $b,d\in\{+,-\}$ and write $L_N=L_N^{++}-L_N^{+-}-L_N^{-+}+L_N^{--}$.
\[cl3.3\] We can choose $R_0>0$, $\eta_0>0$ and $\epsilon,\delta>0$, so that $\gamma_3^++\gamma_3^-+\Gamma_3^++\Gamma_3^-$ lies in a neighbourhood of $z_c^{\pm}$ which is included in $\Omega_{R_0/2,2\eta_0}$ and for all $N\ge
1$, $\tau\in K$, $y\in Y_{r/2,2\eta}$ and $|u|\le\sqrt{1/2+2a^2}$, $$\label{3.11}
\left|\frac{1}{N\rho(u)}\mathcal{K}_N(u_N,u_N+\frac{\tau}{N\rho(u)};y)-
L_N(\tau;y)\right|\le Ce^{-cN}$$ with $c>0$
The claim will be proved below. We will now use the claim to finish the proof of lemma \[lem3.2\]. It follows from (\[3.10\]) that there are critical points $z_N^\pm=S(w_N^\pm)$ for $f_N(z)$ such that $$\label{3.12}
|z_N^\pm-z_c^\pm|\le C(N^{-\xi}+|u-u_N|).$$ We can deform $\gamma_3^\pm$ ($\Gamma_3^\pm$) into contours $\gamma_N^\pm$ ($\Gamma_N^\pm$) such that the endpoints are unchanged, $\gamma_N^\pm(0)=\Gamma_N^\pm(0)=z_N^\pm$ and $\gamma_N^\pm$ ($\Gamma_N^\pm$) have $C^1$-distance $\le C(N^{-\xi}+|u-u_N|)$ to $\gamma_3^\pm$ ($\Gamma_3^\pm$). We can also asume that these contours are chosen so that $\gamma_N^\pm(t)=S(w_N^\pm e^{\pm it})$ and $\Gamma_N^\pm(t)=S(w_N^\pm(1+t))$ for $|t|\ll\epsilon$.
We can now proceed in the standard way with a local saddle point argument in (\[3.10’\]) and prove that there is a constant $C$ such that $$\begin{aligned}
\label{3.12'}
&\left|L_N^{bd}(\tau;y)-h(z_N^b,z_N^d)g_N(z_N^b,z_N^d)
\frac{2\pi}{(2\pi i)^2}\frac{(\gamma_N^b)'(0)(\Gamma_N^d)'(0)
e^{N(f_N(z_N^b)-f_N(z_N^d))} }{\sqrt{f_N''(z_N^b)(\gamma_N^b)'(0)^2}
\sqrt{-f_N''(z_N^d)(\Gamma_N^d)'(0)^2}}\right|\\&\le\frac{C}{\sqrt{N}}
\notag\end{aligned}$$ for all $N\ge
1$, $\tau\in K$, $y\in Y_{R_0,\eta_0}$ and $|u|\le\sqrt{1/2+2a^2}$. Note that $z_N^+=\overline{z_N^-}$ and $f_N(z_N^+)-f_N(z_N^-)$ is purely imaginary. Now, $(\gamma_N^b)'(0)=biS'(w_N^b)$, $(\Gamma_N^b)'(0)=w_N^bS'(w_N^b)$ and a computation shows that $$f_N''(z_N^b)(\gamma_N^b)'(0)^2=-f_N''(z_N^b)(\Gamma_N^b)'(0)^2=
-f_N''(z_N^b)S'(w_N^b)^2(w_N^a)^2,\notag$$ which has a positive real part by (\[3.10\]) and the fact that $f''(z_c^b)S'(w_c^b)^2(w_c^a)^2$ has a positive real part. From (\[2.21\]) we see that $g_N(z_N^b, z_N^d)=0$ if $b\neq d$ and $g_N(z_N^b, z_N^b)=f_N''(z_N^b)$. It follows that $$\frac{g_N(z_N^b, z_N^b)(\gamma_N^b)'(0)(\Gamma_N^b)'(0)}
{\sqrt{f_N''(z_N^b)(\gamma_N^b)'(0)^2}
\sqrt{-f_N''(z_N^b)(\Gamma_N^b)'(0)^2}}=-bi.\notag$$ Also, from (\[3.12\]) it follows that $|h(z_N^b,z_N^b)-h(z_c^b,z_c^b)|\le C(N^{-\xi}+|u-u_N|)$, and thus (\[3.12’\]) yields $$\label{3.13}
|L_N^{bd}(\tau;y)|\le\frac{C}{\sqrt{N}}$$ if $b\neq d$ and $$\label{3.14}
\left|L_N^{bb}(\tau;y)+\frac{bh(z_c^b,z_c^b)}{2\pi i}\right|\le
C(N^{-\xi}+|u-u_N|).$$ Combining (\[3.10’\]), (\[3.13\]) and (\[3.14\]) we obtain $$\label{3.15}
\left|L_N(\tau;y)+\frac{h(z_c^+,z_c^+)-h(z_c^-,z_c^-)}{2\pi
i}\right|\le
C(N^{-\xi}+|u-u_N|).$$ Now, $$h(z_c^\pm,z_c^\pm)=\frac{e^{\omega_0\tau}}{\tau}\left(e^{-\tau
z_c^\pm/a^2\rho(u)}-1\right)\notag$$ and a computation shows that $$\label{3.16}
\frac{z_c^\pm}{a^2\rho(u)}=\pi\frac{1+2a^2}{2a^2}\cot\theta_c\pm\pi
i\doteq\omega_0\pm \pi i.$$ Thus (\[3.15\]) becomes $$\left|L_N(\tau;y)-\frac{\sin\pi \tau}{\pi\tau}\right|\le
C(N^{-\xi}+|u-u_N|).
\notag$$ If we combine this estimate with (\[3.11\]) we see that the lemma is proved.
It remains to prove claim \[cl3.3\].
Let $\gamma_\ast^\pm=\sum_{j\neq 3}\gamma_j^\pm$ and $\Gamma_\ast^\pm=\sum_{j\neq 3}\Gamma_j^\pm$. We have to estimate $$I_1^{bd}=N\int_{\gamma_\ast^b}|dz|\int_{\Gamma^d}|dw||h(z,w)||g_N(z,w)|
e^{N{\text{Re\,}}(f_N(w)-f_N(z_c^d))-N{\text{Re\,}}(f_N(z)-f_N(z_c^b))},\notag$$ and $$I_2^{bd}=N\int_{\gamma^b}|dz|\int_{\Gamma_\ast^d}|dw||h(z,w)||g_N(z,w)|
e^{N{\text{Re\,}}(f_N(w)-f_N(z_c^d))-N{\text{Re\,}}(f_N(z)-f_N(z_c^b))},\notag$$ where $b,d\in\{+,-\}$. Note that $f_N(z_c^+)-f_N(z_c-)$ is purely imaginary. We will concentrate on $I_1^{++}$ since the other cases are similar.
Using the inequality $$\left|\frac{w-y_j}{z-y_j}\right|=\left|1+\frac{w-z}{z-y_j}\right|\le
1+C(|w|+|z|)\notag$$ it is not difficult to see that there are constants $C_1$ and $C_2$ such that $$\label{3.16'}
|h(z,w)||g_N(z,w)|e^{N{\text{Re\,}}(f_N(w)-f_N(z))}\le
C_1E^{C_2N(|z|+|w|)+N({\text{Re\,}}(w^2-2uw)-{\text{Re\,}}(z^2-2uz))/2a^2}$$ for all $y\in\mathbb R^N$, $\tau\in K$ and $|u|\le\sqrt{1/2+2a^2}$. Note that $|{\text{Im\,}}z|\ge c>0$ for all $z\in\gamma$. (The constant $c$ depends on the $\delta$ in the definition of $\gamma$, but as we will see below $\delta$ depends only on the parameter $a$ in the problem.) From the estimate (\[3.16’\]) it follows that by picking $R=R_0$ sufficiently large, the contribution to $I_1^{++}$ from $z$ and/or $w$ outside $\Omega_{R_0,0}$ is $\le e^{-N}$. Thus we can assume that $z,w\in \Omega_{R_0,0}$. Next, we will derive the other estimates we will need to prove the claim.
Assume that $z\in\Omega_{R_0,\eta}$ and $w\in\Gamma_1^+$. Then, $$\begin{aligned}
&|g_N(z,w)e^{Nf_N(w)}|\notag\\
&\le C\left(1+\frac 1N\sum_{k=1}^N\frac
1{|w-y_k|}\right)\prod_{j=1}^N|w-y_j| e^{N{\text{Re\,}}(w^2-2uw)/2a^2}\notag\\
&\le C\left(1+\frac 1N\sum_{j=1}^N\frac 1{|\alpha+i\eta-y_j|}\right)
\prod_{j=1}^N|\alpha+i\eta-y_j|e^{N{\text{Re\,}}(w^2-2uw)/2a^2}\notag\\
&\le Ce^{N[{\text{Re\,}}f_N(\alpha+i\eta)+{\text{Re\,}}(w^2-2uw)-((\alpha+i\eta)^2-2u
(\alpha+i\eta))]/2a^2}.\notag\end{aligned}$$ If we use (\[3.7\]) and the definition of $f_N$ we obtain $$\begin{aligned}
\label{3.17}
&|g_N(z,w)e^{N(f_N(w)-f_N(z_c^+))}|\notag\\
&\le
Ce^{cN(N^{-\xi}+|u-u_N|)+N\eta^2/2a^2+N{\text{Re\,}}(f(\alpha+i\eta)-f(z_c^+))/2a^2}\end{aligned}$$ for $z\in\Omega_{R_0,\eta}$ and $w\in\Gamma_1^+$.
We will now compute how ${\text{Re\,}}f(z)$ changes along $\gamma$. Assume that $\theta_c\ge 0$, the other case is analogous. Consider $\gamma(\theta)= S(\sqrt{1+4a^2}e^{i\theta})$, $\delta\le\theta\le\pi-\delta$. A computation, using the fact that $f'(\gamma(\theta_c))=0$ gives ${\text{Re\,}}\frac
d{d\theta}f(\gamma(\theta))=\frac {1+2a^2}{2a^2}\sin\theta
(\cos\theta_c -\cos\theta)$. From this we see that there is a constant $c_0>0$ such that $$\label{3.18}
{\text{Re\,}}(f(\sqrt{1+4a^2}e^{i\theta})-f(z_c^+))\ge c_0(\theta-\theta_c)^2.$$ Next, consider $\gamma_1(t)=S(\sqrt{1+4a^2}e^{i\delta}-t)$, $t\le
0$. If we write $\omega_\delta=\sqrt{1+4a^2}e^{i\delta}$, then $$\frac {d}{dt}f(\gamma_1(t))=-\frac
1{4a^2}[\omega_\delta-t-2u+\frac{1+4a^2}{\omega_\delta-t}][1-\frac
1{(\omega_\delta-t)^2}] .
\notag$$ Set $\omega_\delta-t=s(t)e^{i\theta(t)}$. A computation shows that $$\begin{aligned}
\label{3.19}
&{\text{Re\,}}\frac {d}{dt}f(\gamma_1(t))
=-\frac{1}{4a^2\sqrt{1+4a^2}}\left\{\left[(s(t)+\frac 1{s(t)})\cos
\theta(t)-2\cos\theta_c\right]\right.
\\&\left.\times\left[1+4a^2-\frac 1{s(t)^2}\cos
2\theta(t) \right]
-\frac 1{s(t)^2}\sin 2\theta(t)(s(t)-\frac
1{s(t)})\sin\theta(t)\right\}.\notag\end{aligned}$$ Note that $\sin\theta(t)=s(t)^{-1}\sqrt{1+4a^2}\sin\delta$. It follows that the right hand side of (\[3.19\]) equals $$\begin{aligned}
\label{3.19'}
&-\frac{1}{4a^2\sqrt{1+4a^2}}\left\{\left[(s(t)+\frac 1{s(t)})
(1+4a^2-\frac 1{s(t)^2}+2\frac{(1+4a^2)\sin^2\delta}{s(t)^4})\right.\right.\\&\left.\left.-
2\frac{(1+4a^2)\sin^2\delta}{s(t)^4}(s(t)-\frac
1{s(t)})\right]\cos\theta(t)
-2(1+4a^2-\frac
1{s(t)^2}+2\frac{(1+4a^2)\sin^2\delta}{s(t)^4})\cos\theta_c \right\}\notag\end{aligned}$$ and this is $$\le -\frac{1}{4a^2\sqrt{1+4a^2}}(1+4a^2-\frac 1{s(t)^2})
\left[(s(t)+\frac
1{s(t)})\cos\theta(t)-2(1+\frac{1+4a^2}{2a^2}\sin^2\delta)
\cos\theta_c\right],\notag$$ since $s(t)\ge 1$. Choose $\delta\le \theta_c/4$ so that $$(1+\frac{1+4a^2}{2a^2}\sin^2\delta)
\cos\theta_c\le\cos\frac{\theta_c}2.\notag$$ Since $s(t)+1/s(t)\ge 2$ and $\theta(t)\le\delta$ we see that there is a constant $c_0>0$ such that $$\label{3.21}
{\text{Re\,}}\frac {d}{dt}f(\gamma_1(t))\le -c_0.$$ For $\gamma_5(t)=S(\sqrt{1+4a^2}e^{i(\pi-\delta)}-t)$, $t\ge 0$, we still have the formula (\[3.19’\]) with $\gamma_1(t)$ replaced by $\gamma_5(t)$ and, since $\pi-\delta\le\theta(t)\le\pi$, we see that the right hand side is $$\ge\frac 1{\sqrt{1+4a^2}}[(s(t)+\frac
1{s(t)})\cos(\pi-\theta(t))+2\cos\theta_c]$$ and consequently there is a constant $c_0>0$ such that $$\label{3.22}
{\text{Re\,}}\frac {d}{dt}f(\gamma_5(t))\ge c_0.$$ Consider now how ${\text{Re\,}}f(w)$ changes along $\Gamma^+$. Set $\Gamma(t)=S(tw_c^+)$, $t\ge t_0$. A computation gives $${\text{Re\,}}\frac
{d}{dt}f(S(tw_c))=\frac{1-t}{2a^2t^2}[1+t(1+4a^2)-(t^2(1+4a^2)+\frac
1t)\cos 2\theta_c].\notag$$ Now, since $|u|\le\sqrt{1/2+2a^2}$, it follows that $\cos 2\theta_c\le 0$ and thus $$\begin{aligned}
\label{3.23}
&{\text{Re\,}}\frac
{d}{dt}f(S(tw_c))\ge \frac{1-t}{2a^2t^2}(1+t(1+4a^2))\quad\text{if
$t_0\le t\le 1$}\notag\\
&{\text{Re\,}}\frac
{d}{dt}f(S(tw_c))\le \frac{1-t}{2a^2t^2}(1+t(1+4a^2))\quad\text{if
$t\ge 1$}.\end{aligned}$$ The first of these estimates can be used to show that if we pick $\eta=\eta_0$ sufficiently small, then $$\eta^2+{\text{Re\,}}(f(\alpha+i\eta)-f(z_c^+))\le -c_0\notag$$ for some positive $c_0$. If we use this in (\[3.17\]) we obtain $$\label{3.24}
|g_N(z,w)e^{N(f_N(w)-f_N(z_c^+))}|\le Ce^{-c_0'N}$$ for some positive $c_0'$. We can now use (\[3.18\]), (\[3.21\]), (\[3.22\]), (\[3.23\]) and (\[3.24\]) to estimate $I_1^{++}$ and see that it is $\le Ce^{-cN}$ for some positive $c$.
Proof of the theorems
=====================
We start with the proof of theorem \[corr\].
By proposition 1.1 and Fubini’s theorem the integral in the left hand side of (\[1.12”\]) can be written $$\label{4.1}
\int_{\mathcal{H}_N}\left(\int_{\mathbb R^N}\rho_N(x,y(H))(Sf)
(N\rho(u)(x_1-u), \dots,N\rho(u)(x_N-u))d^Nx\right)dP^{(N)}(H)$$ Note that $||S(f)||_\infty\le N^m||f||_\infty$. Since $\rho_N(x,\cdot)$ is a probability density on $\mathbb R^N$ we can use lemma 3.1 to replace the expression in (\[4.1\]) by $$\label{4.2}
\int_{\mathcal{H}_N}\left(\int_{\mathbb R^N}\rho_N(x,y(H))(Sf)
(N\rho(u)(x_1-u), \dots,N\rho(u)(x_N-u))d^Nx\right)d\tilde{P}^{(N)}(H)$$ with an error $\le CN^m||f||_\infty N^{2-p(1/2-\xi)}=o(1)$, since $p>2(m+2)$, provided we choose $\xi$ small enough. Now, since $\rho_N(x,\cdot)$ is symmetric it follows from (\[1.12’\]), (\[2.4\]), (\[2.14\]) and (\[2.19\]) that the expression in (\[4.2\]) can be written $$\begin{aligned}
\label{4.3}
\int_{\mathcal{H}_N}\int_{\mathbb R^m} &f(t_1,\dots,t_m)\\
&\times\det(\frac
1{N\rho(u)}
\mathcal{K}(u+\frac{t_i}{N\rho(u)},u+\frac{t_j}{N\rho(u)};y(H)))_{i,j=1}^m
d^mtd\tilde{P}^{(N)}(H).\notag\end{aligned}$$ Since $f$ has compact support and we know that (\[3.1”\]) holds a.s. $[\tilde{P}^{(N)}]$ it follows from lemma \[lem3.2\], with $u_N=u+t_i/N\rho(u)$, $\tau=t_j-t_i$, that $$\left|\mathcal{K}(u+\frac{t_i}{N\rho(u)},u+\frac{t_j}{N\rho(u)};y(H))
-\frac{\sin\pi(t_i-t_j)}{\pi(t_i-t_j)}\right|\le CN^{-\xi},\notag$$ for a.a. $[\tilde{P}^{(N)}]$ and all $(t_1,\dots,t_m)$ in the support of $f$. Thus we can take the limit as $N\to\infty$ in (\[4.3\]) and obtain the right hand side of (\[1.12”\]). This completes the proof.
Before proving theorem 1.3 we need some preliminary results on the level spacing distribution. Let $\rho_N(x)$ be a symmetric probability density on $\mathbb R^N$ with correlation functions defined by (\[1.1\]). Assume that $R_1^{(N)}/N\to\rho(t)$ (weakly) as $N\to\infty$, so that $\rho(t)$ is the asymptotic density. Let $u$ be a given point such that $\rho(u)>0$, and let $t_N$ be a sequence such that $t_N\to\infty$ but $t_N/N\to 0$ as $N\to\infty$. Set, for $|r|\le
1/2$, $$\mathcal{R}_m^{(N)}(\sigma_1,\dots,\sigma_m;r)=
\frac 1{(N\rho(u))^m}R_m^{(N)}(u+\frac{2t_Nr+\sigma_1}
{N\rho(u)},u+\frac{2t_Nr+\sigma_m}{N\rho(u)})\notag$$ and let $\mathcal{R}_m(\sigma_1,\dots,\sigma_m)$ be the limiting correlation functions, which we assume are continuous, symmetric and translation invariant. Assume that, for each $s\ge 0$, $$\label{4.4}
D_N(s)=\sum_{m=N+1}^\infty\frac{s^m}{m!}\sup_{|\sigma_j|\le s}
|\mathcal{R}_m(\sigma_1,\dots,\sigma_m)|<\infty.$$ Set $$H(s)=\sum_{m=0}^\infty\frac{(-1)^m}{m!}\int_{[0,s]^m}
\mathcal{R}_m(\sigma_1,\dots,\sigma_m)d^m\sigma\notag$$ (the probability of no particle in $[0,s]$), which is well defined by (\[4.4\]). Also, set $$\label{4.5}
\epsilon_m^{(N)}=\sup_{|\sigma_j|\le s, |r|\le 1/2}
|\mathcal{R}_m^{(N)}(\sigma_1,\dots,\sigma_m;r)-
\mathcal{R}_m(\sigma_1,\dots,\sigma_m)|.$$
\[prop4.1\] Let $S_N(s,x)$ be defined by (\[1.5\]). Then $$\label{4.6}
\left|\int_{\mathbb R^N}S_N(s,x)\rho_N(x)d^Nx-\int_0^s H''(u)du\right|\le
D_N(s) +\sum_{m=2}^N\frac{s^{m-1}}{(m-1)!}\epsilon_m^{(N)}.$$
We first show that $$\label{4.7}
\int_0^s H''(u)du=\sum_{m=2}^N\frac{s^{m-1}}{(m-1)!}\int_{[0,s]^{m-1}}
\mathcal{R}_m(0,\tau_2,\dots,\tau_m)d\tau_2\dots d\tau_m,$$ see [@DKMVZ]. Since $\mathcal{R}_m$ is translation invariant and symmetric by assumption, we have $$\begin{aligned}
\label{4.7'}
H'(u)&=\lim_{\epsilon\to 0}\frac 1{\epsilon}\sum_{m=0}^\infty
\frac{(-1)^m}{m!} \int_{[-\epsilon,u]^m\setminus [0,u]^m}
\mathcal{R}_m(x_1,\dots,x_m)d^mx\\
&=\lim_{\epsilon\to 0}\sum_{m=0}^\infty
\frac{(-1)^m}{m!}\frac 1{\epsilon}\left( m\int_{[-\epsilon,0]\times
[0,u]^{m-1}}\mathcal{R}_m(x_1,\dots,x_m)d^mx\right)\notag \\
&=\sum_{m=1}^\infty\frac{(-1)^m}{(m-1)!}\int_{[0,u]^{m-1}}
\mathcal{R}_m(0, x_2,\dots,x_m)d^{m-1}x\notag ,\end{aligned}$$ where we have also used (\[4.4\]) and the continuity of $\mathcal{R}_m$. Continuing in the same way we see that $H(u)$ is actually a $C^\infty$ function, in particular $H''(u)$ is well defined and continuous. From (\[4.7’\]) we get $$H'(s)=-\mathcal{R}_m(0)+\sum_{m=2}^\infty
\frac{(-1)^m}{(m-1)!}\int_{[0,s]^{m-1}}
\mathcal{R}_m(0, x_2,\dots,x_m)d^{m-1}x.\notag$$ Hence $H'(0)=-\mathcal{R}_m(0)$ and we see that the right hand side of (\[4.7\]) equals $H'(u)-H'(0)$, which is what we wanted to prove.
It is proved in [@DKMVZ], using a result from [@KS], that $$\begin{aligned}
&\int_{\mathbb R^N}S_N(s,x)\rho_N(x)d^Nx\notag\\&=
\sum_{m=2}^N\frac{(-1)^m}{(m-1)!}\int_{-1/2}^{1/2}dr\int_{[0,\min(s,
(1-2r)t_N)]^{m-1}}
\mathcal{R}_m^{(N)}(0,\sigma_2,\dots,\sigma_m;r)d^{m-1}\sigma.\notag \end{aligned}$$ Hence, the estimate (\[4.6\]) follows from (\[4.4\]), (\[4.5\]) and (\[4.7\]).
We turn now to the proof of theorem \[spac\].
Just as in the proof of theorem \[corr\] above we see that since $P\in\mathcal{W}^{6+\epsilon}$ and $||S_N||_\infty\le N/2t_N$, $$\begin{aligned}
\label{4.8}
&\left|\int_{\mathcal{H}_N}S_N(s,x(M))dQ^{(N)}(M)-
\int_{\mathcal{H}_N}\left(\int_{\mathbb
R^N}S_N(s,x)\rho_N(x;y(H))d^Nx\right) d\tilde{P}^{(N)}(H)\right|\\
&\le C\frac N{t_N}N^{2-(6+\epsilon)(1/2-\xi)}\le \frac{C}{t_N},\notag\end{aligned}$$ if we take $\xi$ sufficiently small, and also that (\[3.1”\]) holds. From proposition \[eigenmeas\], (\[2.4\]) and proposition \[prop2.4\] we know the correlation functions of $\rho_N(x;y)$, and if we take $u_N=u+(2t_Nr+\sigma_i)(N\rho(u))^{-1}$ in lemma \[lem3.2\] we see that $$\begin{aligned}
\label{4.9}
\left|\frac
1{N\rho(u)}
\mathcal{K}(u+\frac{2t_Nr+\sigma_i}{N\rho(u)},
u+\frac{2t_Nr+\sigma_j}{N\rho(u)};y(H))-
\frac{\sin\pi(\sigma_i-\sigma_j)}{\pi(\sigma_i-\sigma_j)} \right|&\le
C(\frac{t_N}N +N^{-\xi})\\&\doteq \omega_N\notag\end{aligned}$$ for a.a. $H$ $[\tilde{P}^{(N)}]$. Thus, the limiting correlation functions are $$\mathcal{R}_m(\sigma_1,\dots,\sigma_m)-
\det\left(\frac{\sin\pi(\sigma_i-\sigma_j)}
{\pi(\sigma_i-\sigma_j)} \right)_{i,j=1}^m.\notag$$ Since the matrix in the determinant is positive definite it follows from the Hadamard inequality that $$D_N(s)\le \sum_{m=N+1}^\infty\frac{s^m}{m!}.\notag$$ Also, since $$\mathcal{R}_m^{(N)}(\sigma_1,\dots,\sigma_m;y)=
\det\left(\frac
1{N\rho(u)}
\mathcal{K}(u+\frac{2t_Nr+\sigma_i}{N\rho(u)},
u+\frac{2t_Nr+\sigma_j}{N\rho(u)};y)\right)_{i,j=1}^m\notag$$ it follows from (\[4.9\]), the multilinearity of the determinant and Hadamard’s inequality that $$|\mathcal{R}_m^{(N)}(\sigma;y)-\mathcal{R}_m(\sigma)|\le
m(1+\omega_N)^{m-1}\omega_N m^{m/2},\notag$$ and hence $\epsilon_m^{(N)}\le m(1+\omega_N)^{m-1}\omega_N
m^{m/2}$. Now, by proposition \[prop4.1\], Stirling’s formula and the fact that $\omega_N\to 0$, $$\begin{aligned}
\label{4.10}
&\left|\int_{\mathbb
R^N}S_N(s,x)\rho_N(x;y(H))d^Nx-\int_0^sH''(u)du\right|\\
&\le\sum_{m=N+1}^\infty\frac{s^m}{m!}+\omega_N\sum_{m=2}^N
\frac{s^m}{(m-1)!} (1+\omega_N)^{m-1}m^{(m+2)/2}=o(1)\notag\end{aligned}$$ as $N\to\infty$, for a.a. $H$ $[\tilde{P}^{(N)}]$. If we combine (\[4.8\]) and (\[4.10\]) we see that the theorem is proved.
[99]{}
=
C. Andréief, [*Note sur une relation les integrales définies des produits de fonctions,*]{} Mém. de la Soc. Bordeaux, [**2**]{} (1883), 1 - 14
Z. D. Bai, [*Methodologies in spectral analysis of large dimensional random matrices: a review,*]{} Statistica Sinica, [**9**]{} (1999), 611 - 661
P. Bleher, A. Its, [*Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model,*]{} Ann. Math., [**150**]{} (1999), 185 - 266
E. Brézin, S. Hikami, [*Correlations of nearby levels induced by a random potential,*]{} Nucl. Phys. B, [**479**]{}, (1996), 697 - 706
E. Brézin, S. Hikami, [*Spectral form factor in a random matrix theory,*]{} Phys. Rev. E, [**55**]{}, (1997), 4067 - 4083
E. Brézin, S. Hikami, S., [*An extension of level-spacing universality ,*]{} cond-mat/9702213
P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, X. Zhou, [*Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory,*]{} Comm. Pure. Appl. Math., [**52**]{} (1999), 1335 - 1425
F. J. Dyson, [*A Brownian-motion Model for the Eigenvalues of a Random Matrix,*]{} J. Math. Phys., [**3**]{} (1962), 1191 - 1198
D. J. Grabiner, [*Brownian motion in a Weyl chamber, non-colliding particles and random matrices,*]{} Ann. Inst. H. Poincaré, [**35**]{} (1999), 177 - 204
A. Guionnet, O. Zeitouni, [*Concentration of the spectral measure for large matrices,*]{}, preprint (2000)
Harish-Chandra, [*Differential operators on a semisimple Lie algebra,*]{} Am J. Math., [**79**]{}, (1957), 87 - 120
D. G. Hobson, W. Werner, [*Non-colliding Brownian motions on the circle,*]{} Bull. London Math. Soc., [**28**]{} (1996), 543 - 650
C. Itzykson, J. -B. Zuber, [*The planar approximation II,*]{} J. Math. Phys., [**21**]{}, (1980), 411 - 421
K. Johansson, [*Random growth and Random matrices,*]{} to appear in the Proceedings of the third European Congress of Mathematics
K. Johansson, [*Non-intersecting paths, random tilings and random matrices,*]{} in preparation
S. Karlin, G. McGregor, [*Coincidence probabilities,*]{} Pacific J. Math, [**9**]{} (1959), 1141 - 1164
N. M. Katz, P. Sarnak, [*Random Matrices, Frobenius Eigenvalues and Monodromy,*]{} AMS Colloquium Publications, Vol. 45, 1999
A. Khorunzhy, [*On smoothed density of states for Wigner random matrices,*]{} Random Oper. and Stoch. Equ., [**5**]{}, (1997), 147 - 162
A. Khorunzhy, B. A. Khoruzhenko, L. A. Pastur, [*On asymptotic properties of large rando matrices with independent entries,*]{} J. Math. Phys., [**37**]{}, (1996), 5033 - 5060
M. L. Mehta, [*Random Matrices,*]{} 2nd ed., Academic Press, San Diego 1991
A. Okounkov, [*Infinite wedge and measures on partitions,*]{} math.RT/9907127
L. A. Pastur, M. Shcherbina, [*Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles,*]{} J. Stat. Phys., [**86**]{}, (1997), 109 - 147
E. J. Pauwels, L. G. G. Rogers, Skew-product decompositions of Brownian motions in [*Geometry of Random Motion,*]{} R. Durrett, M. A. Pinsky, eds., AMS Contemporary Mathematics, Vol. 73, 1988
R. G. Pinsky, [*On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes,*]{} Ann. Prob., [**13**]{} (1985), 363 - 378
C. E. Porter, ed., [*Statistical Theories of spectra: Fluctuations,*]{} Academic Press, New York, 1965
A. Soshnikov, [*Universality at the edge of the spectrum in Wigner random matrices,*]{} Commun. Math. Phys., [ **207**]{} (1999), 697 - 733
Ya. Sinai, A. Soshnikov, [*Central limit theorem for traces of large random symmetric matrices with independent matrix elements,*]{} Bol. Soc. Brasil. Mat., [**29**]{} (1998), 1 - 24
Ya. Sinai, A. Soshnikov, [*A refinment of Wigner’s semicircle law in a neighborhood of the spectrum edge for symmetric matrices*]{}, Funct. Anal. Appnl., [**32**]{} (1998), 114 - 131
C. A. Tracy, H. Widom, [*Correlation Functions, Cluster Functions, and Spacing Distributions for Random Matrices,*]{} J. Statist. Phys., [**92**]{}, (1998), 809 - 835
|
---
abstract: |
We report the discovery of the near-infrared and optical afterglow of the short-duration gamma-ray burst GRB070724A. The afterglow is detected in $iJHK_s$ observations starting 2.3 hr after the burst with $K_s=19.59\pm 0.16$ mag and $i=23.79\pm 0.07$ mag, but is absent in images obtained 1.3 years later. Fading is also detected in the $K_s$-band between 2.8 and 3.7 hr at a $4\sigma$ significance level. The optical/near-IR spectral index, $\beta_{\rm
O,NIR}\approx -2$, is much redder than expected in the standard afterglow model, pointing to either significant dust extinction, $A_V^{\rm host}\approx 2$ mag, or a non-afterglow origin for the near-IR emission. The case for extinction is supported by a shallow optical to X-ray spectral index, consistent with the definition for “dark bursts”, and a normal near-IR to X-ray spectral index. Moreover, a comparison to the optical discovery magnitudes of all short GRBs with optical afterglows indicates that the near-IR counterpart of GRB070724A is one of the [*brightest*]{} to date, while its observed optical emission is one of the faintest. In the context of a non-afterglow origin, the near-IR emission may be dominated by a mini-supernova, leading to an estimated ejected mass of $M\sim 10^{-4}$ M$_\odot$ and a radioactive energy release efficiency of $f\sim 5\times 10^{-3}$ (for $v\sim 0.3$c). However, the mini-SN model predicts a spectral peak in the UV rather than near-IR, suggesting that this is either not the correct interpretation or that the mini-SN models need to be revised. Finally, the afterglow coincides with a star forming galaxy at $z=0.457$, previously identified as the host based on its coincidence with the X-ray afterglow position ($\sim 2''$ radius). Our discovery of the optical/near-IR afterglow makes this association secure, and furthermore localizes the burst to the outskirts of the galaxy, with an offset of $4.8\pm 0.1$ kpc relative to the host center. At such a large offset, the possible large extinction points to a dusty environment local to the burst and rules out a halo or intergalactic origin.
author:
- 'E. Berger, S. B. Cenko, D. B. Fox and A. Cucchiara'
title: 'Discovery of the Very Red Near-Infrared and Optical Afterglow of the Short-Duration GRB 070724A'
---
Introduction {#sec:intro}
============
The determination of sub-arcsecond positions for short-duration gamma-ray bursts (GRBs) is of the utmost importance for our growing understanding of their redshift distribution, energy scale, host galaxies, and local environments. Such localizations require the detection of optical, near-infrared, or radio afterglows; or alternatively an X-ray detection with the [*Chandra*]{} X-ray Observatory. As of August 2009, only 15 short GRBs have been precisely localized in this manner, and all were detected in the optical band [@bpc+05; @ffp+05; @hwf+05; @ltf+06; @pcg+06; @rvp+06; @sbk+06; @gcn6739; @sap+07; @gcn8190; @pac+08; @gcn8934; @gcn8933; @amc+09; @gfl+09; @gcn9342; @pmg+09].
Follow-up observations of some of these bursts have led to the detection and characterization of host galaxies, most of them star forming, and a small proportion ($\sim 20\%$) non-star forming (e.g., @ber09). The precise positions also allow us to study the local environment of the bursts within their hosts, and current observations point to offsets of $\sim 1-15$ kpc (e.g., @bpc+05 [@ffp+05; @sbk+06; @amc+09]). However, since these are projected positions, and since we generally lack detailed afterglow observations that can shed light on the circumburst environment, little is known about whether the bursts originate within the inner or halo regions of their hosts (e.g., @sbk+06 [@lbb+09]).
Rapid optical observations have also been used to place limits on emission from radioactive material synthesized in a putative sub-relativistic outflow associated with a compact object binary merger, a so-called Li-Paczynski mini-supernova [@lp98]. Such emission is theorized to have a typical peak time of $\sim 1$ d, and a peak luminosity of $\sim 10^{42}$ erg s$^{-1}$, corresponding to $m_{\rm AB}\sim 21$ mag at $z\sim 0.5$. No such emission has been detected to date (e.g., @hsg+05 [@bpp+06]). Similarly, no late-time emission from Type Ib/c supernova associations have been detected (e.g., @hwf+05 [@bpp+06; @sbk+06]).
Most recently, near-IR and optical non-detections of have been used to place limits on emission from a putative mini-SN associated with this burst [@ktr+09]. Here we report the detection of near-IR and optical counterparts of about 2.3 hr after the burst, and show that the afterglow is actually one of the brightest near-IR short GRB afterglows detected to date, but is one of the faintest in the optical. We use the observed fluxes and the unusually red color to investigate the properties of the afterglow and/or mini-SN, and to precisely measure the location of the burst relative to its host galaxy. Our discovery of the afterglow of suggests that recovery of a substantial fraction of short GRB optical/near-IR afterglows requires observations to $m_{\rm AB}\sim 24$ mag within about $0.5$ d.
GRB070724A {#sec:obs}
==========
was discovered by the [*Swift*]{} satellite on 2007 July 24 at 10:53:50 UT with a duration of $0.40\pm 0.04$ s [@gcnr74]. The X-ray afterglow was detected with the on-board X-ray Telescope (XRT) beginning 72 s after the burst, while no counterpart was detected with the UV/Optical Telescope [@gcnr74]. The X-ray position was subsequently determined to a precision of $1.7''$ radius ($90\%$ containment). An apparently extended source was detected in coincidence with the XRT error circle in Digitized Sky Survey images, and tentatively proposed as a possible host [@gcn6658]. Subsequent near-IR and optical observations from UKIRT, Gemini-North, the Palomar 60-inch telescope, and the VLT revealed that the source was indeed an extended galaxy, but did not uncover an afterglow [@gcn6662; @gcn6664; @gcn6666]. A recent analysis using image subtraction on UKIRT, NOT, CTIO 1.3-m, Keck, and VLT data similarly reveals no afterglow to limits of $F_{\nu,K}\lesssim 30$ $\mu$Jy and $F_{\nu,i}\lesssim 0.3$ $\mu$Jy at 3.2 and 22.2 hr after the burst, respectively [@ktr+09].
Spectroscopy of the galaxy within the XRT error circle revealed that it is located at a redshift of $z=0.4571$, is undergoing active star formation at a rate of 2.5 M$_\odot$ yr$^{-1}$, has a luminosity of $L_B\approx 1.4$ L$^*$, and a metallicity of $12+{\rm log(O/H)}\approx
8.9$ (@ber09; see also @ktr+09 for similar results). The resulting isotropic $\gamma$-ray energy in the observed $15-150$ keV range is $E_{\rm \gamma,iso}\approx 1.6\times 10^{49}$ erg.
Discovery of the Near-IR and Optical Afterglow {#sec:disc}
==============================================
We observed the field centered on with the Near Infra-Red Imager and Spectrometer (NIRI) mounted on the Gemini-North 8-m telescope in the $JHK_s$ bands starting on 2007 July 24.566 UT (2.69 hr after the burst); see Table \[tab:obs\]. The observations were obtained in excellent seeing conditions, $\approx 0.35''$ in the $K_s$-band. The data were reduced using the [gemini]{} package in IRAF, and individual stacks were created in each filter. Inspection of the images reveals a point source coincident with the south-east edge of the putative host galaxy; see Figure \[fig:fig1\].
Optical observations were obtained with the Gemini Multi-Object Spectrograph (GMOS) mounted on the Gemini-North 8-m telescope in the $gi$ bands starting on 2007 July 24.551 UT (2.33 hr after the burst; Table \[tab:obs\]). The data were reduced using the [gemini]{} package in IRAF. The combined $i$-band image centered on the location of is shown in Figure \[fig:fig4\]. An apparent extension is seen in the same location as the near-IR source.
We obtained follow-up near-IR observations of the GRB with Persson’s Auxiliary Nasmyth Infrared Camera (PANIC) mounted on the Magellan/Baade 6.5-m telescope on 2008 November 18.15 UT in the $K_s$ band (Table \[tab:obs\]). The individual images were dark-subtracted, flat-fielded, and corrected for bad pixels and cosmic rays. We then created object masks, which were used to construct improved flat fields for a second round of reduction. The data were finally registered, shifted, and co-added. The resulting combined image is shown in Figure \[fig:fig1\], and clearly reveals that the point source visible in the NIRI images has faded away.
Similarly, late-time optical $gi$-band observations were obtained with the Low Dispersion Survey Spectrograph (LDSS3) mounted on the Magellan/Clay 6.5-m telescope on 2008 December 7.14 UT (Table \[tab:obs\]). The data were reduced using standard procedures in IRAF. The resulting $i$-band image is shown in Figure \[fig:fig4\]. As in the case of the late-time near-IR observations, the extension seen in the early GMOS observations is no longer detected.
To confirm the fading afterglow and to obtain accurate photometry and astrometry we perform digital image subtraction on the NIRI and PANIC $K_s$-band images and on the GMOS and LDSS3 $gi$ band images with the ISIS package [@ala00], which accounts for variations in the stellar point-spread function (PSF). We adopt the PANIC and LDSS3 images as templates with zero afterglow contribution since they were obtained about 1.3 yr after the burst. The resulting residual $K_s$- and $i$-band images are shown in Figures \[fig:fig1\] and \[fig:fig4\], respectively, and clearly demonstrate that the point source coincident with the host galaxy has faded away. We therefore conclude that this source is the afterglow of . We additionally performed image subtraction on the two NIRI $K_s$-band observations and find that the source has faded between mid-epochs of 2.832 and 3.696 hr after the burst at a $4\sigma$ confidence level (Figure \[fig:fig1\]). No residual is detected in the subtracted $g$-band image.
Absolute and Differential Astrometry
------------------------------------
We determine the absolute position of the afterglow from the NIRI and GMOS residual images using the [SExtractor]{} software package. Since our NIRI images do not contain any 2MASS stars, we first perform an astrometric tie of the GMOS $i$-band image relative to USNO-B (using 13 common objects with a resulting rms of $0.15''$) and then tie the NIRI astrometry to the $i$-band image (using 20 common objects with a combined total rms of $0.18''$). The optical afterglow is located at $\alpha$=, $\delta$=, while the near-IR afterglow position is $\alpha$=, $\delta$= (J2000). These positions are consistent within the uncertainty of the astrometric tie. The optical/near-IR afterglow is offset by about $0.5''$ and $0.9''$ relative to the X-ray positions from @ebp+09 and @but07, which have $90\%$ containment errors of $1.6''$ and $1.7''$, respectively.
The detection of the optical/near-IR afterglow makes the association of with the previously-proposed host galaxy secure [@ber09], and allows us to precisely measure the offset between the GRB and host center. We perform differential astrometry on the NIRI images and find that the offset is $0.34\pm 0.01''$ east and $0.75\pm 0.01''$ south of the host center, corresponding to a radial offset of $0.82\pm 0.01''$. The uncertainty reflects the centroiding accuracy of both the afterglow and host, which we determine using [SExtractor]{}. At a redshift of $z=0.4571$ the scale is 5.785 kpc arcsec$^{-1}$, and the offset is therefore $4.76\pm 0.06$ kpc. This is similar to the offsets measured for previous short GRBs with optical afterglows [@bpc+05; @ffp+05; @sbk+06; @amc+09]. The location of the burst in the late-time PANIC and LDSS3 images does not exhibit any excess emission; see Figure \[fig:fig2\].
Photometry
----------
Photometry of the afterglow was performed on all residual images using photometric standard stars that were observed in conjunction with the PANIC and LDSS3 observations. We find that the afterglow had $K_s=19.59\pm 0.16$ mag and $K_s=19.64\pm 0.17$ mag in the first and second NIRI epochs, respectively, and $i=23.79\pm 0.07$ mag in the GMOS observation. The $3\sigma$ limit on the $g$-band magnitude is $g\gtrsim 23.5$ mag, determined by placing $\sim 10^3$ random apertures on the residual image and using the width of the resulting Gaussian flux distribution as $1\sigma$. The near-IR magnitudes are quoted in the Vega system, while the optical magnitudes are given in the AB system. Since the $g$-band limit is shallower than the detected $i$-band magnitude, it provides no meaningful constraints on the properties of the afterglow.
The observed $K_s$- and $i$-band magnitudes correspond to fluxes of $9.3\pm 1.5$ $\mu$Jy, $8.9\pm 1.5$ $\mu$Jy, and $1.1\pm 0.1$ $\mu$Jy, respectively. We stress that the uncertainty in the flux of the near-IR afterglow is dominated by the convolution with the PANIC image, which was obtained under worse seeing conditions than the NIRI images. To assess the statistical uncertainty in the afterglow flux we note that a stellar point source with $K_s=19.62$ mag, identical to the afterglow brightness, located near the afterglow position has a $1\sigma$ uncertainty of $0.03$ mag in the NIRI images. This greater depth in the NIRI data allows us to detect a significant fading between the two NIRI observations despite an uncertainty of $0.16$ mag relative to the PANIC observation.
We note that the afterglow is also clearly detected in the $J$- and $H$-band images from NIRI. However, due to the lack of late-time template images we cannot robustly measure its brightness in these bands. Still, a color-composite image reveals that the afterglow is redder in the near-IR bands than the rest of the host galaxy; see Figure \[fig:fig3\].
Afterglow/Mini-Supernova Properties {#sec:ag}
===================================
A comparison of our afterglow near-IR flux measurements to contemporaneous limits from UKIRT observations by @ktr+09 reveals that the afterglow is about a factor of three times fainter than the UKIRT upper limits. These authors also find a limit on the optical emission at 0.93 d after the burst of $F_{\nu,i}\lesssim 0.3$ $\mu$Jy. A comparison to our detected $i$-band flux at 0.12 d indicates that the afterglow temporal decay index is $\alpha<-0.6$ ($F_\nu\propto t^\alpha$), typical of GRB afterglows.
On the other hand, a comparison of our contemporaneous $K_s$- and $i$-band fluxes reveals an unusually steep spectral index, $\beta=-2.0\pm 0.2$ ($F_\nu\propto \nu^\beta$). Typically we expect $\beta\approx -0.6$ to $-1.2$ for a wide range of electron power law indices and values of the synchrotron cooling frequency [@spn98].
The unusually red afterglow can be explained in two ways. First, the optical emission may be suppressed by extinction within the host galaxy. To reconcile the observed fluxes with a typical spectral index of $\beta\approx -0.6$ requires $E(i-K_s)_{\rm obs}\approx 1$ mag, or a rest-frame $A_V^{\rm host}\approx 2$ mag for a Milky Way extinction curve. Such a large extinction seems unlikely given the location of the afterglow at the edge of the host galaxy. However, a large average value of $E(B-V)\approx 1.2$ mag (i.e., $A_V\approx 4$ mag) has been inferred for the host galaxy based on its ratio of H$\gamma$ and H$\beta$ emission lines [@ktr+09], indicating that extinction may indeed play a role in suppressing the optical emission.
We further investigate this possibility by comparison to the X-ray afterglow brightness at the time of the optical observations, $F_{\nu,X}(1\,{\rm keV})\approx 0.04$ $\mu$Jy [@gcnr74]. This leads to an optical to X-ray spectral index of $\beta_{\rm O,X}\approx
-0.5$, which marginally qualifies as a “dark burst” [@jhf+04; @ckh+09]. On the other hand, the near-IR to X-ray spectral index, $\beta_{\rm NIR,X}\approx -0.7$, is consistent with a typical afterglow. Thus, the comparison of the optical/near-IR and X-ray afterglow emission is consistent with a standard afterglow origin and significant dust extinction. We note that @ktr+09 find excess absorption in the early X-ray data, but attribute this result to rapid variations in the X-ray flux and spectral hardness. In light of the possible significant dust extinction, the excess photoelectric absorption may indeed be real.
An alternative explanation is that the near-IR flux is dominated by a different source of emission than the afterglow. In particular, in the context of a compact object merger, the emission may be due to the decay of radioactive material synthesized in a sub-relativistic outflow, the so-called Li-Paczynski mini-supernova (@lp98 [@rr02]; see also @ktr+09). In the formulation of @lp98, the emission from such a mini-SN is described by a peak luminosity ($L_p$): $$L_p\approx 2\times 10^{44}\,f_{-3} M_{-2}^{1/2} (3\beta)^{1/2}
(\kappa/\kappa_e)^{-1/2}\,\,\,\,{\rm erg\,\,\, s^{-1}},
\label{eqn:lp}$$ a peak effective temperature ($T_{{\rm eff},p}$): $$T_{{\rm eff},p}\approx 2.5\times 10^4\,f_{-3}^{1/4} M_{-2}^{-1/8}
(3\beta)^{-1/8} (\kappa/\kappa_e)^{-3/8}\,\,\,\,{\rm K},
\label{eqn:Teff}$$ and a peak time ($t_p$): $$t_p\approx 1\,M_{-2}^{1/2} (3\beta)^{-1/2} (\kappa/\kappa_e)^{1/2}
\,\,\,\,{\rm d}
\label{eqn:tp}$$ where $f$ is the fraction of rest mass energy released by the radioactivity, $M$ is the ejecta mass in units of M$_\odot$, $\beta\equiv v/c$ is the ejecta velocity, $\kappa$ is the average opacity, $\kappa_e\approx 0.2$ cm$^2$ g$^{-1}$ is the electron scattering opacity, and we use the notation $X\equiv 10^nX_{n}$.
For our detected source we use the near-IR luminosity and observed time as proxies for $L_p$ and $t_p$, respectively, leading to $L_p\approx 10^{43}$ erg s$^{-1}$ and $t_p\approx 0.1$ d. Using the constraint that $3\beta\lesssim 1$ and assuming that $\kappa=
\kappa_e$, we find from Equation \[eqn:tp\] that $M_{-2}\lesssim
10^{-2}$ (i.e., $M\lesssim 10^{-4}$ M$_\odot$). In conjunction with Equation \[eqn:lp\] this provides a lower limit of $f_{-3}\gtrsim
5$. The resulting lower limit on the effective temperature is $T_{{\rm eff},p}\gtrsim 7\times 10^4$ K, corresponding to a peak in the UV rather than in the near-IR. The apparent discrepancy in the spectral peak may be viewed as an indication that the observed emission is not due to a mini-SN. However, we note that Equations \[eqn:lp\]-\[eqn:tp\] correspond to the case of a power law decay model with an assumed contribution from elements with a wide range of decay timescales. An exponential decay model, in which a single element dominates the release of energy, may lead to distinctly different luminosity and evolution [@lp98].
To summarize, the unusually red optical/near-IR counterpart of , can be explained as a typical afterglow with significant dust extinction, $A_V^{\rm host}\approx 2$ mag. The alternative explanation of a mini-SN leads to an expected peak in the UV, but this may suggest that the mini-SN models should be revised.
Discussion and Conclusions
==========================
Optical afterglow emission has now been detected from 16 short GRBs, including . In Figure \[fig:ag\] we plot the flux of each optical afterglow at the time of its discovery. For we show both the optical and near-IR fluxes, as well as the expected $i$-band flux extrapolated from the $K_s$-band using a typical spectral index of $\beta=-0.6$. While the observed $i$-band flux is one of the faintest to date, the near-IR flux indicates that the afterglow of is actually one of the brightest at the time of its discovery. Indeed, only the optical afterglows of GRBs 050724, 060313, 070714, and 090510 were brighter, and of these only the afterglow of GRB050724 was discovered on a comparable timescale; the optical afterglows of GRBs 060313, 070714, and 090510 were all discovered $\lesssim 20$ min after the burst. On a timescale of 1 hr to 1 d after the burst, the optical afterglows of short GRBs generally have fluxes of $\sim 1-10$ $\mu$Jy, about two orders of magnitude lower than the typical brightness of long GRB afterglows (e.g, @kkz+08 [@ckh+09]). From the existing distribution we conclude that the detection of a substantial fraction of short GRB afterglows requires optical/near-IR observations to $m_{\rm AB}\sim 24$ mag within $\sim 0.5$ d.
The unusually red afterglow of can be explained with a substantial rest-frame dust extinction, $A_V^{\rm host}\approx 2$ mag. This value is larger than the typical extinction inferred for most long-duration GRBs, $A_V\sim 0.1-1$ mag (e.g., @pcb+09), and indeed the optical to X-ray spectral index, $\beta_{\rm OX}\approx
-0.5$, marginally qualifies as a dark burst [@jhf+04; @ckh+09]. Since the GRB is located on the edge of its host galaxy, it is likely that the extinction arises in the local environment of the burst. This implies that the progenitor system was not ejected from the host galaxy into the halo or intergalactic medium. Instead, the large extinction may point to an explosion site within a star forming region, or alternatively that the progenitor system itself produced the dust (for example, a binary system with an evolved AGB star). The possibility that some short GRBs are obscured by dust has important ramifications for the nature of the progenitors, and can also serve to localize the bursts to the galactic disk environments. Thus, rapid and deep near-IR observations are of crucial importance.
Alternatively, in the context of a compact object merger model, the near-IR emission may arise from radioactive decay in a sub-relativistic outflow produced during the merger process – a mini-SN. In this scenario, we find that the required ejected mass is $M\lesssim 10^{-4}$ M$_\odot$, with a radioactive energy release efficiency of $f\gtrsim 5\times 10^{-3}$. We note, however, that in the standard formulation the spectral peak at this time is expected to be in the UV rather than in the near-IR. This may indicate that the detected source is completely due to afterglow emission, or that the mini-SN models need to be revised.
We thank Alicia Soderberg for assistance with the digital image subtraction. This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. Observations were also obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil) and CONICET (Argentina)
, C. 2000, , 144, 363
, E. 2009, , 690, 231
, E., & [Kelson]{}, D. 2009, GRB Coordinates Network, 8934, 1
, E., et al. 2005, , 438, 988
, J. S. 2007, GRB Coordinates Network, 6658, 1
, J. S., et al. 2006, , 638, 354
, N. R. 2007, , 133, 1027
, S. B., [Cobb]{}, B. E., [Perley]{}, D. A., & [Bloom]{}, J. S. 2009a, GRB Coordinates Network, 8933, 1
, S. B., et al. 2009b, , 693, 1484
, S. B., [Rau]{}, A., [Berger]{}, E., [Price]{}, P. A., & [Cucchiara]{}, A. 2007, GRB Coordinates Network, 6664, 1
, S., [Piranomonte]{}, S., [Vergani]{}, S. D., [D’Avanzo]{}, P., & [Stella]{}, L. 2007, GRB Coordinates Network, 6666, 1
, P., et al. 2009, , 498, 711
, A., et al. 2006, , 648, L83
, P. A., et al. 2009, , 397, 1177
, D. B., et al. 2005, , 437, 845
, J. F., et al. 2009, , 698, 1620
, J., et al. 2005a, , 630, L117
, J., et al. 2005b, , 437, 859
, P., [Hjorth]{}, J., [Fynbo]{}, J. P. U., [Watson]{}, D., [Pedersen]{}, K., [Bj[ö]{}rnsson]{}, G., & [Gorosabel]{}, J. 2004, , 617, L21
, D. A., et al. 2008, ArXiv e-prints
, D., et al. 2009, ArXiv e-prints
, N. P. M., & [Hoversten]{}, E. A. 2009, GRB Coordinates Network, 9342, 1
, A. J., [Tanvir]{}, N. R., & [Davis]{}, C. 2007, GRB Coordinates Network, 6662, 1
, A. J., et al. 2006, , 648, L9
, E. M., et al. 2009, ArXiv e-prints
, L.-X., & [Paczy[ń]{}ski]{}, B. 1998, , 507, L59
, D., et al. 2008, GRB Coordinates Network, 8190, 1
, D. A., et al. 2009a, ArXiv e-prints
, D. A., et al. 2009b, , 696, 1871
, D. A., [Thoene]{}, C. C., [Cooke]{}, J., [Bloom]{}, J. S., & [Barton]{}, E. 2007, GRB Coordinates Network, 6739, 1
, S., et al. 2008, , 491, 183
, P. W. A., et al. 2006, , 651, 985
, S., & [Ramirez-Ruiz]{}, E. 2002, , 336, L7
, R., [Piran]{}, T., & [Narayan]{}, R. 1998, , 497, L17
, A. M., et al. 2006, , 650, 261
, G., et al. 2007, , 474, 827
, H., [Barthelmy]{}, S. D., [Parsons]{}, A., [Page]{}, K. L., [de Pasquale]{}, M., & [Schady]{}, P. 2007, GCN Report, 74, 2
[lcccccccc]{} 0.05in 2007 July 24.549 & 0.094 & Gemini-N & GMOS & $g$ & $2\times 180$ & $0.71$ & $\gtrsim 23.5$ & $\lesssim 1.5$\
2007 July 24.551 & 0.097 & Gemini-N & GMOS & $i$ & $2\times 180$ & $0.53$ & $23.79\pm 0.07$ & $1.1\pm 0.1$\
2007 July 24.572 & 0.118 & Gemini-N & NIRI & $K_s$ & $15\times 60$ & $0.35$ & $19.59\pm 0.16$ & $9.3\pm 1.5$\
2007 July 24.585 & 0.131 & Gemini-N & NIRI & $J$ & $15\times 60$ & $0.45$ & $^a$ & $^a$\
2007 July 24.596 & 0.142 & Gemini-N & NIRI & $H$ & $15\times 30$ & $0.46$ & $^a$ & $^a$\
2007 July 24.608 & 0.154 & Gemini-N & NIRI & $K_s$ & $15\times 60$ & $0.35$ & $19.64\pm 0.17$ & $8.9\pm 1.5$\
2008 Nov 18.17 & 481.6 & Magellan & PANIC & $K_s$ & $54\times 20$ & $0.47$ & $^b$ & $^b$\
2008 Dec 7.13 & 500.6 & Magellan & LDSS3 & $g$ & $2\times 240$ & $0.94$ & $^b$ & $^b$\
2008 Dec 7.14 & 500.6 & Magellan & LDSS3 & $i$ & $3\times 120$ & $0.58$ & $^b$ & $^b$
|
---
abstract: |
**Introduction** The *tau statistic* uses geolocation and, usually, symptom onset time to assess global spatiotemporal clustering from epidemiological data. We explore how computation and analysis methods may bias estimates.
**Methods** Following a previous review of the statistic, we tested several aspects that could affect graphical hypothesis testing of clustering or bias clustering range estimates, by comparison with a baseline analysis of an open access measles dataset: these aspects included bootstrap sampling method and confidence interval (CI) type. Correct practice of hypothesis testing of no clustering and clustering range estimation of the tau statistic are explained.
**Results** Our re-analysis of the dataset found evidence against no spatiotemporal clustering $p\textnormal{-value} \in [0,0\textnormal{\textperiodcentered}014]$ (global envelope test). We developed a tau-specific modification of the Loh & Stein bootstrap sampling method, whose more precise bootstrapped tau estimates led to the clustering endpoint estimate being 20% higher than previously published (36[]{.nodecor}0m, 95% bias-corrected and accelerated (BCa) CI (14[]{.nodecor}9, 46[]{.nodecor}6), vs 30m). The estimated bias reduction led to an increase in the clustering area of elevated disease odds by 44%. We argue that the BCa CI is essential for asymmetric sample bootstrap distributions of tau estimates.
**Discussion** Bootstrap sampling method and CI type can bias the clustering range estimated. Moderate radial bias to the range estimate are more than doubled when considered on the areal scale, which public health resources are proportional to. We advocate proper implementation of this useful statistic, ultimately to reduce inaccuracies in control policy decisions made during disease clustering analysis.
address:
- 'MathSys CDT, University of Warwick, UK'
- 'Zeeman Institute (SBIDER), School of Life Sciences and Mathematics Institute, University of Warwick, UK'
- 'Big Data Institute, Li Ka Shing Centre for Health Information and Discovery, University of Oxford, UK'
- 'London School of Hygiene & Tropical Medicine, UK'
author:
- 'Timothy M Pollington[{height="8pt"}](https://orcid.org/0000-0002-9688-5960)'
- 'Michael J Tildesley[{height="8pt"}](https://orcid.org/0000-0002-6875-7232)'
- 'T Déirdre Hollingsworth[{height="8pt"}](https://orcid.org/0000-0001-5962-4238)'
- 'Lloyd AC Chapman[{height="8pt"}](https://orcid.org/0000-0001-7727-7102)'
bibliography:
- 'sample.bib'
nocite:
- '[@Myllymakitalk]'
- '[@January2017]'
- '[@Myllymaki2019]'
- '[@Myllymaki2019]'
title: Measuring spatiotemporal disease clustering with the tau statistic
---
second order dependence ,pointwise confidence interval ,bias corrected accelerated BCa ,percentile confidence interval ,spatial bootstrap ,graphical hypothesis test
Introduction {#S:Intro}
============
[^1]Assessing if *spatiotemporal clustering* is present and measuring its magnitude and range is informative for epidemiologists working to control infectious diseases. The *tau statistic*[^2] is more appropriate than most spatiotemporal statistics for this task as it specifically measures spatiotemporal rather than just spatial clustering, produces non-parametric estimates without previous process beliefs and offers a relative magnitude in the difference of risk, rate or odds versus the background level, which the K function is unable to do [@Lessler2016] [@Pollingtonreview][@Gabriel2009]. This study is motivated by a review of its use that found that its current implementation inflates type I errors (incorrectly rejecting a true null hypothesis) when testing for clustering, and may bias estimates of the range of clustering [@Pollingtonreview]. We investigate the role of these by analysing a well-studied open dataset containing variables with the necessary geolocation and times of onset of symptoms. This dataset represents a *spatially discrete process* since infection is only recorded and can only occur at discrete household locations so the (statistical) support is not spatially continuous [@Diggle2010].
We follow an ordered approach: testing for clustering (§\[S:hypothesistesting\]) and then, conditional on finding evidence against no clustering, estimating the clustering range $\hat{D}$ (§\[S:parameterest\]) separately. We also provide the first precision estimate for $\hat{D}$ (Fig. Graphical abstract). This approach is contrary to the current methods applied to the tau statistic and similar statistics [@Pollingtonreview], which incorrectly combine graphical hypothesis testing for clustering and estimation of the clustering range. We hope these improved methods will contribute to the proper application of this burgeoning statistic.
Its context within this issue
-----------------------------
Myllimäki’s presentation on *global envelope testing* inspired our correction of pointwise confidence intervals for graphical hypothesis testing (Myllymäki, 2019b) (§\[S:hypothesistesting\]); while Pebesma’s talk stressing the need for *reproducibility* [@Pebesmatalk] encouraged us to provide our analysis code in `R Markdown` with the random seeds recorded (\[sec:code\]). In addition to modellers or epidemiologists working on real-time outbreak analysis or post-study analysis, we hope statisticians are inspired to apply this statistic to spatiotemporal branching processes in new fields.
The tau statistic {#S:thetaustatistic}
=================
The tau statistic is a non-parametric global clustering statistic which takes a disease frequency measure (risk, odds or rate) within a certain annulus around an average case and compares it to the background measure (at any distance) [@Salje2012; @Lessler2016; @Pollingtonreview]. It measures the tendency of case pairs to spatially cluster while implicitly accounting for their transmission relation temporally, making it a *spatiotemporal* statistic.
Tau statistic (odds ratio estimator)
------------------------------------
We describe the most common tau estimator $\hat{\tau}_{\textnormal{odds}}$, sourced from a more detailed description and commentary including other tau estimators and a new rate estimator @Pollingtonreview [@Lessler2016].
*Subsection abridged from @Pollingtonreview*\
The distance form of the tau statistic $\hat{\tau}_{\textnormal{odds}}$ is the ratio of the odds $\theta(d_1,d_2)$ of finding any case $j$ that is related to any other case $i$, within a half-closed[^3] annulus $[d_1, d_2)$ around case $i$, to the odds $\theta(0,\infty)$ of finding any case $j$ related to any case $i$ at any distance separation ($d_{ij}\geq 0$) for $n$ total cases (Equation \[eq:tauodds\] & Fig. \[fig:annulus\]). The main computation of Equation \[eq:tauodds\] is effectively a double sum over pairs’ ‘relatedness’ indicator functions $\mathds{1}(\cdot)$. $\tau(d_1,d_2)$ is then evaluated over a distance band set . Sometimes an expanding disc is described by setting $d_1 = 0$, relabelling $d=d_2$ to give $\tau(d)$ instead. $$\label{eq:tauodds}
\begin{split}
\hat{\tau}_{\textnormal{odds}}(d_1, d_2) &:= \frac{\hat{\theta}(d_1, d_2)}{\hat{\theta}(0,\infty)}\\
\textnormal{ where }&\hat{\theta}(d_1, d_2) = \frac{\sum_{i=1}^n\sum_{j=1, j\neq i}^n\mathds{1}(z_{ij} = 1, d_1\leq d_{ij}<d_2)}{\sum_{i=1}^n\sum_{j=1, j\neq i}^n\mathds{1}(z_{ij} = 0, d_1\leq d_{ij}<d_2)}
\end{split}$$ Tau values signify either the presence of spatiotemporal clustering ($\tau>1$), no clustering ($\tau = 1$) or inhibition ($\tau < 1$). The odds estimate $\hat{\theta}$ in Equation \[eq:tauodds\] is the ratio of the number of related case pairs within $[d_1, d_2)$, versus the number of unrelated case pairs within $[d_1, d_2)$. The relatedness of a case pair $z_{ij}$ is commonly determined using temporal information (e.g. onset time difference of cases $i,j$ i.e. $t_j-t_i$) [@Pollingtonreview]. The *serial interval* is the period between the onset times of symptoms in the infector $t_i$ and their infectee $t_j$. Typically temporal relation is defined when case onset times are within a single serial interval of each other. It can be calculated in the `IDSpatialStats` R package [@LesslerGiles].
\* \* \*
In the following sections (§\[S:methods\]-\[S:resanddisc\]) we provide a descriptive analysis of the dataset, before systematically testing several aspects of the tau statistic’s implementation and their impact on the estimated clustering range and bias in this estimate.
Methods {#S:methods}
=======
The dataset and computation methods
-----------------------------------
The dataset is sourced from the `surveillance` R package under a GPL-2 licence [@Meyer2017], as provided by Niels Becker via [@Neal2004] from a re-analysis [@Oesterle1992] of the original study [@Pfeilsticker1863]. We have checked it for errors and inconsistencies. Like Lessler et al.’s analysis (unpublished code shared with us) we take the start of the prodromal period as the date of onset of symptoms. The baseline result of their analysis of the same dataset has been reproduced (Fig. \[fig:taurepro\]) and is very similar, but not completely matching as their random number generator seed was unknown. Using their interpretation of the graph, spatiotemporal clustering is reported up to 30m [@Lessler2016].
The `spatstat` library [@spatstat] in R was used for useful spatial functions, `purrr` for resampling [@purrr], `fields` for image plots [@fields] and `latex2exp` & `scales` for graph notation [@latex2exp; @scales] and the code of ‘January’ (2017) for figure labelling. Computations were run in R using RStudio [@R; @RStudio]. The `IDSpatialStats::get.tau()` and `get.tau.bootstrap()` functions were optimised by re-implementing them in C, which sped up $\tau_{\textnormal{odds}}$ calculations by $\sim$29 times [@Pollington2019]. The associated code can be accessed through this !GitHub link TBA. (In the meantime please download the repo from https://warwick.ac.uk/fac/sci/mathsys/people/students/2015intake/ pollington\_tim/measles-master.zip Make sure the underscore between “pollington” and “tim” isn’t missed). We used @Lessler2016 distance band set throughout, i.e. a mixture of non-overlapping and overlapping bands: $\underline{\Delta} = ${\[0, 10\], \[0, 12\], \[0,14\], …, \[0, 50\], \[2, 52\], \[4, 54\], …, \[74, 124\]}.
Our approach to hypothesis testing and parameter estimation
-----------------------------------------------------------
Our graphical hypothesis test (§\[S:hypothesistesting\]) and parameter estimation (§\[S:parameterest\]) methods (Fig. Graphical abstract) are in contrast to many reviewed papers[^4] (using the tau statistic or similar statistic) which incorrectly used an *envelope*[^5] about the point estimate constructed from piecewise confidence intervals which amounts to multiple hypothesis testing and inflates type I errors (Fig. \[fig:incorrect\]b), and estimated the clustering endpoint $D$ as the distance at which the lower bound of the first piecewise percentile confidence interval that is above $\tau =1$ touches $\tau =1$ (Fig. \[fig:incorrect\]a) [@Pollingtonreview].
Graphical hypothesis test of no clustering {#S:hypothesistesting}
------------------------------------------
Instead we construct a *global envelope* around the distribution of the null hypothesis ($H_0$: $\tau=1$, no spatiotemporal clustering). This is generated by randomly permuting the time marks $t_i$ of the data points $X_i$ = ($\textnormal{x-coordinate}_i$, $\textnormal{y-coordinate}_i$, $\textnormal{onset time}_i$) to scramble any spatiotemporal clustering present and simulate what $\hat{\tau}$ would be under $H_0$. We assess if a subset of distance bands $\underline{\delta}$ exists (as singular or disjoint regions) where the tau point estimate $\hat{\tau}(d)$ is ever above/below the upper/lower bound of this *null envelope*, respectively, anywhere in the distance band set $\underline{\Delta}$, using the `GET` R package (Myllymäki et al., 2019a) (Fig. Graphical abstract). This particular global envelope is an extreme rank type “defined as the minimum of pointwise ranks” with a 95% significance level and extreme rank length p-value interval (note this is a range, not a single value) (Myllymäki et al., 2019a). The test is two-tailed (alternative hypothesis $H_1:\tau\neq 1$) which is necessary as only when the graph is plotted is the presence of clustering or inhibition known. We compute 2,500 bootstrap tau simulations for an optimal test [@Myllymaki2017]. For bootstrap tau calculations containing infinite values we repeat the estimate as `GET` can only accept finite values.
Parameter estimation of the clustering range {#S:parameterest}
--------------------------------------------
If hypothesis testing establishes that spatiotemporal clustering is present within the set of distance bands $\underline{\Delta}$ (§\[S:hypothesistesting\]), it is then sensible to estimate the endpoint of spatiotemporal clustering $D$ for the clustering range $[d_1=0 \textnormal{(assumed)}, d_2=D)$ where the point estimate intercepts $\tau = 1$, i.e. $D:= \{d:\hat{\tau}(d)=1\}$; due to finite distance bands we interpolate between the last tau value above one and first value below one, to obtain $\hat{D}$.
To obtain its uncertainty as a single confidence interval we use bootstrapped tau estimates $\hat{\underline{\tau}}^*$: for each bootstrapped simulation (that represents a connected line of simulated tau estimates for increasing $d$ i.e. $\{\tau^*(d_1,d_2):[d_1,d_2)\in\underline{\Delta}\}$), we record those that originate from above $\tau = 1$ and then intersect $\tau = 1$ at some greater distance $D$, i.e. those for which there exists $D$ satisfying $\hat{\tau}^*(D)=1$. We then take this horizontal set of values $\underline{D}$ and can obtain a confidence interval to describe the uncertainty in $\hat{D}$ (Fig. Graphical abstract). We experiment with different numbers of bootstrap estimates $N$, spatial bootstrap methods, confidence interval construction and distance band sets.
However caution is needed as these simulations are not a random sample of the population of simulations, which is an important prerequisite for confidence interval construction; as we selectively choose those that cross $\tau = 1$ from above and ignore those that start at or below $\tau=1$, or above it but never reach $\tau = 1$. Computing confidence intervals at a 95% confidence level on any random sample with a small 5% dropout shows that the effective confidence level can reduce substantially [@Gorard2014]. This selection bias is also $\underline{\Delta}$ dependent since if we choose a large enough $\underline{\Delta}$, we may find that simulations that start above $\tau=1$ eventually cross $\tau = 1$ and then contribute to the confidence interval.
Although we cannot account for this bias, we report the proportion of simulations that construct the confidence intervals and extend the distance range as computation time permits, to limit this bias.
If inhibition is present at greater distances we ignore estimating its range as it is not of interest. However if the reader wished, it would involve a similar algorithm for assessing clustering at shorter distances, but instead one should capture simulation lines that exit the global envelope lower bound into $\tau < 1$ values for increasing $d$.
### Number of bootstrap estimates {#S:nbootstrapmethod}
The @Lessler2016 measles analysis used only $N=$ 100 bootstrap samples. We repeated their *resampled-index* analysis with $N=$ 2,500 samples which is more than sufficient for a typical bootstrap sample [@Efron1998].
### Bootstrap sampling method {#S:bstrapmethod}
The tau statistic’s variance cannot be calculated analytically so we generate a non-parametric bootstrap distribution of tau estimates, $\hat{\underline{\tau}}^*$. We start with a dataset $\mathbf{X} = (X_1, \ldots, X_n)$ of n cases, where again $X_i$ = ($\textnormal{x-coordinate}_i$, $\textnormal{y-coordinate}_i$, $\textnormal{onset time}_i$). We resample the data’s indices $\underline{i} = \{1, \ldots, n\}$ repeatedly $n$ times (equal to the number of cases), according to the Uniform distribution and with replacement to produce a new empirical bootstrap sample of indices $\underline{i}^* = \{i_k^*\}_{k=1}^n$ and data $\mathbf{X}^*$; $\underline{i}^*$ has the same length of $\underline{i}$ however it is bound to contain duplicated indices due to ‘with replacement’ sampling. We compute the tau odds estimator[^6] on each bootstrap sample $\mathbf{X}^*$ to get $N$ bootstrapped $\tau$ estimates $\hat{\underline{\tau}}^* = (\hat{\tau}^*_1, \ldots , \hat{\tau}^*_N)$. Through bootstrap theory, the sampling distribution $\hat{\underline{\tau}}^*$ may serve as a proxy for the actual distribution of $\tau$ on the data; and further the envelopes constructed from $\hat{\underline{\tau}}^*$ may approximate the envelope of $\tau$ on $\mathbf{X}$ [@Efron1979]. The bootstrap is non-parametric because it randomly resamples the data without imposing a distribution [@Loh2008]. We will call this method the *resampled-index* bootstrap. Loh critiques this “naive” sampling with replacement of the points of a spatial dataset to produce the bootstrap sample, because “the spatial dependence structure has to be preserved as much as possible” [@Loh2008] …“to reflect properties of the original process” [@LohStein2004]. Lessler et al use this method and additionally drop any pairwise evaluation from resampled indices that represent the same point to avoid ‘self comparisons’ [@Lessler2016], i.e. drop pairs $(i_p^*,i_q^*)$ for $p\neq q$.
Instead the *marked point bootstrap* is a fast, non-parametric method to obtain a bootstrap distribution of a second-order correlation function [@LohStein2004]. For a clustered process simulated by a Matérn process, the confidence intervals constructed using it had a higher empirical coverage than other methods, and computed faster.
In essence the difference with the marked point bootstrap is that the bootstrap estimate $\hat{\tau}^*$ is not computed from a resampled (smaller) dataset $\underline{X}^*$ which has some duplicated pairs from duplicate indices $\underline{i}^*$, but from a bootstrap sample of the points’ locally-evaluated $\tau$-functions $\underline{\tau}_i$ (Equation \[eq:lohandstein\]) that are formed for each $i^*\in \underline{i}^*$, but across all points $\underline{j}, j\neq i^*$; so at least each local $\tau_i$ covers all points in $\underline{X}$ unlike the resampled index: $$\label{eq:lohandstein}
\begin{split}
\hat{\tau}_{\textnormal{i}}(d_1, d_2) &:= \frac{\hat{\theta_i}(d_1, d_2)}{\hat{\theta_i}(0,\infty)}\\
\textnormal{ where }&\hat{\theta_i}(d_1, d_2) = \frac{\sum_{j=1, j\neq i}^n\mathds{1}(z_{ij} = 1, d_1\leq d_{ij}<d_2)}{\sum_{j=1, j\neq i}^n\mathds{1}(z_{ij} = 0, d_1\leq d_{ij}<d_2)}
\end{split}$$ These local $\tau$-functions are similar to an application of a spatial bootstrap to the K-function [@Baddeley2015], which like $\tau$ is a second-order correlation function. However we **do not recommend** this literal interpretation of Loh & Stein’s method of averaging localised $\tau$-functions for the tau statistic as we shall explain, but provide it for completeness (Equation \[eq:loh6\]).
$$\label{eq:loh6}
\tau^*(d_1,d_2) = \frac{1}{n}\mathlarger{\mathlarger{\mathlarger{\sum_{i^*}}}}\frac{\theta_{i^*}(d_1,d_2)}{\theta_{i^*}(0,\infty)} = \frac{1}{n}\mathlarger{\mathlarger{\mathlarger{\sum_{i^*}}}}\frac{\Big(\frac{m_{i*}(d_1,d_2,k=1)}{m_{i*}(d_1,d_2,k=0)}\Big)}{\Big(\frac{m_{i*}(k=1)}{m_{i*}(k=0)}\Big)}$$
Our *modified marked point bootstrap* (MMPB) method differs slightly to Loh & Stein’s: rather than forming the bootstrap estimate over local $\tau$-functions (Equation \[eq:loh6\]) we go deeper and compute the number of locally-related or locally-unrelated mark functions $\underline{m}_i(k)$, according to their Boolean time-relatedness $\underline{k}=\{0, 1\}$. This assumes that the mean of the bootstrap distribution of local mark functions asymptotically approximates the (global) tau statistic as Loh & Stein only provided experimental evidence to support this [@LohStein2004; @Loh2008].
The number of time-related cases ($\#\textnormal{related}$) within a distance $[d_1, d_2)$ around a case $i^*$ chosen in the bootstrap sample is: $$\label{eq:loh1}
\textnormal{\#related}^*(d_1,d_2,i^*) = m_{i^*}(k=1) \equiv \sum_{\underline{j},j\neq i^*}\mathds{1}(d_1\leq d_{i^*j} < d_2, z_{i^*j}=1)$$ and then an average is taken over the required $n$ cases in the bootstrap sample of indices $\underline{i}^*$: $$\label{eq:loh2}
\overline{\textnormal{\#related}^*(d_1,d_2)} = \frac{1}{n}\sum_{\underline{i}^*}\sum_{\underline{j},j\neq i^*}\mathds{1}(d_1\leq d_{i^*j} < d_2, z_{i^*j}=1)$$ and similar steps for time-unrelated cases yield: $$\label{eq:loh3}
\overline{\textnormal{\#unrelated}^*(d_1,d_2)} = m(k=0) \equiv \frac{1}{n}\sum_{\underline{i}^*}\sum_{\underline{j},j\neq i^*}\mathds{1}(d_1\leq d_{i^*j} < d_2, z_{i^*j}=0)$$ and finally the odds and tau statistic can be calculated as before: $$\label{eq:loh4}
\theta^*(d_1,d_2) = \frac{\overline{\textnormal{\#related}^*(d_1,d_2)}}{\overline{\textnormal{\#unrelated}^*(d_1,d_2)}} = \frac{\sum_{\underline{i}^*}\sum_{\underline{j},j\neq i^*}\mathds{1}(d_1\leq d_{i^*j} < d_2, z_{i^*j}=1)}{\sum_{\underline{i}^*}\sum_{\underline{j},j\neq i^*}\mathds{1}(d_1\leq d_{i^*j} < d_2, z_{i^*j}=0)}$$ $$\label{eq:loh5}
\tau_{\textnormal{odds}}^*(d_1,d_2) = \frac{\theta^*(d_1,d_2)}{\theta^*(0,\infty)}$$ It turns out that this schema (Equations \[eq:loh1\]-\[eq:loh5\]) is more robust than the original Loh & Stein method (Fig. \[fig:loh\]) when cases $i^*$ have no time-unrelated cases to pair with in their local distance band, i.e. $(m_{i^*}(d_1,d_2,k=0) = 0)$ causes infinite values for $\theta_{i*}(d_1,d_2)$, or `NaN` values when also $m_{i^*}(d_1,d_2,k=1) = 0$, under their approach; the MMBP simply characterises these null events as zeroes and their addition (in Equations \[eq:loh2\] & \[eq:loh3\] separately) does not affect the rest of the calculation. Potential remedies to Loh & Stein’s approach such as dropping these contributions or merging contiguous distance bands were fruitless—the envelope diverged greatly for short distances and was biased above for larger distances and only 773% of simulations contributed to its CI compared to 100% for MMPB (Fig. \[fig:loh\]). It appears dropping these inconvenient $i^*$ cases removes important spatial information which the tau bootstrap estimator in Equation \[eq:loh6\] is sensitive to.
Our method solves the numerical challenges but is not exactly the Loh & Stein method as we indirectly obtain the tau estimate via calculation of the bootstrapped odds $\theta^*$, so it is unclear if the validation of their results automatically transfers to our modified form.
### Confidence interval construction {#S:CItype}
Applying a percentile confidence interval to the sample bootstrap distribution $\underline{D}$ (previously defined in §\[S:parameterest\]) assumes it is symmetric which is not the case, especially at short distances (Fig. \[fig:bootstraphists\]) [@Carpenter2000].
Asymmetry is a property that BCa confidence intervals can cope with better than percentile confidence intervals—@Carpenter2000 compared a range of confidence interval methods and for non-parametric problems consistently found Efron’s bias-corrected and accelerated (BCa) method best due to its low theoretical coverage errors for approximating the exact confidence interval, i.e. $O(N)\sim N^{-1}$ under some assumptions, also known as “*second-order correct coverage*”, while a percentile confidence interval is first-order correct at best [@Efron1987]. The BCa method is an automatic algorithm that transforms a distribution of bootstrap calculations by normalisation to stabilise its variance so that a confidence interval can be constructed, then back-transforms it [@Efron1987]. It can be calculated using the `coxed` R package [@coxed].
### Distance band sets {#S:dband}
Although the tau statistic is non-unique as it is dependent on the distance band set chosen [@Pollingtonreview], the potential variation in $\tau$ estimates from this choice is of interest. From analysing cases’ pairwise distances we propose a reasonable distance band set, i.e. $\underline{\Delta}=$ {0-7, 7-15, 15-20, 20-25, 25-30, …, 195-200m}[^7] as a comparison to Lessler et al.’s overlapping set {0-10, 0-12, 0-14, …, 0-50, 2-52, 4-54, …, 74-124m} and test these using $N=$ 2,500 under the MMPB method.
Results & discussion {#S:resanddisc}
====================
Dataset description {#S:datasetdesc}
-------------------
We analyse an infectious disease dataset of measles in children from case households in Hagelloch, Germany in 1861 [@Meyer2017]. The epidemic over a small $\sim$280 x 240m$^2$ area lasted nearly three months and five distinct generations can be discerned from the epidemic curve (Fig. \[fig:Re\]). Out of the 197 under-14 year olds, 185 became infected, along with three teenagers, leaving 377 remaining teenagers and adults uninfected [@Neal2004]. There is weak signal of direct transmission between cases because some nearby households share similar plot colours (Fig. \[fig:STplot\]).
Graphical hypothesis tests: global envelopes vs pointwise CIs {#S:ResDiscGET}
-------------------------------------------------------------
There is moderately strong evidence against the hypothesis of no spatiotemporal clustering (p-value $\in [0, 0\text{\textperiodcentered}014]$) based on constructing the global envelope around $\tau=1$ under the null hypothesis (Fig. \[fig:get\]), and thus we conclude that the data $\underline{X}$ is inconsistent with the null model ($H_0: \tau=1$). So we turn to the alternative hypothesis, which is that there is clustering and/or inhibition. Fig. \[fig:get\] suggests there is clustering at short distances and inhibition at long distances.
Since previous papers[^8] used the incorrect pointwise CI approach to assess whether there was clustering for the tau statistic or similar statistic, for which a p-value is not available, it is not possible to compare our results to theirs.
Number of bootstrap simulations {#S:nbootstrapsres}
-------------------------------
The endpoint estimate for clustering is $\hat{D}=$360m with a (145, 580m) 95% percentile confidence interval over 100 bootstrapped simulations or (146, 585m) over 2,500 simulations (both CIs used 100% of simulations)—more bootstrapped simulations do not appear to affect the precision, indeed in this instance, $N=100$ had a slightly smaller length (Fig. \[fig:nbootstraps\]).
The point estimate $\hat{D}=$360m is only 20% higher than the baseline clustering range (30m). However, for the first time uncertainty can be quoted with this value. It is likely that previous estimates derived via the improper method of finding the distance at which the lower bound of the central envelope (around $\hat{\tau}$) touches $\tau = 1$ underestimate this range. The plateauing shape of $\hat{\tau}(d)$ before it reaches $\tau = 1$ contributes to the increased imprecision in the estimate of $\hat{D}$. This highlights the utility of a human assessing the graph rather than rigidly using a $\tau = 1$ threshold since for control purposes it is likely that control over a 60m radius around an average case would have the biggest gains over its first 15 metres with diminishing returns with the non-linear increase in area for radii 15-60m (Fig. \[fig:nbootstraps\]). The indifference to the number of bootstrap estimates $N$ may be due to the low number of cases ($n=188$), so that $N=100$ bootstrap samples $\hat{\underline{\tau}}^*$ can adequately represent the data $\underline{X}$. We conjecture that a rule-of-thumb for determining the number of bootstrap samples $N$ is to match it to the number of cases $n$ in the data; choosing more than this may be unnecessary. However, this requires validation with a larger dataset.
Bootstrap sampling: modified marked point vs resampled-index
------------------------------------------------------------
Using the modified marked point bootstrap (MMPB) (§\[S:bstrapmethod\]) yields a narrower envelope than the resampled-index bootstrap, leading to a 95% BCa CI for $\hat{D}$ of (149, 466m); both CIs used 100% of simulations (Fig. \[fig:nbootstrapsv2\]).
If the tau point estimate had been shallower near the $\tau=1$ intercept then the range of spatiotemporal clustering would be far larger and the benefit of MMPB more apparent. Given the reasons why this method is better (§\[S:bstrapmethod\]), we believe the resampled-index method will generally underestimate this range.
The MMPB outperforms the resampled-index bootstrap because the latter loses a lot of pair information from resampling indices and avoiding self-comparisons. This was checked empirically for the measles dataset: the tau point estimate was computed on 188 x 187 = 35,156 unique pairs (we ignore the fact that pairs are undirected i.e. $|t_j-t_i|=|t_i-t_j|$, without loss of generality). On average from 1,000 simulations, the resampled-index bootstrap sampled from 119 unique people, leading to 119 x 118 unique pairs evaluated or $\sim 39\text{\textperiodcentered}9$% of the original pairs. Of course many additional duplicate pairs are used in the resampled-index bootstrap but we are only interested in unique pair information that is retained. The MMPB only has 119 unique mark functions, but each of them is compared with the other 187 cases, leading to 633% of pairs being retained.
Confidence interval: BCa vs percentile {#S:bcaresults}
--------------------------------------
Histograms of the asymmetric distribution of $\underline{D} = \{D_i: \hat{\tau}^*_i(D_i)=1, i=1,\dots,N\}$ by number of bootstrapped samples indicate for both $N=$ 100 or 2,500 samples that a percentile confidence interval gives a less precise estimate; both CIs used 100% of simulations (Fig. \[fig:bootstraphists\]). The BCa method takes a few minutes extra to compute and provides slightly narrower confidence intervals than the original percentile confidence intervals (Fig. \[fig:bootstraphists\]). The resampled-index method appears to introduce positive skew (mean $>$ median) in $\underline{D}$ whereas MMPB with sufficient samples ($N=2500$) introduces a slight negative skew. MMPB reduces the bias$(\underline{\bar{D}},\hat{D})$ (between mean/median estimates of $\underline{D}$ and the point estimate $\hat{D}$) from $\sim$10m to $\sim$5m, or $\sim$17% of $\hat{D}$.
Distance bands
--------------
Overlapping distance band sets appear to produce $\hat{D}$ estimates with more variance 95% BCa CI(149, 466m) than non-overlapping sets CI(154, 261m) (Fig. \[fig:distband\]) but a clearer and smoother trend in tau with increasing distance; both CIs used 100% of simulations. The non-overlapping confidence interval also struggles to contain $\hat{D}$ (Fig. \[fig:distband\]) because the simulations are more erratic about $\tau = 1$, the distribution of $\underline{D}$ is strongly bi-modal which even the BCa technique cannot handle; increased volatility also results in multiple intercepts with $\tau = 1$, however for usability we prefer a single range of clustering, given in this case by the overlapping $\underline{\Delta}$.
Public health importance of these results
-----------------------------------------
The 20% increase in the radial parameter $\hat{D}$ (§\[S:nbootstrapsres\]) from using the corrected parameter estimation algorithm (§\[S:parameterest\]) may not seem an important difference for public health interventions, but their time and cost is proportional to area and the areal increase is 44% since (assuming $d_1=0$) $\pi(1\text{\textperiodcentered}2\hat{D})^2/\pi \hat{D}^2=1\text{\textperiodcentered}44$.
Conclusion and recommendations for improved use {#S:concl}
===============================================
We have shown that for a measles dataset, clustering ranges estimated by the tau statistic can be biased—mostly by the bootstrap sampling method and to some degree the confidence interval type. Using a modified marked point bootstrap and BCa confidence intervals resulted in bias reductions equivalent to increasing the clustering area of elevated odds by 44%. We are keen to contribute these improvements in future versions of `IDSpatialStats` package. The results and explanations (given in §\[S:resanddisc\]) supports these recommendations:
- the modified marked point bootstrap should be used to simulate $\hat{\tau}$ instead of the resampled-index method that could lead to underestimation of the clustering range.
- BCa rather than percentile confidence intervals should obtain better coverage when the distribution of tau simulations $\hat{\underline{\tau}}^*$ is non-symmetric.
#### Tau statistic limitations
If geolocations do not spatially coincide with the infection event, then like childhood influenza that is commonly spread in schools as well as households, the signal of clustering is likely to be weakened. It is unclear how second-order correlation functions like the tau statistic and Ripley’s K function, originally founded in spatiotemporal point processes with continuous support in $\mathbb{R}^2$, behave for spatially discrete data [@Gabriel2009].
Distance band set choice $[d_1,d_2)$ within $\underline{\Delta}$ clearly affects the smoothness of the point estimate, $\hat{D}$ and its precision. A better understanding of how to choose distance bands for a given purpose is now needed. It is also not possible to tell the number of bootstrap samples $N$ required as a function of the number of cases $n$, until more studies are analysed; this is hampered by the lack of modern open access datasets containing geolocation and disease onset times because of (valid) privacy concerns. It is also unknown how the time-relatedness interval choice $[T_1,T_2]$ (where $z_{ij} = \mathds{1}((t_j-t_i)\in [T_1,T_2])$) biases the tau statistic through inclusion of extraneous co-primary or secondary cases—we will investigate these limitations in future work.
\* \* \*
We encourage the adoption of the statistical protocol described (Fig. Graphical abstract) to properly test for clustering, and if appropriate estimate its range. Control programmes are being informed by the tau statistic and applying these bias-reduction methods will improve its accuracy and future health decisions.
Acknowledgements {#S:ack}
================
TMP would like to thank:
- Justin Lessler and Henrik Salje for sharing their unpublished analysis code so we could reproduce their methods [@Lessler2016], and openly answering questions about their work via email or Skype.
- Peter Diggle (PJD) for highlighting an earlier spurious result and also a mistake in mixing up parameter estimation with hypothesis testing, and for reviewing the second draft.
- Mari Myllimäki for rapidly answering `GET` package questions.
- Shaun Truelove who replied to questions by email.
TMP, LACC & TDH gratefully acknowledge funding of the NTD Modelling Consortium by the Bill & Melinda Gates Foundation (BMGF) (grant number OPP1184344) and LACC acknowledges funding of the SPEAK India consortium by BMGF (grant number OPP1183986). Views, opinions, assumptions or any other information set out in this article should not be attributed to BMGF or any person connected with them.
TMP’s PhD and laptop for computations were supported by the Engineering & Physical Sciences Research Council, Medical Research Council and University of Warwick (grant number EP/L015374/1). TMP would like to thank Big Data Institute for hosting them during this analysis under TDH’s guidance.
All funders had no role in the study design, collection, analysis, interpretation of data, writing of the report, or decision to submit the manuscript for publication.
Competing interests
===================
All authors declare no competing interests.
Contributions: CRediT statement
===============================
**TMP:** Conceptualisation, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing - original draft & editing, Visualisation **MJT:** Conceptualisation, Writing - review & editing, Supervision **PJD:** Methodology, Validation (see §\[S:ack\]), Writing - review & editing **TDH:** Conceptualisation, Writing - review & editing, Supervision, Funding acquisition **LACC:** Conceptualisation, Software, Validation, Data curation, Writing - review & editing, Supervision.
Open access
===========
The analysis code are available from this Elsevier Data online repository using link (!TBA, pending acceptance).
This article is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives Works 4·0 International Licence (CCBY-NC-ND4·0). Anyone can copy and distribute this article unchanged and unedited but only for non-commercial purposes, provided the user gives credit by providing this article’s DOI and a link to the licence (creativecommons.org/licences/by-nc-nd/4·0). The use of this material by others does not imply endorsement by the authors.
Figures
=======
![A single distance band half-closed annulus of radii $[d_1, d_2)$ with another case $j$ in it, around an average case $i$ with distance separation $d_{ij}$.[]{data-label="fig:annulus"}](figs/annulus.pdf){width="0.5\linewidth"}
![The incorrect methods employed by most reviewed papers (see footnote \[footnote:prevpapers\]) are either ‘central envelopes’ (a, left), or ‘null envelopes’ (b, right) to test the hypothesis of clustering and estimate the range of clustering parameter $\hat{D}$ simultaneously [@Pollingtonreview]. The single red line $\tau = 1$ represents no spatiotemporal clustering. Grey lines indicate a) negative exponential lines with Normal noise to characterise a series of bootstrapped estimates $\hat{\tau}^*$ of a typical tau function, or b) a line at $\tau=1$ with Normal noise to represent simulations of $\tau = 1$ for null envelope construction; black lines mark out the envelope bounds. The blue line represents a simulated empirical tau point estimate $\hat{\tau}(d)$. We separate these into hypothesis testing and parameter estimation in $\S\ref{S:hypothesistesting}$ & $\S\ref{S:parameterest}$, respectively.[]{data-label="fig:incorrect"}](figs/centralnull.pdf){width="1.0\linewidth"}
![Epidemic curve of the 188 childhood measles cases in Hagelloch, Germany[]{data-label="fig:Re"}](figs/Re.pdf){width="0.99\linewidth"}
![Baseline result: a reasonable reproduction of Lessler et al [@Lessler2016 Fig. 4C]. Note that the end of the clustering range identified by Lessler et al uses their standard convention where the lower bound intersects $\tau = 1$ (145m or $\sim$15m) (however we do not endorse this). But as the midpoint of the distance band is shown (i.e. $\nicefrac{1}{2}(d_1+d_2)$), means \[0, 30m\] is the actual range that would be interpreted rather than “up to 15m” [@Lessler2016] (Lessler has kindly confirmed this by email communication on 26 August 2019). Note how both envelopes superimpose—validating our program against the established `IDSpatialStats` package.[]{data-label="fig:taurepro"}](figs/taureproduction.pdf){width="1.0\linewidth"}
![Spacetime points of cases’ locations with onset times as colour marks. It reveals multiple case households. There is a weak indication of cases in nearby households ($\sim$50m apart) having a similar colour (and thus onset) which may indicate direct transmission up to this distance. Cases jittered by the Uniform distribution up to 2m, separately in $x$ and $y$ dimensions.[]{data-label="fig:STplot"}](figs/STplot.pdf){width="1.0\linewidth"}
![Global envelope test, ‘extreme rank’ type, two-sided at 95% significance level using 2,500 simulations of the null hypothesis ($H_0: \text{no spatiotemporal clustering i.e.\ }\tau= 1$). Note there is a region where $\hat{\tau}$ just exits the global envelope lower bound (suggesting inhibition at far distances) as well as the obvious departure above the upper bound (suggesting clustering at close distances). We are confident that we are simulating $H_0$ because the median simulation stays close to $\tau = 1$ throughout. Distance band start points $d_1 = 0\enskip \forall d_2 \in$ \[0, 50m\], then$\forall d_2>$50m, $(d_2 - d_1) = 50$m.[]{data-label="fig:get"}](figs/get.pdf){width="1.0\linewidth"}
![Effect on $\hat{D}$ precision using resampled index bootstrap sampling. Both CIs used 100% of simulations. $\hat{D}=36\text{\textperiodcentered}0$m; $N=100$: 95% BCa CI(145, 580m); $N=2500$: CI(146, 585m). Distance band set:={0-10, 0-12, 0-14, …, 0-50, 2-52, 4-54, …, 74-124m}.[]{data-label="fig:nbootstraps"}](figs/nbstrap.pdf){width="1.0\linewidth"}
![Effect of bootstrap sampling method on $\hat{D}$ precision. Resampled index 95% BCa CI(147, 600m); modified marked point bootstrap CI(149, 466m); both CIs used 100% of simulations. Distance band set:={0-10, 0-12, 0-14, …, 0-50, 2-52, 4-54, …, 74-124m}.[]{data-label="fig:nbootstrapsv2"}](figs/nbstrapv2.pdf){width="1.0\linewidth"}
![Distribution of $\underline{D}$, by number of bootstrapped samples (N=100 top row or N=2500 bottom row) or by bootstrap sampling method (left = resampled-index, right = modified marked point(MMPB)). $\underline{D}$ is defined as the set of samples from the sampling distribution of values $\hat{D}$ i.e $\underline{D}=\{D_i:\hat{\tau}^*_i(D_i)=1,i=1,\dots,N\}$. Vertical dotted lines indicate the $\hat{\tau}$ point estimate (red), mean (green) and median (blue) of the bootstrapped tau estimates. The resampled-index both has positive skew as the mean estimate is greater than the median estimate, whereas the MMPB both has a negative skew. All bootstrap estimations have a negative bias with respect to mean or median summary measures versus the point estimate, of approximately $\sim$10m for the resampled-index or approximately $\sim$5m for the MMPB. The data points that constructed Figure a’s confidence interval are from Fig. \[fig:nbootstraps\] (n=100 simulations) while figures c & d are from Figure \[fig:nbootstrapsv2\]; all four CIs used 100% of simulations. Distance band set as Figure \[fig:nbootstrapsv2\].[]{data-label="fig:bootstraphists"}](figs/bootstraphistv2.pdf){width="1.0\linewidth"}
![Effect of distance band set on $\hat{D}$ precision. Overlapping set (Lessler et al.):={0-10, 0-12, 0-14, …, 0-50, 2-52, 4-54, …, 74-124m} and non-overlapping:={0-, 7-, 15-, 20-, 25-, …, 195m-}. Non-overlapping sets yield a more erratic point estimate $\hat{\tau}$ yet tighter 95% BCa CI (154, 261m) versus (149, 466m) however on further investigation the distribution of $\underline{D}$ is heavily bimodal; both CIs used 100% of simulations.[]{data-label="fig:distband"}](figs/distband.pdf){width="1.0\linewidth"}
![Modified marked point bootstrap (MMPB) method compared against the original Loh & Stein marked point bootstrap for the tau statistic. The latter’s envelope $\hat{\underline{\tau}}^*$ poorly covers $\hat{\tau}$ at short distances and leads to over-bias in $\hat{\tau}$ at large distances; note that only 774% of tau bootstrap simulations $\hat{\underline{\tau}}^*$ contribute to the Loh & Stein’s CI compared to 100% for MMPB.[]{data-label="fig:loh"}](figs/loh.pdf){width="0.79\linewidth"}
Code analysis {#sec:code}
=============
[^1]: Abbreviations in the paper: BCa = bias-corrected and accelerated; CI = confidence interval; MMPB = modified marked point bootstrap.
[^2]: The tau statistic discussed herein is different from the ‘Kendall’s tau statistic’ or ‘Kendall’s rank correlation coefficient’ which is a bivariate statistic for ordinal data [@bland2000introduction].
[^3]: This is a correction to Lessler et al.’s appendices that originally used an open interval [@Lessler2016]. It has been updated in their GitHub repository [@LesslerGiles] following email communication on 6 December 2018.
[^4]: \[footnote:prevpapers\][@Salje2012; @Grabowski2014; @Bhoomiboonchoo2014; @Levy2015; @Salje2016; @Salje2016social; @Lessler2016; @Grantz2016; @HoangQuoc2016; @Salje2017; @Salje2018; @Succo2018; @Rehman2018; @Azman2018; @Truelove2019] from @Pollingtonreview review
[^5]: An envelope can be loosely defined as a series of function lines bounded above and below: central/null envelopes describe the line function, i.e. bootstrapped point estimate or null distribution, respectively; whereas global envelope or pointwise confidence interval (syn. confidence band) refers to the way function lines are bounded. A *global envelope* is a confidence interval for a function not at a fixed $[d_1,d_2]$ (i.e. pointwise), but over all $\underline{\Delta}$; at a 95% significance level say, in 95% of outcomes of constructing a global envelope, the random envelope would contain the true value of $\tau([d_1,d_2]), \forall [d_1,d_2]\in\underline{\Delta}$ [@Baddeley2015].
[^6]: but without loss of generality for other $\tau$ estimators
[^7]: each distance band is still a half-closed interval
[^8]: see footnote \[footnote:prevpapers\]
|
---
abstract: 'Expressing a matrix as the sum of a low-rank matrix plus a sparse matrix is a flexible model capturing global and local features in data. This model is the foundation of robust principle component analysis [@Candes2011robust; @Chandrasekaran2009ranksparsity], and popularized by dynamic-foreground/static-background separation [@Bouwmans2016decomposition] amongst other applications. Compressed sensing, matrix completion, and their variants [@Eldar2012compressed; @Foucart2013a] have established that data satisfying low complexity models can be efficiently measured and recovered from a number of measurements proportional to the model complexity rather than the ambient dimension. This manuscript develops similar guarantees showing that $m\times n$ matrices that can be expressed as the sum of a rank-$r$ matrix and a $s$-sparse matrix can be recovered by computationally tractable methods from $\mathcal{O}(r(m+n-r)+s)\log(mn/s)$ linear measurements. More specifically, we establish that the restricted isometry constants for the aforementioned matrices remain bounded independent of problem size provided $p/mn$, $s/p$, and $r(m+n-r)/p$ reman fixed. Additionally, we show that semidefinite programming and two hard threshold gradient descent algorithms, NIHT and NAHT, converge to the measured matrix provided the measurement operator’s RIC’s are sufficiently small. Numerical experiments illustrating these results are shown for synthetic problems, dynamic-foreground/static-background separation, and multispectral imaging.'
address:
- 'Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK'
- 'The Alan Turing Institute, The British Library, London NW1 2DB, UK'
author:
- Jared Tanner
- Simon Vary
bibliography:
- 'library.bib'
title: 'Compressed sensing of low-rank plus sparse matrices'
---
matrix sensing ,low-rank plus sparse matrix ,restricted isometry property,non-convex methods ,robust PCA 15A29,41A29,62H25 ,65F10 ,65J20 ,68Q25 ,90C22 ,90C26
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank Robert A. Lamb and David Humphreys for useful discussions around the applications of low-rank plus sparse model to multispectral imaging.
|
---
abstract: 'In this paper we review recent work that has been done on quantum many-particle systems on metric graphs. Topics include the implementation of singular interactions, Bose-Einstein condensation, sovable models and spectral properties of some simple models in connection with superconductivity in wires.'
bibliography:
- 'Literature.bib'
---
[**Many-particle quantum graphs: A review**]{}\
[Jens Bolte]{}[^1]
Department of Mathematics\
Royal Holloway, University of London\
Egham, TW20 0EX\
United Kingdom\
[and]{}
[Joachim Kerner]{}[^2]
Department of Mathematics and Computer Science\
FernUniversität in Hagen\
58084 Hagen\
Germany\
Introduction
============
Quantum graph models describe the motion of particles along the edges of a metric graph. They have become popular models in various areas of physics and mathematics as they combine the simplicity of one-dimensional models with the potential complexity of graphs. One-particle quantum graphs and their applications are describe in detail in [@KotSmi99; @GNUSMY06; @Exnetal08; @Berkolaiko:2013].
Many-particle quantum systems are of fundamental importance in condensed matter as well as in statistical physics, see [@MR04; @SchwablSM; @cazalilla2011one]. In particular, phenomena like Bose-Einstein condensation, Anderson localisation and superconductivity have attracted much attention both in a phenomenological and a mathematical context. However, those phenomena are notoriously difficult to address, so that models that are promising to yield interesting results while still being sufficiently accessible are in demand. This was a major reason to develop and study quantum many-particle models on graphs. Another reason lies in the growing importance of one-dimensional, nano-technological devices [@QuantumWellsWiresDots; @GG08].
Among early quantum graph examples are models of two particles with singular interactions on simple graphs [@MP95; @CauCra07; @Ha07; @Ha08], where some basic spectral properties were studied. Other approaches involve quantum field theory on graphs (see, e.g., [@BelMin06; @Sch09]) where, due to the presence of vertices and the finite lengths of edges, translation invariance is broken. This leads to the presence of symmetry algebras that are of interest in their own right [@MinRag04]. In the context of quantum integrability, these symmetry algebras play a role in the construction of many-particle quantum models on graphs in which one can represent eigenfunctions in terms of a Bethe-ansatz [@CauCra07; @BolGar17; @BolGar17a]. In this context also extensive studies of non-linear Schrödinger equations (see, e.g., [@Noj14; @Cau15]) are of particular interest.
The phenomenon of Anderson localisation, which is known to occur in a large class of systems governed by random Schrödinger operators [@SimonBookSchrödinger; @stollmann2001caught], has been investigated for interacting particles on graphs in [@Sab14].
When particles are indistinguishable, the particle exchange symmetry has to be implemented. In three or more dimensions this leads to the well-known Fermi-Bose alternative. However, in lower dimensions more options may become available including, e.g., the possibility of anyons in two dimensions [@LeiMyr77]. In models of discrete quantum graphs the possible exchange symmetry representations were identified in [@HarKeaRob11; @HarKeaRobSaw14], and a whole range of exotic options were found.
In this paper we mainly review our own contributions to many-particle quantum graphs. This includes the construction of two types of singular pair interactions. The first one [@BKSingular] is closely related to vertices and can be seen as a model of interactions between particles that is mediated by the presence of an impurity (thought of as being located in a vertex); this type of interactions is similar to the one introduced in [@MP95]. The second type of singular interactions [@BKContact] are the more familiar $\delta$-pair interactions. They are models of very short-range, or contact interactions. When implemented for bosons, these interactions lead to a Lieb-Liniger gas [@LL63] on a graph, and in the limit of hardcore interaction they lead to a Tonks-Giradeau gas [@G60]. For all of these models it has been shown that they can be rigorously implemented with self-adjoint Hamiltonians and it has been proven that their spectra are discrete and the eigenvalues follow a Weyl law.
Due to a well-known theorem of Hohenberg [@Hohenberg], free Bose gases in one dimension are often said to not display Bose-Einstein condensation (BEC). This statement, however, is only true if in finite volume one imposes Dirichlet or other standard boundary conditions. It has been known though that non-standard boundary conditions may lead to cases where a finite number of eigenvalues are negative and remain so in the thermodynamic limit, such that in this limit a spectral gap below the continuum develops. Such a scenario then leads to BEC into the negative-energy ground state [@LanWil79; @VerbeureBook]. A similar behaviour occurs for free bosons on graphs, and it is possible to fully characterise all boundary conditions where this is the case [@BolteKernerBEC]. For a gas of bosons with pairwise repulsive hardcore interactions, a suitable Fermi-Bose mapping, however, shows that no condensation can be expected. Furthermore, it can be shown that arbitrarily small repulsive pair interactions prohibit a Bose gas on a graph to condense into the free ground state [@BolKer16].
In statistical mechanics solvable models play a significant role. In this context solvability refers to the fact that eigenfunctions of the Hamiltonan have a simple representation in terms of a so-called Bethe ansatz [@Bet31; @Gau14]. This form of the eigenfunctions also leads to a characterisation of eigenvalues through finitely many secular equations. The Lieb-Liniger gas [@LL63] of $N$ bosons with $\delta$-interactions on a circle is an example of a solvable model and its version on an interval [@Gau71] can be seen as a first example where vertices play a role. Vertices of degree two (and higher) present obstacles to the solvability of models with $\delta$-interactions. A modification of the interactions that preserves solvability when one vertex of degree two is present was found in [@CauCra07]. The basic idea behind this construction can be extended to arbitrarily (finitely) many vertices of any (finite) degree [@BolGar17], as well as to any (finite) number of particles [@BolGar17a].
In the final section we are concerned with a two-particle model on a simple non-compact quantum graph, namely the half-line $\mathbb{R}_+$, which can be thought of as a quantum wire. Besides singular interactions localised on the vertex at zero [@KM16; @EggerKerner] and contact interactions of the Lieb-Liniger type, we introduce a binding potential that leads to a pairing of the two particles [@KMBound; @KernerElectronPairs; @KernerInteractingPairs]. We also provide generalisations of this model by considering singular two-particle interactions whose locations are randomly distributed along the half-line [@KernerRandomI], and by taking into account surface defects in coupling the continuous half-line to a discrete graph [@KernerSurfaceDefects]. In all of these cases we are mainly interested in describing spectral properties of the associated Hamiltonians. Using the acquired knowledge about the spectrum we are able to investigate Bose-Einstein condensation of pairs. These results can be seen as statements about superconductivity in quantum wires.
Preliminaries {#Sec1}
=============
One-particle quantum graphs
---------------------------
A quantum graph is a metric graph $\Gamma$ with a differential operator that serves as Hamiltonian operator describing the motion of a particle along the edges of the graph, see [@GNUSMY06; @Berkolaiko:2013]. A metric graph is a (finite) combinatorial graph with a metric structure that arises from assigning lengths to edges. Let $\mathcal{V}$ be the set of vertices and $\mathcal{E}=\mathcal{E}_\mathrm{int}\cup\mathcal{E}_\mathrm{ext}$ be the set of edges. Then every $e\in\mathcal{E}_\mathrm{int}$, an internal edge, is adjacent to two distinct vertices, and every $e\in\mathcal{E}_\mathrm{ext}$, an external edge, is adjacent to a single vertex. A metric structure is introduced by assigning finite lengths to internal edges; external edges are considered to be of infinite length. In this way every $e\in\mathcal{E}_\mathrm{int}$ is identified with an interval $[0,l_e]$ whereas every $e\in\mathcal{E}_\mathrm{ext}$ is identified with a copy of the real semi-axis $[0,\infty)$. Graphs without external edges, $\mathcal{E}=\mathcal{E}_\mathrm{int}$, are compact.
One now introduces functions on $\Gamma$, $$\psi = (\psi_1,\dots,\psi_E)\ ,$$ where $E=|\mathcal{E}|$ and $\psi_e :[0,l_e]\to\mathbb{C}$ for internal edges and $\psi_e :[0,\infty)\to\mathbb{C}$ for external edges. In this way one defines the Hilbert space $$L^2(\Gamma) := \bigoplus_{e\in\mathcal{E}_\mathrm{int}} L^2(0,l_e)
\bigoplus_{e\in\mathcal{E}_\mathrm{ext}} L^2(0,\infty)\ ,$$ as well as the Sobolev spaces $$H^m(\Gamma) := \bigoplus_{e\in\mathcal{E}_\mathrm{int}} H^m(0,l_e)
\bigoplus_{e\in\mathcal{E}_\mathrm{ext}} H^m(0,\infty)\ .$$ A Hamiltonian operator $H=-\Delta+V$ is a self-adjoint operator (in many cases with domain $\mathcal{D}\subset H^2(\Gamma)$) that acts on functions on an edge as $$(H\psi)_e = -\psi_e'' + V_e\psi_e \ ,$$ where $V=(V_1,\dots,V_E)$ is a potential function. In many quantum graph models, however, one considers the case $V=0$.
In the following we shall restrict our attention to compact graphs, although the examples in Section \[Sec3\] will be non-compact; the necessary modifications are more or less obvious.
In order to characterise domains $\mathcal{D}$ of self-adjointness one has to impose boundary conditions at the vertices on functions in the domain. We denote boundary values of functions as $$\psi_{bv} = \bigl(\psi_1(0),\dots\psi_E(0),\psi_1(l_1),\dots,
\psi_E(l_E)\bigr) \ ,$$ and of inward derivatives as $$\psi'_{bv} = \bigl(\psi'_1(0),\dots\psi'_E(0),-\psi'_1(l_1),\dots,
-\psi'_E(l_E)\bigr) \ .$$ Self-adjoint realisations of $H$ can be obtained as maximal symmetric extensions of the operator with minimal domain $C_0^\infty(\Gamma)$ (see, e.g., [@KosSch99]). Their domains can be uniquely parametrised in terms of an orthogonal projector $P$ and a self-adjoint map $L$, such that $P^\perp LP^\perp=L$, on the space $\mathbb{C}^{2E}$ of boundary values, as [@Kuc04] $$\label{1Pdomain}
\mathcal{D}(P,L) = \bigl\{ \psi\in H^2(\Gamma):\ (P+L)\psi_{bv}+
P^\perp\psi'_{bv}=0 \bigr\} \ .$$ It is often useful to work with quadratic forms instead of self-adjoint operators, making use of the fact that a semi-bounded (from below) self-adjoint operator defines a unique, semi-bounded and closed quadratic form, and vice versa [@BEH08]. The form associated with a quantum graph Laplacian $-\Delta$ on the domain is [@Kuc04] $$\mathcal{Q}[\psi] = \int_\Gamma |\nabla\psi|^2\ {\mathrm{d}}x
-\langle\psi_{bv},L\psi_{bv}\rangle_{\mathbb{C}^{2E}} \ ,$$ with form domain $$\label{1Pformdomain}
\mathcal{D}_{\mathcal{Q}} = \bigl\{ \psi\in H^1(\Gamma):
\ P\psi_{bv}=0 \bigr\} \ .$$ The boundary conditions prescribed in and do not necessarily respect the connectivity of the combinatorial graph. The latter will, however, be the case for local boundary conditions, where $$P=\bigoplus_{v\in\mathcal{V}}P_v \quad\text{and}\quad
L=\bigoplus_{v\in\mathcal{V}}L_v \ ,$$ and $P_v$, $L_v$ act on the subspace $\mathbb{C}^{d_v}$ of boundary values on the edge ends adjacent to the vertex $v\in\mathcal{V}$. Here $d_v$ is the degree of the vertex $v$.
A quantum graph Hamiltonian $H=-\Delta+V$ defined on a domain is self-adjoint, bounded from below, and has compact resolvent (note that the latter fails to hold for non-compact graphs). Hence its spectrum is real, bounded from below, discrete and eigenvalues accumulate only at infinity. In the most relevant case of $V=0$, one can characterise eigenvalues through a secular determinant. One first defines a (vertex) scattering matrix $$S(k) : = -P-(L+{\mathrm{i}}kP^\perp)^{-1}(L-{\mathrm{i}}kP^\perp)\ ,$$ where $k\in\mathbb{C}$ is such that $k^2$ is a spectral parameter for $-\Delta$, and then a matrix $$T(k) : = \begin{pmatrix} 0 & {\mathrm{e}}^{{\mathrm{i}}k\boldsymbol{l}} \\
{\mathrm{e}}^{{\mathrm{i}}k\boldsymbol{l}} & 0 \end{pmatrix}$$ encoding the metric information about $\Gamma$. Here ${\mathrm{e}}^{{\mathrm{i}}k\boldsymbol{l}}$ is a diagonal $E\times E$ matrix with diagonal entries ${\mathrm{e}}^{{\mathrm{i}}kl_e}$, $e=1,\dots,E$. Defining $U(k):=S(k)T(k)$, one can show [@KS06] that $k^2$ is a non-zero eigenvalues of $-\Delta$ of multiplicity $m(k)$, iff $k$ is a zero of $$\label{1Psecular}
\det\bigl({\mathds{1}}-U(k)\bigr)$$ of order $m(k)$. An eigenvalue zero has to be treated separately, see [@KS06; @BolEnd09; @BolEggSte15]. A similar, slightly more complicated condition can be obtained for operators of the form $H=-\Delta+V$, see [@BolEggRue15].
The secular equation can be used to derive a trace formula [@Rot83; @KotSmi99; @BolEnd09] that expresses spectral functions in terms of sums over periodic orbits on the graph.
Many-particle kinematics
------------------------
Following the general construction of systems of several (distinguishable) particles from given one-particle systems in quantum mechanics, the Hilbert space of $N$ distinguishable particles on a metric graph $\Gamma$ is $$\label{NHilbert}
\mathcal{H}_N = L^2(\Gamma) \otimes\cdots\otimes L^2(\Gamma) \ .$$ Vectors in the tensor product are collections of functions $\psi_{e_1\dots e_N}\in L^2([0,l_{e_1}]\times\dots\times [0,l_{e_N}])$. These are functions of $N$ variables describing the positions of the particles on the $N$ edges $e_1,\dots,e_N$, which do not need to be all different. In a slight abuse of notation we shall view these collections of functions as functions on the domain $$\label{Ndomain}
D^{(N)}_\Gamma := \bigcup_{e_1\dots e_N}
(0,l_{e_1})\times\dots\times (0,l_{e_N}) \ ,$$ such that we shall also use the notation $\mathcal{H}_N = L^2(D^{(N)}_\Gamma)$.
$N$-particle observables are self-adjoint operators on $\mathcal{H}_N$. An operator $O$ that respects the tensor product structure of the Hilbert space, $$\label{freeop}
O = \sum_{i=1}^N {\mathds{1}}\otimes\dots\otimes{\mathds{1}}\otimes O_i
\otimes{\mathds{1}}\dots\otimes{\mathds{1}}\ ,$$ does not detect any correlations or interactions between particles. A Hamiltonian describing particle interactions, therefore, cannot be of this product form. In other words, particle interactions will be implemented by choosing a Hamiltonian that does not have the product structure. This can be achieved either in the form of, say, a potential $$\label{2Ppotential}
V(x_1,\dots,x_N) = \sum_{i,j=1}^{N} V_p (x_i ,x_j)$$ with explicit pair interactions. However, one can also implement interactions by choosing an operator domain for an $N$-particle Laplacian $-\Delta_N$, acting as $$(-\Delta_N \psi)_{e_1\dots e_N} =
-\frac{\partial^2\psi_{e_1\dots e_N}}{\partial x_{e_1}^2}-\dots
-\frac{\partial^2\psi_{e_1\dots e_N}}{\partial x_{e_N}^2}\ ,$$ that does not respect the tensor product structure for the operator.
When the $N$ particles are indistinguishable, the exchange symmetry has to be implemented in the kinematic set up of the quantum system. If one adopts the Bose-Fermi alternative, the only relevant representations of the symmetric group will be the totally symmetric one (for bosons) and the totally anti-symmetric one (for fermions). The quantum state spaces then are the totally symmetric and the totally anti-symmetric subspaces $\mathcal{H}_{N,B}$ and $\mathcal{H}_{N,F}$, respectively, of
Singular pair interactions {#Sec2}
==========================
A possible way of introducing interactions is to violate the tensor product structure with boundary conditions, either at the boundaries of the domains $[0,l_{e_1}]\times\dots\times [0,l_{e_N}]$, or at additional boundaries introduced for the purpose of generating other types of interactions. Typically, such boundary conditions will lead to singular interactions that can formally be expressed in terms of $\delta$-functions, see e.g. [@BrascheExnerKuperinSeba92; @Behrndt2013].
Vertex-induced singular interactions {#Sec2.1}
------------------------------------
Boundary conditions imposed at the boundaries of $[0,l_{e_1}]\times\dots\times [0,l_{e_N}]$ alone correspond to interactions that act when at least one particle sits in a vertex (corresponding to $x_{e_j}=0$ or $x_{e_j}=l_{e_j}$). Hence we say that such interactions are vertex induced. An example for a pair of particles on the same edge (of length $l$) would be the two-dimensional Laplacian plus a formal potential of the form $$v(x_1,x_2)\bigl[ \delta(x_1) + \delta(x_1-l) + \delta(x_2) + \delta(x_2-l) \bigr] \ .$$ A version of such an interaction on a $Y$-shaped graph can be found in [@MP95].
Constructing $N$-particle Laplacians with boundary conditions is not as straight forward as for one-particle Laplacians. The reason for this is that the minimal symmetric operator, which is an $N$-particle Laplacian with domain $C_0^\infty(D^{(N)}_\Gamma)$, does not have finite deficiency indices. For that reason it is more appropriate to construct self-adjoint realisations of the $N$-particle Laplacian via their associated sesqui-linear forms.
In the following we restrict our attention to $N=2$, noting that this case contains all the essential steps in order to construct $N$-particle Hamiltonians with pair interactions. As a first step we simplify the notation in that we define $$\label{lscale}
\psi_{e_1 e_2}(x_{e_1},y_{e_2}) = \psi_{e_1 e_2}(l_{e_1}x,l_{e_2}y)$$ with $x,y\in (0,1)$. The $4E^2$ boundary values of functions $\psi\in H^1(D^{(2)}_\Gamma)$ and derivatives of functions $\psi\in H^2(D^{(2)}_\Gamma)$ then are $$\label{graphbv}
\psi_{bv}(y) =
\begin{pmatrix}\sqrt{l_{e_2}}\psi_{e_1 e_2}(0,l_{e_2}y) \\
\sqrt{l_{e_2}}\psi_{e_1 e_2}(l_{e_1},l_{e_2}y) \\
\sqrt{l_{e_1}}\psi_{e_1 e_2}(l_{e_1}y,0) \\
\sqrt{l_{e_1}}\psi_{e_1 e_2}(l_{e_1}y,l_{e_2})
\end{pmatrix} \qquad\text{and}\qquad
\psi'_{bv}(y) =
\begin{pmatrix}\sqrt{l_{e_2}}\psi_{e_1 e_2,x}(0,l_{e_2}y) \\
-\sqrt{l_{e_2}}\psi_{e_1 e_2,x}(l_{e_1},l_{e_2}y)\\
\sqrt{l_{e_1}}\psi_{e_1 e_2,y}(l_{e_1}y,0) \\
-\sqrt{l_{e_1}}\psi_{e_1 e_2,y}(l_{e_1}y,l_{e_2})\end{pmatrix} \ .$$ Here $y\in [0,1]$ and the indices $e_1 e_2$ run over all $E^2$ possible pairs with $e_1 ,e_2 =1,\dots,E$.
With the one-particle form domain in mind we now introduce bounded and measurable maps $P,L: [0,1] \to {\mathrm{M}}(4E^2,{{\mathbb C}})$ such that for a.e. $y \in [0,1]$,
1. $P(y)$ is an orthogonal projector,
2. $L(y)$ is a self-adjoint endomorphism on $\ker P(y)$.
With these maps we can define the quadratic form, $$\label{Qform2graph}
\begin{split}
Q^{(2)}_{P,L}[\psi]
&:= \langle \nabla\psi,\nabla\psi \rangle_{L^2 (D_\Gamma)} -
\langle \psi_{bv},L(\cdot)\psi_{bv} \rangle_{L^2(0,1)\otimes{{\mathbb C}}^{4E^2}} \\
&=\sum_{e_1 ,e_2 =1}^E \int_{0}^{l_{e_2}}\int_0^{l_{e_1}}\Bigl( \bigl|
\psi_{e_1 e_2,x}(x,y) \bigr|^2 + \bigl| \psi_{e_1 e_2,y}
(x,y) \bigr|^2 \Bigr)\ {\mathrm{d}}x\,{\mathrm{d}}y \\
&\qquad\qquad -\int_0^1 \langle\psi_{bv}(y),L(y)\psi_{bv}(y)
\rangle_{{{\mathbb C}}^{4E^2}} \ {\mathrm{d}}y \ ,
\end{split}$$ and prove the following result [@BKSingular].
\[2quadformgraph\] Given maps $P,L:[0,1]\to {\mathrm{M}}(4E^2,{{\mathbb C}})$ as above that are bounded and measurable, the quadratic form with domain $$\label{Defquadgraph}
{{\mathcal D}}_{Q^{(2)}} = \{ \psi \in H^1(D_\Gamma):\ P(y)\psi_{bv}(y)=0\ \text{for a.e.}\
y\in [0,1] \}$$ is closed and semi-bounded (from below).
The semi-bounded, self-adjoint operator associated with this form via the representation theorem for quadratic forms [@BEH08] can be identified as a self-adjoint realisation of the Laplacian when its domain is contained in $H^2(D^{(2)}_\Gamma)$; in this case the form is said to be regular.
In order to identify regular forms we need to impose further restrictions on the maps $P$ and $L$. The first one is that they are block-diagonal, in the form $$\label{2block}
M(y) = \begin{pmatrix} \tilde M(y) & 0 \\ 0 & \tilde M(y) \end{pmatrix} \ ,$$ with respect to an arrangement of the components of where the upper two components for all pairs $e_1,e_2$ are separated from the lower two components. Then we obtain the following result [@BKSingular].
\[TheoremPL(y)2\] Let $L$ be Lipschitz continuous on $[0,1]$ and let $P$ be of the block-diagonal form . Assume that the matrix entries of $\tilde P$ are in $C^3(0,1)$ and possess extensions of class $C^3$ to some interval $(-\eta,1+\eta)$, $\eta>0$. Moreover, when $y\in [0,\varepsilon_{1}]\cup [l-\varepsilon_{2},l]$, with some $\varepsilon_{1},\varepsilon_{2} > 0$, suppose that $L(y)=0$ and that $\tilde P(y)$ is diagonal with diagonal entries that are either zero or one. Then the quadratic form $Q^{(2)}_{P,L}$ is regular. The associated semi-bounded, self-adjoint operator is a Laplacian with domain $$\label{bcgraph}
{{\mathcal D}}_2 (P,L) := \{ \psi\in H^2(D^{(2)}_\Gamma):\ (P(y)+L(y))\psi_{bv}(y)+
P^\perp(y)\psi'_{bv}(y)=0\ \text{for a.e.}\ y\in [0,1] \}\ .$$
Note the similarity of with .
As one would expect from a quantum systems with a configuration space of finite volume, the spectrum of a two-particle Laplacian with domain is discrete. Moreover, a Weyl law for the eigenvalue count holds: Let $\lambda_n$, $n\in\mathbb{N}$, denote the eigenvalues of the operator, then $$\label{Weylsing}
N(\lambda) := \# \{n\in\mathbb{N}:\ \lambda_n\leq\lambda\} \sim
\frac{\mathcal{L}^2}{4\pi}\ ,\quad \lambda\to\infty\ ,$$ where $\mathcal{L}=l_{e_1}+\cdots+l_{e_E}$ is the total length of the metric graph, see [@BKSingular].
The constructions above can be carried over to bosonic or fermionic systems in a straight forward manner; for details see [@BKSingular].
Contact interactions {#sec2.2}
--------------------
Realistic two-particle interactions are often of the form . When the range of the interaction is small one can model the pair potential with a Dirac-$\delta$, so that the formal $N$-particle Hamiltonian is $$\label{contactint}
H_N = -\Delta_N +\alpha\sum_{i<j}\delta(x_i-x_j) \ .$$ Here $\Delta_N$ denotes a non-interacting, self-adjoint realisation of the $N$-particle Laplacian and $\alpha\in\mathbb{R}$ is a constant determining the interaction strength. In this way an interaction takes place when (at least) two particles are at the same position and, therefore, one speaks of a contact interaction. In order to implement contact interactions in a self-adjoint operator one has to impose boundary conditions along hyperplanes in the configuration space of $N$ particles that are characterised by equations $x_i=x_j$. Contact interactions for bosons on a circle have, e.g., been studied in much detail in the form of the Lieb-Liniger model [@LL63], and for distinguishable particles on infinite star graphs in [@Ha07; @Ha08].
A self-adjoint operator representing the formal expression can be defined as an extension of the $N$-particle Laplacian with domain $C_0^\infty(D_\Gamma^{(N)})$. This can be done much in the same way as above for the vertex-induced singular interactions after additional boundaries have been introduced to the domain . As contact interactions require two particles to be on the same edge, components in where $e_1,\dots,e_N$ are $N$ distinct edges do not contribute. Taking the example of $N=2$ as for the vertex-induced singular interactions above, one introduces the subdivision $$\label{diagsquare}
D_{ee} := [0,l_e]\times [0,l_e] = D_{ee}^+\cup D_{ee}^-\ ,$$ of diagonal domains, where $$\label{squarecut}
D_{ee}^+ := \{(x,y)\in D_{ee}:\ x\geq y\}\quad\text{and}\quad
D_{ee}^- := \{(x,y)\in D_{ee}:\ x\leq y\}\ .$$ These subdivisions modify the total domain $D^{(2)}_\Gamma$, see . The resulting domain with the additional boundaries is denoted as $D^{\ast(2)}_\Gamma$.
Boundary values of components $\psi_{e_1e_2}$ of functions $\psi\in H^2(D^{\ast(2)}_\Gamma)$ and their derivatives are as in when $e_1\neq e_2$. For the remaining components, however, the additional boundaries lead to the boundary values $$\label{bvdiag}
\psi_{ee,bv}(y) := \begin{pmatrix} \sqrt{l_e}\psi^{-}_{ee}(0,l_e y) \\
\sqrt{l_e}\psi^{+}_{ee}(l_e,l_e y) \\ \sqrt{l_e}\psi^{+}_{ee}(l_e y,0) \\
\sqrt{l_e}\psi^{-}_{ee}(l_e y,l_e) \\ \sqrt{l_e} \psi^{+}_{ee}(l_e y,l_e y) \\
\sqrt{l_e}\psi^{-}_{ee}(l_e y,l_e y) \end{pmatrix}
\qquad\text{and}\qquad
\psi'_{ee,bv}(y) := \begin{pmatrix} \sqrt{l_{e}}\psi^{-}_{ee,x}(0,l_{e}y) \\
-\sqrt{l_e}\psi^{+}_{ee,x}(l_e,l_e y) \\ \sqrt{l_e}\psi^{+}_{ee,y}(l_e y,0) \\
-\sqrt{l_e}\psi^{-}_{ee,y}(l_e y,l_e) \\ \sqrt{2l_e}\psi^{+}_{ee,n}(l_e y,l_e y)\\
\sqrt{2l_e}\psi^{-}_{ee,n}(l_e y,l_e y) \end{pmatrix}\ ,$$ for $y \in [0,1]$. Here $\psi_{ee}^\pm:D_{ee}^\pm\to\mathbb{C}$ and $$\label{nderdiag}
\psi^\pm_{ee,n} :=
\frac{\pm 1}{\sqrt{2}}\bigl(\psi^\pm_{ee,x}-\psi^\pm_{ee,y}\bigr)$$ is the normal derivative along the diagonal part of the boundary.
The space $\mathbb{C}^{n(E)}$, $n(E)=4E^2+2E$, of boundary values decomposes into a $4E^2$-dimensional subspace $W_{vert}$ of vertex-induced boundary values as in Section \[Sec2.1\], and a $2E$-dimensional subspace $W_{cont}$ of boundary values on diagonals associated with contact interactions. Introducing maps $P$ and $L$ on $[0,1]$ that take values in the orthogonal projectors and self adjoint maps on $W_{vert}\oplus W_{cont}$, respectively, in the same way as in Section \[Sec2.1\], their restrictions to $W_{vert}$ should satisfy the same properties as above. The restrictions to the edge-$e$ subspace of $W_{cont}$ should take the form $$P_{cont,e}=\frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}
\quad\text{and}\quad
L_{cont,e}=-\frac{1}{2}\alpha(y){\mathds{1}}_2\ ,$$ where $\alpha:[0,1]\to\mathbb{R}$ is a possibly varying, Lipschitz-continuous interaction strength. With boundary conditions as described in this choice implies continuity of functions across diagonals, $$\label{fctcont}
\psi^{+}_{ee}(l_e y,l_e y) = \psi^{-}_{ee}(l_e y,l_e y) \ ,$$ and satisfies jump conditions for the normal derivatives, $$\label{derjump}
\psi^+_{ee,n}(l_e y,l_e y) + \psi^-_{ee,n}(l_e y,l_e y) =
\frac{1}{\sqrt{2}}\alpha(y)\psi^\pm_{ee}(l_e y,l_e y) \ .$$ These conditions ensure a rigorous, self-adjoint realisation of the $\delta$-type contact interactions . The operator is a two-particle Laplacian with domain , where now $$\label{bc4contact}
P = P_{vert}\oplus P_{cont} \quad\text{and}\quad
L = L_{vert}\oplus L_{cont} \ .$$ Hardcore contact interactions correspond to Dirichlet conditions along all diagonal boundaries. These conditions follow from $\delta$-type interactions by taking the limit $\alpha\to\infty$. For more detail see [@BKContact].
As in the case of vertex-induced singular interactions, the spectrum of the two-particle Laplacian with domain and is discrete and the Weyl law holds [@BKContact].
A Lieb-Linger model on graphs
-----------------------------
The contact interactions of Section \[sec2.2\] offer an opportunity to extend the Lieb-Linger model of $N$ bosons with $\delta$-interactions on a circle to arbitrary metric graphs. Implementing bosonic symmetry first requires to restrict the $N$-particle Hilbert space $\mathcal{H}_N$, see , to its totally symmetric subspace $\mathcal{H}_{N,B}$. The projector $\Pi_B$ to that subspace acts on vector $\psi\in\mathcal{H}_N$ as $$\label{Boseproject}
(\Pi_B\psi)_{e_1\dots e_N} = \frac{1}{N!}\sum_{\pi\in S_N}
\psi_{e_{\pi(1)}\dots e_{\pi(N)}}\ ,$$ where $S_N$ denotes the symmetric group. In order to implement $\delta$-type contact interactions one has to dissect the hyper-rectangles $(0,l_{e_1})\times\dots\times (0,l_{e_N})$ with at least two coinciding edges, $e_i=e_j$ with $i\neq j$, along the hyperplanes $x_{e_i}=x_{e_j}$. The resulting configuration space is $D_\Gamma^{\ast(N)}$. On the hyperplanes we impose boundary conditions that are equivalent to –.
The remaining, vertex related boundary values can be simplified by making use of the particle exchange symmetry. For functions $\Psi\in H^1_B(D^{\ast(N)}_\Gamma)$ they are $$\label{BVI}
\psi_{bv,vert}({\boldsymbol{y}}) = \begin{pmatrix} \sqrt{l_{e_{2}}\dots l_{e_{N}}}
\psi_{e_{1}\dots e_{N}}(0,l_{e_{2}}y_{1},\dots,l_{e_{N}}y_{N-1}) \\
\sqrt{l_{e_{2}}\dots l_{e_{N}}}
\psi_{e_{1}\dots e_{N}}(l_{e_{1}},l_{e_{2}}y_{1},\dots,l_{e_{N}}y_{N-1}) \end{pmatrix} \ ,$$ and for derivatives, $$\label{BVII}
\psi^{'}_{bv,vert}({\boldsymbol{y}}) = \begin{pmatrix} \sqrt{l_{e_{2}}\dots l_{e_{N}}}
\psi_{e_{1}\dots e_{N},x^{1}_{e_{1}}}(0,l_{e_{2}}y_{1},\dots,l_{e_{N}}y_{N-1}) \\
-\sqrt{l_{e_{2}}\dots l_{e_{N}}}
\psi_{e_{1}\dots e_{N},x^{1}_{e_{1}}}(l_{e_{1}},l_{e_{2}}y_{1},\dots,l_{e_{N}}y_{N-1})
\end{pmatrix} \ ,$$ where ${\boldsymbol{y}}=(y_{1},\dots,y_{N-1})\in [0,1]^{N-1}$.
Introducing maps $L_{vert},P_{vert}:[0,1]^{N-1}\to M(2E^N,\mathbb{C})$ in analogy to , we are now in a position to introduce the quadratic form $$\label{QuadFormContactI}
\begin{split}
Q^{(N)}_{B}[\psi]
&= N \sum_{e_{1}\dots e_{N}}\int_{0}^{l_{e_{1}}}\dots\int_{0}^{l_{e_{N}}}
|\psi_{e_{1}\dots e_{N},x_{e_1}}(x_{e_{1}},\dots,x_{e_{N}})|^{2}\ {\mathrm{d}}x_{e_N}\dots
{\mathrm{d}}x_{e_1} \\
&\quad -N\int_{[0,1]^{N-1}}\langle\psi_{bv,vert},L_{vert}({\boldsymbol{y}})\psi_{bv,vert}
\rangle_{{{\mathbb C}}^{2E^{N}}} {\mathrm{d}}{\boldsymbol{y}}\\
&\quad +\frac{N(N-1)}{2}\sum_{e_{2}...e_{N}}\int_{[0,1]^{N-1}} \alpha(y_1)\
|\sqrt{l_{e_{2}}\dots l_{e_{N}}}\psi_{e_{2}e_{2}\dots e_{N}}
(l_{e_2}y_1,{\boldsymbol{l}}{\boldsymbol{y}})|^{2}\ {\mathrm{d}}{\boldsymbol{y}}\ ,
\end{split}$$ where ${\boldsymbol{l}}{\boldsymbol{y}}=(l_{e_2}y_1,l_{e_3}y_2,\dots,l_{e_N}y_{N-1})$, with form domain $$\begin{split}
\label{QNformdomain}
{{\mathcal D}}_{Q^{(N)}_{B}} = \{\psi \in H^{1}_{B}(D^{\ast(N)}_\Gamma);\ P_{vert}({\boldsymbol{y}})
\psi_{bv,vert}({\boldsymbol{y}})=0\ \text{for a.e.}\ {\boldsymbol{y}}\in [0,1]^{N-1}\}\ .
\end{split}$$ The first two lines in define a bosonic $N$-particle Laplacian with vertex-related boundary conditions, whereas the last line yields pairwise, $\delta$-type contact interactions.
The hardcore limit, $\alpha\to\infty$ (see above), of the Lieb-Liniger gas is the so-called Tonks-Girardeau gas [@G60].
Bose-Einstein condensation {#secBEC}
==========================
One of the most interesting questions arising for bosonic many-particle system is whether they show the phenomenon of Bose-Einstein condensation (BEC). This occurs when below a critical temperature the particles condense into the same one-particle state [@PO56]. The original version of BEC [@EinsteinBEC] was found for free, i.e., non-interacting bosons in three dimensions that are confined to box of finite volume and whose wave functions satisfy standard conditions at the boundary of the box; it occurs in the thermodynamic limit of increasing the particle number and the volume of the box while keeping the particle density fixed. One can readily show that this form of BEC does not occur in one dimension as long as standard boundary conditions are imposed. However, it has long been known that BEC for free bosons does occur in one dimension when the boundary conditions are changed in such a way that the free, one-particle Hamiltonian has a negative eigenvalue and in the thermodynamic limit a gap remains in the spectrum between the ground state and the continuum above zero [@LanWil79; @VerbeureBook].
Free bosons
-----------
In a many-particle system of $N$ free bosons, the Hamiltonian is a symmetrised version of an operator with the tensor product structure . Its eigenvalues are of the form $k_{n_1}^2+\cdots +k_{n_N}^2$, where $k_n^2$ is an eigenvalue of the one-particle Hamiltonian, which we assume to be a Laplacian with domain . The number of negative eigenvalues is controlled by the self-adjoint map $L$ in the characterisation of the domain [@BL10], and this determines whether or not BEC is found in the thermodynamic limit. In this limit the volume growth is achieved by stretching all edge lengths with the same factor, $l_e\mapsto\eta l_e$, $\eta > 0$. Hence, the thermodynamic limit can be performed by sending the total length $\mathcal{L}=\sum_e l_e$ to infinity.
The first result required in order to prove BEC establishes a gap in the spectrum [@BolteKernerBEC].
Let $-\Delta$ be a one-particle Laplacian on a compact metric graph with domain . Assume that $L$ has at least one positive eigenvalue and let $L_{max}$ be the largest eigenvalue. Then the ground state eigenvalue $k^2_0(\mathcal{L})$ of the Laplacian at total length $\mathcal{L}$ converges to $-L_{max}$ in the thermodynamic limit $\mathcal{L}\to\infty$.
In the grand canonical ensemble of statistical mechanics (see, e.g., [@SchwablSM] for details), the density of particles $\rho_{n}(\beta,\mu_{\mathcal{L}})$ in an eigenstate with eigenvalue $k^2_n(\mathcal{L})$ is $$\rho_{n}(\beta,\mu_{\mathcal{L}})=\frac{1}{\mathcal{L}}
\frac{1}{{\mathrm{e}}^{\beta(k^2_n(\mathcal{L})-\mu_{\mathcal{L}})}-1}\ ,$$ where $\beta=1/k_BT$ is the inverse temperature and $\mu_{\mathcal{L}}\leq k^2_0(\mathcal{L})$ is the so-called chemical potential which itself depends on $\mathcal{L}$. More explicitly, $\mu_{\mathcal{L}}$ is chosen such that $$\rho=\frac{1}{\mathcal{L}}\sum_{n=0}^{\infty}
\frac{1}{{\mathrm{e}}^{\beta(k^2_n(\mathcal{L})-\mu_{\mathcal{L}})}-1}$$ is the fixed density of particles on the graph for all values of $\mathcal{L}$.
\[BECdef\] We say that an eigenstate with eigenvalue $k^2_n(\mathcal{L})$ is macroscopically occupied in the thermodynamic limit if $$\limsup_{\mathcal{L}\to\infty} \rho_{n}(\beta,\mu_{\mathcal{L}}) > 0\ .$$ If such an eigenstate exists we say that there is BEC into this eigenstate.
With these observations one is able to obtain a complete characterisation of free Bose gases on compact graphs in terms of BEC [@BolteKernerBEC].
\[freeBEC\] Let $\Gamma$ be a compact metric graph with one-particle Laplacian defined on the domain . If $L$ is negative semi-definite, no BEC occurs at finite temperature in the thermodynamic limit.
If, however, $L$ has at least one positive eigenvalue, there exists a critical temperature $T_c >0$ such that BEC occurs below $T_c$ in the thermodynamic limit.
The one-particle ground state eigenfunction into which all particles condense below the critical temperature is peaked around the vertices and hence is not homogeneous, as it would be in the classical case of particles in a box with Dirichlet boundary conditions, see also [@LanWil79].
Interacting bosons
------------------
For interacting bosons it is much harder to prove that BEC either holds or is absent [@LiebSeiringerProof; @LVZ03]. In the case of the Tonks-Girardeau gas [@G60] of particles with hardcore interactions on a graph, however, one can use a Fermi-Bose mapping in order to prove the absence of phase transitions which then indicates an absence of BEC. The Fermi-Bose mapping is a bijection between the set of bosonic many-particle Laplacians with hardcore interactions and the set of free fermionic Laplacians on the same compact, metric graph. The fermionic $N$-particle Hilbert space $\mathcal{H}_{N,F}$ is the totally antisymmetric subspace of $\mathcal{H}_N$, i.e., the image of the projector $$\label{Fermiproject}
(\Pi_F\psi)_{e_1\dots e_N} = \frac{1}{N!}\sum_{\pi\in S_N}(-1)^{\operatorname{sgn}\pi}
\psi_{e_{\pi(1)}\dots e_{\pi(N)}}\ ,$$ compare . One notices that the antisymmetry implies that continuous fermionic functions vanish along diagonal hyperplanes $x_{e_i}=x_{e_j}$, where $e_i=e_j$ but $i\neq j$, as do functions in the domain of a bosonic Laplacian with hardcore interactions. Using appropriate permutations of edges one can construct a bijection between bosonic and fermionic functions in such a way that the latter are in the domain of a fermionic quadratic form that is associated with a free fermionic Laplacians. As the forms coincide, the Fermi-Bose mapping is isospectral. For details of the construction we refer to [@BolteKernerBEC]. In fermionic systems BEC is well known to be absent. In the present case one calculates the free-energy density of free fermions (with Dirichlet boundary conditions in the vertices) in the thermodynamic limit, $$f_{D,F}(\beta,\mu) = \limsup_{\mathcal{L}\to\infty}\frac{1}{\beta\mathcal{L}}
\operatorname{Tr}{\mathrm{e}}^{-\beta H_N}
= -\frac{1}{\beta}\int_0^\infty
\log(1+{\mathrm{e}}^{-\beta(k^2-\mu)})\,{\mathrm{d}}k,$$ see [@BolteKernerBEC]. This energy density is smooth and has no singularities in $\beta$ which shows that there is no phase transition, consequently indicating an absence of BEC.
Other forms of (repulsive) interactions can be modelled by pair potentials of the type . On a metric graph this takes the form $$\label{reppairint}
(V_{N,\mathcal{L}}\psi)_{e_1\dots e_N}(x_{e_1},\dots,x_{e_N}) =
\sum_{i<j} V_{p,\mathcal{L}}(x_{e_i}-x_{e_j})
\psi_{e_1\dots e_N}(x_{e_1},\dots,x_{e_N})\ ,$$ and gives rise to the (bosonic) $N$-particle Hamiltonian $$H_N = -\Delta_N + V_{N,\mathcal{L}}\ .$$ The pair potentials are repulsive when the functions $V_{p,\mathcal{L}}$ are non-negative, and for technical reasons we assume that for all $\mathcal{L}>0$ there exist $A_{\mathcal{L}}>0$ and $\epsilon_{\mathcal{L}}>0$ such that $V_{p,\mathcal{L}}(x)\geq\epsilon_{\mathcal{L}}$ for all $x\in[-A_{\mathcal{L}},A_{\mathcal{L}}]$. Moreover, the $L^1$-norm of $V_{p,\mathcal{L}}$ is assumed to be independent of $\mathcal{L}$. These assumptions are consistent with choosing functions $V_{p,\mathcal{L}}$ that are a $\delta$-series in the thermodynamic limit $\mathcal{L}\to\infty$. One can, e.g., take $V_{p,\mathcal{L}}(x)=\mathcal{L}v(\mathcal{L}x)$ with $v\in C_0^\infty(\mathbb{R})$, $v\geq 0$ and $\|v\|_1=\alpha$ so that $\lim_{\mathcal{L}\to\infty}V_{p,\mathcal{L}}(x)=\alpha\delta(x)$. With this choice the Lieb-Liniger model will be recovered in the thermodynamic limit.
A Gibbs state at inverse temperature $\beta>0$ is defined via $$\omega_\beta (O) := \frac{\operatorname{Tr}\left(O\,{\mathrm{e}}^{-\beta H_N}\right)}{\operatorname{Tr}{\mathrm{e}}^{-\beta H_N}}\ ,$$ where $O$ is a (bounded) observable, i.e., a (bounded and) self-adjoint operator. If now $\psi_0$ is the ground state of the free bosonic system, i.e., composed of the ground state eigenfunction $\phi_0$ of the one-particle Laplacian and $N(\phi_0)$ is the particle number operator in this ground state one can infer from Theorem \[freeBEC\] whether or not the non-interacting system shows BEC. Assuming this to be the case, one can ask what the effect of adding a repulsive interaction is. It can be shown [@BolKer16] that in the theromodynamic limit the occupation of this ground state vanishes, $$\limsup_{\mathcal{L}\to\infty}\frac{\omega_\beta(N(\phi_0))}{\mathcal{L}} =0\ .$$ According to a direct analogue of Definition \[BECdef\], this means that there is no BEC into the free ground state. Hence, although BEC into the ground state was present in the free bosonic system, even the smallest perturbation by repulsive pair interactions of the type make this condensation disappear.
Summarising, although free bosons on a compact metric graph may display BEC, an addition of repulsive interactions is likely to destroy the condensate. The BEC that can occur is caused by $\delta$-type, attractive, one-particle potentials in the vertices and the associated condensate is not homogeneous, but concentrated around the vertices.
Exactly solvable many-particle quantum graphs
=============================================
Much of the success of one-particle quantum graph models relies on the fact that eigenvalues possess a simple characterisation in terms of a secular equation based on the finite-dimensional determinant . On the one hand this enables one to compute eigenvalues by searching for zeros of a low-dimensional determinant, and on the other hand it leads to a trace formula that is an identity [@Rot83; @KotSmi99; @BolEnd09] rather than a semiclassical approximation as in other, typical models of quantum systems (see, e.g., [@Gut90]).
The secular equation rests on the fact that the edge-$e$ component of an eigenfunction must be of the form $$\psi_e (x_e) = a_e\,{\mathrm{e}}^{{\mathrm{i}}kx_e} + b_e\,{\mathrm{e}}^{-{\mathrm{i}}kx_e}\ ,$$ with some coefficients $a_e,b_e\in\mathbb{C}$. It provides a sufficient condition that the $2E$ coefficients must satisfy in order to yield an eigenfunction. Components of $N$-particle eigenfunctions with eigenvalue $\lambda$ are functions of $N$ variables, $x_{e_1},\dots,x_{e_N}$, so that, in general, they are of the form $$\label{Neigenfct}
\psi_{e_1\dots e_n}(x_{e_1},\dots,x_{e_N}) = \int_{\mathbb{R}^N}
a_{e_1\dots e_n} (k_1,\dots,k_N)\,\delta(k_1^2+\cdots +k_N^2 -\lambda)\,
{\mathrm{e}}^{{\mathrm{i}}(k_1x_{e_1}+\cdots +k_N x_{e_N})}\,{\mathrm{d}}^N k \ .$$ Hence, instead of the need to determine constants, in generic cases with $N\geq 2$ a replacement for the secular equation needs to determine coefficient functions $a_{e_1\dots e_n} (\cdot)$. This would therefore be a condition imposed on elements of an infinite dimensional space.
However, under certain circumstances such conditions may collapse to a finite dimensional subspace. This, indeed, will be the case if certain integrability conditions are satisfied which imply that eigenfunctions can be represented by a so-called Bethe-ansatz [@Gau14]. In essence, a Bethe-ansatz is a finite sum of plane waves, $$\label{Bethean}
\psi_{\mathrm{Bethe}}(x_1,\dots,x_N) = \sum_{\alpha\in J}A_\alpha\,
{\mathrm{e}}^{{\mathrm{i}}(k^\alpha_1 x_1+\cdots+k^\alpha_N x_N)}\ ,$$ where $J$ is a finite index set, such that the vectors $(k^\alpha_1,\dots,k^\alpha_N)$ with $(k^\alpha_1)^2+\cdots+(k^\alpha_N)^2=\lambda$ are drawn from a finite subset of $\mathbb{R}^N$. Contrasting this with the general form of an $N$-particle eigenfunction suggests that a Bethe-ansatz will only be possible under some strict conditions. These integrability conditions (see, e.g., [@Gau14; @CauCra07]) are also behind the Lieb-Liniger model, for which it has long been known that eigenfunctions can be characterised in terms of a finite number of coefficients [@LL63] and take a Bethe-ansatz form . The first example of a quantum graph with a non-trivial vertex where a Bethe-ansatz was shown to work is a particle on a line or ring with one vertex, where non-Kirchhoff conditions are imposed [@CauCra07]. Since $N$ particles on a graph have a configuration space that is composed of subsets of $\mathbb{R}^N$, a further class of example in this spirit where a Bethe-ansatz for the eigenfunctions is known to exist is given by the Dirichlet- or Neumann Laplacian on a fundamental domain for the action of a Weyl group [@Ber80]. Indeed, the mechanism behind these examples can be carried over to a class of quantum graph models, generalising the approach of [@CauCra07]. This has been done in [@BolGar17; @BolGar17a], and in the following we will review those results.
The simplest example is that of two bosons on an interval $[0,l]$ with Dirichlet boundary conditions at the interval ends and a $\delta$-interaction between the particles. This is a modification of the Lieb-Liniger model first studied by Gaudin [@Gau71]. The two-particle Hilbert space $$\mathcal{H}_2 = L^2(0,l)\otimes L^2(0,l) \cong L(D)\ ,$$ where $D$ is the square that will be dissected as in . Accordingly, $\psi^\pm\in L^2(D^\pm)$, for which the Bethe-ansatz $$\label{BetheGaudin}
\psi^\pm(x_1,x_2) = \sum_{P\in\mathcal{W}_2}A_P^\pm\,
{\mathrm{e}}^{{\mathrm{i}}(k_{P(1)} x_1 + k_{P(2)}x_2)}\ ,$$ can be shown to lead to eigenfunctions. Here $\mathcal{W}_2$ is a Weyl group, which is a finite group with eight elements. The fact that the ansatz is consistent comes from the conditions an eigenfunction has to satisfy:
- $-\Delta\psi=\lambda\psi$;
- $\psi(x_1,x_2)=\psi(x_2,x_1)$;
- $\left(\frac{\partial}{\partial x_1}-\frac{\partial}{\partial x_2}
\right)\psi(x,x)=\alpha\psi(x,x)$;
- $\psi(0,x)=\psi(l,x)=0$.
These conditions are compatible with the plane-wave form $A\,{\mathrm{e}}^{{\mathrm{i}}(k_1 x_1 + k_2 x_2)}$ and only require substitutions of the wave vectors $(k_1,k_2)$ with either $(k_2,k_1)$, $(-k_1,k_2)$, or combinations thereof. These operations, seen as an action of a group on $\mathbb{R}^2$, generate the action of the Weyl group $\mathcal{W}_2=\mathbb{Z}/2\mathbb{Z}\rtimes S_2$. An interesting interpretation of this in terms of reflected rays can be found in [@McG64]. The conditions (i)–(iv) also yield a restriction on the allowed wave vectors, $${\mathrm{e}}^{-2{\mathrm{i}}k_n l}=\frac{k_n+k_m-{\mathrm{i}}\alpha}{k_n+k_m+{\mathrm{i}}\alpha}\,
\frac{k_n-k_m-{\mathrm{i}}\alpha}{k_n-k_m+{\mathrm{i}}\alpha}\ ,$$ for all $n\neq m\in\{1,2\}$. Solutions $(k_1,k_2)\neq(0,0)$ with $0\leq k_1\leq k_2$ then give eigenvalues $\lambda=k_1^2+k_2^2$.
The above model, for $N$ bosons, was first studied by Gaudin [@Gau71; @Gau14]. The original Lieb-Liniger model [@LL63], however, was formulated for particles on a circle. Instead of the Dirichlet conditions (iv) one then has to require periodic boundary conditions, which renders the reflection $(k_1,k_2)\mapsto (-k_1,k_2)$ expendable. The Bethe ansatz for the Lieb-Liniger model, therefore, only requires a summation over the symmetric group $S_2$, rather than over the Weyl group $\mathcal{W}_2=\mathbb{Z}/2\mathbb{Z}\rtimes S_2$ as in . Hence, one concludes that the boundaries of the interval are responsible the additional reflections necessary in the Bethe ansatz. In a graph language, the interval ends are vertices of degree one. Adding a vertex of degree two in the context of a Bethe ansatz was first done in [@CauCra07], where is was found that this is incompatible with $\delta$-pair interactions. Instead, the interactions were modified to include another contribution that formally looks like $\delta(x_1+x_2)$. This means that the particles do not only interact when they touch, but also when they are the same distance away from the vertex on either of the edges connected by the vertex. If then this interaction is provided with a variable strength that is supported in a neighbourhood of the vertex, this will still be a localised interaction.
An extension of this method to arbitrary metric graphs with generalisations of the interactions introduced in [@CauCra07] has been done in [@BolGar17], and an extension to $N$ particles can be found in [@BolGar17a]. The first step is to define the singular pair interactions, and this is most clearly done on a star graph of $d$ half-lines. Then a given metric graph is first converted into its star representation, consisting of $|\mathcal{V}|$ star graphs, i.e., one for each $v\in\mathcal{V}$ of degree $d_v$. A Bethe ansatz is made for each star graph, and then the boundary conditions that represent the pair interactions on each star, as well as the matching conditions that allow to recover the original graph from its star representation imply conditions that characterise eigenvalues of the Laplacian with the singular pair interactions.
If now $\Gamma_v$ is the star graph with $d_v$ half-lines that is associated with the vertex $v\in\mathcal{V}$, the Hilbert space is $\oplus_{ee'}L^2(D_{ee'})$; here $e$ and $e'$ are edge labels and $D_{ee'}=\mathbb{R}_+^2$ is the two-particle configuration space when one particle is on edge $e$ and the other one on $e'$. These configuration spaces are dissected into $D_{ee'}^+$ and $D_{ee'}^-$, which are defined in analogy to , and the restrictions of functions $\psi_{ee'}$ to $D_{ee'}^\pm$ are denoted as $\psi_{ee'}^\pm$. One then requires that $$\label{deltatildecond}
\begin{split}
\psi^+_{ee'}(x,x) &=\psi^-_{e'e}(x,x)\ ,\\
\left(\frac{\partial}{\partial x_1}-\frac{\partial}{\partial x_2}
-2\alpha\right)\psi^+_{ee'}(x,x)
&=\left(\frac{\partial}{\partial x_1}-
\frac{\partial}{\partial x_2}\right)
\psi^-_{e'e}(x,x)\ .
\end{split}$$ These conditions are similar to those generating $\delta$-interactions. However, they apply to all pairs of edges, not only the diagonal ones. Hence there is a singular interaction, also across edges, whenever two particles are the same distance away from the vertex. A Bethe ansatz is then introduced for the functions $\psi_{ee'}^\pm$, with the yet to be determined coefficients $A^\pm_{P,ee'}$. In a next step one has to cut the edges of the stars to the finite lengths that are required and then glue the stars to finally yield the original compact graph. In this glueing process it has to be ensured that the interactions only take place when two edges are connected in the same vertex, and not arbitrarily across the graph. In addition to , this yields conditions to be imposed on the coefficients $A^\pm_{P,ee'}$. These conditions can be formulated in terms of secular equations involving determinants $$\label{sectildedelta}
Z(k_1,k_2) = \det\bigl({\mathds{1}}-U(k_1,k_2)\bigr),$$ where $$U(k_1,k_2) = E(k_2)Y(k_2-k_1)({\mathds{1}}_2\otimes S(k_2)\otimes{\mathds{1}}_{2E})Y(k_1+k_2),$$ and $$\begin{split}
Y(k)&=\frac{1}{k+i\alpha}\begin{pmatrix}-i\alpha&k\\k&-i\alpha\end{pmatrix}\otimes
\boldsymbol{\alpha}+\begin{pmatrix}0&1\\1&0\end{pmatrix}
\otimes({\mathds{1}}_{E^2}-\boldsymbol{\alpha})\mathbb{T}_{E^2} \\
E(k)&={\mathds{1}}_{4E}\otimes\begin{pmatrix}0&1\\1&0\end{pmatrix}\otimes
{\mathrm{e}}^{{\mathrm{i}}k\boldsymbol{l}};
\end{split}$$ Here $\mathbb{T}_{E^2}$ is a permutation matrix, and $\boldsymbol{\alpha}$ is a diagonal matrix with the interaction strengths (which could, in principal, be different for each pair of edges) $\alpha_{ee'}$ on the diagonal. More details can be found in [@BolGar17]. The final result is the following statement.
\[deltatildesecularThm\] Let $-\Delta_2$ be a two-particle Laplacian on a compact metric graph with pair interactions as decribed above. Then the zeros $(k_1,k_2)$, where $0\leq k_1\leq k_2$, of $Z(k_i,k_j)$ for $i\neq j\in\{1,2\}$ of order $m$ correspond to eigenvalues $k_1^1+k_2^2$ of $-\Delta_2$.
We note that, with in mind, the secular equations are reminiscent of the one-particle case . Some special cases and numerical results in some example can be found in [@BolGar17]. The generalisation to $N$ particles follows the same lines and is contained in [@BolGar17a].
Many-particle models on a simple non-compact graph {#Sec3}
==================================================
In this section we are concerned with interacting two-particle systems on a simple non-compact quantum graph, namely the positive half-line $\mathbb{R}_+=[0,\infty)$. More specifically, the Hamiltonian has several contributions: a hard-wall binding potential and two singular contributions, one of which is of the vertex-induced type defined in Section \[Sec2.1\] and the other one representing the contact interactions introduced in Section \[sec2.2\]. The Hamiltonian is formally given by $$\label{FormalHamiltonian}
H=-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}+v_{b}(|x-y|)+v(x,y)\left[\delta(x)+\delta(y)\right]\ + \alpha(y) \delta(x-y)\ ,$$ where $v_{b}:\mathbb{R}_+ \rightarrow \overline{\mathbb{R}_+}$ is a (hard-wall) binding potential that is (formally) defined via $$v_{b}(x):=\begin{cases} 0 \quad \text{for} \quad x\leq d \ , \\
\infty \quad \text{otherwise}\ ,
\end{cases}$$ where $d > 0$ characterises the size of the pair. We realise this formal potential by requiring Dirichlet boundary conditions at $|x-y|=d$. Furthermore, $v:\mathbb{R}^2_+ \rightarrow \mathbb{R}$ is supposed to be a real-valued, symmetric and bounded potential, $v \in L^{\infty}(\mathbb{R}^2_+)$. Note that setting $d=\infty$ corresponds to the case where no binding potential in is added. Also note that we always assume $\alpha(\cdot) \in L^{\infty}(\mathbb{R}_+)$.
It is important to note that interactions of the form generically break translation invariance, even with potentials $v(x,y)=v(|x-y|)$, and consequently lead to non-separable many-body problems. Although only rarely discussed in the literature, they have important applications in various areas of physics [@glasser1993solvable; @glasser2005solvable].
Other important situations in which singular interactions of the above form are expected can be found in solid-state physics. For example, similar to the Cooper pairing mechanism of superconductivity [@CooperBoundElectron], two electrons in a metal can effectively interact with each other through the interaction of each individual particle with the lattice via electron-phonon-electron interactions. Hence, if a metal exhibits spatially localised defects, there will be effective, spatially localised two-particle interactions. Furthermore, the idea to consider a binding potential in , which effectively leads to a ‘molecule’, or a pair of particles, also originated from Cooper’s work [@CooperBoundElectron]; another example can be found in [@QUnruh], where the scattering of a bound pair of particles at mirrors is investigated. As a matter of fact, it was Cooper who realised that electrons in a metal will form pairs (Cooper pairs), if the metal is in the superconducting phase, i.e., is cooled below some critical temperature [@BCSI; @MR04]. Hence, the Hamiltonian , or versions thereof, provide toy models to investigate bound pairs of particles in a quantum wire with defects [@KMBound; @KernerRandomI]. Most importantly, in this model one can derive rigorous results related to the superconducting behaviour of quantum wires [@KernerElectronPairs; @KernerSurfaceDefects; @KernerInteractingPairs].
The model without hard-wall binding potential
---------------------------------------------
In this subsection we present results regarding the Hamiltonian without binding potential, i.e., $v_{b}\equiv 0$. For more detail, we refer to [@BKSingular; @BKContact; @KM16; @EggerKerner] from which most of the results are taken.
In a first step one has to give a rigorous meaning to the Hamiltonian , which is only formally defined due the $\delta$-distributions. This requires a suitable variant of Theorem \[2quadformgraph\] and Section \[sec2.2\]. (Note that here the two-particle configuration space without binding potential is $\mathbb{R}^2_+$). In order to do this one constructs a quadratic form on $L^2(\mathbb{R}^2_+)$, $$\label{FormNoBinding}
q^{\infty}_{\alpha,\sigma}[\varphi]:=\int_{\mathbb{R}^2_+}|\nabla \varphi|^2 \ {\mathrm{d}}x - \int_{0}^{\infty}
\sigma(y)|\gamma(\varphi)|^2 {\mathrm{d}}y + \int_{0}^{\infty}\alpha(y)|\varphi(y,y)|^2 \ {\mathrm{d}}y\ ,$$ where $\sigma(y):=-v(0,y)$ is a real-valued boundary potential and $\gamma(\varphi):=\left(\varphi(y,0),
\varphi(0,y)\right)^T$ are the boundary values of $\varphi \in H^1(\mathbb{R}^2_+)$, which are well-defined in $L^2(\partial \mathbb{R}^2_+)$ due to the trace theorem for Sobolev functions [@Dob05]. In the same way one defines $\varphi|_{x=y}$ as the trace of $\varphi \in H^1(\mathbb{R}^2_+)$ along the diagonal $x=y$.
For any given $\sigma,\alpha \in L^{\infty}(\mathbb{R}_+)$ the form $\left(q^{\infty}_{\alpha,\sigma},H^1(\mathbb{R}^2_+) \right)$ is bounded from below and closed.
Hence, according to the representation theorem for quadratic forms [@BEH08] there exists a unique self-adjoint operator associated with the form $q^{\infty}_{\alpha,\sigma}$. We denote this operator by $-\Delta^{d=\infty}_{\sigma,\alpha}$. Since the only volume term in is associated with the $\nabla$-operator, this operator acts as the standard two-dimensional Laplacian $-\Delta$ on functions $\varphi \in {{\mathcal D}}(-\Delta^{d=\infty}_{\sigma,\alpha}) \subset H^1(\mathbb{R}^2_+)$. The boundary integrals in , on the other hand, reflect boundary conditions. More explicitly, one has coordinate-dependent Robin conditions along $\partial \mathbb{R}^2_+$, and coordinate-independent jump conditions along the diagonal $x=y$, see [@BKContact; @KM16] for more detail.
In a next step we characterise the spectrum of the self-adjoint operator $-\Delta^{d=\infty}_{\sigma,\alpha}$.
\[TheoremEssentialSpectrumI\] For any given $\sigma,\alpha \in L^{\infty}(\mathbb{R}_+)$ one has $[0,\infty) \subset \sigma_{ess}(-\Delta^{d=\infty}_{\sigma,\alpha})$. Furthermore, if $\sigma(y),\alpha(y) \rightarrow 0$ as $y \rightarrow \infty$ one has $\sigma_{ess}(-\Delta^{d=\infty}_{\sigma,\alpha=0})=[0,\infty)$.
See the proof of \[Theorem 3.1,[@KM16]\] for the case where $\alpha=0$. An inspection of this proof then allows one to conclude the statement above.
The discrete part of the spectrum, i.e., isolated eigenvalues with finite multiplicity, is characterised in the following statement.
Assume that $\sigma,\alpha \in L^1(\mathbb{R}_+)$ and that $\inf \sigma_{ess}(-\Delta^{d=\infty}_{\sigma,\alpha})=0$. Then, if $$\int_{\mathbb{R}_+} \left[2\sigma(y)-\alpha(y)\right] \ {\mathrm{d}}y > 0\ ,$$ negative eigenvalues will exist.
As in the proof of \[Theorem 3.3,[@KM16]\] one picks the test function $\varphi_{\varepsilon}(r):=e^{-r^{\varepsilon}}$, $\varepsilon > 0$, defined in polar coordinates. Evaluating $q^{\infty}_{\alpha,\sigma}[\varphi_{\varepsilon}]$ one performs the limit $\varepsilon \rightarrow 0$ to conclude that $q^{\infty}_{\alpha,\sigma}[\varphi_{\varepsilon}] < 0$ for small enough $\varepsilon$. The statement then follows by the variational principle [@BEH08]. Note that the factor of $2$ is due to the fact the there are two boundary segments of $\mathbb{R}^2_+$.
Assume that $\sigma,\alpha \in L^{\infty}(\mathbb{R}_+)$ have bounded support. Then there exist only finitely many negative eigenvalues.
The statement follows from a bracketing argument, see [@BEH08] for a general discussion and [@KM16; @KMBound] for applications of this technique.
In a first step one writes $\mathbb{R}^2_+=B_R(\mathbb{R}^2_+) \cup \ (\mathbb{R}^2_+ \setminus B_R(\mathbb{R}^2_+))$ where $B_R(\mathbb{R}^2_+):=\{(x,y)\in \mathbb{R}^2_+: x^2+y^2 < R^2 \}$. The comparison operator then is a direct sum of two two-dimensional Laplacians, i.e., $$-\Delta_{B_R(\mathbb{R}^2_+)} \oplus -\Delta_{\mathbb{R}^2_+ \setminus B_R(\mathbb{R}^2_+)}$$ with the same boundary conditions as $-\Delta^{d=\infty}_{\sigma,\alpha}$, except for additional Neumann boundary conditions along the dissecting line. We then choose $R$ large enough so that $\sigma=\alpha=0$ in $\mathbb{R}^2_+ \setminus B_R(\mathbb{R}^2_+)$. Accordingly, $-\Delta_{\mathbb{R}^2_+ \setminus B_R(\mathbb{R}^2_+)} $ is a positive operator. On the other hand, $-\Delta_{B_R(\mathbb{R}^2_+)}$ is defined on a bounded Lipschitz domain and hence has purely discrete spectrum, i.e., its essential spectrum is empty and there are only finitely many negative eigenvalues. Consequently, the operator bracketing $$-\Delta_{B_R(\mathbb{R}^2_+)} \oplus -\Delta_{\mathbb{R}^2_+ \setminus B_R(\mathbb{R}^2_+)}
\leq -\Delta^{d=\infty}_{\sigma,\alpha}$$ implies the statement.
The model with hard-wall binding potential
------------------------------------------
The model with non-vanishing binding potential, but vanishing contact interaction, was first studied in [@KMBound]. The first important difference to the case where $v_{b}\equiv 0$ is that the two-particle configuration space is reduced from $\mathbb{R}^2_+$ to the ‘pencil-shaped’ domain $$\Omega:=\{(x,y)\in \mathbb{R}^2_+:\ |x-y| \leq d \}\ .$$ Hence, the underlying Hilbert space is $L^2(\Omega)$ rather than $L^2(\mathbb{R}^2_+)$. As before, a rigorous realisation of is obtained via the form $$q^{d}_{\alpha,\sigma}[\varphi]:=\int_{\Omega}|\nabla \varphi|^2 \ {\mathrm{d}}x - \int_{0}^{d}\sigma(y)|
\gamma(\varphi)|^2 {\mathrm{d}}y + \int_{0}^{\infty}\alpha(y)|\varphi(y,y)|^2 \ {\mathrm{d}}y\ ,$$ which is defined on ${{\mathcal D}}_{q}:=\{\varphi \in H^1(\Omega): \varphi|_{\partial \Omega_D}=0 \}$, where $\partial \Omega_D:=\{(x,y)\in \mathbb{R}^2_+:\ |x-y|=d \}$. Note that the Dirichlet boundary conditions along $\partial \Omega_D$ are due to the choice of the hard-wall binding potential.
For every $\sigma, \alpha \in L^{\infty}(\mathbb{R}_+)$ the form $\left(q^{d}_{\alpha,\sigma},{{\mathcal D}}_{q}\right)$ is bounded from below and is closed.
As before, the representation theorem of forms assures the existence of a unique self-adjoint operator associated with $q^{d}_{\alpha,\sigma}$ which shall be denoted by $-\Delta^{d}_{\sigma,\alpha}$. Again, this operator acts as the standard two-dimensional Laplacian with coordinate dependent Robin boundary conditions along the boundary segments with $x=0$ or $y=0$ as well as a jump condition along the diagonal $x=y$ as before, see \[Remark 1,[@KMBound]\] for a more detail.
So far the presence of a binding potential made no difference. However, as soon as we characterise the spectrum of $-\Delta^{d}_{\sigma,\alpha}$, the effect of the binding potential becomes obvious.
\[EssentialSpectrumII\] Assume that $\sigma,\alpha \in L^{\infty}(\mathbb{R}_+)$ are given. Then $[2\pi^2/d^2,\infty) \subset \sigma_{ess}(-\Delta^{d}_{\sigma,\alpha})$. Furthermore, if $\alpha(y) \rightarrow 0$ as $y \rightarrow \infty$ one has $\sigma_{ess}(-\Delta^{d}_{\sigma,\alpha})=[\pi^2/2d^2,\infty)$.
We only add some remarks, see \[Theorem 2,[@KMBound]\] and the proof of Theorem \[TheoremEssentialSpectrumI\] for more detail.
In order to show the first part, one takes a Weyl sequence which consists of (normalised) ground states of the Dirichlet-Laplacian on rectangles $[0,L] \times [0,d/\sqrt{2}]$ that are placed in $\Omega$ such that the ‘L-boundary segments’ touch $\partial \Omega_D$ as well as the diagonal $x=y$. The form $q^{d}_{\alpha,\sigma}$ evaluated for these states gives $2\pi^2/d^2+\pi^2/L^2$. Hence, letting $L$ tend to any limit (including infinity) the statement follows. Note that the integral along the diagonal does not contribute due to the Dirichlet boundary conditions.
Regarding the second part one takes basically the same ground states but on rectangles $[0,L] \times [0,\sqrt{2}d]$. Now, the contribution of the integral along the diagonal does not vanish but, since $\alpha(y) \rightarrow 0$ as $y \rightarrow \infty$, it can be made arbitrarily small. This proves the statement.
Theorem \[EssentialSpectrumII\] illustrates that, as long as the contact interaction strength converges to zero, the binding potential leads to a shift of the essential spectrum by at least $\pi^2/2d^2$. As for the effect on the discrete spectrum we first note that whenever $d=\infty$, by Theorem \[TheoremEssentialSpectrumI\] this is trivial for $\sigma=\alpha=0$. From a physical point of view this seems reasonable since there is no attractive potentials that could lead to bound states. However, quite surprisingly, for $d < \infty$ and $\sigma=\alpha=0$ we have the following result \[Theorem 3,[@KMBound]\].
\[TheoremDiscreteSpectrumPencil\] Consider the self-adjoint operator $-\Delta^{d}_{\sigma=0,\alpha=0}$, i.e., we assume that $\sigma=\alpha=0$. Then $$\sigma_{d}(-\Delta^{d}_{\sigma=0,\alpha=0}) \neq \emptyset\ .$$ In other words, there exist eigenvalues below $\pi^2/2d^2$.
Note that the existence of eigenvalues smaller than $\pi^2/2d^2$ for vanishing boundary and contact potential is a purely quantum mechanical effect. Furthermore, it is a geometrical effect since no non-trivial discrete spectrum would exist if one considered the two-particle system on the full line $\mathbb{R}$ instead of the half-line $\mathbb{R}_+$, see \[Remark 4,[@KMBound]\].
Of course, if one assumes that $\sigma(y) \geq 0$ for a.e. $y \in [0,d]$, and $\alpha(y)\leq 0$ for a.e. $y \in [0,\infty)$, the discrete spectrum will also be non-empty since the boundary integrals in $q^{d}_{\alpha,\sigma}$ are negative (note here the minus sign in the definition of the boundary potential $\sigma$). However, one may ask what happens when a positive boundary potential $\sigma$ becomes large. Since this implies a strong repulsive singular two-particle interaction localised at the origin, bound states may no longer exist. Indeed, we have the following result.
\[AbsenceDiscreteSpectrum\] There exists a constant $\gamma < 0$ such that $\sigma_{d}(-\Delta^{d}_{\sigma,\alpha})=\emptyset$ whenever $\sigma(y) \leq \gamma$ for a.e. $y \in [0,d]$, $\alpha(y) \geq 0$ for a.e. $y \in [0,\infty)$ and $\alpha(y) \rightarrow 0$ as $y \rightarrow \infty$.
Withouth contact potential $\alpha$ this result has been shown in \[Theorem 4,[@KMBound]\].
Now, by Theorem \[EssentialSpectrumII\] we conclude that $\inf \sigma_{ess}(-\Delta^{d}_{\sigma,\alpha})=\pi^2/2d^2$. Furthermore, since $\alpha$ is assumed to be strictly positive, the corresponding operator is larger (in the sense of an operator bracketing) than the operator with same boundary potential $\sigma$, but without contact potential. Consequently, if there existed an eigenvalue smaller than $\pi^2/2d^2$ the same would hold for the operator without contact potential. This, however, is in contradiction with \[Theorem 4,[@KMBound]\].
Theorem \[AbsenceDiscreteSpectrum\] shows that strong singular interactions at the origin (without contact interaction) destabilise the system in the sense that no discrete spectrum is present anymore when compared to the free system with $\sigma=\alpha=0$.
Random singular pair interactions
---------------------------------
In this subsection we consider a generalisation of the Hamiltonian in the sense that the singular, vertex-induced pair interactions are not only present in the origin or the vertex of the graph. This seems desirable since, as described previously, localised two-particle interactions can be associated with defects in the metal and such defects occur, of course, not only at the origin but everywhere in the wire. We note that this model was formulated in [@KernerRandomII] to which we also refer for more detail.
Since the spatial positions of defects in a real metal varies from metal to metal it seems reasonable not to work with a specific (deterministic) two-particle Hamiltonian, but with a random one. In other words, in this section we enter the realm of random Schrödinger operators which in recent years have become an important research area [@stollmann2001caught; @KirschInvitation]. Most importantly, using the language of random Schrödinger operators, one has been able to give a rigorous description of various phenomena in physics such as Anderson localisation [@AndersonLocalisatioin; @SimonBookSchrödinger].
Turning to our model, we consider a system of two particles on the half-line $\mathbb{R}_+$ whose random Hamiltonian shall formally be given by $$\label{RandomHamiltonian}
H_{\omega}=-\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial y^2}+v_b(|x-y|)+\sum_{i=1}^{\infty}
v_{i}(x,y)\left[\delta(x-a_i(\omega))+\delta(y-a_i(\omega)) \right]\ ,$$ where $(a_i(\omega))_{i \in \mathbb{N}}$ are the random positions of the defects, called atoms in the sequel. As before we assume that $(v_i)_{i \in \mathbb{N}}$ are real-valued, bounded and symmetric, $v_i(x,y)=v_i(y,x)$. Furthermore, we define $l_i(\omega):=a_{i}(\omega)-a_{i-1}(\omega)$ for $i \geq 1$ and set $a_0(\omega):=0$. In other words, $l_i(\omega)$ is the random distance between the $i-1$-st and the $i$-th atom.
Now we consider the lengths $(l_i(\omega))_{i \in \mathbb{N}}$ as a family of independent random variables over some probability space $(\Pi,\xi,\mathbb{P})$ generated by a Poisson process, see [@StolzPoisson] for more detail. More explicitly, we assume that the probability for the length $l_i(\omega)$ to be in the interval $[a,b]$ is given by $$\mathbb{P}\left[l_i \in [a,b] \right]=\nu \int_{a}^{b}{\mathrm{e}}^{-\nu l} {\mathrm{d}}l \ ,$$ with $\nu > 0$ denoting the Poisson density.
Again, due to the presence of $\delta$-potentials in we shall again use a suitable quadratic form to rigorously construct a self-adjoint operator that is associated with the formal expression . We introduce $$\label{RandomForm}
q_{\omega}[\varphi]=\int_{\Omega}|\nabla \varphi|^2 \ {\mathrm{d}}x+ \sum_{i=1}^{\infty}
\int_{\Gamma_i(\omega)}\sigma_i(y)|\gamma_{i}(\varphi)|^2 \ {\mathrm{d}}y\ ,$$ where $\gamma_{i}(\varphi)$ denotes the restriction (in the sense of traces of Sobolev functions) to $$\Gamma_i(\omega):=\{(x,y)\in \Omega: x=a_i(\omega) \ \text{or}\ y=a_i(\omega) \}\ .$$ Furthermore, we set $\sigma_i(y):=-v_{i}(0,y)$. Due to the infinite sum appearing in it may not be possible to define $q_{\omega}$ on all of ${{\mathcal D}}_{q} \subset H^1(\Omega)$. Since we want to find a closed realisation of the form $q_{\omega}$ we have to guess a suitable sub-domain. Indeed, one has the following result \[Theorem 2.1,[@KernerRandomII]\].
Let $(\sigma_i(\omega))_{i \in \mathbb{N}} \subset L^{\infty}(\mathbb{R}_+)$ be given. Then the form $q_{\omega}$ on the domain $${{\mathcal D}}_{q}(\omega)=\{\varphi \in H^1(\Omega): q_{\omega}[\varphi] < \infty \}$$ is positive and closed for almost every $\omega \in \Pi$.
We denote the unique self-adjoint operator associated with the form $q_{\omega}$ as $-\Delta_{\sigma}(\omega)$.
The random Schrödinger operators usually considered in the literature have an astonishing property, namely that the spectrum is almost surely non-random [@PasturFigotin; @KirschInvitation], which is due to a certain ergodicity property of the models. For our model, we will see that only the essential part of the spectrum is non-random. The discrete part, however, is random. Indeed we have the following results \[Theorem 3.1, Lemma 3.4, Theorem 3.5,[@KernerRandomII]\].
Let $(\sigma_i(\omega))_{i \in \mathbb{N}} \subset L^{\infty}(\mathbb{R}_+)$ be given. Then $$\sigma_{ess}(-\Delta_{\sigma}(\omega))=[\pi^2/2d^2,\infty)$$ almost surely.
The discrete part of the spectrum, on the other hand, is random. More explicitly, we obtain the following result.
\[TheoremAbsenceDiscreteSpectrum\] Let $(\sigma_i(\omega))_{i \in \mathbb{N}} \subset L^{\infty}(\mathbb{R}_+)$ be given. Then $$\mathbb{P}[\sigma_{d}(-\Delta_{\sigma}(\omega))\neq\emptyset] > 0\ .$$ Furthermore, there exists a constant $\gamma=\gamma(d) > 0$ such that if $\inf \sigma_k > \gamma$ for one $k\in \mathbb{N}$ then $$\mathbb{P}[\sigma_{d}(-\Delta_{\sigma}(\omega))=\emptyset] > 0\ .$$
Theorem \[TheoremAbsenceDiscreteSpectrum\] tells us that the discrete part of the spectrum is destroyed with finite probability as well as conserved with finite probability. This leads to an interesting physical implication: In general, disorder is associated with a suppression of transport as in the Anderson localisation phenomenon. However, assuming that no dense pure point spectrum is created in $[\pi^2/2d^2,\infty)$ and that the density of states does not change, Theorem \[TheoremAbsenceDiscreteSpectrum\] implies that disorder may lead to an improvement of transport with finite probability according to ‘Fermi’s golden rule’.
The condensation of electron pairs in a quantum wire {#SectionCondensationPairs}
----------------------------------------------------
In this subsection we want to report on the results that were obtained in [@KernerElectronPairs; @KernerSurfaceDefects; @KernerInteractingPairs]. Since we are interested in pairs of particles, we consider the case where $d< \infty$, i.e., we assume that a hard-wall binding potential $v_{b}$ is present. In the previous sections we worked on the full Hilbert space $L^2(\Omega)$ describing two distinguishable and spinless particles. However, since we are interested in applying the Hamiltonian to understand superconductivity, which involves electrons, we need to implement the exchange symmetry of identical particles.
In this review we restrict ourselves to the case considered in [@KernerElectronPairs] where the two electrons are assumed to have the same spin. This leads to the requirement that the two-particle wave function has to be anti-symmetric. The case of opposite spin, which is realised in actual Cooper pairs, is considered in [@KernerInteractingPairs]. We only mention here that the results regarding the condensation there are comparable.
In order to ensure anti-symmetry of the wave function we work in the anti-symmetric subspace $$L^2_{a}(\Omega):=\{\varphi \in L^2(\Omega):\ \varphi(x,y)=-\varphi(y,x) \}\ .$$ We then introduce the quadratic form $$\label{FormCondensation}
q^{d}_{\sigma}[\varphi]:=\frac{\hbar^2}{2m_e}\int_{\Omega}|\nabla \varphi|^2 \ {\mathrm{d}}x - \int_{0}^{d}\sigma(y)|\gamma(\varphi)|^2 {\mathrm{d}}y \ ,$$ where $\sigma \in L^{\infty}(\mathbb{R}_+)$, on this subspace. Here we added physical constants with $m_e$ denoting the electron mass. The domain of the form is given by ${{\mathcal D}}_{q}:=\{\varphi \in H^1(\Omega)\cap L^2_{a}(\Omega): \ \varphi|_{\partial \Omega_D}=0 \}$. Again, this form is closed and bounded from below, and hence there exists a unique self-adjoint operator associated with this form. This is the Hamiltonian of our two-particle system. We denote this operator, which again acts as the standard two-dimensional Laplacian, as $-\Delta^{d}_{\sigma}$.
\[TheoremBoundElectrons\] One has $$\sigma_{ess}(-\Delta^{d}_{\sigma})=[\hbar^2 \pi^2 /m_e d^2,\infty)\ .$$ Furthermore, if $\sigma=0$ then $$\sigma_{d}(-\Delta^{d}_{\sigma=0})=\{E_0 \}\ ,$$ i.e., there is exactly one eigenvalue with multiplicity one below the bottom of the essential spectrum. In addition, one has $$0.25 \cdot \frac{\hbar^2 \pi^2}{m_ed^2}\leq E_0 \leq 0.93 \cdot \frac{\hbar^2 \pi^2}{m_ed^2}\ .$$
Theorem \[TheoremBoundElectrons\] has an interesting physical consequence: one important measurable quantity associated with the superconducting phase of a metal is the so-called spectral gap $\Delta > 0$, see [@MR04]. This spectral gap is responsible, for example, for the exponential decay of the specific heat at temperatures lower than the critical one. It is one of the successes of the BCS-theory that the spectral gap can be interpreted as the binding energy of a single Cooper pair. In other words, the spectral gap measures the energy necessary to break up one Cooper pair. Due to the choice of the hard-wall binding potential, in our model the pair cannot be broken up. However, it is possible to excite a pair. Since, as we will see later, the pairs condense into the ground state it seems reasonable to identify the spectral gap as the excitation of a pair from the ground state to the first excited states. In other words, in our model one obtains the relation $$\label{SpectralGap}
\Delta=\Delta(d) \sim \frac{\hbar^2 \pi^2}{m_ed^2}$$ for the spectral gap. This relation establishes a direct link between the spatial extension of a pair and the spectral gap. In particular, since the spectral gap in superconducting metals is of order $10^{-3}$eV [@MR04], the relation implies that $d$ is of the order $10^{-6}$m. Interestingly, this agrees with Cooper’s estimate as presented in [@CooperBoundElectron].
In order to study the condensation phenomenon (similar to BEC as in Section \[secBEC\]) of electron pairs one has to employ methods from quantum statistical mechanics (see, e.g., [@SchwablSM]). In particular, one has to perform a thermodynamic limit as in Section \[secBEC\], and this requires to restrict the system from the half-line to the interval $[0,L]$. The underlying Hilbert space then is $L^2_{a}(\Omega_L)$, with $$\Omega_L:=\{(x,y) \in \Omega: 0 \leq x,y \leq L\}\ .$$ The natural generalisation of is defined on the domain ${{\mathcal D}}_{q_{L}}:=\{\varphi \in H^1(\Omega_L)\cap L^2_{a}(\Omega_L):\ \varphi|_{\partial \Omega_{L,D}}=0\}$ with $\partial \Omega_{L,D}:=\{(x,y) \in \partial \Omega_L:\ |x-y|=d \ \text{or}\ x=L \ \text{or}\ y=L \}$. In other words, one introduces additional Dirichlet boundary conditions along the dissecting lines $x=L$ and $y=L$. We denote the associated self-adjoint operator by $-\Delta^{d}_{\sigma,L}$.
Since $\Omega_L$ is a bounded Lipschitz domain, $-\Delta^{d}_{\sigma,L}$ has purely discrete spectrum. We denote its corresponding eigenvalues, counted with multiplicity, by $\{E^{\sigma}_n(L) \}_{n\in \mathbb{N}_0}$.
\[ConvergenceEigenvalues\] Assume that $\sigma=0$. Then $$\lim_{L \rightarrow \infty}E^{\sigma=0}_0(L)=E_0\ .$$ Furthermore, $E^{\sigma=0}_n(L) \geq \frac{\hbar^2\pi^2}{m_ed^2}$ for all $n \geq 1$ and $L > d$.
Lemma \[ConvergenceEigenvalues\] implies the existence of a finite spectral gap in the thermodynamic limit which eventually is responsible for the condensation of the pairs.
Recalling Definition \[BECdef\], we can now establish the main result of this section.
\[CondensationPairsTheorem\] For $\sigma=0$ there exists a critical density $\rho_{crit}(\beta)$ such that the ground state is macroscopically occupied in the thermodynamic limit for all pair densities $\rho > \rho_{crit}(\beta)$. Furthermore, there exists a constant $\gamma < 0$ such that, for all pair densities $\rho > 0$, no eigenstate is macroscopically occupied if $\|\sigma\|_{\infty} < \gamma$.
For the proof see the proofs of \[Theorem 3.3 and Theorem 3.6,[@KernerElectronPairs]\] as well as \[Theorem 4.4,[@KernerInteractingPairs]\].
Theorem \[CondensationPairsTheorem\] shows that the pairs condense in the quantum wire given that there are no repulsive singular two-particle interactions localised at the origin. However, if the singular interactions are strong enough, the condensate in the ground state will be destroyed. Hence, if one identifies the superconducting phase with the presence of a condensate of pairs (here in an eigenstate for non-interacting pairs), Theorem \[CondensationPairsTheorem\] shows that the superconducting phase in a quantum wire can be destroyed by singular two-particle interactions.
The impact of surface defects on the superconducting phase
----------------------------------------------------------
In this final section we report on yet another application of the two-particle model introduced above which was presented in [@KernerSurfaceDefects]. More explicitly, we extend the model characterised by the form and the associated Hamiltonian $-\Delta^{d}_{\sigma}$ defined on the anti-symmetric Hilbert space $L^2_{a}(\Omega)$. However, we will only consider the case where there are no singular, vertex-induced two-particle interactions at the origin, i.e., we set $\sigma=0$.
In Section \[SectionCondensationPairs\] we investigated the (Bose-Einstein) condensation of pairs of electrons. Theorem \[CondensationPairsTheorem\] shows that the pairs condense into the ground state if no singular interactions are present and given the pair density $\rho > 0$ is large enough. Also, the presence of condensation is paramount for the existence of the superconducting phase. Real metals are never perfect and there exist defects that affect the behaviour of electrons in the bulk. However, besides defects in the bulk, a real metal will also exhibit defects on the surface, i.e., a real surface will not be arbitrarily smooth. Note that the existence of a surface is, to a first approximation, not taken into account in most discussions in solid state physics, since the solid is modelled to be infinitely extended in order to conserve periodicity. However, it has also long become clear that surface effects cannot be neglected altogether [@FossheimSuperconducting]. It is aim of this section to introduce a model to investigate the effect of surface defects on the superconducting phase in the bulk of a quantum wire by investigating their effect on the condensation of electron pairs in the bulk.
In order to take surface defects into account we have to extend our Hilbert space. More explicitly, we set $$\label{HilbertSpaceSurface}
{{\mathcal H}}:=L^2_{a}(\Omega) \oplus \ell^2(\mathbb{N})\ ,$$ where $\ell^2(\mathbb{N})$ is the space of square-summable sequences. Consequently, a given pair of electrons is described by a state of the form $\left(\varphi,f\right)^T$, with $\varphi \in L^2_{a}(\Omega)$ and $f \in \ell^2(\mathbb{N})$. This means that we model the surface defects as the vertices of the discrete graph $\mathbb{N}$, which seems reasonable in a regime where the spatial extension of those defects is small compared to the bulk.
The Hamiltonian of a free pair of electrons on ${{\mathcal H}}$ is given by $$H_{p}:=-\Delta^{d}_{\sigma=0} \oplus \mathcal{L}(\gamma)\ ,$$ where $\mathcal{L}(\gamma)$ is the (weighted) discrete Laplacian acting via $$(\mathcal{L}(\gamma)f)(n)=\sum_{m=1}^{\infty}\gamma_{mn}\left(f(m)-f(n)\right)\ ,$$ with $(\gamma)_{mn}=:\gamma_{mn}=\delta_{|n-m|,1}e_n$ and $(e_n)_{n \in \mathbb{N}} \subset \mathbb{R}_+$.
Now, since we are interested in the condensation phenomenon we have to restrict the system to a finite volume as we have done in the previous section. More explicitly, the finite volume Hilbert space is given by $${{\mathcal H}}_L:=L^2_{a}(\Omega_L) \oplus \mathbb{C}^{n(L)}\ ,$$ where $n(L) \in \mathbb{N}$ denotes the number of surface defects in the interval $[0,L]$. On this Hilbert space one considers $H^L_p$, i.e., the restriction of $H_p$ to the finite-volume Hilbert space ${{\mathcal H}}_L$. This operator has purely discrete spectrum and the eigenvalues are the union of those coming from $-\Delta^{d}_{\sigma}|_{L^2_{a}(\Omega_L)}$ (where this operator is defined as in the previous section) and $\mathcal{L}(\gamma)|_{\mathbb{C}^{n(L)}}$.
In order to formulate the model it is convenient to use the formalism of second quantisation [@MR04]. This means that one works on the Fock space over ${{\mathcal H}}_L$, rather than on ${{\mathcal H}}_L$ itself. The second quantisation of $H^L_p$ is given by $$\label{SecondQuantisationFreeHamiltonian}
\Gamma(H^L_p)=\sum_{n=0}^{\infty}E^{\sigma=0}_{n}(L)a^{\ast}_{n}a_{n}+\sum_{k=1}^{n(L)}\lambda_k(L)b^{\ast}_{k}b_{k}\ ,$$ where $(E^{\sigma=0}_{n}(L))_{n \in \mathbb{N}_0}$ are the eigenvalues of $-\Delta^{d}_{\sigma=0}|_{L^2_{a}(\Omega_L)}$ and $(\lambda_k(L))_{k=1,...,n(L)}$ are the eigenvalues of $\mathcal{L}(\gamma)|_{\mathbb{C}^{n(L)}}$, counted with multiplicity. Furthermore, $(a^{\ast}_n,a_n)$ are the creation and annihilation operators of the states $\varphi_n \oplus 0$, where $\varphi_n \in L^2_a(\Omega_L)$ are the corresponding eigenstate of $-\Delta^{d}_{\sigma=0}|_{L^2_{a}(\Omega_L)}$. In contrast, $(b^{\ast}_{k},b_{k})$ are the creation and annihilation operators of the states $0 \oplus f_n$, where $f_n \in \mathbb{C}^{n(L)}$ are the corresponding eigenstates of $\mathcal{L}(\gamma)|_{\mathbb{C}^{n(L)}}$. To obtain the full Hamiltonian of the model we extend the free Hamiltonian and write $$\label{ModelHamiltonianSurface}
H_L(\rho_s,\alpha,\lambda)=\Gamma(H^L_p)-\alpha\sum_{k=1}^{n(L)}b^{\ast}_{k}b_{k}+
\lambda\rho_s(\mu_L,L)\sum_{k=1}^{n(L)}b^{\ast}_{k}b_{k}\ ,$$ where $\alpha \geq 0$ describes the surface tension; $\lambda \geq 0$ is an interaction strength associated with the repulsion of the pairs in the surface defects and $\rho_s(\mu_L,L)$ is the density of pairs on $\mathbb{C}^{n(L)}$, see the equation below. Note here that $\sum_{k=1}^{n(L)}b^{\ast}_{k}b_{k}$ is the (surface-) number operator whose expectation value equals the number of pairs in the surface defects. Also, the third term in is added to take into account repulsive interactions between electron pairs accumulating in the surface defects which are expected since the surface defects are imagined to be relatively small. The explicit form of this term follows from a simplification of standard mean-field considerations where the interaction term is generally of the form $\lambda \hat{N}^2/V$, where $\hat{N}$ is the number operator and $V$ is the volume of the system. In other words, we have replaced $\hat{N}/V$ by the density $\rho_s(\mu_L,L)$ for which $$\rho_s(\mu_L,L):=\frac{\omega^{H_L(\rho_s,\alpha,\lambda)}_{\beta,\mu_L}
\left(\sum_{k=1}^{n(L)}b^{\ast}_{k}b_{k}\right)}{n(L)}$$ holds with $\omega^{H_L(\rho_s,\alpha,\lambda)}_{\beta,\mu_L}(\cdot)$ denoting the Gibbs state of the grand-canonical ensemble at inverse temperature $\beta=1/T$ and chemical potential $\mu_L$.
The advantage of the Hamiltonian $H(\rho_s,\alpha,\lambda)$ is that it can be rewritten as $$\label{RewrittenHamiltonian}
H_L(\rho_s,\alpha,\lambda)=\sum_{n=0}^{\infty}E^{\sigma=0}_{n}(L)a^{\ast}_{n}a_{n}+
\sum_{k=1}^{n(L)}\left(\lambda_k(L)+\lambda\rho_s(\mu_L,L)-\alpha\right) b^{\ast}_{k}b_{k} \ ,$$ which yields an effective, non-interacting many-pair model with shifted eigenvalues for the discrete part. Note that, in particular, implies $\mu_L < \min\{\lambda\rho_s(\mu_L,L)-\alpha,E_0(L) \}$, taking into account that $\lambda_1(L)=0$.
The goal then is to investigate the macroscopic occupation of the ground state $\varphi_0 \oplus 0$ in a suitable thermodynamic limit (see [@KernerSurfaceDefects] for details) for the Hamiltonian . It turns out that a key quantity is the inverse density of surface defects $\delta > 0$ defined as $$\delta:=\lim_{L \rightarrow \infty}\frac{L}{n(L)}\ .$$ One obtains the following result.
\[DestructionSurface\] If $$\label{ConditionSurfaceI}
2\lambda \cdot \delta\cdot \rho < E_0+\alpha$$ holds, no eigenstate $\varphi_n \oplus 0$ is macroscopically occupied in the thermodynamic limit. This means, in particular, that the condensate in the bulk is destroyed for arbitrary pair densities whenever $\lambda=0$.
Theorem \[DestructionSurface\] has the remarkable consequence that the condensation in the bulk is destroyed for all pair densities $\rho > 0$ in the following cases: the pairs do not repel each other which allows them to accumulate in the surface defects or the number of surface defects is very large.
Finally, we obtain the following result.
\[MainTheorem\] Assume that $\delta,\lambda > 0$. Then there exists a critical pair density $\rho_{crit}=\rho_{crit}(\beta,\alpha,\lambda) > 0$ such that for all pair densities $\rho > \rho_{crit}$ the state $\varphi_0 \oplus 0$ is macroscopically occupied in the thermodynamic limit.
Theorem \[MainTheorem\] shows that the superconducting phase in the bulk can be recovered given the interaction strength $\lambda > 0$ is non-zero and, most importantly, given the number of surface impurities is not too large.
[^1]: E-mail address: [jens.bolte@rhul.ac.uk]{}
[^2]: E-mail address: [Joachim.Kerner@fernuni-hagen.de]{}
|
---
abstract: 'The loop structure plays an important role in many aspects of complex networks and attracts much attention. Among the previous works, Bianconi et al find that real networks often have fewer short loops as compared to random models. In this paper, we focus on the uneven location of loops which makes some parts of the network rich while some other parts sparse in loops. We propose a node removing process to analyze the unevenness and find rich loop cores can exist in many real networks such as neural networks and food web networks. Finally, an index is presented to quantify the unevenness of loop location in complex networks.'
author:
- 'An Zeng, Yanqing Hu, Zengru Di[^1]'
title: Unevenness of Loop Location in Complex Networks
---
**Introduction.** In the last decade, the researches on complex networks have rapidly developed. At the same time, the loop structure has attracted much attention. Loops are very important in complex networks. They can not only characterize the detail structure of networks but also relate to the structural correlations, motifs, robustness and redundancy of pathways, and affect some dynamical as well as equilibrium critical phenomena of the networks\[1,2\]. Recently, to avoid the effect of the loop structure, some researchers even study the acyclic networks\[3,4\].
For the self-avoiding loop, researchers focus mainly on two aspects: the total number of loops and the the dynamic effect of the loop structure. In the former case, many counting methods have been proposed\[5-11\]. In undirected networks, short loops can be exactly counted in terms of powers of the adjacency matrix\[8\]. This method can not deal with long loops, because the counting equation will become very complicated when facing long loops. In directed networks, short loops can be estimated by using $N_{L}\simeq \frac{1}{L} Tr A^{L}$ while long loops can be calculated by the entropy\[9,10\]. Moreover, some researchers analyze statistics of loops with different length $L$. They use the Monte Carlo sampling method to get the frequency and find that the loop number is sharply peaked around a characteristic loop length $L^{*}$. Also, they use $L^{*}$ and the relevant index to characterize the networks\[11\]. On the other hand, the dynamic effect of the loop structure has been studied frequently. It has been pointed out that the loop structure is related to the activity in neural networks such as self-sustained activities\[12,13,14\] and synchronization\[4,15,16\]. Specifically, the self-sustained activity can not survive without the loop structure and the synchronization will be weakened when emerging a dominant loop in the network. What’s more, a scaling behavior of loops is used to explain some critical phenomenon in percolation\[17\] and loop number is also used as a ranking method to quantify the role of both nodes and links\[18\].
However, many problems about loops still remain unnoticed. In the ref.\[10\], Bianconi, Gulbahce and Motter find that many real networks have fewer loops than the counterpart random networks which are a kind of random networks with the same number of in- and out-links in each node as the real networks. Actually, the loops number in different parts of a network varies according to the function of the regions. For example, the feed-forward part of the neural networks are sparse of loops\[3\] while other parts in the brain need loops to carry out self-sustained oscillation for precessing information\[13,14\]. For the food web networks, the loops number in the metazoan part are relatively small while there are many short loops among the microorganisms, called microbial loops, for fixed carbon repacking and recovery path of ecosystem\[19\]. Obviously, loops locate unevenly in many real networks. Some communities of these networks will be rich in loops while loops will be sparse in other parts. This leads us to an interesting question: what is the detail organization of loops location like in the networks? In this paper, we focus on the unevenness of loops location. We first study the distribution of loops on single nodes. Then we analyze the rich loop core phenomenon of uneven loops location by a node removing process in some real networks. Finally, we propose an index to measure the unevenness.
**Heterogenous distribution of loops on single nodes.** In the first step, we should study the loops on each single node to help us understand how the loops locate in the network. For a given network with size $N$, if we want to obtain how many loops passing through a specific node, we can simply remove the node from the network and count how many loops decreases, the decrement is the number of loops on this node. As $N_{L}$ is the loops number with the length $L$ of a network, we denote the $\hat{N}_{L}(i)$ as the number of loops with length $L$ in the network after the node $i$ is removed. So the node has $C_{L}(i)=N_{L}-\hat{N}_{L}(i)$ loops with length $L$ passing through. The number of short loops in directed network can be expressed in terms of powers of the adjacency matrix. In particular, $N_{L}\simeq \frac{1}{L} Tr A^{L}$, provided that $\kappa\equiv
max_{i}\sum_{j}\sum^{'}_{m}(\begin{array}{c}l\\m\end{array})|\lambda^{-m}_{j}P_{ij}P^{-1}_{j+m,i}|\ll1$\[9\]. Because $C_{L}(i)=N_{L}-\hat{N}_{L}(i)$, we can count the short loops on single nodes in any networks.
Here, we focus on the distribution of $C_{L}(i)$ in different networks. We can compare $C_{L}^{r}(i)$ of real networks with $C_{L}^{c}(i)$ of the counterpart random networks and $C_{L}^{e}(i)$ of the corresponding ER random networks. The counterpart random networks are a kind of uncorrelated random networks with the same number of in- and out-links in each node as the real networks. The corresponding ER random networks is given with the same size and the total number of links as the real networks. Of course, $C_{L}^{r}(i)$, $C_{L}^{c}(i)$ and $C_{L}^{e}(i)$ can be obtained by $C_{L}(i)=N_{L}-\hat{N}_{L}(i)$. Actually, the expected value of $C_{L}^{c}(i)$ can be gained by the formula based on the degree sequence. Motivated by the formula of the expected number of loops in the uncorrelated random network\[10,20\], we derive the expected number of loops on single nodes in undirected and directed random networks.
For undirected random networks, the expected number $E(N_{L})$ of short loops with length L is given by\[20\] $$E(N_{L})=\frac{1}{2L}(\frac{<k(k-1)>}{<k>})^{L},$$ where $k$ is the degree sequence of the network and $<.>$ represents the average value of a sequence. We can obtain the expected number of short loops on a specific node as $$\begin{aligned}
E(C_{L}(i))=&
\frac{1}{2L}(\frac{a}{b})^{L}-\frac{1}{2L}(\frac{(a-k_{i}(k_{i}-1))(b-4k_{i})}{(b-2k_{i})^{2}})^{L},\end{aligned}$$ where $a=\sum\limits_{h=1}\limits^{N}k_{h}(k_{h}-1)$ and $b=\sum\limits_{h=1}\limits^{N}k_{h}$.
For directed random networks, the expected number $E(N_{L})$ of short loops with length $L$ can be obtained by\[10\] $$E(N_{L})=\frac{1}{2L}(\frac{<k_{in}k_{out}>}{<k_{in}>})^{L}.$$ Like the undirected network, we also deduce a formula to estimate the expected number of short loops on a specific node $i$. The formula of $E(C_{L}(i))$ is $$\begin{aligned}
E(C_{L}(i))=&\frac{1}{L}(\frac{c}{d})^{L}-\frac{1}{L}(\frac{(c-k^{in}_{i}k^{out}_{i})(1-\frac{k^{in}_{i}}{d-k^{out}_{i}}-\frac{k^{out}_{i}}{d-k^{in}_{i}})}{d-k^{in}_{i}-k^{out}_{i}})^{L}.\quad\end{aligned}$$ where $c=\sum\limits_{h=1}\limits^{N}k^{in}_{h}k^{out}_{h}$ and $d=\sum\limits_{h=1}\limits^{N}k^{in}_{h}$. To examine the validity of our formula, we calculate the exact short loops number\[8\] in directed and undirected random networks with prearranged poisson degree sequences and compare them with the expected values from our formulas. The result shows our formulas can perfectly predict the $C_{L}(i)$ of both undirected and directed random networks with given degree sequences. So far, given the degree sequence of directed and undirected networks, we can use these two formulas to predict the loops number on each node in the uncorrelated random networks. That is to say the distribution $C_{L}^{c}(i)$ can be represented by the distribution of $E(C_{L}^{c}(i))$ which can be simply calculated by our formulas.
{width="8cm"}
{width="8cm"}
In fig.1, we compare the zipf plots of $C_{L}^{r}(i)$, $C_{L}^{c}(i)$, $C_{L}^{e}(i)$ and $E(C_{L}^{c}(i))$ in four different networks including the C.elegans’ neural network, the littlerock food web network, high technology company employees’ friendship network and the stmark food web network. If a node has no loop passing through, $C_{L}(i)=0$. In the zipf plot, this node will not appear in the log axis. Hence, the shorter tail of the line means all the loops are inclined to locate in several specific nodes. In addition, the steeper slope of the line in zipf plots indicates the distribution of $C_{L}(i)$ is more skewed, which means that nodes are quiet different from each other in loops number. If these two features are more significant, the distribution of the $C_{L}(i)$ will be more heterogenous. Comparing the real networks to ER random networks, we find that the loops are more heterogenous in some real networks such as neural networks and some food web networks. Moreover, from the $C_{L}^{r}(i)$ and $C_{L}^{c}(i)$, we can easily find that the degree sequence are not sufficient to describe the heterogeneity of loops distribution on nodes. In many cases, $C_{L}^{r}(i)$ performs a more significant heterogeneity than $E(C_{L}^{c}(i))$ and $C_{L}^{c}(i)$. However, loops are distributed in some social networks almost the same as in the counterpart networks. A typical example is given in fig.1(c).
In particular, we study $C_{L}(i)$ with different lengths in the C.elegans’ neural networks. In fig.2, we use the rank clocks to test whether the rank of loops on each node varies dramatically with different loops length $L$\[21\]. In fig.2, the rank clocks show perfect pentagon in the circumference which means that the top ranks of $C_{L}(i)$ do not change. Hence, we can clearly see that although for different length $L$, the heterogenous loops distributions share the same top rank.
From the heterogenous distribution of the loops in each node, we can easily find that loops locate more uneven in some real networks than in counterpart random ones. That is to say, some nodes of the real network are relatively rich in loops while loops are sparse in some other nodes. This phenomenon indicates that besides the total number, the detail organization of these loops in the real networks is quiet different from the counterpart random networks. If these nodes with many loops tightly connect with each other in the same community, of course this community will be extremely rich in loops. Combined with the result in fig.2, the same top rank in the heterogenous loops distribution will enhance the richness in loops in the community. In the following section, we will discuss the phenomenon of uneven loops location by studying the short loops in a specific kind of community of networks.
**The rich loop core phenomenon.** To investigate the detail organization of the self-avoiding loops in a network, we will study the loops number of different communities. In this paper, we consider that a loop belongs to a community only if all the nodes of the loops are included in this community. For these real networks we are about to analyze in this paper, because long loops are strongly related to the size of a chosen community, short loops are the main elements in the community. Therefore, we only consider short loops in this paper. It has been investigated that many real networks have fewer loops than the randomized counterparts in both short loops and long loops such as C.elegans’ neural network, Food Web networks, Power-grid networks etc\[10\]. Here, we take some of these real networks to make a further study of the loops in their communities.
{width="7cm"}\
{width="7cm"}\
First, we introduce a node removing process. In each step $t$ of the node removing process, we remove the node with the smallest $C_{L}(i)$ in the network. It means that the node with fewer loops passing through will be removed first. Specifically, the $C_{L}(i)$ should be updated in each step. After $N$ steps, we can remove all the nodes from the network. If we want to obtain a community with size $n$, the number of the removed nodes should be $t=N-n$. Because the node about to be removed in every step has the fewest loops among the remaining nodes, the community obtained by this process has the richest short loops among all the communities with the same size. The community obtained by the node removing process is called the selected community in this paper, the loops number in this community is denoted as $M_{L}(n)$. Obviously, $M_{L}(N)=N_{L}$ and $M_{L}(0)=0$. If a network is greatly uneven in the loop location, $M_{L}(n)$ will decrease slowly in the beginning while decline dramatically in the end through out the node removing process.
{width="8cm"}\
We analyze the C.elegans’ neural network by the node removing process as an example. In fig.3, we compare the C.elegans’ neural network, the counterpart random networks and the corresponding ER random networks. It can be seen that although the counterpart random networks have more short loops than C.elegans’ neural network, it can not compete with the C.elegans’ neural network in some specific selected communities. For instance, the selected community with 50 nodes in the C.elegans’ neural network has more short loops than that in the counterpart random networks as in fig.3.
Furthermore, we define the loop density as the proportion of the loops number and the community size. So we can easily get the loop density of the original network $N_{L}/N$. To compare the loop density of the selected communities and the original network, we define the loop density ratio $\eta_{L}(n)$ in each selected community during the node removing process as
$$\eta_{L}(n)=\frac{M_{L}(n)N}{N_{L}n}.$$
The loop density ratio $\eta_{L}(n)$ is the proportion of the selected community’s loop density and the original network’s loop density. If the loops locate unevenly in a network, $\eta_{L}(n)$ will become larger than $1$ during the node removing process. It indicates some communities in the network have bigger loop density than the original network. Obviously, in the C.elegans’ neural network, some of the selected communities enjoy larger loop density than the original network, so $\eta_{L}(n)$ is larger than $1$ as shown in Fig.4. Although the counterpart random networks can also have a $\eta_{L}(n)$ larger than $1$, its value is always lower than that of C.elegans’ neural network. This feature indicates that loops locate more unevenly in C.elegans’ neural network than in the counterpart random networks, which means that lots of loops are limited in some specific nodes of the C.elegans’ neural network.
So the unevenness may result in a rich loop core phenomenon, which means some selected communities with far higher loop density compared with the original networks. Although many real networks have fewer short loops compared to counterpart random models, the rich loop phenomenon will make some communities in real networks more loopy than the corresponding communities in the counterpart random networks. It can be detected by the ratio as: $$\rho_{L}(n)=\frac{M_{L}^{real}(n)}{M_{L}^{rand}(n)},$$ where $n$ is the size of the selected community. For example, the index will turn from $\rho_{L}(n)<1$ to $\rho_{L}(n)>1$ as nodes removed in the C.elegans’ neural network as show in fig.3. Also, we also investigate many other real networks, some typical results are shown in fig.4.
As in fig.4 (a), (c) and (e), although these real networks, such as neural networks and some food web networks, have fewer total loops than the counterpart random networks, some selected communities of them are more loopy than that of counterpart random ones. Moreover, we find that short loops with different lengths $L$ perform the same trend in the rich loop core phenomenon. If the loops with specific length $L$ locate unevenly in the networks, loops with other lengths locate uneven as well. However, not all the networks have this kind of phenomenon as shown in Fig.4(b). Some real networks with far fewer loops than the counterpart random networks have $\rho_{L}(n)<1$ for all $n$. For example, the C.elegans’ metabolic network belongs to this category. In addition, we find that some social networks have more short loops than the counterpart random ones as in fig.4(d). These networks are not discussed in ref\[10\]. This category includes the prisoners’ friendship network, high technology employees’ friendship network, the family visit network, the flying-team partner choosing network, the dining table partner choosing network and so on\[22\]. If these networks enjoy more uneven loops location, their selected communities can only be more and more loopy than that of counterpart random networks as the community size $n$ varies, see fig.4(d). Finally, we use the index $\rho_{L}(n)$ to detect a ER random network with 100 nodes and 800 links. The result in fig.4(f) shows that $\rho_{L}(n)=1$ approximately.
Besides the rich loop core phenomenon, how to find the rich loop core in a network is an interesting question. In this paper, we simply consider the rich loop core appears at the maximum $\eta_{L}(n)$, which means the rich loop core will have highest loop density than any other community in the networks. Some typical results are shown in fig.6. The littlerock food web and the C.elegans’ metabolic network have significant rich loop cores which indicates they have much higher loop density community compared with the original networks. The C.elegans’ neural networks and the StMark food web have such rich loop cores as well. On the contrary, this phenomenon is not so obvious in company employees networks and large degree ER random networks. For the C.elegans’ metabolic network, the high loop density ratio is due to the small number of total loops. These loops unavoidably locate in several specific nodes, so the loops density in the selected community will be very large compared with the original network. Furthermore, we analyze the rich loop cores of the food web and neural networks. It is interesting that most of the nodes in the cores are from interneurons in C.elegans’ neural network and from microorganisms in food web networks. Specifically, the rich loop cores of C.elegans’ neural networks have different sizes $n$ for different loop lengths $L$. For example, $n=31$ when $L=3$, $n=38$ when $L=4$, $n=50$ when $L=5$ and $n=51$ when $L=6$. These rich loop cores share $30$ nodes and $24$ of them are interneurons. The C.elegans’ neural networks are composed by sensory neurons, interneurons and motor neurons. Most interneurons are in the nerve ring ganglia. Their main function is to process signals\[23,24\]. So the loops number in these interneurons is relatively larger than the others. Likewise, the rich loop cores in stmark food web networks are $n=20$ when $L=3$, $n=22$ when $L=4$, $n=24$ when $L=5$ and $n=24$ when $L=6$. These rich loop cores share $14$ nodes and $10$ of them are microorganisms. In the food web networks, the microorganisms are the main element in microbial loops which are strongly related to fixed carbon repacking and recovery path of ecosystem\[19\].
{width="8cm"}\
**Measurement for unevenness of loop location.** In order to quantify how unevenly loops locate in the networks, we present an index which bases on the node removing process. In order to Simplify the computing complexity, we do not update $C_{L}(i)$ in each step during the node removing process in this section. We test and find the result obtained in way is sufficient to represent that by updating $C_{L}(i)$ in each step statistically. Moreover, we remove the nodes based on the attacking rate $p$. For example, if $p=0$, no node is removed and the network is all the same with the original network and the loop number is $N_{L}$. If $p=0.1$, we just remove $[pN]_{ceil}$ nodes from the network and the loop number is $M_{L}(N-[pN]_{ceil})$. Here, $[.]_{ceil}$ represents the operation of rounding upward. Then, we use $A_{L}(p)=\frac{M_{L}(N-[pN]_{ceil})}{N_{L}}$ to normalize the loop number of each community so that $A_{L}(p)$ which is corresponding to the loop number declines from 1 to 0 through out the node removing process as shown in fig.7. Again, we use $L=5$ as an example in fig.7.
If a network is significantly uneven in the loop location, $A_{L}(p)$ will decline slowly in the beginning while dramatically in the end during the node removing process. On the contrary, if loops locate evenly in the network, the $A_{L}(p)$ will decline almost the same as the corresponding ER random networks. Therefore, the unevenness of loop location can be measured by the difference between the real network and the corresponding ER random network. Here, we use $I_{L}^{r}(p)=A_{L}^{r}(p)-A_{L}^{e}(p)$ to estimate the difference. So the unevenness of loops location can be represented by the index as $$R_{L}=\int_{0}^{1} I_{L}(p)dp,$$ where $-1<R_{L}<1$. The severer the unevenness is, the larger the index $R_{L}$ is, which means the loop location departs more largely from the corresponding ER random network. Of course, the index $R_{L}$ can also be used in analyzing the counterpart random network.
Actually, the index $R_{L}$ is the area between the lines of the real network and the ER random network in fig.7. It can be seen that the real networks and the counterpart random networks can be different in the unevenness, as the index $R_{L}^{r}\neq R _{L}^{c}$. Typically, if the real network is very sparse in the links, the network has only small number of loops. This will lead to a phenomenon that some part of the network has some loops, while other part has no loop at all. However, the corresponding ER random network has the same condition too. For the $R_{L}$ is the area between the lines of the real network and the ER random network, the $R_{L}$ will be a small value under this circumstance. Whether this uneven location of loops results from the specific structure of the real network or from the small degree as the ER random network can be estimated from the index $R_{L}$. As the figure(d) in the fig.7, we use the index $R_{L}$ to investigate a ER random network with small degree as $K=2$. The result shows that the three lines are almost the same meaning that $R_{L}\approx0$.
{width="4cm"} {width="4cm"}\
{width="4cm"} {width="4cm"}\
=0.4em
network $size$ $links$ $\bar{R}_{r}$ $\bar{R}_{c}$ $\bar{R}_{r}-\bar{R}_{c}$
---------------------- -------- --------- --------------- --------------- --------------------------- -- --
C.elegans’ neural $306$ $2359$ $0.436$ $0.317$ $0.119$
C.elegans’ metabolic $453$ $2040$ $0.468$ $0.337$ $0.131$
E.coli’s metabolic $896$ $958$ $-0.021$ $-0.002$ $-0.019$
Mondego FW $46$ $400$ $0.231$ $0.257$ $-0.026$
Michigan FW $39$ $221$ $0.194$ $0.207$ $-0.013$
Littlerock FW $183$ $2494$ $0.672$ $0.439$ $0.233$
StMarks FW $54$ $356$ $0.378$ $0.256$ $0.122$
Prisoners $67$ $182$ $0.169$ $0.004$ $0.165$
Flying-teamers $48$ $351$ $0.108$ $0.060$ $0.048$
Company employees $36$ $147$ $0.219$ $0.163$ $0.056$
ER random $--$ $--$ $0$ $0$ $0$
: Results of the analysis of networks based on index $\bar{R}$
Additionally, we consider several more directed real networks\[22\]. As mentioned above, although each network has different kinds of short loops based on length $L$, these loops perform almost the same. We use average $R_{L}$ to represent the unevenness in loops location. It can be gained by $\bar{R}=<R_{L}>$ and the $L=3,4,...,L_{max}$ where $L_{max}=8$ according to Ref\[9\]. The index $R$ for these real networks are given in table 1. From the $\bar{R}_{r}$, how significant the unevenness in loops location is can be known. By comparing the $\bar{R}_{r}$ and the $\bar{R}_{c}$, we can distinguish whether this unevenness results from the degree sequence. In table 1, it can be seen from the $\bar{R}_{r}$ that the C.elegans neural network, C.elegans metabolic network and some food web networks are really uneven in loop location. The social networks, Escherichia Coli’s metabolic network and some other food web networks do not have such significant unevenness. In fact, the number of different species will affect the loop location in food web networks. For example, too many microorganisms will make the loops more even and too many metazoans will reduce the total loops number, at this time the rich loop core will be more obvious. Additionally, both the C.elegans’ metabolic network and the Escherichia Coli’s metabolic network have very few loops, but the degree sequence and the total links of the former one allows the counterpart random and the ER random networks to have much more loops while the latter one does not. Hence, the $\bar{R}_{r}$ of these two metabolic networks are different. Moreover, comparing the $\bar{R}_{r}$ and the $\bar{R}_{c}$ in table 1, it can be found that the degree sequence is not sufficient to describe the unevenness in loops location. It is clear that the C.elegans’ neural network, littlerock and stmark food web networks are more uneven in loops location than the counterpart networks. That is why they can have some communities more loopy than the corresponding communities in the counterpart random models despite the fewer total loops number.
**Conclusion.** The previous works on the loops mainly focus on the total number of loops and the dynamic effect of the loop structure. However, the loop location is also very important in networks. Generally, loops tend to locate in some specific nodes in some real networks, which means some communities of the network are extremely rich in loops while the loops are relatively sparse in other parts. If this uneven location is significant enough, the rich loop core phenomenon can be formed in some real networks.
The rich loop core phenomenon is meaningful for the typical function of real networks. For instance, the loop structure is strongly related to the self-sustained activities in neural networks, so the rich loop core may help to understand the functional regions in the neural networks. For the food web networks, almost all the loops in rich loop cores of food web networks are microbial loops and plays an important part in fixed carbon repacking and recovery path of ecosystem. In addition, this uneven location of loops may provide a new way to study the community detection in directed networks, which asks for further research.
**Acknowledgement.** The authors would like to thank Dong Zhou, Hongzhi You, Prof. Ying Fan and Prof. Yiming Ding for many useful suggestions. This work is supported by NSFC under Grants No. 70974084, No. 60534080 and No. 70771011.
[99]{}
R. Pastor-Satorras, A. Vazquez, A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001). S. N. Dorogovtsev et al., Rev. Mod. Phys. 80, 1275 (2008). B. Karrer and M. E. J. Newman, Phys. Rev. Lett. 102, 128701 (2009) A.Zeng, Y.Hu and Z.Di, Europhys. Lett. 87, 48002 (2009) K. Klemm and P. F. Stadler, Phys. Rev. E 73, 025101(R) (2006) E. Marinari, G. Semerjian and V. VanKerrebroeck, Phys. Rev. E 75, 066708 (2007) E. Marinari, R. Monasson and G. Semerjian, Europhys. Lett. 73, 8-14 (2006) G. Bianconi, Eur. Phys. J. B 38, 223¨C230 (2004) G. Bianconi and N. Gulbahce, J. Phys. A: Math. Theor. 41, 224003 (2008) G. Bianconi, N. Gulbahce and A. E. Motter, Phys. Rev. Lett. 100, 118701 (2008) H. D. Rozenfeld et al., J. Phys. A: Math. Gen. 38, 4589¨C4595 (2005) A. Roxin, H. Riecke and S. A. Solla, Phys. Rev. Lett. 92, 198101 (2004) X. Liao et al., arXiv:0906.2356v1 L. F. Lago-Fernandez, R. Huerta, F. Corbacho, J. A. Siguenza, Phys. Rev. Lett. 84, 2758-2761 (2000) X. Ma,L. Huang, Y. C. Lai and Z. Zheng, Phys. Rev. E 79, 056106 (2009) A. Arenas, A. D¨ªaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Physics Reports 469, 93-153 (2008) J.D. Noh, Eur. Phys. J. B 66, 251-257 (2008) V. VanKerrebroeck and E. Marinari, Phys. Rev. Lett. 101, 098701 (2008) E. Sherr and B. Sherr, Limnology and Oceanography, 33, 1225-1227 (1988) Z. Burda, J. Jurkiewicz, A. Krzywicki, Phys. Rev. E 70, 026106 (2004) M. Batty, Nature 444, 592-596 (2006) The network datas are available on line at http://vlado.fmf.unilj.si/pub/networks/data/, http://www.imsc.res.in/ sitabhra/research/neural/celegans/, http://www.casos.cs.cmu.edu/index.php, www.cosinproject.org/. J. G. White, E. Southgate, J. N. Thomson and S. Brenner, Philos Trans R Soc London B, Biol Sci 314, 1-340 (1986) E. L. Tsalik, O. Hobert, J Neurobiol 56(2),178-97 (2003)
[^1]: zdi@bnu.edu.cn
|
---
abstract: 'Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. A particularly interesting feature of this development is the ability to construct (scalar and vector) linearly solvable networks using certain classes of matroids. Furthermore, it was shown through the connection between network coding and matroid theory that linear network coding is not always sufficient for general network coding scenarios. The current work attempts to establish a connection between matroid theory and network-error correcting and detecting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of network-error detecting codes to arrive at the definition of a *matroidal error detecting network* (and similarly, a *matroidal error correcting network* abstracting from network-error correcting codes). An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error detecting (correcting) network code if and only if it is a matroidal error detecting (correcting) network associated with a representable matroid. Therefore, constructing such network-error correcting and detecting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa. We then present algorithms which enable the construction of matroidal error detecting and correcting networks with a specified capability of network-error correction. Using these construction algorithms, a large class of hitherto unknown scalar linearly solvable networks with multisource multicast and multiple-unicast network-error correcting codes is made available for theoretical use and practical implementation, with parameters such as number of information symbols, number of sinks, number of coding nodes, error correcting capability, etc. being arbitrary but for computing power (for the execution of the algorithms). The complexity of the construction of these networks is shown to be comparable to the complexity of existing algorithms that design multicast scalar linear network-error correcting codes. Finally we also show that linear network coding is not sufficient for the general network-error correction (detection) problem with arbitrary demands. In particular, for the same number of network-errors, we show a network for which there is a nonlinear network-error detecting code satisfying the demands at the sinks, while there are no linear network-error detecting codes that do the same.'
author:
-
title: 'A Matroidal Framework for Network-Error Correcting Codes'
---
Introduction {#sec1}
============
Network coding, introduced in [@ACLY], is a technique to increase the rate of information transmission through a network by coding different information flows present in the network. One of the chief problems in network coding is to find whether a given network with a set of sources and sink demands is solvable using a scalar linear network code. Much work has been done on the existence and construction of scalar linear network coding techniques in several papers including [@CLY; @KoM; @JSCEEJT].
Matroids are discrete objects which abstract the notions of linear dependence among vectors. They arise naturally in several discrete structures including graphs and matrices. The relationship between network coding and matroid theory was first introduced in [@DFZ]. The authors of [@DFZ] showed that the scalar linear solvability of a network with a given set of demands was related to the existence of a *representable matroid* (matroids which arise from matrices over fields) satisfying certain properties. This connection was further developed and strengthened in [@DFZ2; @KiM; @SHL; @LiS; @RSG]. Using the techniques of [@DFZ; @KiM; @RSG], it is known that several network instances which are scalar (or vector) linearly solvable can be constructed using representable matroids and their generalisations. Using the equivalence between networks and matroids, it was shown in [@DFZ3] that linear network codes are not always sufficient for solving network coding problems where the sinks have arbitrary demands (i.e., not necessarily multicast). An explicit network was demonstrated which had a nonlinear network coding solution but no linear network coding solutions.
Linear network-error correcting codes were introduced in [@YeC1; @YeC2] as special kinds of linear network codes which could correct errors that occurred in the edges of the network. Linear network-error detection codes are simply linear network-error correction codes where the sinks are able to decode their demands in the presence of errors at edges known to the sinks. Together with the subsequent works [@Zha; @Mat; @YaY], the bounds and constructions similar to classical block coding techniques were carried over to the context of linear network-error correction and detection. As network-error correcting (detecting) codes are essentially special kinds of network codes, the issues of network coding especially with respect to existence and construction have their equivalent counterparts in network-error correction (detection). Network-error correction was extended to case of non-multicast in [@VHEKE]. In [@KTT], linear network-error correction schemes were found to be incapable of satisfying the demands for networks with node adversaries rather than edge adversaries. Nonlinear error correction schemes are also found to perform better than linear error correction in networks with unequal edge capacities [@KHEA].
In the current work, we present the connection between matroids and network-error correcting and detecting codes. The results of this work may be considered as the network-error correction and detection counterparts of some of the results of [@DFZ; @KiM; @DFZ3]. The organisation and the chief contributions of our work are as follows.
- After reviewing linear network-error correcting and detecting codes in Section \[sec2\] and matroid theory in Section \[sec3\], in Section \[sec4\] we define the notion of a *matroidal error detecting network* associated with a particular matroid. Using this definition, we show that an acyclic network has a scalar linear network-error detecting code (satisfying general demands) if and only if there exists a representable matroid $\cal M$ such that the given network is a matroidal error detecting network associated with $\cal M.$ Therefore, networks with scalar linear network-error detecting codes are shown to be analogous to representable matroids satisfying a certain set of properties. Because of the equivalence between network-error detection and network-error correction, all these results have their counterparts for network-error correcting codes also.
- In Section \[sec5\], we give algorithms which construct multisource multicast and multiple-unicast matroidal error correcting networks associated with general matroids (not necessarily representable) satisfying the required properties. If the matroids associated with such networks are representable over finite fields, then these networks are obtained along with their corresponding scalar linear network-error correcting codes. Therefore, our results generate a large class of hitherto unknown networks which have scalar linear network-error correcting codes, a few of which are shown in this paper by implementing the representable matroids version of our algorithms in MATLAB. Though the implementation the nonrepresentable matroids version of our algorithm is difficult, we do give a small result as a first step in this direction in Subsection \[mecnnonrepresentable\]. The complexity of the construction of multicast and multiple-unicast networks associated with representable matroids is shown to be comparable to the complexity of existing algorithms that design multicast scalar linear network-error correcting codes for given networks in Section \[seccomplexity\].
- Based on the results from [@DFZ3], in Section \[secinsufficiency\], we prove the insufficiency of linear network coding for the network-error detection problem on networks with general demands (i.e., not necessarily multicast). In particular, we demonstrate a network (adapted from the network used in [@DFZ3] to demonstrate the insufficiency of linear network coding for the general network coding problem) for which there exists a nonlinear single edge network-error detecting code that satisfies the sink demands, while there are no linear network-error detecting codes that do the same.
- In Subsection \[insuffnetwork\], we show that this network, for which linear network-error detection is insufficient, is a matroidal error detecting network with respect to a nonrepresentable matroid. Thus our definition of matroidal error detecting networks is not limited to networks with linear network-error detecting schemes alone, instead has a wider scope, accommodating nonlinear error detection schemes also.
Though algorithms for constructing network-error correcting codes are known for given single source multicast networks [@YeC1; @YeC2; @YaY], there is no general characterisation of networks and demands for which scalar linear network-error correction codes can be designed. The authors believe that the algorithm given in this paper could provide useful insights in this regard. Furthermore, it could also prove useful in the design of practical network topologies in which network coding and network-error correction (detection) have advantages over routing and classical error correction (detection). We also highlight that though there are many papers in network coding literature which discuss network coding for multiple-unicast networks, the results obtained in our paper are some of the first in network-error correction literature which talk about network-error correction codes for multiple-unicast networks.
*Notations:* The following notations will be followed throughout the paper. The disjoint union of any two sets $A$ and $B$ is denoted by $A\uplus B.$ For a finite set $A,$ the power set of $A$ is denoted by $2^A.$ A finite field is denoted by the symbol $\mathbb{F}.$ For some positive integer $k,$ the identity matrix of size $k$ over $\mathbb{F}$ is denoted by $I_k.$ The rank of a matrix $A$ over $\mathbb{F}$ is denoted by $rank(A),$ and its transpose is denoted by $A^T.$ The $\mathbb{F}$-vector space spanned by the columns of a matrix $A$ over $\mathbb{F}$ is denoted by $\langle A\rangle.$ The set of columns of $A$ is denoted by $cols(A).$ The support set of a vector $\boldsymbol{x}$ and its Hamming weight are denoted by $supp(\boldsymbol{x})$ and $w_H(\boldsymbol{x})$ respectively. The symbol $\boldsymbol{0}$ represents an all zero vector or matrix of appropriate size indicated explicitly or known according to the context. For some matrix $A,$ we denote by $A^l$ the $l^{th}$ column of $A,$ and for a subset ${\cal L}$ of the column indices of $A,$ we denote by $A^{\cal L}$ the submatrix of $A$ with columns indexed by $\cal L.$ Likewise, we denote by $A_{j}$ the $j^{th}$ row of $A,$ and by $A_{\cal J}$ the submatrix of $A$ with rows given by the subset ${\cal J}$ of the row indices.
Network-Error Correcting and Detecting Codes {#sec2}
============================================
As in [@KoM; @YeC1], we model the directed acyclic network as a directed acyclic multigraph (one with parallel edges) ${\cal G}({\cal V},{\cal E})$ where ${\cal V}$ is the set of vertices of $\cal G$ representing the nodes in the network and ${\cal E}$ is the set of edges representing the links in the network. An ancestral ordering is assumed on $\cal E$ as the network is acyclic. Each edge is assumed to carry at most one finite field symbol at any given time instant. A non-empty subset ${\cal S} \subseteq {\cal V}$, called the set of sources, generates the information that is meant for the sinks in the network, represented by another non-empty subset ${\cal T} \subseteq {\cal V}$. Each sink demands a particular subset of the information symbols generated by the sources. Any node in the network can be a source and a sink simultaneously, however not generating and demanding the same information. Let $n_{s_i}$ be the number of information symbols (from some finite field $\mathbb{F}$) generated at source $s_i.$ Let ${\mu} = \left\{1,2,...,\sum_{i=1}^{|{\cal S}|} n_{s_i}=n\right\}$ denote the ordered index set of messages (each corresponding to a particular information symbol) generated at all the sources. For each edge $e\in {\cal E},$ we denote by $tail(e)$ the node from which $e$ is outgoing, and by $head(e)$ the node to which $e$ is incoming. Also, for each node $v\in {\cal V},$ let $In(v)$ denote the union of the messages (a subset of $\mu$) generated by $v$ and the set of incoming edges at $v.$ Similarly, let $Out(v)$ denote the union of the subset of messages demanded by $v$ and the set of outgoing edges from $v.$ Further, for any $e\in {\cal E},$ we denote by $In(e)$ the set $In(tail(e)).$
A network code on $\cal G$ is a collection of functions, one associated with each node of the network mapping the incoming symbols at that node to its outgoing symbols. When these functions are scalar linear, the network code is said to be a scalar linear network code. To be precise, a scalar linear network code is an assignment to the following matrices.
- A matrix $A_{s_i}$ of size $n_{s_i} \times |{\cal E}|,$ for each source $s_i \in {\cal S}, i=1,2,...,|{\cal S}|,$ denoting the linear combinations taken by the sources mapping information symbols to the network, with non-zero entries (from $\mathbb{F}$) only in those columns which index the outgoing edges from $s_i.$
- A matrix $K$ of size $|{\cal E}| \times |{\cal E}|$ which indicates the linear combinations taken by the nodes in the network to map incoming symbols to outgoing symbols. For $i<j,$ the $(i,j)^{th}$ element of $K,$ $K_{i,j},$ is an element from $\mathbb{F}$ representing the network coding coefficient between edge $e_i$ and $e_j.$ Naturally $K_{i,j}$ can be non-zero only if $e_j$ is at the downstream of $e_i.$
Also, to each sink $t \in {\cal T},$ we associate a matrix $B_{t}$ of size $|{\cal E}| \times n_{t},$ where $n_{t}$ is the number of incoming edges at $t.$ Corresponding to the $n_t$ rows that index these incoming edges, we fix the $n_t \times n_t$ submatrix of $B_{t}$ as an identity submatrix. The other entries of $B_t$ are fixed as zeroes.
For $i=1,2,...,|{\cal S}|,$ let $\boldsymbol{x_{s_i}} \in \mathbb{F}^{n_{s_i}}$ be the row vector representing the information symbols at source $s_i$. Let $\boldsymbol{F}=\left(I_{|{\cal E}|}-K\right)^{-1}$ and $A_{s_i}\boldsymbol{F}B_{t} = \boldsymbol{F_{s_i,t}}.$ Let ${\cal A}$ be the $n \times |{\cal E}|$ row-wise concatenated matrix $$\label{formofA}
\left(
\begin{array}{c}
A_{s_1}\\
A_{s_2}\\
.\\
.\\
A_{s_{|{\cal S}|}}
\end{array}
\right).$$ The columns of ${\cal A}\boldsymbol{F}$ are known as the *global encoding vectors* corresponding to the edges of the network, indicating the particular linear combinations of the information symbols which flow in the edges. We assume that no edge is assigned an all zero global encoding vector, for then it can simply be removed from the network and a smaller graph can be assumed. The global encoding vector corresponding to the $n$ messages are fixed to be the $n$ standard basis vectors over $\mathbb{F},$ the concerned field. A network code can also be specified completely by specifying global encoding vectors for all edges in the network, provided that they are valid assignments, i.e., global encoding vectors of outgoing edges are linear combinations of those of the incoming edges.
Let $\boldsymbol{x}=\left(\boldsymbol{x_{s_1}}~~~\boldsymbol{x_{s_2}}~~ ... ~~\boldsymbol{x_{s_{|{\cal S}|}}}\right)$ be the vector of all information symbols. Let $\boldsymbol{{\cal D}_{t}} \subseteq {\mu}$ denote the set of demands at sink $t,$ and let $\boldsymbol{x_{s_{{\cal D}_t}}}$ denote the subvector of the super-vector $\boldsymbol{x}$ corresponding to the information symbols indexed by $\boldsymbol{{\cal D}_{t}}.$ An edge is said to be in error if its input symbol (from $tail(e)$) and output symbol (to $head(e)$), both from $\mathbb{F}$, are not the same. We call this as a *network-error*. We model the network-error as an additive error from $\mathbb{F}$. A *network-error vector* is a $|{\cal E}|$ length row vector over $\mathbb{F}$, whose components indicate the additive errors on the corresponding edges. The case of multicast network-error correction, where a single source multicasts all its symbols to all sinks in the presence of errors, has been discussed in several papers (see for example, [@YeC1; @YeC2; @YaY]) all being equivalent in some sense.
Now we briefly review the results for network-error correcting and detecting codes in the case of arbitrary number of sources and sinks with arbitrary demands. Let $\boldsymbol{z}$ be the network-error vector corresponding to a particular instance of communication in the network. Let $\boldsymbol{F_{{\cal S},t}}$ be the matrix ${\cal A}\boldsymbol{F}B_t.$ Let $\boldsymbol{F}B_{t}=\boldsymbol{F_{t}}.$ Then a sink $t$ receives the $n_t$ length vector $$\label{networkinoutrelationship}
\boldsymbol{y_{t}}=\boldsymbol{x}\boldsymbol{F_{{\cal S},t}}+\boldsymbol{z}\boldsymbol{F_{t}}.$$ One way to interpret the input-output relationship shown by (\[networkinoutrelationship\]) is to think of the network as a finite state machine whose states are the symbols flowing on the edges. The matrix $\boldsymbol{F_{{\cal S},t}}$ then describes the transfer matrix of this state machine between the sources and sink $t$. Some of the states of this network could be in error (i.e. the network-errors at the edges), which is captured by the network-error vector $\boldsymbol{z}.$ These errors are also reflected at the sink outputs, in their appropriate linear combinations, given by the matrix $\boldsymbol{F_{t}}.$ For more details the reader is referred to [@YaY].
A network code which enables every sink to successfully recover the desired information symbols in the presence of any network-errors in any set of edges of cardinality at most $\alpha$ is said to be a $\alpha$-*network-error correcting code*. A network code which enables the sink demands to be recovered in the presence of errors in at most $\beta$ edges which are *known* to the sinks, is called a $\beta$-*network-error detecting code*. It is not difficult to see that a scalar linear network code is a scalar linear $\alpha$-network-error correcting code if and only if the following condition holds at each sink $t\in{\cal T}$. $$\begin{aligned}
\nonumber
\boldsymbol{y_{t}}& = \boldsymbol{x}\boldsymbol{F_{{\cal S},t}} +\boldsymbol{z}\boldsymbol{F_{t}} \neq \boldsymbol{0} \in \mathbb{F}^{n_t},\\
\label{eqn2}
&\forall ~ \boldsymbol{x}\in \mathbb{F}^{n}: \boldsymbol{x_{s_{{\cal D}_{t}}}} \neq \boldsymbol{0},~\forall~ \boldsymbol{z}\in \mathbb{F}^{|{\cal E}|}: w_H(\boldsymbol{z}) \leq 2\alpha.\end{aligned}$$ Similarly, for a $\beta$-network-error detecting code, we must have the following condition holding true for all sinks. $$\begin{aligned}
\nonumber
\boldsymbol{y_{t}}& = \boldsymbol{x}\boldsymbol{F_{{\cal S},t}} +\boldsymbol{z}\boldsymbol{F_{t}} \neq \boldsymbol{0} \in \mathbb{F}^{n_t},\\
\label{eqn2a}
&\forall ~ \boldsymbol{x}\in \mathbb{F}^{n}: \boldsymbol{x_{s_{{\cal D}_{t}}}} \neq \boldsymbol{0},~\forall~ \boldsymbol{z}\in \mathbb{F}^{|{\cal E}|}: w_H(\boldsymbol{z}) \leq \beta.\end{aligned}$$ The proof that (\[eqn2\]) indeed implies a $\alpha$-network-error correcting code follows from the fact that we can always demonstrate a pair of information vectors $\boldsymbol{x}$ and $\boldsymbol{x'}$ with $\boldsymbol{x_{s_{{\cal D}_{t}}}} \neq\boldsymbol{x'_{s_{{\cal D}_{t}}}}$ and a corresponding pair of error vectors $\boldsymbol{z}$ and $\boldsymbol{z'}$ with $w_H(\boldsymbol{z}) \leq \alpha$ and $w_H(\boldsymbol{z'})\leq \alpha$ such that the corresponding outputs $\boldsymbol{y_{t}}$ and $\boldsymbol{y'_{t}}$ are equal, if and only if the sink $t$ is not able to distinguish between $\boldsymbol{x_{s_{{\cal D}_{t}}}}$ and $\boldsymbol{x'_{s_{{\cal D}_{t}}}}$ in the presence of errors. A similar argument can be given for (\[eqn2a\]).
Thus, by (\[eqn2\]) and (\[eqn2a\]), it is clear that a $\beta$-network-error detecting code is also a $\lfloor \frac{\beta}{2}\rfloor$-network-error correcting code, while an $\alpha$-network-error correcting code is also a $2\alpha$-network-error detecting code.
The *error pattern* corresponding to a network-error vector $\boldsymbol{z}$ is defined as its support set $supp(\boldsymbol{z}),$ which we shall also alternatively refer to using the corresponding subset of $\cal E.$ Let $\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}$ denote the submatrix of $\boldsymbol{F_{t}}$ consisting of those rows of $\boldsymbol{F_{t}}$ which are indexed by $supp(\boldsymbol{z}).$ The condition (\[eqn2\]) can then be rewritten as $$\begin{aligned}
\nonumber
\boldsymbol{y_{t}}= & \left(\boldsymbol{x}~~\boldsymbol{\bar{z}}\right)
\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right) \neq \boldsymbol{0} ,~\forall ~ \boldsymbol{x}\in \mathbb{F}^{n}: \boldsymbol{x_{s_{{\cal D}_{t}}}} \neq \boldsymbol{0},\\
\label{eqn3}
&~\forall~ \boldsymbol{\bar{z}}\in \mathbb{F}^{2\alpha},~\forall~supp(\boldsymbol{z})\in \left\{{\cal F} \subseteq {\cal E}:|{\cal F}|=2\alpha \right\}.\end{aligned}$$ Similarly condition (\[eqn2a\]) becomes $$\begin{aligned}
\nonumber
\boldsymbol{y_{t}}= & \left(\boldsymbol{x}~~\boldsymbol{\bar{z}}\right)
\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right) \neq \boldsymbol{0} ,~\forall ~ \boldsymbol{x}\in \mathbb{F}^{n}: \boldsymbol{x_{s_{{\cal D}_{t}}}} \neq \boldsymbol{0},\\
\label{eqn3a}
&~\forall~ \boldsymbol{\bar{z}}\in \mathbb{F}^{\beta},~\forall~supp(\boldsymbol{z})\in \left\{{\cal F} \subseteq {\cal E}:|{\cal F}|=\beta \right\}.\end{aligned}$$ For the special case of a single source multicast, the condition (\[eqn3\]) becomes $$\begin{aligned}
\nonumber
\boldsymbol{y_{t}}= & \left(\boldsymbol{x}~~\boldsymbol{\bar{z}}\right)
\left(
\begin{array}{c}
\boldsymbol{F_{s,t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right) \neq \boldsymbol{0} \in \mathbb{F}^{n_{t}},~\forall ~ \boldsymbol{x} \neq \boldsymbol{0},\\
\label{eqnmulticastcondition}
&~\forall~ \boldsymbol{\bar{z}}\in \mathbb{F}^{2\alpha},~\forall~supp(\boldsymbol{z})\in \left\{{\cal F} \subseteq {\cal E}:|{\cal F}|=2\alpha \right\},\end{aligned}$$ which is known from [@YeC1; @YeC2; @Zha; @YaY]. Some of these papers also discuss the case of unequal error correcting capabilities at different sinks, but in our paper we only consider $\alpha$-network-error correction at all sinks uniformly. The extension to the unequal error capabilities is natural and therefore omitted.
For the multiple-unicast case, where each source has only one symbol to unicast to some sink and each sink has only one information symbol to receive from some source, the condition (\[eqn2\]) becomes $$\begin{aligned}
\nonumber
\boldsymbol{y_{t}}= & x_{s_{\boldsymbol{{\cal D}_{t}}}}\boldsymbol{F_{s_{{\cal D}_{t}},t}} + \left(\sum_{i=1,i\neq \boldsymbol{{\cal D}_{t}}}^{|{\cal S}|}x_{s_i}\boldsymbol{F_{s_i,t}}+\boldsymbol{z}\boldsymbol{F_{t}}\right) \neq \boldsymbol{0},\\
\label{multipleunicastcondition}
&~\forall ~ \boldsymbol{x}: x_{s_{\boldsymbol{{\cal D}_{t}}}} \neq 0,~\forall~ \boldsymbol{z}\in \left\{\boldsymbol{z}\in \mathbb{F}^{|{\cal E}|}: w_H(\boldsymbol{z}) \leq 2\alpha\right\},\end{aligned}$$ where the first term above represents the signal part of the received vector and the second term denotes the interference plus noise part. Note that $ x_{s_{\boldsymbol{{\cal D}_{t}}}}$ denotes the demanded information symbol at sink $t,$ while $x_{s_i}$ denotes the information symbol generated at source $s_i.$ Equations similar to (\[eqnmulticastcondition\]) and (\[multipleunicastcondition\]) can be obtained for $\beta$-network-error detecting codes also, by simply replacing $2\alpha$ by $\beta.$
A technical lemma
-----------------
We now present a technical lemma, which will be used in Section \[sec4\]. The result of the lemma can be inferred from the results of [@VHEKE], but we give it here for the sake of completeness.
\[lemmadecoding\] Let $I_{\boldsymbol{{\cal D}_{t}}}$ denote the $(n+\beta) \times |\boldsymbol{{\cal D}_{t}}|$ matrix with a $|\boldsymbol{{\cal D}_{t}}|\times |\boldsymbol{{\cal D}_{t}}|$ identity submatrix in $|\boldsymbol{{\cal D}_{t}}|$ of the first $n$ rows corresponding to the demands $\boldsymbol{{\cal D}_{t}}$ at sink $t,$ and with all other elements being zero. For some $supp(\boldsymbol{z})\in \left\{{\cal F} \subseteq {\cal E}:|{\cal F}|=\beta \right\},$ the condition $$\begin{aligned}
\label{eqn4}
\left(\boldsymbol{x}~~\boldsymbol{\bar{z}}\right)
\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right) \neq \boldsymbol{0},~\forall\boldsymbol{x}\in\mathbb{F}^n:\boldsymbol{x_{s_{{\cal D}_{t}}}} \neq \boldsymbol{0},~\forall\boldsymbol{\bar{z}}\in \mathbb{F}^{\beta}\end{aligned}$$ holds if and only if the following condition holds $$\label{eqnlemma}
cols(I_{\boldsymbol{{\cal D}_{t}}}) \subseteq \left\langle\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right)\right\rangle.$$ Therefore a given network code is $\beta$-network-error detecting (or $\lfloor\frac{\beta}{2}\rfloor$-network-error correcting) if and only if the condition (\[eqnlemma\]) holds for all $supp(\boldsymbol{z})\in \left\{{\cal F} \subseteq {\cal E}:|{\cal F}|=\beta \right\}$ at all sinks $t\in{\cal T}.$
We first prove the *If* part. Since $cols(I_{\boldsymbol{{\cal D}_{t}}})$ is in the subspace $\left\langle\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right)\right\rangle,$ linear combinations of the columns of $\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right)$ should generate the columns of $I_{\boldsymbol{{\cal D}_{t}}}.$ Thus, we must have for some matrix $X$ of size $n_{t}\times |\boldsymbol{{\cal D}_{t}}|,$ $$\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right)X = I_{\boldsymbol{{\cal D}_{t}}}.$$ Now suppose for some $\left(\boldsymbol{x}~~\boldsymbol{\bar{z}}\right)$ with $\boldsymbol{x_{s_{{\cal D}_{t}}}} \neq \boldsymbol{0}$ and some $~\boldsymbol{\bar{z}}\in \mathbb{F}^{\beta}$ we have $$\left(\boldsymbol{x}~~\boldsymbol{\bar{z}}\right)
\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right) = \boldsymbol{0}.$$
Multiplying both sides by $X,$ we then have $\boldsymbol{x_{s_{{\cal D}_{t}}}} = \boldsymbol{0},$ a contradiction. This proves the If part.
Now we prove the *only if* part. Let $\boldsymbol{F_{{\cal S},t,{\cal D}_{t}}}$ denote the submatrix of $\boldsymbol{F_{{\cal S},t}}$ consisting of the $|\boldsymbol{{\cal D}_{t}}|$ rows corresponding to the symbols demanded by $t.$ Let $\boldsymbol{F_{{\cal S},t,\overline{{\cal D}_{t}}}}$ denote the submatrix of $\boldsymbol{F_{{\cal S},t}}$ with rows other than those in $\boldsymbol{F_{{\cal S},t,{\cal D}_{t}}}.$ Then because (\[eqn4\]) holds, we must have $$\begin{aligned}
rank&\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(z),{t}}}
\end{array}
\right)\\
&= rank(\boldsymbol{F_{{\cal S},t,{\cal D}_{t}}}) + rank
\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t,\overline{{\cal D}_{t}}}} \\
\boldsymbol{F_{supp(z),{t}}}
\end{array}
\right).\end{aligned}$$ The above equation follows because (\[eqn4\]) requires that the rows of $\boldsymbol{F_{{\cal S},t,{\cal D}_{t}}}$ and $\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t,\overline{{\cal D}_{t}}}} \\
\boldsymbol{F_{supp(z),{t}}}
\end{array}
\right)$ be linearly independent. Thus, $$\begin{aligned}
\label{eqnno33}
rank\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(z),{t}}}
\end{array}
\right)= |\boldsymbol{{\cal D}_{t}}| + rank
\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t,\overline{{\cal D}_{t}}}} \\
\boldsymbol{F_{supp(z),{t}}}
\end{array}
\right).\end{aligned}$$ Let the concatenated matrix $$\left(
\begin{array}{cc}
\boldsymbol{F_{{\cal S},t}} \vspace{-0.2cm}& \\
\vspace{-0.2cm} & I_{\boldsymbol{{\cal D}_{t}}}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}} & \\
\end{array}
\right)$$ be denoted by $Y.$ Again, it is easy to see that $$\begin{aligned}
&rank(Y)\\
&=
rank
\left(
\begin{array}{cc}
\boldsymbol{F_{{\cal S},t,{\cal D}_{t}}} & I_{|\boldsymbol{{\cal D}_{t}}|}
\end{array}
\right)
+
rank
\left(
\begin{array}{cc}
\boldsymbol{F_{{\cal S},t,\overline{{\cal D}_{t}}}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right)\\
&= |\boldsymbol{{\cal D}_{t}}|
+
rank
\left(
\begin{array}{cc}
\boldsymbol{F_{{\cal S},t,\overline{{\cal D}_{t}}}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right) \\
&= rank\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(z),{t}}}
\end{array}
\right),\end{aligned}$$ where the last equality follows from (\[eqnno33\]). This proves the only if part. Together with (\[eqn3a\]), the lemma is proved.
Matroids {#sec3}
========
In this section, we provide some basic definitions and results from matroid theory that will be used throughout this paper. For more details, the reader is referred to [@Oxl].
\[matroiddefnindp\] Let $E$ be a finite set. A *matroid* $\cal{M}$ on $E$ is an ordered pair $(E,\cal{I}),$ where the set $\cal{I}$ is a collection of subsets of $E$ satisfying the following three conditions
1. $\phi \in \cal{I}.$
2. If $X \in \cal{I}$ and $X' \subseteq X,$ then $X' \in \cal{I}.$
3. If $X_1$ and $X_2$ are in $\cal{I}$ and $|X_1|<|X_2|,$ then there is an element $e$ of $X_2-X_1$ such that $X_1 \cup e\in \cal{I}.$
The set $E$ is called the *ground set* of the matroid and is also referred to as $E(\cal{M}).$ The members of set $\cal{I}$ (also referred to as ${\cal I}(\cal{M})$) are called the *independent sets* of $\cal{M}.$ A maximal independent subset of $E$ is called a *basis* of $\cal M$, and the set of all bases of ${\cal M}$ is denoted by ${\cal B}({\cal M}).$ The set ${\cal I}({\cal M})$ is then obtained as ${\cal I}({\cal M})=\left\{X\subseteq B: B \in {\cal B}({\cal M})\right\}.$ A subset of $E$ which is not in $\cal{I}$ is called a *dependent set*. A minimal dependent set of $E$ (any of whose proper subsets is in $\cal{I}$) is called a *circuit* and the set of circuits of $E$ is denoted by $\cal{C}$ or $\cal{C}(\cal{M}).$ With ${\cal M},$ a function called the *rank* function is associated, whose domain is the power set $2^E$ and codomain is the set of non-negative integers. The rank of any $X \subseteq E$ in $\cal{M},$ denoted by $r_{\cal{M}}(X)$, is defined as the maximum cardinality of a subset of $X$ that is a member of $\cal{I}(\cal{M}).$ We denote $r_{\cal{M}}\left(E({\cal M})\right)=r({\cal M}).$
The set of circuits of a matroid $\cal M$ satisfy the property that if $C_1,C_2 \in {\cal C}({\cal M}),$ and $e\in C_1 \cap C_2,$ then there exists a circuit $C_3 \subseteq \left(C_1\cup C_2\right) - e.$ This is known as the *circuit-elimination axiom*.
Besides using the independent sets, a matroid on $E$ can defined by several other ways, including by specifying the set of circuits, the set of bases or the rank function. We now give the definition of a matroid based on the properties satisfied by the rank function for our use in Section \[secinsufficiency\].
\[matroiddefnrank\] Let $E$ be a finite set. A function $r:2^E\rightarrow \mathbb{Z}^+\cup\left\{0\right\}$ is the *rank* function of a matroid on $E$ if and only if $r$ satisfies the following conditions.
1. If $X\subseteq E,$ then $0\leq r(X) \leq |X|.$
2. If $X\subseteq Y \subseteq E,$ then $r(X)\leq r(Y).$
3. If $X$ and $Y$ are subsets of $E,$ then $$r(X\cup Y)+r(X\cap Y) \leq r(X)+r(Y).$$
Two matroids ${\cal M}_1$ and ${\cal M}_2$ are said to be *isomorphic*, denoted as ${\cal M}_1 \widetilde{=} {\cal M}_2,$ if there is a bijection $\varphi$ from $E({\cal M}_1)$ to $E({\cal M}_2)$ such that, for all $X \subseteq E({\cal M}_1),$ $\varphi(X)$ is independent in ${\cal M}_2$ if and only if $X$ is independent in ${\cal M}_1.$
The *vector matroid* associated with a matrix $A$ over some field $\mathbb{F}$, denoted by ${\cal M}[A],$ is defined as the ordered pair $(E,\cal{I})$ where $E$ consists of the set of column labels of $A,$ and $\cal{I}$ consists of all the subsets of $E$ which index columns that are linearly independent over $\mathbb{F}$. An arbitrary matroid $\cal M$ is said to be $\mathbb{F}$-*representable* if it is isomorphic to a vector matroid associated with some matrix $A$ over some field $\mathbb{F}.$ The matrix $A$ is then said to be a *representation* of $\cal M.$ The rank function of a representable matroid $\cal{M}$, given by $r_{\cal{M}}(X),X\subseteq E$ is therefore equal to the rank of the submatrix of columns corresponding to $X$ in the matrix $A$ to which the matroid is associated. A matroid which is not representable over any finite field is called a *nonrepresentable* matroid.
\[exm1\] Let $A =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1
\end{array}
\right)
$ with elements from $\mathbb{F}_2.$ Then the matroid ${\cal M}[A]$ over the set $E=\left\{1,2,3,4\right\}$ of column indices of $A$ is defined by $${\cal I}({\cal M}) = \left\{\left\{1\right\},\left\{2\right\},\left\{4\right\},\left\{1,2\right\},\left\{1,4\right\},\left\{2,4\right\}\right\}.$$
Let $E = \left\{1,2,...,m\right\}$ for some positive integer $m.$ For some non-negative integer $k \leq m,$ let ${\cal I} = \left\{I \subseteq E: |I|\leq k\right\}.$ The set $\cal I$ satisfies the axioms of independent sets of a matroid on $E,$ referred to as the *uniform matroid* ${\cal U}_{k,m}.$
The vector matroid of a generator matrix of an MDS code of length $m$ and with number of information symbols $k$ is isomorphic to the uniform matroid $U_{k,m}.$
Let $\{{\cal M}_i:i=1,2,..,m\}$ be a collection of matroids defined on the disjoint groundsets $\left\{E_i:i=1,2,..,m\right\}$ respectively. The *direct sum* of the matroids, denoted by $\boxplus_{i=1}^m{\cal M}_i,$ over the groundset $\uplus_{i=1}^m E_i$ is the matroid with the independent sets as follows. $${\cal I}=\left\{\uplus I_i: I_i\in{\cal I}({\cal M}_i)\right\}.$$
\[lemmamatroidequivalence\] Let ${\cal M}={\cal M}[A],$ $A$ being a matrix over some field $\mathbb{F}.$ The matroid $\cal M$ remains unchanged if any of the following operations are performed on $A$
- Interchange two rows.
- Multiply a row by a non-zero member of $\mathbb{F}$.
- Replace a row by the sum of that row and another.
- Adjoin or delete a zero row.
- Multiply a column by a non-zero member of $\mathbb{F}$.
By the row operations of Lemma \[lemmamatroidequivalence\], it is clear that any $\mathbb{F}$-representable matroid can be uniquely expressed as the vector matroid of a matrix of the form $\left(I_{r({\cal M})}~~~A_{r({\cal M})\times (|E({\cal M})|-r({\cal M}))}\right),$ with elements from $\mathbb{F}.$
Let $\cal{M}$ be the matroid $(E,{\cal I})$ and suppose that $X\subseteq E.$ Let ${\cal I}|X=\left\{I\subseteq X: I \in {\cal I}\right\}.$ Then the ordered pair $(X,{\cal I}|X)$ is a matroid and is called the *restriction* of $\cal{M}$ to $X$ or the *deletion* of $E-X$ from $\cal{M}.$ It is denoted as ${\cal M}|X$ or ${\cal M}\backslash(E-X).$ It follows that the circuits of ${\cal M}|X$ are given by ${\cal C}({\cal M}|X)=\left\{C\subseteq X : C\in {\cal C}(\cal M)\right\}.$
The restriction of a $\mathbb{F}$-representable matroid is also $\mathbb{F}$-representable. The restriction of a vector matroid ${\cal M}[A]$ to a subset $T$ of the column indices of $A$ is also obtained as the vector matroid of a matrix $A'$ where $A'$ is obtained from $A$ by considering only those columns of $A$ indexed by $T.$
Let ${\cal M}={\cal M}[A]$ be the matroid from Example \[exm1\]. Let $T=\left\{1,2,3\right\}\subseteq E({\cal M}).$ The matroid ${\cal M}|T$ is given by ${\cal I}({\cal M}|T) = \left\{\left\{1\right\},\left\{2\right\},\left\{1,2\right\}\right\} = {\cal I}({\cal M}[A']),$ where $A' =
\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0
\end{array}\right).$
Let $\cal M$ be a matroid and ${\cal B}^*({\cal M})$ be $\left\{E({\cal M})-B:B\in {\cal B}({\cal M})\right\}$. Then the set ${\cal B}^*({\cal M})$ forms the set of bases of a matroid on $E({\cal M}),$ defined as the *dual matroid* of $\cal M,$ denoted as ${\cal M}^*.$ Clearly $({\cal M}^*)^* = {\cal M}.$ We also have $$r_{{\cal M}^*}(X)=|X|-r({\cal M})+r_{\cal M}(E({\cal M})-X),$$ for any $X\subseteq E({\cal M}).$
\[exmdual\] The dual matroid of the matroid ${\cal M}[A]$ given in Example \[exm1\] is given by the vector matroid ${\cal M}[A']$ corresponding to the matrix $A'=
\left(\begin{array}{cccc}
0 & 0 & 1 & 0 \\
1 & 1 & 0 & 1
\end{array}\right).
$
\[contraction\] Let $\cal M$ be a matroid on $E$ and $T \subseteq E.$ The *contraction* of $T$ from $\cal M,$ denoted as ${\cal M}/T,$ is given by the matroid $({\cal M}^*\backslash T)^*$ with $E-T$ as its ground set. The set of independent sets of ${\cal M}/T$ is as follows. $$\label{independentcontraction}
{\cal I}({\cal M}/T) =\left\{I\subseteq E-T : I \cup B_T \in {\cal I}({\cal M}) \right\} \\$$ where $B_T$ is some basis of ${\cal M}|T.$ The set of circuits of ${\cal M}/T$ consists of the minimal non-empty members of $\left\{C-T:C\in {\cal C}({\cal M})\right\}.$
In Section \[sec4\], we show that for a network to be a matroidal error detecting (or correcting) network associated with a matroid $\cal M,$ the circuits of $\cal M$ have to satisfy certain conditions. Thus the concept of circuits of a matroid is the gateway for our results concerning matroidal error detecting (correcting) networks. This is in contrast with the theory of matroidal networks developed in [@DFZ; @KiM], where any arbitrary matroid can give rise to a corresponding matroidal network.
Let $\cal M$ be the matroid with ground set $E=\left\{a,b,c,d,e\right\}$ and with set of bases ${\cal B}$ being the set of all subsets of $E$ of size four. We wish to find ${\cal M}/\left\{d,e\right\}.$ It can be seen that the dual matroid ${\cal M}^*$ has the set of all singletons of $E$ as its set of bases ${\cal B}^*.$ Then, the matroid ${\cal M}^*\backslash\left\{d,e\right\}$ has the ground set $E'=\left\{a,b,c\right\}$ and the set of bases $${\cal B}' = \left\{\{a\},\{b\},\{c\}\right\}.$$ The dual matroid of ${\cal M}^*\backslash\left\{d,e\right\}$ is the matroid ${\cal M}/\left\{d,e\right\}$ with the ground set $\left\{a,b,c\right\}$ and the set of bases $${\cal B}'' = \left\{\{a,b\},\{a,c\},\{b,c\}\right\}.$$
[@Oxl] \[remarkcontraction\] The contraction of a $\mathbb{F}$-representable matroid is also $\mathbb{F}$-representable. Let ${\cal M}[A]$ be the vector matroid associated with a matrix $A$ over $\mathbb{F}.$ Let $e$ be the index of a non-zero column of $A.$ Suppose using the elementary row operations listed in Lemma \[lemmamatroidequivalence\], we transform $A$ to obtain a matrix $A'$ which has a single non-zero entry in column $e.$ Let $A''$ denote the matrix which is obtained by deleting the row and column containing the only non-zero entry of column $e.$ Then $${\cal M}[A]/\left\{e\right\} = \left({\cal M}[A]^*\backslash\left\{e\right\}\right)^* = {\cal M}[A''],$$ where ${\cal M}[A]^*$ is the dual matroid of ${\cal M}[A].$
Let ${\cal M}={\cal M}[A]$ be the matroid from Example \[exm1\]. We want to find ${\cal M}[A]/\left\{4\right\}.$ We first obtain ${\cal M}[A]/\left\{4\right\}$ in a straightforward manner according to the definition of contraction. The dual matroid of ${\cal M}[A]$ is the vector matroid corresponding to the matrix $$A_d=
\left(
\begin{array}{cccc}
0 & 0 & 1 & 1 \\
1 & 1 & 0 & 1
\end{array}
\right).$$ Now ${\cal M}[A_d]\backslash\left\{4\right\}$ is the vector matroid corresponding to the matrix $$A'_d=
\left(
\begin{array}{ccc}
0 & 0 & 1 \\
1 & 1 & 0
\end{array}
\right).$$ According to the definition of contraction, ${\cal M}[A'_d]^*={\cal M}[A]/\left\{4\right\}.$ The set of bases of ${\cal M}[A'_d]^*$ is $\left\{\{1\},\{2\}\right\}.$ Thus the matroid $${\cal M}[A]/\left\{4\right\} = \left(E=\left\{1,2,3\right\},{\cal I}=\left\{\phi,\{1\},\{2\}\right\}\right).$$ We can also obtain ${\cal M}[A]/\left\{4\right\}$ using the technique shown in Remark \[remarkcontraction\]. Towards that end, using row operations on $A,$ we obtain the matrix $$A' =
\left(
\begin{array}{cccc}
1 & 0 & 0 & 1 \\
1 & 1 & 0 & 0
\end{array}
\right).$$ By removing the row corresponding to the only non-zero entry in the $4^{th}$ column of $A'$ and the $4^{th}$ column itself, we obtain the matrix $A''=(1~ 1~ 0).$ It is easily verified that ${\cal M}[A'_d]^* = {\cal M}[A''].$
Let $\cal M$ be a matroid on $E$ and $X$ be a subset of $E$. The *closure* of $X$ is then defined to be the set $cl_{\cal M}(X)=\left\{x\in E:r_{\cal M}(X\cup x)=r_{\cal M}(X)\right\}.$ If $X=cl_{\cal M}(X),$ then $X$ is said to be a *flat* of $\cal M.$ A flat $H$ such that $r_{{\cal M}}(H) = r({\cal M})-1$ is called a *hyperplane* of $\cal M.$ Moreover, $X \subset E$ is a hyperplane of $\cal M$ if and only if $E-X$ is a circuit of ${\cal M}^*.$
Consider the matroid ${\cal M}[A]$ of Example \[exm1\]. Let $X=\left\{1\right\},$ then $cl_{\cal M}(X)=\left\{1,3\right\}$ is a flat. Moreover it is also a hyperplane of $\cal M.$ Also, it can be easily verified that the set $$E(\{{\cal M}[A]\})-\left\{1,3\right\} = \left\{2,4\right\}$$ is a circuit of the dual matroid ${\cal M}[A]^*,$ given in Example \[exmdual\].
Let $\cal N$ be a matroid on $E.$ If for some $e\in E,$ $\left\{e,f\right\}\in {\cal C}({\cal N})$ for some $f \in E,$ then the matroid $\cal N$ is said to be a *parallel extension* of ${\cal M}={\cal N}\backslash \{e\},$ and is denoted by ${\cal M}+^p_f e.$ The element $e$ is said to be *added in parallel* with element $f.$ Also, a parallel extension ${\cal N}^*$ of ${\cal M}^*$ is said to be a *series extension* of ${\cal M},$ in which case ${\cal M}={\cal N}/\{e\}$ and $\cal N$ is denoted by ${\cal M}+^s_f e.$ The element $e$ is then said to be *added in series* with element $f.$
The following two lemmas summarise equalities which can be proved easily from the definitions of the series and parallel matroids and the duality relations between them. We state them here without proof so that we may use them later in Section \[sec5\].
\[remparallel\] Let $f\in E({\cal M})$ such that $\{f\}\notin{\cal C}({\cal M}).$ In a parallel extension ${\cal N}={\cal M}+^p_f e$ of $\cal M.$ The following statements are true. $$\begin{aligned}
\label{eqn201}
&r_{\cal N}(X)=r_{\cal M}(X), \forall X\subseteq E({\cal M}).\\
\label{eqn202}
&r_{\cal N}(X-f+e)=r_{\cal M}(X), \forall X\subseteq E({\cal M})~\text{with}~f\in X.\\
\label{eqn203}
&r({\cal N})=r({\cal M}).\\
\label{eqn204}
&{\cal M}={\cal N}\backslash\{e\}.\end{aligned}$$
\[remseries\] Let $f\in E({\cal M})$ such that $\{f\}\notin{\cal C}({\cal M}).$ In a series extension ${\cal N}={\cal M}+^s_f e$ of $\cal M,$ The following statements are then true. $$\begin{aligned}
\label{eqn301}
&{\cal B}({\cal N})=\{B\cup\{e\}:B \in {\cal B}({\cal M})\}.\\
\label{eqn302}
&r_{\cal M}(X)=r_{\cal N}(X), \forall X\subseteq E({\cal M})~\text{such that}~f\notin X.\\
\label{eqn303}
&{\cal M}={\cal N}/\{e\}.\end{aligned}$$
We now present two lemmas, which will be useful for describing the construction of matroidal error detecting (correcting) networks in Section \[sec5\]. They also serve as examples for parallel and series extensions of a matroid. To the best of our knowledge they are not explicitly found in existing matroid literature. Therefore, we prove them here for the sake of completeness.
\[parallelrepresentation\] Let $A$ be an $n\times N$ matrix over $\mathbb{F}.$ For some $1\leq i \leq n,$ let ${A^i}$ be a non-zero column of $A.$ Let $B$ be the $n\times (N+1)$ matrix $$\left(
\begin{array}{ccccc}
{A^1} & {A^2} & ... & {A^{N}} & {A^{i}}
\end{array}
\right).$$ Then, ${\cal M}[B]={\cal M}[A]+^p_i \left\{N+1\right\},$ i.e., ${\cal M}[B]$ is a parallel extension of the vector matroid associated with $A.$
Clearly ${\cal M}[B]\backslash \left\{N+1\right\} = {\cal M}[A] = {\cal M}.$ Moreover, in ${\cal M}[B],$ the $\left(N+1\right)^{th}$ column of $B$ is equal to the $i^{th}$ column, thus $\left\{i,N+1\right\} \in {\cal C}({{\cal M}[B]}).$ Thus, by definition, ${\cal M}[B] = {\cal M}[A]+^p_i \left\{N+1\right\},$ the parallel extension of ${\cal M}[A]$ at $i.$ This proves the lemma.
\[seriesrepresentation\] Let $
A = \left(
\begin{array}{cccc}
{A^1} & {A^2} & ... & {A^N}
\end{array}
\right)
$ be an $n\times N$ matrix over $\mathbb{F},$ where ${A^j}$ denotes the $j^{th}$ column of $A.$ For some $1\leq i \leq n,$ let ${A^i}$ be a non-zero column of $A$ such that $A^i \in \left\langle\left(A^{\left\{1,...,N\right\}-i}\right)\right\rangle$ . Let $B$ be the $(n+1)\times (N+1)$ matrix $$\left(
\begin{array}{ccccccccc}
{A^1} & {A^2} & ...& {A^{i-1}} & {A^{i}} & {A^{i+1}} & ... & {A^{N}} & \boldsymbol{0} \\
0 & 0 & ... & 0 & 1 & 0 & .... & 0 & 1
\end{array}
\right),$$ where $\boldsymbol{0} \in \mathbb{F}^n.$ Then the vector matroid associated with $B,$ ${\cal M}[B],$ is a series extension of the vector matroid associated with $A,$ ${\cal M}[A]$ at $i,$ i.e., ${\cal M}[B]={\cal M}[A]+^s_i \left\{N+1\right\}.$
Because $A^i \in \left\langle\left(A^{\left\{1,...,N\right\}-i}\right)\right\rangle,$ we must have $B^i \in \left\langle\left(B^{\left\{1,...,N,N+1\right\}-i}\right)\right\rangle.$ Also from the form of $B,$ we have $B^i \notin \left\langle\left(B^{\left\{1,...,N\right\}-i}\right)\right\rangle.$ Thus, $\left\{1,2,..,N+1\right\}-\left\{i,N+1\right\}$ of columns forms a hyperplane of ${\cal M}[B].$ Therefore, $\left\{i,N+1\right\}$ is a circuit in ${\cal M}[B]^*.$ Also, as ${\cal M}[B]/\left\{N+1\right\}={\cal M}[A],$ we must have ${\cal M}[B]^*\backslash\left\{N+1\right\}={\cal M}[A]^*.$ Thus ${\cal M}[B]^*$ is a parallel extension of ${\cal M}[A]^*,$ i.e., ${\cal M}[B]^*={\cal M}[A]^*+_i^p\left\{N+1\right\}.$ Hence ${\cal M}[B]={\cal M}[A]+_i^s\left\{N+1\right\},$ i.e., ${\cal M}[B]$ is a series extension of ${\cal M}[A].$ This proves the lemma.
If a matroid $\cal M$ is obtained from a matroid $\cal N$ by deleting a non-empty subset $T$ of $E({\cal N}),$ then $\cal N$ is called an *extension* of $\cal M.$ In particular, if $|T|=1$, then ${\cal N}$ is said to be a *single-element extension* of ${\cal M}.$
\[singleelement\] Let $\cal K$ be a set of flats of $\cal M$ satisfying the following conditions.
- If $F\in {\cal K}$ and $F'$ is a flat of $\cal M$ containing $F,$ then $F' \in{\cal K}.$
- If $F_1,F_2\in{\cal K}$ are such that $r_{\cal M}(F_1)+r_{\cal M}(F_2)=r_{\cal M}(F_1\cup F_2)+r_{\cal M}(F_1\cap F_2),$ then $F_1 \cap F_2 \in {\cal K}.$
Any set $\cal K$ of flats of $\cal M$ which satisfies the above conditions is called a *modular cut* of $\cal M.$ There is a one-one correspondence between the set of all modular cuts of a matroid and the set of all single-element extensions of a matroid. We denote the single-element extension ${\cal N}$ corresponding to the modular cut $\cal K$ as ${\cal M}+_{_{\cal K}} e,$ where $e$ is the new element that is added. Also, the set ${\cal K}$ consists precisely of those flats of $\cal M$ such that for each $F\in {\cal K},$ we have $r_{\cal N}(F\cup e) = r_{\cal N}(F).$
\[examplesingle\] Let ${\cal M}$ be the vector matroid of the matrix over $\mathbb{F}_2$ $$B=\left(
\begin{array}{ccccc}
1&0&0&0&1\\
0&1&0&0&1\\
0&0&1&0&1
\end{array}
\right).$$ Consider the flats $F_1=\{3,4,5\}$ and $F_2=\{1,2,3,4,5\}.$ Note that the flats $F_1$ and $F_2$ form a modular cut $\cal K$ satisfying the conditions in Definition \[singleelement\]. Thus there exists a single-element extension of $\cal M$ which corresponds to this modular cut. Let ${\cal M}'$ be this matroid. It can be verified that ${\cal M}'$ is the vector matroid of the matrix over $\mathbb{F}_3$ $$B'=\left(
\begin{array}{cccccc}
1&0&0&0&1&1\\
0&1&0&0&1&1\\
0&0&1&0&1&2
\end{array}
\right).$$ However, ${\cal M}'$ does not have a representation over the field $\mathbb{F}_2.$
Let $\cal M$ be a matroid. For a flat $F$ in the set of flats of $\cal M,$ let ${\cal K}_{F}$ denote the set of all flats of $\cal M$ which contain $F.$ Then ${\cal K}_{F}$ can be easily verified to be a modular cut of $\cal M$ and is defined as the *principal modular cut* of $\cal M$ *generated by the flat* $F.$ The single-element extension of $\cal M$ corresponding to this principal modular cut is then defined as the *principal extension of* $\cal M$ *generated by the flat* $F,$ and is denoted by ${\cal M}+_{_{{\cal K}_F}} e,$ where $e$ is the new element added.
The single-element extension shown in Example \[examplesingle\] is a principal extension of the matroid ${\cal M}$ generated by the flat $F_1.$ The principal modular cut corresponding to this extension is then ${\cal K}.$
Matroidal Error Correcting and Detecting Networks {#sec4}
=================================================
In this section, we define *matroidal error correcting and detecting networks* and establish the link between matroids and network-error correcting and detecting codes. The contents of this section are logical extensions of the concept of the matroidal networks defined in [@DFZ] which gave the connection between matroids and network codes. The definition of a matroidal network is as follows.
\[matroidalnetworkdefinition\] Let ${\cal G}({\cal V},{\cal E})$ be a network with a message set $\mu.$ Let ${\cal M}=(E,{\cal I})$ be a matroid. The network ${\cal G}$ is said to be a *matroidal network* associated with ${\cal M}$ if there exists a function $f:\mu\cup{\cal E}\rightarrow E$ such that the following conditions are satisfied.
1. $f$ is one-one on $\mu.$
2. $f(\mu) = \cup_{m \in \mu}f(m) \in{\cal I}.$
3. $r_{\cal M}(f(In(v)))=r_{\cal M}(f(In(v)\cup Out(v))),$ $\forall v\in {\cal V}.$
Suppose $\cal M$ is a representable matroid. Then the first two conditions of Definition \[matroidalnetworkdefinition\] can be interpreted as associating independent global encoding vectors with the information symbols. The last condition will then ensure that flow conservation holds throughout the network, and also that the sinks are able to decode the demanded information symbols. Thus Definition \[matroidalnetworkdefinition\] can be looked at as the matroidal generalization of a scalar linear network code, which is confirmed by the following theorem proved in parts in [@DFZ] and [@KiM].
\[matroidalnetworkthm\] A network $\cal G$ is matroidal in association with a representable matroid if and only if it has a scalar linear network coding solution.
Let ${\cal G}({\cal V},{\cal E})$ be an acyclic network with a collection of sources $\cal S$ with message set $\mu$ (with $n$ elements) and sinks $\cal T,$ and a given topological order on $\cal E.$ Let $\beta < |{\cal E}|$ be a non-negative integer, and $\mathfrak{F}=\left\{{\cal F} \subseteq {\cal E}:|{\cal F}| = \beta \right\}$ be the collection of error patterns of size $\beta.$ Let $\cal M$ be a matroid over a ground set $E$ with $n + 2|{\cal E}|$ elements, and with $r({\cal M})= n+|{\cal E}|.$ We now define *matroidal error detecting and correcting networks* by extending the definition of matroidal networks of [@DFZ] for the case of networks where errors occur.
\[matroidalerrornetworkdefinition\] The network ${\cal G}$ is said to be a *matroidal* $\beta$-*error detecting network* associated with $\cal M,$ if there exists a function $f:\mu\cup{\cal E} \rightarrow E({\cal M})$ such that the following conditions are satisfied.
(A) **Independent inputs condition**: $f$ is one-one on $\mu,$ where $f(\mu) = \cup_{m \in \mu}f(m) \in {\cal I}({\cal M})$.
(B) **Flow conservation condition**: For some basis $B$ of ${\cal M}$ obtained by extending $f(\mu)$ (where $B-f(\mu)=\left\{b_{n+1},...,b_{n+|{\cal E}|}\right\}$ is ordered according to the given topological order on $\cal E$), the following conditions should hold for all $e_i \in {\cal E}.$
1. $f(e_i) \notin cl_{\cal M}(B-f(\mu))$
2. $r_{\cal M}\left(f\left(In(e_i)\right)\cup f(e_i)\cup b_{n+i}\right)$ $$\begin{aligned}
%\nonumber
& = r_{\cal M}\left(f\left(In(e_i)\right)\cup b_{n+i}\right)\\
%\nonumber
& = r_{\cal M}\left(f\left(In(e_i)\right)\right) + r_{\cal M}(b_{n+i}) \\
& = r_{\cal M}\left(f\left(In(e_i)\right)\right) + 1.\end{aligned}$$
(C) **Successful decoding condition**: For each error pattern ${\cal F}=\left\{e_{i_1},e_{i_2},...,e_{i_{\beta}}\right\}\in\mathfrak{F},$ let $B_{\overline{{\cal F}}} = B-f(\mu)-\left\{b_{n+i_1},b_{n+i_2},...,b_{n+i_{\beta}}\right\}.$ Let ${\cal M}_{\cal F}$ be the $n+\beta+|{\cal E}|$ element matroid ${\cal M}/B_{\overline{\cal F}}.$ Then, at every sink $t\in {\cal T},$ for each ${\cal F} \in \mathfrak{F},$ we must have $$r_{{\cal M}_{\cal F}}\left(f\left(In_{\cal E}(t)\right)\cup f\left(\boldsymbol{{\cal D}_t}\right)\right)=r_{{\cal M}_{\cal F}}\left(f\left(In_{\cal E}(t)\right)\right),
%r_{{\cal M}_{\cal F}}\left(f_{\cal F}\left(In_{\cal E}(t)\right)\cup f_{\cal F}\left(\boldsymbol{{\cal D}_t}\right)\right)=r_{{\cal M}_{\cal F}}\left(f_{\cal F}\left(In_{\cal E}(t)\right)\right),$$ where $In_{\cal E}(t) \subseteq In(t)$ denotes the set of incoming edges at sink $t$ and $\boldsymbol{{\cal D}_t}$ is the set of demands at $t.$
\[matroidalerrorcorrectionnetworkdefinition\] The network ${\cal G}$ is said to be a *matroidal* $\alpha$-*error correcting network* associated with a matroid $\cal M,$ if it is a matroidal $2\alpha$-*error detecting network* associated with $\cal M.$
As with Definition \[matroidalnetworkdefinition\], Definitions \[matroidalerrornetworkdefinition\] and \[matroidalerrorcorrectionnetworkdefinition\] can be viewed as the matroidal abstractions of a scalar linear network-error detecting and correcting codes (Theorem \[matroidalerrornetworkthm\] will present the formal statement and proof of this abstraction). If $\cal M$ is a representable matroid, then as in Definition \[matroidalnetworkdefinition\], Condition (A) is equivalent to saying that the global encoding vectors corresponding to the information symbols are linearly independent. Condition (B1) is equivalent to saying that the symbol flowing on any edge in the network is a *non-zero* linear combination of the information symbols, added with a (not necessarily non-zero) linear combination of the network-errors in the network. Such a condition is not a restriction, because if an edge carries an all-zero linear combination of the input symbols, then such an edge can simply be removed from the network. Condition (B2) is equivalent to a modified flow conservation condition in networks with errors, implying that the symbol flowing through any edge $e$ in the network is a linear combination of the incoming symbols at $In(e)$ and the network-error in that particular edge. Condition (C) ensures that the sinks can decode their demands. Although our definitions are abstracted from scalar linear network-error detecting and correcting codes, we will show in Section \[secinsufficiency\] that it applies to nonlinear schemes also.
The Condition (C) of Definition \[matroidalerrornetworkdefinition\] requires that $f(x), \forall x\in\mu\cup{\cal E}$ exist in $E({\cal M}_{\cal F})$ in the first place. However, this is ensured by Condition (B1). To see this, first we note that $f(\mu)\subset E({\cal M}_{\cal F})$ because these elements are in $B$ and are not contracted out of $\cal M.$ Now consider the set $f(e)\cup(B-f(\mu))$ for any $e\in {\cal E},$ which is independent in $\cal M$ because of Condition (B1). By (\[independentcontraction\]) in the definition of the contraction of a matroid, we have that $f(e)$ exists and is also not dependent in ${\cal M}_{\cal F}.$ Therefore, $f(x)$ is well defined in ${\cal M}_{\cal F}$ also.
Although Definition \[matroidalnetworkdefinition\] and Definition \[matroidalerrornetworkdefinition\] in the case of no network-errors do not immediately appear to agree, it can be shown that a network is a matroidal network associated with some matroid $\cal M,$ if and only if it is a matroidal error detecting network with $\beta=0,$ with respect to another matroid derived using extensions of $\cal M.$ This can be inferred easily from the remainder of this paper, therefore we leave it without an explicit proof.
We now present the main result of this paper which is the counterpart of the results from [@DFZ; @KiM] which relate networks with scalar linearly solvable network codes to representable matroids.
\[matroidalerrornetworkthm\] Let ${\cal G}({\cal V},{\cal E})$ be an acyclic communication network with sources $\cal S$ and sinks $\cal T.$ The network ${\cal G}$ is a *matroidal* $\beta$-*error detecting network* associated with a $\mathbb{F}$-representable matroid if and only if it has a scalar linear network-error detecting code over $\mathbb{F}$ that can correct network-errors at any $\beta$ edges which are known to the sinks.
*If part:* Suppose there exists a scalar linear $\beta$-network-error detecting code over $\mathbb{F}$ for $\cal G$ with the matrices $A_{s_i} (i=1,2,...,|{\cal S}|), \boldsymbol{F}$ and $B_{t},~t\in{\cal T},$ as defined in Section \[sec2\], according to the given topological ordering on $\cal E$. Let $\cal A$ be the matrix as in (\[formofA\]).
Let $\cal X$ be the row-wise concatenated matrix $
\left(
\begin{array}{c}
{\cal A}\boldsymbol{F} \\
\boldsymbol{F}
\end{array}
\right)
$ of size $(n+|{\cal E}|)\times |{\cal E}|,$ and $\cal Y$ be the column-wise concatenated matrix $
\left(I_{n+|{\cal E}|}~~~{\cal X}\right).
$ Also, let ${\cal M}={\cal M}[{\cal Y}],$ the vector matroid associated with $\cal Y,$ with $E({\cal M})$ being the set of column indices of $\cal Y.$ Let $
f:{\cal E}\cup\mu\rightarrow E({\cal M})
$ be the function defined as follows. $$\begin{aligned}
& f(m_i) = i,~~m_i \in \mu, i=1,2,...,n. \\
& f(e_i) = n+|{\cal E}|+i,~\forall~e_i \in {\cal E}~\text{in the given ordering}. \end{aligned}$$ We shall consider the basis for $\cal M$ as $B=\left\{1,2,...,n+|{\cal E}|\right\},$ i.e., the first $n+|{\cal E}|$ columns of ${\cal Y}$. This basis will be used repeatedly in the proof. We shall now prove that the matroid $\cal M$ and function $f$ satisfy the conditions of Definition \[matroidalerrornetworkdefinition\]. Towards this end, first we see that Condition (A) holds by the definition of function $f.$
We first prove that Condition (B1) holds. We have that ${\cal Y}^{n+|{\cal E}|+i} \notin \left\langle({\cal Y}^{B-f(\mu)})\right\rangle,$ because no edge is assigned a zero-global encoding vector, i.e., no column of ${\cal A}\boldsymbol{F}$ is zero. Thus Condition (B1) holds.
To show Condition (B2), first note that because the given set of coding coefficients for the network is a (valid) network code, $\boldsymbol{F}$ is such that $$\label{eqn5}
\boldsymbol{F}^{j} = \left(\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}K_{i,j}\boldsymbol{F}^{i}\right)+\boldsymbol{1_j},$$ where $\boldsymbol{1_j}$ is a column vector in $\mathbb{F}^{|{\cal E}|}$ with all zeros except for the $j^{th}$ entry which is $1 \in \mathbb{F}.$ Also, (\[eqn5\]) implies that $$\begin{aligned}
\nonumber
\left({\cal A}\boldsymbol{F}\right)^{j} & = {\cal A}\boldsymbol{F}^{j} \\
\nonumber
& = {\cal A}\left(\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}K_{i,j}\boldsymbol{F}^{i}\right)+{\cal A}\boldsymbol{1_{j}}\\
\label{eqn6}
&= \left(\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}K_{i,j}\left({\cal A}\boldsymbol{F}\right)^{i}\right)+{\cal A}^j.\end{aligned}$$ Thus, combining (\[eqn5\]) and (\[eqn6\]), we have $$\begin{aligned}
{\cal X}^j &= {\cal Y}^{n+|{\cal E}|+ j} \\
& = \left(\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}K_{i,j}{\cal X}^i\right) + {\cal Y}^{f(\mu)}{\cal A}^j + {\cal Y}^{n+j} \\
& = \left(\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}K_{i,j}{\cal Y}^{n+|{\cal E}|+ i}\right) + {\cal Y}^{f(\mu)}{\cal A}^j + {\cal Y}^{n+j},\\\end{aligned}$$ where ${\cal Y}^{n+j}$ corresponds to $b_{n+j} \in B-f(\mu)$ and the non-zero coefficients of ${\cal A}^j$ can occur only in those positions corresponding to the set of messages generated at $tail(e_j),$ if any, which is a subset of $In(tail(e_j))=In(e_j)$. Also, for any $e_i \in {\cal E}$ with $tail(e_j)=head(e_i),$ the vector ${\cal Y}^{n+|{\cal E}|+ i}$ is some column of the matrix ${\cal Y}^{f(In(e_j))}.$ Thus $$\label{eqnno20}
{\cal Y}^{n+|{\cal E}|+j} \in \left\langle\left({\cal Y}^{f(In(e_j))\cup b_{n+j}}\right)\right\rangle.$$ We also note that the $(n+j)^{th}$ row of ${\cal Y}^{n+j}$ contains $1$ (indicating the error at the edge $e_j$) while the $(n+j)^{th}$ row of ${\cal Y}^{f(In(e_j))}$ is all-zero because of the topological ordering in the acyclic network (as symbols flowing in any edge can have contribution only from upstream errors). Therefore ${\cal Y}^{n+|{\cal E}|+j} \notin \left\langle{\cal Y}^{f(In(e_j))}\right\rangle.$ Along with (\[eqnno20\]), this proves that Condition (B2) holds.
Now we prove that Condition (C) also holds. Let $I({\cal F})=\left\{i_1,i_2,...,i_{\beta}\right\}$ be the index set following the topological ordering corresponding to an arbitrary error pattern ${\cal F}\in\mathfrak{F}$ and let the set $\left\{n+i_1,n+i_2,...,n+i_{\beta}\right\}$ be denoted as $n+I({\cal F}).$ First we note that by definition, ${\cal M}_{\cal F}$ is the vector matroid of the matrix $$\label{eqnno21}
{\cal Z} = {\cal Y}_{f(\mu)\cup (n+I({\cal F}))} = \left(I_{n+\beta}~~ {\cal X}_{f(\mu)\cup (n+I({\cal F}))}\right),$$ where $
{\cal X}_{f(\mu)\cup (n+I({\cal F}))} = \left(
\begin{array}{c}
{\cal A}\boldsymbol{F} \\
\boldsymbol{F}_{I({\cal F})}
\end{array}
\right).
$ Now for a sink $t\in{\cal T},$ $${\cal Z}^{f(In_{\cal E}(t))}={\cal X}_{f(\mu)\cup (n+I({\cal F}))}^{f(In_{\cal E}(t))}=
\left(
\begin{array}{c}
{\cal A}\boldsymbol{F}^{f(In_{\cal E}(t))} \\
\boldsymbol{F}_{I({\cal F})}^{f(In_{\cal E}(t))}
\end{array}
\right).$$ But according to Section \[sec2\], we have, ${\cal A}\boldsymbol{F}^{f(In_{\cal E}(t))} = \boldsymbol{F_{{\cal S},t}},$ and $\boldsymbol{F}_{I({\cal F})}^{f(In_{\cal E}(t))} = \boldsymbol{F_{supp(\boldsymbol{z}),t}},$ where $supp(\boldsymbol{z})={\cal F}.$ By Lemma \[lemmadecoding\], as the given network code is $\beta$-network-error detecting, we must have $$cols(I_{\boldsymbol{{\cal D}_{t}}}) \subseteq \left\langle\left(
{\cal Z}^{f(In_{\cal E}(t))}\right)\right\rangle,$$ where $\boldsymbol{{\cal D}_{t}}\subseteq \mu$ is the set of demands at $t$. But then $I_{\boldsymbol{{\cal D}_{t}}}={\cal Z}^{f(\boldsymbol{{\cal D}_{t}})}$ by (\[eqnno21\]). This proves Condition (C) for sink $t.$ The choice of error pattern and sink being arbitrary, this proves the If part of the theorem.
*Only If part:* Let $\cal M$ be the given $\mathbb{F}$-representable matroid, along with the function $f,$ and basis $B=f(\mu)\uplus\left\{b_{n+1},b_{n+2},...,b_{n+|{\cal E}|}\right\}$ that satisfy the given set of conditions. Let ${\cal Y}=(I_{n+{|{\cal E}|}}~~{\cal X})$ be a representation of $\cal M$ over $\mathbb{F},$ such that $B=\left\{1,2,...,n+|{\cal E}|\right\}.$ First we prove the following claim.
*Claim:* There exists an $n\times |{\cal E}|$ matrix ${\cal A},$ and a $|{\cal E}|\times |{\cal E}|$ matrix $\boldsymbol{F}$ of the form $\boldsymbol{F}=(I_{|{\cal E}|}-K)^{-1}$ for some strictly upper-triangular matrix $K$, such that $$\label{formofX}
{\cal X}=\left(
\begin{array}{c}
{\cal A}\boldsymbol{F} \\
\boldsymbol{F}
\end{array}
\right).$$ *Proof of claim*:
Consider an edge $e_j\in {\cal E}.$ Let $\mu_{tail(e_j)}$ denote indices of the set of messages generated at $tail(e_j).$ As Condition (B2) holds, ${\cal Y}^{f(e_j)}$ is such that $$\begin{aligned}
\nonumber
&{\cal Y}^{f(e_j)} \\
\label{eqn7}
&= \hspace{-0.4cm}\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}a'_{i,j}{\cal Y}^{f(e_i)} +\hspace{-0.3cm} \sum_{m_i \in {\mu}_{tail(e_j)}}\hspace{-0.5cm}c'_{i,j}{\cal Y}^{f(m_i)}+a'_{j,j}{\cal Y}^{n+j},\end{aligned}$$ for some $a'_{i,j}$ and $c'_{i,j}$ in $\mathbb{F}.$ Note that if $e_j$ is such that $In(e_j)\subseteq \mu,$ then by (\[eqn7\]), ${\cal Y}^{f(e_j)}$ is just a linear combination of ${\cal Y}^{{\mu}_{tail(e_j)}}$ and ${\cal Y}^{n+j}.$ Following the ancestral ordering for $j,$ it can be seen that for any edge $e_j,$ ${\cal Y}^{f(e_j)}$ is a linear combination of ${\cal Y}^{\left\{1,2,...,n+j\right\}}$ and ${\cal Y}^{\mu}.$ Thus we have, $${\cal Y}^{f(e_j)} = \sum_{\scriptsize e_i \in {\cal E}: i \leq j}a_{i,j}{\cal Y}^{n+i} +\hspace{-0.2cm} \sum_{m_i \in {\mu}}c_{i,j}{\cal Y}^{f(m_i)}.$$ As Condition (B1) holds, we must have at least one $c_{i,j}\neq 0, \forall i=1,2,...,n$ and because of Condition (B2), we must have $a_{j,j} = a'_{j,j}\neq 0.$ This structure of ${\cal Y}^{f(e_j)}$ also implies that ${\cal Y}^{f(e_j)}\neq {\cal Y}^b,$ for any $b\in B.$ Moreover, we also see that ${\cal Y}^{f(e_i)}\neq {\cal Y}^{f(e_j)},$ for any distinct pair $e_i,e_j$ of edges in $\cal E.$ Arranging all the ${\cal Y}^{f(e_i)}$s in the given topological order (i.e., with $f(e_j)=n+|{\cal E}|+j$), we get ${\cal Y}^{f({\cal E})} = {\cal X},$ and $${\cal X} =
\left(\begin{array}{c}
J_{n\times|{\cal E}|} \\ L_{|{\cal E}|\times|{\cal E}|}
\end{array}\right),$$ where $J$ comprises of the elements $c_{i,j}, 1\leq i \leq n,1\leq j \leq |{\cal E}|$ and $L$ is the matrix $$\scriptsize
L = \left(\begin{array}{ccccc}
a_{1,1} & a_{1,2} & .& .& a_{1,|{\cal E}|}\\
0 & a_{2,2} & .& .& a_{2,|{\cal E}|}\\
. & 0 & .& .& .\\
. & . & .& .& .\\
. & . & .& .& .\\
0 & 0 & .& 0 & a_{|{\cal E}|,|{\cal E}|}
\end{array}\right).$$
By Lemma \[lemmamatroidequivalence\], the matroid $\cal M$ does not change if some row or some column of ${\cal Y}=(I_{n+{|{\cal E}|}}~~{\cal X})$ is multiplied by a non-zero element of $\mathbb{F}.$ Let ${\cal Y}'$ be the matrix obtained from ${\cal Y}$ by multiplying the rows $\left\{n+1,n+2,...,n+|{\cal E}|\right\}$ by the elements $\left\{a_{1,1}^{-1},a_{2,2}^{-1},...,a_{|{\cal E}|,|{\cal E}|}^{-1}\right\}$ respectively, and then multiplying the columns $\left\{n+1,n+2,...,n+|{\cal E}|\right\}$ by $\left\{a_{1,1},a_{2,2},...,a_{|{\cal E}|,|{\cal E}|}\right\}$ respectively. The matrix ${\cal Y}'$ is then of the form $(I_{n+{|{\cal E}|}}~~{\cal X}'),$ where $
{\cal X}' =
\left(\begin{array}{c}
J \\ L'_{|{\cal E}|\times|{\cal E}|}
\end{array}\right),
$ $L'$ being the upper-triangular matrix obtained from $L,$ i.e., $$\scriptsize
L' = \left(\begin{array}{ccccc}
1 & a_{1,2}a_{1,1}^{-1} & . & .& a_{1,|{\cal E}|}a_{1,1}^{-1}\\
0 & 1 & . & .& a_{2,|{\cal E}|}a_{2,2}^{-1}\\
. & 0 & . & .& .\\
. & . & . & .& .\\
. & . & . & .& .\\
0 & 0 & . & 0& 1
\end{array}\right).$$ As $\cal M$ is the vector matroid of ${\cal Y}'$ also, without loss of generality we assume that ${\cal Y}={\cal Y}',$ with $a_{1,1}=a_{2,2}=...=a_{|{\cal E}|,|{\cal E}|}=1.$
Now let $H$ be the $n\times |{\cal E}|$ matrix whose columns are populated as follows. For all $j=1,2,...,|{\cal E}|,$ $$H^j=J^j - \hspace{-0.2cm}\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}a'_{i,j}J^i= \sum_{m_i \in {\mu}_{tail(e_j)}}\hspace{-0.5cm}c'_{i,j}{\cal Y}^{f(m_i)}_{f(\mu)}.$$ We shall now show that $J^j=HL^j,~\forall~j=1,2,...,|{\cal E}|.$ Clearly for any edge $e_j$ such that $In(e_j)\subset \mu,$ (such edges exist because of acyclicity of $\cal G$), we have $J^j=HL^j,$ as $L^j$ is the basis vector which picks the $j^{th}$ column of $H,$ which is equal to $J^j.$ We now use induction on $j$ (according to the topological order) to show that $J^j=HL^j,~\forall~j=1,2,...,|{\cal E}|$. Now assume that for some $e_j,$ all $e_i\in In(e_j)$ are such that $J^i=HL^i.$ By (\[eqn7\]), we have $$\begin{aligned}
J^j &= \hspace{-0.2cm}\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}a'_{i,j}J^i \hspace{-0.1cm} + \sum_{m_i \in {\mu}_{tail(e_j)}}\hspace{-0.5cm}c'_{i,j}{\cal Y}^{f(m_i)}_{f(\mu)}\\
&= \hspace{-0.15cm}\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}a'_{i,j}HL^i + H^j \\
& = H\left(\hspace{-0.15cm}\sum_{\scriptsize \begin{array}{c} e_i \in {\cal E}:\\ tail(e_j)=head(e_i) \end{array}}\hspace{-1cm}a'_{i,j}L^i+\boldsymbol{1_j}\right)\\
&= HL^j,\end{aligned}$$ where the second equality above follows from the induction assumption and the definition of $H^j,$ $\boldsymbol{1_j}$ is a column vector of length $|{\cal E}|$ with all zeros except for the $1$ at $j^{th}$ position, and the last equality follows from (\[eqn7\]). Thus we have $J^j=HL^j.$ Continuing the induction on $j,$ we have that $J^j=HL^j,~\forall~j=1,2,..,|{\cal E}|.$ Therefore, we have $
{\cal X}=\left(
\begin{array}{c}
HL \\
L
\end{array}
\right).
$ Thus, with ${\cal A}=H,$ and $\boldsymbol{F}=L,$ we have that $\cal X$ is of the form as in (\[formofX\]). This proves the claim. We finally show that there is a scalar linear $\beta$-network-error detecting code for ${\cal G}.$ Let the matrices $A_{s_i},i=1,2,...,|{\cal S}|$ be obtained according to (\[formofA\]) with $H={\cal A},$ and let the network coding matrix $K = I-L^{-1}.$ Then, the columns of the matrix $HL$ denote the global encoding vectors of the edges of $\cal E$ in the given topological order. Clearly this is a valid network code for $\cal G,$ by the structure of the matrices $H$ and $L.$
For some arbitrary error pattern, ${\cal F}\in\mathfrak{F},$ ${\cal M}_{\cal F}$ (as in Condition (C)) is clearly the vector matroid of the matrix $${\cal Z} = {\cal Y}_{f(\mu)\cup (n+I({\cal F}))} = \left(I_{n+\beta}~~ {\cal X}_{f(\mu)\cup (n+I({\cal F}))}\right),$$ where $I({\cal F})=\left\{i_1,i_2,...,i_{\beta}\right\}$ is the index set corresponding to $\cal F,$ and $
{\cal X}_{f(\mu)\cup (n+I({\cal F}))} = \left(
\begin{array}{c}
HL \\
L_{I({\cal F})}
\end{array}
\right).
$ Now for a sink $t\in{\cal T},$ $${\cal Z}^{f(In_{\cal E}(t))}={\cal X}_{f(\mu)\cup (n+I({\cal F}))}^{f(In_{\cal E}(t))}=
\left(
\begin{array}{c}
HL^{f(In_{\cal E}(t))} \\
L_{I({\cal F})}^{f(In_{\cal E}(t))}
\end{array}
\right).$$ By Condition (C), we have $cols({\cal Z}^{f(\boldsymbol{{\cal D}_{t}})}) \subseteq \left\langle({\cal Z}^{f(In_{\cal E}(t))})\right\rangle.$ But we have by the notations of Section \[sec2\], for $supp(\boldsymbol{z})={\cal F}$ $$\begin{aligned}
&{\cal Z}^{f(\boldsymbol{{\cal D}_{t}})} = I_{\boldsymbol{{\cal D}_{t}}}\\
&HL^{f(In_{\cal E}(t))} = \boldsymbol{F_{{\cal S},t}}\\
&L^{f(In_{\cal E}(t))}_{I({\cal F})}=\boldsymbol{F_{supp(\boldsymbol{z}),t}}.\end{aligned}$$ Thus, $cols(I_{\boldsymbol{{\cal D}_{t}}})\subseteq \left\langle\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right)\right\rangle.$ As the choice of sink and error pattern was arbitrary, using Lemma \[lemmadecoding\] it is seen that the network code given by the column vectors of $HL$ is $\beta$-network-error detecting. This completes the proof of the theorem.
Theorem \[matroidalerrornetworkthm\] has the following corollary which is easy to prove.
\[matroidalerrorcorrectingnetworkthm\] Let ${\cal G}({\cal V},{\cal E})$ be an acyclic communication network with sources $\cal S$ and sinks $\cal T.$ The network ${\cal G}$ is a *matroidal* $\alpha$-*error correcting network* associated with a $\mathbb{F}$-representable matroid if and only if it has a scalar linear network-error correcting code over $\mathbb{F}$ that can correct network-errors at any $\alpha$ edges in the network.
Constructions of Multisource Multicast and Multiple-Unicast Error Correcting networks {#sec5}
=====================================================================================
In the theory of matroidal networks developed in [@DFZ; @KiM], we could start with any matroid and obtain a network which is matroidal with respect to that matroid. In particular, if we start with a representable matroid, we always obtain a network which has a scalar linear network code. On the other hand, to obtain matroidal error detecting (correcting) networks, the matroid has to satisfy the conditions of Definition \[matroidalerrornetworkdefinition\], in particular Condition (C) which puts restrictions on the choice of the matroid according to the nature of its contractions. If we are looking for networks with scalar linear network-error correcting codes, such matroids should be representable. Thus, unlike [@DFZ; @KiM], it is not straightforward how to obtain or construct such matroids (representable or otherwise). In this section, we propose algorithms for constructing such matroids (not necessarily representable) along with their corresponding networks (in particular multisource multicast and multiple-unicast), such that these networks are matroidal error correcting networks associated with the constructed matroids. The matroidal $\alpha$-error correcting networks constructed by our algorithms naturally are also matroidal $2\alpha$-error detecting networks. The construction of matroidal $\beta$-error detecting networks (for general $\beta$) can be done in a similar fashion, and therefore we omit it.
Each such matroidal error correcting network is obtained by constructing a series of networks and a corresponding series of matroids associated with which the networks are matroidal error correcting. The series of networks are constructed using two types of nodes defined as follows.
- *N*odes which have a single incoming edge from a coding node and multiple outgoing edges to other coding nodes or sinks are known as *forwarding nodes*. We denote the set of all forwarding nodes as ${\cal V}_{fwd}.$
- Nodes which combine information from several incoming edges from the forwarding nodes and transmit the coded information to their corresponding forwarding nodes are known as *coding nodes*.
If the series of matroids constructed are representable matroids, then the networks constructed are obtained along with scalar linear network-error correcting codes that satisfy the sink demands successfully.
Let $In({\cal V}_{fwd})$ be the set of all incoming edges of all forwarding nodes ${\cal V}_{fwd}.$ In a network with the property that coding and forwarding nodes alternate in any path from a source to a sink in the network, it is sufficient to consider error patterns that are subsets of $In({\cal V}_{fwd})$ to define the error correcting capability of the network, rather than subsets of all the edges in the network. If errors corresponding to such error patterns are correctable, then in such networks other errors are also correctable, as symbols flowing through edges other than $In({\cal V}_{fwd})$ are only copies of symbols flowing through $In({\cal V}_{fwd}).$ The networks that we design using our algorithms are restricted to have these properties, and therefore it is sufficient to construct a matroid $\cal M$ with ${\cal E}=In({\cal V}_{fwd})$ that satisfies the conditions in Definition \[matroidalerrornetworkdefinition\].
The goal of the construction algorithms is to generate a network defined by the following parameters that are to be given as inputs to the algorithms.
- *Number of sources* ($|{\cal S}|$): The number of sources in the multisource multicast network or in the multiple-unicast network.
- *Number of information symbols* ($n=\sum_{s_k\in{\cal S}}n_{s_k}$): For multicast, $n_{s_k}$ is the number of information symbols generated by $s_k,$ while $n$ is the total number of information symbols generated by all sources. For the multiple-unicast case, $n$ represents the number of non-collocated sources present in the network, each generating one information symbol.
- *Number of correctable network-errors* ($\alpha$): This fixes the number of outgoing edges from the source(s). For multicast, the number of outgoing edges from the source $s_k$ is fixed as $N_k = n_{s_k}+2\alpha.$ For multiple-unicast, the number of outgoing edges from each source is fixed as $1+2\alpha.$ These edges and their head nodes are for the sake of clearly presenting our algorithm, and can be absorbed back into the corresponding sources once the algorithm is completed.
- *Number of network-coding nodes* ($N_C$): At each iteration in our algorithm, one network-coding node and one forwarding node will be added to the network, and a corresponding matroid constructed associated with which the extended network will be a matroidal error correcting network. The algorithm will run until $N_C$ forwarding nodes have been added.
- *Number of multicast sinks* ($|{\cal T}|$): This value indicates the number of sinks to which the information symbols is to be multicast. For the multiple-unicast case, we assume that the number of sinks is equal to the number of sources (i.e. messages).
Sketch of Construction and Illustrative Examples {#sketchandillustrexamples}
------------------------------------------------
Fig. \[fig:flowchart\] presents a sketch of our algorithm for constructing acyclic matroidal $\alpha$-error correcting multisource multicast and multiple-unicast networks. The full description of the algorithm for multisource multicast is given in Section \[subsec5b\] and for multiple-unicast in Section \[subsec5c\]. We now present a couple of illustrative examples before we give the full description of our algorithm.
\[multicastex\] Fig. \[fig:multicastconstructionexample0\]-\[fig:multicastconstructionexample4\] describe the stages of a two source multicast network with input parameters $n_{s_1}=2, n_{s_2}=1, \alpha = 1, |{\cal T}|=2,$ and $N_C=4,$ as it evolves through the iterations in the construction shown in the sketch. The network shown in Fig. \[fig:multicastconstructionexample0\] is the initial naive network. A representation of the initial matroid corresponding to this naive network is shown in (\[eqnstage0multicast\]) in Fig. \[fig:multicasteqns\] and is obtained from two MDS codes over $\mathbb{F}_8$, one of length $n_{s_1}+2\alpha=4$ implemented at source $s_1$ and another at source $s_2$ with length $n_{s_2}+2\alpha=3.$ Both codes have minimum distance $3.$ Each successive iteration in the construction adds a new coding node to the network, and a new column and row to the matrix representing the matroid. The equations (\[eqnstage1multicast\])-(\[eqnstage4multicast\]) shown in Fig. \[fig:multicasteqns\] indicate the matrices representative of the representable matroids which correspond to the networks shown in Fig. \[fig:multicastconstructionexample1\]-\[fig:multicastconstructionexample4\], respectively.
Let $e_i$ be the incoming edge at forwarding node $i.$ The function $f$ for each corresponding pair of network and matroid is defined as follows. $$\begin{aligned}
&f(\mu) = \left\{1,2,3\right\}.\\
&f(e_i) = 3+|In({\cal V}_{fwd})|+i,~\forall~e_i \in In({\cal V}_{fwd}).\end{aligned}$$ For reasons mentioned in the beginning of this section, it is sufficient to define $f$ for the input indices $\mu$ and the set of edges $In({\cal V}_{fwd}).$ Each network is seen to be matroidal $1$-error correcting with respect to the corresponding matroid along with the function $f.$
Fig. \[fig:unicastconstructionexample0\]-\[fig:unicastconstructionexample3\] show the stages of the network evolution of a multiple-unicast network with parameters $n=3, \alpha = 1,$ and $N_C=3.$ For $i=1,2,3$, the $k^{th}$ sink demands the information symbol generated by the $k^{th}$ source. The representative matrices of the corresponding matroids are shown in (\[eqnstage0unicast\])-(\[eqnstage3unicast\]) in Fig. \[fig:unicasteqns\]. The initial matroid represented by the matrix in (\[eqnstage0unicast\]) is obtained from a repetition code of length $3$ and minimum distance $3.$ The function $f$ is defined in the same way as in the multicast example. Again, every network is matroidal $1$-error correcting with the corresponding matroid and function $f.$
The example networks shown in this paper which are obtained using our construction algorithms (executed in MATLAB) are matroidal error correcting networks with respect to a representable matroid, i.e., all the example networks have a scalar linear solution. The reason for presenting networks associated only with representable matroids is that obtaining matroidal error correcting networks associated with nonrepresentable matroids seems to be a computationally difficult problem. This is because our algorithms have to repeatedly compute various types of matroid extensions satisfying different kinds of properties. Computations and descriptions of the extensions of nonrepresentable matroids is a computationally intensive task. We further elaborate on the difficulty of obtaining networks associated with representable matroids in Subsection \[mecnnonrepresentable\]. Using stronger mathematical machinery with respect to nonrepresentable matroids and their minors, the complexity of obtaining associated networks could be reduced and our algorithms can then be used to obtain examples of the same. In Subsection \[mecnnonrepresentable\], we present a result which can be considered as a first step towards obtaining matroidal error correcting networks which are associated with nonrepresentable matroids.
\
\
\
\
\
\
\
\
$$\label{eqnstage0multicast}
%\scriptsize
\left(
\begin{array}{cccccccc}
%%%
&1&1&1&1&0&0&0\\
&1&2&4&3&0&0&0\\
&0&0&0&0&1&1&1\\
&&&&&&&\\
I_{10}&1&0&0&0&0&0&0\\
&0&1&0&0&0&0&0\\
&0&0&1&0&0&0&0\\
&0&0&0&1&0&0&0\\
&0&0&0&0&1&0&0\\
&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage1multicast}
%\scriptsize
%%%%%
\left(
\begin{array}{ccccccccc}
%%%
&1&1&1&1&0&0&0&1\\
&1&2&4&3&0&0&0&4\\
&0&0&0&0&1&1&1&6\\
&&&&&&&&\\
&1&0&0&0&0&0&0&0\\
I_{11}&0&1&0&0&0&0&0&0\\
&0&0&1&0&0&0&0&1\\
&0&0&0&1&0&0&0&0\\
&0&0&0&0&1&0&0&6\\
&0&0&0&0&0&1&0&0\\
&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage2multicast}
%\scriptsize
\left(
\begin{array}{cccccccccc}
%%%
&1&1&1&1&0&0&0&1&1\\
&1&2&4&3&0&0&0&4&1\\
&0&0&0&0&1&1&1&6&5\\
&&&&&&&&&\\
&1&0&0&0&0&0&0&0&1\\
&0&1&0&0&0&0&0&0&0\\
I_{12}&0&0&1&0&0&0&0&1&0\\
&0&0&0&1&0&0&0&0&0\\
&0&0&0&0&1&0&0&6&0\\
&0&0&0&0&0&1&0&0&0\\
&0&0&0&0&0&0&1&0&5\\
&0&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage3multicast}
%\scriptsize
\left(
\begin{array}{ccccccccccc}
%%%
&1&1&1&1&0&0&0&1&1&1\\
&1&2&4&3&0&0&0&4&1&2\\
&0&0&0&0&1&1&1&6&5&1\\
&&&&&&&&&&\\
&1&0&0&0&0&0&0&0&1&0\\
&0&1&0&0&0&0&0&0&0&1\\
I_{13}&0&0&1&0&0&0&0&1&0&0\\
&0&0&0&1&0&0&0&0&0&0\\
&0&0&0&0&1&0&0&6&0&1\\
&0&0&0&0&0&1&0&0&0&0\\
&0&0&0&0&0&0&1&0&5&0\\
&0&0&0&0&0&0&0&1&0&0\\
&0&0&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage4multicast}
%\scriptsize
\left(
\begin{array}{cccccccccccc}
%%%
&1&1&1&1&0&0&0&1&1&1&1\\
&1&2&4&3&0&0&0&4&1&2&3\\
&0&0&0&0&1&1&1&6&5&1&1\\
&&&&&&&&&&&\\
&1&0&0&0&0&0&0&0&1&0&0\\
&0&1&0&0&0&0&0&0&0&1&0\\
I_{14}&0&0&1&0&0&0&0&1&0&0&0\\
&0&0&0&1&0&0&0&0&0&0&1\\
&0&0&0&0&1&0&0&6&0&1&1\\
&0&0&0&0&0&1&0&0&0&0&0\\
&0&0&0&0&0&0&1&0&5&0&0\\
&0&0&0&0&0&0&0&1&0&0&0\\
&0&0&0&0&0&0&0&0&1&0&0\\
&0&0&0&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&0&0&0&1\\
%%
\end{array}
\right)$$
\
\
\
\
\
\
\
\
\[fig:multicasteqns\]
------------------------------------------------------------------------
\[fig:unicastnetworkstages\]
$$\label{eqnstage0unicast}
%\footnotesize
\left(
\begin{array}{cccccccccc}
%%%
&1&1&1&0&0&0&0&0&0\\
&0&0&0&1&1&1&0&0&0\\
&0&0&0&0&0&0&1&1&1\\
%%%
&&&&&&&&&\\
&1&0&0&0&0&0&0&0&0\\
&0&1&0&0&0&0&0&0&0\\
&0&0&1&0&0&0&0&0&0\\
I_{12} &0&0&0&1&0&0&0&0&0\\
&0&0&0&0&1&0&0&0&0\\
&0&0&0&0&0&1&0&0&0\\
&0&0&0&0&0&0&1&0&0\\
&0&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage1unicast}
%\footnotesize
\left(
\begin{array}{ccccccccccc}
%%%
&1&1&1&0&0&0&0&0&0&1\\
&0&0&0&1&1&1&0&0&0&4\\
&0&0&0&0&0&0&1&1&1&4\\
%%%
&&&&&&&&&&\\
&1&0&0&0&0&0&0&0&0&0\\
&0&1&0&0&0&0&0&0&0&0\\
&0&0&1&0&0&0&0&0&0&1\\
I_{13} &0&0&0&1&0&0&0&0&0&0\\
&0&0&0&0&1&0&0&0&0&4\\
&0&0&0&0&0&1&0&0&0&0\\
&0&0&0&0&0&0&1&0&0&0\\
&0&0&0&0&0&0&0&1&0&4\\
&0&0&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage2unicast}
%\footnotesize
\left(
\begin{array}{cccccccccccc}
%%%
&1&1&1&0&0&0&0&0&0&1&1\\
&0&0&0&1&1&1&0&0&0&4&4\\
&0&0&0&0&0&0&1&1&1&4&3\\
%%%
&&&&&&&&&&&\\
&1&0&0&0&0&0&0&0&0&0&0\\
&0&1&0&0&0&0&0&0&0&0&1\\
&0&0&1&0&0&0&0&0&0&1&0\\
I_{14} &0&0&0&1&0&0&0&0&0&0&0\\
&0&0&0&0&1&0&0&0&0&4&4\\
&0&0&0&0&0&1&0&0&0&0&0\\
&0&0&0&0&0&0&1&0&0&0&0\\
&0&0&0&0&0&0&0&1&0&4&3\\
&0&0&0&0&0&0&0&0&1&0&0\\
&0&0&0&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage3unicast}
%\footnotesize
\left(
\begin{array}{ccccccccccccc}
%%%
&1&1&1&0&0&0&0&0&0&1&1&2\\
&0&0&0&1&1&1&0&0&0&4&4&7\\
&0&0&0&0&0&0&1&1&1&4&3&1\\
%%%
&&&&&&&&&&&&\\
&1&0&0&0&0&0&0&0&0&0&0&0\\
&0&1&0&0&0&0&0&0&0&0&1&3\\
&0&0&1&0&0&0&0&0&0&1&0&1\\
I_{15} &0&0&0&1&0&0&0&0&0&0&0&0\\
&0&0&0&0&1&0&0&0&0&4&4&7\\
&0&0&0&0&0&1&0&0&0&0&0&0\\
&0&0&0&0&0&0&1&0&0&0&0&0\\
&0&0&0&0&0&0&0&1&0&4&3&5\\
&0&0&0&0&0&0&0&0&1&0&0&4\\
&0&0&0&0&0&0&0&0&0&1&0&0\\
&0&0&0&0&0&0&0&0&0&0&1&3\\
&0&0&0&0&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
------------------------------------------------------------------------
\[fig:unicasteqns\]
Multisource Multicast Construction {#subsec5b}
----------------------------------
We now give the full description of our construction for the case of multisource multicast. The construction generates a multisource multicast network with the given parameters $|{\cal S}|, \left\{n_{s}:s\in {\cal S}\right\}, \alpha,N_C$, and $|{\cal T}|,$ along with a matroid (not necessarily representable) with respect to which the network is matroidal $\alpha$-error correcting. For the sake of the completeness of the description of our construction algorithm, we present a simple lemma.
\[seriesextensionlemma2\] Let $\cal N$ be a series extension of the matroid ${\cal M} = {\cal N}/e_2$ at $e_1,$ i.e., ${\cal N}={\cal M}+_{e_1}^s e_2.$ Let $C$ be a circuit of $\cal M$ containing $e_1,$ then $C\cup e_2$ is a circuit of $\cal N.$
As $C \in {\cal C}({\cal M}),$ $E({\cal M})-C$ is a hyperplane of ${\cal M}^*$ not containing $e_1.$ To prove $C\cup e_2 \in {\cal C}({\cal N}),$ we prove that $E({\cal N})-C\cup e_2 = E({\cal M})-C$ is a hyperplane (obviously not containing $e_1$ or $e_2$) in ${\cal N}^*$ also.
Note that ${\cal N}^*$ is a parallel extension of ${\cal M}^*.$ In a parallel extension ${\cal N}^*$ of ${\cal M}^*,$ the rank of any subset $X\subseteq E({\cal M}^*)$ does not change in the extension. Therefore $r_{{\cal N}^*}(E({\cal M})-C)=r_{{\cal M}^*}(E({\cal M})-C)=r_{{\cal M}^*}-1=r_{{\cal N}^*}-1.$
Now all that we have to prove is that $E({\cal M})-C$ is a flat in ${\cal N}^*$ also. Suppose not, then we must have that $cl_{{\cal N}^*}(E({\cal M})-C) = E({\cal N}^*).$ Thus, as $e_1 \notin (E({\cal M})-C),$ there should be a circuit $C'$ such that $C' \subseteq \left(E({\cal M})-C\right)\cup e_1,$ with $e_1\in C'.$ But then this means $C'\in {\cal C}({\cal M}^*)$ also, which implies that $e_1\in cl_{{\cal M}^*}(E({\cal M})-C) = E({\cal M})-C.$ But this is not the case. Hence $E({\cal M})-C$ is a flat, and hence a hyperplane, in ${\cal N}^*.$ Therefore $C\cup e_2 = (E({\cal N})-(E({\cal M})-C)) \in {\cal C}({\cal N}).$ This proves the lemma.
We now present our construction as an elaboration of the algorithm sketch shown in Fig. \[fig:flowchart\]. The details of the functionality of the algorithm sketch, such as the method of updating the incoming edges to the sinks, the method of updating the matroid, field size issues which govern the possibility of adding new coding nodes and representability of matroidal extensions, etc., can be inferred through the description of our algorithm and the discussion that follows. The construction is based on matroids which need not always be representable. However, at all the appropriate junctures, the equivalent scenario for representable matroids is given as remarks. Throughout the remainder of this section we will assume that a matroid remains unchanged when its elements are reordered according to some permutation, as this implies only a relabeling of the matroid elements. \
\
***Step 1: Initializing the network:*** \
The network ${\cal G}$ is initialized by creating the collection of source nodes ${\cal S}$ and a collection of sink nodes ${\cal T}.$
Corresponding to each source $s_k \in {\cal S},$ create a set of $N_{s_k}=n_{s_k}+2\alpha$ forwarding nodes, each with one incoming edge from $s_k.$ Let the collection of these incoming edges be $e_1,...,e_{N},$ where $N=\sum_{s_k}N_{s_k}$ is the total number of forwarding nodes added.
For each sink $t,$ create $N$ temporary incoming edges $In(t)$ originating from the $N$ forwarding nodes. Because it is sufficient to consider error patterns on the incoming edges at the forwarding nodes, we abuse our notation to say that $In(t)=\left\{e_1,...,e_N\right\}={\cal E},~\forall~t\in{\cal T}.$ This initialized network is represented in Fig. \[fig:multicastinit\].
![The initialization of the multisource multicast network[]{data-label="fig:multicastinit"}](multicastinit.pdf)
\
\
***Step 2: Initializing the matroid***\
We now obtain a matroid $\cal M$ such that the network $\cal G$ is a matroidal $\alpha$-error correcting network with respect to this matroid $\cal M.$ Towards that end, we consider the direct sum $${\cal U}=\boxplus_{k=1}^{|{\cal S}|}{\cal U}_{n_{s_k},N_{s_k}},$$ where ${\cal U}_{n_{s_k},N_{s_k}}$ is the uniform matroid of rank $n_{s_k}$ with the groundset with $N_{s_k}$ elements given as follows. $$E({\cal U}_{n_{s_k},N_{s_k}})=\left\{u_1^{k},u_2^{k},...,u_{N_{s_k}}^{k}\right\}.$$ The matroid ${\cal U}$ has rank $n=\sum_{k=1}^{|{\cal S}|}n_{s_k}.$ Let the ground set of this matroid be $$\label{no18eqn}
E({\cal U})=\{u_1,u_2,...,u_N\}=\uplus_{k=1}^{|{\cal S}|}\{u_1^{k},u_2^{k},...,u_{N_{s_k}}^{k}\},$$ where $$\{u_1,u_2,...,u_n\}=\uplus_{k=1}^{|{\cal S}|}\{u_1^k,u_2^k,...,u_{n_{s_k}}^k\}$$ is a basis for $\cal U.$
If an MDS code of length $N_{s_k}$ and with $n_{s_k}$ information symbols exists, then ${\cal U}_{n_{s_k},N_{s_k}}$ corresponds to the vector matroid of a generator matrix of an $N_{s_k}$-length MDS code which has minimum distance $2\alpha+1$. If such an MDS code exists, let this generator matrix be the $n_{s_k}\times N_{s_k}$ matrix of the form $U_{s_k} = \left(I_{n_{s_k}}~~~A_{s_k}\right).$ If such MDS codes exist for each source, then a representation of the matroid ${\cal U}$ is given as $$\left(\begin{matrix}
U_{s_1} & \boldsymbol{0} & \ldots & \boldsymbol{0}\\
\boldsymbol{0} & U_{s_2} & \ldots & \boldsymbol{0}\\
\vdots & \vdots & \ddots & \vdots\\
\boldsymbol{0} & \boldsymbol{0} &\ldots & U_{s_{|{\cal S}|}}
\end{matrix}
\right).$$ Rearranging the columns of the above representation, we have the alternative representation for ${\cal U}$ which we shall use in the description of our algorithm. $$\label{MDSrep}
U=\left(I_n~~~A\right),$$ where $$A=\left(\begin{matrix}
A_{s_1} & \boldsymbol{0} & \ldots & \boldsymbol{0}\\
\boldsymbol{0} & A_{s_2} & \ldots & \boldsymbol{0}\\
\vdots & \vdots & \ddots & \vdots\\
\boldsymbol{0} & \boldsymbol{0} &\ldots & A_{s_{|{\cal S}|}}
\end{matrix}
\right).$$
Corresponding to the elements $u_i,i=1,2,...,n,$ we add the elements $u_i^p,i=1,2,...,n$ respectively in parallel. By definition of a parallel extension, it can be seen that the order in which these elements are added does not matter. Let the resultant matroid be ${\cal U}_p.$ The set $$\begin{aligned}
E({\cal U}_p)=\{u_1^p, u_2^p,...,u_n^p, u_1,u_2,...,u_N\}\end{aligned}$$ is the ground set of ${\cal U}_p$ such that $\{u_i^p,u_i\}, \forall i=1,2,..,n$ are circuits in ${\cal U}_p.$ By repeatedly using (\[eqn202\]) for the succession of parallel extensions, it can be seen that the set $\{u_1^p, u_2^p,...,u_n^p\}$ forms a basis of ${\cal U}_p$.
If ${\cal U}$ is representable, by Lemma \[parallelrepresentation\] a representation of the matroid ${\cal U}_p$ is then the matrix $U' = \left(I_n~~~I_n~~~A\right).$
Corresponding to the elements $u_i,i=1,2,...,N,$ we now add the elements $u_i^s,i=1,2,...,N$ respectively in series. Again, the order in which these elements are added does not matter. Let ${\cal U}_{p,s}$ be the resultant matroid. We then have $$\begin{aligned}
E({\cal U}_{p,s})&=\{u_1^p, u_2^p,...,u_n^p,u^s_1,u^s_2,...,u^s_N,u_1,u_2,...,u_N\}\\
&=\uplus_{k=1}^{|{\cal S}|}\{u_1^{k},u_2^{k},...,u_{N_{s_k}}^{k}\}\cup\{u_1^p,...,u_n^p,u^s_1,...,u^s_N\}.\end{aligned}$$ By repeatedly using Lemma \[seriesextensionlemma2\], we see that all the circuits of ${\cal U}_{p,s}$ containing $u_i$ will also contain $u_i^s$ for all $i=1,2,..,N.$ In particular, the set of circuits include $\{u_i^p,u_i,u^s_i\}, \forall i=1,2,..,n.$ Moreover, by repeatedly using (\[eqn301\]), we also see that the set $\{u_1^p, u_2^p,...,u_n^p,u^s_1,u^s_2,...,u^s_N\}$ forms a basis for ${\cal U}_{p,s}.$
Let $\cal M$ be the matroid ${\cal U}_{p,s}.$ Consider the initialized network ${\cal G}$ with edges ${\cal E}=\left\{e_1,e_2,...,e_{|{\cal E}|}\right\}$ and with $\cal E$ being the $N$ incoming edges (abusing the notation) at all sinks. For $k=0,1,...,|{\cal S}|-1,$ we define $R_k=\sum_{j=1}^{k}N_{s_j},$ where $R_0=0.$ Let $$f:{\cal E}\cup\mu\rightarrow E({\cal M})$$ be a function such that
- $f(e_{R_k+j})=u_j^{k+1},~j=1,2,...,N_{s_k},~k=0,1,...,|{\cal S}|-1.$
- $f(m_j)=u_j^p, m_j\in\mu,~j=1,2,..,n.$
Let $$B=\left\{b_1,b_2,..,b_{n+{|{\cal E}|}}\right\}=\{u_1^p, u_2^p,...,u_n^p,u^s_1,u^s_2,...,u^s_N\},$$ taken in the following one-one correspondence. $$\begin{aligned}
b_i=&u_i^p,& i=&1,2,..,n \\
b_{n+R_k+j}=&u_i^s~(\text{where}~u_i=u_j^{k+1})& j=&1,2,...,N_{s_k},\\
&&k=&0,1,...,|{\cal S}|-1.\end{aligned}$$ Thus, the basis vector corresponding to the $i^{th}$ input ($1\leq i \leq n$) is $b_i=u_i^p$ and the basis vector corresponding to the error at the edge $e_{R_k+j}$ (for some $k$ and $j$ as above) is $b_{n+R_k+j}=u_i^s$ (for some $i$ such that $u_i=u_j^{k+1}$).
Suppose ${\cal U}$ is representable, by Lemma \[seriesrepresentation\] a representation of the matroid ${\cal U}_{p,s}$ is $$\label{eqn9}
U'' =
\left(
\begin{array}{ccccc}
I_n & \boldsymbol{0} & \boldsymbol{0} & I_n & A \\
\boldsymbol{0} & I_n & \boldsymbol{0} & I_n & \boldsymbol{0}\\
\boldsymbol{0} & \boldsymbol{0} & I_{N-n} & \boldsymbol{0} & I_{N-n}
\end{array}
\right),$$ where $N=|{\cal E}|.$ Thus $U''$ is of the form $(I_{n+{|{\cal E}|}}~~~{\cal X}),$ where ${\cal X}$ is the appropriate $(n+{|{\cal E}|})\times {|{\cal E}|}$ matrix in (\[eqn9\]). It is not difficult to see that with the assignment $f$ to $\mu\cup{\cal E},$ and basis $B,$ the network $\cal G$ is a matroidal $\alpha$-error correcting network in association with the representable matroid $\cal M,$ as $\left(I_n~~~A\right)$ corresponds to a matrix defined as in (\[MDSrep\]), whose columns correspond to the columns of generator matrices of MDS codes implemented at each source.
However, we claim that even when $\cal U$ is not representable, the network $\cal G$ is still a matroidal $\alpha$-error correcting network in association with $\cal M,$ with this assignment $f$ to $\mu\cup{\cal E},$ and with basis $B.$ We now prove this claim by verifying the conditions of Definition \[matroidalerrornetworkdefinition\] as follows. **Condition (A):** Condition (A) is verified as $$f(\mu)=\{u_i^p:~i=1,2,..,n\} \subseteq B$$ and therefore is independent in ${\cal U}_{p,s}.$
**Condition (B1):** Suppose for some $e\in {\cal E},$ Condition (B1) is not satisfied, i.e., $$f(e)=u_j^{k+1}=u_i \in cl_{{\cal U}_{p,s}}(B-f(\mu)).$$ This means that there is a circuit $C_1 \subseteq (B-f(\mu))\cup\{u_i\}$ with $u_i \in C_1.$ Note that in ${{\cal U}_{p,s}},$ the set $C_2=\{u_i^p,u_i,u^s_i\}$ is also a circuit. Thus applying the circuit elimination axiom to the circuits $C_1$ and $C_2$ with $u_i\in C_1\cap C_2,$ we have that there is some circuit $$C_3\subseteq (B-f(\mu))\cup\{u_i^p,u^s_i\} \subseteq B.$$ However, $B$ is an independent set in ${\cal U}_{p,s}.$ Thus $$f(e)=u_i \notin cl_{{\cal U}_{p,s}}(B-f(\mu)), \forall i=1,2,..,N.$$ Hence Condition (B1) is satisfied.
**Condition (B2):** Consider $e_{R_k+j}\in {\cal E}$ such that $f(e_{R_k+j})=u_j^{k+1}=u_i$ (for some $i$). As $\{u_1^p, u_2^p,...,u_n^p\}$ is a basis in ${\cal U}_p,$ we must have some circuit $C_i \subseteq \{u_1^p, u_2^p,...,u_n^p,u_i\},$ with $u_i\in C_i,$ for each $u_i,i=1,2,..,N.$ Therefore, in ${\cal U}_{p,s},$ by Lemma \[seriesextensionlemma2\], $C'_i=C_i\cup\{u_i^s\}$ is a circuit. Thus $$u_i \in cl_{{\cal U}_{p,s}}(\{u_1^p, u_2^p,...,u_n^p\}\cup\{u_i^s\}).$$ In other words, $f(e_{R_k+j})\in cl_{{\cal U}_{p,s}}(f(\mu)\cup\{b_{n+R_k+j}\}).$ As $f(\mu)=f(In(e_{R_k+j})),$ $$\begin{aligned}
\label{eqnno23}
f(e_{R_k+j})\in cl_{{\cal U}_{p,s}}(f(In(e_{R_k+j}))\cup\{b_{n+R_k+j}\}).\end{aligned}$$ Moreover, $$\begin{aligned}
\label{eqnno24}
f(e_{R_k+j})=u_i \notin cl_{{\cal U}_{p,s}}(f(\mu))=cl_{{\cal U}_{p,s}}(f(In(e_{R_k+j}))),\end{aligned}$$ where $u_i \notin cl_{{\cal U}_{p,s}}(f(\mu))$ follows from the fact that any circuit containing $u_i$ in ${\cal U}_{p,s}$ must also contain $u_i^s,$ by Lemma \[seriesextensionlemma2\]. Thus, by (\[eqnno23\]) and (\[eqnno24\]), Condition (B2) is satisfied.
**Condition (C):** Let ${\cal F}=\{e_{R_{k_1}+j_1},...,e_{R_{k_{2\alpha}}+j_{2\alpha}}\}\in\mathfrak{F}$ be an arbitrary error pattern with $$B_{\overline{\cal F}}=B-f(\mu)-\{u_{i_1}^s,...,u_{i_{2\alpha}}^s\},$$ where $\{u_{i_1}^s,...,u_{i_{2\alpha}}^s\}$ corresponds to the basis vectors of the errors at ${\cal F}.$ The contraction ${\cal M}/B_{\overline{\cal F}}$ then has the ground set $$E({\cal M}/B_{\overline{\cal F}})=\{u_1^p,...,u^p_n,u_1,u_2,..,u_N,u_{i_1}^s,...,u_{i_{2\alpha}}^s\}.$$ By repeatedly using (\[eqn303\]), we see that this matroid is precisely the matroid obtained from ${\cal U}_p$ by adding the elements $\{u_{i_1}^s,...,u_{i_{2\alpha}}^s\}$ in series with $\{u_{i_1},...,u_{i_{2\alpha}}\}$ respectively. Now to verify Condition (C), we have to show that $$\label{eqn100}
\{u_1^p,...,u^p_n\} \subset cl_{{\cal M}/B_{\overline{\cal F}}}(\{u_1,u_2,..,u_N\}),$$ as $f(\mu)=\{u_1^p,...,u^p_n\}$ and $ f(In_{\cal E}(t))=\{u_1,u_2,..,u_N\}, \forall t\in{\cal T}.$ To show (\[eqn100\]), we consider the set $$U_{\cal F}=\{u_1,...,u_N\}-\{u_{i_1},...,u_{i_{2\alpha}}\}=\uplus_{k=1}^{|{\cal S}|}U_{\cal F}^k,$$ where $U_{\cal F}^k=\{u_1^{k},u_2^{k},...,u_{N_{s_k}}^{k}\}-\{u_{i_1},...,u_{i_{2\alpha}}\}.$ For each $k,$ the set $U_{\cal F}^k$ contains at least $n_{s_k}$ elements. Thus, $U_{\cal F}^k$ contains a basis of ${\cal U}_{n_{s_k},N_{s_k}}.$ Therefore, $U_{\cal F}$ contains a basis of ${\cal U}.$ This means that $U_{\cal F}$ contains a basis of ${\cal U}_p$ also. This is seen by repeatedly using (\[eqn201\]), given the fact that $U_{\cal F}$ contains a basis of ${\cal U}.$ Moreover as $u_j \notin (U_{\cal F}\cup f(\mu)), \forall j=i_1,...,i_{2\alpha},$ again by repeatedly using (\[eqn302\]), we have $$\begin{aligned}
\label{eqn101}
&r_{{\cal M}/B_{\overline{\cal F}}}(U_{\cal F})=r_{{\cal U}_p}(U_{\cal F}) = n,\\
\label{eqn102}
&r_{{\cal M}/B_{\overline{\cal F}}}(U_{\cal F}\cup f(\mu))=r_{{\cal U}_p}(U_{\cal F}\cup f(\mu))=n,\end{aligned}$$ where the final equalities in both (\[eqn101\]) and (\[eqn102\]) follow from the fact that $U_{\cal F}$ has a basis of ${\cal U}_p.$ Equations (\[eqn101\]) and (\[eqn102\]) together prove (\[eqn100\]), which proves that Condition (C) also holds.
Thus we have verified all the conditions of Definition \[matroidalerrornetworkdefinition\]. Therefore the matroid ${\cal U}_{p,s}$ is a candidate matroid for the initial matroidal error correcting network $\cal G.$ In the forthcoming steps, both the network $\cal G$ and the matroid $\cal M$ are together made to evolve so as to preserve the matroidal nature of $\cal G$ in association with $\cal M.$ \
\
***Step 3: Extending the network*** \
Let ${\cal G}_{temp}={\cal G},$ ${\cal M}_{temp} = {\cal M},$ $B_{temp}=B,$ ${\cal E}_{temp}={\cal E},$ ${\cal X}_{temp}={\cal X},$ and $In_{temp}(t)=In(t),~ \forall~ t\in {\cal T}.$ Let $f_{temp}:{\cal E}_{temp}\cup \mu \rightarrow {\cal E}({\cal M}_{temp})$ be the function defined as $f_{temp}(a) = f(a), \forall a\in\mu \cup {\cal E}_{temp}.$
Choose a random subset ${\cal E}_C\subseteq {\cal E}_{temp}$ of size at least $2.$ Add a new coding node to ${\cal G}_{temp}$ having incoming edges from the forwarding nodes whose incoming edges correspond to those in ${\cal E}_C.$ Add a new forwarding node, which has an incoming edge denoted as $e_{|{\cal E}_{temp}|+1}$ coming from the newly added coding node.
\
\
***Step 4: Extending the matroid*** \
Let $cl$ be the closure operator in ${\cal M}_{temp}.$ Let ${\cal K}$ be a modular cut which contains $cl(f_{temp}({\cal E}_C))$ but does not contain $cl\left(B_{temp}-f_{temp}(\mu)\right).$ If such a modular cut does not exist, the algorithm goes back to ***Step 3*** and proceeds with a different choice for ${\cal E}_C.$ If such a modular cut does not exist for any choice of ${\cal E}_C,$ then the algorithm ends without producing the appropriate output network.
Let $r$ being the rank function in ${\cal M}_{temp}+_{_{\cal K}}x,$ the single-element extension of ${\cal M}_{temp}$ corresponding to the modular cut $\cal K.$ Then, in the matroid ${\cal M}_{temp}+_{_{\cal K}}x,$ the set $f_{temp}({\cal E}_C)\cup x$ contains a circuit with $x$, as $r(cl(f_{temp}({\cal E}_C))\cup x)=r(cl(f_{temp}({\cal E}_C)))$ by definition of a single-element extension.
If ${\cal M}_{temp}+_{_{\cal K}}x$ is a representable extension, it has a representation of the form $$(I_{n+|{\cal E}_{temp}|}~~~{\cal X}'~~~\boldsymbol{x}),$$ over some finite field such that the following hold.
- The submatrix ${\cal X}'$ is such that the matrix $(I_{n+|{\cal E}_{temp}|}~~~{\cal X}')$ is also a representation for ${\cal M}_{temp},$ as $$({\cal M}_{temp}+_{_{\cal K}}x)\backslash x = {\cal M}_{temp}.$$
- The vector $\boldsymbol{x}$ is a column vector of size $n+|{\cal E}_{temp}|$ and can be obtained as a linear combination of the column vectors of ${\cal X}'$ corresponding to $f_{temp}({\cal E}_C).$
- Moreover, the first $n$ components of $\boldsymbol{x}$ are not all zero because $x\notin cl\left(B_{temp}-f_{temp}(\mu)\right),$ as $cl\left(B_{temp}-f_{temp}(\mu)\right) \notin {\cal K}.$
We now add element $y$ in series with element $x$ to get the matroid $\left({\cal M}_{temp}+_{_{\cal K}} x\right) +^s_x y.$ Now the updates to the temporary variables are made as follows.
- ${\cal M}_{temp}=\left({\cal M}_{temp}+_{_{\cal K}} x\right) +^s_x y.$
- $B_{temp}= B_{temp} \cup b_{n+|{\cal E}_{temp}|+1},~\text{where}~ b_{n+|{\cal E}_{temp}|+1} = y.$
- $f_{temp}(e_{|{\cal E}_{temp}|+1})=x \in E({\cal M}_{temp}).$
- Let ${\cal G}_{temp}$ be updated by adding the two new nodes (coding node and forwarding node) to the node set, and with ${\cal E}_{temp}={\cal E}_{temp}\cup e_{|{\cal E}_{temp}|+1}.$ Thus the edge $e_{|{\cal E}_{temp}|+1}$ is now referred to as $e_{|{\cal E}_{temp}|}.$
If ${\cal M}_{temp}+_{_{\cal K}}x$ is representable, then by Lemma \[seriesrepresentation\], so is $\left({\cal M}_{temp}+_{_{\cal K}} x\right)+^s_x y$, with the corresponding representation $$\label{eqn8}
\left(
\begin{array}{cccc}
I_{n+|{\cal E}_{temp}|} & \boldsymbol{0} & {\cal X}' & \boldsymbol{x}\\
\boldsymbol{0} & 1 & \boldsymbol{0} & 1
\end{array}
\right),$$ where the $\boldsymbol{0}$s represent zero row and column vectors of the appropriate sizes. The column corresponding to the new element $y$ is then $\left(
\begin{array}{c}
\boldsymbol{0}\\
1
\end{array}
\right).
$ We also make the following update $${\cal X}_{temp}=
\left(
\begin{array}{cc}
{\cal X}' & \boldsymbol{x}\\
\boldsymbol{0} & 1
\end{array}
\right).$$
\
***Step 5: Updating the incoming edges at the sinks*** \
For each sink $t,$ we update the set $In_{temp}(t)$ at most once as follows.
- For some $e_i \in In_{temp}(t),$ if there is some circuit ${\cal C}_{e_i} \subseteq \left(f_{temp}({\cal E}_C) \cup x \cup y\right)$ such that $\left(x\cup f_{temp}(e_i) \subseteq {\cal C}_{e_i}\right),$ then let $In_{temp}(t)= (In_{temp}(t)-e_i)\cup e_{|{\cal E}_{temp}|}.$
The update is based on the rationale that if the flow on $e_i$ has been encoded into the flow in the newly added edge $e_{|{\cal E}_{temp}|},$ then in any sink which has $e_i$ as an incoming edge, the edge $e_i$ can be replaced by $e_{|{\cal E}_{temp}|}$ in the set of incoming edges. Such an update is only the most natural one possible. It is possible to update the incoming edges at the sinks more interestingly, however requiring more computations (such an optional update is described in ***Step 6*** of this algorithm). An example instance of the extended network (from Fig. \[fig:multicastinit\]), along with the updated incoming edges at the sinks is shown in Fig. \[fig:multicastextended\].
![Example of an extension of the network in Fig. \[fig:multicastinit\] with ${\cal E}_C=\left\{e_{N_{s_1}},e_{R_{|{\cal S}|-1}+1}\right\}.$ The newly added nodes and edges are indicated in blue and in bold. The unremoved incoming edges to the sinks are dimmed as they criss-cross with the newly added nodes and edges.[]{data-label="fig:multicastextended"}](multicastnetcod.pdf)
\
\
***Step 6: Checking the conditions of Definition \[matroidalerrornetworkdefinition\]*** \
The matroid ${\cal M}_{temp}$ along with function $f_{temp}$ and basis $B_{temp}$ satisfies the conditions (A) and (B) of Definition \[matroidalerrornetworkdefinition\] with respect to the network ${\cal G}_{temp}$ for the following reasons.
- Condition (A) is satisfied because $f_{temp}(\mu) = \left\{b_1,b_2,...,b_n\right\}\in B_{temp}.$
- Condition (B1) is satisfied because $f_{temp}(e_{|{\cal E}_{temp}|})=x\notin cl\left(B_{temp}-f_{temp}(\mu)\right),$ as $cl\left(B_{temp}-f_{temp}(\mu)\right)\notin {\cal K}.$
- We know that ${\cal M}_{temp}$ is the series extension of the matroid ${\cal M}_{temp}/y$ at $x.$ Using this fact, and by applying Lemma \[seriesextensionlemma2\] (with ${\cal N}$ being the updated matroid ${\cal M}_{temp}$, and with $e_1=x$ and $e_2=y$), we have that any circuit containing $x$ in ${\cal M}_{temp}$ also contains $y$. Therefore, we have, $$x\in cl(f_{temp}({\cal E}_C)\cup y) \text{~but~} x\notin cl(f_{temp}({\cal E}_C)),$$ where $cl$ is the closure operator in ${\cal M}_{temp}.$ Thus it is seen that Condition (B2) is satisfied as $f_{temp}(e_{|{\cal E}_{temp}|})=x$ and $y=b_{n+|{\cal E}_{temp}|}.$
Condition (C) of Definition \[matroidalerrornetworkdefinition\] is not ensured by ***Step 4*** and therefore has to be checked independently.
Suppose ${\cal M}_{temp}$ is representable before extension, and we also wish to obtain a representable extension. This corresponds to a scalar linear network-error correcting code for ${\cal G}_{temp}$. In other words, the vector $\boldsymbol{x}$ of (\[eqn8\]), which corresponds to a linear combination of the global encoding vectors from existing nodes, has to be designed such that the error correcting capability of the scalar linear network-error correcting code is maintained. Using the techniques of [@YeC1; @YeC2; @Zha; @Mat; @YaY], this can always be done as long as the field size is large enough (discussed in Section \[seccomplexity\]). Once the vector $\boldsymbol{x}$ is found, the matroid is also updated as the vector matroid of the matrix in (\[eqn8\]). Thus, we can find a suitable extension of the initial matroid such that the updated ${\cal M}_{temp}$ is a representable matroid that maintains Condition (C). However, in this case the field size demanded by the algorithms in [@YeC1; @YeC2; @Zha; @Mat; @YaY] is in general quite high, and therefore the scalar linear network-error correcting code obtained operates over such a large field.
In general, ${\cal M}_{temp}$ need not be representable. Therefore we simply check Condition (C) by brute-force. If Condition (C) does not hold, then the algorithm returns to ***Step 4*** to search for an extension of the matroid which satisfies all the conditions of Definition \[matroidalerrornetworkdefinition\].
If Condition (C) of Definition \[matroidalerrornetworkdefinition\] holds for all sinks and for all error patterns on the incoming edges of the forwarding nodes, then all the concerned variables are updated as follows.
- $In(t)=In_{temp}(t), ~\forall~ t\in {\cal T} .$
- *Optional Update:* Optionally, for any sink $t,$ the set $In(t)$ can be updated as the set $I\cup e_{|{\cal E}_{temp}|},$ where $I$ is the smallest subset of $(In_{temp}(t)-e_{|{\cal E}_{temp}|})$ such that upon fixing $In(t)=I\cup e_{|{\cal E}_{temp}|},$ Condition (C) is still satisfied. This involves further brute-force checking of Condition (C) for each such subset of $In_{temp}(t).$ However, it can generate networks where there are no unnecessary incoming edges at any sink. The implementation of this optional update in our MATLAB program is illustrated in Example \[multicastex\] of Subsection \[sketchandillustrexamples\] in the transition between Fig. \[fig:multicastconstructionexample2\] and Fig. \[fig:multicastconstructionexample3\], and also in Example \[unicastncexample\] in Section \[sec6\].
- ${\cal M}={\cal M}_{temp}.$
- ${\cal B}={\cal B}_{temp}.$
- If ${\cal M}_{temp}$ is representable, let ${\cal X}={\cal X}_{temp}.$ (Thus the matroid $\cal M$ is again the vector matroid of the matrix of the form $(I_{n+|{\cal E}|}~~~{\cal X})$.)
- ${\cal G}={\cal G}_{temp}.$
- ${\cal E}={\cal E}_{temp}.$
- $f(a) = f_{temp}(a)~\forall a\in \mu\cup{\cal E}.$
If $N_C$ coding nodes have already been added, then the algorithm ends with the output of all the above variables. Otherwise, the algorithm returns back to ***Step 3***, to find a new extension to the graph and the matroid. Note that as the network $\cal G$ is maintained to be a matroidal $\alpha$-error correcting network over the matroid $\cal M$ at each addition of a coding node, the resultant network after the final extension is also a matroidal $\alpha$-error correcting network in association with the matroid $\cal M.$ If ${\cal M}$ is a representable matroid, then a scalar linear network-error correcting code is obtained according to the proof of Theorem \[matroidalerrornetworkthm\].
Multiple-Unicast Construction {#subsec5c}
-----------------------------
We now present a similar algorithm as that of multicast for the construction of multiple-unicast network instances. As this algorithm follows the same pattern as that of the multicast algorithm, we only point out the differences between the two. \
\
***Step 1: Initializing the multiple-unicast network*** \
The network is initialized by creating $n$ source nodes (each of which generate one message), and $1+2\alpha$ forwarding nodes corresponding to each source node, each with one incoming edge from the corresponding source. Let these edges be $\left\{e_1, e_2, ... , e_{n(1+2\alpha)}\right\}={\cal E}.$ Let ${\cal T}$ be the collection of $n$ sink nodes created.
For the sink $t_i$ which demands the message from source $s_i,$ $1+2\alpha$ imaginary incoming edges are drawn from the forwarding nodes corresponding to that particular source. Again, we abuse our notation and denote by $In(t_i)$ the incoming edges of these forwarding nodes. This initialized network is represented in Fig. \[fig:unicastinit\].
![Initial network of the multiple-unicast algorithm[]{data-label="fig:unicastinit"}](unicastinit.pdf){width="3.5in"}
\
\
***Step 2: Initializing the matroid*** \
As before, we obtain a matroid $\cal M$ such that the network $\cal G$ is a matroidal $\alpha$-error correcting network with respect to this matroid $\cal M.$ Let $A$ be the $n \times n(1+2\alpha)$ matrix $$\left(\begin{array}{cccc}
\boldsymbol{1_{1+2\alpha}} & \boldsymbol{0_{1+2\alpha}} & ... & \boldsymbol{0_{1+2\alpha}}\\
\boldsymbol{0_{1+2\alpha}} & \boldsymbol{1_{1+2\alpha}} & ... & \boldsymbol{0_{1+2\alpha}}\\
. & . & ... & .\\
. & . & ... & . \\
\boldsymbol{0_{1+2\alpha}} & \boldsymbol{0_{1+2\alpha}} & ... & \boldsymbol{1_{1+2\alpha}}
\end{array}
\right),$$ where $\boldsymbol{1_{1+2\alpha}}$ and $\boldsymbol{0_{1+2\alpha}}$ represent the all-ones and all-zeros row vectors of size $1+2\alpha$ over some finite field. Let $\cal M$ be the vector matroid of the following matrix, $$\left(\begin{array}{cccc}
I_n & \boldsymbol{0} & A \\
\boldsymbol{0} & I_{n(1+2\alpha)} & I_{n(1+2\alpha)}
\end{array}
\right),$$ where the $\boldsymbol{0}$s represent zero matrices of appropriate sizes. Note that the above matrix is of the form $(I_{n+|{\cal E}|}~~~{\cal X})$ with $|{\cal E}| = n(1+2\alpha).$
Let $B=\left\{1,2,3,...,n+|{\cal E}|\right\}$ be the basis of $\cal M$ considered. Let $
f:{\cal E}\cup\mu\rightarrow E({\cal M})
$ be the function defined as follows. $$\begin{aligned}
& f(m_i) = i,~~m_i \in \mu, i=1,2,...,n. \\
& f(e_i) = n+|{\cal E}|+i,~\forall~e_i \in {\cal E}.\end{aligned}$$ Then it can be seen that this matroid $\cal M$ with the basis $B$ and function $f$ satisfy the conditions of Definition \[matroidalerrornetworkdefinition\], as each source is simply employing a repetition code of length $1+2\alpha$. \
\
***Step 3**(extending the network)* and ***Step 4**(extending the matroid)* are the same as the multicast construction. Therefore we proceed to ***Step 5***. \
\
***Step 5: Updating the incoming edges at the sinks*** \
In multiple-unicast (or more generally, in the networks with arbitrary demands), there arises the issue of interference from other undesired source symbols with the desired symbols at any sink, thereby necessitating the presence of side information besides the sufficient error correction capability in order to decode correctly. Therefore, unlike the multicast case, simply replacing the encoded edge with the newly formed edge will not suffice to update $In_{temp}(t),$ as the newly formed edge can include additional interference not present in the encoded edge.
The following procedure is therefore adopted to update the incoming edges at each of the sinks.
1. This is the same as in multicast and done at most once for a sink. For some $e_i \in In_{temp}(t),$ if there is some circuit ${\cal C}_{e_i} \subseteq f_{temp}({\cal E}_C) \cup x\cup y$ such that $x\cup f_{temp}(e_i) \subseteq {\cal C}_{e_i},$ then let $In_{temp}(t)= (In_{temp}(t)-e_i)\cup e_{|{\cal E}_{temp}|+1}.$ If no such $e_i$ exists, there is no need to update $In_{temp}(t)$ and this entire step can be skipped.
2. Let $e_i$ be the element that is replaced in $In_{temp}(t).$ Let $e_j \in {\cal E}_{temp}$ such that the following conditions hold.
- $e_j \notin In_{temp}(t)$ but $f_{temp}(e_j)\in ({\cal C}_{e_i}-f_{temp}(e_i)).$ \
- $$\begin{aligned}
r_{{\cal M}_{temp}}\left(\hspace{-0.15cm}\right.&\left.f\left(In_{temp}(t)-e_{|{\cal E}_{temp}|+1}\right)\cup f(e_j)\right) \\
& > r_{{\cal M}_{temp}}\left(f\left(In_{temp}(t)-e_{|{\cal E}_{temp}|+1}\right)\right).\end{aligned}$$
This means that the flow in $e_j$ has been encoded as additional new interference into the flow in the newly added edge $e_{|{\cal E}_{temp}|+1},$ thus creating the necessity of additional side information at the sink $t$ to cancel out this interfering flow. We thus update $In_{temp}(t)$ as $In_{temp}(t)=In_{temp}(t)\cup e_j.$ Thus for each $e_j$ such that the above two conditions hold at sink $t,$ $e_j$ is included in $In_{temp}(t)$ so that sufficient side information is available at the sink to decouple any newly introduced interference and decode the necessary information. This is also to be repeated at each sink.
An example instance of an extension of the network of Fig. \[fig:unicastinit\], along with the updated incoming edges at the sinks is shown in Fig. \[fig:unicastextended\]. As with the multicast algorithm, it is possible to update the sink incoming edges after Condition (C) has been checked. Thus such an update can be optionally included at the end of ***Step 6***.
![Example of an extension of the network in Fig. \[fig:unicastinit\]. The newly added nodes and edges are indicated in blue and in bold.[]{data-label="fig:unicastextended"}](unicastnetcod.pdf)
\
\
***Step 6**(checking the conditions of Definition \[matroidalerrornetworkdefinition\])* is the same as that of the multicast construction, therefore we don’t elaborate further. The optional update to the incoming edges to the sinks can also be done in a similar fashion as in ***Step 6*** of the multicast construction.
As in the multicast construction, at each step the matroidal property of the network is preserved, thus the output of the algorithm is a matroidal $\alpha$-error correcting network which unicasts the set of messages in the presence of at most $\alpha$ network-errors.
On constructing matroidal error correcting networks associated with nonrepresentable matroids {#mecnnonrepresentable}
---------------------------------------------------------------------------------------------
One of the major results of [@DFZ] was that nonrepresentable matroids can be used to construct matroidal networks for which Shannon-type information inequalities (the most widely used collection of information inequalities in information theory) cannot bound their capacity as tightly as the non-Shannon-type information inequalities do. In other words, networks connected with nonrepresentable matroids can prove to be very useful in obtaining insights on the general theory of network coding. It can therefore be expected that matroidal error correcting and detecting networks associated with nonrepresentable matroids will be useful in obtaining similar insights for network-error detection and correction. It was already mentioned in the beginning of Section \[sec5\] that it is not straightforward to obtain representable or nonrepresentable matroids from which we can construct matroidal network-error correcting or detecting networks directly. The difficulty is that, unlike [@DFZ], Definitions \[matroidalerrornetworkdefinition\] and \[matroidalerrorcorrectionnetworkdefinition\] for matroidal error detecting and correcting networks require matroids whose contractions have to satisfy specific properties which enable the decoding of the demanded symbols at sinks. Since this is a fundamental requirement of error detecting and correcting networks, it is clear that such a requirement cannot avoided. This motivated the method used in our algorithms to construct such networks, i.e., starting with simple matroids and their counterpart networks and then extending them together while keeping the conditions of error correction intact. The chief reasons for the inability of using our algorithms to obtain example networks which are associated with nonrepresentable matroids are as follows.
- Descriptions of nonrepresentable matroids with many elements in its groundset is not an easy task, even on a computing device. More importantly, computing the extensions (in particular single-element extensions, which involve computation of the flats and the modular cuts) of such nonrepresentable matroids with many elements is computationally intensive. Furthermore, there are a large number of possible single-element extensions for any matroid with many elements in its groundset. Checking the representability or nonrepresentability of such extensions is not easy.
- Evaluating the error correcting property of a given linear network-error correcting code involves going through all possible error patterns and checking if the error correction holds for each of them. To the best of the authors’ knowledge, such a brute-force technique is used in all available coherent linear network-error correction literature (see [@YeC1; @Zha; @Mat; @YaY], for example) to construct linear network-error correction codes. Thus checking Condition (C) of Definitions \[matroidalerrornetworkdefinition\] and \[matroidalerrorcorrectionnetworkdefinition\] demands brute-force analysis of all the contractions corresponding to all possible error patterns. Compared to representable matroids, computing the contractions of nonrepresentable matroids is computationally intensive.
- In [@DFZ], Shannon and non-Shannon information inequalities were used to capture the uniqueness of the *Vamos network* obtained from the nonrepresentable Vamos matroid (see [@DFZ] for more details). In our case, even if we suppose that a matroidal error detecting (or correcting) network associated on a nonrepresentable matroid is obtained through our algorithm, such an analysis seems rather complicated, again the issue being the number of possible error patterns. Verifying that the best possible linear error correction schemes have rates of information transmission less than the best possible nonlinear schemes once again implies going through each of the error-patterns and evaluating the maximum possible rates of transmission. The number of these calculations grows linearly with the number of possible error-patterns and can quickly become unwieldy.
Though we do not present examples of networks obtained using our algorithms which are associated with nonrepresentable matroids from our algorithms because of the above reasons, we present a proposition in this subsection as a first step towards reducing the search-space of matroidal extensions in order to obtain nonrepresentable matroids which satisfy the properties in Definition \[matroidalerrornetworkdefinition\]. Also, in Section \[secinsufficiency\], using ideas from [@DFZ3], we present an example network which is a matroidal 1-error detecting network associated with a nonrepresentable matroid, using which we show that linear network-error detection and correction schemes are not always sufficient to satisfy network demands in the presence of network-errors.
Proposition \[propprincipal\] below shows that if we are to use the constructions of Section \[sec5\] to obtain matroidal error correcting or detecting networks associated with nonrepresentable matroids, then the extension of the matroid considered in ***Step 4*** of the multicast and the multiple-unicast constructions must necessarily be a non-principal extension, i.e., the modular cut corresponding to the extension must not be a principal modular cut. The proof of the following proposition is given for the sake of completeness as to the best of the authors’ knowledge it seems to be unavailable in matroid theory literature.
\[propprincipal\] Let $A$ be a matrix of size $k\times m$ ($k\leq m$) with elements from some field $\mathbb{F}_q,$ and let ${\cal M}={\cal M}[A].$ Let ${\cal K}_F$ be the principal modular cut of $\cal M$ generated by flat $F$ of ${\cal M}.$ Then the principal extension ${\cal M}+_{_{{\cal K}_F}} e$ of the matroid $\cal M$ is representable over an extension of $\mathbb{F}_q.$
Let $X = A^{F},$ the submatrix of $A$ with respect the column indices given by $F.$ Let $\langle X\rangle_q$ denote the space spanned by the columns of $X$ over $\mathbb{F}_q.$ Let $X_{(0)}, X_{(1)},..,X_{{(M-1)}}$ be the submatrices corresponding to all the flats $F_0, F_1,..., F_{M-1}$ of $\cal M$ which do not contain $F.$ Thus for each $i=0,1,2,...,M-1,$ there exists at least one non-zero vector $v_i \in \mathbb{F}_q^k$ such that $v_i \in \langle X\rangle_q$ but $v_i \notin \langle X_{(i)}\rangle_q.$
Consider the extension field $\mathbb{F}_{Q}, Q=q^M.$ Let $\beta$ be the primitive element of $\mathbb{F}_Q,$ with respect to $\mathbb{F}_q$ as the base field. Thus any element of $\mathbb{F}_Q$ can be uniquely expressed as a polynomial of degree at most $M-1$ in $\beta.$
Let $$v=\sum_{i=0}^{M-1}v_i\beta^{i} \in \mathbb{F}_Q^k.$$ Let $\tilde{A} = \left(A~|~v\right)$ be the matrix over $\mathbb{F}_Q$ where the elements of the submatrix $A$ are viewed as elements from the basefield $\mathbb{F}_{q}$ embedded in $\mathbb{F}_Q.$ We claim that $\tilde{A}$ is the required representation for the matroid extension ${\cal M}+_{_{{\cal K}_F}} e.$ Let $\langle X\rangle_Q$ denote the vector space spanned by the columns of $X$ over $\mathbb{F}_Q.$ According to Definition \[singleelement\], to show that $\tilde{A}$ is the required representation, it is enough to show that $v \in \langle X\rangle_Q$ but $v \notin \langle X_{(i)}\rangle_Q, i=0,1,2,..,M-1.$ For each $i=0,1,...,M-1,$ as $v_i \in \langle X\rangle_q$ it is clear that $v_i \in \langle X\rangle_Q,$ also. Thus $v \in \langle X\rangle_Q.$ Now, for some $r$ such that $0\leq r \leq M-1,$ consider a $\mathbb{F}_Q$ linear combination of the column vectors in $X_{(r)}$ as follows. $$\begin{aligned}
\nonumber
\sum_{j}g_jX_{(r)}^j & = \sum_{j}\left(\sum_{j'=0}^{M-1}g_{j,j'}\beta^{j'}\right)X_{(r)}^j \\
& = \sum_{j'=0}^{M-1}\left(\sum_jg_{j,j'}X_{(r)}^j\right)\beta^{j'},\end{aligned}$$ where $g_j=\sum_{j'=0}^{M-1}g_{j,j'}\beta^{j'} \in \mathbb{F}_Q$ with $g_{j,j'} \in \mathbb{F}_q,~\forall j'.$ As $v_r \notin \langle X_{(r)}\rangle_q,$ we must have that for any $j'=0,1,2,...,M-1,$ $$\sum_jg_{j,j'}X_{(r)}^j \neq v_r.$$ For the same reason, we must have $$\sum_{j}g_jX_{(r)}^j= \sum_{j'=0}^{M-1}\left(\sum_jg_{j,j'}X_{(r)}^j\right)\beta^{j'} \neq \sum_{i=0}^{M-1}v_i\beta^{i} = v,$$ for any $r = 0,1,2,...,M-1$ and for any linear coefficients $g_j \in \mathbb{F}_Q~\forall j.$
Thus $v \notin \langle X_{(i)}\rangle_Q,~\forall i=0,1,2,...,M-1.$ Thus $\tilde{A}$ satisfies the conditions to be a representation for ${\cal M}+_{_{{\cal K}_F}} e.$ This proves the proposition.
Complexity {#seccomplexity}
==========
We now calculate upper bounds on the complexity of the algorithms for the case of scalar linear network-error correcting codes (i.e., representable matroids). These calculations are for the implementation of our algorithms without the execution of the optional update to the incoming edges to the sinks in ***Step 6***. Including this optional update step will certainly increase the complexity of the algorithms. However, the calculations that follow capture the essential running time of our algorithms in the representable case. In the case of nonrepresentable matroids, the complexity of our algorithms will depend heavily on the matroidal operations involved to obtain the extensions, computing the contractions and checking the ranks of subsets in the computed contractions in order to verify the error correcting properties of the matroidal network so formed. As such matroidal operations are involved, it is not clear how to proceed in this direction. Hence we take up on computing the complexity of our algorithms in generating networks associated only with representable matroids. In any case, constructing network associated with nonrepresentable matroids using our algorithm can be expected to be at least as difficult as the representable case, since in the representable case all the matroids have matrix representations and all matroid operations are implementable as operations based on linear algebra.
For obtaining the complexity of our multisource multicast algorithm, we shall directly use the complexity of the construction algorithm for single source multicast scalar linear network-error correcting codes given in [@Mat]. Further, we shall also show that our multiple-unicast algorithm (in the case of representable matroids) is equivalent to a variant of the algorithm in [@Mat] and therefore the complexity of the algorithm of [@Mat] can be used to obtain that of our multiple-unicast algorithm also.
Network-Error Correcting Codes - Algorithm of [@Mat]
----------------------------------------------------
Algorithm \[alg:necc\] is a brief version of the algorithm given in [@Mat] for constructing an scalar linear $\alpha$-network-error correcting code for a given single source, acyclic network that meets the network Singleton bound given in [@YeC1]. The construction of [@Mat] is based on the network code construction algorithm of [@JSCEEJT]. The algorithm constructs a network code such that all network-errors in upto $2\alpha$ edges will be corrected as long as the sinks know where the errors have occurred. Such a network code is then shown [@Mat] to be equivalent to an $\alpha$-network-error correcting code. Other equivalent (in terms of complexity) network-error correction algorithms can be found in [@Zha] [@YaY].
------------------------------------------------------------------------
$(1)$ Let $\cal F$ be the set of all subsets of $\cal E$ of size $2\alpha.$ Add an imaginary source $s'$ and draw $n=N-2\alpha$ edges from $s'$ to $s.$
$(2)$
It is shown in [@Mat] that Algorithm \[alg:necc\] results in a network code which is a $\alpha$-network-error correcting code meeting the network Singleton bound, as long as the field size $$\label{eqn15}
q > |{\cal T}||{\cal F}|= |{\cal T}|
\left(
\begin{array}{c}
|{\cal E}| \\
2\alpha
\end{array}
\right).$$ The complexity of the algorithm is then $O\left(|{\cal F}||{\cal T}|N\left(|{\cal E}||{\cal F}||{\cal T}|+|{\cal E}|+N+2\alpha\right)\right).$
Multicast
---------
We use the complexity of Algorithm \[alg:necc\] to calculate the complexity of our multisource multicast algorithm. This requires converting the multisource multicast network to the single source multicast network, as Algorithm \[alg:necc\] works only on a single source multicast network. This can be done after ***Step 1*** of the algorithm, where we can add a super-source to the network from which edges flow into the actual set of sources ${\cal S}.$ After ***Step 1***, the network is clearly matroidal $\alpha$-error correcting with respect to the direct sum of the uniform matroids. And thus the network after ***Step 1*** has a multicast scalar linear $\alpha$-network-error correcting code if the direct sum is representable. Constructing the $N_C$ nodes and their global encoding vectors while preserving the error correcting property, i.e. generating the network and appropriate matroid extensions, can be done using Algorithm \[alg:necc\], once all the variables have been initialized and the super source has been added.
We consider errors only at the incoming edges of the forwarding nodes, and there are at most $|{\cal E}|=N+N_C$ such edges at any iteration of our algorithm. Let $\eta = \left(\begin{array}{c} |{\cal E}|\\ 2\alpha \end{array} \right).$ If the field size of operation assumed is greater than $|{\cal T}|\eta$, then by Algorithm \[alg:necc\], a suitable extension to the representable matroid (i.e., a suitable global encoding vector to the edge of the newly added incoming node) exists at each iteration of our algorithm, and the total complexity of obtaining the network and the representable matroid (equivalently, the linear network-error correcting code) will be $O\left(\eta|{\cal T}|N\left(|{\cal E}|\eta|{\cal T}|+|{\cal E}|+N+2\alpha\right)\right),$ assuming that the other steps in the algorithm can be done in constant time or with negligible complexity compared to **Step 4** and **Step 6**. With a smaller field size, the complexity of obtaining the network and the matroid will continue to be bounded similarly, provided the suitable vectors exist at all iterations. At the end of using Algorithm \[alg:necc\] to obtain the coding nodes and the linear network-error correction code, the super-source and the outgoing edges from the super-source can be removed to give our required network.
Multiple Unicast
----------------
Unlike multicast, there exist no known algorithms to construct network-error correcting codes for multiple unicast networks which we can use to compute the complexity according to the requirements of our algorithm. Therefore, we take an indirect approach. At each iteration in our multiple unicast algorithm (omitting the optional update in ***Step 6***), we show that the construction of a suitable global encoding vector (for the current edge under processing) for satisfying the multiple-unicast conditions is equivalent to the construction of a suitable global encoding vector such that certain matrices are full-rank as in Step $2(iii)$ of Algorithm \[alg:necc\] for each error pattern in $\cal F$. Thus, the complexity of our multiple-unicast algorithm can be obtained from the complexity of Algorithm \[alg:necc\] after suitable changes.
Let ${\cal G}(i)$ be the state of the multiple unicast network at the iteration $i$ ($i=0$ representing the initial state and $i=N_C$ representing the final iteration) of our multiple-unicast algorithm. That is, in the network ${\cal G}(i),$ $i-1$ coding nodes have already been added and the global encoding vectors corresponding to their incoming edges have been fixed. Also, a particular subset of the forwarding nodes have been picked and the $i^{th}$ coding and the corresponding forwarding node have been added according to **Step 3** of the algorithm. We also update the incoming edges at the sinks according to **Step 5** even before fixing the global encoding vector of the newly added edge by simply adding edges containing all possible interfering flows as the new side information for the sinks. So the steps that remain to be executed are **Step 4** and **Step 6**, i.e., picking a suitable global encoding vector for the newly added edge $e_{n(2\alpha+1)+i}$ (from the newly added coding node) so that the error correction capability and decoding continue to hold at the sinks. After achieving this goal, those edges which carry side information that are not used for the decoding process at the sinks can be removed.
Let $n_t(i)$ be the number of incoming edges at sink $t$ and $\boldsymbol{F_{{\cal S},t}(i)}$ be the transfer matrix of size $n\times n_t(i)$ from the sources to sink $t$ at the end of iteration $i$ of our multiple-unicast algorithm (i.e., after fixing a suitable global encoding vector for $e_{n(2\alpha+1)+i}$). Towards obtaining a bound on the complexity of our algorithm, we first prove the following lemma.
\[lemma4\] For each sink $t$ in ${\cal G}(i),$ there exists some full rank square matrix $A_t(i)$ of size $n_t(i)$ such that $$\boldsymbol{F_{{\cal S},t}(i)}A_t(i)=\left(I^j~I^j~..~I^j~|~C(i)\right),$$ where $I^j$ is the $j^{th}$ basis vector corresponding to the input $x_j$ demanded by sink $t$ and is repeated $2\alpha+1$ times in the above matrix.
The claim holds for ${\cal G}(0)$ with $C(0)$ being an empty matrix. We assume that the claim holds for ${\cal G}(i)$ and will prove that it holds for ${\cal G}(i+1)$ as well. Because of the network code and the way the incoming edges at the sinks are updated, we have for some nonsingular square matrix $L$ of size $n_t(i+1),$ $$\boldsymbol{F_{{\cal S},t}(i+1)}=\left(\boldsymbol{F_{{\cal S},t}(i)}~|~V\right)L,$$ where $V$ is a matrix with $n$ rows, consisting of the global encoding vectors of the newly added incoming edges (at iteration $i+1$) with interfering flows. Because the claim holds for ${\cal G}(i),$ we must have $$\begin{aligned}
\boldsymbol{F_{{\cal S},t}}&\boldsymbol{(i+1)}\\
&=\left(\left(I^j~I^j~..~I^j~|~C(i)\right)A_t(i)^{-1}~|~V\right)L\\
&=\left(I^j~I^j~..~I^j~|~C(i)~|~V\right)\left(\begin{array}{cc}A_t(i)^{-1} & \boldsymbol{0}\\ \boldsymbol{0} & I_{V}\end{array}\right)L,\end{aligned}$$ where the $\boldsymbol{0}$s represent zero matrices of appropriate sizes, and $I_{V}$ is the identity matrix such that $V=VI_{V}.$
The matrix $$B=\left(\begin{array}{cc}A_t(i)^{-1} & \boldsymbol{0} \\ \boldsymbol{0} & I_{V}\end{array}\right)L$$ is invertible. Let $C(i+1)=\left(C(i)~|~V\right).$ Let $A_t(i+1)=B^{-1}.$ $$\boldsymbol{F_{{\cal S},t}(i+1)}A_t(i+1)=\left(I^j~I^j~..~I^j~|~C(i+1)\right).$$ By induction on $i$ ($i=1,2,...,N_C$) the lemma is proved.
Let $\boldsymbol{F_{t}(i)}$ denote the matrix $\boldsymbol{F_{t}}$ at the end of the $i^{th}$ iteration. Let $\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}$ denote the submatrix of $\boldsymbol{F_{t}(i)}$ consisting of those rows of $\boldsymbol{F_{t}}$ which are indexed by $supp(\boldsymbol{z}),$ for some error vector $\boldsymbol{z}.$
The following lemma is now a direct consequence of Lemma \[lemma4\] and Lemma \[lemmadecoding\] and will help us to connect our multiple-unicast algorithm to Algorithm \[alg:necc\].
\[multipleunicastequivalence\] Let $\bar{A}_t(i)$ be the matrix consisting of the first $2\alpha+1$ columns of $A_t(i).$ The sink $t$ can successfully decode its demanded $j^{th}$ information symbol ($\boldsymbol{{\cal D}_{t}}=j$) in ${\cal G}(i)$ if the square matrix $$\left(
\begin{array}{c}1~~1~~.~.~.~~1\\
\hline\vspace{-0.3cm}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}\bar{A}_t(i)
\end{array}\right)$$ is full-rank for each error vector $\boldsymbol{z}$ such that $supp(\boldsymbol{z}) \in {\cal F},$ the set of all possible error patterns.
If the given matrix is full-rank for all possible errors, then we must have for any such error vector $\boldsymbol{z}$ $$cols(I_{\boldsymbol{{\cal D}_{t}}}) \subseteq \left\langle\left(
\begin{array}{c}I^j~~I^j~~.~.~.~~I^j\\
\hline\vspace{-0.3cm}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}\bar{A}_t(i)
\end{array}\right)\right\rangle,$$ as $I_{\boldsymbol{{\cal D}_{t}}}=
\left(
\begin{array}{c}I^j\\
\hline\vspace{-0.3cm}\\
\boldsymbol{0}
\end{array}\right)$ and as $\left(
\begin{array}{c}I^j~~I^j~~.~.~.~~I^j\\
\hline\vspace{-0.3cm}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}\bar{A}_t(i)
\end{array}\right)$ has exactly $2\alpha+1$ non-zero rows. But then, this means $$\begin{aligned}
\nonumber
cols(I_{\boldsymbol{{\cal D}_{t}}}) & \subseteq \left\langle\left(
\begin{array}{c}I^j~~I^j~~.~.~.~~I^j~|~C(i)\\
\hline\vspace{-0.3cm}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}A_t(i)
\end{array}\right)\right\rangle\\
\label{eqn103}
& \subseteq \left\langle\left(\left(
\begin{array}{c}\boldsymbol{F_{{\cal S},t}}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}
\end{array}\right)A_t(i)\right)\right\rangle\\
\label{eqn104}
& \subseteq \left\langle\left(
\begin{array}{c}\boldsymbol{F_{{\cal S},t}}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}
\end{array}\right)\right\rangle,\end{aligned}$$ where (\[eqn103\]) is because of Lemma \[lemma4\] and (\[eqn104\]) is because $A_t(i)$ is full-rank. By Lemma \[lemmadecoding\], this means that the demand $\boldsymbol{{\cal D}_{t}}=j$ can be successfully decoded by the sink $t.$
Lemma \[multipleunicastequivalence\] connects the problem of designing a multiple-unicast network-error correcting code for ${\cal G}(i)$ with maintaining the full-rankness of a set of matrices as in Algorithm \[alg:necc\]. Thus, Algorithm \[alg:necc\] can be used to design a multiple-unicast network-error correcting code for ${\cal G}(i)$ by modifying Step $2(iii)$ to fix the local encoding kernels at the new coding node such that the following condition is satisfied.
- The matrix $\left(
\begin{array}{c}1~~1~~.~.~.~~1\\
\hline\vspace{-0.3cm}\\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}(i)}\bar{A}_t(i)
\end{array}\right)$ is full-rank for each sink $t$ and for each error pattern $supp(\boldsymbol{z}) \in {\cal F},$ at each iteration $i=1,2,...,N_C.$
As in the multicast case, we have that the maximum number of edges at any particular iteration is less than $|{\cal E}|=N+N_C.$ With $\eta=\left(\begin{array}{c}|{\cal E}| \\ 2\alpha \end{array}\right),$ we invoke the result from [@Mat] to note that a suitable choice of the local encoding kernels is possible if $q \geq |{\cal T}|\eta=n\eta.$ The complexity of our multiple-unicast algorithm is $O\left(nN\eta\left(|{\cal E}|n\eta+|{\cal E}|+N+2\alpha\right)\right),$ again assuming that the other steps in the algorithm can be done in constant time or with negligible complexity compared to **Step 4** and **Step 6**.
Insufficiency of Linear Network-Error Detecting and Correcting Codes {#secinsufficiency}
====================================================================
In [@DFZ3], it was shown that there exist networks for which linear network codes (linearity in a very general sense) are insufficient to achieve the maximum rate of information transmission to the sinks, when compared to general network coding (including nonlinear schemes). In other words, the *network coding capacity* of a network could be strictly greater than the *linear network coding capacity* of the network. A network for which linear network coding cannot achieve network coding capacity was explicitly constructed in [@DFZ3]. The network in [@DFZ3] was constructed by ‘conjoining’ two subnetworks, of which one is linearly solvable over fields of characteristic two, and the other is linearly solvable over fields of odd characteristic. The two subnetworks were constructed based on results from matroid theory, in particular the Fano and the non-Fano matroids [@Oxl]. The matrix $A$ shown below considered over any field of characteristic two (for example, $\mathbb{F}_2$) is a representation for the Fano matroid. $$\label{fanononfanorepresentation}
A=\left(
\begin{array}{ccccccc}
1 & 0 & 0 & 1 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 & 1 & 1 & 1
\end{array}
\right).$$ The matrix $A$ is also a representation for the non-Fano matroid except that it is over a field with characteristic not equal to two (for example, $\mathbb{F}_3$). Combining the two subnetworks, the conjoined network is shown to be linearly unsolvable. We refer the reader to [@DFZ3] for more details.
Because of the fact that network coding is a special case of network-error correction (or equivalently network-error detection), it is to be expected that linear network-error correcting (detecting) codes must be insufficient for solving network-error correction (detection) problems on general networks. In Subsection \[insuffnetwork\], we present an explicit example network for which linear network-error detection (for the case of single edge errors) is not sufficient, using simple extensions of the networks shown in [@DFZ3]. The reason for choosing such simple extensions is two fold. Firstly, the networks chosen are sufficient to prove the insufficiency claim. The second reason, as the verification of the linear nonsolvability of the chosen networks will make it clear, is that rigorously proving that linear network-error correcting codes are not sufficient for a particular network can require many times the computations necessary for showing linear network coding is insufficient. Choosing extensions of the networks shown in [@DFZ3] to demonstrate the insufficiency of linear network coding makes our job easier. For these two reasons, we work with the chosen networks which are simple extensions of those from [@DFZ3]. Nevertheless, it is certainly possible to construct more complicated networks for which linear network-error correction and detection are insufficient.
In the following subsections, we construct the network for which linear network-error detection is insufficient, while a nonlinear scheme is shown to provide the required error detection. We combine simple extensions of the networks shown in [@DFZ3] to create the network that we are looking for.
A network solvable only on alphabets of characteristic two
----------------------------------------------------------
Consider the network $\tilde{\cal N}_1$ shown in Fig. \[fig:subnet1\].
![The network-error detection network $\tilde{\cal N}_1$ which is solvable only over fields of characteristic two. It is a matroidal $1$-error detecting network associated with the matroid ${\cal M}_{\tilde{\cal N}_1}$ whose representation is shown in (\[representationn1\]).[]{data-label="fig:subnet1"}](nonlinearsubnetwork11.pdf){width="3.4in"}
The nodes $v_4,$ $v_5$ and $v_6$ generate the messages $a,$ $b$ and $c$ (over some finite field) respectively. The sinks $v_{37},$ $v_{38},$ and $v_{39}$ demand the symbols $c,$ $b,$ and $a$ respectively. Some of the edges in the network are marked by the values $M_i$ which are coefficients of some arbitrary scalar linear network code for the network. Any edge which is not marked by a coefficient is assumed to have the identity element as its coefficient, meaning it just forwards the information from its tail node to the head node. It can be easily seen that these $M_i$s are sufficient to characterise any scalar linear network code for $\tilde{\cal N}_1.$ Each of the sinks have a direct edge from the corresponding node generating their demands, indicated by a duplicate node along with the edges $e_1,$ $e_2$ and $e_3$. The network ${\cal N}_1$ from [@DFZ3] is simply the network obtained from $\tilde{\cal N}_1$ by the deletion of the edges $e_1, e_2,$ and $e_3.$ Thus $\tilde{\cal N}_1$ is a simple extension of the network ${\cal N}_1$ from [@DFZ3]. We now prove the following lemma.
\[lemmanetworkn1\] A single edge network-error detection code over a finite field exists for $\tilde{\cal N}_1$ if and only if the finite field used has characteristic two.
*Only if part:*
Let the network coding coefficients $M_i$s define a single edge network-error detecting code over some field $\mathbb{F}.$ Note that there are exactly two paths from any source to the corresponding sink, one through the network coded portion of the network and the other through the direct edges $e_1, e_2,$ and $e_3.$ Therefore it is clear that for detecting single-edge errors, we require $M_{15}, M_{16}, M_{17}$ to be nonzero. Thus, we see that the sinks $v_{37}, v_{38}$ and $v_{39}$ can decode the required symbols by observing the symbols on the direct edges $e_3, e_2$ and $e_1$ from $v_6, v_5$ and $v_4$ respectively, as long as these edges are not in error.
In order to show that the characteristic of the field used should be two for the network code defined using $M_i$s to be a single edge network-error detecting code, we consider the single edge errors at the edges $e_1, e_2$ and $e_3.$
Consider that the only error in the network occurs in edge $e_3.$ Then the matrix $\left(
\begin{array}{c}
\boldsymbol{F_{{\cal S},t}} \\
\boldsymbol{F_{supp(\boldsymbol{z}),{t}}}
\end{array}
\right)
$ corresponding to $supp(\boldsymbol{z})=e_3$ at the sink $t=v_{37}$ is $$\boldsymbol{F_{v_{37}}^{e_3}} =
\left(
\begin{array}{ccc}
M_9 & M_1M_5M_{10} & 0 \\
0 & (M_2M_5+M_3M_6)M_{10} & 0 \\
0 & M_4M_6M_{10} & M_{15} \\
\hline
0 & 0 & 1
\end{array}
\right),$$ where the ordering of the columns adopted in the above matrix corresponds to the incoming edges at the sink given as follows. $$In(v_{37})=\{v_4\rightarrow v_{37},v_{29}\rightarrow v_{37},e_3\}.$$ By Lemma \[lemmadecoding\], for some $x_1,x_2,$ and $x_3$ belonging to the finite field, we must have $$\boldsymbol{F_{v_{37}}^{e_3}}(x_1~x_2~x_3)^T =
\left(
\begin{array}{c}
0 \\
0 \\
1 \\
0
\end{array}
\right).$$ Thus we must have $$\begin{aligned}
M_9x_1 + M_1M_5M_{10}x_2 &= 0. \\
M_2M_5M_{10}x_2+M_3M_6M_{10}x_2 & = 0. \\
M_4M_6M_{10}x_2+M_{15}x_3 &=1. \\
x_3 & = 0.\end{aligned}$$ Let $M_9x_1=M_9'$, and $M_{10}x_2=M_{10}'.$ Then we have $$\begin{aligned}
\label{eqnno1}
M_9' + M_1M_5M_{10}' &= 0. \\
\label{eqnno2}
M_2M_5M_{10}'+M_3M_6M_{10}' & = 0. \\
\label{eqnno3}
M_4M_6M_{10}'&=1. \end{aligned}$$
The transfer matrix corresponding to error at $e_2$ at the sink $t=v_{38}$ ($In(t)=\{v_{29}\rightarrow v_{38},v_{30}\rightarrow v_{38},e_2\}$) is $$\boldsymbol{F_{v_{38}}^{e_2}} =
\left(
\begin{array}{ccc}
M_1M_5M_{11} & M_1M_7M_{12} & 0 \\
(M_2M_5+M_3M_6)M_{11} & M_2M_7M_{12} & M_{16} \\
M_4M_6M_{11} & M_8M_{12} & 0 \\
\hline
0 & 0 & 1
\end{array}
\right).$$ As before, by Lemma \[lemmadecoding\], for some finite field coefficients $y_1,y_2,$ and $y_3,$ we must have $$\begin{aligned}
M_1M_5M_{11}y_1 + M_1M_7M_{12}y_2 & = 0. \\
(M_2M_5+M_3M_6)M_{11}y_1 + M_2M_7M_{12}y_2+M_{16}y_3 & = 1.\\
M_4M_6M_{11}y_1+M_8M_{12}y_2 & = 0.\\
y_3& = 0.\end{aligned}$$ Letting $M_{11}y_1 = M_{11}'$ and $M_{12}y_2=M_{12}',$ we have $$\begin{aligned}
\label{eqnno4}
M_1M_5M_{11}' + M_1M_7M_{12}' & = 0. \\
\label{eqnno5}
M_2M_5M_{11}'+M_3M_6M_{11}'+M_2M_7M_{12}' & = 1. \\
\label{eqnno6}
M_4M_6M_{11}'+M_8M_{12}' &=0.\end{aligned}$$ The transfer matrix corresponding to error at $e_1$ at the sink $t=v_{39}$ ($In(t)=\{v_{30}\rightarrow v_{39},v_{18}\rightarrow v_{39},e_1\}$) is $$\boldsymbol{F_{v_{39}}^{e_1}} =
\left(
\begin{array}{ccc}
M_1M_7M_{13} & 0 & M_{17} \\
M_2M_7M_{13} & M_3M_{14} & 0 \\
M_8M_{13} & M_4M_{14} & 0 \\
\hline
0 & 0 & 1
\end{array}
\right).$$ Again, by Lemma \[lemmadecoding\], for some finite field coefficients $z_1,z_2,z_3,$ we must have $$\begin{aligned}
M_1M_7M_{13}z_1 + M_{17}z_3 & = 1. \\
M_2M_7M_{13}z_1 + M_3M_{14}z_2 & = 0. \\
M_8M_{13}z_1 + M_4M_{14}z_2 &= 0. \\
z_3&=0.\end{aligned}$$ Letting $M_{13}z_1 = M_{13}'$ and $M_{14}z_2=M_{14}',$ we have $$\begin{aligned}
\label{eqnno7}
M_1M_7M_{13}'& = 1. \\
\label{eqnno8}
M_2M_7M_{13}' + M_3M_{14}' & = 0.\\
\label{eqnno9}
M_8M_{13}' + M_4M_{14}' &= 0.\end{aligned}$$ Equations similar to (\[eqnno1\])-(\[eqnno9\]) were derived in [@DFZ3] for the network ${\cal N}_1.$ Mimicking the arguments in [@DFZ3], we now show that the characteristic of the finite field used must be two. From (\[eqnno3\]) and (\[eqnno7\]), we must have that the matrices $M_1, M_4, M_6, M_7, M_{10}',$ and $M_{13}'$ are all invertible. By (\[eqnno2\]), we must then have $M_2M_5+M_3M_6=0.$ Thus by (\[eqnno5\]), we must have $$\label{eqnno10}
M_2M_7M_{12}'=1.$$ and therefore $M_2$ and $M_{12}'$ are invertible. By (\[eqnno4\]), $M_5M_{11}'=-M_7M_{12}'$ and thus $M_5$ and $M_{11}'$ are invertible. Furthermore, $M_3M_{14}'=-M_2M_7M_{13}'$ by (\[eqnno8\]), and $M_9'=-M_1M_5M_{10}'$ by (\[eqnno1\]). Thus $M_3, M_{14}',$ and $M_9'$ are invertible. As $M_8=-M_4M_{14}'M_{13}'^{-1}$ by (\[eqnno9\]), the matrix $M_8$ is invertible too. Thus all the matrices in the equations (\[eqnno1\])-(\[eqnno9\]) are invertible.
From (\[eqnno4\]), we have $$\begin{aligned}
0 &= M_5M_{11}'+M_7M_{12}'\\
&= M_5M_{11}'+M_2^{-1}(M_2M_7M_{12}')\\
& = M_5M_{11}'+M_2^{-1}.\end{aligned}$$ where the last equality follows from (\[eqnno10\]).
Thus we have $$\begin{aligned}
\label{eqnno11}
M_2M_5M_{11}' = -1.\end{aligned}$$ From (\[eqnno6\]), we have $$\begin{aligned}
\nonumber
0 & = M_4M_6M_{11}'+M_8M_{12}' \\
& = M_4M_3^{-1}M_3M_6M_{11}'-M_4M_{14}'M_{13}'^{-1}M_{12}', \end{aligned}$$ where the last equality follows from (\[eqnno9\]). Now, using (\[eqnno2\]) and (\[eqnno8\]), we have $$\begin{aligned}
\nonumber
0& = -M_4M_3^{-1}M_2M_5M_{11}'+M_4M_3^{-1}M_2M_7M_{13}'M_{13}'^{-1}M_{12}' \\
\nonumber
& = M_4M_3^{-1}(M_2M_7M_{12}'-M_2M_5M_{11}') \\
\nonumber
&= M_4M_3^{-1}(1-M_2M_5M_{11}'), \end{aligned}$$ where the last equality follows from (\[eqnno10\]). Thus we must have $$\begin{aligned}
\label{eqnno12}
M_2M_5M_{11}'=1.\end{aligned}$$ Thus, from (\[eqnno11\]) and (\[eqnno12\]), we see that we require $1=-1.$ This is true only in a field of characteristic two. *If part:*
It is easy to verify that using $M_i=1 \in \mathbb{F}_{2^m}$ (for any $m$) for all $i$ results in a single edge network-error detecting code for $\tilde{\cal N}_1.$
In the case of a network code with all $M_i=1 \in \mathbb{F}_{2^m},~\forall i,$ we now argue that the network $\tilde{\cal N}_1$ is a matroidal $1$-error detecting network with respect to the vector matroid ${\cal M}_{\tilde{\cal N}_1}$ of the matrix over $\mathbb{F}_{2^m}$ shown below. $$\label{representationn1}
\left(
\begin{array}{cccccccc}
& 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
& 0 & 1 & 0 & 1 & 1 & 0 & 1 \\
& 0 & 0 & 1 & 0 & 1 & 1 & 1 \\
& & & & & & & \\
& 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
I_{10} & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
& 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
& 0 & 0 & 0 & 1 & 0 & 1 & 1 \\
& 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
& 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
& 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right)$$ Let the function with respect to which the matroid ${\cal M}_{\tilde{\cal N}_1}$ is associated be $$\begin{aligned}
\label{f1}
f_1:\mu_{\tilde{\cal N}_1}\cup{\cal E}_{{\tilde{\cal N}_1}}\rightarrow E({\cal M}_{\tilde{\cal N}_1}).\end{aligned}$$ The function $f_1$ maps the input symbols ($\mu_{\tilde{\cal N}_1}=\{a,b,c\}$) and the edges of $\tilde{\cal N}_1$ to the elements of the groundset $E({\cal M}_{\tilde{\cal N}_1}).$ The labeling on the columns (i.e., the mapping given by $f_1$) of the matrix given in (\[representationn1\]) is as follows. The first three columns correspond to the inputs $\mu_{\tilde{\cal N}_1}$. The next seven columns constitute the basis elements of the errors at $\{e_i:i=1,2,..,7\}$ as shown in Fig. \[fig:subnet1\]. The last seven columns correspond to the linear combination of the input symbols and the errors flowing on these edges. Though there are a total of $21$ edges in $\tilde{\cal N}_1,$ these seven edges are sufficient to characterise the matroid associated with the single edge network-error detecting code on $\tilde{\cal N}_1.$ It is easy to verify that the function $f_1$ and the matroid $\tilde{\cal M}_{\tilde{\cal N}_1}$ satisfy all the requirements of Definition \[matroidalerrornetworkdefinition\] for a single edge network-error detecting code. We list the elements of the ground set of ${\cal M}_{\tilde{\cal N}_1}$ in the ordering of the columns shown in (\[representationn1\]) as follows. $$\begin{aligned}
\nonumber
E({\cal M}_{\tilde{\cal N}_1}) = \left\{x_i:i=1,2,3\right\}&\cup\left\{y_i:i=1,2,...,7\right\} \\
\label{groundsetn1}
&\cup\left\{y_i':i=1,2,...,7\right\}.\end{aligned}$$ Finally, we have the following lemma which follows from Lemma \[lemmanetworkn1\] and the discussion above.
\[n1representation\] The network $\tilde{\cal N}_1$ is a matroidal 1-error detecting network associated with a $\mathbb{F}_2$-representable matroid.
A network not solvable on alphabets of characteristic two
---------------------------------------------------------
Consider the network $\tilde{\cal N}_2$ shown in Fig. \[fig:subnet2\]. The network has five sources $v_7, v_8, v_3, v_{11}$ and $v_{12}$ generating the information symbols $a, b, c, d,$ and $e$ respectively. There are seven sinks $v_{40}, v_{41}, v_{42}, v_{43}, v_{44}, v_{45}$ and $v_{46}$ demanding the symbols $c, b, a, c, e, d,$ and $c$ respectively. The network ${\cal N}_2$ of [@DFZ3] is the subnetwork of $\tilde{\cal N}_2$ consisting of all nodes and edges except the direct edges from $v_3, v_8, v_7, v_3, v_{12}, v_{11},$ and $v_3$ to the sinks. We seek the conditions to be satisfied by the finite field over which a single edge network-error detection code can be designed for $\tilde{\cal N}_2.$
Again, it is easy to verify that assuming all $1$s from a finite field with characteristic not equal to two as the network coding coefficients of $\tilde{\cal N}_2$ results in a single edge network-error detection code. The network $\tilde{\cal N}_2$ is then a matroidal $1$-error detecting network associated with the matroid ${\cal M}_{\tilde{\cal N}_2}$ whose representation (over any field with characteristic not equal to two) is shown in (\[representationn2\]) at the top of the next page.
$$\label{representationn2}
\left(
\begin{array}{c|c|ccccccccccccccc}
& & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
& & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\
I_5 & \boldsymbol{0} & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\
& & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
& & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \\
%& & & & & & & & & & & & & & & & & \\
\hline
%& & & & & & & & & & & & & & & & & \\
\boldsymbol{0} & I _{15} & & & & & & & I_{15} & & & & & & & &
\end{array}
\right)$$
------------------------------------------------------------------------
The corresponding function $f_2$ is given as $$\begin{aligned}
\label{f2}
f_2:\mu_{\tilde{\cal N}_2}\cup{\cal E}_{{\tilde{\cal N}_2}}\rightarrow E({\cal M}_{\tilde{\cal N}_2}),\end{aligned}$$ where $\mu_{\tilde{\cal N}_2}=\{a,b,c,d,e\}$ is the collection of the input symbols. As with $\tilde{\cal N}_1,$ not all the edges of $\tilde{\cal N}_2$ are considered in the representation of ${\cal M}_{\tilde{\cal N}_2}.$ The columns of the matrix shown in (\[representationn2\]) (and therefore the mappings of the function $f_2$) are indexed as follows. The first five columns correspond to the five input symbols. The next $15$ columns correspond to the error basis elements at the edges $\{e_i:i=1,2,..,15\}$ as shown in Fig. \[fig:subnet2\]. The final $15$ columns correspond to the linear combination of the inputs and error symbols flowing at these $15$ edges. We list the elements of the ground set of ${\cal M}_{\tilde{\cal N}_2}$ in the ordering of the columns shown in (\[representationn2\]) as follows. $$\begin{aligned}
\nonumber
E({\cal M}_{\tilde{\cal N}_2}) = \left\{x_i:i=1,2,..,5\right\}&\cup\left\{z_i:i=1,2,...,15\right\} \\
\label{groundsetn2}
&\cup\left\{z_i':i=1,2,...,15\right\}.\end{aligned}$$ As with $\tilde{\cal N}_1,$ it can be seen that in the absence of errors in the additional direct edges to the sinks (those not in ${\cal N}_2$), the sinks of $\tilde{\cal N}_2$ can straight away decode their required demands. Assuming single edge network-errors on these additional edges and using arguments equivalent to those in [@DFZ3] (as was done in the proof of Lemma \[lemmanetworkn1\]), we have the following lemma, which we state without proof.
\[lemmanetworkn2\] The network $\tilde{\cal N}_2$ has a single edge network-error detecting code if and only if the finite field used has characteristic not equal to two.
The following lemma follows directly from Lemma \[lemmanetworkn2\] and the preceding discussion.
\[n2representation\] The network $\tilde{\cal N}_2$ is a matroidal 1-error detecting network associated with a $\mathbb{F}_3$-representable matroid.
{width="6in"}
------------------------------------------------------------------------
It can be seen from the proof of Lemma \[lemmanetworkn1\] that particular error patterns were considered in order to verify whether the linear network code defined over a particular alphabet satisfies the required network-error detection (correction) properties. Given an arbitrary network, it may be necessary to consider all possible error patterns, i.e., $\left(\begin{array}{c}{\cal E} \\ \beta \end{array} \right)$ of them to verify the $\beta$ network-error detection capability. This is why proving insufficiency of linear network coding for network-error correction or detection could be computationally much harder than proving insufficiency of linear network codes for network coding with no errors.
A network for which linear network-error detection is insufficient {#insuffnetwork}
------------------------------------------------------------------
{width="7in"}
\
------------------------------------------------------------------------
We now present the network $\tilde{\cal N}_3$ shown in Fig. \[fig:nonlinearnet\] for which linear network coding is insufficient to achieve the sinks demands in the presence of network-errors. The network $\tilde{\cal N}_3$ is a conjoining of the network $\tilde{\cal N}_1$ and $\tilde{\cal N}_2$ with the exception of a few additional dummy edges. Thus, we assume ${\cal E}_{\tilde{\cal N}_3}={\cal E}_{\tilde{\cal N}_1}\cup {\cal E}_{\tilde{\cal N}_2}.$ We ignore the dummy edges for the sake of the clarity. The network ${\cal N}_3$ shown in [@DFZ3] is equivalent to $\tilde{\cal N}_3$ except for the direct edges to the sinks from the corresponding sources. Because of Lemmas \[lemmanetworkn1\] and Lemma \[lemmanetworkn2\], the network $\tilde{\cal N}_3$ does not have a linear single edge network-error detecting code over any field.
However, there is a nonlinear single edge network-error detecting code over an alphabet $\cal A$ of size $4$ , the corresponding edge functions of which are shown along the edges of $\tilde{\cal N}_3$ in Fig. \[fig:nonlinearnet\]. Except for the additional direct edges from the sources to the corresponding sinks, the network coding functions on $\tilde{\cal N}_3$ are adopted from the network code for ${\cal N}_3$ in [@DFZ3]. All the missing edge functions are considered to be identity. The symbols $+$ and $-$ indicate the addition and subtraction in the ring ${\mathbb Z}_4,$ while the symbols $\oplus$ indicates the bitwise XOR operation in ${\mathbb Z}_2 \oplus {\mathbb Z}_2.$ In other words, for any two elements $a,b \in {\cal A}$, the element $a+b$ and $a-b$ indicate the sum of $a$ and $b$ and the difference between $a$ and $b$ viewing them as elements from ${\mathbb Z}_4.$ The element $a\oplus b$ indicates the bitwise XOR between $a$ and $b$ viewing them as elements from ${\mathbb Z}_2 \oplus {\mathbb Z}_2.$ For some $a\in {\cal A},$ $t(a)$ is the element of ${\cal A}$ obtained by switching the components of $a$ considered as element of ${\mathbb Z}_2 \oplus {\mathbb Z}_2.$ The nonlinearity of the network-error correction code comes from the nonlinearity of the function $t,$ and because $\oplus$ is linear in ${\mathbb Z}_2 \oplus {\mathbb Z}_2$ but nonlinear in ${\mathbb Z}_4,$ while $+$ and $-$ are linear in ${\mathbb Z}_4$ but nonlinear in ${\mathbb Z}_2 \oplus {\mathbb Z}_2.$ Using the arguments developed in [@DFZ3], it is straightforward to show that these coding functions define a single edge network-error detection code for $\tilde{\cal N}_3.$
We can now ask the question - *Is the network $\tilde{\cal N}_3$ a matroidal 1-error detecting network?* If the answer is yes, then it would mean that our definition of a matroidal error detecting network (Definition \[matroidalerrornetworkdefinition\]) has a wider scope and is not limited to linear network-error detection and representable matroids. Also, an equivalent question can be raised about the network ${\cal N}_3$ shown in [@DFZ3] - *Is the network ${\cal N}_3$ a matroidal network?* This second question is left unanswered in both [@DFZ3] (where the insufficiency results for linear network coding in ${\cal N}_3$ was first presented) and in [@DFZ] (where the matroidal connections to the construction of ${\cal N}_1$ and ${\cal N}_2$ were discussed). We answer these questions in the affirmative. In the rest of this Subsection, we obtain a matroid ${\cal M}_{\tilde{\cal N}_3}$ associated with which the network $\tilde{\cal N}_3$ is a matroidal 1-error detecting network. That the network ${\cal N}_3$ of [@DFZ3] is matroidal follows easily.
We first prove the following lemma.
\[conditionsn3matroidal\] Let $E({\cal M}_{\tilde{\cal N}_3})=E({\cal M}_{\tilde{\cal N}_1})\cup E({\cal M}_{\tilde{\cal N}_2})$ be the groundset of a matroid ${\cal M}_{\tilde{\cal N}_3}.$ If the matroid ${\cal M}_{\tilde{\cal N}_3}$ satisfies the following two conditions $$\begin{aligned}
\label{cond1}
{\cal M}_{\tilde{\cal N}_3}|_{E({\cal M}_{\tilde{\cal N}_1})} = {\cal M}_{\tilde{\cal N}_1}.\\
\label{cond2}
{\cal M}_{\tilde{\cal N}_3}|_{E({\cal M}_{\tilde{\cal N}_2})} = {\cal M}_{\tilde{\cal N}_2},\end{aligned}$$ then the network $\tilde{\cal N}_3$ is matroidal 1-error detecting associated with ${\cal M}_{\tilde{\cal N}_3}.$
Let $\mu_{\tilde{\cal N}_3}=\mu_{\tilde{\cal N}_1}\cup \mu_{\tilde{\cal N}_2}.$ Clearly, $\mu_{\tilde{\cal N}_3}=\mu_{\tilde{\cal N}_2}.$ Let $f_3:\mu_{\tilde{\cal N}_3}\cup {\cal E}_{\tilde{\cal N}_3} \rightarrow E({\cal M}_{\tilde{\cal N}_3})$ be a function such that $$\begin{aligned}
&f_3(\mu_{\tilde{\cal N}_3})=f_2(\mu_{\tilde{\cal N}_2}),\\
&f_3(e) = f_1(e), \forall e\in {\cal E}_{\tilde{\cal N}_1}, \\
&f_3(e) = f_2(e), \forall e\in {\cal E}_{\tilde{\cal N}_2},\end{aligned}$$ where $f_1$ and $f_2$ are defined as in (\[f1\]) and (\[f2\]) respectively. Since $\tilde{\cal N}_3$ is a conjoining of the networks ${\tilde{\cal N}_1}$ and ${\tilde{\cal N}_2},$ i.e. as ${\cal E}_{\tilde{\cal N}_3}={\cal E}_{\tilde{\cal N}_1}\cup {\cal E}_{\tilde{\cal N}_2},$ it is clear that the function $f_3$ is well defined.
Now, since the networks $\tilde{\cal N}_1$ and $\tilde{\cal N}_2$ are already matroidal 1-error detecting networks associated to ${\cal M}_{\tilde{\cal N}_1}$ (with respect to $f_1$) and ${\cal M}_{\tilde{\cal N}_2}$ (with respect to $f_2$) respectively, by the definition of $f_3$ it follows that $\tilde{\cal N}_3$ is a matroidal 1-error detecting network associated with $\tilde{\cal N}_3$ with respect to $f_3.$
In order to show that $\tilde{\cal N}_3$ is matroidal 1-error detecting, we have to demonstrate a matroid which satisfies the conditions in Lemma \[conditionsn3matroidal\]. In the rest of this subsection, we show that such a matroid can be obtained. We use Definition \[matroiddefnrank\] of a matroid based on its rank function to obtain our matroid ${\cal M}_{\tilde{\cal N}_3}.$
Let $r:2^{E({\cal M}_{\tilde{\cal N}_1})\cup E({\cal M}_{\tilde{\cal N}_2})}\rightarrow \mathbb{Z}^+\cup\left\{0\right\}$ be a function defined as $$r(X) = r_{{\cal M}_{\tilde{\cal N}_1}}(X_1)+r_{{\cal M}_{\tilde{\cal N}_2}}(X_2)-r_{{\cal M}_{\tilde{\cal N}_2}}(X_{1,2}),$$ where $X_1=X\cap E({\cal M}_{\tilde{\cal N}_1}), X_2=X\cap E({\cal M}_{\tilde{\cal N}_2}),$ and $X_{1,2}=X\cap E({\cal M}_{\tilde{\cal N}_1})\cap E({\cal M}_{\tilde{\cal N}_2})=X\cap\{x_1,x_2,x_3\} = X_1\cap X_2.$ The functions $r_{{\cal M}_{\tilde{\cal N}_1}}$ and $r_{{\cal M}_{\tilde{\cal N}_2}}$ are the rank functions of the matroids ${\cal M}_{\tilde{\cal N}_1}$ and ${\cal M}_{\tilde{\cal N}_2}$ respectively. Clearly the function $r$ is well defined. Also, as $r_{{\cal M}_{\tilde{\cal N}_2}}(X_2)\geq r_{{\cal M}_{\tilde{\cal N}_2}}(X_{1,2}),$ we must have $r(X)\geq 0,~\forall X.$ Also, for any $X\subseteq E({\cal M}_{\tilde{\cal N}_1})\cup E({\cal M}_{\tilde{\cal N}_2}),$ we note that $$\begin{aligned}
\label{simplifyeqn}
r_{{\cal M}_{\tilde{\cal N}_2}}(X_{1,2})=r_{{\cal M}_{\tilde{\cal N}_1}}(X_{1,2})=|X_{1,2}|.\end{aligned}$$ Now, suppose there is a matroid with the above function $r$ as its rank function. Then it can be seen that from the definition of the function $r$ that such a matroid satisfies the requirements of Lemma \[conditionsn3matroidal\]. This is because for any $X\subseteq E({\cal M}_{\tilde{\cal N}_1}), r(X)=r_{{\cal M}_{\tilde{\cal N}_1}}(X),$ and for any $X\subseteq E({\cal M}_{\tilde{\cal N}_2}), r(X)=r_{{\cal M}_{\tilde{\cal N}_2}}(X).$ Thus the network $\tilde{\cal N}_3$ would be a matroidal $1$-error detecting network associated with such a matroid. We now prove the following lemma which shows that the function $r$ defines a matroid.
\[functionrmatroid\] The function $r$ is the rank function of a matroid.
We have to show that the function $r$ satisfies the properties **R1**, **R2**, and **R3** of Definition \[matroiddefnrank\].
We first consider the condition **R1**. We have by the definition of $r,$ for $X\subseteq E({\cal M}_{\tilde{\cal N}_1})\cup E({\cal M}_{\tilde{\cal N}_2}),$ $$r(X) = r_{{\cal M}_{\tilde{\cal N}_1}}(X_1)+r_{{\cal M}_{\tilde{\cal N}_2}}(X_2)-r_{{\cal M}_{\tilde{\cal N}_2}}(X_{1,2}),$$ where $X_1=X\cap E({\cal M}_{\tilde{\cal N}_1}), X_2=X\cap E({\cal M}_{\tilde{\cal N}_2}),$ and $X_{1,2}=X\cap E({\cal M}_{\tilde{\cal N}_1})\cap E({\cal M}_{\tilde{\cal N}_2})=X\cap\{x_1,x_2,x_3\}.$ Because $r_{{\cal M}_{\tilde{\cal N}_1}}$ and $r_{{\cal M}_{\tilde{\cal N}_2}}$ are rank functions and by (\[simplifyeqn\]), we must have $$\begin{aligned}
\nonumber
r(X) &\leq |X_1|+|X_2|-|X_{1,2}|\\
\nonumber
&\leq |X_1|+|(X_2-X_{1,2})\uplus X_{1,2}|-|X_{1,2}|\\
\label{no9eqn}
r(X) &\leq |X_1|+|X_2-X_{1,2}|=|X|.\end{aligned}$$ We have already seen that $r(X)\geq 0, \forall X.$ Along with (\[no9eqn\]), this means that the function $r$ satisfies **R1**. Now we prove that **R2** holds.
Let $X\subseteq Y\subseteq E({\cal M}_{\tilde{\cal N}_1})\cup E({\cal M}_{\tilde{\cal N}_2}).$ Then $X_1=X\cap E({\cal M}_{\tilde{\cal N}_1})\subseteq Y_1 = Y\cap E({\cal M}_{\tilde{\cal N}_1}).$ Similarly, $X_2\subseteq Y_2,$ and $X_{1,2}\subseteq Y_{1,2}.$
Let $B_{X_1}$ be a subset of $X_1$ of the largest size which is independent in ${\cal M}_{\tilde{\cal N}_1}$. Similarly let $B_{X_2}\subseteq X_2, B_{X_{1,2}}\subseteq X_{1,2}, B_{Y_1}\subseteq Y_1, B_{Y_2}\subseteq Y_2, B_{Y_{1,2}}\subseteq Y_{1,2}$ be some maximal independent subsets in the appropriate matroids. Because $X_{1,2}\subseteq X_1\subseteq Y_1,$ we can always find $B_{X_{1,2}}, B_{X_1}, B_{Y_1}$ such that $B_{X_{1,2}}\subseteq B_{X_1}\subseteq B_{Y_1},$ by repeated application of **I3** in Definition \[matroiddefnindp\]. Similarly, we assume $B_{X_{1,2}}\subseteq B_{X_2}\subseteq B_{Y_2}$ and $B_{X_{1,2}}\subseteq B_{Y_{1,2}}.$
By the definition of $r,$ we have $$\begin{aligned}
\nonumber
r(X)&=|B_{X_1}|+|B_{X_2}|-|B_{X_{1,2}}|\\
\nonumber
&=|B_{X_{1,2}}\uplus (B_{X_1}-B_{X_{1,2}})|+|B_{X_2}|-|B_{X_{1,2}}|\\
\nonumber
&=|B_{X_{1,2}}|+|B_{X_1}-B_{X_{1,2}}|+|B_{X_2}|-|B_{X_{1,2}}|\\
\label{no2eqn}
r(X)&=|B_{X_1}-B_{X_{1,2}}|+|B_{X_2}|.\end{aligned}$$ As in (\[no2eqn\]), we have $$\begin{aligned}
\nonumber
r(Y)&=|B_{Y_1}-B_{Y_{1,2}}|+|B_{Y_2}|\\
\nonumber
&\geq |B_{X_1}-B_{Y_{1,2}}|+|B_{Y_2}|~~~~~(\text{as}~B_{X_1}\subseteq B_{Y_1}) \\
\nonumber
&\geq |B_{X_1}-(B_{X_{1,2}}\uplus (B_{Y_{1,2}}-B_{X_{1,2}}))|+|B_{Y_2}|\\
\label{no3eqn}
r(Y)&\geq |B_{X_1}-B_{X_{1,2}}|-|B_{Y_{1,2}}-B_{X_{1,2}}|+|B_{Y_2}|.\end{aligned}$$ We also have the following equations. $$\begin{aligned}
\nonumber
|B_{Y_2}|&=|B_{X_2}\uplus (B_{Y_2}-B_{X_2})|\\
\nonumber
&=|B_{X_2}|+|B_{Y_2}-B_{X_2}|\\
\nonumber
&\geq |B_{X_2}|+|(B_{Y_2}-B_{X_2})\cap E({\cal M}_{\tilde{\cal N}_1})|\\
\label{no4eqn}
|B_{Y_2}|&\geq |B_{X_2}|+|B_{Y_{1,2}}-B_{X_{1,2}}|.\end{aligned}$$ By (\[no3eqn\]) and (\[no4eqn\]), we have $$\begin{aligned}
\label{no5eqn}
r(Y)\geq |B_{X_1}-B_{X_{1,2}}|+|B_{X_2}|.\end{aligned}$$ Comparing (\[no2eqn\]) and (\[no5eqn\]), we have $r(X)\leq r(Y).$ Hence **R2** holds. Finally, we prove the condition **R3** also holds.
Let $X,Y \subseteq E({\cal M}_{\tilde{\cal N}_1})\cup E({\cal M}_{\tilde{\cal N}_2}).$ By the definition of $r$ and (\[simplifyeqn\]), we have $$\begin{aligned}
\nonumber
r&(X)+r(Y)-r(X\cap Y)\\
\nonumber
&=r_{{\cal M}_{\tilde{\cal N}_1}}(X_1)+r_{{\cal M}_{\tilde{\cal N}_2}}(X_2)-|X_{1,2}|\\
\nonumber
&~~+r_{{\cal M}_{\tilde{\cal N}_1}}(Y_1)+r_{{\cal M}_{\tilde{\cal N}_2}}(Y_2)-|Y_{1,2}|\\
\label{no6eqn}
&~~-r_{{\cal M}_{\tilde{\cal N}_1}}(X_1\cap Y_1)-r_{{\cal M}_{\tilde{\cal N}_2}}(X_2\cap Y_2)+|X_{1,2}\cap Y_{1,2}|.\end{aligned}$$ Also, we have $$\begin{aligned}
\nonumber
r&(X\cup Y)\\
\nonumber
&=r_{{\cal M}_{\tilde{\cal N}_1}}(X_1\cup Y_1)+r_{{\cal M}_{\tilde{\cal N}_2}}(X_2\cup Y_2)-|X_{1,2}\cup Y_{1,2}|\\
\nonumber
&\leq r_{{\cal M}_{\tilde{\cal N}_1}}(X_1)+r_{{\cal M}_{\tilde{\cal N}_1}}(Y_1)-r_{{\cal M}_{\tilde{\cal N}_1}}(X_1\cap Y_1)\\
\nonumber
&~~+r_{{\cal M}_{\tilde{\cal N}_2}}(X_2)+r_{{\cal M}_{\tilde{\cal N}_2}}(Y_2)-r_{{\cal M}_{\tilde{\cal N}_2}}(X_2\cap Y_2)\\
\label{no7eqn}
&~~-|X_{1,2}\cup Y_{1,2}|,\end{aligned}$$ where the last inequality follows from the fact that $r_{{\cal M}_{\tilde{\cal N}_1}}$ and $r_{{\cal M}_{\tilde{\cal N}_2}}$ are rank functions.
From (\[no6eqn\]) and (\[no7eqn\]), to show that $r(X\cup Y)\leq r(X)+r(Y)-r(X\cap Y),$ we must prove $$\begin{aligned}
\label{no8eqn}
|X_{1,2}\cup Y_{1,2}|\geq |X_{1,2}|+|Y_{1,2}|-|X_{1,2}\cap Y_{1,2}|.\end{aligned}$$ But (\[no8eqn\]) holds with equality by the law of unions of sets, and thus the condition **R3** holds for the function $r$.
Thus from Lemma \[functionrmatroid\], the function $r$ defines a matroid. Let this matroid be the candidate matroid ${\cal M}_{\tilde{\cal N}_3}$ as in Lemma \[conditionsn3matroidal\]. Note that ${\cal M}_{\tilde{\cal N}_3}$ satisfies the conditions of Lemma \[conditionsn3matroidal\], as explained in the discussion preceding Lemma \[functionrmatroid\]. Thus, if ${\cal M}_{\tilde{\cal N}_3}$ is representable over some field $\mathbb{F},$ then the matroids ${\cal M}_{\tilde{\cal N}_1}$ and ${\cal M}_{\tilde{\cal N}_2}$ must also be $\mathbb{F}$-representable, as restrictions of $\mathbb{F}$-representable matroids are $\mathbb{F}$-representable. However, the matroids ${\cal M}_{\tilde{\cal N}_1}$ and ${\cal M}_{\tilde{\cal N}_2}$ can never have representations over the same field because of Lemma \[lemmanetworkn1\] and Lemma \[lemmanetworkn2\]. Thus ${\cal M}_{\tilde{\cal N}_3}$ is nonrepresentable. We thus have the following lemma.
\[n3representation\] The network $\tilde{\cal N}_3$ is a matroidal 1-error detecting network associated with the nonrepresentable matroid ${\cal M}_{\tilde{\cal N}_3}.$
Thus Definition \[matroidalerrornetworkdefinition\] applies to error detecting networks associated with nonrepresentable matroids also. A similar argument can be given for Definition \[matroidalerrorcorrectionnetworkdefinition\] also.
\[amalgam\] A matroid ${\cal M}$ on the groundset $E=E_1\cup E_2$ is said to be an *amalgam* of the matroids ${\cal M}_1={\cal M}|E_1$ and ${\cal M}_2={\cal M}|E_2$ (the reader is referred to [@Oxl] for more details). Thus the matroid ${\cal M}_{\tilde{\cal N}_3}$ is an amalgam of ${\cal M}_{\tilde{\cal N}_1}$ and ${\cal M}_{\tilde{\cal N}_2}.$
By Lemma \[n3representation\] and because of the connection between the network $\tilde{\cal N}_3$ and the network ${\cal N}_3$ shown in [@DFZ3], it is easy to prove that ${\cal N}_3$ is a matroidal network associated with a nonrepresentable matroid, one which is constructed as an amalgam of the matroids $M_{\tilde{\cal N}_1}/\left\{y_i:i=1,...,7\right\}$ and $M_{\tilde{\cal N}_2}/\left\{z_i:i=1,...,15\right\}.$ We leave the details of this proof to the reader.
More Examples {#sec6}
=============
In this section, we present some examples of networks with scalar linear network codes and network-error correcting codes to illustrate our construction algorithms. Each example shown in this section is obtained by running an instance of the corresponding algorithm fixing the number of sources ($|{\cal S}|$), number of messages ($n$), number of correctable errors ($\alpha$), number of coding nodes to be added ($N_C$), number of sinks $|{\cal T}|$ (necessary for multicast) and the finite field used. Furthermore, for ease of computation, we also fix the number of edges whose symbols are to be encoded at any iteration in the construction algorithm to the new coding node, i.e., $|{\cal E}_C|$ is fixed. These examples are obtained by randomly picking existing forwarding nodes at any iteration in the algorithm to combine their information flows, and then checking if the resultant network code (or the equivalent matroid) satisfies the necessary properties. The MATLAB codes that generate these examples will be provided by the authors on request. All the figures and the corresponding matroid representations (or network coding coefficients) are shown at the end of the manuscript.
Multicast
---------
Fig. \[fig:bigmulticastexm\] shows a single source multicast network with a scalar linear $3$-error correcting network code and $N_C=10$. Table \[tab1\] shows all the relevant parameters using which the algorithm designs the network and the linear network coding coefficients obtained as outputs of the algorithm. The global encoding vectors of the $N$ outgoing edges from the source in the network correspond to the columns of a generator matrix of an MDS code with minimum distance $2\alpha+1=7$ and length $N=n+2\alpha=9.$ The values in the last column of Table \[tab1\] represent the particular linear combination using which the information flows from the existing forwarding nodes (specified by the first column in Table \[tab1\]) are combined at the new coding node formed (the corresponding forwarding node is given by the second column of Table \[tab1\]). These linear encoding coefficients are represented by the decimal equivalents of the polynomial representations of the respective finite field elements. Also in Fig. \[fig:bigmulticastexm\], the direct links from the source to the sinks are indicated by incoming edges from the corresponding duplicate nodes (which are unconnected to the rest of the network).
This example also illustrates the ability of our multicast algorithm to construct scalar linear network-error correcting codes for multicast networks over smaller fields when compared with existing algorithms in [@Zha; @Mat; @YaY]. To see this, suppose that the network shown in Fig. \[fig:bigmulticastexm\] was given as the input network to the algorithms in [@Zha; @Mat; @YaY] in order to design a multicast $3$-network-error correcting code. These algorithms require a field size $q$ such that $$q \geq
\sum_{t\in{\cal T}}\left(
\begin{array}{c}
|{\cal E}| \\
2\alpha
\end{array}
\right) \geq \sum_{t\in{\cal T}}\left(
\begin{array}{c}
N_C \\
2\alpha
\end{array}
\right) = 3\left(
\begin{array}{c}
10 \\
6
\end{array}
\right) = 630$$ to design a multicast linear network-error correcting code that can correct any $3$ network-errors in the given network. Thus only if $q\geq 630,$ the algorithms in [@Zha; @Mat; @YaY] guarantee the construction of a suitable network-error correcting code for our final network. However, our algorithm obtains a network-error correcting code for this network over $\mathbb{F}_{16}$ because it designs the network and the associated matroids together and representations of these associated matroids can be given over $\mathbb{F}_{16}$. The topology of the network is controlled by our algorithm. This is in contrast with the algorithms in [@Zha; @Mat; @YaY], which take a given network as the input and design the network-error correcting code for that network. The field size demands of [@Zha; @Mat; @YaY] are less dependent on the actual topology of the network and depend more on its size.
Multiple-Unicast
----------------
\[unicastncexample\] Fig. \[fig:unicastncnetwork0\]-\[fig:unicastncnetwork5\] show the stages of the network evolution of a multiple-unicast network with parameters $n=3, \alpha=0$ (no error correction) and $N_C=5$. The direct links from the different sources to the sinks are indicated by incoming edges from the corresponding duplicate nodes. Every sink demands the information symbol generated by the corresponding source. The representative matrices of the corresponding matroids are shown in (\[eqnstage0unicastnc\])-(\[eqnstage3unicastnc\]) in Fig. \[fig:unicastnceqns\]. Every network is a matroidal $0$-error correcting network with the corresponding matroid and function $f,$ as defined in Example \[multicastex\]. Note the reduction in the number of incoming edges at Sink $T_2$ from three in Fig. \[fig:unicastncnetwork4\] to two in Fig. \[fig:unicastncnetwork5\]. This is a result of using the optional update in ***Step 6*** of our multiple-unicast algorithm. The transfer matrix from the sources to sink $T_2$ at the end of the final iteration is $$\boldsymbol{F_{{\cal S},T_2}}=
\left(
\begin{array}{cc}
1 & 2\\
4 & 1\\
3 & 6
\end{array}
\right),$$ where the matrix is over $\mathbb{F}_8$ (with modulo polynomial $x^3+x+1$), with the entries being the decimal equivalents of the polynomial representations of elements from $\mathbb{F}_8.$ The demanded symbol at $T_2$ is generated by $s_2,$ and corresponds to the second row above. The interference from source $s_3$, corresponding to the third row, is seen to be a scaled version of the interference from $s_1$, corresponding to the first row. Thus in this case, our multiple-unicast algorithm generates a network for which the interference is aligned by the network rather than canceled. However, the sink itself is enabled to cancel the interference. It is easily seen that a linear combination of the two columns of $\boldsymbol{F_{{\cal S},T_2}}$ generates the basis vector $(0~1~0~0)^T,$ enabling the sink $T_2$ to decode the demanded information symbol generated by source $s_2.$
Fig. \[fig:bigmultipleunicastexm\] shows a multiple-unicast network with a $2$-error correcting code, with all relevant parameters of which are shown in Table \[tab2\]. The $i^{th}$ sink demands the information symbol generated by the $i^{th}$ source. Each source employs a repetition code of length $2\alpha+1=5$ on its outgoing edges. As in Table \[tab1\], the values in the last column of Table \[tab2\] represent the decimal equivalents of the field elements in their polynomial representation. The direct links from the different sources to the sinks are indicated by incoming edges from the corresponding duplicate nodes.
Concluding remarks and Discussion {#sec7}
=================================
The matroidal connections to network-error correction and detection have been analysed in this paper. It was shown that networks with scalar linear network-error correcting and detecting codes correspond to representable matroids with certain special properties. We also presented algorithms which can construct matroidal error correcting networks. The same algorithms can also be used to construct matroidal error detecting networks. By restricting ourselves to the class of representable matroids, we can therefore obtain a large number of networks with scalar linear network-error correcting and detecting codes, some of which were presented as examples. Further restricting ourselves to the matroids which are representable over particular fields, we can obtain networks which have scalar linear network-error correcting codes over those particular fields. This may facilitate some intuition towards finding the minimum field size requirement for scalar linear network-error correcting codes to exist, which is known to be a hard problem. Also, running our algorithms along with the optional update of sink incoming edges in ***Step 6*** may provide insight on the solvability and capacity of general multisource multicast and multiple-unicast networks in the presence of errors. In particular, the multiple-unicast algorithm can then be used to generate multiple-unicast networks where interference from other sources is not always canceled by the network nodes, as shown by Example \[unicastncexample\]. Following techniques similar to [@DFZ3], it was also shown that linear network codes prove are not always sufficient to provide the demanded error correction.
It is known [@Oxl] that characterising all possible modular cuts of a matroid, and therefore all possible extensions of a matroid is in general a difficult task. Moreover, we require extensions which satisfy certain constraints for the resultant network to be matroidal, and have to satisfy even more constraints if they have to be associated with representable matroids. Characterising such extensions could be a particularly rewarding exercise. As a first step towards characterising such extensions and also towards obtaining matroidal error correcting networks associated with nonrepresentable matroids, we proved Proposition \[propprincipal\] regarding the principal extensions of a representable matroid. It can be expected that deeper theoretical insights on the theory of network coding and error correction can be gained with more powerful machinery from matroid theory.
[160]{} R. Ahlswede, N. Cai, R. Li and R. Yeung, “Network Information Flow”, IEEE Transactions on Information Theory, vol.46, no.4, July 2000, pp. 1204-1216.
N. Cai, R. Li and R. Yeung, “Linear Network Coding”, IEEE Transactions on Information Theory, vol. 49, no. 2, Feb. 2003, pp. 371-381.
R. Koetter and M. Medard, “An Algebraic Approach to Network Coding”, IEEE/ACM Transactions on Networking, vol. 11, no. 5, Oct. 2003, pp. 782-795.
S. Jaggi, P. Sanders, P.A. Chou, M. Effros, S. Egner, K. Jain and L.M.G.M. Tolhuizen, “Polynomial time algorithms for multicast network code construction”, IEEE Trans. Inf. Theory, vol. 51, no. 6, June 2005, pp. 1973-1982.
R. Dougherty, C. Freiling, and K. Zeger, “Networks, Matroids, and Non-Shannon Information Inequalities”, IEEE Transactions on Information Theory, Vol. 53, No. 6, June 2007.
A. Kim and M. Medard, “Scalar-linear Solvability of Matroidal Networks Associated with Representable Matroids”, International Symposium on Turbo Codes and Iterative Information Processing (ISTC), Sep. 6-10, 2010, pp. 452 - 456.
S. El Rouayheb, A. Sprintson, and C. Georghiades, “A new construction method for networks from matroids,” ISIT 2009, June 28 - July 2009, pp. 2872-2876.
R. Dougherty, C. Freiling, and K. Zeger, “Linear Network Codes and Systems of Polynomial Equations”, ISIT 2008, Toronto, Canada, July 6-11, pp. 1838 - 1842.
Q. Sun, S. T. Ho, S.Y.R. Li, “On Network Matroids and Linear Network Codes”, ISIT 2008, Toronto, Canada, July 6-11, 2008, pp. 1833-1837.
S. Y. R. Li and Q. T. Sun, Network coding theory via commutative algebra, IEEE Transactions on Information Theory, vol. 57, no. 1, Jan 2011, pp. 403-415.
R. Dougherty, C. Freiling, and K. Zeger, “Insufficiency of Linear Coding in Network Information Flow”, IEEE Transactions on Information Theory, Vol. 51, No. 8, August 2005.
R.W. Yeung and N. Cai, “Network error correction, part I: basic concepts and upper bounds ”, Comm. in Inform. and Systems, vol. 6, 2006, pp. 19-36.
N. Cai and R. W. Yeung, “Network error correction, part II: lower bounds", Comm. in Inform. and Systems, vol. 6, 2006, pp. 37-54.
Z. Zhang, “Linear network-error Correction Codes in Packet Networks”, IEEE Transactions on Information Theory, vol. 54, no. 1, Jan. 2008, pp. 209-218.
R. Matsumoto, “Construction Algorithm for Network Error-Correcting Codes Attaining the Singleton Bound”, IEICE Trans. Fundamentals, Vol. E90-A, No. 9, September 2007, pp. 1729-1735.
S. Yang, R. W.Yeung, C. K. Ngai, “Refined Coding Bounds and Code Constructions for Coherent Network Error Correction”, IEEE Transactions on Information Theory, vol. 57, no. 9, Sep. 2011, pp. 1409-1424.
S. Vyetrenko, T. Ho, M. Effros, J. Kliewer, E. Erez, “Rate regions for Coherent and Noncoherent Multisource Network Error Correction”, Proceedings of ISIT 2009, Seoul, Korea, June 28 - July 3, pp. 1001-1005.
O. Kosut, L. Tong, D. Tse, “Nonlinear Network Coding is Necessary to Combat General Byzantine Attacks”, Proceedings of the Forty-Seventh Annual Allerton Conference, Illinois, USA, Sep. 30 - Oct.2, pp. 593-599.
S. Kim, T. Ho, M. Effros, A. S. Avestimehr, “Network Error Correction With Unequal Link Capacities”, IEEE Transactions on Information Theory, vol. 57, no. 2, Feb. 2011, pp. 1144-1164.
J. G. Oxley, “Matroid Theory”, Oxford University Press, 1992.
------------------------------ ------------------------- --------------------------------------------------------------
**Nodes used to form new** **New forwarding node** $\boldsymbol{\mathbb{F}}$ **linear combination of nodes of**
**coding node (see figure)** **formed (see figure)** **column $1$ used to form new node**
(4,8) 10 (1,2)
(6,10) 11 (1,5)
(1,11) 12 (1,9)
(5,7) 13 (1,2)
(3,8) 14 (1,3)
(2,12) 15 (1,8)
(7,9) 16 (1,13)
(13,16) 17 (1,1)
(15,17) 18 (1,2)
(14,18) 19 (1,1)
------------------------------ ------------------------- --------------------------------------------------------------
\
\
\
------------------------------------------------------------------------
$$\label{eqnstage0unicastnc}
%\footnotesize
\left(
\begin{array}{cccc}
&1&0&0\\
&0&1&0\\
&0&0&1\\
&&&\\
I_{6}&1&0&0\\
&0&1&0\\
&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage1unicastnc}
%\footnotesize
\left(
\begin{array}{ccccc}
&1&0&0&0\\
&0&1&0&1\\
&0&0&1&2\\
&&&&\\
I_{7}&1&0&0&0\\
&0&1&0&1\\
&0&0&1&2\\
&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage2unicastnc}
%\footnotesize
\left(
\begin{array}{cccccccc}
&1&0&0&0&1&6&1\\
&0&1&0&1&0&1&4\\
&0&0&1&2&2&7&3\\
&&&&&&&\\
&1&0&0&0&1&6&1\\
&0&1&0&1&0&1&4\\
I_{10}&0&0&1&2&2&7&3\\
&0&0&0&1&0&0&4\\
&0&0&0&0&1&6&0\\
&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&1\\
\end{array}
\right)$$
$$\label{eqnstage3unicastnc}
%\footnotesize
\left(
\begin{array}{ccccccccc}
&1&0&0&0&1&6&1&2\\
&0&1&0&1&0&1&4&1\\
&0&0&1&2&2&7&3&6\\
&&&&&&&&\\
&1&0&0&0&1&6&1&2\\
&0&1&0&1&0&1&4&1\\
I_{11}&0&0&1&2&2&7&3&6\\
&0&0&0&1&0&0&4&1\\
&0&0&0&0&1&6&0&2\\
&0&0&0&0&0&1&0&0\\
&0&0&0&0&0&0&1&0\\
&0&0&0&0&0&0&0&1\\
\end{array}
\right)$$
------------------------ -------------------------------- --------------------------------------------------------------
**Nodes used to form** **New forwarding node formed** $\boldsymbol{\mathbb{F}}$ **linear combination of nodes of**
**new coding node** **column $1$ used to form new node**
(7,8) 26 (1,4)
(12,24) 27 (1,2)
(4,20) 28 (1,5)
(3,19) 29 (1,7)
(21,26) 30 (1,1)
(11,18) 31 (1,3)
(17,26) 32 (1,2)
(30,32) 33 (1,3)
(8,16) 34 (1,6)
(10,28) 35 (1,3)
(6,22) 36 (1,3)
(17,27) 37 (1,2)
(13,27) 38 (1,2)
(20,25) 39 (1,6)
(33,38) 40 (1,6)
------------------------ -------------------------------- --------------------------------------------------------------
|
---
abstract: 'Let $G$ be a connected linear algebraic group over an algebraically closed field $k$, and let $H$ be a connected closed subgroup of $G$. We prove that the homogeneous variety $G/H$ is a rational variety over $k$ whenever $H$ is solvable, or when $\dim(G/H) \leqslant 10$ and ${{\operatorname{char}}}(k)=0$. When $H$ is of maximal rank in $G$, we also prove that $G/H$ is rational if the maximal semisimple quotient of $G$ is isogenous to a product of almost-simple groups of type $A$, type $C$ (when ${{\operatorname{char}}}(k) \neq 2$), or type $B_3$ or $G_2$ (when ${{\operatorname{char}}}(k) = 0$).'
address:
- 'Department of Mathematics National University of Singapore 10 Lower Kent Ridge Road, Singapore 119076, Singapore'
- 'Department of Mathematics National University of Singapore 10 Lower Kent Ridge Road, Singapore 119076, Singapore'
author:
- CheeWhye Chin
- 'De-Qi Zhang'
title: Rationality of homogeneous varieties
---
[^1]
Introduction
============
Let $k$ be an algebraically closed field of arbitrary characteristic $p \ge 0$. Throughout this paper, we work only with the Zariski topology on varieties or algebraic groups over $k$. By a result that goes back to Chevalley when ${{\operatorname{char}}}(k)=0$ (cf. [@Ch §2, Cor. 2 to Th. 1]) and to Rosenlicht for arbitrary ${{\operatorname{char}}}(k)$ (cf. [@Ro57 end of §3]), a connected linear algebraic group $G$ over $k$ is a rational variety: the field $k(G)$ of rational functions on $G$ is a purely transcendental extension of $k$. If $H \subseteq G$ is any closed subgroup, the homogeneous variety $G/H$ is thus unirational: its field $k(G/H)$ of rational functions is contained in a purely transcendental extension of $k$ (namely, $k(G)$). It is thus natural to consider the following:
\[RatPbm\] Let $G$ be a connected linear algebraic group over the algebraically closed field $k$, and let $H \subseteq G$ be a closed subgroup. Is the homogeneous variety $G/H$ rational? Equivalently: is the field $k(G/H)$ of rational functions on $G/H$ a purely transcendental extension of $k$?
This is a long-standing question in algebraic geometry, known as the *rationality problem*. When $H$ is not assumed to be connected, the answer is negative in general; see remark \[rmkSaltman\] below. When $H$ is connected, the rationality problem in the generality of \[RatPbm\] is open even when ${{\operatorname{char}}}(k)=0$. The problem was mentioned (possibly for the first time) in [@Hab], in which it was suggested that the answer is negative in general even when $H$ is connected, although no counter-example is known to date. However, affirmative answers have been established in several cases; for instance, the rationality of $G/H$ when $\dim(G/H) \leqslant 4$ and ${{\operatorname{char}}}(k)=0$ was proved in [@MU Lemma 1.15].
A more general form of the rationality problem is concerned with the rationality of the field $k(X)^H$ when $X$ itself is a rational variety on which a linear group $H$ acts. Two typical versions of this problem have been considered in the literature: namely, when $X$ is the underlying vector space of a finite dimensional representation of $H$, and when $X$ is the underlying variety of a connected group $H$ acting on itself by conjugation; see [@CTS] for a survey of the former, and [@CTKPR] for works on the latter. The variant \[RatPbm\] of the rationality problem we consider in this paper amounts to the case when $X$ is the underlying variety of a connected group $G$ on which a subgroup $H$ acts by right multiplication. Our goal is to give several more criteria under which one can establish an affirmative answer to this variant of the rationality problem, and to the extent possible, in a characteristic-free manner.
Here, we give an overview of the main results of this paper. We refer the reader to theorems \[Th0\], \[ThA\], \[ThB0\] and \[ThB\] for the precise statements.
In theorem \[Th0\], we show that $G/H$ is rational for any closed subgroup $H$ contained in a Borel subgroup $B$ of $G$; in particular, this is so whenever $G$ or $H$ is connected and solvable (for then $H$ is necessarily contained in some Borel subgroup of $G$). This result holds for any characteristic of $k$ and is probably folklore to the experts, although it seems not to have appeared in the literature. It is analogous to a theorem of Miyata (cf. [@Mi]) asserting that the field $k(V)^H$ is a purely transcendental extension of $k$ when $H$ is a linear group and $V$ is a finite dimensional representation of $H$ which is triangularizable. We establish theorem \[Th0\] via a certain splitting principle for the quotient map $G \dasharrow G/B$ (cf. cor. \[solv split\]) which was “morally speculated” by Prof. V. Popov in a conversation with the second author. The main tools needed are some classical results of Rosenlicht (cf. lemmas \[tSS\] and \[Rosenlict generic section\]).
In theorem \[ThA\], we show that $G/H$ is rational for any connected closed subgroup $H$ of maximal rank in $G$, if the maximal semisimple quotient of $G$ is isogenous to a product of almost-simple groups of type $A$, type $C$ (when ${{\operatorname{char}}}(k) \neq 2$), or type $B_3$ or $G_2$ (when ${{\operatorname{char}}}(k) = 0$). Here, the main ingredient is the Borel-de Siebenthal algorithm (cf. [@BdS Th. 6] and the algebraic group version in [@Le prop. 6.6]) classifying maximal connected reductive subgroups of maximal rank in a given connected semisimple group. We also employ general structural results of algebraic groups such as the Bruhat decomposition (cf. lemma \[parabolic section\]), the theorem of Borel-Tits (cf. lemma \[Borel-Tits\]), as well as properties of special groups (in the sense of Serre). Our theorem \[ThA\] may be compared with [@CTKPR Th. 0.2], where the authors prove, among other things, that for an almost-simple group $G$ of type $A$ or $C$, acting on itself by conjugation, the field extension $k(G)/k(G)^{G}$ is purely transcendental.
In sections \[low dim homog\] and \[low dim homog ctd\], we assume ${{\operatorname{char}}}(k)=0$ and use geometric arguments (cf. lemma \[finbir\]) to show the rationality of certain low dimensional homogeneous varieties $G/H$, our starting point being the classical theorems of Lüroth and Castelnuovo (cf. lemma \[dim1or2\]) asserting the rationality of unirational varieties of dimension $\leqslant 2$ (over our algebraically closed field $k$ of characteristic zero). We show in theorem \[ThB0\] that $G/H$ is rational whenever $\dim(G/H) \leqslant 5$, and in theorem \[ThB\] that if $H$ is connected, $G/H$ is rational whenever $\dim(G/H) \leqslant 10$. We follow the approach in [@MU Lemma 1.15] of reducing to the case when $G$ is semisimple, but we utilize new ingredients such as the geometry of the big cell and the structure of the centralizers of subtori in a connected reductive group (cf. [@Hu §28.5, §22.4, §26.2]).
\[rmkSaltman\] Our results provide affirmative answers to the rationality problem \[RatPbm\] in the respective situations, but except for low dimensional homogeneous varieties (i.e. when $\dim(G/H) \leqslant 5$ as in theorem \[ThB0\]), we require that the closed subgroup $H \subseteq G$ be connected. This is not surprising, since there are examples of finite subgroups $H$ in a product $G$ of general linear groups for which $G/H$ is not rational. Indeed, by the work of Saltman [@Sa Th. 3.6] in the context of the Noether problem, one knows that for each prime $p$, there exist finite $p$-groups $H$ with the following property: for any algebraically closed field $k$ with ${{\operatorname{char}}}(k)\neq p$, if $V$ denotes the vector space of the regular representation of $H$ over $k$, and if $k(V)^H$ denotes the subfield of $H$-invariants in the field $k(V)$ of rational functions on $V$ over $k$, then $k(V)^H$ is not a purely transcendental extension of $k$, so the variety $V/H$ is not rational. If we decompose $V \cong \bigoplus_i V_i^{\oplus m_i}$ into irreducible representations $V_i$ of $H$, then $m_i = \dim_k V_i$, and the linear representation of $H$ on the isotypic component $V_i^{\oplus m_i}$ yields an action of $H$ by right multiplication on $G_i := {{\operatorname{GL}}}_{m_i}(k)$. Thus $V$ contains an $H$-stable open subvariety isomorphic to $G := \prod_i G_i$. Since $V/H$ is not rational, $G/H$ is not rational either.
As we are mainly interested in the birational properties of homogeneous varieties, we consider only rational actions and rational quotients (unless explicitly stated otherwise). A (right) *rational action* of a linear algebraic group $H$ on an irreducible variety $X$ (cf. [@BSU §2.3]) is a group homomorphism $H \to {{\operatorname{Aut}}}(k(X))$ from $H$ to the group ${{\operatorname{Aut}}}(k(X))$ of birational automorphisms of $X$, such that the resulting rational map $a : X \times H \dasharrow X$ is (defined and) regular on an open dense subset of $X \times H$. Given such a rational action, the *rational quotient of $X$ by $H$* (in the sense of Rosenlicht, cf. [@Ro56 Th. 2]) is any irreducible variety whose field of rational functions is identified with $k(X)^H$; i.e. it is a $k$-variety $X/H$ characterized up to birational equivalence by the equality $k(X/H) = k(X)^H$ of rational function fields. [^2] The $k$-inclusion $k(X/H) \subseteq k(X)$ of fields induces a dominant rational map $X \dasharrow X/H$, called the *rational quotient map*, which is $H$-equivariant with respect to the trivial action on $X/H$, and has the universal property that any dominant rational map from $X$ which is constant on general $H$-orbits in $X$ factors through it. Thus, if $G$ is a connected linear algebraic group and $H \subseteq G$ is a closed subgroup, the homogeneous space $G/H$ of $H$-cosets in $G$ is regarded as the rational quotient of the variety $G$ by the right multiplication action of $H$. Similarly, if $K \subseteq G$ is another closed subgroup, the space of double cosets $K {{\backslash}}G / H$ is both the rational quotient of the variety $K {{\backslash}}G$ by the right multiplication action of $H$, as well as the rational quotient of the variety $G/H$ by the left multiplication action of $K$.
[**Acknowledgement.**]{}
The authors heartily thank our friends and colleagues, and especially the referee of an earlier version of this paper, for the remarks \[rmkSaltman\] and \[rmkBGBHrational\], and also prop. \[PropB\](d), which allows the case of type $C$ in theorem \[ThA\] to be included. The second author would like to offer his thanks to M. Brion for valuable discussions about the rationality of $G/H$ when $G$ is solvable and for reminding him of the reference to [@GIT] in the argument for Lemma \[Borel-Tits\]. He also thanks M. Chen for the warm hospitality during his visit to Fudan University while working on this paper. He is partially supported by an ARF of NUS.
Quotients by subgroups contained in a Borel
===========================================
In this section, we work over an algebraically closed base field $k$ of arbitrary characteristic, and we consider the rationality problem \[RatPbm\] for $G/H$ when $H$ is contained in a Borel subgroup of $G$. Our goal is to establish theorem \[Th0\]. We first review some classical results and arguments.
\[Rosenlicht\] \[tSS\] Let $H$ be a linear algebraic group acting rationally on a variety $X$, and let $K \lhd H$ be a closed normal subgroup. Then $H' := H/K$ acts rationally on the quotient variety $X' := X/K$, and $X'/H'$ is (naturally) birational to $X/H$.
See [@Ro56 Th. 5]. In particular, if $K \lhd H$ lies in the kernel of the action of $H$ on the field $k(X)$ (i.e. $K$ acts trivially on an open dense subset of $X$), then $X' = X/K$ is (birational to) $X$ itself, and hence $X/H$ is birational to $X/H'$. In other words, in forming the rational quotient $X/H$, we may replace $H$ by its image in the birational automorphism group ${{\operatorname{Aut}}}(k(X))$ of $X$. This is also clear from the characterization of the rational quotient by its rational function field.
Let $H$ be a linear algebraic group acting rationally on a variety $X$. The action is called *generically free* iff there exists an open dense subset $U \subseteq X$ such that for every $x \in U$, the stabilizer subgroup ${}_xH := \{h \in H \,|\, x \cdotp h = x\}$ is trivial. An easy example of a generically free action is the right multiplication action of $H$ on a connected linear algebraic group $G$ which contains $H$ as a closed subgroup. A generically free action is necessarily *generically faithful*, i.e. the homomorphism $H \to {{\operatorname{Aut}}}(k(X))$ is injective; but the converse does not hold in general. However, one has the following:
\[Demazure generic freeness\] Suppose $H$ is a torus acting rationally on a variety $X$. If the action is generically faithful, then it is generically free. In particular, one has $\dim X = \dim(X/H) + \dim H$.
See [@De §1, Prop. 10].
\[spQ\] [^3] Let $M$ be a linear algebraic group acting rationally on a variety $X$. Suppose that the action of $M$ on $X$ is generically free, and that the rational quotient map $\pi : X \dasharrow X/M$ admits a rational section $s : X/M \dasharrow X$. Then there exists a birational map $$f
\ :\
(X/M) \times M
\dashrightarrow
X
\qquad
\text{which is $M$-equivariant}$$ with respect to the natural right action of $M$ on the domain ${{\operatorname{Dom}}}(f)$ of $f$ via multiplication on the second factor $M$. In particular, for any closed subgroup $H \subseteq M$, $X/H$ is birational to $(X/M) \times (M/H)$.
This is the birational analogue of the classical statement that a principal bundle with a section is trivial.
We assume that $M$ acts on $X$ from the right. Replacing $X$ by another birational model, we may assume (cf. [@Ro56 Th. 1]) that the action is biregular, so that one has a quotient morphism $\pi : X \to X/M$. Since the singular locus of $X$ is stable under the action of $M$, and the action is generically free, we may replace $X$ by an open dense subset and assume that $X$ is non-singular and that the action is in fact free, i.e. the stabilizer subgroup ${}_xM$ is trivial for every $x \in X$. Likewise, shrinking $X/M$ (and $X$ correspondingly) if necessary, we also assume that $X/M$ is non-singular. Further replacing $X/M$ by the domain ${{\operatorname{Dom}}}(s)$ of the rational section $s$ and $X$ by $\pi^{-1}({{\operatorname{Dom}}}(s))$, we may assume that $s : X/M \to X$ is a regular section, i.e. a morphism such that $\pi \circ s = {{\operatorname{id}}}_{X/M}$. Thus we have a bijective morphism $$f
\ :\
(X/M) \times M
\longrightarrow
X
,
\qquad
(y,m)
\mapsto
s(y) \cdotp m$$ which is clearly $M$-equivariant: $f(y,m) \cdotp m'
=
f(y, m \cdotp m')
$. Since $X/M$ and $X$ are normal varieties, we may apply [@Bo Lemma 6.14(ii)] to infer that $f$ is in fact an isomorphism. This shows the main assertion of the lemma. The final claim follows by passing to the quotient by $H$.
\[Rosenlicht’s generic section theorem\] \[Rosenlict generic section\] Let $M$ be a connected solvable group acting rationally on a variety $X$. Then the rational quotient map $\pi : X \dasharrow X/M$ admits a rational section $s : X/M \dasharrow X$.
See [@Ro56 Th. 10].
\[solv split\] Let $G$ be a connected linear algebraic group, and let $H \subseteq G$ be any closed subgroup contained in a Borel subgroup $B$ of $G$. Then $G/H$ is birational to $(G/B) \times (B/H)$.
Since $B$ is connected and solvable, the quotient map $G \dasharrow G/B$ has a rational section by lemma \[Rosenlict generic section\]. The right-multiplication action of $B$ on $G$ is generically free, so lemma \[spQ\] is applicable and yields the corollary.
\[Sol\] Let $M$ be a connected solvable group acting rationally on a variety $X$. Then $X$ is birational to $(X/M) \times {{\mathbb{P}}}^d$ for some $d \leqslant \dim M$. In particular, if $X/M$ is rational, then so is $X$.
Since $M$ is connected solvable, we can find a sequence of connected normal closed subgroups $M = M_r \rhd M_{r-1} \rhd \cdots \rhd M_1 \rhd M_0 = \{1\}$ such that the subquotients $M_{i+1}/M_i$ are 1-dimensional groups, isomorphic to either ${{{{{\mathbb{G}}}}_{\textup{m}}}}$ or ${{{{{\mathbb{G}}}}_{\textup{a}}}}$. For each $i \in \{0,\ldots,r{-}1\}$, we will show that $X/M_i$ is birational to $X/M_{i+1}$ or $(X/M_{i+1}) \times {{\mathbb{P}}}^1$; by descending induction on $i$, it would then follow that $X = X/M_0$ is birational to $(X/M_r) \times {{\mathbb{P}}}^d$ for some $d \leqslant r = \dim M$, whence the lemma.
Consider the rational action of $H_i := M_{i+1}/M_i$ on the variety $X_i := X/M_i$, and let $H_i'$ denote the image of $H_i$ in ${{\operatorname{Aut}}}(k(X_i))$; by lemma \[tSS\], $X/M_{i+1}$ is naturally birational to $X_i/H_i'$, and $H_i'$ is either trivial or isomorphic to ${{{{{\mathbb{G}}}}_{\textup{m}}}}$ or ${{{{{\mathbb{G}}}}_{\textup{a}}}}$. If $H_i' \cong {{{{{\mathbb{G}}}}_{\textup{m}}}}$, by lemma \[Demazure generic freeness\], the action of $H_i'$ on $X_i$ is generically free, and by lemma \[Rosenlict generic section\], the rational quotient map $X_i \dasharrow X_i/H_i'$ admits a rational section $X_i/H_i' \dasharrow X_i$, whence by lemma \[spQ\], $X_i$ is birational to $(X_i/H_i') \times H_i'$. If $H_i' \cong {{{{{\mathbb{G}}}}_{\textup{a}}}}$, we replace $X$ by a suitable birational model and assume that it is affine, and then apply [@Sp Prop. 14.2.2] to see that $X_i$ is birational to $(X_i/H_i') \times H_i'$, which proves what we want.
\[Sol quotient\] Let $M$ be a connected solvable group. Then for any closed subgroup $H \subseteq M$, the quotient variety $M/H$ is rational.
Apply lemma \[Sol\] to the natural left action of $M$ on $X := M/H$, and note that the rational quotient $M \backslash X$ of $X$ by $M$ is a point.
In lemma \[Sol\], the connected solvable group $M$ acts on a variety $X$ which is not necessarily a group; this mildly generalizes [@Ro56 Cor. 1 to Th. 10], and will be very convenient for us later on. However, cor. \[Sol quotient\] which is deduced from lemma \[Sol\] does not give the best result: the quotient variety $M/H$ there is in fact isomorphic to a product of copies of ${{{{{\mathbb{G}}}}_{\textup{a}}}}$ and ${{{{{\mathbb{G}}}}_{\textup{m}}}}$; see [@Ro63 Theorem 5]. Both results, as well as lemma \[Rosenlict generic section\], hold for split connected solvable linear algebraic groups over an arbitrary base field, by essentially the same argument.
\[Th0\] Let $G$ be a connected linear algebraic group, and let $H \subseteq G$ be any closed subgroup contained in a Borel subgroup of $G$. Then $G/H$ is a rational variety.
Let $B$ be a Borel subgroup of $G$ containing $H$. Then $G/H$ is birational to $(G/B) \times (B/H)$ by cor. \[solv split\]. The quotient flag variety $G/B$ is rational (well-known; see also lemma \[parabolic section\] below), and $B/H$ is rational by cor. \[Sol quotient\]. Hence $G/H$ is rational.
We record below some reduction arguments which will be useful later on.
\[mod radical\] Let $G$ be a connected linear algebraic group and let $H \subseteq G$ be any closed subgroup. Let $R := R(G)$ be the solvable radical of $G$, let $G' := G/R$ be the maximal semisimple quotient of $G$, and let $H' := H/(H \cap R)$ be the image of $H$ in $G'$. Then $G/H$ is birational to $G'/H' \times {{\mathbb{P}}}^s$ for some $s \leqslant \dim R$. In particular, if $G'/H'$ is rational then so is $G/H$.
By lemma \[Sol\] applied to the natural left action of $R$ on $G/H$, we see that $G/H$ is birational to $(G/HR) \times {{\mathbb{P}}}^s$ for some $s \leqslant \dim R$. The result follows from the observation that $G/HR$ is isomorphic to $(G/R)/(H/H \cap R) = G'/H'$.
\[no-name-lemma\] For $i=1,2$, let $M_i$ be a connected solvable group acting rationally on a variety $X_i$. Assume that $X_1/M_1$ is birational to $X_2/M_2$, and that $\dim X_1 \leqslant \dim X_2$. Then $X_2$ is birational to $X_1 \times {{\mathbb{P}}}^d$ where $d := \dim X_2 - \dim X_1$. In particular, if $X_1$ is rational, then so is $X_2$.
By lemma \[Sol\], $X_i$ is birational to $(X_i/M_i) \times {{\mathbb{P}}}^{d_i}$ for some $d_i \geqslant 0$. Since $X_1/M_1$ is birational to $X_2/M_2$ by assumption, it follows that $d = d_2 - d_1$ and that $X_2$ is birational to $X_1 \times {{\mathbb{P}}}^d$.
Quotients by connected subgroups of maximal rank
================================================
We continue to work over an algebraically closed base field $k$, of arbitrary characteristic unless otherwise stated. Let $G$ be a connected linear algebraic group. The maximal semisimple adjoint quotient ${{\overline{G}}}$ of $G$ decomposes as a direct product ${{\overline{G}}}= {{\overline{G}}}_1 \times \cdots \times {{\overline{G}}}_\ell$ of connected adjoint almost-simple groups; we refer to these ${{\overline{G}}}_i$’s as the *adjoint factors of $G$*. Our main result in this section is:
\[ThA\] Let $G$ be a connected linear algebraic group, and let $H \subseteq G$ be a connected closed subgroup of maximal rank in $G$. Assume that each adjoint factor of $G$ is of type $A$, type $C$ (when ${{\operatorname{char}}}(k) \neq 2)$, or type $B_3$ or $G_2$ (when ${{\operatorname{char}}}(k) = 0)$. Then $G/H$ is a rational variety.
We prove this in \[ThAproof\]; the essential case is when $G$ is simply-connected and almost-simple. To reduce to this case, let $G_i$ denote the simply-connected cover of the adjoint factor ${{\overline{G}}}_i$, let ${{\overline{H}}}_i$ denote the image of $H$ in ${{\overline{G}}}_i$, and let $H_i$ denote the preimage of ${{\overline{H}}}_i$ in $G_i$.
\[reduction to adjoint factors\] With the above notation, each $H_i$ is a subgroup of maximal rank in $G_i$, and $G/H$ is birational to $(G_1/H_1) \times \cdots \times (G_\ell/H_\ell) \times {{\mathbb{P}}}^s$ for some $s \leqslant \dim R$, where $R := R(G)$ is the solvable radical of $G$. In particular, if each $G_i/H_i$ is rational then so is $G/H$.
Let $G' := G/R$ be the maximal semisimple quotient of $G$. The image $H'$ of $H$ in $G'$ is a subgroup of maximal rank in $G'$. By lemma \[mod radical\], $G/H$ is birational to $(G'/H') \times {{\mathbb{P}}}^s$ for some $s \leqslant \dim R$.
Next, let $Z(G')$ be the centre of $G'$, let ${{\overline{G}}}:= G'/Z(G')$ be the adjoint quotient of $G'$, and consider the quotient isogeny $q : G' \to {{\overline{G}}}$. The map ${{\overline{H}}}\mapsto H' := q^{-1}({{\overline{H}}})$ is a bijection between the collection of connected subgroups of maximal rank of ${{\overline{G}}}$ and the collection of connected subgroups of maximal rank of $G'$ (cf. [@SGA3-2 Exp. XII Cor. 7.12]). Since $H'$ contains $Z(G')$, one obtains an isomorphism between $G'/H'$ with ${{\overline{G}}}/{{\overline{H}}}$. Applying the same argument to the quotient isogeny ${{\tilde{G}}}\to {{\overline{G}}}$ from the simply-connected cover ${{\tilde{G}}}$ of ${{\overline{G}}}$, one obtains an isomorphism between ${{\tilde{G}}}/{{\tilde{H}}}$ with ${{\overline{G}}}/{{\overline{H}}}$, where ${{\tilde{H}}}$ is the preimage of ${{\overline{H}}}$ in ${{\tilde{G}}}$.
In our notation above, we can thus write ${{\tilde{G}}}$ as the direct product ${{\tilde{G}}}= G_1 \times \cdots \times G_\ell$; the subgroup ${{\tilde{H}}}$, which is of maximal rank in ${{\tilde{G}}}$, is then of the form ${{\tilde{H}}}= H_1 \times \cdots \times H_\ell$, and each $H_i$ is a subgroup of maximal rank in $G_i$ (cf. [@BdS §3], or the algebraic group version in [@Le Prop. 4.1]). Thus ${{\tilde{G}}}/{{\tilde{H}}}$ is isomorphic to $(G_1/H_1) \times \cdots \times (G_\ell/H_\ell)$. The lemma follows.
To proceed further, let us first review some preliminary results pertaining to the rationality of $G/H$ in the greater generality when $G$ is a connected reductive group and $H \subseteq G$ is any closed (connected) subgroup.
\[parabolic section\] Let $G$ be a connected reductive group, and let $P \subseteq G$ be a parabolic subgroup. Then $G/P$ is rational, and the rational quotient map $G \dasharrow G/P$ admits a rational section.
This is a standard consequence of the Bruhat decomposition; we give a detailed proof here for the sake of clarity.
Let $B = T \cdot U \subseteq G$ be a Borel subgroup contained in $P$, with maximal torus $T$ and unipotent radical $U$; write $B^- = U^- \cdot T$ for the opposite Borel subgroup. The parabolic subgroup $P$ is then the standard parabolic subgroup $P_I$ associated to a subset $I$ of the set of simple roots of $G$ relative to $T$. By the Bruhat decomposition, $G$ contains the Zariski-dense open subset (big cell) $U^-\cdot T \cdot U$, and $P$ contains the Zariski-dense open subset $U_I^- \cdot T \cdot U$, where $U_I^-$ denote the subgroup of $U^-$ generated by the 1-dimensional unipotent subgroups associated to the negative roots belonging to the sub-root lattice spanned by $I$ (cf. [@Hu §30.1]). Write $V_I^-$ for the product variety of the 1-dimensional unipotent subgroups associated to the negative roots in the complement of the sub-root lattice spanned by $I$; thus $V_I^-$ is a rational variety, and one has $U^- \cong V_I^- \times U_I^-$ as varieties. Then the Bruhat decomposition above shows that the natural multiplication map $$f
\ :\
V_I^- \times P
\dasharrow
G
,
\qquad
(v,p)
\mapsto
v \cdotp p$$ is a birational map. With respect to the trivial action on $V_I^-$ and the natural right action of $P$ on $P$ and on $G$, the map $f$ is $P$-equivariant. Thus it induces the birational map $V_I^- \dasharrow G/P$, showing that $G/P$ is rational. The inverse birational map yields the desired rational section of $G \dasharrow G/P$.
\[subparabolic\] Let $G$ be a connected reductive group, and let $H \subseteq G$ be any closed subgroup contained in a parabolic subgroup $P$ of $G$. Then $G/H$ is birational to $(G/P) \times (P/H)$. In particular, if $P/H$ is rational, then so is $G/H$.
The first assertion is deduced from lemma \[spQ\] by the same argument as in the proof of cor. \[solv split\], using lemma \[parabolic section\] above instead of lemma \[Rosenlict generic section\]. The second assertion then follows by another use of lemma \[parabolic section\].
\[Borel-Tits\] \[Borel-Tits\] Let $G$ be a connected reductive (resp. connected semisimple) group. Suppose $H \subseteq G$ is a connected closed subgroup which is not contained in any proper parabolic subgroup of $G$. Then $H$ is reductive (resp. semisimple).
If $H$ is not reductive, its unipotent radical $U := R_u(H)$ is a non-trivial normal subgroup of $H$. Since $G$ is connected reductive, there exists (cf. [@Hu §30.3 Cor. A]) a parabolic subgroup $P$ of $G$ with $N_G(U) \subseteq P$ and $U \subseteq R_u(P)$. The first inclusion gives $H \subseteq P$; the second inclusion forces $P$ to be a proper parabolic subgroup, contradicting our hypothesis on $H$. Hence $H$ is reductive.
Now suppose $G$ is connected semisimple. If $H$ is not semisimple, its center $Z(H)$ is of positive dimension, and we may choose a non-trivial 1-parameter subgroup $\lambda : {{{{{\mathbb{G}}}}_{\textup{m}}}}\to G$ of $G$ with image in $Z(H)$. By [@GIT Def. 2.3/Prop. 2.6], there is a unique closed subgroup $P(\lambda) \subseteq G$ characterized by the property that $$\gamma \in P(\lambda)
\ \iff\
\lambda(t)
\,
\gamma
\,
\lambda(t^{-1})
\quad
\begin{array}[t]{l}
\text{has a specialization in $G$}
\\
\text{when $t \in {{{{{\mathbb{G}}}}_{\textup{m}}}}$ specializes to $0$}
;
\end{array}$$ moreover, one knows that $P(\lambda)$ is a parabolic subgroup of $G$, and that the image of $\lambda$ is contained in the solvable radical of $P(\lambda)$. As $G$ is semisimple, this last fact forces $P(\lambda)$ to be a proper parabolic subgroup of $G$. But since $H$ centralizes $\lambda$ by construction, the characterizing property of $P(\lambda)$ shows that $H \subseteq P(\lambda)$, contradicting our hypothesis on $H$. Thus $H$ is semisimple.
Recall that an algebraic group $M$ over $k$ is called *special* (in the sense of Serre) iff $H^1(K,M) = \{1\}$ for every field $K$ containing $k$. By the classification theorem of Serre and Grothendieck for special groups (see for instance [@Re Th. 5.4]), one knows that a connected semisimple group $M$ is special if and only if it is a direct product of simply-connected almost-simple groups of type $A_n$ or $C_n$ (i.e. ${{\operatorname{SL}}}_{n+1}$ or ${{\operatorname{Sp}}}_{2n}$). Special groups enjoy the following important property: if $X$ is an irreducible variety on which a special group $M$ acts generically freely, the rational quotient map $X \dasharrow X/M$ admits a rational section (cf. [@Re Lemma 5.2 and Prop. 5.3]).
\[PropB\] Let $G$ be isomorphic to ${{\operatorname{Sp}}}_{2n}$ for some $n \geqslant 2$, and let $M \subsetneq G$ be a maximal connected proper subgroup which is semisimple and of maximal rank. Then:
- $M$ is a product of two simply-connected almost-simple groups of type $C$ (i.e. $M$ is $G$-conjugate to ${{\operatorname{Sp}}}_{2m} \times {{\operatorname{Sp}}}_{2n-m}$ for some $0 < m < n$), and hence it is a special group;
- the rational quotient map $G \dasharrow G/M$ admits a rational section;
- for any closed subgroup $H \subseteq G$ contained in $M$, $G/H$ is birational to $(G/M) \times (M/H)$;
- if ${{\operatorname{char}}}(k) \neq 2$, then $G/M$ is a rational variety.
By the Borel-de Siebenthal algorithm (cf. [@BdS Th. 6] and the algebraic group version in [@Le Prop. 6.6]), one knows that the Dynkin diagram of $M$ is obtained from the extended Dynkin diagram of $G$ by removing a vertex corresponding to a simple root $\alpha$ of $G$ whose corresponding coefficient $n_\alpha$ for the longest root $\alpha_0$ of $G$ is a prime number. Moreover, one has the exact sequence $$1
\rightarrow
\mu(n'_\alpha)
\rightarrow
Z({{\tilde{M}}})
\rightarrow
Z(M)
\rightarrow
1
,$$ where ${{\tilde{M}}}$ is the simply-connected cover of $M$, $n'_\alpha := \frac{|\alpha|^2}{|\alpha_0|^2} \cdot n_\alpha$, and $\mu(n'_\alpha)$ is the group of roots of unity of order $n'_\alpha$. Since $G$ is simply-connected of type $C$, one has $n_\alpha = 2$ and $n'_\alpha = 1$, and the Dynkin diagram of $M$ consists of two connected components both of type $C$. Hence $M = {{\tilde{M}}}$ is simply-connected, and is a product of two simply-connected almost-simple groups of type $C$. Thus $M$ is a special group. The right-multiplication action of $M$ on $G$ is generically free, and hence by the property of $M$ being a special group, (cf. [@Re Lemma 5.2 and Prop. 5.3]), the rational quotient map $G \dasharrow G/M$ has a rational section. Hence by lemma \[spQ\], for any closed subgroup $H$ contained in $M$, $G/H$ is birational to $(G/M) \times (M/H)$. This proves parts (a), (b) and (c).
For part (d), using [@Le Prop. 6.6] again, we see that $M$ is the centralizer of an element of order $2$ (an involution) in $G$; therefore, $G/M$ is a symmetric variety (which makes sense since ${{\operatorname{char}}}(k) \neq 2$ by assumption). It is well-known (cf. [@Sp Th. 4.2, Cor. 4.3]) that a symmetric variety is a spherical variety: the natural left action on $G/M$ by a Borel subgroup $B$ of $G$ gives rise to a Zariski-dense open orbit of $G/M$. Consequently, $G/M$ is birational to a quotient variety of $B$, and hence by cor. \[Sol quotient\], it is a rational variety.
\[Proof of theorem \[ThA\]\] \[ThAproof\] We can now establish our main theorem in this section. By lemma \[reduction to adjoint factors\], we are reduced to showing that when $G$ is a connected simply-connected and almost-simple group of type $A$, type $C$ (when ${{\operatorname{char}}}(k) \neq 2)$, or type $B_3$ or $G_2$ (when ${{\operatorname{char}}}(k) = 0)$, and $H \subsetneq G$ is a connected proper closed subgroup of maximal rank in $G$, then $G/H$ is rational. We proceed by induction on the common rank $n$ of $G$ and $H$, the case of $n = 0$ being trivial. Henceforth assume that $n \geqslant 1$, and that our conclusion holds for groups of the stated types of lower ranks.
Suppose $H$ is contained in some proper parabolic subgroup $P \subsetneq G$. By cor. \[subparabolic\], $G/H$ is birational to $(G/P) \times (P/H)$, and by lemma \[parabolic section\], $G/P$ is rational. If $G$ is of type $A$, $C$ or $G_2$ (resp. type $B_3$), the adjoint factors of $P$ are all of type $A$ (resp. type $A_1$ or $C_2$), and the ranks of these factors are strictly lower than that of $G$. By lemma \[reduction to adjoint factors\] applied to $H \subseteq P$ and our induction hypothesis, we see that $P/H$ is rational, and hence $G/H$ is rational.
If $G$ is of type $A_n$ for $n \geqslant 1$, the Borel-de Siebenthal algorithm shows that every connected proper subgroup $H \subsetneq G$ of maximal rank in $G$ is contained in some proper parabolic subgroup of $G$; our proof of theorem \[ThA\] is therefore complete in this case.
If $G$ is of type $C_n$ for $n \geqslant 2$ and ${{\operatorname{char}}}(k) \neq 2$, we are reduced to the case when $H \subsetneq G$ is not contained in any proper parabolic subgroup of $G$ and is therefore semisimple by lemma \[Borel-Tits\]. We let $M \subsetneq G$ be a maximal connected proper subgroup containing $H$; thus $M$ is also of maximal rank in $G$, and is not contained in any proper parabolic subgroup of $G$, and by lemma \[Borel-Tits\], $M$ is semisimple. By prop. \[PropB\], $G/H$ is birational to $(G/M) \times (M/H)$, and $G/M$ is rational (because ${{\operatorname{char}}}(k) \neq 2$); moreover, the adjoint factors of $M$ are all of type $C$, and the ranks of these factors are strictly lower than that of $G$. By lemma \[reduction to adjoint factors\] applied to $H \subseteq M$ and our induction hypothesis, we see that $M/H$ is rational, and hence $G/H$ is rational.
In the remaining cases, $G$ is of type $B_3$ or $G_2$ with ${{\operatorname{char}}}(k) = 0$, and $H \subsetneq G$ is not contained in any proper parabolic subgroup of $G$; again, $H$ is semisimple by lemma \[Borel-Tits\]. The rationality of $G/H$ is then established directly using results in section \[low dim homog ctd\], and we defer the proof of these cases to cor. \[B23C3G2\]. The proof of theorem \[ThA\] is thus completed — modulo the use of cor. \[B23C3G2\] for the low dimensional cases.
It is possible that the assumptions in theorem \[ThA\] on the adjoint factors of $G$ can be removed altogether. This would be the case if the assertion of prop. \[PropB\](b) can be established for almost-simple groups of any type; our induction argument in \[ThAproof\] would then yield the stable-rationality of $G/H$ in general. In turn, the rationality of $G/H$ in general would be reduced to the assertions of prop. \[PropB\](d) for almost-simple groups of any type; i.e. to the rationality of $G/M$ when $G$ is almost-simple (of any type) and $M \subsetneq G$ is a maximal connected proper subgroup of maximal rank in $G$ but which is not contained in any proper parabolic subgroup of $G$. As explained in \[ThAproof\], such an $M$ is semisimple, and its (finitely many) possibilities are determined by the Borel-de Siebenthal algorithm.
Low dimensional homogeneous varieties {#low dim homog}
=====================================
From now on, we work over an algebraically closed base field $k$ of characteristic 0. In this and the next section, we apply geometric methods to study the rationality problem \[RatPbm\]. Our goal is to establish theorems \[ThB0\] and \[ThB\] asserting the rationality of all homogeneous varieties $G/H$ of sufficiently low dimensions, thereby answering the rationality problem \[RatPbm\] affirmatively in these cases. In this section, we place no restriction on the connectedness of $H$, while in section \[low dim homog ctd\], we extend our rationality results further when $H$ is assumed to be connected. The following argument will be used several times in both sections.
\[finbir\] Let $H_1 \subseteq H_2$ be two connected algebraic groups such that $H_2$ (and hence $H_1$) act rationally on an algebraic variety $X$, and let $f : X/H_1 \dasharrow X/H_2$ be the dominant rational quotient map. The following are equivalent:
- $f$ is birational;
- $f$ is generically injective;
- $f$ is generically finite;
- $\dim(X/H_1) = \dim(X/H_2)$;
- for all points $x \in X$ in general position, its orbits $x \cdotp H_1$ and $x \cdotp H_2$ under the action of $H_1$ and $H_2$ have the same Zariski closure.
The implications $\text{(a)}
\Rightarrow
\text{(b)}
\Rightarrow
\text{(c)}
\Rightarrow
\text{(d)}
$ are clear. To show the other implications, we replace $X$ by another birational model and assume that both $H_1 \subseteq H_2$ act regularly on $X$ (cf. [@Ro56 Th. 1]). For a point $x \in X$ in general position, its orbit $x \cdotp H_i$ under the action of $H_i$ is an irreducible locally closed subvariety in $X$ of dimension $\dim(x \cdotp H_i) = \dim X - \dim(X/H_i)$. If $\dim(X/H_1) = \dim(X/H_2)$, these orbits are of the same dimension, and since we have the inclusion $x \cdotp H_1 \subseteq x \cdotp H_2$, these orbits have the same Zariski closure in $X$; hence $\text{(d)} \Rightarrow \text{(e)}$. The fiber of $f$ over the point $x \cdotp H_2$ in $X / H_2$ consists of those points $x' \cdotp H_1$ in $X / H_1$ which, when regarded as orbits in $X$, belong to the same Zariski closure as $x \cdotp H_2$ in $X$; hence $\text{(e)} \Rightarrow \text{(b)}$. Finally, recall that (cf. [@Hu §4.6, Th.]) a dominant injective morphism between irreducible varieties induces a finite purely inseparable extension of their function fields. As our base field $k$ is of characteristic 0, we infer that when $f$ is generically injective, it induces a trivial extension of function fields and $f$ is therefore birational; hence $\text{(b)} \Rightarrow \text{(a)}$.
Recall that a unirational curve over any field is rational by Lüroth’s theorem, and a unirational surface over an algebraically closed field of characteristic 0 is rational by Castelnuovo’s rationality criterion. Hence over our algebraically closed base field $k$ of characteristic 0, a unirational variety is rational if its dimension is $\leqslant 2$.
\[dim1or2\] Let $G$ be a connected linear algebraic group, and let $B \subseteq G$ be a Borel subgroup of $G$. For any closed subgroup $H \subseteq G$, if $\dim(B \backslash G/H) \leqslant 2$, then $G/H$ is rational.
The underlying variety of $G$ is rational (cf. [@Ch]); the space of double cosets $B \backslash G/H$, being dominated by $G$, is therefore a unirational variety. Hence our hypothesis on its dimension implies that $B \backslash G/H$ is rational. By lemma \[Sol\] applied to the left action of $B$ on $G/H$, it now follows that $G/H$ is rational.
\[TB3\] Let $G$ be a connected semisimple group with maximal torus $T$, and let $B \subseteq G$ be a Borel subgroup of $G$ containing $T$. For any closed subgroup $H \subseteq G$, one has $\dim(B \backslash G/H) \leqslant \dim(T \backslash G/H)$; and if equality holds, then $G/H$ is rational.
Let $U$ be the unipotent radical of $B$. The inclusion of $T$ in $B = T \ltimes U$ induces a dominant rational map $T \backslash G/H \dasharrow B \backslash G/H$, so one always has $\dim(B \backslash G/H) \leqslant \dim(T \backslash G/H)$. Assume that equality holds. By lemma \[finbir\] applied to the left action of $T$ and $B$ on $G/H$, we see that for a point $x \in G/H$ in general position, its orbits $T \cdotp x$ and $B \cdotp x$ under the action of $T$ and $B$ have the same Zariski closure in $G/H$. Since $B = U \cdot T$ and since $G$ contains the Zariski-dense open subset (big cell) $U^-\cdotp T \cdotp U$ (cf. [@Hu §28.5]), $$G/H = G \cdotp x =
\overline{U^- B \cdotp x} =
\overline{U^- T \cdotp x} =
\overline{B^- \cdotp x} .$$ Hence $G/H$ is birational to $B^-/B^-_{x}$ where $B^-_{x}$ the stabilizer of $x$ in $B^-$; and since $B^-$ is connected solvable, it follows from cor. \[Sol quotient\] that this is rational.
\[ThB0\] Let $G$ be a connected linear algebraic group, and let $H \subseteq G$ be any closed subgroup. If $\dim(G/H) \leqslant 5$, then $G/H$ is a rational variety.
By lemma \[mod radical\], we may replace $G$ by its maximal semisimple quotient $G'$ and $H$ by its image $H'$ in $G'$; the rationality of $G'/H'$ implies that of $G/H$, but the dimension of $G'/H'$ can only be at most that of $G/H$. Henceforth, we assume $G$ is semisimple.
Let $T$ be a maximal torus of $G$, and let $B$ be a Borel subgroup of $G$ containing $T$. If $\dim T \leqslant 1$, the semisimple group $G$ is either trivial or of rank 1, isogenous to ${{\operatorname{SL}}}_2$; in either case, we see that $$\dim(B \backslash G/H)
\ \leqslant\
\dim(B \backslash G)
\ \leqslant\
1
,$$ and the rationality of $G/H$ follows from lemma \[dim1or2\]. Henceforth we assume $\dim T \geqslant 2$.
Replacing $G$ further by its image in ${{\operatorname{Aut}}}(G/H)$ if necessary, we may also assume that the natural left action of $G$ on $G/H$ is generically faithful. Applying lemma \[Demazure generic freeness\] to the left action of $T$ on $G/H$, we have $\dim(G/H)
=
\dim(T \backslash G/H) + \dim T
$. This is $\leqslant 5$ by hypothesis, so $\dim(T \backslash G/H) \leqslant 3$. But as one always has $\dim(B \backslash G/H) \leqslant \dim(T \backslash G/H)$, this means that $$\text{either}
\qquad
\dim(B \backslash G/H) \leqslant 2
\qquad
\text{or}
\qquad
\dim(B \backslash G/H) = \dim(T \backslash G/H) = 3
,$$ and the rationality of $G/H$ follows from lemma \[dim1or2\] and lemma \[TB3\] respectively.
Low dimensional homogeneous varieties, continued {#low dim homog ctd}
================================================
In this section, we consider the rationality of $G/H$ when $H \subseteq G$ is a *connected* closed subgroup. We still work over an algebraically closed base field $k$ of characteristic 0. In the series of lemmas below leading up to the main theorem \[ThB\] of this section, we adopt the following hypotheses and notation.
\[notA\] Let $G$ be a connected semisimple group, and let $H \subseteq G$ be a connected closed subgroup. We fix once and for all:
\
------------- ------------------------------------------------------------------------------------------------------
$U(H)$ the unipotent radical of $H$,
$S$ a (reductive) Levi subgroup of $H$, so that $H = S \ltimes U(H)$;
$T_H$ a maximal torus of $S$ (and hence of $H$),
$B_S^{\pm}$ a pair of opposite Borel subgroups of $S$ containing $T_H$,
$U_S^{\pm}$ the unipotent radical of the corresponding $B_S^{\pm}$, so that $B_S^{\pm} = T_H \ltimes U_S^{\pm}$;
$B_H^{\pm}$ the preimage in $H$ of $B_S^{\pm}$, so that $B_H^{\pm} = B_S^{\pm} \ltimes U(H)$,
$U_H^{\pm}$ the preimage in $H$ of $U_S^{\pm}$ so that $U_H^{\pm} = U_S^{\pm} \ltimes U(H)$.
------------- ------------------------------------------------------------------------------------------------------
\
Here, $B_H^{\pm}$ are Borel subgroups of $H$, with unipotent radicals $U_H^{\pm}$, and $T_H$ is a maximal torus of $H$ contained in $B_H$. Having fixed these, we choose:
\
----------- ----------------------------------------------------------------------------------------
$B = B^+$ a Borel subgroup of $G$ containing $B_H$;
$T$ a maximal torus of $B$ (and hence of $G$) containing $T_H$;
$B^-$ the opposite Borel subgroup of $G$ containing $T$, such that $B^- \cap B = T$;
$U^\pm$ the unipotent radical of the corresponding $B^\pm$, so that $B^\pm = T \ltimes U^\pm$.
----------- ----------------------------------------------------------------------------------------
\
We also set $$\begin{aligned}
u(H)
&
\ :=\
\dim U(H)
,
\\
u_G
&
\ :=\
\dim U = \dim U^-
,
&
\quad
t_G
&
\ :=\
\dim T
\quad
\text{(the rank of $G$)}
,
\\
u_H
&
\ :=\
\dim U_H = \dim U_H^-
,
&
\quad
t_H
&
\ :=\
\dim T_H
\quad
\text{(the rank of $H$)}
.
\end{aligned}$$ Thus: $$\begin{aligned}
\dim S
&
\ =\
\dim T_H + 2\dim U_S
&
=\
&
t_H + 2(u_H - u(H))
,
\\
\dim H
&
\ =\
\dim S + \dim U(H)
&
=\
&
t_H + 2 u_H - u(H)
,
\\
\dim G
&
\ =\
\dim T + 2\dim U
&
=\
&
t_G + 2u_G
,
\end{aligned}$$ and hence $$\dim G/H
\ =\
\dim G - \dim H
\ =\
(t_G - t_H) + 2(u_G - u_H) + u(H)
.
\tag{$*$}
\label{dimformula}$$ The subgroup inclusion maps $U_H \subseteq B_H$, $B_H \subseteq H$ and $U^- \subseteq B^-$ induce the dominant rational maps $\alpha$, $\gamma$ and $\varphi$ between the respective spaces of double cosets: $$\begin{matrix}
B^- \backslash G / U_H
&
\ \overset{\textstyle\alpha}{\dashrightarrow}\
&
B^- \backslash G / B_H
&
\ \overset{\textstyle\varphi}{\dashleftarrow}\
&
U^- \backslash G / B_H
\\[-1ex]
&
&
\shortmid
&
&
\\[-1ex]
&
&
\downarrow^{\rlap{$\ \gamma$}}
&
&
\\
&
&
B^- \backslash G / H
&
&
\\
\end{matrix}
\tag{$**$}
\label{dblcosetsdiagram}$$ We will consider these rational maps in the series of lemmas below leading up to theorem \[ThB\].
\[uGH3\] In the situation of \[notA\]:
- $B^- \backslash G/U_H$ is rational, of dimension $u_G - u_H$.
- One has $\dim(B^- \backslash G/H)
\leqslant
\dim(B^- \backslash G/U_H)
$; and if equality holds, then $G/H$ is rational.
- If $u_G - u_H \leqslant 3$, then $G/H$ is rational.
The Bruhat (big cell) decomposition of $G$ shows that $B^- \backslash G / U_H$ contains a Zariski-dense constructible subset $B^- \backslash B^- U / U_H$ which is birational to $U/U_H$; in turn, this is rational by cor. \[Sol quotient\]. Since $\dim(U/U_H) = u_G - u_H$, we see that $B^- \backslash G / U_H$ is rational and of that dimension; this shows part (a).
The asserted inequality of part (b) follows from the existence of the dominant rational map $\gamma \circ \alpha$ in the diagram of \[notA\]. If equality holds, lemma \[finbir\] applied to the right action of $U_H$ and $H$ on $B^- \backslash G$ shows that $\gamma \circ \alpha$ is a birational map, and hence $B^- \backslash G / H$ is rational; by lemma \[Sol\] applied to the left action of $B^-$ on $G/H$, it then follows that $G/H$ is rational.
For part (c), if $\dim(B^- \backslash G/H) \leqslant 2$, the rationality of $G/H$ follows from lemma \[dim1or2\]; henceforth, assume that $\dim(B^- \backslash G / H) \geqslant 3$. Our hypothesis together with parts (a) and (b) then yield $$3
\ \leqslant\
\dim(B^- \backslash G / H)
\ \leqslant\
\dim(B^- \backslash G / U_H)
\ \leqslant\
3
,$$ whence equality holds throughout, and the rationality of $G/H$ follows from part (b) again.
\[rmkBGBHrational\] Although we do not need it below, it is of interest to note that $B^- \backslash G / B_H$ is in fact rational. Indeed, $B^- \backslash G /U_H$ contains the Zariski-open subset $B^- \backslash B^- U / U_H \cong U/U_H$, and with respect to the (regular) action of $T_H$ on $U/U_H$ by conjugation, the isomorphism is $T_H$-equivariant. Thus the quotient $B^- \backslash G /B_H$ of $B^- \backslash G /U_H$ by $T_H$ is birational to the quotient of $U/U_H$ by $T_H$. Moreover, $U/U_H$ is $T_H$-equivariantly isomorphic to the quotient $V := {{\operatorname{Lie}}}(U) / {{\operatorname{Lie}}}(U_H)$ of Lie algebras, on which $T_H$ acts linearly via its adjoint actions on ${{\operatorname{Lie}}}(U)$ and ${{\operatorname{Lie}}}(U_H)$. So $B^- \backslash G /B_H$ is birational to the quotient $V/T_H$. Now choose a basis of $V$ for which the action of $T_H$ is diagonal, and let $V_0 \subseteq V$ denote the open subset on which all coordinates are nonzero. Then $V_0$ is isomorphic to a torus, on which $T_H$ acts by multiplication; thus $V_0/T_H$ is a torus as well. This shows that $V_0/T_H$ and hence $V/T_H$ and $B^- \backslash G /B_H$ are all rational.
\[orbit\] In the situation of \[notA\], one has $\dim(B^- \backslash G / H)
\leqslant
\dim(B^- \backslash G / B_H)
$; and if equality holds, then $G/H$ is rational.
The asserted inequality follows from the existence of the dominant rational map $\gamma$ in the diagram of \[notA\]. If equality holds, then by lemma \[finbir\] applied to the right action of $B_H$ and $H$ on $B^- \backslash G$, we see that $\gamma
:
B^- \backslash G / B_H
\dasharrow
B^- \backslash G / H
$ is birational. By lemma \[uGH3\], the variety $X_1 := B^- \backslash G /U_H$ is rational, of dimension $u_G - u_H$. The torus $M_1 := B_H/U_H$ acts by right multiplication on $X_1$ with quotient $X_1/M_1 = B^- \backslash G / B_H$. On the other hand, the variety $X_2 := G/H$ is of dimension $(t_G - t_H) + 2(u_G - u_H) + u(H)$ by the formula in \[notA\]. The solvable group $M_2 := B^-$ acts by left multiplication on $X_2$ with quotient $M_2 \backslash X_2 = B^- \backslash G / H$. Thus $X_1/M_1$ is birational to $M_2 \backslash X_2$ via $\gamma$; since $\dim X_1 \leqslant \dim X_2$, lemma \[no-name-lemma\] is applicable and shows that the rationality of $X_1$ implies that of $X_2 = G/H$. This completes the proof of the lemma.
\[BU\] In the situation of \[notA\], set $$d
\ :=\
\dim(U^- \backslash G / B_H)
-
\dim (B^- \backslash G / B_H)
.$$
- Let $L$ denote the identity component of the kernel $L'$ of the natural left action of $T$ on $U^- \backslash G/B_H$. Then $d = t_G - t_L$, where $t_L := \dim L$.
- If $d \leqslant 1$, then $G/H$ is rational.
First note that $d \geqslant 0$ by the existence of the dominant rational map $\varphi$ in the diagram of \[notA\]. The torus $L$ is of finite index in the diagonalizable group $L'$ contained in $T$; hence $\dim(T/L') = \dim(T/L) = t_G - t_L$. By construction, the induced left action of $T/L'$ on $U^- \backslash G/B_H$ is generically faithful, and its quotient is $TU^- \backslash G/B_H
=
B^- \backslash G/B_H
$. Hence by lemma \[Demazure generic freeness\], we have $$\dim(U^- \backslash G/B_H)
\ =\
\dim(B^- \backslash G/B_H)
+
\dim(T/L')
,$$ from which it follows that $d = \dim(T/L') = t_G - t_L$. This shows part (a) of the lemma.
Let $D := C_G(L)$ denote the centralizer of $L$ in $G$; it is a connected reductive subgroup of $G$ (cf. [@Hu §26.2, Cor. A; §22.3, Th.]) containing the maximal torus $T$, and a Borel subgroup is given by $B \cap D$ (cf. [@Hu §22.4, Cor.]), whose unipotent radical is $U_D := U \cap D$. We set $u_D := \dim U_D$.
We claim that $U$ is contained in the image $U_D \cdotp U_H$ of multiplying $U_D$ and $U_H$ in $G$. Assuming this for the moment, we infer that $u_G \leqslant u_D + u_H$, and part (b) of the lemma can be deduced from this as follows. If $d = 0$, then $t_L = t_G$, so $L = T$ is the maximal torus of $G$, and it is self-centralizing in $G$ (cf. [@Hu §26.2, Cor. A]), whence $D = T$, and we have $u_D = 0$. If $d = 1$, then $L$ is a subtorus of codimension 1 in $T$; if $L$ is a regular subtorus, then $D = T$ and we have $u_D = 0$ as before; if $L$ is a singular subtorus, then $D$ is isogenous to $L \times {{\operatorname{SL}}}_2$ (cf. [@Hu §26.2, Cor. B]), and we have $u_D = 1$. In any case, we see that $d \leqslant 1$ implies $u_G - u_H \leqslant u_D \leqslant 1 \leqslant 3$, and the rationality of $G/H$ follows from lemma \[uGH3\].
We now proceed to prove our claim that $U \subseteq U_D \cdotp U_H$. The torus $T$ normalizes $U^-$, and so it acts regularly from the left on $U^- \backslash G/B_H$, which contains $U^- \backslash U^- B/B_H$ (isomorphic to $B/B_H$) as a Zariski-dense $T$-stable open subset, by the Bruhat (big cell) decomposition of $G$. Hence $L$ acts trivially from the left on $U^- \backslash U^- B/B_H$. This means that for any $b \in B$ and any $\ell \in L$, one has $U^- \cdotp \ell \cdotp b \cdotp B_H
=
U^- \cdotp b \cdotp B_H
$, whence $$\ell \cdotp b
\ =\
v_1 \cdotp b \cdotp b_1
\qquad
\text{for some}
\quad
v_1 \in U^-
,
\
b_1 \in B_H
.$$ Thus $v_1 \in U^- \cap B = \{1\}$, and if we write $b_1 = t_1 \cdotp u_1$ (with $t_1 \in T_H$, $u_1 \in U_H$) and $b = t \cdotp u$ (with $t \in T$, $u \in U$), then reducing modulo $U$ shows that $\ell = t_1$ in $T$. Hence, for any $b \in B$ and $\ell \in L$, there exists $u_1 \in U_H$ such that $$b^{-1} \cdotp \ell \cdotp b
\ =\
\ell \cdotp u_1
.$$ Specializing this relation to the case when $b$ lies in $U_H$, we see that $L$ normalizes $U_H$, and hence $L \cdotp U_H$ is a connected subgroup of $B$ containing $L$ as a maximal torus. Specializing the relation to the case when $b$ equals $u \in U$, we see that $u^{-1} \cdotp L \cdotp u$ is also a maximal torus in $L \cdotp U_H$, so it is $U_H$-conjugate to $L$: there exists $u_2 \in U_H$ such that $(u \cdotp u_2)^{-1} \cdotp L \cdotp (u \cdotp u_2)
=
L
$, or equivalently, $u \cdotp u_2 \in U \cap N_G(L)$. But since $U$ is connected and solvable, by [@Hu §19.4, Prop.]), $U \cap N_G(L) = N_U(L)$ is equal to $C_U(L) = U \cap C_G(L) = U \cap D = U_D$. We have thus shown that for any $u \in U$, there exists $u_2 \in U_H$ (depending on $u$) such that $u \cdotp u_2 \in U_D$. Hence $U$ is contained in $U_D \cdotp U_H$, and our claim follows.
\[tu6\] In the situation of \[notA\]:
- $U^- \backslash G/B_H$ is rational, of dimension $(t_G - t_H) + (u_G - u_H)$.
- If $(t_G - t_H) + (u_G - u_H) \leqslant 5$, then $G/H$ is rational.
The Bruhat (big cell) decomposition of $G$ shows that $U^- \backslash G/B_H$ contains a Zariski-dense constructible subset $U^- \backslash U^- B / B_H$ which is birational to $B/B_H$; in turn, this is rational by cor. \[Sol quotient\]. Since $\dim(B/B_H) = (t_G - t_H) + (u_G - u_H)$, we see that $U^- \backslash G / B_H$ is rational and of that dimension; this shows part (a).
For part (b), consider the dominant rational maps $\gamma$ and $\varphi$ in the diagram of \[notA\], which give the inequalities $$\dim(B^- \backslash G / H)
\ \leqslant\
\dim(B^- \backslash G / B_H)
\ \leqslant\
\dim(U^- \backslash G / B_H)
.$$ If $\dim(B^- \backslash G/H) \leqslant 2$, the rationality of $G/H$ follows from lemma \[dim1or2\], while if one has $\dim(B^- \backslash G / H)
=
\dim(B^- \backslash G / B_H)
$, the rationality of $G/H$ follows from lemma \[orbit\]. In the remaining cases, our hypothesis yields $$4
\ \leqslant\
\dim(B^- \backslash G / H) + 1
\ \leqslant\
\dim(B^- \backslash G / B_H)
\ \leqslant\
\dim(U^- \backslash G / B_H)
\ \leqslant\
5
,$$ whence $d := \dim(U^- \backslash G / B_H) - \dim(B^- \backslash G / B_H)$ is $\leqslant 1$; the rationality of $G/H$ now follows from lemma \[BU\].
\[Hbu\] In the situation of \[notA\], suppose $H$ is reductive; set $$e
\ :=\
\dim (B^- \backslash G / U_H)
-
\dim (B^- \backslash G / B_H)
.$$
- Let $K$ denote the identity component of the kernel $K'$ of the natural right action of $T_H$ on $B^- \backslash G/U_H$. Then $e = t_H - t_K$, where $t_K := \dim K$.
- If the natural left action of $G$ on $G/H$ has zero-dimensional kernel, then $t_K = 0$ and hence $e = t_H$.
- If $e = 0$, which is to say $\dim (B^- \backslash G / U_H)
=
\dim (B^- \backslash G / B_H)
$, then $G/H$ is rational.
Part (a) of the lemma is established along the same lines as part (a) of lemma \[BU\], using the generically faithful right action of $T_H/K'$ on $B^- \backslash G / U_H$, passing to the quotient and applying lemma \[Demazure generic freeness\] to get $$\dim(B^- \backslash G/U_H)
\ =\
\dim(B^- \backslash G/B_H)
+
\dim(T_H/K')
.$$
Let $E := C_G(K)$ denote the centralizer of $K$ in $G$; it is a connected reductive subgroup of $G$ (cf. [@Hu §26.2, Cor. A; §22.3, Th.]) containing the maximal torus $T$, and a Borel subgroup is given by $B \cap E$ (cf. [@Hu §22.4, Cor.]), whose unipotent radical is $U_E := U \cap E$. Let $E_H := C_H(K) = E \cap H$ denote the centralizer of $K$ in $H$; it is a connected reductive subgroup of $H$ containing the maximal torus $T_H$, and a Borel subgroup is given by $B_H \cap E_H = B \cap E \cap H$, whose unipotent radical is $U_{E_H} := U_H \cap E_H = U \cap E \cap H$. We set $u_E := \dim U_E$ and $u_{E_H} := \dim U_{E_H}$; hence $\dim E = t_G + 2u_E$ and $\dim E_H = t_H + 2u_{E_H}$.
We claim that $U$ is equal to the Zariski closure $\overline{U_E \cdotp U_H}$ of the image of multiplying $U_E$ and $U_H$ in $G$. Assuming this for the moment, we infer that $u_G \leqslant u_E + u_H$. More precisely, since the multiplication map $U_E \times U_H \to \overline{U_E \cdotp U_H} = U$ has a general fiber isomorphic to $U_{E_H} = U_E \cap U_H$, we see that $u_G + u_{E_H} = u_E + u_H$. We have the natural inclusion map $$E/E_H
\ =\
E/(E \cap H)
\ \hookrightarrow\
G/H
.$$ Here, since $H$ is reductive by hypothesis, one has $u(H) = 0$, and so by the dimension formula in \[notA\], we have $$\dim(G/H)
\ =\
(t_G - t_H) + 2(u_G - u_H)
.$$ On the other hand, $$\dim(E/E_H)
\ =\
(t_G + 2u_E) - (t_H + 2u_{E_H})
\ =\
(t_G - t_H) + 2(u_G - u_H)
.$$ Thus $\dim E/E_H = \dim G/H$, which shows that the locally closed subvariety $E/E_H$ of $G/H$ is in fact a Zariski-open subset, whence $G/H$ and $E/E_H$ are birational to each other. This is the key fact needed for showing parts (b) and (c) of the lemma. For part (b), we note that $K$ is a normal subgroup of both $E$ and $E_H$; if we let $\overline{E} := E/K$ and $\overline{E_H} := E_H/K$ denote the respective quotient groups, then $E/E_H$ is naturally isomorphic to $\overline{E}/\overline{E_H}$, and the morphisms $$\overline{E}/\overline{E_H}
\ \cong\
E/E_H
\ \hookrightarrow\
G/H$$ are $E$-equivariant with respect to the natural left actions of $E$. But $K$ acts trivially on $\overline{E}/\overline{E_H}$ by construction, so it also acts trivially on $G/H$. Hence $K$ is contained in the kernel of the natural left action of $G$ on $G/H$; if this kernel is zero-dimensional, so is $K$, which is to say $t_K = 0$. For part (c), if we have $e = 0$, then $t_K = t_H$, so $K = T_H$ is the maximal torus of $H$, and it is therefore self-centralizing in $H$ (cf. [@Hu §26.2, Cor. A]), whence $E_H = E \cap H = C_H(K)$ is equal to $T_H$. Then $E/E_H = E/T_H$ is rational by theorem \[Th0\], and the rationality of $G/H$ follows.
To prove our claim, it suffices to show that a point in general position in $U$ belongs to (the Zariski closure of) $U_E \cdotp U_H$, since the reverse inclusion is clear. The torus $T_H$ normalizes $U_H$, and so it acts regularly from the right on $B^- \backslash G/U_H$, which contains $B^- \backslash B^- U/U_H$ (isomorphic to $U/U_H$) as a Zariski-dense open subset, by the Bruhat (big cell) decomposition of $G$. Hence $K$ acts trivially from the right on $B^- \backslash B^- U/U_H$. This means that for a point $u \in U$ in general position, and for any $k \in K$, one has $B^- \cdotp u \cdotp k \cdotp U_H
=
B^- \cdotp u \cdotp U_H
$, whence $$u \cdotp k
\ =\
v_1 \cdotp t_1 \cdotp u \cdotp u_1
\qquad
\text{for some}
\quad
v_1 \in U^-
,
\
t_1 \in T
,
\
u_1 \in U_H
.$$ Thus $v_1 \in U^- \cap B = \{1\}$, and reducing modulo $U$ shows that $k = t_1$ in $T$; hence $u^{-1} \cdotp k \cdotp u
=
k \cdotp u_1^{-1}
$ lies in $K \cdotp U_H$. Note that $K \cdotp U_H$ is a connected subgroup of $B_H$ containing $K$ as a maximal torus; the above discussion shows that $u^{-1} \cdotp K \cdotp u$ is also a maximal torus in $K \cdotp U_H$, so it is $U_H$-conjugate to $K$: there exists $u_2 \in U_H$ such that $(u \cdotp u_2)^{-1} \cdotp K \cdotp (u \cdotp u_2)
=
K
$, or equivalently, $u \cdotp u_2 \in U \cap N_G(K)$. But since $U$ is connected and solvable, by [@Hu §19.4, Prop.]), $U \cap N_G(K) = N_U(K)$ is equal to $C_U(K) = U \cap C_G(K) = U \cap E = U_E$. We have thus shown that for a point $u \in U$ in general position, there exists $u_2 \in U_H$ (depending on $u$) such that $u \cdotp u_2 \in U_E$. This establishes our claim and hence completes the proof of the lemma as well.
\[uGH4\] In the situation of \[notA\], suppose $H$ is reductive; if $u_G - u_H \leqslant 4$, then $G/H$ is rational.
Consider the dominant rational maps $\gamma$ and $\alpha$ in the diagram of \[notA\], which give the inequalities $$\dim(B^- \backslash G / H)
\ \leqslant\
\dim(B^- \backslash G / B_H)
\ \leqslant\
\dim(B^- \backslash G / U_H)
.$$ If $\dim(B^- \backslash G / H) \leqslant 2$, the rationality of $G/H$ follows from lemma \[dim1or2\]; henceforth, assume that $\dim(B^- \backslash G / H) \geqslant 3$. We have $\dim(B^- \backslash G / U_H) \leqslant 4$ by our hypothesis and lemma \[uGH3\]. Hence among the two inequalities in the above display, equality holds for at least one of them. The rationality of $G/H$ then follows from lemma \[orbit\] or lemma \[Hbu\] respectively.
We are now in a position to show our main result of this section.
\[ThB\] Let $G$ be a connected linear algebraic group, and let $H \subseteq G$ be a connected closed subgroup. If $\dim(G/H) \leqslant 10$, then $G/H$ is a rational variety.
By lemma \[mod radical\], we may replace $G$ by its maximal semisimple quotient; henceforth we assume that $G$ is semisimple and adopt the notation of \[notA\]. By the dimension formula in \[notA\], we have $$\dim(G/H)
\ =\
(t_G - t_H) + 2(u_G - u_H) + u(H)
.$$ If $(t_G - t_H) + (u_G - u_H) \leqslant 5$, the rationality of $G/H$ follows from lemma \[tu6\], while if $u_G - u_H \leqslant 3$, the rationality of $G/H$ follows from lemma \[uGH3\]. In the remaining cases, our hypothesis $\dim(G/H) \leqslant 10$ together with the above dimension formula imply that $$u(H) = 0
,
\quad
u_G - u_H = 4
,
\quad
\text{and}
\quad
t_G - t_H = 2
.$$ Hence $H$ is reductive, and the rationality of $G/H$ now follows from lemma \[uGH4\].
With a bit more work, we can establish a slightly technical but also more applicable result in theorem \[ThBrank\]. First, we note that the dimension formula $$\dim(U^- \backslash G / B_H)
\ =\
(t_G + u_G) - (t_H + u_H)
,
\qquad
\dim(B^- \backslash G / U_H)
\ =\
u_G - u_H$$ of lemmas \[tu6\] and \[uGH3\] yields:
\[equiv\] In the situation of \[notA\], the following are equivalent:
- $\dim (B^- \backslash G / B_H)
=
u_G - u_H - t_H
$.
- $\dim (U^- \backslash G / B_H)
-
\dim (B^- \backslash G / B_H)
=
t_G$ (i.e. $d = t_G$ in lemma \[BU\]).
- $\dim (B^- \backslash G / U_H)
-
\dim (B^- \backslash G / B_H)
=
t_H$ (i.e. $e = t_H$ in lemma \[Hbu\]).
\[ThBrank\] Let $G$ be a connected semisimple group, and let $H \subseteq G$ be a connected reductive closed subgroup. Suppose the natural left action of $G$ on $G/H$ has zero-dimensional kernel. If $\dim(G/H) < t_G + t_H + 8$, then $G/H$ is a rational variety.
We adopt the notation of \[notA\]. Our hypothesis on the action of $G$ on $G/H$ together with lemma \[Hbu\] gives $e = t_H$, which by lemma \[equiv\] means that $\dim(B^- \backslash G / B_H)
=
u_G - u_H - t_H
$. Since $H$ is reductive by hypothesis, one has $u(H) = 0$, and so by the dimension formula in \[notA\], we have $$\dim(G/H)
\ =\
(t_G - t_H) + 2(u_G - u_H)
.$$ Hence our assumption that this is $< t_G + t_H +8$ amounts to the inequality $$\dim(B^- \backslash G / B_H)
\ <\
4
.$$ If $\dim(B^- \backslash G/H) \leqslant 2$, the rationality of $G/H$ follows from lemma \[dim1or2\]. In the remaining case, by the inequality in lemma \[orbit\], we must have $$3
\ \leqslant\
\dim(B^- \backslash G / H)
\ \leqslant\
\dim(B^- \backslash G / B_H)
\ \leqslant\
3
,$$ whence equality holds throughout, and the rationality of $G/H$ follows from lemma \[orbit\] again.
\[B23C3G2\] Let $G$ be a connected group which is almost-simple of type $B_3$ or $G_2$, and let $H \subseteq G$ be a connected semisimple subgroup of maximal rank in $G$. Then $G/H$ is rational.
Since the case when $H = G$ is trivial, we shall assume that the connected semisimple closed subgroup $H \subsetneq G$ is properly contained in $G$. The natural left action of $G$ on $G/H$ is thus non-trivial, and since $G$ is almost-simple by hypothesis, the kernel of the action is zero-dimensional; hence theorem \[ThBrank\] is applicable whenever the required bound on $\dim(G/H)$ holds. As $H$ is semisimple, we have the crude lower bound $\dim H \geqslant 3n$ where $n$ denotes the common rank of $G$ and $H$. From the following table of values:
$G$ of type $B_3$ $G_2$
------------------------ ------- -------
$\dim G = $ 21 14
$\dim H \geqslant $ 9 6
$\dim(G/H) \leqslant $ 12 8
$n + n + 8 =$ 14 12
we see that $\dim(G/H) < n + n + 8$ in each of the cases considered, whence theorem \[ThBrank\] yields the rationality of $G/H$.
We note that cor. \[B23C3G2\] completes the proof of theorem \[ThA\] in \[ThAproof\].
Concluding remarks
==================
In this final section, we indicate a few other cases in which our results yield an affirmative answer to the rationality problem \[RatPbm\]. We still work over an algebraically closed base field $k$ of characteristic 0.
Let $G$ be a connected linear algebraic group with $\dim G \leqslant 13$. Then for any connected closed subgroup $H \subseteq G$, $G/H$ is a rational variety.
If $\dim(G/H) \leqslant 10$, the rationality of $G/H$ follows from theorem \[ThB\]. The remaining cases are when $11 \leqslant \dim(G/H) \leqslant 13$, but this means $\dim H \leqslant 2$ and thus $H$ is solvable, whence the rationality of $G/H$ follows from theorem \[Th0\].
\[dimG=14\] Let $G$ be a connected linear algebraic group with $\dim G = 14$. Then for any connected closed subgroup $H \subseteq G$, $G/H$ is a rational variety — except possibly when $G$ is the simple group of type $G_2$ and $H \subseteq G$ is semisimple of type $A_1$ (in which case $\dim(G/H) = 11$).
As in the proof of the previous result, when $\dim H \geqslant 4$ or when $\dim H \leqslant 2$, the rationality of $G/H$ follows from theorem \[ThB\] or theorem \[Th0\] respectively. Henceforth we assume that $\dim H = 3$ and that $H$ is not solvable. This means that $H$ is semisimple of type $A_1$; in particular, $\dim(G/H) = 11$.
Let $R = R(G)$ be the solvable radical of $G$. If $\dim(G/HR) < \dim(G/H) = 11$, then $G/HR$ is rational by theorem \[ThB\], and so $G/H$ is rational by lemma \[mod radical\]. Hence we may assume that $\dim(G/HR) = \dim(G/H) = 11$, which means $H = HR$ is a connected closed subgroup of $G$ containing $R$ in its radical; since $H$ is semisimple, this forces $R$ to be trivial, and hence $G$ is also semisimple. By the classification of semisimple groups, $\dim G = 14$ implies that $G$ is either of type $A_2+2A_1$ or of type $G_2$. In the former case, we have $u_G - u_H
= (3 + 2) - 1 = 4
$ and the rationality of $G/H$ follows from lemma \[uGH4\]. In the latter case, we are in the possible exceptional situation of the proposition.
When the homogeneous variety $G/H$ is of dimension $\leqslant 10$, the rationality problem \[RatPbm\] has been answered affirmatively in theorem \[ThB\]. We consider the cases when $G/H$ is of dimension 11 and 12 in prop. \[dimG/H=11\] and \[dimG/H=12\] below. To put the homogeneous variety $G/H$ in a somewhat “reduced form”, we impose the hypothesis that $G$ acts on $G/H$ with a zero-dimensional kernel; by lemma \[tSS\], this can always be achieved without changing the birational type of $G/H$ by replacing $G$ and $H$ by their images in ${{\operatorname{Aut}}}(k(G/H))$.
\[dimG/H=11\] Let $G$ be a connected semisimple group, and let $H \subseteq G$ be a connected closed subgroup. Suppose the natural left action of $G$ on $G/H$ has zero-dimensional kernel. If $\dim (G/H) = 11$, then $G/H$ is a rational variety — except possibly when $G$ is semisimple of type $G_2$ and $H$ is semisimple of type $A_1$.
We adopt the notation of \[notA\]. If $H$ is contained in some proper parabolic subgroup $P$ of $G$, then by cor. \[subparabolic\], $G/H$ is birational to $(G/P) \times (P/H)$, and by lemma \[parabolic section\], $G/P$ is rational. Since $\dim(P/H) \leqslant \dim(G/H)-1 = 10$, theorem \[ThB\] shows that $P/H$ is rational, and the rationality of $G/H$ follows. Henceforth we assume $H$ is not contained in any proper parabolic subgroup $P$ of $G$; thus by lemma \[Borel-Tits\], $H$ is semisimple.
Proceeding as in the proof of theorem \[ThB\], we see that $G/H$ is rational if $(t_G - t_H) + (u_G - u_H) \leqslant 5$ or $u_G - u_H \leqslant 3$, or even $u_G - u_H = 4$ (by lemma \[uGH4\]). In the remaining cases, our hypothesis $\dim(G/H) = 11$ together with the dimension formula in \[notA\] imply that $$(\,
u_G-u_H
\,,\,
t_G-t_H
\,)
\quad
\text{is equal to}
\quad
(5,1)
.$$ By lemma \[uGH3\], we have $\dim(B^- \backslash G / U_H) = u_G - u_H$ which is $= 5$ here, so we may argue as in the proof of lemma \[uGH4\] to see that $G/H$ is rational except possibly when $$\dim (B^- \backslash G / U_H) = 5
,
\quad
\dim (B^- \backslash G / B_H) = 4
,
\quad
\dim (B^- \backslash G / H) = 3
.$$ In this case, our hypothesis on the action of $G$ on $G/H$ together with lemma \[Hbu\] gives $e = t_H$ in the notation there, which by lemma \[equiv\] means that $\dim(B^- \backslash G / B_H)
=
u_G - u_H - t_H
$. Thus $H$ is a semisimple group of rank $t_H = 1$ and hence of type $A_1$, and $G$ is a semisimple group of rank $t_G = t_H + 1 = 2$, with $\dim G = \dim(G/H) + \dim H = 14$. By the classification of semisimple groups, this implies $G$ is of type $G_2$, and we are in the possible exceptional situation of the proposition.
\[dimG/H=12\] Let $G$ be a connected semisimple group, and let $H \subseteq G$ be a connected reductive closed subgroup. Suppose the natural left action of $G$ on $G/H$ has zero-dimensional kernel. If $\dim (G/H) = 12$, then $G/H$ is a rational variety — except possibly when $G$ is semisimple of type $A_3$ and $H$ is semisimple of type $A_1$.
Again we adopt the notation of \[notA\], but note that $u(H) = 0$ by hypothesis here. As in the proof of the previous result, we see that $G/H$ is rational if $(t_G - t_H) + (u_G - u_H) \leqslant 5$ or $u_G - u_H \leqslant 4$. In the remaining cases, our hypothesis $\dim(G/H) = 12$ together with the dimension formula in \[notA\] imply that $$(\,
u_G-u_H
\,,\,
t_G-t_H
\,)
\quad
\text{is equal to}
\quad
(6,0)
\ \text{ or }\
(5,2)
.$$ Our hypothesis on the action of $G$ on $G/H$ together with theorem \[ThBrank\] shows that $G/H$ is rational if $12 = \dim (G/H) < t_G + t_H + 8$; henceforth we assume that $t_G + t_H \leqslant 4$.
In the first case above, we have $t_G = t_H \leqslant 2$. By the classification of semisimple groups, this implies $\dim G \leqslant 14$, and hence $\dim H = \dim G - \dim(G/H) \leqslant 2$. This means that $H$ is solvable, and so $G/H$ is rational by theorem \[Th0\].
In the second case above, we must have $t_H = 1$ and $t_G = t_H + 2 = 3$. Again, if $H$ is solvable, then $G/H$ is rational by theorem \[Th0\]; henceforth we assume that $H$ is not solvable. This means that $H$ is semisimple of rank $t_H = 1$ and hence of type $A_1$, and $G$ is a semisimple group of rank $t_G = 3$, with $\dim G = \dim(G/H) + \dim H = 15$. By the classification of semisimple groups, this implies $G$ is of type $A_3$, and we are in the possible exceptional situation of the proposition.
In view of the possible exceptional situations in prop. \[dimG=14\] and \[dimG/H=11\], our rationality results do not entirely cover the case when $G$ is the simple group of type $G_2$, and we are thus lead to pose the following:
Let $G$ be the 14-dimensional connected semisimple group of type $G_2$. Let $H \subseteq G$ be a connected semisimple closed subgroup of type $A_1$ which is not contained in a proper parabolic subgroup of $G$. Is the $11$-dimensional homogeneous variety $G/H$ a rational variety?
The question can be made a bit more precise. By the Jacobson-Morozov theorem, the semisimple closed subgroups of type $A_1$ in a semisimple group $G$ are in bijection with the non-trivial unipotent elements in $G$. For $G \cong G_2$, there are four such unipotent conjugacy classes, of which the regular and the subregular unipotent classes are the ones corresponding to ($G$-conjugacy classes of) subgroups $H$ of type $A_1$ which are not contained in any proper parabolic subgroup of $G$. If $H$ corresponds to the subregular unipotent class, one knows that $H \cong {{\operatorname{PGL}}}_2 \cong {{\operatorname{SO}}}_3$ is contained in a maximal connected semisimple subgroup $M$ of type $A_2$ in $G \cong G_2$, and that $M \cong {{\operatorname{SL}}}_3$, which is a special group (in the sense of Serre); by lemma \[spQ\], this implies that $G/H$ is birational to $(G/M) \times (M/H)$, and since both $G/M$ and $M/H$ are of dimension $\leqslant 10$ and hence rational by theorem \[ThB\], it follows that $G/H$ is also rational in this case. Thus the only outstanding case of the question, yielding the possible exceptional situation in prop. \[dimG=14\] and \[dimG/H=11\], is when $H$ corresponds to the regular unipotent class of $G \cong G_2$, i.e. when $H \cong {{\operatorname{PGL}}}_2$ arises as the image of ${{\operatorname{SL}}}_2$ under its irreducible 7-dimensional representation to $G_2 \subseteq {{\operatorname{SO}}}_7$. [^4]
We conclude by the following result indicating the nature of a potential “minimal counter-example” for the rationality of homogeneous variety $G/H$.
Let $G$ be a connected linear algebraic group, and let $H \subseteq G$ be a connected closed subgroup such that the natural left action of $G$ on $G/H$ is faithful. Suppose that the homogeneous variety $G/H$ is not rational, and that $\dim(G/H)$ is minimal among these non-rational varieties. Then:
- $G$ is semisimple; $H$ is semisimple, and is not contained in any proper parabolic subgroup of $G$.
- $H$ is of finite index in its normalizer $N_G(H)$ in $G$; consequently, the Tits fibration $G/H \to G/N_G(H)$ is a finite morphism.
- Let $X$ be a smooth projective $G$-equivariant compactification of $G/H$ with $D := X \smallsetminus (G/H)$ a simple normal crossing divisor; then the rational map $\Phi_{|{-}(K_X+D)|}$ given by the complete linear system $|{-}(K_X+D)|$ is a generically finite map. In particular, $-(K_X+D)$ is a big divisor.
Let $R := R(G)$ be the radical of $G$. By lemma \[mod radical\], $G/H$ is birational to $G'/H' \times {{\mathbb{P}}}^s$ for some $s \leqslant \dim R$, where $G' = G/R$ is the maximal semisimple quotient of $G$, and $H' = H/(H \cap R)$ is the image of $H$ in $G'$. Since $G/H$ is non-rational, so is $G'/H'$, whence the minimality of $\dim(G/H)$ implies that $s = 0$, i.e. $G/H$ is birational to $G'/H'$. Since $G$ acts faithfully on $G/H$ by assumption, while $R$ acts trivially on $G'/H'$, it follows that $R$ is trivial and $G$ is semisimple.
Suppose $H$ is contained in a proper parabolic subgroup $P$ of $G$. Since $\dim(P/H)$ is strictly smaller than $\dim(G/H)$, the minimality of $\dim(G/H)$ forces $P/H$ to be rational. But $G/P$ is rational by lemma \[parabolic section\], while by cor. \[subparabolic\], $G/H$ is birational to $(G/P) \times (P/H)$ and hence is rational, contradicting our assumption of the non-rationality of $G/H$. Hence $H$ is not contained in any proper parabolic subgroup of $G$. By lemma \[Borel-Tits\], it follows that $H$ is semisimple. This proves part (a).
Suppose the closed subvariety $N_G(H)/H$ of $G/H$ has positive dimension. Choose a connected 1-dimensional closed subgroup ${{\overline{M}}}$ in $N_G(H)/H$, and let $M \subseteq N_G(H)$ be its preimage in $N_G(H)$. By lemma \[Sol\] applied to the natural right action of ${{\overline{M}}}$ on $G/H$, we see that $G/H$ is birational to $G/M$ or to $(G/M) \times {{\mathbb{P}}}^1$. Since $\dim(G/M)$ is strictly smaller than $\dim(G/H)$, the minimality of $\dim(G/H)$ forces $G/M$ to be rational; but this implies that $G/H$ is also rational, contradicting our assumption. Hence $N_G(H)/H$ is 0-dimensional, which proves part (b).
Let $(X, D)$ be a smooth projective $G$-equivariant compactification of the positive-dimensional homogeneous variety $G/H$, and let $D = X \smallsetminus (G/H)$. As observed in [@Br §2.1, Proof of Prop. 3.3.5 (iii)], the Tits fibration $G/H \to G/N_G(H)$ (as a rational map from $X \supseteq G/H$) is given by $\Phi_{|V|}$, where $$V
\ :=\
{{\operatorname{Im}}}(\,
\wedge^{\dim G} \, ({{\operatorname{Lie}}}G)
\longrightarrow
H^0(X, -(K_X + D))
\,)$$ is a non-zero vector space over $k$. The generical finiteness of $\Phi_{|V|}$ then implies the same for $\Phi_{|{-}(K_X+D)|}$, which proves part (c).
[SGA3-II]{}
A. Borel, *Linear algebraic groups*, 2nd enlarged ed., Springer-Verlag, New York, 1991, Graduate Texts in Mathematics, No. **126**.
A. Borel and J. De Siebenthal, *Les sous-groupes fermés de rang maximum des groupes de [L]{}ie clos*, Comment. Math. Helv. **23** (1949), 200–221.
M. Brion, *Log homogeneous varieties*, Proceedings of the [XVI]{}th [L]{}atin [A]{}merican [A]{}lgebra [C]{}olloquium ([S]{}panish), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 1–39.
M. Brion, P. Samuel, and V. Uma, *Structure of algebraic groups and geometric applications*, Chennai Mathematical Institute Lecture Series in Mathematics, vol. 1, American Mathematical Society, Providence, RI, 2012.
C. Chevalley, *On algebraic group varieties*, J. Math. Soc. Japan **6** (1954), 303–324.
J. -L. Colliot-Thélène and J. J. Sansuc, *The rationality problem for fields of invariants under linear algebraic groups (with special regards to the [B]{}rauer group)*, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res., Mumbai, 2007, pp. 113–186.
J. -L. Colliot-Thélène, B. Kunyavskii, V. Popov and Z. Reichstein, *Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?*, Compos. Math. **147** (2011), no. 2, 428–466.
M. Demazure, *Sous-groupes algébriques de rang maximum du groupe de [C]{}remona*, Ann. Sci. École Norm. Sup. (4) **3** (1970), 507–588.
D. Mumford, J. Fogarty and F. Kirwan, *Geometric invariant theory*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), **34**, Springer-Verlag, Berlin, 1994.
W. Haboush, *Brauer groups of homogeneous spaces. [I]{}*, Methods in ring theory ([A]{}ntwerp, 1983), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 129, Reidel, Dordrecht, 1984, pp. 111–144.
J. E. Humphreys, *Linear algebraic groups*, Springer-Verlag, New York, 1975, Graduate Texts in Mathematics, No. **21**.
S. P. Lehalleur, *Subgroups of maximal rank of reductive groups*, retrieved from: [http://www.math.ens.fr/[\~]{}gille/pepin.pdf]{}, to appear in “Autour des schémas en groupes”, Panoramas et Synthèses.
T. Miyata, *Invariants of certain groups. [I]{}*, Nagoya Math. J. **41** (1971), 69–73.
S. Mukai and H. Umemura, *Minimal rational threefolds*, Algebraic geometry ([T]{}okyo/[K]{}yoto, 1982), Lecture Notes in Math., vol. **1016**, Springer, Berlin, 1983, pp. 490–518.
R. Piene and M. Schlessinger, *On the Hilbert scheme compactification of the space of twisted cubics*, Amer. J. Math. **107** (1985), no. 4, 761–774.
Z. Reichstein, *On the notion of essential dimension for algebraic groups*, Transform. Groups **5** (2000), no. 3, 265–304.
M. Rosenlicht, *Some basic theorems on algebraic groups*, Amer. J. Math. **78** (1956), 401–443.
M. Rosenlicht, *Some rationality questions on algebraic groups*, Ann. Mat. Pura Appl. **43** (1957), 25–50.
M. Rosenlicht, *Questions of rationality for solvable algebraic groups over nonperfect fields*, Ann. Mat. Pura Appl. **61** (1963), 97–120.
D. J. Saltman, *Noether’s problem over an algebraically closed field*, Invent. math. **77** (1984) 71–84
*Schémas en groupes. [II]{}: [G]{}roupes de type multiplicatif, et structure des schémas en groupes généraux*, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. **152**, Springer-Verlag, Berlin, 1970.
T. A. Springer, *Some results on algebraic groups with involutions*, in: Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Studies in Pure Math., vol. **6**, North-Holland, Amsterdam, 1985, pp. 525–543.
[^1]:
[^2]: Given a rational action of $H$ on $X$, we may replace $X$ by another birational model and assume that $H$ acts on $X$ regularly (cf. [@Ro56 Th. 1]), and it then follows (essentially by [@Ro56 Th. 2]) that there exists a geometric quotient of (an open subset of) $X$ by $H$ (in the sense of GIT); the rational quotient $X/H$ can thus be regarded as the birational equivalence class of the moduli space of “general $H$-orbits in $X$”.
[^3]: After the second author gave a talk at Fudan University on a preliminary version of this paper, the PhD student JinSong Xu of the host Professor Meng Chen informed us of an independent proof of this result in the case when $X$ is a homogeneous variety.
[^4]: Likewise, a similar analysis shows that the only possible exceptional situation in prop. \[dimG/H=12\] is when the semisimple group $H$ of type $A_1$ corresponds via the Jacobson-Morozov theorem to the regular unipotent class of $G$ of type $A_3$, i.e. $H$ is the isomorphic image of ${{\operatorname{SL}}}_2$ under its irreducible 4-dimensional representation to ${{\operatorname{SL}}}_4$ — in the “adjoint” case, the homogeneous space ${{\operatorname{PGL}}}_4/{{\operatorname{PGL}}}_2$ is known (cf. [@PS]) to be rational.
|
---
abstract: 'An exact, axially symmetric solution to the Einstein-Klein-Gordon field equations is employed to model the dark matter in spiral galaxies. The extended rotation curves from a previous analysis are used to fit the model and a very good agreement is found. It is argued that, although our model possesses three parameters to be fitted, it is better than the non-relativistic alternatives in the sense that it is not of a phenomenological nature, since the dark matter would consist entirely of a scalar field.'
author:
- 'F. Siddhartha Guzmán[^1], Tonatiuh Matos'
- 'Hugo Villegas-Brena'
title: 'Scalar dark matter in spiral galaxies.'
---
[**Key words:**]{} galaxies: haloes – cosmology: dark matter.
Introduction.
=============
Since the pioneering works by Oort and Zwicky, back in the 1930’s ([@Oort]; [@Zwicky]), the existence of dark matter in the Universe has been firmly established by astronomical observations at very different length-scales, ranging from the neighbourhood of the Solar System to the clusters of galaxies. What it means is that a large fraction of the mass needed to produce, within the framework of Newtonian mechanics, the observed dynamical effects in all these very different systems, is not seen. This puzzle has stimulated the exploration of lots of proposals, and very imaginative explanations have been put forward, from exotic matter to nonrelativistic modifications of Newtonian dynamics.\
In particular, the measurement of rotation curves (RC) in galaxies shows that the coplanar orbital motion of gas in the outer parts of these galaxies keeps a more or less constant velocity up to several luminous radii. The discrepancy arises when one applies the usual Newtonian dynamics to the observed luminous matter and gas, since then the circular velocity should decrease as we move outwards. The most widely accepted explanation is that of a spherical halo of dark matter, its nature being unknown, which would surround the galaxy and account for the missing mass needed to produce the flat RC. Another possibility, considered much less often, is the so called Modified Newtonian Dynamics (MOND), which was put forward by Milgrom (1983); in this model, the usual second Newton law would broke at “small” accelerations, as compared to some (in principle universal) acceleration parameter, $a_0$. Although it seems to provide for a very good phenomenological description of the RC, it lacks, at least until now, a more sound theoretical basis.\
Our aim here is to give another explanation to the dark matter problem in spiral galaxies, this time using a fully relativistic approach, and making use of the well known scalar fields. Combining an exact solution of the Einstein-Klein-Gordon field equations with the observations from luminous matter and gas, we are able to reproduce the flat extended RC of spiral galaxies. Scalar fields are the simplest generalization of General Relativity and they can be introduced on very fundamental grounds, as in the case of the Brans-Dicke, the Kaluza-Klein and the Super-Strings theories. They appear also in cosmological models, like inflation, and in general in all modern unifying theories. Recently, it has been suggested ([@Cho]; [@peebles]) that a massive scalar field could account for the dark matter at cosmological scales, and a previous work ([@MatGuzb]) has shown a preliminary analysis in the context of spiral galaxies. This is encouraging since it makes models like the one proposed here to seem more plausible.\
The paper is organized as follows: in §\[sec:metric\] we introduce the field equations and the explicit solution for an axially symmetric configuration; in §\[sec:model\] the geodesic equations are written and the model for the dark matter in a spiral galaxy is introduced; section §\[sec:results\] gives the main results concerning the fitting of the model to the observations, and in §\[sec:end\] some concluding remarks are done. Finally, two appendices describe some geometrical properties of our metric and make a brief conceptual comparison with the dark halo and the MOND hypotheses.\
The field equations and their solution. {#sec:metric}
=======================================
As mentioned in the Introduction, scalar fields appear in a natural way within the framework of unifying theories. As example we mention Kaluza-Klein (KK) and Super-Strings (SS), where the scalar field appears in the effective action after dimensional reduction. Let us begin with the most general scalar-tensor theory of gravity, as given by the action:
$$\tilde{{\cal S}}=\int d^{4}x\sqrt{-\tilde{g}}\left( -\frac{F}{\kappa _{0}}%
\tilde{R}+(\tilde{\nabla}\Phi )^{2}G+\tilde{V}(\Phi )\right) .$$
where $F,$ $\tilde{V}$ and $G$ are functions of $\Phi $ only, $%
\tilde{g}$ is the determinant of the metric $\tilde{g}_{\mu\nu}$ and $\kappa
_{0}=16\pi G/c^{4}$. We can perform a conformal transformation to some other frame by means of the redefinitions ([@damour2]; [@Frolov]),
$$g_{\mu \nu }=F\tilde{g}_{\mu \nu },$$
$$\phi =\frac{1}{\sqrt{2}}\int d\Phi \left[ \frac{3}{2}\left( \frac{F^{\prime }%
}{F}\right) ^{2}+\frac{G}{F}\right] ^{1/2},$$
where the prime denotes $d/d\Phi $. The action then takes the form:
$$S=\int d^{4}x\sqrt{-g}[-\frac{R}{\kappa _{0}}+2(\nabla \phi )^{2}+V(\phi )],
\label{eq:action}$$
where $R$ is the four dimensional scalar curvature and $g$ the determinant of the metric $g_{\mu \nu }$. $\phi $ and $V(\phi )$ repectively are the scalar field and the scalar potential in the new frame, and all the other quantities are also calculated using the new metric. The choice of the Einstein frame, implicit in this form of the action, is made because the field equations are more easily solved in this frame. As is well known, since the coupling of a scalar field with gravity is defined up to a conformal transformation, there is some ambiguity in regarding a specific frame as the “physical” frame (see, e.g., [@Damour]). We shall consider in detail what happens both to the action above and to the equations of motion in section \[sec:model\].\
The next step is to decide what kind of potential $V(\phi )$ is the most convenient for modeling a galaxy. Let us reason as follows. It is known that the energy density of the dark matter in the halo of the galaxies goes like $%
1/r^{2}$. The energy momentum tensor of the scalar field is basically the sum of quadratic terms of the scalar field derivatives plus the scalar potential, $i.e.$, $T_{\mu \nu }\sim \phi _{,\mu }\phi _{,\nu }+V(\phi )\sim
1/r^{2}.$ If we assume that the term $\phi _{,\mu }\phi _{,\nu }\sim 1/r^{2}$ as well as the term $V(\phi )\sim 1/r^{2}$, from the first assumption we infer that $\phi \sim \ln (r)$, and from the second assumption we arrive at $%
V(\phi )\sim \exp (-2\phi )$. So, in what follows we will take the potential $V(\phi )=\Lambda \exp (-2\alpha \phi )$ and we shall consider the four dimensional action:
$$S=\int d^{4}x\sqrt{-g}[-\frac{R}{\kappa _{0}}+2(\nabla \phi
)^{2}+e^{-2\alpha \phi }\Lambda ].$$
For the time being, from this very general setting we shall look for an exact solution to the field equations that will serve us as a model for a spiral galaxy. Now, since the velocity of the gas and the red shift measurements in a galaxy are made over the equatorial plane, it is reasonable to impose axial symmetry on the solution we are looking for, in contrast with the usual spherical dark halo profile used in most studies ([@Begeman]). Moreover, the fact that a substantial amount of the total mass in these galaxies is in the form of dark matter suggests that, in a first approximation, the observed baryonic mass (both stars and gas) will not contribute significantly to the total energy density of the system, at least in the region outside the luminous disk; instead, the scalar matter will determine the space-time curvature, and the material particles will move on geodesics determined (almost) by the energy density of the scalar field. Finally, the RC exhibits a constant velocity of the order of 100-350 km/s, which, compared to the velocity of light, clearly allows one to consider the galaxy as a static system.\
With the above simplifying assumptions, the most general axially symmetric, static metric can be written as:
$$\label{eq:metric}
ds^2 = \frac{1}{f}[e^{2k}(dz d\overline{z})+W^2d\varphi^2]-fc^2dt^2,$$
where $z:=\rho+i\zeta$, the bar means complex conjugation, and the real valued functions $f,W$ and $k$ depend only on $z$ and $\overline{z}$ (or equivalently on $\rho$ and $\zeta$).\
After varying the action in equation (\[eq:action\]), one obtains the following field equations:
$$\phi _{;\mu }^{;\mu }-\frac{1}{4}\frac{dV}{d\phi }=0,$$
$$R_{\mu \nu }=\kappa _{0}[2\phi _{,\mu }\phi _{{,\nu }}+\frac{1}{2}g_{\mu \nu
}V(\phi )], \label{eq:feqs}$$
which are the Klein-Gordon and Einstein field equations, respectively, and $\mu,\nu=0,1,2,3$. A very powerful technique, known as the harmonic maps ansatz, can be employed to find families of solutions to the equations (\[eq:feqs\]), starting with the metric (\[eq:metric\]). The details can be found in Matos (1989; 1994; 1995) and Guzmán & Matos (1999), so we shall only describe it very briefly here.\
The harmonic maps ansatz.
-------------------------
In a few words, the main idea behind the method is to re-parameterize the functions in metric (\[eq:metric\]) with convenient auxiliary functions which will obey a generalization of the Laplace equation, along with some consistency relationships; the latter are usually quite difficult to fulfill, and great care and intuition must be taken in order to get a system of equations both workable with and interesting enough. In this case we shall take $f=e^{\lambda }$, and assume that $f$ and $\phi $ are functions of $W$, which in turn is a function of $z$ and $\overline{z}$ alone. After lengthy but straightforward calculations one is left with the system:
$$\begin{aligned}
\hat{\Delta}\lambda &=&\kappa _{0}\sqrt{-g}V(\phi ),
\nonumber \\
2\hat{\Delta}\phi &=&-\frac{1}{4}\sqrt{-g}\frac{dV}{d\phi }, \nonumber \\
W_{,z\overline{z}} &=&\frac{1}{2}\kappa _{0}\sqrt{-g}V(\phi ), \nonumber\\
k_{,z} &=&\frac{W_{,zz}}{2W_{,z}}+\frac{W}{4}\lambda _{,z}^{2}W_{,z}+\kappa
_{0}W\phi _{,z}^{2}W_{,z}, \label{eq:set}\end{aligned}$$
and a similar equation for $k_{,\overline{z}}$, replacing $z$ by $%
\overline{z}$. The symbol $\hat{\Delta}$ stands for a generalized Laplace operator, such that for every function $h(z,\overline{z})$, $\hat{\Delta}h
:= (Wh_{,z})_{,\overline{z}}+(Wh_{,\overline{z}})_{,z}$.
The model for the galaxy.
-------------------------
Regarding the set of equations (\[eq:set\]), it can be noted that the last equation and its complex conjugate are integrable once the functions $%
\lambda ,\phi $ and $W$ have been integrated. The first three equations, however, are highly coupled since $\sqrt{-g}=W\exp {(2k-\lambda )}/2$; moreover, the operator $\hat{\Delta}$ itself contains $W$. In spite of this, we have been able to obtain a not too restrictive solution, which can be written as:
$$\begin{aligned}
\lambda &=&\ln {M}+\ln {f_{0}}, \nonumber \nonumber \\
\phi &=&\phi _{0}+\frac{1}{2\sqrt{\kappa _{0}}}\ln {M}, \nonumber \\
V &=&-\frac{4f_{0}}{\kappa _{0}M}, \nonumber\\
e^{2k} &=&M_{,z\overline{z}}M, \label{eq:sol}\end{aligned}$$
where $f_{0}$ and $\phi _{0}$ are integration constants with
$$e^{-2\sqrt{\kappa _{0}}\phi _{0}}=4f_{0}\Lambda /\kappa _{0} \label{const}$$
and $M\equiv W$ is, as stated before, a function of $z$ and $\overline{z}$, restricted only by the condition
$$\label{eq:condM}
MM_{,z\overline{z}} = M_{,z}M_{,\overline{z}},$$
but otherwise arbitrary. Observe that $\Lambda $ and $%
\kappa _{0}$ are fundamental constants of the theory but $f_{0}$ and $\phi_{0}$ are integration constants, $i.e.$, they are different for each space-time (each galaxy), fulfilling the relation (\[const\]). The values of these constants will determine the characteristics of a particular galaxy.\
We shall take the solution (\[eq:sol\]) as the general relativistic description of the galaxy, considering a particular choice of $M$, namely, $%
M=z\overline{z}/r_0$, where $r_0$ is a constant with dimensions of length.\
Geodesic motion along the equatorial plane. {#sec:model}
===========================================
Since the particles compossing the gas from which observations arise are small compared to the whole galaxy, they can be considered as test particles moving on the background metric (\[eq:metric\]). Therefore, the next step is to study the geodesic motion of test particles along the equatorial plane. From metric (\[eq:metric\]) we can write, for material particles:
$$\frac{1}{f}\left[ e^{2k}\left( \left( \frac{d\rho }{d\tau }\right)
^{2}+\left( \frac{d\zeta }{d\tau }\right) ^{2}\right) +W^{2}\left( \frac{%
d\varphi }{d\tau }\right) ^{2}\right] -fc^{2}\left( \frac{dt}{d\tau }\right)
^{2}=c^{2}. \label{eq:geod}$$
As the solution is axially symmetric and static, there will be two constants of motion, namely, the angular momentum per unit mass,
$$B=\frac{W^{2}}{f}\frac{d\varphi }{d\tau },$$
and the total energy of the test particle,
$$A = fc^2\frac{dt}{d\tau},$$
where $\tau$ is the proper time of the test particle. In order to obtain useful information from these constants, it is convenient to write the line element as:
$$\begin{aligned}
ds^{2} &=&\left[ \frac{1}{fc^{2}}\left( e^{2k}(\dot{\rho}^{2}+\dot{\zeta}%
^{2})+W^{2}\dot{\varphi}^{2}\right) -f\right] c^{2}dt^{2}
\label{eq:interval} \\
&=&\left( \frac{v^{2}}{c^{2}}-f\right) c^{2}dt^{2},\end{aligned}$$
since the squared three-velocity, $v^2$, is given by:
$$v^{2}=g_{ab}v^{a}v^{b}=\frac{e^{2k}}{f}(\dot{\rho}^{2}+\dot{\zeta}^{2})+%
\frac{W^{2}}{f}\dot{\varphi}^{2}$$
where $a,b=1,2,3$. On the other hand, for a material freely falling observer (i.e., an observer in geodesic motion) we must have $ds^2 =
-c^2d\tau^2$, and equating this expression with equation (\[eq:interval\]), we can arrive to:
$$A^{2}=\frac{c^{4}f^{2}}{f-v^{2}/c^{2}}. \label{eq:A}$$
If we now identify the equatorial plane of the galaxy with the plane $\zeta
=0$, the geodesic equation (\[eq:geod\]) reduces to the following, after inserting the constants of motion $A$ and $B$:
$$\frac{1}{f}e^{2k}\left(\frac{d\rho}{d\tau}\right)^2+\frac{B^2f}{W^2} -\frac{%
A^2}{c^2 f} = c^2.$$
This equation describes the motion of test particles in the equatorial plane of the galaxy, and in every particular trajectory the constants $A$ and $B$ remain so, i.e., constant. However, a key point here is that, shall we change of trajectory, the values for $A$ and $B$ will change accordingly for the new trajectory. Since the RC give an average of the circular velocity of particles in the galaxy, we shall consider circular orbits only, for which $\dot{\rho} = 0$, so that $v$ in equation (\[eq:interval\]) can be identified with $v_c\equiv v_{circular}$. From this, an expression for $B$ in terms of $v^2$ can be written down:
$$B^2 = \frac{v^2}{f-v^2/c^2}\frac{W^2}{f},$$
and since, as stated before, $v \ll c$, this gives:
$$\label{eq:B}
B^2 \cong v^2\frac{W^2}{f^2}.$$
Using now the form of $f$ given by equations (\[eq:sol\]), one arrives at the following remarkably simple relation between $v$ and $B$:
$$v_{DM}=f_{0}B, \label{eq:vdm}$$
where we have written $v_{DM}$ instead of $v$ to stress the fact that this velocity for test particles is due to the scalar dark matter. Formula (\[eq:vdm\]) is the main result of this model, it states the way how the circular velocity due to the dark matter is determined by the angular momentum from each orbit. It is remarkable that formula (\[eq:vdm\]) is invariant under conformal transformation $g_{\mu \nu }=F\tilde{g}_{\mu
\nu }$ of the metric, $i.e.$, this formula is valid for the metrics $g_{\mu
\nu }$ and $\tilde{g}_{\mu \nu }.$\
In order to gain some insight into the physical meaning of the solution (\[eq:sol\]), we write it in Boyer-Lindquist-like coordinates (Schwarzschild coordinates) $(r,\theta ,\varphi )$, related to $(\rho ,\zeta ,\varphi )$ by $\rho =\sqrt{r^{2}+b^{2}}\sin {\theta }$, $\zeta =r\cos {\theta }$; metric (\[eq:metric\]) then reads:
$$ds^{2}=\frac{1+b^{2}\cos ^{2}{\theta }/r^{2}}{f_{0}r_{0}}\left( \frac{dr^{2}%
}{1+b^{2}/r^{2}}+r^{2}d\theta ^{2}\right) +\frac{r^{2}+b^{2}\sin ^{2}{\theta
}}{f_{0}r_{0}}d\varphi ^{2}-f_{0}c^{2}\frac{r^{2}+b^{2}\sin ^{2}{\theta }}{%
r_{0}}dt^{2}. \label{eq:metricBL}$$
On the other hand, the effective energy density $\mu_{DM}$ is given by:
$$\mu _{DM}=\frac{1}{2}V(\phi )=-\frac{2f_{0}r_{0}}{\kappa
_{0}(r^{2}+b^{2}\sin ^{2}{\theta )}}. \label{eq:mudm}$$
The fact that this energy density is negative does not constitute a serious drawback since, as mentioned before, we most perform a conformal transformation of the metric (\[eq:metric\]) in order to obtain the action corresponding to a theory with a more physical interpretation.\
To be able to obtain more quantitative results and to compare this model with the most usual approaches, one further assumption must be made, regarding the constant of motion $B$. The observed luminous matter in a galaxy behaves in accord to Newtonian dynamics to a good approximation, so that its angular momentum per unit mass will be $B=v_{L}\times D$, where $%
v_{L}$ is the contribution of the luminous matter, and $D$ is the interval from the metric as written in equation (\[eq:metricBL\]); as we are in the equatorial plane, on circular orbits and at one particular instant, we have $%
d\rho =d\zeta =dt=0$, and it is easy to check that $D=\sqrt{%
(r^{2}+b^{2})/f_{0}}$. It is now reasonable to substitute this value for $B$ in equation (\[eq:vdm\]), since the expression for $v_{DM}$ represents the velocity of test particles due to the presence of the scalar field; we get:
$$v_{DM}=f_{0}v_{L}\sqrt{(r^{2}+b^{2})/f_{0}}, \label{eq:vdmb}$$
Noting that the total kinetic energy will be well approximated by the sum of the individual contributions, i.e., $%\onehalf
1/2 mv_C^2 \simeq
%\onehalf
1/2 mv_{L}^2+
%\onehalf
1/2 mv_{gas}^2+
%\onehalf
mv_{DM}^2$, we arrive at the final form of the velocity along circular trajectories in the equatorial plane of the galaxy:
$$v_{C}^{2}=v_{L}^{2}(1+f_{0}(r^{2}+b^{2}))+v_{gas}^{2}, \label{eq:vctot}$$
where the constants $f_0$ and $b$ will be parameters to adjust to the observed RC. To this end we shall proceed as follows: we take the photometric and RC data for 6 spiral galaxies from Begeman et al. (1991) and Kent (1987), as listed in Table \[tab:sample\]. This data is fitted using a non-linear least squares routine adding a third parameter, namely, the usual mass-luminosity ratio $M/L$, which is taken to be constant in each particular galaxy; when there exist disk and bulge observations, two $M/L$ ratios are assumed. The total luminous mass at a distance $r$ from the center of the galaxy will be $M_L(r) = (M/L) \times L(r)$, i.e.:
$$\label{eq:vlum}
v_{L}^2(r) = \frac{GM_L(r)}{r}.$$
Combining equations (\[eq:vctot\]) and (\[eq:vlum\]), and including the 21 cm data from gas, we are led to:
$$v_{C}^{2}(r)=\frac{GM_{L}(r)}{r}(1+f_{0}(r^{2}+b^{2}))+v_{gas}^{2}.
\label{eq:modeltofit1}$$
In the next section \[sec:results\] we shall show how this compares with the actual observational data. It can also be useful to compare this derivation with more common explanations for the RC, two of which are briefly described in appendix \[sec:dhmond\].
Results. {#sec:results}
========
The main results are shown in Fig. \[fig:fits\] and in Table \[tab:fits\]. Figure \[fig:fits\] shows the observational RC (for simplicity we have omited the error bars) as well as the fitted curves using equation (\[eq:modeltofit1\]). Shown are also the individual contributions from luminous matter, gas and the scalar field. It can be noted that the agreement is quite good (within 5% in all cases), which could have been expected since there are three parameters to be adjusted. However, it should be noted that this approach is made on a very solid theoretical basis, because we have begun with a relativistic description of the galaxy. Moreover, the dark matter in this model would be entirely constituted by the scalar field.\
\[fig:fits\]
For the parameter $M/L$ our results are in very good agreement with previous analyses which employ the dark-halo and the MOND approaches (e.g. [@Begeman]). The remaining two parameters, $f_0$ and $b$, serve only to determine completely the metric, and they do not have a direct physical interpretation other than as part of the scalar field energy density. In Table \[tab:fits\] the best fit parameters are listed, along with the formal $1\sigma$ error in the fitted parameter.
---------- ----------- ---------- ------------------- ----------
Galaxy Type Distance Luminosity $R_{HI}$
(Mpc) ($10^9M_{\odot}$) (kpc)
NGC 2403 Sc(s)III 3.25 7.90 19.49
NGC 2903 Sc(s)I-II 6.40 15.30 24.18
NGC 6503 Sc(s)II.8 5.94 4.80 22.22
NGC 3198 Sc(rs)I-I 9.36 9.00 29.92
NGC 2841 Sb 9.46 20.50 42.63
NGC 7331 Sb(rs)I-I 14.90 54.00 36.72
---------- ----------- ---------- ------------------- ----------
: Sample of galaxies \[tab:sample\]
---------- ---------------- ----------------- ------- ---------------------
Galaxy (M/L)$_{disk}$ (M/L)$_{bulge}$ b f$_0$
(kpc) (kpc$^{-1}$)
NGC 2403 1.75 – 1.63 0.0116
0.04 – 0.003 $5.7\times 10^{-4}$
NGC 2903 2.98 – 8.33 0.0043
0.12 – 0.03 $4.0\times 10^{-4}$
NGC 6503 2.12 – 1.79 0.013
0.09 – 0.01 $8.7\times 10^{-4}$
NGC 3198 2.69 – 7.83 0.0054
0.08 – 0.02 $3.0\times 10^{-4}$
NGC 2841 5.39 3.25 13.85 0.0039
0.34 0.36 0.16 $2.5\times 10^{-4}$
NGC 7331 5.06 1.11 0.845 0.0013
0.23 0.06 0.002 $8.9\times 10^{-5}$
---------- ---------------- ----------------- ------- ---------------------
: Best-fit parameters \[tab:fits\]
Conclusions. {#sec:end}
============
In this work we have obtained an exact, axially symmetric and static solution to the field equations of gravity coupled with a scalar field. This solution has been successfully employed as a model for a spiral galaxy, and in particular, we have been able to reproduce the RC of matter in these galaxies with an excellent agreement, both with the observations themselves and with previous analyses of this kind of data.\
It should be stressed that, although our model has three parameters to be fitted, which in general allows for a great flexibility, they are obtained from a theory that is of a fundamental nature, namely, the low energy limit of a family of unification theories. This makes the calculations herein shown to be natural since no [*ad hoc*]{} hypotheses are needed, in contrast to the dark halo or the MOND models. We conclude that this work enables us to state that scalar fields are strong candidates to constitute the dark matter, not only at a cosmological scale, but also within spiral galaxies.
Some geometrical aspects of the metric. {#app:geom}
=======================================
In order to gain some insight into the solution (\[eq:metricBL\]), we shall consider a couple of issues related to the geometrical and topological aspects of the metric. By defining: $$x_{1}=\ln \sqrt{r^{2}+b^{2}\sin ^{2}\theta },$$ $$x_{2}=\arctan \left( \frac{\cot \theta }{\sqrt{1+b^{2}/r^{2}}}\right) ,$$ $$x_{3}=\varphi ,$$ $$x_{4}=f_{0}t,$$ we have:
$$\label{eq:metricflat}
ds^2=\frac{1}{f_0r_0}e^{2x_1}(dx_1^2+dx_2^2+dx_3^2-c^2dx_4^2).$$
From this, it can be argued that our coordinates ($\theta ,\varphi
$) are not really angles, but in fact just cartesian coordinates. This, however, can only be stated if we know the global topology of our space-time, which we do not. As a counterexample, we can consider the 2-torus, ${\cal T}^{2}$; in this case, the metric is not only conformally flat but in fact flat altogether, i.e., $ds^{2}=dx^{2}+dy^{2}$; with a proper rescaling, this can be written $ds^{2}=d\theta _{1}^{2}+d\theta
_{2}^{2}$, where now $\theta _{1}$ and $\theta _{2}$ vary from $0$ to $2\pi $ and can be thought of effectively as angles. A similar example is the 2-sphere, ${\cal S}^{2}$. Therefore, our point of view is that, since interpreting ($r,\theta ,\varphi $) as spherical-like coordinates allows us to reproduce the rotation curves for galaxies, we can consider them in such way, and in particular, $\varphi $ does represent angles about the axial direction.
Dark-halo profiles and the MOND hypothesis. {#sec:dhmond}
===========================================
The most commonly accepted approach to explain the RC is to assume that there is some kind of unseen matter around the visible part of the galaxy, forming what is usually called a ‘halo’; then, a mass density profile for this dark matter is formulated and combined with data from the visible and 21 cm observations, and the model is fitted to the RC as obtained from red-shifts in the galaxy. A quite broad family of density profiles is given by ([@Zhao]):
$$\rho_{halo}(r) = \frac{\rho_c}{(r/r_c)^\gamma[1+(r/r_c)^\alpha]^{(\beta-
\gamma)/\alpha}},$$
where $\rho_0$ is the central density and $r_c$ is the ‘core’ radius, both of the halo. In particular, the simplest and most widely used profile is the so called modified isothermal sphere (MIS), for which $%
(\alpha,\beta,\gamma)=(2,2,0)$. This model is attractive because there are cosmological arguments which seem to suggest that, under the suitable conditions, astronomical objects of this kind might actually evolve, the dark matter being cold or hot depending on the evolutionary arguments (see, e.g., [@Kravtsov] and references therein); however, a completely satisfactory evolution scenario remains to be derived. In the actual fitting procedure, it is usually more convenient to work with the asymptotic circular velocity $v_h$ obtained from the isothermal sphere halo, by applying the virial theorem and Newton’s law:
$$v_h = \sqrt{4\pi G\rho_0r_c^2}.$$
Beginning with the luminosity observations, $L(r)$, there are three parameters to be adjusted (four in the case of galaxies with separate observations from the disk and the bulge): the ratio $M/L$, usually assumed to be constant over the whole optical disk, the core radius $r_c$ and the asymptotic velocity $v_h$.\
The other approach we shall consider here is known as the modified Newtonian dynamics (MOND), which was proposed by Milgrom (1983); in this case there is no dark matter at all, rather a deviation from the usual Newton’s second law would occur when one is dealing with very ‘small’ accelerations, where ‘small’ means small with respect to some (in principle, universal) critical acceleration parameter, $a_0$. Instead of ${\bf F}= m_g {\bf a}$, one would write:
$$\label{eq:mond}
{\bf g}_N = \mu(a/a_0){\bf a},$$
where ${\bf g}_N$ is the conventional gravitational acceleration, $%
{\bf a}$ is the true acceleration of a particle with respect to some fundamental frame ($a\equiv|{\bf {a}|}$), and $\mu$ is a function of $a/a_0$ of which only the asymptotic forms $\mu(a/a_0\gg1)\approx 1$, $\mu(a/a_0\ll 1)\approx a/a_0$ are known. It can then be seen that for accelerations much larger than the acceleration parameter $a_0$, $\mu\approx 1$ and we recover the Newtonian dynamics. For the rotation law, the usual expression remains to be valid: $v_C^2/r=a$, and also $g_N\approx MGr^{-2}$; combining this with equation (\[eq:mond\]), we get the asymptotic velocity:
$$v_{C}^{4}=GMa_{0}.$$
In this case the acceleration parameter $a_0$ can be taken as fixed so there is only one free parameter, again the ratio $M/L$. Alternatively, $a_0$ can also be considered as a free parameter. Although this approach works very well when fitting the rotation curves, there is no [*a priori*]{} reason to believe that a deviation of this kind could indeed occur, so this is usually considered to be a purely phenomenological description.
[99]{} Begeman K. G., Broeils A. H. Sanders R. H., 1991, MNRAS, 249, 523
Cho Y. M., Keum Y. Y., 1988, Class. Quantum Grav., 15, 907
Damour T., Esposito-Farèse G., 1992, Class. Quantum Grav., 9, 2093
Damour T., Esposito-Farèse G., 1993, Phys. Rev. Lett., 70, 2220
Frolov A. V., 1999, Class. Quantum Grav., 16, 407
Guzmán F. S., Matos T., 1999, in Proceedings of the VIII Marcel Grossman Meeting, ed. D. Ruffini et al. (Singapore: World Scientific), 333
Guzmán F. S., Matos T., 2000, Class. Quantum Grav., 17, L9
Kent S. M., 1987, AJ, 93, 816
Kravtsov A. V., Klypin A. A., Bullock J. S., Primack J. R., 1998, ApJ, 502, 48
Matos T., 1989, Ann. Phys. (Leipzig), 46, 462
Matos T., 1994, J. Math. Phys., 35, 1302
Matos T., 1995, Math. Notes, 58, 1178
Milgrom M., 1983, ApJ, 270, 371
Peebles P. J .E. 1999, preprint astro-ph/9910350.
Oort J., 1932, Bull. Astron. Inst. Neth., 6, 249
Zhao H. S., 1996, MNRAS, 278, 488
Zwicky F., 1933, Helv. Phys. Acta, 6, 110
[^1]: E-mail: siddh@fis.cinvestav.mx
|
---
abstract: |
A novel functional integral formulation of quantum mechanics for non-Lagrangian systems is presented. The new approach, which we call “stringy quantization,” is based solely on classical equations of motion and is free of any ambiguity arising from Lagrangian and/or Hamiltonian formulation of the theory. The functionality of the proposed method is demonstrated on several examples. Special attention is paid to the stringy quantization of systems with a general $A$-power friction force $-\kappa\dot{q}^A$. Results for $A = 1$ are compared with those obtained in the approaches by Caldirola-Kanai, Bateman and Kostin. Relations to the Caldeira-Leggett model and to the Feynman-Vernon approach are discussed as well.
Dedicated to my father on the occasion of his 60th birthdays.
author:
- Denis Kochan
title: 'Functional integral for non-Lagrangian systems'
---
Introduction {#1}
============
Quantization is a phenomenon that changes our bright classical perspective into a bit uncertain and at first sight rather nonintuitive picture. This picture, however, is more rigorous than the classical one, possesses many fascinating features and has produced a lot of successful predictions.
The subtle problem of transition from classical to quantal attracts attention from the early days of quantum mechanics. Over the years various techniques and methods for solving this puzzle have been invented. Our aim is not to trace back the complete (hi)story of the milestone ideas in this field (for the review we refer to [@englis]). What we want to do is to give a concise exposition of the method we have developed. However, since we have generalized the original Feynman’s path integral approach, we will recapitulate this approach shortly in section \[2\].
The main goal of our paper is to obtain a functional integral formula for the quantum propagator which would not refer to Lagrangian and/or Hamiltonian function. Quantum propagator is the probability amplitude $\mathbf{A}(q_1,t_1\,|\,q_0,t_0)$ for the transition of the system from the initial configuration $(q_0,t_0)$ to the final configuration $(q_1,t_1)$. We will derive a closed expression for this quantity starting from the given set of classical dynamical equations of motion.
The proposed method uses functional integration in the extended phase space, but instead of integration over path histories we introduce integration over stringy surfaces. This crucial element of our approach is explained in full detail in sections \[3\] and \[4\] and in appendices \[A\] and \[B\]. We also make sure that whenever the system under consideration becomes Lagrangian, the stringy description reduces to the standard one with the Feynman path integral.
In sections \[4.1\] and \[4.2\] some simple examples are scrutinized. It is well known that there are classical dynamical systems that cannot be described within the traditional Lagrangian or Hamiltonian framework. The stringy approach enables us to quantize them straightforwardly. In section \[4.1\] the quantization of a weakly non-Lagrangian system is performed. The transition amplitude is computed for a particle in a conservative field, whose motion is damped by a general $A$-power friction force $F=-\kappa\dot{q}^{A}$. The stringy results for $A=1$ are compared with the results obtained in the approaches by Caldirola-Kanai [@CK], Bateman [@bateman] and Kostin [@kostin].
One can argue against the stringy quantization of dissipative systems that it is unable to describe decoherence phenomena. This is true, but the same objection can be raised against the generally accepted heuristic approaches by the authors cited above, as well as those by Dekker [@dekker2], Razavy [@razavy2], Geicke [@geicke] and others, simply because their kinematical and dynamical prerequisites are different from the prerequisites of the particle-plus-environment quantum models. However, a possible argument for the stringy quantization is that the particle-plus-environment models are not able to handle satisfactorily the case with the friction force $-\kappa\dot{q}^A$ for the general power $A$. Moreover, they describe a rather different phenomenon, namely the quantum Brownian motion for which the total force equals $-\kappa\dot{q}+\mbox{\emph{stochastic
term}}$.
Section \[4.2\] contains the analysis of a curious two-dimensional Douglas system [@douglas] which is strongly non-Lagrangian, i.e. not derivable from any sort of Lagrangian or Hamiltonian. This causes a fundamental problem for all conventional quantization methods, but can be dealt with in a rather transparent way in the stringy approach.
In section \[5\] conclusion, discussion and outlook are collected. The section also includes some comments on the relation between the stringy quantization and the Caldeira-Leggett model of quantum Brownian motion [@particle+environment], as well as the influence functional technique by Feynman and Vernon [@feynman-vernon].
Some rather technical material is left to appendices. Appendix \[A\] is devoted to the stringy variational principle, which plays an important role in the motivation of our approach. In appendix \[B\] computational details concerning the surface functional integral for quantum friction force systems are presented.
Historically the first attempt at quantization based on the dynamical equations of motion belongs to Feynman, see [@dyson]. A similar problem was considered by Wigner, Yang $\&$ Feldman, Nelson, Okubo and others, see [@w-yf-o]. Among the recent investigations in this field let us mention the work of Lyakhovich $\&$ Sharapov [@sharapov] and Gitman $\&$ Kupriyanov [@gitman]. They consider the same problem as us, but their strategy is different. In our opinion, their approach fits much better the context of gauge field dynamics.
As Ludwig Faddeev noted during the Edward Witten’s talk at the **Mathematical Physics Conference: From XX To XXI Century**[^1], *quantization is not a science, quantization is an art*. Let us believe that the quantization method proposed here will fit the Ludwig’s dictum and will be meaningful enough to be considered artistic.
Feynman Quantum Mechanics {#2}
=========================
According to Feynman [@feynman], the probability amplitude of the transition of the system from the space-time configuration $(q_0,t_0)$ to another space-time configuration $(q_1,t_1)$ is $$\label{FeynmanI}
\mathbf{A}(q_1,t_1\,|\,q_0,t_0)=\frac{1}{\mathbf{N}}\int[\mathscr{D}\tilde{\gamma}]
\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{\tilde{\gamma}}\hspace{-1mm}pdq-Hdt\Bigr\}}\,.$$ Here the integral is taken over all paths (histories) $\tilde{\gamma}(t)=(\tilde{q}(t),\tilde{p}(t),t)$ in the extended phase space [^2], satisfying the conditions $\tilde{q}(t_0)=q_0$ and $\tilde{q}(t_1)=q_1$.
The preexponential factor $1/\mathbf{N}$ in the expression (\[FeynmanI\]) serves just the normalization. To fix it properly we impose two physical conditions on the transition amplitude. First we introduce an integral condition that ensures that the total probability is conserved,
\[normalization\] $$\int
dq_1\,\overline{\boldsymbol{\mathbf{A}}}(q_1,t_1\,|\,q_0,t_0)\,\mathbf{A}(q_1,t_1\,|\,q'_0,t_0)
=\delta(q_0-q'_0)\,.$$ This specifies the absolute value of $\mathbf{N}$. Then we add a constraint that expresses the obvious fact that no evolution takes place if the final time $ t_1$ approaches the initial time $t_0$, $$\lim\limits_{t_1\rightarrow t_0}\mathbf{A}(q_1,t_1\,|\,q_0,t_0)=\delta(q_1-q_0)\,.$$
This determines the phase of $\mathbf{N}$.
A miraculous consequence of the definition of the transition amplitude (\[FeynmanI\]) (not an additional requirement!) is that it satisfies the evolutionary chain rule, or Chapman-Kolmogorov equation, $$\label{Ch-K}
\mathbf{A}(q_1,t_1\,|\,q_0,t_0)=\hspace{-1mm}\int\hspace{-1mm}
dq'\mathbf{A}(q_1,t_1\,|\,q',t)\,\mathbf{A}(q',t\,|\,q_0,t_0)\,.\hspace{-1mm}$$ The infinitesimal version of this formula is the celebrated Schrödinger equation.
Quantum states of the system are described by the square integrable functions with the standard Hilbert space structure. Physical observables are hermitian operators acting on such functions. Given the state $\Psi_0(q)$ at the initial moment $t_0$ one is able to predict the state $\Psi_1(q)$ at any later moment $t_1$ according to the formula $$\label{evolution}
\Psi_0(q)\ \rightsquigarrow\ \Psi_1(q)=\int
dq'\,\mathbf{A}(q,t_1\,|\,q',t_0)\Psi_0(q')\,.$$ In what follows it is assumed that the above concept of states, observables and quantum evolution is valid for the stringy quantization of non-Lagrangian systems as well. The only new element is a modified prescription for the evolutionary integral kernel $\mathbf{A}(q_1,t_1\,|\,q_0,t_0)$. The same kinematical prerequisites can be found also in other phenomenological approaches [@CK]-[@geicke].
One step beyond Feynman {#3}
=======================
A possible step beyond the theory summarized above consists in the elimination of the Hamiltonian function $H$ from formula (\[FeynmanI\]). The price to be paid is the replacement of the path integration by the surface functional integration.
Our aim is to construct the amplitude for the transition between $(q_0,t_0)$ and $(q_1,t_1)$ starting from the classical equations of motion (and not from the Hamiltonian function which provides them) $$\label{eq.}
\dot{q}^a=\frac{\partial H}{\partial p_a} \equiv \frac {p^a}m\,,
\quad \dot{p}_a=-\frac{\partial H}{\partial q^a} \equiv F_a\,.$$ In the first set of equations, $m$ is the mass of the particle. We restricted ourselves to the simplest case of one particle, although it is trivial to generalize the theory to a system with an arbitrary number of particles. Note also that if the particle is unconstrained and we make use of Cartesian coordinates, the momenta $p^a$ defined in (\[eq.\]) reduce to $p_a$.
Suppose that there exists a unique classical trajectory in the extended phase space $\gamma_{cl}(t)=(q_{cl}(t),p_{cl}(t),t)$, connecting the points $(q_0,t_0)$ and $(q_1,t_1)$. Then we can assign to any other trajectory $\tilde{\gamma}(t)=(\tilde{q}(t),\tilde{p}(t),t)$, which enters the path integral in (\[FeynmanI\]), two auxiliary curves $$\begin{aligned}
\lambda_0(s)\hspace{-2mm}&=&\hspace{-2mm}\bigl(q_0,\pi(s),t_0\bigr),\
\mbox{where}\ \pi(0)=p_{cl}(t_0),\ \pi(1)=\tilde{p}(t_0);\\
\lambda_1(s)\hspace{-2mm}&=&\hspace{-2mm}\bigl(q_1,\phi(s),t_1\bigr),\
\mbox{where}\ \phi(0)=p_{cl}(t_1),\ \phi(1)=\tilde{p}(t_1).\end{aligned}$$ The curves are parameterized by the parameter $s\in[0,1]$ and live in the momentum subsectors of the extended phase space with fixed $(q_0,t_0)$ and $(q_1,t_1)$, see Figure \[umbilic\].
![Schematic picture of two auxiliary curves $\lambda_0(s)$ and $\lambda_1(s)$ which connect the classical history $\gamma_{cl}(t)$ with the given history $\tilde{\gamma}(t)$ in the extended phase space. $\lambda$-curves are located in the $n$-dimensional subspaces of the extended phase space in which the momenta are varying while the coordinates and time are kept fixed. The contour $\partial\Sigma=\tilde{\gamma}-\lambda_1-\gamma_{cl}+\lambda_0$ forms a boundary for plenty of extended phase space surfaces. One of them, denoted as $\Sigma$, is drawn in the figure.[]{data-label="umbilic"}](figure1.pdf){height="11.18cm" width="8.5cm"}
Using these definitions one can write [^3] $$\label{loop}
\int\limits_{\tilde{\gamma}} p_adq^a-Hdt=\int\limits_{\gamma_{cl}}
p_adq^a-Hdt + \oint\limits_{\partial\Sigma} p_adq^a-Hdt\,,$$ where $\partial\Sigma=\tilde{\gamma}-\lambda_1-\gamma_{cl}+\lambda_0$ is a contour in the extended phase space consisting of four curves $\tilde{\gamma}(t)$, $\gamma_{cl}(t)$, $\lambda_0(s)$, $\lambda_1(s)$. The first integral on the right hand side is the classical action $\mathbf{S}_{cl}(q_1,t_1\,|\,q_0,t_0)$ (its equivalent for the non-Lagrangian case will be specified later), while the second integral can be rewritten as $$\oint\limits_{\partial\Sigma} p_adq^a-Hdt=\int\limits_{\Sigma}
dp_a\wedge\Bigl(dq^a-\frac{
\partial H}{\partial p_a}dt\Bigr)-\frac{\partial H}{\partial q^a}dq^a\wedge dt\,,$$ where $\Sigma$ is a surface spanning the contour $\partial\Sigma$, i. e. a map from the parametric space $(t,s)\in[t_0,t_1]\times[0,1]$ to the extended phase space, $$\Sigma: (t,s)\mapsto
\Sigma(t,s)\equiv\bigl(q^a(t,s),p_a(t,s),t(t,s)=t\bigr)\,,$$ satisfying $${\Sigma(t,0)=\gamma_{cl}(t) \atop \Sigma(t,1)=\tilde{\gamma}(t)\
\,}\ \ \ \ \mbox{and}\ \ \ \ {\Sigma(t_0,s)=\lambda_0(s)\atop
\Sigma(t_1,s)=\lambda_1(s)}\,.$$ The surface $\Sigma$ can be viewed as a worldsheet of a string, therefore we will call the quantization method using such surfaces “stringy.” Partial derivatives of the Hamiltonian function entering the integral over $\Sigma$ can be eliminated with the help of the equations of motion (\[eq.\]). By doing so, one transforms (\[loop\]) into the following form: $$\label{loop2}
\int\limits_{\tilde{\gamma}}
p_adq^a-Hdt=\mathbf{S}_{cl}(q_1,t_1\,|\,q_0,t_0)+\int\limits_{\Sigma}\Omega\,,$$ where the two-form $\Omega$ is defined as $$\label{Omega}
\Omega=dp_a\wedge dq^a-\Bigl(\frac{p^a}{m}dp_a -
F_adq^a\Bigr)\wedge dt\,.$$ This two-form is an object in the extended phase space and its structure can be read out from the underlying equations of motion. Note that for non-potential forces the expression $(p^a/m) dp_a -
F_adq^a$ does not reduce to $dH$, so that the two-form $\Omega$ is not closed. This becomes essential in the next subsection.
It is obvious that for a given pair of histories $(\tilde{\gamma},\gamma_{cl})$ there exist infinitely many $\Sigma$-surfaces such that $\tilde{\gamma}-\gamma_{cl}\subset\partial\Sigma$. All of them form a set which we will call $\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$ [^4]. Since no $\Sigma$ is preferred and $\int_\Sigma\Omega$ is only boundary dependent, it is natural to average the exponent of (\[loop2\]) over the whole stringy set $\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$. After doing so we obtain the identity $$\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{\tilde{\gamma}}p_adq^a-Hdt\Bigr\}}
=\frac{\mbox{\large{e}}^{\frac{i}{\hslash}\mathbf{S}_{cl}}}{\mathrm{N}_{\tilde{\gamma}}}\hspace{-1.4mm}
\int\limits_{\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}}\hspace{-3mm}[\mathscr{D}\Sigma]
\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{\Sigma}\Omega\Bigr\}},$$ where $\mathrm{N}_{\tilde{\gamma}}$ is the cardinality of the stringy set $\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$, $\mathrm{N}_{\tilde{\gamma}}:=\#\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$, and $[\mathscr{D}\Sigma]$ is a functional integration measure specified in Appendix \[B\]. If no topology-related problems arise in the extended phase space, the infinite constant $\mathrm{N}_{\tilde{\gamma}}$ is independent of the history $\tilde{\gamma}$. Taking all this into account we can rewrite (\[FeynmanI\]) as $${\label{FeynmanIII}}\boxed{
\vspace{5mm} \hspace{1mm}\mathbf{A}(q_1,t_1\,|\,q_0,t_0)\,=\,
\frac{\mbox{\large{e}}^{\frac{i}{\hslash}\mathbf{S}_{cl}}}{\mathbf{N}}
\hspace{-1mm}\int\limits_{\mathscr{U}_{\gamma_{cl}}}^{^{^{}}}[\mathscr{D}\Sigma]
\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{\Sigma}\Omega\Bigr\}},\hspace{2mm}
}$$ where the set $\mathscr{U}_{\gamma_{cl}}$ over which the functional integration is carried out contains all strings in the extended phase space which are anchored to the given classical trajectory $\gamma_{cl}$. Some surfaces from $\mathscr{U}_{\gamma_{cl}}$ are depicted in Figure \[string\]. In (\[FeynmanIII\]), the undetermined constant $1/\mathrm{N}_{\tilde{\gamma}}$ was absorbed into the overall preexponential factor $1/\mathbf{N}$ and the path integral over $\tilde{\gamma}$’s was converted into the surface functional integral, as promised earlier, using the identities $$\label{surface}
\int[\mathscr{D}\tilde{\gamma}]\hspace{-3.5mm}\int\limits_{\ \ \
\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}}
\hspace{-5mm}[\mathscr{D}\Sigma]\cdots\ =
\hspace{-5.5mm}\int\limits_{\ \ \
\bigcup_{\tilde{\gamma}}\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}}
\hspace{-6.7mm}[\mathscr{D}\Sigma]\cdots\ =
\hspace{-1.6mm}\int\limits_{\mathscr{U}_{\gamma_{cl}}}\hspace{-1.0mm}[\mathscr{D}\Sigma]\cdots\
\,.$$
![Four elements (two open, one closed and one shrunk to $\gamma_{cl}$) of $\mathscr{U}_{\gamma_{cl}}$, anchored to the given classical history $\gamma_{cl}$ in the extended phase space. The two open surfaces belong to two different stringy classes $\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$, while the closed and the shrunk surface are both elements of the same stringy class $\mathscr{U}_{(\gamma_{cl},\gamma_{cl})}$. As the worldsheet $\Sigma$ varies, the front and rear boundary curves, denoted in the text $\lambda_0$ and $\lambda_1$, vary as well.[]{data-label="string"}](figure2.pdf){height="11.18cm" width="8.5cm"}
One more step beyond – non-Lagrangian systems {#4}
=============================================
In case the Hamiltonian function $H$ is given, the formula (\[FeynmanIII\]) for the propagator is equivalent to the Feynman formula (\[FeynmanI\]) we have started with. The surface functional integral (\[FeynmanIII\]) is however completely free of $H$ and requires just the knowledge of the classical equations of motion. This observation allows us to postulate (\[FeynmanIII\]) as a quantization tool in situations in which one cannot use the standard Hamiltonian approach. We just have to relax the requirement of the closedness of the two-form $\Omega$, following from the definition of $\Omega$ for Hamiltonian systems. This relaxation is what is hidden behind the slightly provocative phrase *one more step beyond* in the title of this subsection.
The problematic part here is the definition of the classical action $\mathbf{S}_{cl}(q_1,t_1\,|\,q_0,t_0)$ for non-Lagrangian systems. Later we will see that in some specific situations this quantity can be read out from the structure of the surface integral $$\int\limits_{\mathscr{U}_{\gamma_{cl}}}[\mathscr{D}\Sigma]\exp{\Bigl\{\frac{i}
{\hslash}\int\limits_{\Sigma}\Omega\Bigr\}}\,.$$ Another possibility which comes to mind is to use the integrating factor of the generally non-closed two-form $\Omega$. This means that one will look for a function $f(q,p,t)$ on the extended phase space such that $d(f\Omega)=0$. If the structure of the dynamical equations allows for such a function (i.e. the forces satisfy the Helmholtz condition) then we can define a local auxiliary Hamiltonian $H_{aux}$ such that $f\Omega=d(p_adq^a-H_{aux}dt)$. Having $H_{aux}$ we can define the auxiliary classical action $\mathbf{S}_{aux}$. The alternative approach which tries to identify $\mathbf{S}_{cl}$ with $\mathbf{S}_{aux}$ has, however, several substantial disadvantages. First of all, the procedure of finding $f$ (and subsequently $H_{aux}$ and $\mathbf{S}_{aux}$) is highly ambiguous. Furthermore, $\mathbf{S}_{aux}$ lacks symmetries that were originally present in the equations of motion. Because of these findings we do not follow this strategy hereinafter.
From the physical point of view the dynamical equations seem to be more fundamental than their compact but ambiguous precursors, Hamiltonian and/or Lagrangian function, see [@guys]. Since (\[FeynmanIII\]) requires just the knowledge of the equations of motion, it determines the transition amplitude in a completely new way.
Our proposal for the transition amplitude has appeared here out of thin air. Actually, we were rewriting (\[FeynmanI\]) in terms of stringy surfaces and then, when realizing that the integrated function can be written without any reference to the Hamiltonian, we postulated formula (\[FeynmanIII\]) to be valid in general. However, this is not the whole story. The stringy functional quantization can be motivated also by the stringy variational principle. Having the dynamical equations and initial and final endpoints, one can form $\mathscr{U}_{\gamma_{cl}}$ and $\Omega$. Then using these objects one can introduce the stringy action functional $$\label{action}
\mathscr{S}:\mathscr{U}_{\gamma_{cl}}\rightarrow \mathbb{R}\,,\ \ \ \ \
\Sigma\mapsto \mathscr{S}(\Sigma):=\int\limits_{\Sigma}\Omega\,.$$ This is a variational problem with varying boundaries, therefore the total variation has two terms. First one specifies the boundary and determines the initial equations of motion for $\gamma_{cl}$. The second one specifies the bulk of the stationary world-sheet $\Sigma$ (which turns out to be shrunk into $\gamma_{cl}$ itself). Moreover, one immediately verifies that in the special case when $\Omega=d(p_adq^a-Hdt)$, the stringy variational principle reduces (up to an additive constant) to the celebrated Hamilton least action principle. Of course, many subtleties were omitted here, but all of them can be found in [@kochan], or in Appendix \[A\]. Thus, our variational principle enables us to perform the limit $\hslash\rightarrow 0$, in which we recover the original classical dynamics as required.
Quantization of friction force systems {#4.1}
======================================
To examine the functionality of the proposed quantization method let us first analyze the simplest friction force system. It consists of a particle with unit mass, moving in one dimension under the action of the conservative force $F=-dU/dq$ and the friction force $-\kappa\dot{q}^{A}$ [^5]. Thus, $$\ddot{q}=-\kappa\dot{q}^{A}+F\, \Rightarrow\,
\Omega=d\bigl(pdq-\tfrac{1}{2}p^2dt-Udt\bigr)-\kappa
p^{A}\,dq\wedge dt\,.$$ In this example the surface functional integral can be calculated explicitly (for more detail see [@kochan], or Appendix \[B\]). In the course of calculation, the surface functional integral in the extended phase space reduces to path integral in the configuration space, $$\begin{aligned}
\label{disi}
&&\hspace{-4mm}\int\limits_{\mathscr{U}_{\gamma_{cl}}}[\mathscr{D}\Sigma]\exp{\Bigl\{\frac{i}{\hslash}
\int\limits_{\Sigma}\Omega\Bigr\}}\propto\nonumber\label{dissipative}\\
&&\propto\exp{\Bigl\{-\frac{i}{\hslash}\int\limits_{t_0}^{t_1}\bigl(\tfrac{1}{2}\dot{q}_{cl}^2-U(q_{cl})-
\kappa q_{cl}p^{A}_{cl}\bigr)dt\Bigr\}}\times\nonumber\\
&&\times\int[\mathscr{D}q]\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{t_0}^{t_1}\bigl(\tfrac{1}{2}\dot{q}^2-U(q)-
\kappa qp^{A}_{cl}\bigr)dt\Bigr\}}\,.\end{aligned}$$ This combined with the formula (\[FeynmanIII\]) suggests that it will be convenient to define the classical action as [^6] $$\label{A-action}
\mathbf{S}_{Acl}(q_1,t_1|q_0,t_0)=\int\limits_{t_0}^{t_1}\bigl(\tfrac{1}{2}
\dot{q}_{cl}^2-U(q_{cl})-\kappa q_{cl}p^{A}_{cl}\bigr)dt\,.$$ When doing so, we obtain a controlled cancelation of $\exp\{\frac{i}{\hslash}\mathbf{S}_{Acl}\}$ with the preexponential factor arising from (\[disi\]). The final probability amplitude then assumes a compact and reasonable form: $$\begin{aligned}
\label{propagator}
&&\hspace{-1cm}\mathbf{A}_{Astr}(q_1,t_1|q_0,t_0)=\nonumber\\
&&\hspace{-1cm}=\frac{1}{\mathbf{N}_{Astr}}\int[\mathscr{D}q]\exp{\Bigl\{\frac{i}{\hslash}
\hspace{-0.5mm}\int\limits_{t_0}^{t_1}\hspace{-0.5mm}\bigl(\tfrac{1}{2}\dot{q}^2-\hspace{-0.2mm}
U(q)-\hspace{-0.2mm}\kappa qp^{A}_{cl}\bigr)dt\Bigr\}}\,.\end{aligned}$$ The preexponential factor $1/\mathbf{N}_{Astr}$ can in principle be obtained by subjecting $\mathbf{A}_{Astr}(q_1,t_1|q_0,t_0)$ to the conditions (\[normalization\]).
Let us explain why one should consider the propagator formula (\[propagator\]) reasonable. In the path integral above there appears an unconventional “external source” term $-\kappa
qp^{A}_{cl}$. Its appearance guarantees that the quantum dynamics governed by $\mathbf{A}_{Astr}(q_1,t_1|q_0,t_0)$ transforms into classical mechanics in the limit $\hslash \to 0$. This follows from the simple fact that the unique solution of the saddle point equation for the path integral (\[propagator\]), $$\ddot{q}=-\frac{dU}{dq}-\kappa\dot{q}^{A}_{cl}\,,$$ which satisfies the given initial and final conditions $q(t_0)=q_0$ and $q(t_1)=q_1$, is the classical trajectory $q_{cl}(t)$.
The external source term in (\[propagator\]) breaks the validity of the Chapman-Kolmogorov (memoryless) equation [^7]. From the physical point of view it is a desired phenomenon. The microscopic origin of friction is some environmental interaction. This, however, was not accounted for here explicitly. What has been considered is some effective (macroscopic, phenomenological) interaction emerging on the classical level only. Microscopically the system is a part of a larger system and hence it should be affected by the memory effect.
It is not difficult to compute $\mathbf{A}_{Astr}(q_1,t_1|q_0,t_0)$ for the general power $A$ and the potentials $U(q)=0$ and $U(q)=\tfrac{1}{2}\omega^2 q^2$. In both cases we can carry out the path integration in (\[propagator\]) explicitly to obtain $$\begin{aligned}
\label{propagator1}
&&\hspace{-1cm}\mathbf{A}_{Astr}(q_1,t_1|q_0,t_0)=
\nonumber\\
&&=\frac{1}{\mathbf{N}_{Astr}}\exp\left\{\frac{i}{\hslash}
\mathbf{S}_{Acl}(q_1,t_1|q_0,t_0)\right\}\,,\end{aligned}$$ where the non-Lagrangian action $\mathbf{S}_{Acl}(q_1,t_1|q_0,t_0)$ is given by the expression (\[A-action\]). An open problem is to determine the preexponential factor $1/\mathbf{N}_{Astr}(t_1-t_0,q_1,q_0)$ for all powers $A$ except for $A = 1$ (at least the author is incapable to do that). If $A\neq 1$, the preexponential factor is apparently dependent not only on the time difference $t_1-t_0$, but also on the endpoint positions $q_0$ and $q_1$. This hypothesis is supported by the results obtained in [@stuckens-kobe], where the path integral for a non-conservative force quadratic in velocity is calculated. The normalization prefactor which appears there is explicitly dependent on the endpoints positions as well as on $t_1-t_0$.
In what follows we will restrict ourself to the case when the power $A$ is equal to 1. This simplified setup enables us to compare the stringy approach with the Caldirola-Kanai, Bateman and Kostin approaches [^8].
Stringy *versus* Caldirola-Kanai approach
-----------------------------------------
The dissipative system under consideration is weakly non-Lagrangian. This means that $d\Omega\neq 0$, but $\Omega$ possesses a local integrator $f\neq 0$. Let us define the auxiliary Lagrangian function (one of plenty) as $$\begin{aligned}
L_{CK}(q,t)&=e^{\kappa
t}\Bigl[\tfrac{1}{2}\dot{q}^2-U(q)\Bigr].\label{L-CK}
%A&=2 & L_{HNT}(q,\dot{q})&=e^{2\kappa q}\,\frac{\dot{q}^2}{2}-\int^q\hspace{-2mm}d\tilde{q}\,
%e^{2\kappa\tilde{q}}\,\frac{dU(\tilde{q})}{d\tilde{q}},\label{L-HNT}\end{aligned}$$ This Lagrangian is usually called CK-Lagrangian after its inventors P. Caldirola and E. Kanai [@CK]. Henceforward a damped free particle is considered only, i.e. the potential energy $U(q)$ is supposed to be zero. It is immediately clear that the transition amplitude computed from (\[propagator1\]) differs from the CK-amplitude $$\begin{aligned}
\mathbf{A}_{CK}(q_1,t_1|q_0,t_0)\hspace{-1mm}&=&\hspace{-1mm}\dfrac{1}{\mathbf{N}_{CK}}
\int[\mathscr{D}q]\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{t_0}^{t_1}\hspace{-1mm}L_{CK}(q,t)dt\Bigr\}}\\
&=&\hspace{-1mm}\dfrac{1}{\mathbf{N}_{CK}}\exp\{\tfrac{i}{\hslash}\,\mathbf{S}_{CK}(q_1,t_1|q_0,t_0)\}
\,.\end{aligned}$$ Indeed, the stringy and CK actions which enter the corresponding transition amplitudes are different, $$\begin{aligned}
&&\mathbf{S}_{1cl}=\frac{\kappa}{4}(q_1-q_0)\frac{(q_0+3q_1)\mathrm{e}^{-\kappa
t_1}-(q_1+3q_0)\mathrm{e}^{-\kappa t_0}}
{\mathrm{e}^{-\kappa t_0}-\mathrm{e}^{-\kappa t_1}},\\
&&\mathbf{S}_{CK}=\frac{\kappa}{2}\,\frac{(q_1-q_0)^2}{\mathrm{e}^{-\kappa
t_0}-\mathrm{e}^{-\kappa t_1}}.\end{aligned}$$ Since both actions are quadratic functions of the endpoints $q_0$ and $q_1$, and since we require that the total probability is conserved, the preexponential factors for both transition amplitudes can be computed from the Van Vleck formula $$%\begin{equation}\label{van-vleck}
\frac{1}{\mathbf{N}}=\sqrt{\frac{i}{2\pi\hslash}\,\frac{\partial^2 \mathbf{S}}{\partial q_1\,\partial q_0}}\,.$$Using this formula we arrive at the normalized transition amplitudes $$\begin{aligned}
\hspace{-7mm}\mathbf{A}_{1str}(q_1,t_1|q_0,t_0)\hspace{-1.8mm}&=&\hspace{-1.8mm}
\sqrt{\frac{\kappa}{4\pi i \hslash\tanh\tfrac{\kappa}{2}(t_1-t_0)}}\,
\mbox{\large{e}}^{\frac{i}{\hslash}\mathbf{S}_{1cl}},\label{damp.A}\\
\hspace{-7mm}\mathbf{A}_{CK}(q_1,t_1|q_0,t_0)\hspace{-1.8mm}&=&\hspace{-1.8mm}
\sqrt{\frac{\kappa}{2\pi i \hslash\,(\mathrm{e}^{-\kappa t_0}-\mathrm{e}^{-\kappa t_1})}}\,
\mbox{\large{e}}^{\frac{i}{\hslash}\mathbf{S}_{CK}},\label{damp.B}\end{aligned}$$ which trivially satisfy (\[normalization\]). Moreover, one immediately verifies that in the frictionless limit $\kappa\rightarrow 0$ both transition amplitudes coincide with the free particle propagator. A short inspection shows that the stringy propagator depends on $t_0$ and $t_1$ only through $t_1-t_0$, but this is not the case for the CK propagator. The same observation holds for $\mathbf{S}_{1cl}$ and $\mathbf{S}_{CK}$. This is a typical feature for all auxiliary actions defined in terms of integrators of $\Omega$, see [@all; @others; @II]. We pay attention to this point because the classical equation of motion $\ddot{q}=-\kappa\dot{q}$ which we have started with is invariant with respect to time translations, and one would naturally expect the same invariance on the quantum level. Since only $\mathbf{A}_{1str}(q_1,t_1|q_0,t_0)$ has this feature, we obtain an efficient argument for the stringy functional integral (\[FeynmanIII\]) when compared to the CK one. The evolution of Gaussian wave packets $$\Psi_0(q)\propto\mbox{\large{e}}^{-q^2+\tfrac{i}{\hslash}p_0q}\,\rightsquigarrow\,
\Psi_1(q)\propto\hspace{-1mm}\int\hspace{-1mm} dq'\mathbf{A}(q,t_1\,|\,q',t_0)\Psi_0(q')\,,$$ whose dynamics is governed by (\[damp.A\]) and (\[damp.B\]), is visualized in Figure \[CK-characteristics\]. For more detail, see [@kochan2].
![Expectation values of position (solid color lines) and momentum (dashed color lines) as functions of time for the Gaussian wave packet (the best fit for the moving classical particle), drawn for the stringy and CK propagators (\[damp.A\]) and (\[damp.B\]), respectively. Initial wave packet characteristics are $\langle q\rangle(t_0)=0\,\mathrm{m}$ and $\langle p\rangle(t_0)=5\,\mathrm{m\,s^{-1}}$, and the friction constant $\kappa$ is set to $0.6\,\mathrm{s^{-1}}$. One realizes that at the time $t_1-t_0=\ln{3}\ \kappa^{-1}$ the stringy mean momentum becomes zero. Classically, for $t_1-t_0>\kappa^{-1}$ the physical relevancy of the solution breaks down. This explains the time bound on the applicability of stringy propagator (\[damp.A\]).[]{data-label="CK-characteristics"}](figure3.pdf){height="11.18cm" width="8.5cm"}
From the figure we can see that the stringy averaged momentum $\langle p\rangle$ becomes negative (i.e. meaningless) for $t_1-t_0>\ln{3}\ \kappa^{-1}$. Let us try to explain this peculiarity. It is well known that the relevancy of the classical solution $q_{cl}(t)$ of $\ddot{q}=-\kappa\dot{q}$ breaks down when the time $t_1-t_0$ exceeds the relaxation time $\kappa^{-1}$. Since $q_{cl}$ is used in the derivation of the formula (\[damp.A\]) for the stringy propagator, its applicability is automatically restricted as well. After this restriction is taken into account, the stringy evolution of $\langle p\rangle$ becomes more acceptable from the physical point of view then the evolution in the CK approach.
Stringy *versus* Bateman approach
---------------------------------
The key element of the Bateman(-Morse-Feshbach) approach [@bateman] are new subsidiary degrees of freedom introduced in addition to the initial ones, which are amplified rather than damped; thus, they evolve according to the time reversed dynamical equations. In our case we have: $$\ddot{q}=-\kappa\dot{q}\ \ \mbox{(damped) \ AND}\ \
\ddot{Q}=+\kappa\dot{Q} \ \ \mbox{(amplified)}\,.$$ These equations of motion can be derived from the least action principle with the quadratic and time independent Bateman(-Morse-Feshbach) Lagrangian: $$\label{Bateman lagrangian}
L_B(q,\dot{q},Q,\dot{Q})=\dot{q}\dot{Q}+\frac{\kappa}{2}(q\dot{Q}-Q\dot{q})\,.$$ The canonical as well as path integral quantization of the theory encounters various difficulties because non-normalizable states of the amplified system must be employed [@ghosh-hasse]. This problem will be discussed later, when the treatment of the auxiliary $Q$-degrees of freedom in this approach will be described. The path integral evaluation of the transition probability amplitude based on $L_B$ is straightforward [@chetouani]. The result is: $$\begin{aligned}
&&\hspace{-1cm}\mathbf{A}_{B}(q_1,Q_1,t_1|q_0,Q_0,t_0)=\frac{\kappa}
{4\pi\hslash\sinh{\frac{\kappa T}{2}}}\times \nonumber\label{bateman propagator}\\
&&\hspace{-5mm}\times\exp\Bigl\{\frac{i}{\hslash}\alpha\bigl[Q_1(q_1-q_0\beta_{-})+
Q_0(q_0-q_1\beta_{+})\bigr]\Bigr\}\,,\end{aligned}$$ where $$\alpha=\frac{\kappa}{2}\,\frac{1}{\tanh{\frac{\kappa T}{2}}}\,,\ \
\ \beta_{\pm}=\frac{\mathrm{e}^{\pm\frac{\kappa
T}{2}}}{\cosh{\frac{\kappa T}{2}}}\,, \ \ \ T=t_1-t_0\,.$$ Our aim, however, is to find the transition amplitude for the damped (sub)system only. In order to obtain it we must project out the nonphysical $Q$-degrees of freedom. When using the standard formula $$\mathbf{A}_{B}^{\mathrm{trial}}(q_1,t_1|q_0,t_0)=\int dQ A_{B}(q_1,Q,t_1|q_0,Q,t_0)\,,$$ we do not reproduce the free particle propagator for $\kappa=0$ as desired. To overcome this trouble the following “repairing prescription” is introduced (for more detail see [@nemes-toledo; @piza]): the amplitude for the damped (sub)system to pass from $|q_0\rangle$ to $|q_1\rangle$ within the time $T$ is equal to the amplitude for the whole system to pass from the nonphysical state $|q_0\rangle|\Psi_{-}\rangle$ to the nonphysical state $|q_1\rangle|\Psi_{+}\rangle$, $$\begin{aligned}
&&\hspace{-1cm}\mathbf{A}_{B}^{\mathrm{eff}}(q_1,t_1|q_0,t_0):=
\mbox{Amp}\Bigl(|q_0\rangle|\Psi_{-}\rangle\rightarrow|q_1\rangle|\Psi_{+}\rangle\Bigr)=\nonumber\label{remedy}\\
&&\hspace{-1cm}=\int dQdQ'\,
\overline{\Psi}_{+}(Q')\,A_{B}(q_1,Q',t_1|q_0,Q,t_0)\,\Psi_{-}(Q)\,,\end{aligned}$$ where the non-normalizible $Q$-system states $|\Psi_{\pm}\rangle$ are chosen as $$\Psi_{\pm}(Q)=\sqrt{i\cosh{\tfrac{\kappa}{2}T}\sqrt{\frac{\mp 2i\alpha}{\pi\hslash}}}
\exp\bigl\{\pm \frac{i}{\hslash}\alpha Q^2\bigr\}\,.$$ Substituting (\[bateman propagator\]) into (\[remedy\]) we obtain the following effective propagator for the damped (sub)system: $$\begin{aligned}
&&\hspace{-1cm}\mathbf{A}_{B}^{\mathrm{eff}}(q_1,t_1|q_0,t_0):=\sqrt{\frac{\kappa}
{4\pi i\hslash\tanh{\frac{\kappa}{2}T}}}\times \nonumber\label{damp.C}\\
&&\hspace{-5mm}\times\exp\Bigl\{\frac{i}{\hslash}\frac{\alpha}{4}\Bigl[\bigl(q_1-q_0
\beta_{-}\bigr)^2+\bigl(q_0-q_1\beta_{+}\bigr)^2\Bigr]\Bigr\}\,.\end{aligned}$$ The form of the auxiliary states $|\Psi_{+}\rangle$ and $|\Psi_{-}\rangle$ guarantees that the effective Bateman propagator for the damped (sub)system satisfies the normalization condition (\[normalization\]).
![Expectation values of the position and momentum as functions of time for the Gaussian wave packet, drawn for the stringy and effective Bateman propagators (\[damp.A\]) and (\[damp.C\]), respectively. The initial wave packet characteristics and the meaning of solid and dashed color lines are the same as in Figure \[CK-characteristics\].[]{data-label="Bateman-characteristics"}](figure4.pdf){height="11.18cm" width="8.5cm"}
The Bateman effective propagator (\[damp.C\]) depends only on the time difference $T$, which favors it in comparison with the CK-propagator (\[damp.B\]). On the other hand, the artificial “repairing procedure” involving nonphysical states discredits the Bateman approach with respect to the stringy one. The Gaussian wave packet characteristics obtained from (\[damp.A\]) and (\[damp.C\]) are drawn in Figure \[Bateman-characteristics\]. As seen from the figure, the stringy quantization possesses again better qualitative features than the approach to which we have compared it.
Stringy *versus* Kostin approach
--------------------------------
A possible incorporation of the classical dynamics into the quantum one can be obtained by using Heisenberg equations for the position and momentum operators [^9]: $$\dot{\hat{q}}=\hat{p}\,,\ \ \ \dot{\hat{p}}=F(\hat{q},\hat{p})\,,$$ where $\hat{q}$ and $\hat{p}$ obey $\bigl[\hat{q},\hat{p}\bigr]=i\hslash$ for all $t$. In the special case when $F(\hat{q},\hat{p})=-dU(\hat{q})/dq-\kappa\hat{p}$, Kostin found an equivalent description of the system by the Schrödinger equation [@kostin]: $$\label{Kostin}
i\hslash\partial_t\Psi(q,t)=\Bigl[-\frac{\hslash^2}{2}\frac{d^2}{dq^2}
+U(q)+\hslash K[\Psi]\Bigr]\Psi(q,t)\,,$$ where $K[\Psi]$ is the $\Psi$-dependent Kostin potential defined as $$K[\Psi]=i\frac{\kappa}{2}\Bigl[\ln{\frac{\overline{\Psi}(q,t)}{\Psi(q,t)}}-
\int\hspace{-1.0mm} dq'\,\overline{\Psi}(q',t)\ln{\frac{\overline{\Psi}(q',t)}{\Psi(q',t)}}\Psi(q',t)\Bigr].$$ This is known as Kostin-Schrödinger(-Langevin) equation. In the Kostin’s paper a complete set of solutions of this equation in the special case of free particle ($U(q)=0$) is presented. The set consists of the states $\Psi_{p_0}(q,t)$ labeled by the continuous quantum number $p_0$, the initial momentum of the particle. The state $\Psi_{p_0}(q,t)$ starts as a momentum eigenstate with the momentum $p_0$, and remains the momentum eigenstate also later, but with the decreasing momentum, $$\hat{p}\Psi_{p_0}(q,t)=p_0\mathrm{e}^{-\kappa(t-t_0)}\Psi_{p_0}(q,t)\,.$$ From the ensemble of non-localized Kostin’s states one would like to form normalized wave packet solutions, localized in configuration as well as momentum space. This is, however, impossible since the Kostin-Schrödinger equation is nonlinear; thus, the Kostin approach belongs to nonlinear quantum mechanics and lacks superposition principle. In Figure \[Kostin-characteristics\] we plot the expectation value of the momentum for the Kostin state $\Psi_{p_0}(q,t_1)$ and the stringy evolved state $\Psi(q,t_1)=\int dq'
\mathbf{A}_{1str}(q,t_1|q',t_0)\Psi_{p_0}(q',t_0)$ for various values of $\kappa$ [^10]. As $\kappa$ tends to zero, the stringy and Kostin results match each other.
![Expectation value of the momentum as a function of time for an eigenstate of initial momentum $\Psi_{p_0}(q,t_0)\propto\exp\{\frac{i}{\hslash}qp_0\}$, drawn for the stringy propagator (\[damp.A\]) (blue dashed lines) and the propagator given by the Kostin-Schrödinger equation (\[Kostin\]) (red dashed lines), respectively. The values of the friction constant $\kappa$ are indicated next to the curves. The initial momentum is $p_0=5\,\mathrm{m\,s^{-1}}$.[]{data-label="Kostin-characteristics"}](figure5.pdf){height="11.18cm" width="8.5cm"}
Quantization of the Douglas system {#4.2}
==================================
After we have dealt with the weak non-Lagrangeanity, let us consider the simplest strongly non-Lagrangian system. It was proposed by Jesse Douglas (one of the two winners of the first Fields Medals awarded in 1936) when studying the inverse problem of variational calculus [@douglas]. The system is governed by the following dimensionless dynamical equations: $$\label{eqs.}
\ddot{x}=-\dot{y},\ \ \ddot{y}=-y\ \ \ \Longleftrightarrow\ \ \ {p_x=\dot{x},
\atop p_y=\dot{y},}\ \ \ {\dot{p}_x=-p_y, \atop \dot{p}_y=-y\,.}$$ A quick calculation shows that the associated two-form $$\Omega=d\bigl(p_xdx+p_ydy-\frac{p^2_x}{2}dt-\frac{p^2_y}{2}dt-\frac{y^2}{2}dt\bigr)-p_ydx\wedge dt$$ does not possess a non-trivial local integrator $f$. Consequently, equations (\[eqs.\]) cannot be obtained as Euler-Lagrange (Hamilton) equations.
From the standard point of view, this curious situation is stalemate: *no Lagrangian $\Rightarrow$ no Quantum Mechanics*. However, the surface integral method offers a way to overcome this deadlock. Using the stringy functional integration in the extended phase space we get $$\begin{aligned}
&&\hspace{-4mm}\int\limits_{\mathscr{U}_{\gamma_{cl}}}\hspace{-1mm}[\mathscr{D}\Sigma]
\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{\Sigma}\Omega\Bigr\}}\propto\nonumber\label{douglas}\\
&&\hspace{-1cm}\propto
\exp{\Bigl\{-\frac{i}{\hslash}\int\limits_{t_0}^{t_1}
\bigl(\tfrac{1}{2}\dot{x}_{cl}^2+\tfrac{1}{2}\dot{y}_{cl}^2-\tfrac{1}{2}y_{cl}^2-x_{cl}p_{ycl}
\bigr)dt\Bigr\}}\hspace{-1mm}\times\nonumber\\
&&\hspace{-1cm}\times\hspace{-1mm}\int[\mathscr{D}x\,\mathscr{D}y]\exp{\Bigl\{\frac{i}{\hslash}
\int\limits_{t_0}^{t_1}\hspace{-1mm}\bigl(\tfrac{1}{2}\dot{x}^2+\tfrac{1}{2}\dot{y}^2-\tfrac{1}{2}y^2-xp_{ycl}
\bigr)dt\Bigr\}}.\end{aligned}$$ Here, obviously, $(x_{cl}(t),p_{xcl}(t))$ and $(y_{cl}(t),p_{ycl}(t))$ stand for the classical solutions of (\[eqs.\]) matching the initial and final endpoints for which the quantum transition probability amplitude is sought for, $x_{cl}(t_0)=x_0$, $y_{cl}(t_0)=y_0$ and $x_{cl}(t_1)=x_1$, $y_{cl}(t_1)=y_1$. In the classical limit $\hslash\rightarrow 0$ we want the stringy propagator (\[FeynmanIII\]) to maintain the properties of the original dynamical system. This guides us to define: $$\mathbf{S}_{Dcl}(x_1,y_1,t_1|x_0,y_0,t_0)\hspace{-0.5mm}=\hspace{-1.5mm}\int\limits_{t_0}^{t_1}\hspace{-1mm}
\bigl(\tfrac{1}{2}\dot{x}_{cl}^2+\tfrac{1}{2}\dot{y}_{cl}^2-\tfrac{1}{2}y_{cl}^2-x_{cl}p_{ycl}\bigr)dt\,.$$ The exponential $\exp\{\frac{i}{\hslash}\mathbf{S}_{Dcl}\}$ of the classical action defined in such a way cancels the prefactor in (\[douglas\]). Finally we arrive at the following transition amplitude for the Douglas system: $$\begin{aligned}
&&\hspace{-4mm} \mathbf{A}_{D}(x_1,y_1,t_1|x_0,y_0,t_0)=\frac{1}{\mathbf{N}_D}\hspace{-1mm}\times\nonumber\\
&&\hspace{-1cm}\times\hspace{-1mm}\int[\mathscr{D}x\mathscr{D}y]
\exp{\Bigl\{\frac{i}{\hslash}\hspace{-1mm}\int\limits_{t_0}^{t_1}\bigl(\tfrac{1}{2}\dot{x}^2+\tfrac{1}{2}\dot{y}^2
-\tfrac{1}{2}y^2-xp_{ycl}\bigr)dt\Bigr\}}.\end{aligned}$$ The path integral in the configuration space we have constructed is quadratic and hence can be computed explicitly. The normalized transition amplitude is $$\mathbf{A}_D(x_1,y_1,t_1|x_0,y_0,t_0)=\frac{\mbox{\large{e}}^{\frac{i}{\hslash}\mathbf{S}_{Dcl}}}{2\pi i\hslash}\,\left|
\begin{array}{cc}
\dfrac{\partial^2 \mathbf{S}_{Dcl}}{\partial x_1\,\partial x_0} & \dfrac{\partial^2 \mathbf{S}_{Dcl}}{\partial x_1\,\partial y_0} \\
\vspace{-2mm}\\
\dfrac{\partial^2 \mathbf{S}_{Dcl}}{\partial y_1\,\partial x_0} & \dfrac{\partial^2 \mathbf{S}_{Dcl}}{\partial y_1\,\partial y_0}
\end{array}
\right|^{\frac 12}\hspace{-2mm},$$ where $$\begin{aligned}
\mathbf{S}_{Dcl}&=&
\frac{(x_1-x_0)^2}{2(t_1-t_0)}+\tan^2{\left(\frac{t_1-t_0}{2}\right)}\,\frac{(y_1+y_0)^2}{2(t_1-t_0)}\\
&+&\frac{\tan{\left(\frac{t_1-t_0}{2}\right)}}{(t_1-t_0)}\,(x_1-x_0)(y_1+y_0)-(x_1y_1-x_0y_0)\\
&+&\frac{3}{4\sin{(t_1-t_0)}}\bigl[(y_1^2+y_0^2)\cos{(t_1-t_0)}-2y_1y_0\bigr]\\
&-&\frac{(t_1-t_0)}{4\sin^2{(t_1-t_0)}}\bigl[(y_1^2+y_0^2)-2y_1y_0\cos{(t_1-t_0)}\bigr]\,.\end{aligned}$$
Conclusion, discussion and outlook {#5}
==================================
In the paper we have developed a new quantization method that generalizes the conventional path integral approach. Throughout the paper we considered only the nonrelativistic quantum mechanics of spinless systems. However, the generalization to the field theory is rather straightforward. We have just to pass from the space of particle positions to the space of field configurations.
The mathematical language of our exposition respects the Vladimir Arnoľd “principle of minimal generality.” Formulas are mostly written down in one (local) chart. From this, however, one can ascend to a global, coordinate free description employing bundles and jet prolongations. In order that we did not cloud up the main idea of the paper, we did not follow such fluffy approach. However, in the future it could be useful to analyze obstructions to the implementation of our method which arise from the global properties of the underlying space-time geometry.
Special attention was paid to the stringy quantization of dissipative systems and of the Douglas system. The latter cannot be quantized by any known quantization technique while the former can. In fact, dissipative (friction force) systems were studied extensively in the past and several approaches to their quantization were proposed. The stringy quantization was compared with three of them, Caldirola-Kanai, Bateman and Kostin, and it was shown that it either gives better results (in the first two cases) or is simpler to apply (in the third case).
Stringy *versus* Caldeira-Leggett model
---------------------------------------
As mentioned in the introduction, the quantization proposed here shares a common property with the well established approaches of Caldirola-Kanai, Bateman, Kostin and others: they all fail to cover decoherence phenomena. On the other hand, as we have seen, the stringy quantization produces a quantum propagator that does not respect the memoryless Chapman-Kolmogorov equation. We believe that this essential feature of the theory is measuring/reflecting the non-Lagrangeanity of the system on the quantum level.
From the author’s point of view one can rise a conceptual objection also against the particle-plus-environment model by Caldeira and Leggett [@particle+environment]. Their approach is microscopical except the way in which the spectral density $\rho_D(\omega)$ for the reservoir degrees of freedom is introduced. This density is not obtained from any kind of microscopical theory and the form it assumes is motivated solely by the necessity to obtain the dissipative term $-\kappa\dot{q}$ in the effective theory. On the top of it, the interaction Hamiltonian is such that it produces, after integrating out the reservoir degrees of freedom, an additional Langevin stochastic force $F(t)$. Thus, the effective classical motion of the particle is governed by the Brownian equation of motion $\dot{p}=-\kappa
p+F(t)$, which is conceptually different from the physical situation considered in section \[4.1\]. It is also worth to point out that the Caldeira-Leggett model cannot describe satisfactorily the friction force proportional to $-p^A$ with $A\neq 1$. There is no doubt that for $A=1$ the model gives better results than all effective theories mentioned before including the stringy one, but it is fair to say that it describes a different physical phenomenon, namely the quantum Brownian motion of a particle which is in thermal equilibrium with a reservoir.
Surface functional integral *versus* Feynman-Vernon
---------------------------------------------------
The goal of our paper was to write down a functional integral formula for the quantum transition amplitude in terms of the underlying classical equation of motion. However, we did not eliminate the artificial notions of pure states and classical action for non-Lagrangian systems. In the surface functional integral (\[FeynmanIII\]), these notions were implicitly present.
A possible alternative to our approach consists in expressing the transition probability (not the probability amplitude!) using the functional integral. The probability that the system evolves from the mixed state $\rho_{_0}(q_0,q_0^\prime,t_0)$ at the time $t_0$ to the mixed state $\rho_{_1}(q_1,q_1^\prime,t_1)$ at some later time $t_1$ is $$\begin{aligned}
\mathbf{P}_{\rho_{_0}\rightarrow\rho_{_1}} &\propto& \int dq_0dq_0^\prime dq_1dq_1^\prime\rho_{_0}(q_0,q_0^\prime,t_0)
\rho_{_1}(q_1,q_1^\prime,t_1)\times\\
&\times& \int[\mathscr{D}\tilde{\gamma}][\mathscr{D}\tilde{\gamma}^\prime]\exp\Bigl\{\frac{i}{\hslash}
\Bigl(\int\limits_{\tilde{\gamma}}-\int\limits_{\tilde{\gamma}^\prime}\Bigr)(pdq-Hdt)\Bigr\}\,.\end{aligned}$$ Here $\tilde{\gamma}$ and $\tilde{\gamma}^\prime$ are curves in the extended phase space whose $q$-projections connect $q_0$ with $q_1$ and $q_0^\prime$ with $q_1^\prime$ respectively. Using the Stokes theorem one is able to convert the difference of the line integrals of the one-form $pdq-Hdt$ into the surface integral of the two-form $\Omega=d(pdq-Hdt)$. Explicitly, $$\label{loop3}
\Bigl(\int\limits_{\tilde{\gamma}}-\int\limits_{\tilde{\gamma}^\prime}\Bigr)pdq-Hdt=-\int\limits_\Sigma\Omega-
\Bigl(\int\limits_{\lambda_1}-\int\limits_{\lambda_0}\Bigr) pdq\,,$$ where $\Sigma$ represents again a map from the parametric space $(t,s)\in[t_0,t_1]\times[0,1]$ to the extended phase space, $$\Sigma: (t,s)\mapsto
\Sigma(t,s)\equiv\bigl(q_\Sigma(t,s),p_\Sigma(t,s),t_\Sigma(t,s)=t\bigr)\,,$$ such that $\partial\Sigma=\tilde{\gamma}^\prime(t)-\lambda_1(s)-\tilde{\gamma}(t)+\lambda_0(s)$. The sideways $\lambda$-boundary curves[^11] of $\Sigma$ $$\lambda_0(s)\equiv\Sigma(t_0,s)\ \ \ \ \mbox{and}\ \ \ \
\lambda_1(s)\equiv\Sigma(t_1,s)$$ live in the instant phase spaces of $\mathbb{R}^{2n+1}[q^a,p_a,t]$, i.e. in the submanifolds $\mathbb{R}^{2n}[q^a,p_a,t_0]$ and $\mathbb{R}^{2n}[q^a,p_a,t_1]$. Moreover, as the intrinsic parameter $s$ varies from $0$ to $1$, the $q_\Sigma$-components of $\lambda_0(s)$ and $\lambda_1(s)$ vary from $q_0$ to $q_0^\prime$ and from $q_1$ to $q_1^\prime$ respectively.
Let us denote by $\mathscr{U}$ the space of all $\Sigma$-maps for which $${q_\Sigma(t_0,0)=q_0, \atop q_\Sigma(t_0,1)=q_0^\prime,}\ \
{q_\Sigma(t_1,0)=q_1, \atop q_\Sigma(t_1,1)=q_1^\prime,}\ \
{\lambda_0(s)\subset\mathbb{R}^{2n}[q,p,t_0], \atop
\lambda_1(s)\subset\mathbb{R}^{2n}[q,p,t_1].}$$ Since the right hand side of (\[loop3\]) depends only on $\Sigma$ and its two boundaries, the double path integral $\int[\mathscr{D}\tilde{\gamma}][\mathscr{D}\tilde{\gamma}^\prime]\cdots$ entering $\mathbf{P}_{\rho_{_0}\rightarrow\rho_{_1}}$ can be rewritten as a surface functional integral $\int[\mathscr{D}\Sigma]\cdots$, $$\begin{aligned}
\label{Feynman-Vernon}
\vspace{5mm}
\mathbf{P}_{\rho_{_0}\rightarrow\rho_{_1}}\hspace{-2mm}
&\propto&\hspace{-2mm} \int\hspace{-1mm} dq_0dq_0^\prime
dq_1dq_1^\prime\rho_{_0}
(q_0,q_0^\prime,t_0)\rho_{_1}(q_1,q_1^\prime,t_1)\times\nonumber\\
\hspace{-2mm}&\times&\hspace{-2mm}
\int\limits_{\mathscr{U}}[\mathscr{D}\Sigma]\exp\Bigl\{-\frac{i}{\hslash}\Bigl[
\int\limits_\Sigma\Omega+
\Bigl(\int\limits_{\lambda_1}\hspace{-1mm}-\hspace{-1mm}\int\limits_{\lambda_0}\Bigr)
pdq\Bigr]\Bigr\}\,.\end{aligned}$$ The formula we have just arrived at was derived under the assumption that $\Omega=d(pdq-Hdt)$. However, it is clear that the expression in the exponent is free of any reference to the Hamiltonian and requires just the classical equations of motion. Therefore it seems reasonable to postulate the probability formula (\[Feynman-Vernon\]) also for non-Lagrangian systems.
The approach we shortly presented here does not use the notion of pure states of a non-Lagrangian system. It resembles the Feynman-Vernon approach [@feynman-vernon], in which one introduces the influence functional in the presence of dissipative forces. Our probability formula (\[Feynman-Vernon\]) uses the surface functional integral in the extended phase space, while in the Feynman-Vernon approach one just has to calculate a double path integral in the configuration space. There is a chance that after a certain discretization we will be able to convert the surface functional integral into the double path integral in the configuration space (see appendix \[B\]), and as a result, we will find the explicit form of the influence functional. Work on this topic is in progress.
This research was supported by MŠ SR CERN-Fellowship Program and VEGA Grant 1/1008/09. Special thanks go to Vladimír Balek for his interest and fruitful discussions we have had in the course of the work.\
\
$\mathscr{A.}$ $\mathscr{M.}$ $\mathscr{D.}$ $\mathscr{G.}$
Stringy variational principle {#A}
=============================
In section \[3\] we shortly outlined how the transition amplitude (\[FeynmanIII\]) can be related to the stringy variational principle, with the action $\mathscr{S}$ defined in (\[action\]). In what follows we will perform a complete variation of $\mathscr{S}$, with the domain extended from $\mathscr{U}_{\gamma_{cl}}$ to a wider class $\mathscr{U}:=\bigcup_{\tilde{\gamma}_0}\bigcup_{\tilde{\gamma}_1}
\mathscr{U}_{(\tilde{\gamma}_1,\tilde{\gamma}_0)}$. The stringy set $\mathscr{U}$ contains all surfaces in the extended phase space trapped between the submanifolds with fixed $(q_0,t_0)$ and $(q_1,t_1)$ (including separate histories, regarded as shrunk surfaces). In the course of variation the classical dynamics (\[eq.\]) we have started from will be recovered.
Suppose we have an extremal surface $\Sigma_{ext}\in\mathscr{U}_{(\tilde{\gamma}_1,\tilde{\gamma}_0)}\subset\mathscr{U}$ and a variational vector field $W$ defined in its neighborhood such that the flow of $W$ preserves $\mathscr{U}$. By definition, $W$ moves $\Sigma_{ext}\in\mathscr{U}_{(\tilde{\gamma}_1,\tilde{\gamma}_0)}$ to some other worldsheet $\Sigma_{ext}^{\delta
W}\in\mathscr{U}_{(\tilde{\gamma}_1^{_{'}},\tilde{\gamma}_0^{_{'}})}$, where $\delta$ is an infinitesimal increment of the parameter of the flow generated by $W$. The situation is depicted schematically in Figure \[flow\].
![Vector field $W$ acting infinitesimally in the extended phase space. As a result, the initial surface $\Sigma_{ext}\in\mathscr{U}_{(\tilde{\gamma}_1,\tilde{\gamma}_0)}$ moves to some nearby surface $\Sigma_{ext}^{\delta
W}\in\mathscr{U}_{(\tilde{\gamma}_1^{_{'}},\tilde{\gamma}_0^{_{'}})}$, and its boundary $\partial\Sigma_{ext}=\tilde{\gamma}_1-\lambda_1-
\tilde{\gamma}_0+\lambda_0$ is transported to a new boundary $\partial\Sigma_{ext}^{\delta
W}=\tilde{\gamma}_1^{_{'}}-\lambda_1^{_{'}}-
\tilde{\gamma}_0^{_{'}}+\lambda_0^{_{'}}$. Since the variational vector field is assumed to preserve $\mathscr{U}$, $W$ stays tangential to the momentum submanifolds with fixed $(q_0,t_0)$ and $(q_1,t_1)$.[]{data-label="flow"}](figure6.pdf){height="11.18cm" width="8.5cm"}
The extremality of $\Sigma_{ext}$ means that for all variational $W$-fields it holds $$0=\lim\limits_{\delta\rightarrow 0}\frac{\mathscr{S}\bigl(\Sigma_{ext}^{\delta W}\bigr)-\mathscr{S}\bigl
(\Sigma_{ext}\bigr)}{\delta}=\hspace{-1.8mm}\int\limits_{\Sigma_{ext}}\hspace{-1.8mm}\mathcal{L}_{W}\Omega\,,$$ where $\mathcal{L}_W$ stands for the Lie derivative. Let us compute the surface integral on the left hand side: $$\begin{aligned}
&&\hspace{-2cm} \label{variation}
\int\limits_{\Sigma_{ext}}\hspace{-1.8mm}\mathcal{L}_{W}\Omega=\hspace{-1.8mm}\int\limits_{\Sigma_{ext}}
\hspace{-1.8mm}d\bigl(W\lrcorner\,\Omega\bigr)+W\lrcorner\,d\Omega=\nonumber\\
&&\hspace{1cm}=\int\limits_{\partial\Sigma_{ext}}\hspace{-2.5mm}W\lrcorner\,\Omega+\hspace{-1.8mm}\int
\limits_{\Sigma_{ext}}\hspace{-1.8mm}W\lrcorner\,d\Omega\,,\end{aligned}$$ where the the symbol $\lrcorner$ denotes the inner product (contraction) of a vector with a differential form. The first term on the right hand side depends only the values of $\Omega$ on the boundary of $\Sigma_{ext}$ and can be recast into the form $$\int\limits_{\partial\Sigma_{ext}}\hspace{-2.5mm}W\lrcorner\,\Omega=\Bigl(\int\limits_{\tilde{\gamma}_1}-
\int\limits_{\tilde{\gamma}_0}+\int\limits_{\lambda_0}-\int\limits_{\lambda_1}\Bigr)\,W\lrcorner\,\Omega\,.$$ The last two terms here give individually zero contributions, as can be seen from the following quick consideration: the vector field $W$ is assumed to preserve $\mathscr{U}$ and therefore its restrictions to the $\lambda$-boundaries are $$W\hspace{-0.35mm}\bigr|_{\lambda_{0}}\hspace{-1.9mm}=W_a\bigl(p,q_0,t_0\bigr)\frac{\partial}{\ \partial_{p_a}}\,,\ \ \
W\hspace{-0.35mm}\bigr|_{\lambda_{1}}\hspace{-1.9mm}=W_a\bigl(p,q_1,t_1\bigr)\frac{\partial}{\ \partial_{p_a}}.$$ The one-form we are integrating is $W\lrcorner\,\Omega=W_a\bigl(dq^a-(p^a/m)dt\bigr)$, but since both $q$ and $t$ stay unchanged on $\lambda$, the integral is zero. As a result, to annihilate the boundary term for all variational fields $W$ we are forced to chose the curves $\tilde{\gamma}_0=(\tilde{q}_0(t),\tilde{p}_0(t),t)$ and $\tilde{\gamma}_1=(\tilde{q}_1(t),\tilde{p}_1(t),t)$ in the extended phase space in such a way that their instant tangent vectors annihilate $\Omega$ at any moment $t\in[t_0,t_1]$. Thus, it must hold
\[boundary\] $$0=\frac{d\tilde{\gamma}}{dt}\lrcorner\,\Omega\,,$$ or, equivalently, $$\begin{aligned}
&&\hspace{-1cm}
0=(F_a-\dot{p}_a)dq^a+\left(\dot{q}^a-\frac{p^a}{m}\right)dp_a+\nonumber\\
&&\hspace{1.7cm}
+\left(\dot{q}^aF_a-\dot{p}_a\frac{p_a}{m}\right)dt\,,\end{aligned}$$
for both ${\tilde{\gamma}_0}$ and ${\tilde{\gamma}_1}$. Here one recognizes the dynamical equations (\[eq.\]) as desired.
From the boundary term analysis and the assumption about the uniqueness of the history $\gamma_{cl}$ which we adopted from the very beginning we can conclude that $\Sigma_{ext}\in\mathscr{U}_{(\gamma_{cl},\gamma_{cl})}\subset\mathscr{U}_{\gamma_{cl}}$. Hence $\Sigma_{ext}$ has the topology of a closed string attached to the classical trajectory $\gamma_{cl}$. To specify its shape the second term in (\[variation\]) must be employed. The resulting variational equation, written in a coordinate-free notation, is
\[bulk\] $$0=d\Omega\,\bigl(\partial_t\,\Sigma_{ext},\,\partial_s\,\Sigma_{ext},\,.\,\bigr)\,.$$ This is equivalent to the following system of partial differential equations for the unknown functions $q^a(t,s)$ and $p_a(t,s)$ (indices $a$ and $k$ run from $1,\dots, n$): $$\begin{aligned}
0&= \frac{\partial q^{k}}{\partial s}\,\frac{\partial F_k}{\partial p_{a}}\,,\\
0&= \frac{\partial q^{k}}{\partial s}\,\Bigl(\frac{\partial F_{a}}{\partial q^{k}}-\frac{\partial F_{k}}{\partial q^{a}}\Bigr)
+\frac{\partial p_{k}}{\partial s}\,\frac{\partial F_{a}}{\partial p_{k}}\,.
\end{aligned}$$
One solution of these equations, satisfying all boundary conditions, is trivial. It is the shrunk surface $\Sigma_{ext}(t,s)=\gamma_{cl}(t)$. After recalling the assumption of uniqueness of $\gamma_{cl}$ once more, we can see that this is the [*only*]{} solution of (\[bulk\]). If there existed a closed unshrunk extremal surface $\Sigma_{ext}$, the initial set of equations of motion (\[eq.\]) would have at least one one-parametrical family of classical solutions between the given pair of endpoints.
The variational principle we described above operates on a wider stringy class $\mathscr{U}$ than it is in fact necessary. The surface integral formula (\[FeynmanIII\]) requires just the restricted subset $\mathscr{U}_{\gamma_{cl}}\subset\mathscr{U}$. The transition to $\mathscr{U}_{\gamma_{cl}}$ is advisable for two reasons. First, in the Lagrangian case with $\Omega=d(pdq-Hdt)$ we obtain an equivalence (modulo additive constant) between the least action principle using $\mathscr{S}$ and the standard Hamilton least action principle. Explicitly, $$\Sigma\in\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}\subset\mathscr{U}_{\gamma_{cl}}\mapsto\mathscr{S}(\Sigma)=
\Bigl(\,\int\limits_{\tilde{\gamma}}-\int\limits_{\gamma_{cl}}\,\Bigr)(pdq-Hdt)\,.$$ Second, the wider class $\mathscr{U}$ contains plenty of degenerate (shrunk) surfaces, the histories $\tilde{\gamma}$. These are obviously stationary surfaces of $\mathscr{S}$ when being varied within $\mathscr{U}$, since $$\mathscr{S}(\tilde{\gamma}^{\delta W})-\mathscr{S}(\tilde{\gamma})=\hspace{-1mm}\int\limits_{\tilde{\gamma}^{\delta W}}
\hspace{-1mm}\Omega\,-\int\limits_{\tilde{\gamma}}\hspace{-0mm}\Omega=0-0=0$$ for any variational vector field $W$. However, only one of these histories, namely $\gamma_{cl}$, satisfies both equations (\[boundary\]) and (\[bulk\]) at the same time. This fictitious problem is avoided when one works from the beginning with the stringy subclass $\mathscr{U}_{\gamma_{cl}}\subset\mathscr{U}$.
Surface functional integral – computational details {#B}
===================================================
Let us explain here in some detail how the surface functional integral is computed. To be as tangible as possible consider a one-dimensional system only, so that the extended phase space will be the three dimensional space $\mathbb{R}^3[q,p,t]$ (more dimensions represent only a technical problem). The particle is supposed to move under the combined action of the potential $U(q)$ and the friction force with a general $A$-power law. The dynamical equations are $$\dot{q}=p\,,\ \ \dot{p}=-\frac{d}{dq}U(q)-\kappa\,\dot{p}^{A}\,,$$ and in addition to them, we require that the particle satisfies the boundary conditions $q(t_0)=q_0$ and $q(t_1)=q_1$. According to the definition (\[Omega\]), the two-form $\Omega$ is $$\label{alpha}
\Omega=d\bigl(pdq-\frac{1}{2}p^2dt-U(q)dt\bigr) -\kappa\, p^{A}dq\wedge dt\,.$$ Our aim is to compute the surface functional integral (\[FeynmanIII\]) over the stringy set $\mathscr{U}_{\gamma_{cl}}=\bigcup_{\tilde{\gamma}}\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$. The direct application of (\[surface\]) and (\[alpha\]) together with the Stokes theorem yields $$\begin{aligned}
& & \hspace{-5mm}\int\limits_{\mathscr{U}_{\gamma_{cl}}}\hspace{-1mm}[\mathscr{D}\Sigma]\exp{\Bigl\{\frac{i}{\hslash}
\int\limits_{\Sigma}\Omega\Bigr\}}=\label{giuseppe}\\
& & \hspace{-2mm}=\int[\mathscr{D}\tilde{\gamma}]\exp{\Bigl\{\frac{i}{\hslash}\Bigl(\,\int\limits_{\tilde{\gamma}}-
\int\limits_{\gamma_{cl}}\,\Bigr)\bigl(pdq-\frac{1}{2}p^2dt-U(q)dt\bigr)\Bigr\}}\times\nonumber\\
& & \hspace{-2mm}\times\hspace{-4mm}\int\limits_{\ \ \ \mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}}\hspace{-5mm}
[\mathscr{D}\Sigma]\exp{\Bigl\{-\frac{i}{\hslash}\kappa\int\limits_{\Sigma}p^{A}dq\wedge dt\Bigr\}}\,.\nonumber\end{aligned}$$ The nontrivial part of this is the functional integral over the stringy subset $\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$. As mentioned earlier, $\Sigma\in\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$ is a map $$\Sigma:[t_0,t_1]\times[0,1]\rightarrow\mathbb{R}^3[q,p,t]\,,\ \ \ (t,s)\mapsto\Sigma(t,s)\,,$$ such that for $\forall t\in[t_0,t_1]$ and $\forall s\in[0,1]$ there holds $$\label{constraints}
{\Sigma(t,0)=\gamma_{cl}(t) \atop \Sigma(t,1)=\tilde{\gamma}(t)\
\,}\ \ \ \ \mbox{and}\ \ \ \
{\Sigma(t_0,s)\in\mathbb{R}^3[q_0,p,t_0] \atop
\Sigma(t_1,s)\in\mathbb{R}^3[q_1,p,t_1]}\,.$$ To proceed further with the functional integral in question, introduce a set of regularly distributed nodal points in the parameter space, $$\bigl\{(t_0+\tau\,\varDelta,\sigma\,\varepsilon)\in\bigr[t_0,t_1]\times[0,1]\}\,.$$ The points of the set are labeled by two discrete indices, the time index $\tau=0,\dots, K$ and the space index $\sigma=0,\dots
L$, see Figure \[grid\]. In this way we obtain for any $\Sigma$-map from $\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$ a discretized $(K+1)\times (L+1)$-tuple $$\Sigma\leftrightsquigarrow\bigl\{\Sigma(t_0+\tau\varDelta,\sigma\varepsilon)\equiv\bigl(q_{(\tau,\sigma)},
p_{(\tau,\sigma)},
t_0+\tau\varDelta\bigr)\bigr\}_{(\tau,\sigma)=(0,0)}^{(K,L)}\,.$$ The boundary values are required to be consistent with (\[constraints\]). Explicitly, for all indices $\tau$ and $\sigma$ it must hold $${q_{(\tau,0)}=q_{cl}(t_0+\tau\varDelta), \atop
p_{(\tau,0)}=p_{cl}(t_0+\tau\varDelta),}\ \ \
{q_{(\tau,L)}=\tilde{q}(t_0+\tau\varDelta), \atop
p_{(\tau,L)}=\tilde{p}(t_0+\tau\varDelta),}\ \ \
{q_{(0,\sigma)}=q_0, \atop q_{(K,\sigma)}=q_1.}$$ The discretization of $\Sigma$ enables us to approximate the integral of $p^{A}dq\wedge dt$ as follows [^12]: $$\begin{aligned}
\fint\limits_\Sigma& &\hspace{-6mm} p^{A}dq\wedge dt:=\sum\limits_{\tau=0}^{K-1}\sum\limits_{\sigma=0}^{L-1}\varDelta\varepsilon\
\Bigl\{p^A_{(\tau,\sigma)} \frac{q_{(\tau,\sigma+1)}-q_{(\tau,\sigma)}}{\varepsilon}\Bigr\}=\nonumber\\
&=&\sum\limits_{\tau=1}^{K-1}\sum\limits_{\sigma=1}^{L-1}\,\varDelta\ \bigl\{q_{(\tau,\sigma)}\bigl(\,p^A_{(\tau,\sigma-1)}-
p^A_{(\tau,\sigma)}\,\bigr)\bigr\}+\nonumber\\
&+&\sum\limits_{\tau=1}^{K-1}\,\varDelta\bigl\{q_{(\tau,L)}\,p^A_{(\tau,L-1)}-q_{(\tau,0)}\,p^A_{(\tau,0)}\bigr\}\label{discr.1}\,.\end{aligned}$$
![Rectangular nodal web in the parametric space $[t_0,t_1]\times[0,1]$ of a surface $\Sigma\in\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$. Each elementary tile encloses the area $\varDelta\cdot\varepsilon=(t_1-t_0)/(KL)$ (at the end, the numbers $K$ and $L$ will be sent to infinity). The points marked by crosses are constrained by the conditions (\[constraints\]).[]{data-label="grid"}](figure7.pdf){height="11.18cm" width="8.5cm"}
Formally, the discretized functional integral over all stringy configurations from $\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}$ is a multiple integral over all unconstrained variables $x_{(\tau,\sigma)}, p_{(\tau,\sigma)}$ which are needed to specify $\Sigma$. The only problematic part, as usual, is the choice of an appropriate integration measure. For the reason that will become clear in a moment, we choose the measure as $$\label{discr.2}
\hspace{-3mm}\fint\limits_{\ \ \
\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}}\hspace{-5mm}[\mathscr{D}\Sigma]:=\hspace{-2mm}
\int\limits_{-\infty}^{+\infty}\
\prod\limits_{\tau=1}^{K-1}\prod\limits_{\sigma=1}^{L-1}\,\Bigl(\frac{A\kappa\varDelta}{2\pi
i\hslash}\,p^{A-1}_{(\tau,\sigma)}\Bigr)\,
dq_{(\tau,\sigma)}\,dp_{(\tau,\sigma)}.\hspace{-5mm}$$ The first step, when dealing with the discretized functional integral $$\mathrm{Int}\equiv\hspace{-4mm}\fint\limits_{\ \ \ \mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}}\hspace{-5mm}[\mathscr{D}\Sigma]
\exp{\Bigl\{-\frac{i}{\hslash}\kappa\fint\limits_{\Sigma}p^{A}dq\wedge dt\Bigr\}}\,,$$ consists in the integration over the internal stringy positions $dq_{(\tau,\sigma)}$. After this integration, the integrand transforms into a chain of delta functions, $$\begin{aligned}
\mathrm{Int}\hspace{-0.5mm}&=&\hspace{-2mm}\int\limits_{-\infty}^{+\infty}\prod\limits_{\tau=1}^{K-1}\prod\limits_{\sigma=1}^{L-1}\,
\bigl(A\,p^{A-1}_{(\tau,\sigma)}\bigr)\,dp_{(\tau,\sigma)}\,\mbox{\large{$\boldsymbol{\delta}$}}\bigl(p^A_{(\tau,\sigma-1)}-
p^A_{(\tau,\sigma)}\bigr)\times\\
&\times& \exp{\Bigl\{-\frac{i}{\hslash}\kappa
\sum\limits_{\tau=1}^{K-1}\,\varDelta\bigl[q_{(\tau,L)}\,p^A_{(\tau,L-1)}-q_{(\tau,0)}\,
p^A_{(\tau,0)}\bigr]\Bigr\}}\,.\end{aligned}$$ The next step is the integration over the internal stringy momenta $dp_{(\tau,\sigma)}$. After performing this trivial integration we arrive at the expression $$\mathrm{Int}=\exp{\Bigl\{-\frac{i}{\hslash}\kappa
\sum\limits_{\tau=1}^{K-1}\,\varDelta\bigl[q_{(\tau,L)}\,p^A_{(\tau,0)}-q_{(\tau,0)}\,
p^A_{(\tau,0)}\bigr]\Bigr\}}\,.$$ In the exponent there appears the discretized version of the integral $$\int\limits_{t_0}^{t_1}\,dt\bigl\{\tilde{q}(t)\,p^A_{cl}(t)-q_{cl}(t)\,p^A_{cl}(t)\bigr\}\,,$$ where $\tilde{q}(t)$ stands for the $q$-projection of $\tilde{\gamma}(t)$. Consequently, after returning back to the continuum limit we obtain the following important result: $$\begin{aligned}
\label{int.}
&&\hspace{-1.2cm}\int\limits_{\ \ \
\mathscr{U}_{(\tilde{\gamma},\gamma_{cl})}}\hspace{-5mm}[\mathscr{D}\Sigma]
\exp{\Bigl\{-\frac{i}{\hslash}\kappa\int\limits_{\Sigma}p^{A}dq\wedge dt\Bigr\}}=\nonumber\\
&&\hspace{-5mm}=\exp{\Bigl\{-\frac{i}{\hslash}\kappa
\int\limits_{t_0}^{t_1}\,dt\bigl[\tilde{q}(t)\,p^A_{cl}(t)-q_{cl}(t)\,p^A_{cl}(t)
\bigr]\Bigr\}}\,.\end{aligned}$$ After substituting (\[int.\]) into the initial formula (\[giuseppe\]) we recover a path integral in the extended phase space. The integral is quadratic in momenta, with the discretized (standard) Liouville measure $$\fint[\mathscr{D}\tilde{\gamma}]\ \cdots
=\int\limits_{-\infty}^{+\infty}\frac{d\tilde{p}_K}{2\pi\hslash}\,\prod\limits_{\tau=1}^{K-1}
\frac{d\tilde{p}_{\tau}\,d\tilde{q}_{\tau}}{2\pi\hslash}\
\cdots\,,$$ The integration over momenta can be carried out explicitly and we finally obtain a path integral in the configuration space only (tildes are removed from the position variables), $$\begin{aligned}
&&\hspace{-8mm}\int\limits_{\mathscr{U}_{\gamma_{cl}}}[\mathscr{D}\Sigma]
\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{\Sigma}\Omega\Bigr\}}=\nonumber\\
&&\hspace{-5mm}=\exp{\Bigl\{-\frac{i}{\hslash}\int\limits_{t_0}^{t_1}\Bigl(\frac{1}{2}
\dot{q}_{cl}^2-U(q_{cl})-\kappa q_{cl}\,p^A_{cl}\Bigr)dt\Bigr\}}\times\nonumber\\
&&\hspace{-5mm}\times\int[\mathscr{D}q]\exp{\Bigl\{\frac{i}{\hslash}\int\limits_{t_0}^{t_1}
\Bigl(\frac{1}{2}\dot{q}^2-U(q)-\kappa
q\,p^A_{cl}\Bigr)dt\Bigr\}}\,.\end{aligned}$$ This formula served us as a guide when introducing the classical action (\[A-action\]).
Let us make one final comment. In our functional measure (\[discr.2\]) no integration over the momenta $p_{(0,\sigma)}$ and $p_{(K,\sigma)}$ was prescribed. These momenta are however needed when the boundary of the discretized surface $\Sigma\in\mathscr{U}_{\gamma_{cl}}$ is specified. They define the auxiliary curves $\lambda_0(s)$ and $\lambda_1(s)$ introduced in section \[3\]. The curves were chosen completely arbitrarily, but as seen from (\[discr.1\]), the result is not affected by them at all. Hence, since everything substantial is independent of $p_{(0,\sigma)}$ and $p_{(K,\sigma)}$, it is justifiable to discard these quantities from the functional measure $[\mathscr{D}\Sigma]$. If we would not do that for some reason, they will integrate into an artificial infinite factor, which will be removed anyway after we apply the normalization conditions (\[normalization\]).
[99]{} S. T. Ali, M. Engliš: *Quantization Methods: A Guide for Physicists and Analysts*, Rev. Math. Phys. **17** (2005), 391-490, arXiv: math-ph/0405065. P. Caldirola: *Forze non conservative nella meccanica quantistica*, Nuovo Cim. **18** (1941), 393-400.
E. Kanai: *On the Quantization of the Dissipative Systems*, Prog. Theor. Phys. **3** (1948), 440-442. H. Bateman: *On Dissipative Systems and Related Variational Principles*, Phys. Rev. **38** (1931), 815-819. M. D. Kostin: *On the Schrödinger-Langevin Equation*, J. Chem. Phys. **57** (1972), 3589-3591. H. Dekker: *On the Quantization of Dissipative Systems in the Lagrange-Hamilton Formalism*, Z. Phys. B **21** (1975) 295-300. M. Razavy: *On the Quantization of Dissipative Systems*, Z. Physik B **26** (1977) 201-206. J. Geicke: *Semi-classical quantisation of dissipative equations*, J. Phys. A: Math. Gen. **22** (1989) 1017-1025. J. Douglas: *Solution of the inverse problem in the calculus of variations*, Trans. Am. Math. Soc. **50** (1941), 71-128. A. O. Caldeira, A. J. Leggett: *Path integral approach to quantum Brownian motion*, Physica **121A** (1983), 587-616.
U. Weiss: *Quantum Dissipative Systems* (second edition), World Scientific 1999, Singapore. R. P. Feynman, F. L. Vernon: *The Theory of a General Quantum System Interacting with a Linear Dissipative System*, Annals of Physics **24** (1963), 118-173. F. J. Dyson: *Feynman’s Proof of the Maxwell Equations*, Am. J. Phys. **58** (1990), 209-211. E. P. Wigner: *Do the Equations of Motion Determine the Quantum Mechanical Commutation Relations?*, Phys. Rev. **77** (1950), 711-712.
C. N. Yang, D. Feldman: *The S-Matrix in the Heisenberg Representation*, Phys. Rev. **79** (1950), 972 - 978.
E. Nelson: *Derivation of the Schrödinger Equation from Newtonian Mechanics*, Phys. Rev. **150** (1966), 1079-1085.
S. Okubo: *Does the Equation of Motion determine Commutation Relations?*, Phys. Rev. D **22** (1980), 919-923. P. O. Kazinski, S. L. Lyakhovich, A. A. Sharapov: *Lagrange structure and quantization*, JHEP 0507 (2005) 076, arXiv: hep-th/0506093.
S. L. Lyakhovich, A. A. Sharapov: *Quantizing non-Lagrangian gauge theories: an augmentation method*, JHEP 0701 (2007) 047, arXiv: hep-th/0612086. D. M. Gitman, V. G. Kupriyanov: *Canonical quantization of so-called non-Lagrangian systems*, Eur. Phys. J. C **50** (2007), 691-700, arXiv: hep-th/0605025. R. P. Feynman: *Space-Time Approach to Non-Relativistic Quantum Mechanics*, Rev. of Mod. Phys. **20** (1948), 367-387.
R. P. Feynman, A. R. Hibbs: *Quantum Mechanics and Path Integrals*, McGraw-Hill Inc., New York, 1965. M. Henneaux: *Equations of motion, Commutation Relations and Ambiguities in the Lagrangian Formalism*, Annals Phys. **140** (1982), 45-64.
M. Henneaux: *On the Inverse Problem of the Calculus of Variations*, J. Phys. A: Math. Gen. **15** (1982), L93-L96.
W. Sarlet: *The Helmholtz Condition Revisited. A New Approach to the Inverse Problem of Lagrangian Dynamics*, J. Phys. A: Math. Gen. **15** (1982), 1503-1517. D. Kochan: *Direct quantization of equations of motion*, Acta Polytechnica **47** No. 2-3 (2007), 60-67, arXiv: hep-th/0703073.
D. Kochan: *How to Quantize Forces(?): An Academic Essay on How the Strings Could Enter Classical Mechanics*, J. Geom. Phys. 60 (2010), 219-229, arXiv: hep-th/0612115.
D. Kochan: *Quantization of Non-Lagrangian systems: some irresponsible speculations* AIP Conf. Proc. **956** (2007), 3-8. C. Stuckens, D. H. Kobe: *Quantization of a particle with a force quadratic in the velocity*, Phys. Rev. A **34** (1986) 3565-3567. H. Dekker: *Classical and quantum mechanics of the damped harmonic oscillator*, Phys. Rep. **80** (1981), 1-112. M. Razavy: *Classical and Quantum Dissipative Systems*; Imperial Colleges Press, London, 2005. I. C. Moreira: *Propagators for the Caldirola-Kanai-Schr" odinger Equation*, Lett. Nuovo Cim. **23** (1978), 294-298.
A. D. Jannussis, G. N. Brodimas, A. Strectlas: *Propagator with friction in Quantum Mechanics*, Phys. Lett. **74A** (1979), 6-10.
Bin Kang Cheng: *Exact evaluation of the propagator for the damped harmonic oscillator*, J. Phys. A: Math. Gen. **17** (1984), 2475-2484.
U. Das, S. Ghosh, P. Sarkar, B. Talukdar: *Quantization of Dissipative Systems with Friction Linear in Velocity*, Physica Scripta **71** (2005), 235-237. D. Kochan: *Quantization of Non-Lagrangian Systems*, Int. J. Mod. Phys. A **24** Nos. 28 $\&$ 29 (2009), 5319-5340. G. Ghosh, R. W. Hasse: *Coherent state and the damped harmonic oscillator*, Phys. Rev. A **24** (1981), 1621-1623. L. Chetouani et al: *Path integral for the damped harmonic oscillator coupled to its dual*, J. Math. Phys. **35** (1994), 1185-1191. M. C. Nemes, A. F. R. de Toledo Piza: *A Quantum Phenomenology of Viscosity*, Revista Brasileira de Física **7** No.2 (1977), 261-270.
M. C. Nemes, A. F. R. de Toledo Piza: *Quantization of a phenomenological viscous force*, Phys. Rev. A **27** (1983), 1199-1202.
[^1]: It took place at EPFL Laussane 16.-17.3.2009.
[^2]: For the record, the extended phase space is locally described by the coordinates in the configuration space $q^a$, the canonically conjugated momenta $p_a$ and the time coordinate $t$ (the index $a$ runs from 1 to the number of degrees of freedom $n$). One assumes the standard Poisson brackets $\{p_a,q^b\}=-\delta_a^b$. In a more elevated language, the extended phase space is the space $T^*M\times\mathbb{R}$, where $M$ stands for the configuration space and $\mathbb{R}$ for time.
[^3]: The integrals over $\lambda_0$ and $\lambda_1$ give trivially zero contributions to the contour integral, since on both $\lambda$’s $dq=0$ and $dt=0$.
[^4]: The precise definition is: suppose there are two histories $\tilde{\gamma}_0(t)$ and $\tilde{\gamma}_1(t)$ in the extended phase space whose projections $\mathit{Proj}(\tilde{\gamma}_0)$ and $\mathit{Proj}(\tilde{\gamma}_1)$ on the extended configuration space connect $(q_0,t_0)$ with $(q_1,t_1)$. Then $\mathscr{U}_{(\tilde{\gamma}_1,\tilde{\gamma}_0)}$ contains all continuous and oriented surfaces $\Sigma$ in the extended phase space such that $$\hspace{5mm}{\Sigma(t,s=0)=\tilde{\gamma}_0(t) \atop
\Sigma(t,s=1)=\tilde{\gamma}_1(t)}\ \ \ \mbox{and}\ \ \
{\mathit{Proj}\bigl(\Sigma(t=t_0,s)\bigr)=(q_0,t_0) \atop
\mathit{Proj}\bigl(\Sigma(t=t_1,s)\bigr)=(q_1,t_1)}\,.$$ If $\tilde{\gamma}_0\neq\tilde{\gamma}_1$, the orientation of $\Sigma$ is supposed to be such that $\tilde{\gamma}_1-\tilde{\gamma}_0$ is a part of the boundary of $\Sigma$. The $\mathit{Proj}$ection is simply the momentum forgetting map, or to put it in a more elevated way, the canonical projection from the cotangent bundle to the base manifold. A special case is the set $\mathscr{U}_{(\tilde{\gamma},\tilde{\gamma})}$, comprised of all closed surfaces in the extended phase space containing $\tilde{\gamma}$. In what follows, we consider any history $\tilde{\gamma}$ as a degenerated (shrunk) closed surface, hence $\tilde{\gamma}\in\mathscr{U}_{(\tilde{\gamma},\tilde{\gamma})}$ by definition.
[^5]: Strictly speaking, the expression for the friction force can be used only for the motion with increasing $q$. The universally applicable expression is $-\kappa$0.5mm sgn$(\dot{q})|\dot{q}|^{A}$.
[^6]: The prefix $A$ in the subscript $Acl$, as well as in the subscript $Astr$ introduced later, refers to the power in the expression for the friction force.
[^7]: The external source in the functional integral (\[propagator\]) can be viewed as a sort of additional time-dependent potential, therefore one would expect that the Chapman-Kolmogorov equation (\[Ch-K\]) will be satisfied. However, this argument is not applicable in the present situation. The point is that if we merge two paths *KL* and *LM* into a new path *KM*, then the time-dependent potentials $-\kappa qp^A_{cl-KL}(t)$ and $-\kappa qp^A_{cl-LM}(t)$ differ, for a general position of $\emph{L}$, from the potential $-\kappa qp^A_{cl-KM}(t)$ in the corresponding time (sub)intervals, $$-\kappa qp^A_{cl-KM}(t)\neq \Bigl\{{-\kappa qp^A_{cl-KL}(t)\
\mbox{for}\ t\in[t_K,t_L] \atop -\kappa qp^A_{cl-LM}(t)\ \mbox{for}\ t\in[t_L,t_M]}$$ Since this differs substantially from the standard case, the argument about the validity of the Chapman-Kolmogorov equation cannot be used.
[^8]: An exhaustive list of various quantization techniques applicable in the presence of linear dissipation, including those mentioned in the introduction, can be found in [@dekker] and [@razavy].
[^9]: The ordering problem and its consequences are not discussed here.
[^10]: Since the traveling wave $|p_0\rangle$ is not a square integrable function, we evolved by $\mathbf{A}_{1str}(q,t_1|q',t_0)$ a modified (normalized) state $|\Psi_\epsilon\rangle=\int
dp\,\mathrm{e}^{-\epsilon(p-p_0)^2}|p\rangle$ ($\epsilon>0$). After the stringy evolution of $|\Psi_\epsilon\rangle$, the expectation value of the momentum was computed and the regularization parameter $\epsilon$ was set to zero.
[^11]: The situation here strongly resembles that discussed in section \[3\]. However, there is one substantial difference between the two cases. In the present case, we allow the $\lambda$-curves to vary in both coordinates $q$ and $p$, only the time $t$ must stay unchanged along them. In the case considered in section \[3\] we were more stringent: the $\lambda$-curves were allowed to vary only with respect to $p$ and had to stay unchanged with respect to both $q$ and $t$.
[^12]: From now on we are using a new symbol $\fint$ for all discretized integrals. In the continuum limit $K$ and $L$ approach independently infinity and $\fint\to\int$. However, we must keep in mind that we must perform simultaneously the limits $\varDelta \to 0$ and $\varepsilon \to 0$ so that the quantities $K\varDelta=t_1-t_0$ and $L\varepsilon=1$ stay finite.
|
---
abstract: 'This paper demonstrates the anisotropic response of quantum critical fluctuations with respect to the direction of the magnetic field $B$ in Ni-doped CeCoIn$_5$ by measuring the magnetization $M$ and specific heat $C$. The results show that $M/B$ at $B=0.1\ {\rm T}$ for both the tetragonal $c$ and $a$ directions exhibits $T^{-\eta}$ dependencies, and that $C/T$ at $B=0$ follows a $-\ln T$ function, which are the characteristics of non-Fermi-liquid (NFL) behaviors. For $B\,||\,c$, both the $M/B\propto T^{-\eta}$ and $C/T \propto -\ln T$ dependencies change into nearly temperature-constant behaviors by increasing $B$, indicating a crossover from the NFL state to the Fermi-liquid state. For $B\,||\,a$, however, the NFL behavior in $C/T$ persists up to $B=7\ {\rm T}$, whereas $M/B$ exhibits temperature-independent behavior for $B\ge 1\ {\rm T}$. These contrasting characteristics in $M/B$ and $C/T$ reflect the anisotropic nature of quantum critical fluctuations; the $c$-axis spin component significantly contributes to the quantum critical fluctuations. We compare this anisotropic behavior of the spin fluctuations to superconducting properties in pure CeCoIn$_5$, especially to the anisotropy in the upper critical field and the Ising-like characteristics in the spin resonance excitation, and suggest a close relationship between them.'
author:
- Makoto Yokoyama
- Kohei Suzuki
- Kenichi Tenya
- Shota Nakamura
- Yohei Kono
- Shunichiro Kittaka
- Toshiro Sakakibara
title: |
Anisotropic magnetic-field response of quantum critical fluctuations\
in Ni-doped CeCoIn$_{\bm{5}}$
---
Introduction
============
The role of spin fluctuations in unconventional superconductivity is a long-standing subject in the physics of strongly correlated electron systems. The unconventional superconducting (SC) phase commonly emerges in the vicinity of magnetic orders in many strongly correlated electron systems, such as high-$T_c$ cuprates, FeAs-based alloys, and heavy fermion compounds. In particular, the heavy fermion compounds often exhibit SC order proximity to a magnetic quantum critical point (QCP), corresponding to a magnetic phase transition at zero temperature. Hence, quantum critical fluctuations that are enhanced around the QCP are expected to play a critical role in the SC order of the heavy fermion compounds.
Among the heavy fermion superconductors, CeCoIn$_5$ has attracted continuous interest because of its anomalous SC properties coupled with magnetic correlations [@rf:Petrovic2001]. This compound has a HoCoGa$_5$-type tetragonal structure \[Fig. 1(b), inset\] and exhibits a SC order below $T_c=2.3\ {\rm K}$. The magnetically mediated pairing mechanism of the SC order is inferred from the $d$-wave ($d_{x^2-y^2}$) symmetry of the SC gap [@rf:Izawa2001; @rf:An2010; @rf:Park2008]. The inelastic neutron scattering experiments have revealed that a resonance excitation involving the tetragonal $c$-axis spin component develops in the SC state [@rf:Stock2008; @rf:Raymond2015; @rf:Song2016; @rf:Mazzone2017; @rf:Stock2018; @rf:Eremin2008; @rf:Chubukov2008; @rf:Michal2011]. Furthermore, applying the magnetic field $B$ yields another SC phase that coexists with an incommensurate antiferromagnetic (AFM) modulation (the so-called $Q$ phase) at very low temperatures below 0.3 K and at high fields just below $H_{c2}$ for $B\,\perp \,c$ [@rf:Bianchi2003-1; @rf:Kakuyanagi2005; @rf:Young2007; @rf:Kenzelmann2008; @rf:Aperis2008; @rf:Agterberg2009; @rf:Yanase2008; @rf:Yanase2009]. All of these features indicate a close coupling between the anisotropic spin correlations and the SC state, but the nature of the spin correlations with respect to CeCoIn$_5$ has not yet been fully uncovered.
A key to clarifying the relationship between the spin correlations and the anomalous SC properties is expected to be found in the field-induced non-Fermi-liquid (NFL) behaviors observed under $B$ when applied along the $c$ axis. At $B\sim 5\ {\rm T}$, the specific heat divided temperature exhibits $-\ln T$ dependence, and both electrical resistivity and magnetization follow nearly $T$-linear functions down to very low temperatures [@rf:Bianchi2003-2; @rf:Paglione2003; @rf:Tayama2002]. It is widely believed that spin fluctuations enhanced near an AFM QCP are responsible for these NFL behaviors [@rf:Paglione2003; @rf:Bianchi2003-2; @rf:Tokiwa2013]. In fact, substituting the ions for elements in CeCoIn$_5$, such as Nd for Ce [@rf:Hu2008; @rf:Raymond2014], Rh for Co [@rf:Zapf2001; @rf:Yoko2006; @rf:Yoko2008; @rf:Ohira-Kawamura2007], and Cd, Hg, and Zn for In [@rf:Pham2006; @rf:Nicklas2007; @rf:Yoko2014; @rf:Yoko2015], can induce long-range AFM orders. Moreover, possible field-induced AFM ordering at extremely low temperatures ($T\le 20\ {\rm mK}$) has been proposed by a recent quantum oscillation measurement for pure CeCoIn$_5$ [@rf:Shishido2018].
In contrast, the substitutions of Sn for In [@rf:Bauer2005; @rf:Bauer2006; @rf:Ramos2010] and Ni for Co [@rf:Otaka2016] do not induce the AFM phase, but simply yield paramagnetic ground states through the suppression of the SC phase. In a recent study, we have revealed that in the mixed compound CeCo$_{1-x}$Ni$_x$In$_5$, the SC transition temperature $T_c$ monotonically decreases from 2.3 ($x=0$) to 0.8 K ($x=0.20$) with increasing $x$; subsequently, the SC order disappears above the critical Ni concentration $x=0.25$ [@rf:Otaka2016]. At this concentration, the NFL behaviors are realized around the zero field, characterized by the $-\ln T$ dependence in the specific heat divided by the temperature, the weak diverging behavior in the magnetization, and the nearly $T$-linear behavior of the electrical resistivity [@rf:Otaka2016]. These NFL features are quite similar to those seen in pure CeCoIn$_5$, strongly suggesting that the NFL anomaly observed in Ni-doped CeCoIn$_5$ also originates from the AFM quantum critical fluctuations. Furthermore, the effective magnetic moment for $x\le 0.3$, estimated from the Curie-Weiss law at high temperatures, is nearly independent of $x$ and coincides well with that calculated from the $J=5/2$ multiplet in the Ce$^{3+}$ ion [@rf:Otaka2016], suggesting that the Ce 4$f$ electrons are mainly responsible for the magnetic properties in pure and Ni-doped CeCoIn$_5$.
The observation of the NFL behavior at the zero field in Ni-doped CeCoIn$_5$ provides an opportunity to investigate the magnetic anisotropy of the quantum critical fluctuations. In pure CeCoIn$_5$, in contrast, it is difficult to perform such an investigation with typical macroscopic measurements, because the quantum critical behavior is suppressed (or masked) by the SC phase at low magnetic fields and is visible only at very low temperatures above $\mu_0H_{c2}$ ($4.9\ {\rm T}$ for $B\,||\,c$ and $11.6\ {\rm T}$ for $B\,||\,a$) [@rf:Ronning2005; @rf:Hu2012]. Consequently, the magnetic anisotropy of the quantum critical fluctuations remains unclear. In this paper we demonstrate the anisotropic changes of the NFL behaviors in the magnetization and specific heat between $B\,||\,c$ and $B\,||\,a$ in CeCo$_{1-x}$Ni$_x$In$_5$, and we discuss the relationship between the anisotropic spin fluctuations and the SC properties in pure and Ni-doped CeCoIn$_5$.
Experiment Details
==================
A single crystal of CeCo$_{1-x}$Ni$_{x}$In$_5$ with $x=0.25$ was grown using an Indium flux technique, the details of which are described elsewhere [@rf:Otaka2016]. The energy dispersive x-ray spectroscopy (EDS) and the inductively coupled plasma mass spectrometry (ICP-MS) measurements for the sample indicated that the actual Ni concentration approximately coincided with the starting (nominal) value within the deviation of $\Delta x/x\sim 17\%$, including the experimental error. Furthermore, through the EDS measurements, we confirmed the homogeneous distributions of the elements in the single crystal prepared for the experiments. The magnetization along the $c$ and $a$ axis was measured in temperatures as low as 0.1 K and in the magnetic field $B$ ($\mu_0H$) at up to 8 T with a capacitively detected Faraday force magnetometer [@rf:Sakakibara94]. A commercial SQUID magnetometer (MPMS, Quantum Design) was used for the magnetization measurements in the temperature range of 2.0–300 K and the magnetic field at up to 5 T. The specific heat $C_p$ was measured in the temperature range of 0.31–4 K and in the field range of 0–7 T with a conventional quasiadiabatic technique.
Results
=======
Figures 1(a) and 1(b) show the temperature dependencies of the $c$- and $a$-axis magnetization divided by the magnetic field $M/B$ respectively. Note that the $M/B$ data are plotted with logarithmic scales for both the vertical and horizontal axes. $M/B$ for both directions showed qualitatively similar features. Namely, $M/B$ at the lowest field ($B=0.1\ {\rm T}$) exhibited diverging behavior with a $T^{-\eta}$ function ($\eta < 1$) as the temperature decreased. The $T^{-\eta}$ dependence in $M/B$ was realized in a very wide temperature range of 0.1–10 K for both directions. It is natural to conclude that this NFL behavior originates from the quantum critical fluctuations, because similar NFL behaviors are also found in various macroscopic quantities in pure CeCoIn$_5$ [@rf:Bianchi2003-2; @rf:Paglione2003; @rf:Tokiwa2013] and its doped alloys [@rf:Bauer2005; @rf:Bauer2006; @rf:Yoko2017]. In both the $c$- and $a$-axis magnetization, the diverging feature was reduced by further applying $B$, and the $M/B$-constant behavior was then realized at low temperatures.
{width="45.00000%"}
From a quantitative viewpoint, however, a significant anisotropy was found in the NFL region between the $c$-axis and $a$-axis magnetization $M_c$ and $M_a$, respectively. The exponent $\eta$ of $M/B\propto T^{-\eta}$ at $B=0.1\ {\rm T}$ in $M_c$ \[$\eta_c=0.20(2)$\] was larger than the value \[$\eta_a=0.12(2)$\] in $M_a$, as the details of those evaluation procedures are described later. Furthermore, the magnitude of $M_c$ at $B=0.1\ {\rm T}$ and $T=0.1\ {\rm K}$ was twice that of $M_a$. The diverging behavior in the temperature variation of $M_c$ was thus stronger than that of $M_a$. Indeed, this feature can be verified by considering the variation of $M_c/M_a$ as a function of temperature (Fig. 2). $M_c/M_a$ exhibited a peak with a magnitude of $1.5$ at $\sim 60\ {\rm K}$. The peak structure was also observed in the other physical quantities, such as the electrical resistivity [@rf:Petrovic2001], and its origin is considered a development of a coherent heavy-fermion state below this temperature. The $M_c/M_a$ value was reduced to $1.32$ with decreasing temperatures, down to $\sim 15\ {\rm K}$. However, the spin fluctuations, associated with the NFL behavior, enhanced the $M_c/M_a$ value at low temperatures again; $M_c/M_a$ for $B=0.1\ {\rm T}$ increased with a decrease in temperature below $\sim 15\ {\rm K}$ and then reached 1.96(10) at 0.11 K. In a high magnetic-field region, in contrast, $M_c/M_a$ for $B=5$ and 8 T exhibited a saturation to the values of 1.55(3) and 1.46(2) at low temperatures, respectively, remaining with magnitudes comparable to those in high temperatures. These experimental results suggest that the NFL anomaly involved mainly the $c$-axis spin component.
![ Temperature variations of the ratio of the $c$-axis and the $a$-axis magnetization $M_c/M_a$ for CeCo$_{0.75}$Ni$_{0.25}$In$_5$. A $T^{-0.08}$ $[=T^{-(\eta_c-\eta_a)}]$ function is represented as a dashed line for comparison. ](fig2.pdf){width="45.00000%"}
Figures 3(a) and 3(b) show the specific heat divided by the temperature $C_p/T$ obtained under various fields along the $c$ and $a$ axis, respectively. For $B\,||\,c$, $C_p/T$ was markedly enhanced below $\sim 0.7\ {\rm K}$ for $B \ge 3\ {\rm T}$, although its temperature dependence became weak with increasing $B$ at high temperatures. This enhancement is considered to be caused by the Zeeman splitting of the nuclear spins. Such an effect should also be included in $C_p/T$ for $B\,||\,a$. To eliminate this contribution, we estimated the nuclear Schottky anomaly $C_{\rm nucl}$ by performing a calculation based on the natural abundance of the nuclear spins in the sample \[Fig. 3(b), inset\]. At $B=7\ {\rm T}$ and 0.4 K, the fraction of $C_{\rm nucl}$ in $C_p$ was estimated to be 14% for $B\,||\,c$ and 12% for $B\,||\,a$. Note that the contribution of Ni and Co nuclear spins was only 10% in $C_{\rm nucl}$; therefore, the ambiguity of the Ni/Co concentration ($\Delta x/x\sim 17\%$) in the sample is negligible in the estimation of $C_{\rm nucl}$.
{width="45.00000%"}
Figures 4(a) and 4(b) display the specific heat data obtained by subtracting the nuclear spin contribution $C/T \equiv (C_p -C_{\rm nucl})/T$ for $B\,||\,c$ and $B\,||\,a$, respectively. $C/T$ for $B=0$ increased with decreasing temperature, with a nearly $-\ln T$ dependence at temperatures as low as 0.31 K. As displayed in Fig. 4(a), this feature was markedly suppressed by applying $B$ along the $c$ axis, and $C/T$ eventually became nearly independent of temperature at $B=7\ {\rm T}$. The feature of suppression in $C/T$ coincides fairly well with that observed in $M_c/B$ \[Fig. 1(a)\]; hence, these behaviors are attributed to a crossover from the NFL to Fermi-liquid (FL) states.
{width="45.00000%"}
However, it was found that the $-\ln T$ diverging behavior in $C/T$ was not suppressed as much by $B$ for $B\,||\,a$. At 0.4 K, the reduction of the specific heat at $B=5\ {\rm T}$, $1-C(5\ {\rm T})/C(0\ {\rm T})$, was estimated to be 5% for $B\,||\,a$, whereas it was 14% for $B\,||\,c$. In addition, $C/T$ at $B=7\ {\rm T}$ continued to increase with the decreasing temperature for $B\,||\,a$, whereas it was nearly independent of temperature for $B\,||\,c$. Similar weak $B$ dependence of $C/T$ for $B\,||\,a$ was also found at a very high $B$ region above $\mu_0H_{c2}=11.6\ {\rm T}$ in pure CeCoIn$_5$ [@rf:Ronning2005]. This weak $B$ dependence in $C/T$ for $B\,||\,a$ is in stark contrast to the rapid reduction of $M_a/B$ with $B$ for the same $B$ direction; $M_a/B$ was markedly suppressed by applying $B$ and then became constant at low temperatures for $B\ge 1\ {\rm T}$ \[Fig. 1(b)\]. These contrasting features in $C/T$ and $M_a/B$ for $B\,||\,a$ strongly suggest that the fluctuating spin component is perpendicular to the applied $B$ direction; that is, the $c$-axis spin component significantly contributes to the quantum critical fluctuations in CeCo$_{1-x}$Ni$_x$In$_5$. This situation is similar to that expected in the Ising model with a transverse magnetic field, in which the transverse magnetic field does not align the spins but yields a quantum paramagnetic state with short-range spin correlations [@rf:Sachdev99]. However, it should be remembered that the anisotropy of magnetic moments in the present system was not so strong that it can be regarded as simply the Ising-like anisotropy. In addition, the spins of the itinerant heavy quasiparticles, rather than the completely localized spins, were likely responsible for the quantum critical fluctuations. Hence, the deviation from the Ising-like characteristics of the magnetic moments would lead to a suppression of the quantum critical fluctuations and would then stabilize the FL state at a high $B$ region above $B=7\ {\rm T}$, even for $B\, ||\,a$.
In Figs. 5(a) and 5(b) we summarize the exponent $\eta$ of $M/B\propto T^{-\eta}$ at low temperatures for $B\,||\,c$ and $B\,||\,a$, respectively. In these plots, $\eta$ was estimated using a simple relation: $\eta=-T/(M/B)\,d(M/B)/dT$. The effect of the Van Vleck susceptibility $\chi_V$ may be included in the estimation of $\eta$ using an alternative formula: $\eta=-T/(M/B-\chi_V)\,d(M/B)/dT$. However, we confirmed that the trends seen in Figs. 5(a) and 5(b) did not depend on the finite $\chi_V$ value, at least up to $\chi_V \sim 2.5\times 10^{-3} \ \mu_B$/T Ce, which is about half of the magnitude of the experimentally observed magnetic susceptibility at 300 K [@rf:Otaka2016].
{width="43.00000%"}
As displayed in Fig. 5(a), the NFL state with $\eta = 0.2$ governed the low $B$ region in the $B-T$ plane for $B\,||\,c$, and the suppression of the NFL state at the high $B$ region was realized as a reduction of $\eta$ toward $\sim 0$. A similar gradual suppression of the NFL behavior with $B$ was also observed in $C/T$. In fact, when the characteristic temperature $T^*$ below which $C/T$ deviates from the $-\ln T$ function \[see Fig. 4(a)\] was plotted onto the image map of $\eta$ in Fig. 5(a), we found that the $T^*(B)$ curve traced the contour of $\eta=0.06$ well. This consistency between $M/B$ and $C/T$ reflected the occurrence of the NFL-to-FL crossover for $B\,||\,c$.
However, the situation for $B\,||\,a$ was very different, as shown in Fig. 5(b). The finite $\eta$ value for $B\sim 0$ was rapidly suppressed by $B$, and a large $\eta\sim 0$ region was then distributed in the $B-T$ plane. In contrast, the $T^*(B)$ curve entered deeply into the $\eta\sim 0$ region. Note that for $B\,||\,a$, $C/T$ did not exhibit the $C/T$-constant behavior ascribed to the FL state, even below $T^*$, although $T^*$ could be defined in the $C/T$ data. It is likely that the $c$-axis spin component of the quantum critical fluctuations, which was not significantly influenced by $B$ for $B\,||\,a$, led to this inconsistency between $M/B$ and $C/T$ for $B\,||\,a$, as argued previously.
Discussion
==========
The present investigation of CeCo$_{0.75}$Ni$_{0.25}$In$_5$ revealed the clear anisotropic response of the NFL behaviors in $M/B$ and $C/T$ with respect to the $B$ direction; the crossover from the NFL state to the FL state occurred for $B\,||\,c$, whereas the NFL behavior persisted at least up to $B=7\ {\rm T}$ for $B\,||\,a$. In this section, we compare this anisotropic NFL behavior to the SC properties in pure CeCoIn$_5$.
First, we find that the anisotropy concerning the stability of the SC phase in pure CeCoIn$_5$ qualitatively coincides with that of the quantum critical fluctuations in Ni25%-doped CeCoIn$_5$. The SC order parameter in the pure compound has a characteristic temperature scale of $\sim 2\ {\rm K}$, corresponding to $T_c$. In addition, this SC state is broken at $\mu_0H_{c2}=4.9\ {\rm T}$ for $B\,||\,c$, although it persists up to $11.6\ {\rm T}$ for $B\,||\,a$ [@rf:Ikeda2001]. In the Ni25%-doped alloy, $C/T$ at $\sim 2\ {\rm K}$ takes on the NFL characteristics up to the same $B$ ranges as $H_{c2}$ in the pure compound. This coincidence of the stability of the NFL and SC states concerning $B$ implies that the spin correlations, similar to those yielding the NFL behavior in the Ni25%-doped alloy, play a critical role in the occurrence of the SC order in the pure compound. If this is the case, such spin correlations should be concerned with the determination of $H_{c2}$ through both the SC condensation energy and the paramagnetic spin susceptibility, yielding a Pauli paramagnetic effect [@rf:Yanase2008], because $H_{c2}$ of CeCoIn$_5$ is considered Pauli limited [@rf:Izawa2001; @rf:Ikeda2001].
Second, it is remarkable that the $c$-axis spin component is primarily responsible for the quantum critical fluctuations in Ni25%-doped CeCoIn$_5$. Indeed, such anisotropic spin fluctuations and excitations are also observed in the SC phase of pure CeCoIn$_5$. The recent inelastic neutron scattering experiments for pure and Nd-doped CeCoIn$_5$ have revealed that the spin resonance excitation emerging in the SC phase has a nearly Ising nature along the $c$ axis [@rf:Raymond2015; @rf:Mazzone2017]. This similarity in the spin polarization suggests that the spin resonance excitation in pure CeCoIn$_5$ and the NFL behavior in Ni-doped CeCoIn$_5$ have similar origins. The spin fluctuations in Ni25%-doped CeCoIn$_5$ may have an energy distribution centered at $\hbar\omega\sim 0$, because the NFL behavior at $B\sim 0$ in $M/B$ and $C/T$ persists down to very low temperatures. However, once the SC order occurs, as in pure CeCoIn$_5$, the spin fluctuations may have gapped energy due to the SC condensation, detected as the spin resonance excitation in the inelastic neutron scattering measurements. In this situation, the coherency and Ising-like polarization of the spin fluctuations may be somewhat enhanced along with the variation of the ground state from the paramagnetic NFL state to the SC ordered phase. In fact, it has been demonstrated that the spin resonance excitation may condensate into AFM ordering [@rf:Song2016; @rf:Stock2018; @rf:Michal2011], supporting the above suggestion that the quantum critical fluctuations and the spin resonance excitation have similar origins because the quantum critical fluctuations likely originate from the AFM instabilities in pure CeCoIn$_5$ and its doped alloys [@rf:Paglione2003; @rf:Pham2006; @rf:Yoko2017].
Despite the aforementioned considerations, the microscopic nature of the quantum critical fluctuations has not yet been uncovered. We believe that the relationship of the spin fluctuations between pure and Ni25%-doped CeCoIn$_5$ would be clarified by comprehensive investigations using the inelastic neutron scattering technique on CeCo$_{1-x}$Ni$_{x}$In$_5$ with a wide $x$ range. Such investigations could provide a key to understanding the anomalous SC properties coupled with the magnetic correlations in CeCoIn$_5$.
Conclusion
==========
Our magnetization and specific heat measurements for CeCo$_{0.75}$Ni$_{0.25}$In$_5$ revealed anisotropic NFL behavior, depending on the $B$ direction. For $B\,||\,c$, the diverging behaviors in the temperature variations of $M/B$ and $C/T$ changed into nearly $T$-constant behaviors, reflecting the NFL-to-FL crossover with increasing $B$. For $B\,||\,a$, however, the NFL behavior in $C/T$ persisted up to $B=7\ {\rm T}$, although $M/B$ was sufficiently reduced with $B$ for $B \ge 1\ {\rm T}$. These anisotropic responses in $M/B$ and $C/T$ indicate that the quantum critical fluctuations are suppressed by the $c$-axis magnetic field more effectively than by the $a$-axis field because they are composed mainly of the $c$-axis spin component. We compared this feature to the SC properties in pure CeCoIn$_5$, especially to the anisotropy in the upper critical field and the Ising-like characteristics in the spin resonance excitation, and suggested a close coupling between them.
We are grateful to Y. Oshima, I. Kawasaki, R. Otaka, and Rahmanto for their experimental support, and to D. Ueta and T. Masuda for their assistance with the specific-heat measurements prior to this study. M.Y. expresses gratitude to Y. Yanase for fruitful discussions. This study was supported in part by JSPS KAKENHI Grant Number 17K05529, and the research completed at ISSP was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “J-Physics" (15H05883) from JSPS.
[99]{} C. Petrovic, P. G. Pagliuso, M. F. Hundley, R. Movshovich, J. L. Sarrao, J. D. Thompson, Z. Fisk, and P. Monthoux, J. Phys.: Condens. Matter [**13**]{}, L337 (2001). K. Izawa, H. Yamaguchi, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. [**87**]{}, 057002 (2001). K. An, T. Sakakibara, R. Settai, Y. Onuki, M. Hiragi, M. Ichioka, and K. Machida, Phys. Rev. Lett. [**104**]{}, 037002 (2010). W. K. Park, J. L. Sarrao, J. D. Thompson, and L. H. Greene, Phys. Rev. Lett. [**100**]{}, 177001 (2008). C. Stock, C. Broholm, J. Hudis, H. J. Kang, and C. Petrovic, Phys. Rev. Lett. [**100**]{}, 087001 (2008). S. Raymond and G. Lapertot, Phys. Rev. Lett. [**115**]{}, 037001 (2015). Y. Song, J. van Dyke, I. K. Lum, B. D. White, S. Jang, D. Yazici, L. Shu, A. Schneidewind, P. Čermák, Y. Qiu, M. B. Maple, D. K. Morr and P. Dai, Nat. Commun. [**7**]{}, 12774 (2016). D. G. Mazzone, S. Raymond, J. L. Gavilano, P. Steffens, A. Schneidewind, G. Lapertot, and M. Kenzelmann, Phys. Rev. Lett. [**119**]{}, 187002 (2017). C. Stock, J. A. Rodriguez-Rivera, K. Schmalzl, F. Demmel, D. K. Singh, F. Ronning, J. D. Thompson, and E. D. Bauer, Phys. Rev. Lett. [**121**]{}, 037003 (2018). I. Eremin, G. Zwicknagl, P. Thalmeier, and P. Fulde, Phys. Rev. Lett. [**101**]{}, 187001 (2008). A. V. Chubukov and L. P. Gor’kov, Phys. Rev. Lett. [**101**]{}, 147004 (2008). V. P. Michal and V. P. Mineev, Phys. Rev. B [**84**]{}, 052508 (2011). A. Bianchi, R. Movshovich, C. Capan, P. G. Pagliuso, and J. L. Sarrao Phys. Rev. Lett. [**91**]{}, 187004 (2003). K. Kakuyanagi, M. Saitoh, K. Kumagai, S. Takashima, M. Nohara, H. Takagi, and Y. Matsuda, Phys. Rev. Lett. [**94**]{}, 047602 (2005). B.-L. Young, R. R. Urbano, N. J. Curro, J. D. Thompson, J. L. Sarrao, A. B. Vorontsov, and M. J. Graf, Phys. Rev. Lett. [**98**]{}, 036402 (2007). M. Kenzelmann, Th. Strassle, C. Niedermayer, M. Sigrist, B. Padmanabhan, M. Zolliker, A. D. Bianchi, R. Movshovich, E. D. Bauer, J. L. Sarrao, J. D. Thompson, Science [**321**]{}, 1652 (2008). A. Aperis, G. Varelogiannis, P. B. Littlewood, and B. D. Simons, J. Phys.: Condens. Matter [**20**]{} 434235 (2008). D. F. Agterberg, M. Sigrist, and H. Tsunetsugu, Phys. Rev. Lett. [**102**]{}, 207004 (2009). Y. Yanase, J. Phys. Soc. Jpn. [**77**]{}, 063705 (2008). Y. Yanase and M. Sigrist, J. Phys. Soc. Jpn. [**78**]{}, 114715 (2009). A. Bianchi, R. Movshovich, I. Vekhter, P. G. Pagliuso, and J. L. Sarrao: Phys. Rev. Lett. [**91**]{}, 257001 (2003). J. Paglione, M. A. Tanatar, D. G. Hawthorn, E. Boaknin, R. W. Hill, F. Ronning, M. Sutherland, L. Taillefer, C. Petrovic, and P. C. Canfield, Phys. Rev. Lett. [**91**]{}, 246405 (2003). T. Tayama, A. Harita, T. Sakakibara, Y. Haga, H. Shishido, R. Settai, and Y. Onuki: Phys. Rev. B [**65**]{}, 180504(R) (2002). Y. Tokiwa, E. D. Bauer, and P. Gegenwart, Phys. Rev. Lett. [**111**]{}, 107003 (2013). R. Hu, Y. Lee, J. Hudis, V. F. Mitrovic, and C. Petrovic, Phys. Rev. B [**77**]{}, 165129 (2008). S. Raymond, S. M. Ramos, D. Aoki, G. Knebel, V. P. Mineev, and G. Lapertot, J. Phys. Soc. Jpn. [**83**]{}, 013707 (2014). V. S. Zapf, E. J. Freeman, E. D. Bauer, J. Petricka, C. Sirvent, N. A. Frederick, R. P. Dickey, and M. B. Maple, Phys. Rev. B [**65**]{}, 014506 (2001). M. Yokoyama, H. Amitsuka, K. Matsuda, A. Gawase, N. Oyama, I. Kawasaki, K. Tenya, and H. Yoshizawa, J. Phys. Soc. Jpn. [**75**]{}, 103703 (2006). S. Ohira-Kawamura, H. Shishido, A. Yoshida, R. Okazaki, H. Kawano-Furukawa, T. Shibauchi, H. Harima, and Y. Matsuda, Phys. Rev. B [**76**]{}, 132507 (2007). M. Yokoyama, N. Oyama, H. Amitsuka, S. Oinuma, I. Kawasaki, K. Tenya, M. Matsuura, K. Hirota, and T. J. Sato, Phys. Rev. B [**77**]{}, 224501 (2008). L. D. Pham, T. Park, S. Maquilon, J. D. Thompson, and Z. Fisk, Phys. Rev. Lett. [**97**]{}, 056404 (2006). M. Nicklas, O. Stockert, T. Park, K. Habicht, K. Kiefer, L. D. Pham, J. D. Thompson, Z. Fisk, and F. Steglich, Phys. Rev. B [**76**]{}, 052401 (2007). M. Yokoyama, K. Fujimura, S. Ishikawa, M. Kimura, T. Hasegawa, I. Kawasaki, K. Tenya, Y. Kono, and T. Sakakibara, J. Phys. Soc. Jpn. [**83**]{}, 033706 (2014). M. Yokoyama, H. Mashiko, R. Otaka, Y. Sakon, K. Fujimura, K. Tenya, A. Kondo, K. Kindo, Y. Ikeda, H. Yoshizawa, Y. Shimizu, Y. Kono, and T. Sakakibara, Phys. Rev. B [**92**]{}, 184509 (2015). H. Shishido, S. Yamada, K. Sugii, M. Shimozawa, Y. Yanase, and M. Yamashita, Phys. Rev. Lett. [**120**]{}, 177201 (2018). E. D. Bauer, C. Capan, F. Ronning, R. Movshovich, J. D. Thompson, and J. L. Sarrao, Phys. Rev. Lett. [**94**]{}, 047001 (2005). E. D. Bauer, F. Ronning, C. Capan, M. J. Graf, D. Vandervelde, H. Q. Yuan, M. B. Salamon, D. J. Mixson, N. O. Moreno, S. R. Brown, J. D. Thompson, R. Movshovich, M. F. Hundley, J. L. Sarrao, P. G. Pagliuso, and S. M. Kauzlarich, Phys. Rev. B [**73**]{}, 245109 (2006). S. M. Ramos, M. B. Fontes, E. N. Hering, M. A. Continentino, E. Baggio-Saitovich, F. D. Neto, E. M. Bittar, P. G. Pagliuso, E. D. Bauer, J. L. Sarrao, and J. D. Thompson, Phys. Rev. Lett. [**105**]{}, 126401 (2010). R. Otaka, M. Yokoyama, H. Mashiko, T. Hasegawa, Y. Shimizu, Y. Ikeda, K. Tenya, S. Nakamura, D. Ueta, H. Yoshizawa, and T. Sakakibara, J. Phys. Soc. Jpn. [**85**]{}, 094713 (2016). F. Ronning, C. Capan, A. Bianchi, R. Movshovich, A. Lacerda, M. F. Hundley, J. D. Thompson, P. G. Pagliuso, and J. L. Sarrao, Phys. Rev. B [**71**]{}, 104528 (2005). T. Hu, H. Xiao, T. A. Sayles, M. Dzero, M. B. Maple, and C. C. Almasan, Phys. Rev. Lett. [**108**]{}, 056401 (2012). T. Sakakibara, H. Mitamura, T. Tayama and H. Amitsuka, Jpn. J. Appl. Phys. [**33**]{}, 5067 (1994). M. Yokoyama, H. Mashiko, R. Otaka, Y. Oshima, K. Suzuki, K. Tenya, Y. Shimizu, A. Nakamura, D. Aoki, A. Kondo, K. Kindo, S. Nakamura, and T. Sakakibara, Phys. Rev. B [**95**]{}, 224425 (2017). S. Sachdev, [*Quantum Phase Transitions*]{}, (Cambridge University Press, Cambridge, 1999). S. Ikeda, H. Shishido, M. Nakashima, R. Settai, D. Aoki, Y. Haga, H. Harima, Y. Aoki, T. Namiki, H. Sato and Y. Onuki: J. Phys. Soc. Jpn. [**70**]{} (2001) 2248.
|
---
abstract: 'The epoch of galaxy assembly from $2\leq z \leq 4$ marks a critical stage during the evolution of today’s galaxy population. During this period the star-formation activity in the Universe was at its peak level, and the structural patterns observed among galaxies in the local Universe were not yet in place. A variety of novel techniques have been employed over the past decade to assemble multiwavelength observations of galaxies during this important epoch. In this primarily observational review, I present a census of the methods used to find distant galaxies and the empirical constraints on their multiwavelength luminosities and colors. I then discuss what is known about the stellar content and past histories of star formation in high-redshift galaxies; their interstellar contents including dust, gas, and heavy elements; and their structural and dynamical properties. I conclude by considering some of the most pressing and open questions regarding the physics of high-redshift galaxies, which are to be addressed with future facilities.'
author:
- 'Alice E. Shapley'
bibliography:
- 'apj-jour.bib'
- 'araa.bib'
title: 'Physical Properties of Galaxies from $z=2-4$'
---
epsf.def
Galaxy Evolution, Galaxy Formation, Galaxy Observations, Galaxy Structure, Interstellar Medium, Stellar Populations
INTRODUCTION {#sec:intro}
============
Understanding the detailed formation and evolution of the galaxies we observe today remains one of the great challenges of modern cosmology. An exceedingly rich variety of galaxy properties exists in terms of luminosity, mass, color, structure, gas content, heavy-element enrichment, and environment, many of which, in turn, are strongly correlated with each other. As reviewed by @blanton2009, based on the latest generation of wide-field surveys of the local Universe, astronomers have constructed an exquisitely detailed and statistically robust description of the galaxy population [*today*]{}. These results are crucial in terms of providing a boundary condition, or endpoint, for our description of the formation and evolution of galaxies. Ultimately, we strive to tell that story from beginning to end. To do so, we must assemble additional observations and theories.
Multiple complementary approaches can be used to construct the history of galaxy formation. These include ab initio analytic models or numerical simulations; examinations of the fossil record contained in the ages, metallicities, and phase-space distributions of stars in nearby galaxies; and direct observations of distant galaxies, for which the cosmologically significant lookback time allows a probe of the Universe at an earlier time. In order to test theoretical models of galaxy formation at every time step, observations of galaxies over a wide range of lookback times are required. Furthermore, a robust translation must be performed between the observer’s empirical quantities of luminosity, color, and velocity dispersion, and the theorist’s physical quantities of stellar and dynamical mass, current star-formation rate and past star-formation history.
In the comparison between observations and theoretical models of galaxy evolution, an important recent development is the establishment of a precision cosmological framework. Observations of the cosmic microwave background radiation, large-scale structure, Type Ia supernova, the abundance of galaxy clusters, and the expansion rate of the Universe, all appear to be well described by a cosmological model in which the Universe is spatially flat, with the dominant component of the mass-energy density in the form of dark energy, and the remainder consisting mostly of cold dark matter with a small fraction of baryons. The initial spectrum of density fluctuations in this model is adiabatic, Gaussian, and nearly scale invariant. The most recent determination of cosmological parameters from the [*Wilkinson Microwave Anisotropy Probe*]{} (WMAP) [@spergel2003] is presented in @komatsu2011, and highlights the fact that most parameters are determined with better than $5-10$% precision.
Constraining the background cosmological parameters is a crucial part of understanding galaxy formation not only for converting apparent quantities such as flux and angular size, respectively, into intrinsic ones such as luminosity, star-formation rate, stellar mass, and physical size. Cosmological parameters are also required for precise theoretical calculations whose predictions can be compared with observations, because, according to leading models, the underlying set of cosmological parameters determine how tiny primordial dark matter density fluctuations evolve under the influence of gravity into the large-scale spatial distribution of matter in the current Universe. In this framework, the collapsed perturbations of dark matter – dark matter halos – serve as the very sites of galaxy formation. Based on recent determinations of cosmological parameters, massive numerical simulations of the growth of dark matter structure have been performed [@springel2005; @boylankolchin2009]. In order to compare more directly with observations of galaxy formation, models (either numerical hydrodynamic or semi-analytic) describing the baryonic processes of gas cooling, star formation, and metal enrichment are also required, and these, too, are advancing with increased spatial resolution and complexity [e.g., @ceverino2010; @somerville2008].
In order to probe the origin of the global patterns observed in the current galaxy population, we must look back to a time before these trends were already in place. Furthermore, the old stellar populations of nearby early-type galaxies in dense environments suggest that the bulk of their stars formed at $z\geq 2$ [@thomas2005]. Therefore, catching the formation of spheroids “in the act" requires observations at such early times. Given the strong correlation between the properties of the spheroidal components of galaxies and their central black holes, the epoch when the spheroids are forming holds special interest for explaining this connection. As described in more detail during the course of this review, the redshift range, $2\leq z\leq 4$, corresponding to a lookback time of $\sim 10-12$ Gyr, is ideal for directly observing the progenitors of today’s fairly luminous spheroidal and disk galaxies while in the very process of attaining the properties that come to define them over the next 10 Gyr up to the present day. Towards the end of this redshift range, the overall level of “activity" in the Universe – both in terms of star formation and black hole accretion - was at its peak value. In contrast to what is observed in the present-day Universe, a significant fraction of the most massive galaxies still sustained active star formation. Furthermore, the Hubble Sequence of disk and elliptical galaxies was not yet in place, and the abundance of rich clusters was vanishingly small. The early Universe looked drastically different from its current state. Therefore, studying this epoch can yield important clues about the evolution of galaxies.
Within the past decade, there has been incredible progress in the study of galaxies in this important redshift range of galaxy assembly. A variety of novel techniques have been used to identify distant galaxies, and the sample of galaxies with spectroscopic redshifts at $2\leq z\leq 4$ now numbers well into the thousands. Although we are far from approaching the overwhelming statistical power of the giant local redshift surveys such as the Sloan Digital Sky Survey [SDSS; @abazajian2009; @blanton2009] and 2dF Galaxy Redshift Survey [2dFGRS; @colless2001], in terms of number of galaxies surveyed spectroscopically, volume probed, and data quality, key features of the galaxy population at these early times are emerging. The physical properties inferred for distant galaxies have provided important inputs and daunting challenges to state of the art theoretical models of galaxy formation. As we look forward to the next generation of instrumentation on current and future large ground-based telescopes, as well as the James Webb Space Telescope ([*JWST*]{}) in space, and ever more sophisticated galaxy formation models, it is worth reviewing what is known about the physical properties of high-redshift galaxies at $2 \leq z \leq 4$.
This primarily observational review is constructed as follows. In Section \[sec:technique\], we provide an overview of the many different techniques that have recently been employed for identifying high-redshift galaxies. We continue in Section \[sec:empirical\] by reviewing the global multiwavelength distributions in luminosity and color for galaxies in this redshift range. In Section \[sec:stellarpop\], we delve into the techniques used to transform empirical quantities such as luminosity and color into physical ones relating to galaxy stellar populations. In particular, we focus here on stellar content and the history of star-formation activity, as well as the relationship between these quantities. Sections \[sec:ISM\] and \[sec:structure\] in turn consider what is known about the interstellar medium (ISM) of distant galaxies – where the interstellar contents include gas, dust, and metals – and their structural properties and dynamics. Although many gaps remain in our knowledge of these fundamental physical properties, as described in Section \[sec:future\], future facilities and instrumentation will guide us in our quest to assemble a comprehensive picture of the galaxy population during this distant yet intriguing epoch in the history of galaxy formation.
HIGH-REDSHIFT GALAXY SELECTION TECHNIQUES {#sec:technique}
=========================================
In this section, we provide a brief historical context for the recent dramatic developments in the study of high-redshift galaxies, as well as reviewing several of the most common and complementary techniques for identifying distant objects.
Historical Context {#subsec:technique-history}
------------------
Over the past 10–15 years, the study of high-redshift galaxies has truly exploded, with an increasing number of surveys for systems at lookback times of order 10 Gyr. The number of galaxies with spectroscopic redshifts at $z>2$ is now well into the thousands, and the number whose multiwavelength photometric properties identify them as such is more than an order of magnitude larger. Due to the difficulties of obtaining optical and near-infrared (near-IR) spectra of faint objects, photometric redshifts have played an increasingly common role in describing the properties of distant galaxies – bringing with them both the advantages of much larger samples, and also the drawbacks of larger uncertainties in derived galaxy properties. Also, with some exceptions like the VIRMOS VLT Deep Survey [VVDS; @lefevre2005], which targets galaxies for spectroscopy down to a given optical magnitude limit, these new results for the most part utilize several novel techniques for efficiently identifying distant galaxies with minimal contamination by systems at lower redshift. Although effective at isolating high-redshift galaxies, all of these selection methods suffer from incompleteness with respect to a sample defined in terms of physical quantities such as stellar or dynamical mass, or star-formation rate. In this section, we describe the landscape of galaxy surveys that have contributed thus far to our picture of the high-redshift galaxy population.
Rest-frame Ultra-Violet Selection {#subsec:technique-LBG}
---------------------------------
One method for selecting distant galaxies is based on their rest-frame UV colors. This method was first applied at $z\sim
3$, and specifically exploits the combined effects of neutral hydrogen opacity within a star-forming galaxy and along the line of sight through the intergalactic medium (IGM). Accordingly, $z\sim 3$ star-forming galaxies with moderate amounts of dust extinction (less than a factor of 100 in the rest-frame UV) will have distinctive colors in a $UG{\cal R}$ filter system, with fairly flat $G-{\cal R}$ and extremely red $U-G$ colors. The “Lyman Break Technique" has been used to identify thousands of galaxies (so-called Lyman Break Galaxies, or LBGs) at $z\sim 3$ [@steidel1996b; @steidel2003], and, using different filter sets, at $z\sim 4$ and $5$ [@ouchi2004a]. At $z\sim
6$ and beyond, even redder sets of three filters have been used to identify star-forming galaxies [@bouwens2007; @oesch2010; @bouwens2010]. In these latter cases, however, the main spectral break between the bluest and middle filter arises due to hydrogen Ly$\alpha$ opacity in the IGM, as opposed to opacity at the Lyman limit. Rest-frame UV selection using the initial set of $UG{\cal R}$ filters has also been extended down to lower redshift [@steidel2004; @adelberger2004], where galaxies at $1.4 \leq z \leq 2.5$ are isolated due to a [*lack*]{} of significant spectral break. Their fairly flat rest-frame UV colors, modulated at $2\leq z \leq 2.5 $ only by Ly$\alpha$ forest line blanketing in the $U$-band, also prove distinctive. Figure \[fig:technique-LBG-steidel2004\], from @steidel2004, provides an illustration of the rest-frame UV selection criteria in $UG{\cal R}$ color space, tuned to find galaxies at $1.5\leq z \leq 3.5$.
Given the criterion of detection in the rest-frame UV, the techniques described above necessarily select galaxies with ongoing star formation and are not sensitive to passive galaxies. Furthermore, the windows in color-color selection space exclude galaxies whose rest-frame UV continuum shape is indicative of significant dust reddening. Ground-based rest-frame UV surveys are also typically characterized by a rest-frame UV (observed optical) flux limit. Detection in rest-frame UV (observed optical) bands results in objects well suited to optical spectroscopic follow-up. Accordingly, successful spectroscopic follow-up of high-redshift galaxies has been weighted towards rest-frame-UV-selected samples, probing galaxies with active ongoing star formation.
Rest-frame Optical/Infrared Selection {#subsec:technique-DRGBzK}
-------------------------------------
Other techniques are tuned to select galaxies on the basis of their rest-frame optical colors and are based on a detection at observed near- or mid-infrared (mid-IR) wavelengths. For these techniques, the relevant spectral break is either the Balmer break at $\sim 3650$Å, which arises when the integrated stellar spectrum from a galaxy at these wavelengths indicates the overall spectral shape of A-stars, or else the 4000 Åbreak, which reflects the absorption from ionized metals in the atmospheres of late-type stars. Although both breaks are indications of maturity in stellar populations, they are by no means equivalent in terms of the underlying stellar populations that cause them. The Balmer break appears in stellar populations featuring ongoing star-formation over sustained timescales ($>100$ Myr), or post-starburst populations $0.3-1$ Gyr since the cessation of star formation. The 4000 Å break is strongest in passive stellar populations in which the current level of star formation has been negligible for more than 1 Gyr. Isolating $z\geq 2$ galaxies on the basis of their rest-frame optical breaks requires deep, near-IR photometry. As described in @franx2003 and [@vandokkum2003], a threshold of $J_{Vega}-K_{Vega} >2.3$ is effective at identifying objects that dominate the high-mass regime of the $z>2$ stellar mass function [@kriek2008a]. Figure \[fig:technique-DRGBzK-franx2003daddi2004\] [left, from @franx2003] demonstrates the sensitivity of the $J_{Vega}-K_{Vega}$ color to mature stellar populations at $z>2$. These Distant Red Galaxies (DRGs) in fact typically have significant dust obscuration ($A_V>1$) and active star-formation rates ($\geq 100 M_{\odot}\mbox{ yr}^{-1}$) [@papovich2006], although some show little evidence for ongoing star formation [@vandokkum2008b]. Regardless, the red rest-frame UV to optical colors of DRGs down to current near-IR limits indicate typical stellar masses in excess of $10^{11} M_{\odot}$. The main limitation of such studies is the limited amount of spectroscopic follow up, due to the optical faintness (typically $R\geq 25$) of DRGs and the difficulty of obtaining large samples of near-IR spectra [but see, e.g., @kriek2008a].
Another common technique for isolating galaxies at $1.4\leq z
\leq 2.5$ consists of the so-called “$BzK$" method [@daddi2004]. As shown in Figure \[fig:technique-DRGBzK-franx2003daddi2004\] (right), $BzK$ galaxies are identified in $K$-selected samples of galaxies on the basis of their colors in the $z-K$ versus $B-z$ plane. Galaxies with fairly blue $B-z$ colors and red $z-K$ colors are selected as star-forming $z\sim 2$ systems (“$sBzK$"), due to the presence of a Balmer break. At the same time, quiescent systems (“$pBzK$") at the same redshift are identified on the basis of red colors in both $B-z$ and $z-K$. Although fairly general in terms of selecting both star-forming and quiescent galaxies, this method misses the youngest star-forming galaxies at $z\sim 2$, which lack a significant Balmer or 4000 Å break [@reddy2005]. While the star-forming $sBzK$ galaxies have rest-frame UV colors that are redder on average than the corresponding $UV$-selected galaxies at $z\sim 2$, there is significant overlap between these two photometric selection technique down to a fixed $K$-band magnitude limit – much more than between the UV-selected and DRG samples, which overlap at only the 10% level [@reddy2005]. Other surveys tuned to find high-redshift galaxies based on their rest-frame optical or near-IR properties include the Gemini Deep Deep Survey [GDDS; @abraham2004] and the Galaxy Mass Assembly ultra-deep Spectroscopic Survey [GMASS; @cimatti2008].
Submillimeter / Mid-Infrared Selection {#subsec:technique-IR}
--------------------------------------
The overall increase in the level of star-formation activity in the Universe at earlier times results in an increased abundance of extreme, bolometrically ultra-luminous systems. These systems emit the bulk of their radiation at rest-frame far-IR wavelengths, because of copious amounts of dust obscuring their star-formation and AGN activity. Systems with bolometric luminosities greater than $L=10^{12} L_{\odot}$ are commonly referred to as Ultra-luminous Infrared Galaxies (ULIRGs). Submillimeter and mid-IR instrumentation sensitive to cool and hot dust, respectively, have enabled the identification of these extreme high-redshift ULIRGs on the basis of their reprocessed emission, while multiwavelength imaging and spectroscopic follow-up has elucidated the range of their properties. The Submillimetre Common-user Bolometer Array (SCUBA) has been used to identify bolometrically-luminous submillimeter galaxies [SMGs; @smail1997], and the largest set of spectroscopically-confirmed such objects was obtained by following up the subset of sources with both $F_{850
\mu m} \geq 5 $ mJy and [*Very Large Array*]{} (VLA) 1.4 GHz fluxes greater than $F_{1.4
\mbox{ GHz}} \sim 30 \mu$Jy [@chapman2003; @chapman2005]. The radio fluxes were used to obtain the precise ($1"-2$“) positions required for spectroscopic observations, which were not achievable with the coarse (15”) SCUBA beam. These luminous SMGs with radio counterparts have a median redshift of $z=2.2$, although, given that the requirement of a radio flux detection recovers $\sim 50$% of the $F_{850 \mu m} \geq 5 $ mJy population, their redshift distribution may not be fully representative of the luminous SMG population as a whole.
The Multiband Imaging Photometer for Spitzer (MIPS) onboard the [*Spitzer Space Telescope*]{} has been used to identify high-redshift dusty sources on the basis of their brightness at $24 \mu$m and faintness at optical wavelengths. @yan2007 and @dey2008 present such samples, identified using slightly different criteria, but both based on similar criteria of detection at $24 \mu$m, with a large ratio of mid-IR to optical flux. The majority of these objects are at $1.5 \leq z \leq 3$, with comparable space densities ($\sim 10^{-5} \mbox{ Mpc}^{-3}$) to those of SMGs. On the other hand, the $24 \mu$m-selected sources are characterized by warmer dust temperatures and a higher frequency of AGN signatures at mid-IR wavelengths than SMGs selected at longer wavelengths (see Section \[subsec:ISM-dustem\]). Quantifying the relative contributions of star-formation and AGN activity in powering the extreme luminosities of both mid-IR and submillimeter-selected ULIRGs will enable us to isolate the underlying nature of these sources. Another key goal consists of constraining the relative importance of major mergers and smooth mass accretion as triggers for the ULIRG phase.
Narrowband Selection {#subsec:technique-LAE}
--------------------
In contrast to the identification of high-redshift objects on the basis of broadband spectral shape, the use of a narrowband filter tuned to the redshifted wavelength of a specific emission line is effective at isolating objects with large emission-line equivalent widths. The most common emission line for which narrowband filters are designed is hydrogen Ly$\alpha$. As shown in Figure \[fig:technique-LAE-gronwall2007\] [from @gronwall2007], based on images through both the narrowband filter and a broadband filter close in wavelength, objects with red broadband minus narrowband colors are flagged as Ly$\alpha$ line emitters (LAEs). More than 2000 LAEs have been identified with ground-based facilities at $2\leq z \leq 8$ [e.g., @cowie1998; @rhoads2000; @ouchi2008; @gronwall2007; @nilsson2011]. The star-forming LAEs tend to be significantly fainter on average than the UV-continuum-selected objects described in Section \[subsec:technique-LBG\], and therefore offer a probe of the faint end of the luminosity function. On the other hand, the faint nature of these objects leads to a challenge in assembling high signal-to-noise (S/N) multiwavelength imaging and spectra for individual LAEs (with information other than a measurement of Ly$\alpha$ emission), hindering the determination of their relationship to other galaxy populations at similar redshifts.
EMPIRICAL PROBES OF THE HIGH-REDSHIFT GALAXY POPULATION {#sec:empirical}
=======================================================
Before reviewing what is known about the stellar and interstellar content of distant galaxies, we must consider the empirical measurements from which these physical properties are inferred. The observables here are distributions in luminosity and color, which offer some of the most basic and fundamental probes of a galaxy population. In addition to traditional optical photometry tracing the rest-frame ultraviolet, our view of the global photometric properties of galaxies at $z\geq 2$ is now based on deep, near-IR surveys from the ground using wide-field imagers on $4$-meter-class telescopes (e.g., KPNO/NEWFIRM, UKIRT/WFCAM, CTIO/ISPI, Palomar/WIRC) and narrow- and wide-field imagers on $8-10$-meter class telescopes (e.g., Keck/NIRC, VLT/ISAAC, Subaru/MOIRCS, VLT/HAWK-I), and mid- and far-IR surveys using [*Spitzer*]{} and [*Herschel*]{} in space. Multiwavelength observations have, therefore, granted us a window into the luminosity and color distributions of high-redshift galaxies spanning from the rest-frame UV through the rest-frame far-IR.
Luminosity Functions {#subsec:empirical-LF}
--------------------
The galaxy luminosity function offers constraints on the overall abundance of objects, as well as the integrated luminosity density at a given wavelength. As such, the luminosity function provides a key observational baseline for each redshift at which it is measured.
### REST-FRAME UV LUMINOSITY FUNCTIONS {#subsubsec:empirical-LF-UV}
Some of the first luminosity functions to be measured for high-redshift galaxies were based on optical observations of rest-frame UV-selected $z\sim 3$ and $z\sim 4$ LBGs [@steidel1999], probing rest-frame wavelengths of $\lambda\sim 1700$Å. This work highlighted the importance of dust extinction for converting the rest-frame UV luminosities of star-forming galaxies into unobscured values, in order to obtain dust-corrected star-formation rates. We will return to this point in Sections \[sec:stellarpop\] and \[subsec:ISM-dustext\], when we consider the star-formation rates and dust content of high-redshift galaxies. Rest-frame UV luminosity functions for large samples of distant galaxies have now been estimated by several different groups, from $z\sim 2$ all the way out to $z\sim 8$ [@bouwens2010]. As in the local Universe, the galaxy luminosity function at high redshift is typically parameterized using the Schechter form of a power law multiplying an exponential function [@schechter1976]. The associated free parameters to constrain are the characteristic luminosity, $L^*$ (or $M^*$ in the space of absolute magnitude), the faint-end slope, $\alpha$, and the overall normalization, $\Phi^*$.
At $z\sim 2-3$, the work of @reddy2008 and @reddysteidel2009 is based on the largest set of spectroscopic redshifts in the literature, and, including a consideration of the systematic variation of dust reddening with UV luminosity, is accordingly the most robust. Specifically, @reddysteidel2009 utilizes $>2000$ spectroscopic redshifts, and $\sim 31,000$ $z\sim 2-3$ photometric candidates in 31 independent fields over $0.9\mbox{ deg}^2$. In this paper, the standard LBG selection limit of ${\cal R}=25.5$ was extended to fainter magnitudes $(\sim 0.1 L^*$) to obtain tighter constraints on the faint-end slope, $\alpha$. Recent luminosity function determinations at $z\sim 4$ include those by @bouwens2007, using 4671 $B$-dropout galaxies selected over $580\mbox{ arcmin}^2$ with deep [*Hubble Space Telescope*]{} ([*HST*]{}) Advanced Camera for Surveys (ACS) imaging, down to $M_{UV,\mbox{AB}}=-16$, and by [@vanderburg2010], based on $\sim 36,000$ $g-$dropout galaxies selected in the CFHT Legacy Survey in four independent $1\mbox{ deg}^2$ fields down to $M_{UV,\mbox{AB}}=-18.7$. Both of these $z\sim 4$ surveys, however, are solely based on photometric selection, without spectroscopic confirmation. Figure \[fig:empirical-LF-UV-reddysteidel2009bouwens2007\] reviews recent determinations of the rest-frame UV luminosity function at $2 \leq z \leq 4$.
The rest-frame UV luminosity functions of star-forming galaxies at $z\sim 2-4$ are characterized by steep faint-end slopes ($\alpha\sim -1.6 - -1.7 $) and characteristic luminosities of $M^*_{\mbox{AB}}=-21$. These steep faint-end slopes are in contrast to the flatter one ($\alpha= -1.22$) determined from the local far-UV (FUV) luminosity function using [*Galaxy Evolution Explorer*]{} (GALEX) data [@wyder2005]. Also, the characteristic luminosities at $z\sim 2-4$ are roughly three magnitudes brighter than those determined from the local GALEX FUV luminosity function. @reddysteidel2009 review other recent luminosity function determinations at $z\sim 2-3$, highlighting some of the discrepancies in the literature, including those determinations with significantly flatter faint-end slopes [@sawicki2006a; @gabasch2004], and/or larger counts at the brightest luminosities [@paltani2007]. To address some of these differences, @reddysteidel2009 point out the pitfalls associated with attempting to measure the luminosity function over small areas $(<50 \mbox{ arcmin}^2)$, making incorrect assumptions about the nature of the intrinsic mean and dispersion in colors of the faintest galaxies, and insufficiently accounting for contamination by low-redshift interlopers. Again, we emphasize the importance of spectroscopy for understanding the redshift selection function of the objects for which the luminosity function is being constructed, and the proper characterization of systematic effects modulating the volume probed by a given galaxy survey.
### REST-FRAME OPTICAL AND NEAR-INFRARED LUMINOSITY FUNCTIONS {#subsubsec:empirical-LF-optnearir}
While the rest-frame UV luminosity of distant galaxies reflects the emission from massive stars, longer wavelengths probe different aspects of galaxy stellar populations and dust content. Specifically, the rest-frame optical luminosity function is more reflective of older stars, although the extent to which emission at these wavelengths reflects the integrated stellar mass depends in detail on the star-formation history of the galaxy [@shapley2001; @shapley2005]. The rest-frame optical ($V$-band) luminosity function was first determined at $z\sim 3$ by @shapley2001, based on the LBG ${\cal R}$-band luminosity function and the distribution of ${\cal R}-K_s$ colors for a sample of 118 LBGs, 81 of which had spectroscopic redshifts. These measurements yielded a steep faint-end slope of $\alpha=-1.85$, which is significantly different from the local determinations with $\alpha \sim -1$, and a characteristic luminosity of $M^*_V=-22.98$, which is 1.5 magnitudes brighter than the local value [@blanton2003].
Most recently, based on a much larger set of $\sim 1000$ $K$-band measurements at $2\leq z\leq 3.5$, selected over a total area of $378\mbox{ arcmin}^2$ from multiple near-IR surveys of varying depths, @marchesini2007 constructed $B$, $V$, and $R$-band luminosity functions. Using mainly photometric redshifts, @marchesini2007 find that the faint-end slopes of these luminosity functions are, within the errors, consistent with the fairly shallow slope determined for the local optical luminosity function, whereas the characteristic magnitudes are significantly brighter ($\gtrsim 1$ magnitude). In contrast to @shapley2001, @marchesini2007 find values of $\alpha$ ranging from $-1.0 - -1.4$. However, it is worth pointing out that, in the region of overlap, the @marchesini2007 rest-frame $V$-band luminosity function for “blue" ($J-K\leq2.3$) galaxies and the @shapley2001 rest-frame $V$-band LBG luminosity function are entirely consistent. Both total and “blue" $V$-band luminosity functions from @marchesini2007 are shown along with the luminosity function from @shapley2001 in Figure \[fig:empirical-LF-optnearir-marchesini2007\], reproduced from @marchesini2007. Potential limitations of the @marchesini2007 analysis are in the small fraction of spectroscopic redshifts ($\sim 4$%) and corresponding reliance on photometric redshifts, and the small area over which the photometry is deep enough to robustly probe the faint end ($\sim 25 \mbox { arcmin}^2$). With ultra-deep $K$-band \[$K_{limit}\sim 23$(Vega)\] surveys over significantly larger areas ($\sim 1000$s of arcmin$^2$), such as the UKIDSS Ultra-Deep Survey (UDS), the much-needed robust constraints on the faint-end slope of the rest-frame optical luminosity function will be within reach. Extensive spectroscopic follow-up is also necessary to avoid some of the biases related to photometric redshifts that are described in @reddy2008 and @reddysteidel2009.
In principle, the rest-frame near-IR luminosity is even more closely tied to stellar mass than the rest-frame optical (despite some of the uncertainties we will discuss in Section \[subsec:stellarpop-SPS\]). The largest study to date of the evolving rest-frame $K$-band luminosity function is based on the UKIDSS UDS First Data release, presented by @cirasuolo2010, updating their earlier work [@cirasuolo2007]. This study features a $K$+$z$-band-selected catalog of $\sim 50,000$ galaxies over $0.7 \mbox{ deg}^2$ ($\sim 10,000$ of which are at $z>1.5$), which is complete down to $K=23$ (AB). Given the location of the UKIDSS UDS in the Subaru/XMM-Newton Deep Survey field, this survey also benefits from extensive multiwavelength coverage spanning from FUV (GALEX) to mid-IR ([*Spitzer*]{}) wavelengths. @cirasuolo2010 estimate photometric redshifts and rest-frame $K$-band luminosities based on spectral-energy distribution (SED) fits to the multiwavelength photometry of their sources. The depth of the @cirasuolo2010 sample is not sufficient to trace the faint-end slope and its evolution past $z\sim 1$, so the evolution to high redshift is simply quantified in terms of the luminosity function normalization and characteristic luminosity. From $z\sim 0$ to $z\sim 2$, @cirasuolo2010 report a brightening in $M^*$ by $\sim 1$ magnitude and a decrease by a factor of $\sim 3.5$ in normalization, $\Phi^*$. Again, deeper near-IR photometry and extensive spectroscopic follow-up for near-IR-selected catalogs will be required to probe the full evolution of the rest-frame near-IR luminosity function at high redshift.
### REST-FRAME MID-INFRARED AND BOLOMETRIC LUMINOSITY FUNCTIONS {#subsubsec:empirical-LF-ir}
At even longer rest-frame wavelengths, we begin to probe the direct emission from dust. Building on the work of earlier missions such as the [*Infrared Astronomical Satellite*]{} (IRAS) and the [*Infrared Space Observatory*]{} (ISO), which traced the evolution of infrared-luminous (IR-luminous) sources out to $z\sim 1$, [ *Spitzer*]{} has played a crucial role in tracing the global dust emission properties for large samples of $z>1$ star-forming galaxies. The most widely-used tool in this endeavor is the $24
\mu\mbox{m}$ channel of the MIPS instrument, which probes a rest-frame wavelength of $\lambda \sim 8\mu\mbox{m}$ at $z\sim
2$, sensitive to the emission from polycyclic aromatic hydrocarbon (PAH) emission. Ground-based submillimeter observatories such as SCUBA have also played a key role in understanding the evolution of most IR-luminous sources, but for much smaller galaxy samples. Overall, out to $z\sim 1$, the evolution of the mid-IR and total IR luminosity function is characterized by both significant luminosity and density evolution [@lefloch2005]. IR-luminous galaxies were more numerous in the past, and the IR luminosity density at $z\sim 1$ is dominated by luminous infrared galaxies (LIRGs), with $L_{IR}>10^{11}L_{\odot}$.
Recent work by @perezgonzalez2005, @caputi2007, and @rodighiero2010 have characterized the luminosity functions of MIPS 24$\mu$m-selected galaxies at $z>1$. We focus here on the results of @caputi2007 and @rodighiero2010, who used more conservative criteria for excluding AGNs from their samples, and adopted conversions between rest-frame $8\mu$m and total IR luminosities that are most consistent with observed constraints (and agree well with each other). These studies select galaxies down to a flux limit of $S(24 \mu\mbox{m})=80 \mu\mbox{Jy}$ in the $0.08 \mbox{
deg}^2$ of the Great Observatories Origins Deep Survey (GOODS) North and South fields, while @rodighiero2010 additionally include a shallower catalog down to a limit of $S(24 \mu\mbox{m})=400 \mu\mbox{Jy}$ in the $0.85 \mbox{ deg}^2$ of the VVDS-SWIRE area. As shown in Figure \[fig:empirical-LF-ir-caputi2007\], @caputi2007 determine mid- and total IR luminosity functions at $z\sim 1$ and $z\sim 2$, while @rodighiero2010 construct the corresponding luminosity functions in nine redshift bins from $z\sim 0$ to $z\sim 2.5$.
Both of these works echo the previously determined increase of an order of magnitude in the bolometric IR luminosity density from $z\sim 0$ to $z\sim 1$, and the heightened abundance of both LIRGs and ULIRGs. @rodighiero2010 find that the IR luminosity density is roughly constant from $z\sim 1$ to $z\sim 2.5$, while @caputi2007 find evidence for a slight decline. For both of these studies, it is worth pointing out that their completeness limits at $z\sim 2$ in $8\mu$m luminosity translate into bolometric IR luminosities of $\sim 10^{12}L_{\odot}$. Therefore, these luminosity functions only directly probe the ULIRG regime. Furthermore, neither study can place constraints on the faint-end slope of the luminosity function, and instead adopt a fixed parameter of $\alpha=-1.2$ based on local observations. Using indirect estimates of the rest-frame $8\mu$m and bolometric IR luminosity functions at $z\sim 2$, inferred from the extinction-corrected luminosities of UV-selected star-forming galaxies, @reddy2008 extend the determination of the IR luminosity function into the LIRG regime, and suggest that a significantly steeper faint-end slope may be required. Accordingly, ULIRGs make a sub-dominant ($\sim 25$%) contribution to the far-IR luminosity density at $z\sim 2$. Future direct observations of this fainter IR regime with [*Herschel*]{}, and even more sensitive planned facilities such as the [*Single Aperture Far-Infrared*]{} (SAFIR) observatory, will be crucial in untangling the evolution of the IR luminosity density and the history of obscured star formation in the Universe.
Color-Magnitude Diagrams {#subsec:empirical-CMD}
------------------------
A standard tool for describing the galaxy population in the local Universe is the optical “color-magnitude" diagram, in which rest-frame optical colors are plotted as a function of rest-frame optical absolute magnitude. The added dimension of color proves very useful for separating different types of galaxies. Indeed, while the division of the galaxy population into different types (e.g., late-type spirals and irregulars, and early-type ellipticals and lenticulars), and the correlation between galaxy structure and color have been well-known for a long time, the advent of large spectroscopic surveys such as SDSS have allowed for a quantitative description of the bimodality in galaxy photometric properties, based on incredibly robust statistics. Both @strateva2001 and @baldry2004 have presented striking evidence that local galaxies occupy a bimodal distribution in the space of $u-r$ colors, and that the fraction of bluer galaxies increases at fainter optical absolute magnitudes. Galaxies occupying the so-called “red-sequence" part of the bimodal distribution primarily consist of morphologically early-type galaxies and follow a very tight relationship between color and magnitude in which more luminous galaxies have redder colors. At the same time, objects residing in the “blue cloud" are predominantly late-type galaxies, and show a looser, though still systematic, variation of color and luminosity in the same sense. The bimodal distribution of galaxy colors as a function of luminosity, along with the correlation of color and other galaxy properties, suggests two distinct types of formation histories. It is therefore of critical interest to trace the galaxy color-magnitude diagram to earlier times, and to determine how far back the bimodality in the galaxy distribution persists.
Based on the COMBO-17 survey, @bell2004 demonstrate that bimodality is detected in the color-magnitude diagram out to $z\sim 1$. Now, using the results from recent near- and mid-IR-selected surveys, other groups have considered the question of galaxy bimodality at even higher redshifts. As shown in Figure \[fig:empirical-CMD-cassata2008\], with a sample of 1021 [*Spitzer*]{}/IRAC $4.5\mu$m-selected objects from the GMASS survey with optical through mid-IR SEDs, 190 of which have spectroscopic redshifts above $z=1.4$, @cassata2008 demonstrate that a bimodality in galaxy rest-frame $U-B$ colors persists up to $z=2$. @brammer2009 uses $\sim 25,000$ objects with $K<22.8$ (AB) selected from the NEWFIRM Medium-Band Survey (NMBS), and detect a bimodality in rest-frame $U-V$ colors out to $z\sim 2.5$. The increasing importance of obscured star formation at higher redshifts tends to cause contamination of the red sequence by dusty, star-forming galaxies, and @brammer2009 show a much cleaner division in dust-corrected colors, or when considering only galaxies whose mid-IR flux limits of $S(24\mu\mbox{m})<20 \mu$Jy indicate the presence of little or no dust. Based on a smaller sample of 28 $K$-selected galaxies with spectroscopic redshifts $2 < z < 3$, @kriek2008b detect a red sequence in rest-frame $U-B$ colors, quantified in terms of a significant overdensity of galaxies within a narrow bin of red rest-frame $U-B$ color. These high-redshift red-sequence galaxies are characterized by little or no ongoing star formation, with strong Balmer breaks indicating that they are likely in a post-starburst phase. While suggestive of the existence of a red sequence, the sample in @kriek2008b is too small to test for the presence of bimodality in galaxy colors.
Both @kriek2008b and @cassata2008 stress the need for precise (and ideally spectroscopic) redshift information when attempting to detect features in the galaxy rest-frame color distribution such as bimodality or the presence of a red sequence. While they lack actual spectroscopic confirmation, galaxies in the NMBS presented by @brammer2009 have extremely accurate photometric redshifts, with $\Delta z / (1+z) < 0.02$ at $z>1.7$. Photometric redshifts typical of other studies, with errors of $\Delta z/(1+z)\sim 0.1$ or worse, will lead to random and systematic errors in inferred rest-frame UV and optical colors that will wash out these trends when a single color is considered. However, using only photometric redshifts but considering [*two*]{} rest-frame colors together, @williams2009 discern a bimodal behavior in the space of rest-frame $U-V$ versus $V-J$ color-color space out to $z\sim 2.5$. The rest-frame colors for this study are inferred from optical, near-IR, and [*Spitzer*]{}/IRAC photometry drawn from the UKIDSS UDS, the Subaru-[*XMM*]{} Deep Survey (SXDS), and the [*Spitzer*]{} Wide-Area Infrared Extragalactic Survey (SWIRE), and effectively separate quiescent, non-star-forming galaxies from their actively star-forming counterparts. The evolving locations in color space and very existence of the “red sequence" and “blue cloud" provide important constraints on models of galaxy formation.
THE STAR-FORMATION RATES AND STELLAR CONTENT OF HIGH-REDSHIFT GALAXIES {#sec:stellarpop}
======================================================================
While the luminosities and colors of galaxies represent basic and fundamental observables, we seek to translate these measurements into physical quantities. Specifically, the SEDs of galaxies are commonly interpreted in terms of the current rate of star formation and its past history, as well as the integrated stellar content of galaxies. In this section, we summarize both the simple methods used to infer star-formation rates from specific luminosities, as well as the techniques used to model the stellar populations of high-redshift galaxies based on multiwavelength SEDs. Along the way, we highlight the results of applying these methods to determine the global history of star formation, the build-up of stellar mass density, and the relationship between star-formation rate and stellar mass in distant galaxies.
Star-formation Rate Indicators {#subsec:stellarpop-SFindicator}
------------------------------
As reviewed by @kennicutt1998, there are several diagnostics of star-formation activity in external galaxies. In the study of distant galaxies at $z>2$, integrated light measurements tracing young stellar populations are used to infer the rate at which stars are being produced. These measurements include rest-frame UV and IR luminosities, hydrogen recombination emission-line luminosities, and stacked X-ray and radio luminosities (due to the sensitivity limits of current X-ray and radio facilities, $z>2$ star-forming galaxies are typically only detected in a statistical sense). In general, these diagnostics are only sensitive to the presence of massive stars. Therefore, an estimate of the total star-formation rate requires the assumption of a particular form for the stellar initial mass function (IMF), which is then used to extrapolate down to the low stellar masses that dominate the integrated stellar mass.
The most commonly used star-formation diagnostic for high-redshift galaxies is the ultraviolet continuum luminosity over the wavelength range $1500-2800$ Å, which is dominated by O and B stars and directly related to the star-formation rate in galaxies where star formation has been proceeding at a roughly constant rate on $\sim 10^8$-year timescales. The rest-frame UV luminosity at $2\leq z \leq 4$ is based on observed-frame optical photometry. As described in Section \[subsec:ISM-dustext\], the colors of star-forming galaxies at high redshift suggest that dust extinction leads to a significant attenuation of the rest-frame UV luminosity. Therefore, the rest-frame UV luminosities based on optical apparent fluxes must be corrected for dust when calculating intrinsic star-formation rates.
Dust absorbing the radiation from massive stars is heated and re-radiates this absorbed energy in the far-IR region of the spectrum. Therefore, the far-IR luminosity from dust also provides a tracer of the rate of massive star formation. The range “far-IR" is commonly defined as $8-1000 \mu$m [@kennicutt1998], and the luminosity spanning this wavelength range is adopted as a tracer of star formation for systems in which star formation has proceeded continuously for at least $10^7-10^8$-year timescales. For galaxies with bolometric IR luminosities large enough ($L_{IR}\gtrsim 10^{12}L_{\odot}$) to be detected with ground-based submillimeter telescopes such as SCUBA, the observed 850$\mu$m flux can be converted into a far-IR luminosity with an assumption of dust temperature (see Section \[subsec:ISM-dustem\]). For $z\sim 2$ galaxies with smaller bolometric luminosities (down to $L_{IR}\sim 10^{11} L_{\odot}$), deep [*Spitzer*]{}/MIPS $24 \mu$m observations have provided a proxy for far-IR luminosity. At $z\sim 2$, the $24 \mu$m channel probes a rest wavelength of $8 \mu$m, the location of strong PAH emission. Local galaxy templates from @dale2002, @chary2001, @elbaz2002, and @rieke2009 are used to convert from the mid-IR to total IR luminosity, which corresponds to an increase by an order of magnitude. Recent [*Herschel*]{} Photodetector Array Camera and Spectrometer (PACS) $160 \mu$m measurements of the far-IR luminosities of star-forming galaxies at $z\sim 2$ suggest that MIPS $24 \mu$m proxies for far-IR luminosities tend to yield overestimates by factors of $\sim 4-7.5$ [@nordon2010]. This apparent discrepancy will require additional study. [*Spitzer*]{}/MIPS $70 \mu$m data probing rest-frame $20-25 \mu$m at $z\sim 2$ have not been used as widely because of the limit of the depth of the existing [*Spitzer*]{} data, in which very few individual $z\geq 2$ galaxies are detected.
As photons from massive stars ionize nearby interstellar gas, the emission from recombining hydrogen gas in these star-forming regions serves as a proxy for the rate of production of ionizing photons, and, by extension, the formation rate of massive stars. The ionizing flux is dominated by emission from the most massive ($>10M_{\odot}$) stars and provides an estimate of the instantaneous rate of star formation. With ground-based observations, H$\alpha$ emission lines have been measured for star-forming galaxies up to $z \sim 2.6$ (at which redshift the thermal background begins to dominate the noise) and used to estimate star formation rates [@erb2006c; @forsterschreiber2009]. The weaker H$\beta$ feature has also been used to estimate star-formation rates at redshifts beyond $z=2.6$ [@pettini2001; @mannucci2009]. In addition to Balmer lines, the Ly$\alpha$ feature has been used as a proxy for star-formation rate. However, due to the resonant nature of the Ly$\alpha$ transition, this line is especially sensitive to the effects of dust extinction and scattering, which can both lead to the preferential destruction of Ly$\alpha$ photons and their diffusion over a large area with reduced surface brightness. Accordingly, there is evidence that even dust-corrected observations of Ly$\alpha$ from individual galaxies tend to underpredict the star formation rate, when compared with other indicators such as rest-frame UV luminosity or H$\alpha$ emission [@hayes2011].
Additional star-formation rate diagnostics are based on X-ray and radio luminosities. High-mass X-ray binaries, young supernova remnants, and hot interstellar gas contribute to the X-ray luminosity in star-forming galaxies. @ranalli2003 calibrate the relation between rest-frame $2-10$ keV and far-IR luminosities for such systems, leading to a relation between hard X-ray luminosity and star-formation rate. Based on the tight linear correlation between far-IR and $1.4$ GHz radio luminosity among nearby star-forming galaxies, @yun2001 derive the relation between star-formation rate and radio luminosity. The deepest current [*Chandra*]{} $0.25-2.0$ keV (observed) and VLA 1.4 GHz radio data (e.g., in the GOODS-N region) are still not sufficient for detecting all but the most luminous star-forming galaxies and AGNs at $z\geq 2$. Therefore, most studies that make use of X-ray or radio estimates of star-formation rates use stacking techniques [e.g., @reddy2004; @daddi2007; @pannella2009], yielding only sample-averaged properties.
Evolution of the Star-formation Rate Density {#subsec:stellarpop-SFD}
--------------------------------------------
The evolution of the star-formation-rate density is considered one of the most fundamental observational descriptions of the galaxy population as a whole. Matching and explaining this observed evolution is often used as a benchmark of success for theoretical models of galaxy formation. The evolution of the star-formation-rate density is constructed by integrating the luminosity function at a specific wavelength sensitive to star formation (e.g., rest-frame UV, H$\alpha$, far-IR, or radio), in order to obtain the associated luminosity density. Then, a conversion between luminosity and star-formation rate (as described in Section \[subsec:stellarpop-SFindicator\]) is used to obtain the associated star-formation-rate density. In such studies, it is crucial to specify the limits down to which the luminosity function is integrated at each redshift [e.g., @bouwens2007]. Also, at shorter wavelengths, such as rest-frame UV and H$\alpha$, corrections for dust extinction must be applied in order to obtain the unobscured star-formation rate density [see Section \[subsec:ISM-dustext\]; @calzetti2000]. These corrections are sometimes applied in an average sense to the integrated luminosity density, or else in a luminosity-dependent fashion, taking into account the relationship between luminosity and dust obscuration [@hopkins2004].
The first attempts to chart the star-formation history of the Universe were presented by @madau1996 and @lilly1996. At the present, an extensive set of measurements has been compiled by @hopkins2004 and @hopkinsbeacom2006 from $z\sim 0$ to $z\sim 6$, using a variety of star-formation rate indicators including rest-frame UV, far-IR, radio, and X-ray continuum luminosities, and both Balmer and forbidden \[OII\] emission lines. These measurements were all converted to a common cosmology ($\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, $H_0=70 \mbox{ km s}^{-1}\mbox{ Mpc}^{-1}$), and rest-frame UV and optical measurements were corrected for dust extinction. In @bouwens2007 [@bouwens2010], rest-frame UV luminosity functions, both corrected and uncorrected for dust extinction, are used to map the evolution of the star-formation-rate density to $z\sim 8$. As shown in Figure \[fig:stellarpop-SFD-hopkinsbeacom2006bouwens2010\], in both of these compilations, the evolution of the star-formation-rate density is characterized by an order-of-magnitude increase from $z\sim 0$ to $z\sim 2$. In more detail, the compilations by @hopkins2004 and @hopkinsbeacom2006 show the dust-corrected star-formation-rate density increasing by an order of magnitude from its local value already by $z\sim 1$ and remaining roughly flat to $z\sim 2$. The evolution in @bouwens2007 [@bouwens2010] is characterized by a smooth rise all the way from $z\sim 0$ to $z\sim 2$, due to a different extinction correction adopted for the rest-frame UV data at $z<2$. In both versions of the cosmic star-formation history, the global star-formation-rate density remains roughly constant between $z\sim 2$ and $z\sim 4$, and then declines towards higher redshift. @hopkinsbeacom2006 propose that the star-formation history is constrained to within $\sim 30-50$% up to $z\sim 1$ and within a factor of $\sim 3$ at higher redshifts. Additionally, recent results from @reddy2008 suggest that $\sim 70-80$% of the star-formation-rate density at $z\sim 2-3$ is produced by galaxies with bolometric luminosities $L_{bol}\leq 10^{12} L_{\odot}$, and that the highly-obscured ULIRGs selected by submillimeter surveys [e.g., @chapman2005], although individually luminous, do not dominate the star-formation-rate density.
Stellar Population Synthesis Models {#subsec:stellarpop-SPS}
-----------------------------------
While luminosities tied to individual specific wavelength ranges are commonly used to infer the current rate of star formation, the multiwavelength SED of a galaxy can be interpreted in terms of its integrated stellar, dust and metal content, and its past history of star formation. This technique is referred to as stellar population synthesis modeling, and was first employed by @tinsley1968. In the intervening decades, stellar population synthesis models have become increasingly sophisticated, and have played a crucial role in inferring the physical properties of galaxies both near and far based on their photometric and spectroscopic properties. Population synthesis models are also used to estimate photometric redshifts, in so far as the observed multiwavelength photometry of a galaxy is fit in terms of not only best-fitting stellar population parameters, but also redshift.
The basic ingredients of stellar population synthesis models consist of a theoretical prescription for all stages of stellar evolution as a function of mass and metallicity, as well as a stellar spectral library of the observed properties of stars at different positions in the Hertzsprung-Russell diagram. The time evolution of the integrated spectrum from a coeval population of stars described by a specific stellar IMF and metallicity can then be predicted. For a given star-formation history, the evolution of the integrated spectrum of stars is computed by summing the contributions of simple stellar populations formed at successive time steps, with the normalization at each time step determined by the corresponding star-formation rate. Thus, stellar population synthesis models can predict the evolution of the integrated spectrum of a stellar population for an arbitrary star-formation history, IMF, and stellar metallicity. Stellar population synthesis models are often coupled with theoretical models [@charlotfall2000] or empirical parameterizations [@calzetti2000] of dust extinction to explain the observed properties of galaxies in terms of both stellar and dust content. The resulting model galaxy spectrum can be passed through photometric filters in order to predict the evolution of luminosities, colors, and mass-to-light ratios in specific photometric bands. Additional properties of the stellar population can also be calculated, including the rate of Type Ia and Type II supernovae explosions, the mass in stellar remnants and recycled gas, and the ionizing photon luminosity.
Currently, there are several different stellar population synthesis models employed for fitting the SEDs of distant galaxies. Perhaps most widely used are the models of @bruzualcharlot2003 and their current version (Charlot & Bruzual 2011, in preparation), though other significant theoretical efforts include the PEGASE and Starburst99 models [@fioc1997; @leitherer1999], and, more recently, the models of @maraston2005 and @conroy2009. There is much ongoing discussion in the literature regarding the systematic uncertainties and differences among these different stellar population synthesis codes, in terms of their descriptions of various stages of stellar evolution off of the main sequence.
There has been a particular focus on the treatment of the thermally-pulsating asymptotic giant branch (TP-AGB). TP-AGB stars are red giants with low- to intermediate-mass main sequence progenitors, which make a significant contribution to the near-IR luminosity of a simple stellar population at ages between $0.5$ and $2.0$ Gyr. As emphasized by @maraston2005, derived properties such as the age and near-IR mass-to-light ratio, and, correspondingly, the inferred stellar mass, will be very sensitive to the treatment of TP-AGB stars in systems in an evolutionary state that prominently features these stars. As a result, the stellar populations in which the correct treatment of the TP-AGB matters the most are post-starburst systems, as opposed to those in which star formation is proceeding at a roughly constant rate [@daddi2007]. As shown in Figure \[fig:stellarpop-SPS-maraston2005\], in the models of @maraston2005, TP-AGB stars make a much more significant contribution to the integrated galaxy luminosity at $\sim 1$ Gyr than in the models of @bruzualcharlot2003, resulting in systematically lower derived stellar masses and ages, in particular for post-starburst systems [@maraston2006]. Rather than adopting a specific prescription for the contribution of the TP-AGB phase, @conroy2009 parameterize the luminosities and effective temperatures of TP-AGB stars as variables to be constrained by the actual data. Systematic uncertainties in TP-AGB parameters therefore translate into systematic errors in other derived properties such as stellar mass and past star-formation history. Recently, @kriek2010 constructed a composite SED over the rest-frame range $1200-40000$ Å for a sample of 62 post-starburst galaxies at $0.7\leq z \leq 2.0$ drawn from the NEWFIRM Medium-Band Survey. In contrast to the results of @maraston2006, this entire SED is fit well by the @bruzualcharlot2003 models, whereas the rest-frame optical and near-IR regions of the spectrum are not simultaneously fit by the models of @maraston2005. The SED-fitting results of @kriek2010 suggest that the @maraston2005 models give too much weight to the TP-AGB phase. Clearly, consensus has yet to be reached about the influence of TP-AGB stars in the integrated spectra of galaxies.
Uncertainties in the detailed nature of the stellar IMF also translate into systematic uncertainties in the conversion between luminosity and stellar mass. For the range of stellar populations typically observed at high redshift (i.e., when the Universe was less than a few Gyr old), and over the rest-frame UV to near-IR wavelength range where current observations probe, models with the same star-formation history and the assumption of either a [@chabrier2003] or @salpeter1955 IMF over the mass range $0.1-100 M_{\odot}$ produce virtually identical colors. However, the model assuming a @chabrier2003 IMF corresponds to a stellar mass a factor of $\sim 1.8$ lower. Although both IMFs are described by power-law functions at $M\geq 1 M_{\odot}$, the @chabrier2003 IMF follows a log-normal distribution below $1 M_{\odot}$, turning over at the so-called “characteristic mass," whereas the @salpeter1955 IMF continues to increase as a power-law all the way down to $0.1 M_{\odot}$.
Evidence in the local and low-redshift Universe suggests that the @chabrier2003 IMF yields stellar $M/L$ ratios for lower-mass elliptical galaxies (with velocity dispersions of $\sigma \sim 200 \mbox{ km s}^{-1}$) in agreement with their luminosities and dynamical mass estimates. On the other hand, elliptical galaxies with larger dynamical masses appear to be characterized by either steeper (i.e., more Salpeter-like) IMFs or higher dark-matter fractions [@treu2010; @graves2010]. There are also extreme environments such as the Galactic Center and surrounding starburst clusters such as the Arches, in which direct evidence for IMF variations has been reported [@stolte2002]. At higher redshift, based on more indirect methods such as measuring the simultaneous $U-V$ color and stellar $B$-band $M/L$ ratio evolution for elliptical galaxies at $z\sim 0$ to $z\sim 0.8$ (with the latter estimated from the observed evolution in the fundamental plane), @vandokkum2008a infers that the stellar IMF for elliptical galaxy progenitors had a flatter slope at $z\sim 4$ in the regime near $1 M_{\odot}$. Furthermore, the characteristic mass at which the log-normal portion of the IMF turns over is inferred to shift to $\sim 2 M_{\odot}$. Adopting a completely independent approach based on a comparison of the star-formation rates and stellar masses of vigorously star-forming galaxies at $z\sim 2$, @dave2008 also infers a higher characteristic mass at this redshift. An upward shift in characteristic mass results in an IMF that is more “bottom light" than the present-day @chabrier2003 function, and a lower conversion factor from light to stellar mass. In contrast, recently @vandokkum2010_nature analyzed the spectra of four elliptical galaxies in the Virgo cluster and, based on the strength of stellar absorption features tracing $M< 0.3 M_{\odot}$ stars, concluded that the IMF in the massive star-forming progenitors of these systems had a “bottom heavy" IMF [*steeper*]{} than Salpeter between $0.1$ and $1.0 M_{\odot}$.
@bastian2010 review evidence for IMF variations both in the local Universe and at significant cosmological distances for the most part with skepticism. Clearly additional study is required to quantify or rule out the evidence for IMF variations as a function of environment. Stellar population synthesis models featuring explicit parameterizations of the most unconstrained phases of stellar evolution, as well as the form of the stellar IMF, will yield confidence intervals for derived stellar population parameters that more accurately reflect the systematic uncertainties associated with stellar population modeling [@conroy2009].
In spite of the significant uncertainties associated with stellar population synthesis modeling, it is now standard practice to use such models to infer basic physical properties of galaxies over a wide range of redshifts. In the local Universe for a sample of $>10^5$ SDSS galaxies, a combination of optical broadband photometric properties and spectral indices (the 4000 Å spectral break and the strength of Balmer absorption lines) have been modeled using the population synthesis code of @bruzualcharlot2003 to infer galaxy properties such as star-formation history, dust attenuation, and stellar mass [@kauffmann2003a]. At higher redshifts, stellar population modeling is typically tuned to broadband photometry alone, as rest-frame optical spectra are not of sufficient quality to measure stellar absorption features or detailed continuum shape at high S/N. At $z\geq2$, stellar population synthesis modeling is only possible if the rest-frame SED is probed at wavelengths both above and below age-sensitive spectral discontinuities such as the Balmer or 4000 Å break. Therefore, optical photometry probing the rest-frame UV must be, at the very least, combined with data probing the rest-frame optical regime (i.e. near-IR observed wavelengths), and, preferably, with additional [*Spitzer*]{}/IRAC photometry probing the rest-frame near-IR. Early models of $z>2$ stellar populations were featured in @sawicki1998, @shapley2001, and @papovich2001 (LBGs) and @forsterschreiber2004 (DRGs). A fairly recent example of $z \sim 2$ stellar population modeling from @muzzin2009 is shown in Figure \[fig:stellarpop-SPS-muzzin2009\]. This analysis is based on extremely well-sampled SEDs for spectroscopically-confirmed galaxies including optical, near-IR, and [*Spitzer*]{}/IRAC photometry as well as binned Gemini/GNIRS near-IR spectra, and features a systematic comparison of the parameters derived from @maraston2005, @bruzualcharlot2003, and updated Charlot & Bruzual models. Stellar population synthesis models are now routinely used to derive high-redshift galaxy physical properties. Such modeling is a standard component of the derivation of global distributions such as the galaxy stellar mass function (Section \[subsec:stellarpop-MstarD\]).
For the modeling of high-redshift galaxy stellar populations, in addition to the adoption of a particular stellar population synthesis code, the stellar IMF and metallicity are assumed parameters. A specific, wavelength-dependent dust extinction law (e.g., starburst, SMC, Milky Way) is also assumed. The star-formation history, $SFR(t)$, is commonly parameterized in the form of an exponential decline, with $SFR(t)=SFR_0 \times \exp(-t/\tau)$. In this case, $\tau$, the e-folding time, and $t$, the time since the onset of star formation, are both parameters to constrain. With such a parameterization, a continuous star-formation history corresponds to $\tau=\infty$ and $SFR(t)=SFR_0$. In addition to single episodes describing the star-formation rate as a function of time, more complex functions for the history of star formation have been considered. In particular, two-component models consisting of the linear combination of an old, high mass-to-light ratio component and a younger population with ongoing star formation, have been used to constrain how much stellar mass from the old stellar population could be “hiding" under the glare of a younger burst of star formation. In addition to $\tau$ and $t$, the parameters commonly derived for high-redshift galaxies are indicators of the degree of dust extinction \[$E(B-V)$ or $A_V$\], the current star-formation rate, and the integrated stellar mass. For a given stellar population synthesis code, the stellar mass has been demonstrated to be the best-constrained parameter [@papovich2001; @shapley2001; @shapley2005], whereas other parameters are more subject to uncertainties in the nature of the star-formation history (i.e., $\tau$), which is difficult to constrain in the absence of external multiwavelength information. Recently, @maraston2010 have in fact argued that so-called “inverted-$\tau$" models \[i.e. models in which the star-formation rate increases with time as $SFR(t)=SFR_0 \times \exp(+t/\tau)$\] provide a better description of the extinction and star-formation rate based on rest-frame UV data alone, as well as the star-formation rates and stellar masses of mock high-redshift galaxies constructed from semi-analytic models. In addition, @papovich2011 demonstrate that the globally averaged relations between the star-formation rates and stellar masses of galaxies at high redshift (see Section \[subsec:stellarpop-SFRM\*\]) appear to favor rising star-formation histories, as opposed to ones that are constant or declining. The best parameterization of the star-formation histories of high-redshift galaxies is clearly still a matter of debate.
The Diversity of High-Redshift Stellar Populations {#subsec:stellarpop-diversity}
--------------------------------------------------
The methods described above have been used to investigate the stellar populations of high-redshift galaxies selected using the various techniques discussed in Section \[sec:technique\]. Based on the results of stellar population syntehsis modeling, we can make some general comments about the range of stellar populations observed, while keeping in mind the uncertainties inherent to the modeling process, and the biases that different selection techniques impose. If the current samples of UV-selected galaxies are roughly characterized by star-formation rates of $10-100 M_{\odot}
\mbox{ yr}^{-1}$, typical stellar masses of $1-5 \times 10^{10} M_{\odot}$ [@erb2006b; @shapley2001; @reddy2004], and moderate amounts of extinction in the rest-frame UV (factor of $\sim 5$), the very complementary sample of rest-frame optically selected DRGs are typically characterized by higher star-formation rates ($\geq 100 M_{\odot}\mbox{ yr}^{-1}$), stellar masses (down to the typical survey limits of $K\sim 21$ Vega) of $\sim 10^{11} M_{\odot}$, and larger amounts of dust extinction in the rest-frame UV and optical ranges of the spectrum [@forsterschreiber2004; @papovich2006]. At the same time, a minority of DRGs show little evidence for ongoing star formation [@vandokkum2008b]. In addition to their prodigious star-formation rates, SMGs appear to be characterized by stellar masses that are comparable to those of the typical DRGs, and several times larger on average than those of UV-selected galaxies [@michalowski2010; @borys2005]. However, it is worth keeping in mind that the effects of dust extinction and AGN contamination on the broadband SED lead to larger systematic uncertainties in the derived stellar populations of SMGs. At the other end of the spectrum, so to speak, the stellar populations of LAEs are characterized by less dust extinction on average even than those of the UV-selected galaxies [@gawiser2007]. Due to the typically faint rest-frame UV luminosities of emission-line selected galaxies, the LAEs also tend to be faint in the rest-frame UV continuum, with lower star-formation rates than those of UV-selected galaxies, which were targeted down to a brighter continuum limit [@kornei2010].
In addition to considering the typical properties of galaxies as a function of selection method – which may not have anything other than historical value – it is also worth mentioning the range of star-formation histories observed as a function of mass. In particular, at high stellar masses $(M>10^{11} M_{\odot}$, assuming a Salpeter IMF from $0.1-100 M_{\odot}$), there exist at $z\sim 2$ not only active star-forming galaxies with $SFR>100 M_{\odot} \mbox{ yr}^{-1}$, but also passive, evolved galaxies with little evidence for ongoing star formation [@kriek2008b; @vandokkum2008b]. Quiescence and mature stellar populations constitute the physical interpretation of the empirically-derived red sequence reported by @kriek2008b and described in Section \[subsec:empirical-CMD\]. These quiescent galaxies appear to consitute $\sim 40-50$% of the most massive galaxies ($M_{star}\geq 10^{11} M_{\odot}$ at $z\sim 2-3$). Based on these results, it is worth noting that, in contrast to the patterns observed in the local Universe, at least half of most massive galaxies at $z\sim 2$ are still actively in the process of forming [@daddi2007; @papovich2006], and that there is an incredible diversity observed among the star-formation histories of these massive galaxies. At the same time, it is a challenge to explain massive galaxies at early times with little evidence for ongoing star formation, given that theoretical models predict copious rates of mass accretion at $z\sim 2$ for such massive systems [e.g., @dekel2009].
Evolution of the Stellar Mass Density {#subsec:stellarpop-MstarD}
-------------------------------------
Stellar population synthesis modeling provides a powerful tool for estimating the global evolution of the stellar content in galaxies, which reflects the combined processes of star formation and mergers. This evolution is described by constructing the galaxy stellar mass function at a range of redshifts. Analogous to the galaxy luminosity function, the stellar mass function is parameterized in terms of a characteristic mass, $M_{star}^*$, low-mass slope, $\alpha$, and overall normalization, $\Phi^*$. At each redshift, the stellar mass function can be integrated to determine the corresponding stellar mass density. Alternatively, the growth in stellar mass as a function of galaxy mass can provide important insights into galaxy formation models. The cosmic stellar mass density should also reflect the integral of past star formation in the Universe, and therefore the integral of the star-formation-rate density described in Section \[subsec:stellarpop-SFD\] can be compared for consistency with the independently-derived stellar-mass density.
In the local Universe, the galaxy stellar mass function has been determined from large samples of galaxies drawn from the 2dFGRS matched to the 2 Micron All-Sky Survey (2MASS), and SDSS [@cole2001; @baldry2008]. Presently, stellar mass functions have been measured out to $z\sim 5$, suggesting that roughly half of the local stellar mass density appears to be in place at $z\sim 1$. Measurements of the stellar mass function at $z>1$ are based on samples with multiwavelength (optical and IR) photometry and primarily photometric redshifts. The first determination of the stellar mass function at $z\sim 2-3$ was presented in @dickinson2003, based on a sample of rest-frame $B$-band-selected objects in the Hubble Deep Field North (HDF-N) with both optical and near-IR photometry. Subsequently, many other groups have measured stellar mass functions at $z\geq 2$, selecting galaxies at optical, near-IR and mid-IR wavelengths [e.g., @fontana2004; @fontana2006; @drory2005; @pozzetti2007; @elsner2008].
Both @perezgonzalez2008 and @marchesini2009 construct stellar mass functions at $z\geq 2$ based on fairly deep (near-IR and mid-IR magnitude limits of $23-25$ AB) and wide-area ($500-700$ arcmin$^2$) surveys, selected, respectively, with IRAC and $K$-band data. The larger areas of these surveys, compared to previous determinations, provide results that are less susceptible to cosmic variance. As shown in @marchesini2009, the $2\leq z\leq 3$ stellar mass functions in the literature, when integrated over a fixed range in stellar mass ($10^8 \leq M_{star}/M_{\odot} < 10^{13}$), vary by a factor of $\sim 3$ in stellar mass density. Figure \[fig:stellarpop-MstarD-marchesini2009\] [from @marchesini2009] features a compilation of global stellar mass density estimates as a function of redshift. At $z\sim 2$, the reported fraction of the local stellar mass density that is in place ranges from $\sim 8-25$%, while that number drops to $\sim 4-12$% at $z\sim 3.5$. @marchesini2009 also offer an in-depth analysis of the random and systematic uncertainties involved in constructing a stellar mass function from the modeling of multiwavelength photometry. These include the errors associated with photometric redshifts, cosmic variance, differences among stellar population synthesis codes [as parameterized by, e.g., @conroy2009], choice of stellar IMF, stellar metallicity, and extinction law. A proper accounting for these sources of random and systematic error significantly increases the uncertainties on derived quantities such as the evolution of the global stellar mass density, as well as the evolution of galaxies as a function of stellar mass. For example, when only taking into account random uncertainties, @marchesini2009 find that the abundance of galaxies below the characteristic mass evolves more strongly with redshift than that of the most massive galaxies $(M_{star}>10^{11.5})$, which show a lack of strong evolution in number density. However, when the full random and systematic error budget is accounted for, more significant evolution in the abundance of massive galaxies cannot be ruled out.
Additional uncertainties in the stellar mass density at high redshift result from uncertainties in the low-mass slope of the stellar mass function. While most works adopt $\alpha$ in the range to $-1.0$ to $-1.4$, the stellar mass regime crucial for constraining this parameter is not well probed with current observations. For example, the dataset featured in @marchesini2009 suffers from incompleteness below $\sim 10^{10}M_{\odot}$. According to @reddysteidel2009, if the stellar mass function has a steeper low-mass slope, as suggested by the steep faint-end slope of the rest-frame UV luminosity function and the relationship between UV luminosity and stellar mass, up to $\sim 50$% of the stellar mass density may be contained in galaxies with stellar masses $\leq 10^{10} M_{\odot}$, as opposed to the $\sim 10-20$% inferred from extrapolating the Schechter fits of @marchesini2009. In fact, using a $K$-selected sample based on significantly deeper near-IR imaging with Subaru/MOIRCS ($K=24.1$ Vega), @kajisawa2009 estimate a steeper low-mass slope of $\alpha=-1.5$ at $z\sim 2$ and $\alpha=-1.6$ at $z\sim 3$, as well as perhaps a trend of $\alpha$ steepening with redshift, also suggested by @fontana2006. This result is intriguing, but the area over which the low-mass slope is adequately probed is only $28 \mbox{ arcmin}^2$. Clearly, data of this depth must be collected over a significantly wider area to minimize the effects of cosmic variance and obtain more robust constraints on the stellar mass density at $z>2$. In particular, constraining the abundance and dust-extinction properties of faint, low-mass galaxies at high redshift will prove very important for comparisons of the past integral of the star-formation-rate density with the stellar mass density at each redshift [@reddysteidel2009]. Careful comparisons of this sort are crucial for determining whether or not the integral of global past star formation indicates a discrepancy with the global stellar mass density [@wilkins2008].
$SFR-M_{star}$ Scaling Relations {#subsec:stellarpop-SFRM*}
--------------------------------
In addition to considering the evolution of star-formation rates and stellar masses in galaxies separately, the evolution of the relationship between these quantities provides important clues as to how stellar mass builds up in galaxies as a function of redshift [*and*]{} mass. At $z\sim 2$, this relationship is highlighted by @daddi2007 in a sample of star-forming $sBzK$ galaxies (passive $pBzK$ galaxies and galaxies with no MIPS $24\mu$m detection were excluded from this analysis). Using star-formation rates estimated from either dust-corrected UV luminosity, or the sum of mid-IR and uncorrected UV luminosities, @daddi2007 find a strong correlation between star-formation rate (SFR) and stellar mass, described by the relation, $SFR\propto M_{star}^{0.9}$, and shown in Figure \[fig:stellarpop-SFRM\*-daddi2007\]. The ultra-luminous SMGs are outliers to the $SFR-M_{star}$ trend, with star-formation rates a factor of $\sim 10$ higher than expected, given their stellar masses. @pannella2009 used 1.4 GHz radio stacking observations to estimate average star-formation rates for non-AGN $sBzK$ objects in the Cosmic Evolution Survey (COSMOS) field, binned by stellar mass. The radio stacking analysis reveals the trend $SFR \propto M_{star}^{0.95}$, consistent with @daddi2007. Accordingly, the specific star-formation rate, i.e. the star-formation rate divided by stellar mass, appears to be roughly constant over an order of magnitude in stellar mass ($10^{10} - 10^{11} M_{\odot}$, assuming a Salpeter IMF from $0.1-100 M_{\odot}$). Similar trends between star-formation rate and stellar mass have been observed among star-forming galaxies at $z\sim 1$ by @elbaz2007 and @noeske2007, but with a lower overall normalization, such that, at a given fixed stellar mass, the expected star-formation rate is a factor of $\sim 4$ lower. A comparison with the trend observed in the local Universe [@brinchmann2004; @elbaz2007] indicates an evolution by a factor of $\sim 30-40$ [@daddi2007; @pannella2009].
On the other hand, perhaps the tightness of the $SFR-M_{star}$ correlation has been overemphasized, as a result of selection effects. Using the same ultra-deep, $K$-selected sample described in Section \[subsec:stellarpop-MstarD\], @kajisawa2010 investigate the $SFR-M_{star}$ relation from $0.5 \leq z \leq 3.5$. At $z\sim 2$, these authors find significantly more scatter at the high-stellar-mass end of the relation. In this study, UV dust-corrected and mid-IR+UV star-formation rates are estimated using the same techniques as in @daddi2007, but there is no requirement for galaxies to be detected at $24 \mu$m. At $M_{star}=10^{11} M_{\odot}$, for example, UV dust-corrected star-formation rates range from $1-1000 M_{\odot} \mbox{ yr}^{-1}$. Furthermore, the power-law slope between star-formation rate and mass, and therefore, the specific star-formation rate, tend to decrease at $M_{star}>10^{10.5} M_{\odot}$. Given that the slope and small scatter of the $SFR-M_{star}$ relation has been interpreted in terms of lending support to galaxy formation models in which smooth gas accretion dominates the growth of galaxies at high redshift [@dave2008], it is crucial to characterize these quantities accurately and in an unbiased manner. The above studies are largely based on photometric redshifts, which tend to increase the uncertainties in all derived physical properties. Therefore, larger spectroscopic samples are needed. Based on such spectroscopic studies, a description of the distribution of star-formation rates as a function of stellar mass for a [*stellar-mass selected*]{} sample will provide the ideal observational probe of this potentially meaningful trend.
THE INTERSTELLAR CONTENT OF HIGH-REDSHIFT GALAXIES {#sec:ISM}
==================================================
The stellar content of galaxies offers an incomplete version of the story of their formation and evolution, which must be filled in by a characterization of their interstellar environments. Indeed, the multiwavelength study of the current and past history of star formation in galaxies cannot be constructed without understanding the nature of dust. In addition to attenuating and reddening the radiation from stars, dust also reradiates the absorbed emission in the IR with a characteristic overall spectral shape that depends on temperature. The detailed emission spectrum in the mid-IR offers a further probe of the energy sources heating dust grains (i.e., radiation from star formation or an AGN). While dust reprocesses the light from stars, it is the cool gas content of galaxies that forms the very fuel for star formation. The elevated rate of star formation in galaxies in the early Universe is a direct result of the large cool gas fractions in these systems. As stars evolve and die, they return gas and heavy elements to the ISM. The patterns of chemical enrichment in both gas and stars therefore reflect the past history of star formation, and of gas inflows and outflows in galaxies. In this section, we review what is known about the interstellar contents of high-redshift galaxies. We begin by discussing both dust extinction and reprocessed emission, and then turn to observations of the molecular gas and metals, all of which provide important insights into the nature of distant galaxies.
Dust Extinction {#subsec:ISM-dustext}
---------------
Starburst galaxies in the local Universe follow a correlation between attenuation and reddening. As these galaxies become more attenuated in the rest-frame UV (as probed by the ratio of far-IR to UV luminosities), their rest-frame UV continua become redder. Based on multiwavelength observations spanning from the UV to far-IR, this correlation has been quantified as a relationship between the rest-frame UV slope, $\beta$ (where $f_\lambda \propto \lambda ^{\beta}$), and $A_{1600}$ (the attenuation at 1600 Å), such that $A_{1600} = 4.43 + 1.99 \beta$ [@meurer1999]. The starburst obscuration curve of @calzetti2000 describes the wavelength dependence of effective attenuation when dust is distributed in a patchy foreground screen, relative to young stars. For reference, the @calzetti2000 curve lacks the 2175 Å bump characteristic of the Milky Way extinction curve, and has a different ratio of total to selective extinction, with $R_V=A_V/E(B-V)=4.05$, as opposed to the average Milky Way value of 3.1 [@cardelli1989]. The @calzetti2000 law is also “grayer" than the SMC law [@prevot1984], rising less steeply in the near-UV, and therefore implying more attenuation for a given reddening in the rest-frame UV. This “grayness" likely stems from a geometrical configuration between gas and stars in which the dust is distributed in a patchy foreground-like screen, and some stars suffer little or no extinction [@calzetti2001]. This starburst attenuation curve predicts an almost identical relation between $A_{1600}$ and $\beta$ to that of @meurer1999, with the assumption of fairly uniform intrinsic rest-frame UV colors for starburst galaxies.
Soon after the discovery of LBGs at $z\sim 3$, their observed range of rest-frame UV colors was interpreted in terms of a range of dust reddening [@steidel1999; @meurer1999; @calzetti2000], in analogy with the description of local starbursts. The @calzetti2000 obscuration curve was used to translate between rest-frame UV color and $E(B-V)$, and, by extension, $A_{1600}$, with the average value of $E(B-V)=0.15$ corresponding to an attenuation factor of $\sim 4.7$ at $1500$ Å. In @steidel1999, this factor was used to correct the observed UV-luminosity densities at $z\sim 3$ and $z\sim 4$ and show that only a small fraction of the intrinsic UV radiation typically escapes from even UV-selected galaxies at high redshift. Furthermore, consistent with the trend observed among starbursts in the local Universe, @adelberger2000 find that objects with greater bolometric luminosities suffer more extinction in the rest-frame UV. However, the trend at $z\sim 3$ is offset from the local one in the sense that, for a given bolometric luminosity (i.e. star-formation rate), the extinction is significantly smaller at higher redshift. @bouwens2009 have investigated the average reddening as a function of rest-frame UV luminosity (uncorrected for dust), demonstrating that, at $z\sim 2.5$ and $z\sim 4.0$, $\beta$ becomes bluer for fainter objects. Furthermore, the average $\beta$ value of the most UV-luminous galaxies is redder at $z\sim 2.5$ than at $z\sim 4.0$. The trend between $\beta$ and UV luminosity was not apparent in the datasets of @adelberger2000 and @reddy2008, most likely due to the smaller dynamic range of UV luminosities probed.
At this point, the @calzetti2000 law is the most commonly adopted extinction law for modeling the stellar populations of high-redshift galaxies (Section \[sec:stellarpop\]). Therefore, it is important to consider the empirical support for this choice, based on comparisons between UV-extinction-corrected and extinction-free estimates of star-formation rates. While individual star-forming galaxies are not detected in deep [*Chandra*]{} X-ray and and VLA 1.4 GHz radio imaging observations, stacking methods have proven very powerful for comparing different multiwavelength star-formation rate indicators. @reddy2004 measure the stacked X-ray and radio fluxes for spectroscopically-confirmed, UV-selected star-forming galaxies at $1.5\leq z \leq 3.0$ in the GOODS-N field. On average, the inferred X-ray and radio-derived star-formation rates are, respectively, $42$ and $56\; M_{\odot}\mbox{ yr}^{-1}$. The UV star-formation rate, extinction corrected based on rest-frame UV colors and assuming the validity of the @calzetti2000 law, is $50\; M_{\odot}\mbox{ yr}^{-1}$ – consistent with the radio and X-ray estimates. Furthermore, the ratio of star-formation rates derived from X-ray and uncorrected UV fluxes implies a factor of $\sim 4.5-5.0$ attenuation in the UV, consistent with the attenuation inferred from the UV colors. @pannella2009 perform a similar radio stacking analysis for star-forming $sBzK$ galaxies with photometric redshifts at $z\sim 2$ in the COSMOS field, finding that the UV attenuation, $A_{1500}$, derived from the ratio between the extinction-free radio-derived star-formation rate and the UV-uncorrected star-formation rate, agrees well with that inferred on the basis of rest-frame UV colors alone, assuming the @calzetti2000 law applies.
[*Spitzer*]{}/MIPS $24 \mu$m observations have allowed for a test of extinction on a per object basis at $z\sim 2$. @reddy2006 [@reddy2010] compute mid-IR luminosities from observed $24 \mu$m fluxes for spectroscopically-confirmed UV-selected galaxies, and extrapolate these to total IR luminosities using local templates [@elbaz2002]. The relation between extinction ($L_{FIR}/L_{1600}$) and reddening ($\beta$) for these UV-selected galaxies is compared with the relation among local starbursts, revealing that the majority of objects (though not all – see below) follow the local trend. Furthermore, @reddy2006 confirm the result of @adelberger2000 of the positive correlation between bolometric luminosity and extinction, but now based on a [*Spitzer*]{}/MIPS estimate of the IR luminosity. The evolution in extinction with redshift is also reproduced using [*Spitzer*]{}, which suggests that extinction in the rest-frame UV is a factor of $\sim 10$ smaller at $z\sim 2$ than at $z\sim 0$, at fixed bolometric luminosity. @daddi2007 perform an analogous test of UV extinction laws, using star-formation rates for star-forming $sBzK$ galaxies derived from both rest-frame UV and [*Spitzer*]{} $24 \mu$m fluxes. @daddi2007 also conclude that the @calzetti2000 law is valid for the the majority of systems at $z\sim 2$.
While the success of the @calzetti2000 law is impressive, in terms of correcting UV luminosities and matching various extinction-free tracers of star-formation rates for large samples of high-redshift galaxies, it is also important to highlight the cases in which it appears to fail. For example, in ultraluminous SMGs at $z\sim 2$, the dust-corrected UV luminosities (assuming the @calzetti2000 law) underpredict the bolometric luminosities suggested by submillimeter and radio luminosities by as much as a factor of $\sim 10- 100$ [@chapman2005; @reddy2006; @daddi2007]. A related yet rather non-intuitive fact is that $\geq 50$% of the SMGs in the @chapman2005 sample have rest-frame UV colors that actually satisfy the UV-selection criteria of @steidel2003 [@steidel2004]. The mismatch between predicted (based on rest-UV color) and observed IR luminosities, shown in Figure \[fig:ISM-dustext-reddy2006siana2009\] (left), may arise in systems where regions of massive star formation are completely opaque to UV radiation, and the UV radiation that does escape is from regions that are disjoint from the dusty ones dominating the bolometric output [@daddi2007]. @reddy2006 find that the @calzetti2000 law appears to break down at $z\sim 2$ for systems more luminous than $\sim 2\times 10^{12} L_{\odot}$. On the other hand, @magdis2010a demonstrate that, at $z\sim 3$, the starburst attenuation curve yields consistent results for galaxies with bolometric luminosities as large as $10^{13} L_{\odot}$. In general, however, the @calzetti2000 law does not appear valid for describing dust extinction in the most luminous sources at $z\geq 2$.
While the local starburst relation appears to underpredict the UV attenuation for the most bolometrically luminous systems, there is a discrepancy in the opposite sense for objects that, based on their stellar population modeling, appear “young," with best-fit ages (assuming constant star-formation histories) less than 100 Myr. This second discrepancy was highlighted most dramatically in the case of the strongly gravitationally-lensed objects MS1512-cB58 (or “cB58") [@pettini2000] and the Cosmic Eye [@smail2007]. Both of these objects have rest-frame UV colors and continuum slopes that suggest dust-corrected star-formation rates several times larger than what is actually measured using [*Spitzer*]{}/MIPS mid-IR photometry and InfraRed Spectrograph (IRS) spectroscopy (a factor of $\sim 3-5$ for cB58, and $\sim 8$ for the Cosmic Eye) [@siana2008; @siana2009]. This discrepancy, shown in Figure \[fig:ISM-dustext-reddy2006siana2009\] (right), was previously noted for cB58 on the basis of millimeter and submillimeter observations [@baker2001; @sawicki2001].
A similar discrepancy is observed by @reddy2006 [@reddy2010] among UV-selected galaxies with best-fit stellar population ages of $t\leq 100$ Myr. As shown in Figure \[fig:ISM-dustext-reddy2006siana2009\] (left), the measured ratio of $L_{FIR}/L_{UV}$ for these young systems falls significantly below the local starburst relation, given their observed rest-frame UV slopes, $\beta$. Both cB58 and the Cosmic Eye are described by similarly young ages as well. Therefore, it appears that these young systems are described by a different extinction law, one that is steeper (as in the case of the SMC curve), such that a given observed amount of reddening corresponds to less attenuation in the rest-frame UV. More work is needed to develop a self-consistent evolutionary scenario to explain the reddening and attenuation properties of star-forming galaxies as a function of the maturity of their stellar populations. Finally, we call attention to recent results based on [*Herschel*]{} PACS $160 \mu$m measurements of star-forming galaxies at $z\sim 2$ [@nordon2010]. A comparison of PACS and UV, dust-corrected star-formation rates indicates that the UV-corrected values are, on average, 0.3 dex higher (with a scatter of 0.35 dex). More extensive comparisons with upcoming PACS far-IR measurements will be vital for further testing of the @calzetti2000 law.
Related to the overall reliability of the @calzetti2000 law, we must also consider the question of differential extinction of the radiation from stars and ionized gas. In local starbursts, emission-line tracers of ionized gas appear to be systematically more attenuated than the stellar continuum at similar wavelengths. The relationship derived between the stellar and nebular extinction is $E(B-V)_{stars} = 0.44 E(B-V)_{nebular}$ [@calzetti2000]. Currently, there is conflicting evidence about the relative extinction of stars and gas at $z\sim 2$. Comparing star-formation rates inferred from rest-frame UV and H$\alpha$ luminosities for a sample of $z\sim 2$ UV-selected galaxies (see Section \[subsec:ISM-metals\]), @erb2006c conclude that $E(B-V)_{stars} \approx E(B-V)_{nebular}$, and that a @calzetti2000 law applied to correct both UV-continuum and H$\alpha$ measurements gives rise to the best agreement between star-formation rate indicators. On the other hand, @forsterschreiber2009 find for a set of star-forming galaxies with VLT/SINFONI integral-field unit (IFU) maps of H$\alpha$ emission at roughly the same redshift as the @erb2006c sample (see Section \[subsubsec:structure-dynamics-ifu\]) that the best agreement between H$\alpha$ and UV-derived star-formation rates results when H$\alpha$ luminosities are corrected by an additional factor of $\sim 2$, and with the assumption $E(B-V)_{stars} = 0.44 E(B-V)_{nebular}$. To settle the question of differential extinction, much larger samples of objects are required with both measurements of multiple Balmer emission lines and rest-frame UV estimates of star-formation rates. Assembling these measurements will be possible with the next generation of multi-object near-IR spectrographs presently coming on-line on $8-10$-meter class telescopes.
Dust Emission {#subsec:ISM-dustem}
-------------
In addition to absorbing ultraviolet and optical radiation from stars, dust re-emits at IR and submillimeter wavelengths. Direct observations of this re-radiated emission at long wavelengths have opened a window into the nature of dust in distant galaxies, in terms of its temperature, composition, and the sources heating it. At far-IR and submillimeter wavelengths, direct observations of dust emission are restricted to the most luminous sources, with $L_{bol} > 10^{12} L_{\odot}$. While mid-IR imaging observations have been obtained for lower-luminosity systems using [*Spitzer*]{}/MIPS, spectroscopy has been limited to the most luminous sources, selected on the very basis of their bright submillimeter or mid-IR emission \[except in cases of strongly gravitationally-lensed systems [@siana2009]\].
Until recently, direct measurements of dust temperatures in $z>2$ ULIRGs only existed for small samples. Indirect estimates of the dust temperature, $T_d$, in SMGs were obtained by measuring the flux at individual rest-frame far-IR (i.e., observed 850 $\mu$m) and radio (i.e., 1.4 GHz) wavelengths, and assuming that the local correlation between far-IR and radio luminosities applies [@condon1992]. Based on the ratio between 850 $\mu$m flux and inferred total far-IR luminosity, $T_d$ can be inferred, assuming that the dust emission follows a single-temperature modified blackbody spectrum of the form $S_{\nu}\propto \frac{\nu^{3+\beta}}{\exp(h\nu/kT_d)-1}$, with the emissivity, $\beta$, set to a value of $1.5$. @chapman2005 use such a method to characterize their sample of 73 SMGs with spectroscopic redshifts at a median redshift of $z=2.2$. For this sample, the median inferred dust temperature is $T_d=36\pm 7$ K, $\sim 5$ K cooler than local ULIRGs with similar IR luminosities. In order to constrain the dust temperature more directly, photometric measurements at multiple rest-frame far-IR wavelengths are required, and the observations between [*Spitzer*]{}/MIPS and SCUBA/850 $\mu$m wavelengths have until recently been limited. These include Caltech Submillimeter Observatory (CSO) Submillimeter High Angular Resolution Camera (SHARC-2) observations at 350 $\mu$m from @kovacs2006 and @coppin2008 for a total of $\sim 30$ SMGs with previous $850 \mu$m detections. In these studies the additional far-IR SED point suggests a median $T_d=30-35$ K, consistent with the earlier, indirect estimate of $T_d=36$ K for SMGs. Furthermore, the total far-IR luminosities of SMGs in these samples are better constrained and typically $L_{FIR}\sim \mbox{ few} \times 10^{12} L_{\odot}$, with dust masses $M_{dust}\sim 10^9 M_{\odot}$, significantly larger than those observed in local starburst galaxies [@coppin2008].
New [*Herschel*]{} observations at $100$ and $160 \mu$m with PACS and at $250$, $350$, and $500 \mu$m with the Spectral and Photometric Imaging Receiver (SPIRE) have recently provided significantly more refined estimates of dust temperatures in high-redshift ULIRGs, and revealed a temperature diversity hinted at by earlier, indirect studies [@chapman2004; @casey2009]. Combining data in the three SPIRE channels with $850\mu$m SCUBA measurements, @chapman2010 find a median of $T_d=34\pm 5$ K for a sample of 31 SMGs and a hotter median $T_d=41\pm 5$ K for 37 radio-selected ULIRGs with fainter submillimeter fluxes (where the errors represent the standard deviations of the samples, not of the median values). These samples are characterized by median far-IR luminosities of $L_{IR}=7.1\times 10^{12} L_{\odot}$ and $3.8\times 10^{12} L_{\odot}$, respectively. Such large luminosities were inferred previously on the basis of much more limited rest-frame far-IR data, with large uncertainties on the total luminosity due to its strong dependence on assumed dust temperature. The constraints on the far-IR SED shape now offer a much more robust indication of the dust luminosity and temperature. @magdis2010b focus on combined PACS and SPIRE observations for a sample of star-forming MIPS $24\mu$m-selected ULIRGs at $z\sim 2$, finding a broad range of dust temperatures spanning from $25 \leq T_d \leq 65$ K, with a median of $T_d=42$ K, and a range of IR luminosities of $1.7\times 10^{12} L_{\odot} \leq L_{IR} \leq 8.7 \times 10^{12} L_{\odot}$. In particular, and as emphasized previously by @chapman2004, it is shown that ULIRGs with hotter dust temperatures and fainter submillimeter fluxes at $850-1200\mu$m would be missed in current ground-based surveys of SMGs, given their sensitivity limits at $\sim 1$mm. Therefore, ground-based surveys for $z>2$ ULIRGs selected on the basis of $850\mu$m fluxes may be biased towards the coolest of the most luminous galaxies. The origin of the diversity in dust temperatures among these most luminous systems is an open question.
While the far-IR SED offers constraints on the thermal properties and total luminosities of the most luminous galaxies at $z>2$, the mid-IR spectral range reveals the nature of their smaller dust grains and PAH molecules. Furthermore, a fundamental issue regarding ULIRGs at high redshift concerns the nature of the sources heating the dust that emits such copious amounts of far-IR radiation. The two basic alternatives are star-formation or AGN activity. While rest-frame UV and optical spectra [@chapman2005; @swinbank2004] and X-ray observations [@alexander2005] of SMGs indicate evidence for AGN activity in these systems, it is challenging to assess the bolometric importance of the AGNs from these data. The rest-frame mid-IR spectral region, from $\sim 5-15 \mu$m, offers strong discriminatory power between star-formation and AGN activity, and constraints on the relative contributions of each to the bolometric luminosity of high-redshift ULIRGs. The energetics of the radiation field are determined by the prominence of PAH emission features at $6.2$, $7.7$, $8.6$, and $11.3 \mu$m, relative to the strength of an underlying power-law continuum from hot dust and silicate dust absorption at $9.7\mu$m. The PAH emission is primarily tied to star formation, whereas the hot dust continuum is associated with AGN activity. The strength of silicate absorption potentially indicates the importance of a buried nuclear component [@sajina2007; @pope2008].
The IRS instrument onboard [*Spitzer*]{} has proven critical for untangling the processes powering the dust emission in ULIRGs at $z\sim 2$. Strikingly, the mid-IR spectra of the majority of SMGs exhibit strong PAH emission features [@menendez2009; @pope2008], indicating the dominance of star-formation over AGN activity in these systems. While many SMGs have X-ray properties suggesting the presence of an AGN, this component does not appear to be energetically dominant in terms of the bolometric luminosity, typically contributing $\leq 30$% of the luminosity in the mid-IR [@menendez2009; @pope2008]. Only 4 out of the 24 sources presented in @menendez2009, and 2 out of 13 of those described in @pope2008, have mid-IR spectra dominated by a power-law continuum, and the composite spectra for both samples exhibit pronounced PAH features. On the other hand, the $z\sim 2$ $24\mu$m-selected ULIRGs analyzed by @sajina2007 are more heterogeneous in the mid-IR. The majority ($\sim 75$%) of these sources have mid-IR spectra dominated by a power-law continuum and therefore AGN activity. At the same time, more than half of these power-law sources have PAH emission features as well, indicating contributions from both AGN and star-formation processes. The minority ($\sim 25$%) of PAH-dominated sources also indicate evidence for AGN activity in the form of hot dust continuum. Furthermore, the significant strength of $9.7\mu$m silicate absorption in $\sim 25$% of the sample indicates the presence of an obscured, compact nuclear component.
Figure \[fig:ISM-dustem-pope2008\], from @pope2008, illustrates the diversity among high-redshift ULIRG mid-IR spectra, including the PAH-dominated flavor common among SMGs and in a minority of $24 \mu$m-selected ULIRGs, and the power-law dominated spectra common in $24 \mu$m-selected ULIRGs, with different amounts of silicate absorption. While SMGs and $24 \mu$m-selected ULIRGs appear to have similar bolometric luminosities, AGNs appear to play a more significant role in the $24 \mu$m-selected ULIRGs, which also have hotter dust temperatures. These differences have been interpreted in terms of an evolutionary scenario, in which SMGs and $24 \mu$m-selected ULIRGs represent, respectively, earlier and later stages of a gas-rich, major merger event [@pope2008; @yan2010]. Through the progression of these stages, obscured nuclear AGN activity grows in importance as the system evolves into an unobscured QSO, and, eventually, a massive elliptical galaxy. While this proposed scenario is intriguing, additional constraints on the number densities, stellar mass and gas content of these various high-redshift samples is necessary to establish robust connections between them [@yan2010].
Molecular Gas Content {#subsec:ISM-gas}
---------------------
For a complete characterization of the process of star formation at high redshift, observations of the molecular phase of the ISM are critical. Stars form directly from this dense interstellar phase, which dominates the cool gas content of the most actively star-forming galaxies [@blitz2006]. In the local Universe, the star-formation rate per unit area, $\Sigma_{SFR}$, is directly related to the total gas surface density, $\Sigma_{gas}$, according to the empirical Schmidt Law, $\Sigma_{SFR} \propto \Sigma_{gas}^N$, with observational determinations of $N$ ranging from $0.9-1.7$ [@kennicutt1998]. Tracing this relationship at high redshift is key to understanding how gas is converted into stars during the epoch of peak star formation. Furthermore, the star-formation rate and gas content can be related to characterize the efficiency of star formation, as well as the timescale on which gas will be depleted. With short (compared to the Hubble time) gas depletion timescales, star formation can only be sustained by the ongoing accretion of gas from the IGM. The relationship between star formation and the available gas reservoir, in terms of its mass, baryonic mass fraction, and spatial extent, therefore offers crucial inputs into models of galaxy formation, which must include a proper description of the balance between gas accretion, the conversion of gas into stars, and the energetic feedback related to the process of star formation.
In the local Universe, rotational transitions of carbon monoxide (CO) serve as excellent tracers of molecular hydrogen gas [@young1991]. These same transitions are used to trace the molecular gas content of high-redshift galaxies, using millimeter-wave and radio telescopes. The first such observations were of extreme sources with known ultraluminous far-IR luminosities, such as QSOs and SMGs, and were suggestive of large amounts of dust and gas [@omont1996; @genzel2003]. Recently, however, CO detections have been achieved for less extreme systems, well into the LIRG regime and falling on the correlation between $SFR$ and $M_{star}$ that appears to describe more quiescently star-forming galaxies over the range $10 M_{\odot}\mbox{yr}^{-1} \leq SFR \leq \sim \mbox{few}
\times 100 M_{\odot}\mbox{yr}^{-1}$ [@daddi2007; @tacconi2008]. For observations of CO at high redshift, the IRAM Plateau de Bure Interferometer (PdBI) has played a dominant role, tuned to detecting various upper-level transitions \[e.g. CO(2-1), CO(3-2), CO(4-3), and so on\] in the millimeter range of the spectrum. Very recently, longer-wavelength observations carried out at the Expanded Very Large Array (EVLA) and Green Bank Telescope (GBT) have been tuned to the ground-state CO(1-0) transition, which is more directly tied to the molecular gas mass [@ivison2011; @harris2010].
In order to infer the mass of molecular gas, $M_{gas}$, associated with detected CO emission, there are two major sources of systematic uncertainty. One is in the conversion factor between CO(1-0) emission luminosity and $M_{gas}$. The second results from the fact that, for the most part, high-redshift CO observations have been of upper-level $J$ transitions, whereas the CO-to-$H_2$ conversion factor is calibrated for the CO(1-0) transition. The ratios between upper-level and ground-state $J$ transitions depend on the excitation and physical conditions in the molecular gas, and incorrect assumptions about these conditions will result in a bias in the inferred CO(1-0) luminosity. In some cases, higher-$J$ transitions, which are sensitive to warmer and denser regions, may even offer an incorrect representation of the spatial distribution of the full molecular gas reservoir [@ivison2011].
As for the CO-to-$H_2$ conversion factor, $\alpha$, it has been calibrated in the Milky Way with a value of $\alpha_{MW}\sim
4-5$, in units of $M_{\odot} (K \mbox{ km s}^{-1}\mbox{
pc}^2)^{-1}$. This value is very close to the theoretical expectation based on the assumption that CO line emission is produced in discrete, virialized clouds obeying scaling relations among their masses, sizes, and linewidths [@young1991]. On the other hand, in galactic nuclei and local starburst galaxies, different dynamical and geometrical conditions apply to the molecular gas, which may reside in a smoother, diskier configuration whose motions are additionally affected by the gravitational potential from stars. These differences result in a lower conversion factor for starbursts of $\alpha=0.8-1.6$ [@tacconi2008]. For observations of high-redshift SMGs, the starburst conversion factor has been adopted [@tacconi2006; @tacconi2008]. In fact, using the higher, Milky Way conversion factor would result in baryonic (gas plus stellar) masses in excess of the measured dynamical masses. On the other hand, for the lower-luminosity LIRGs (both UV-selected and $BzK$ sources), a Milky Way type conversion factor is favored on the basis of similar dynamical arguments [@daddi2010] and the idea that CO emission in these LIRGs arises in virialized clouds with densities similar to those observed in local quiescent disk galaxies [@tacconi2008].
We now consider the excitation of the molecular gas, which will affect the conversion between higher-$J$ transitions and CO(1-0). While the CO levels in SMGs appear to be thermally populated up to $J\geq 3$, simultaneous observations of CO(1-0) (VLA), and CO(2-1), and CO(3-2) (PdBI) transitions in a $BzK$ LIRG at $z\sim 1.5$ suggests that the CO(3-2) level is significantly subthermally excited, similar to what is observed in the Milky Way and other local disk galaxies [@dannerbauer2009]. These low-excitation physical conditions require a larger conversion factor from CO(3-2) to CO(1-0), and therefore, a larger inferred molecular gas mass for a given CO(3-2) line luminosity.
CO observations of both ULIRGs and LIRGs at $z\sim 2$ have yielded many important insights into the nature of star formation in different regimes of total galaxy luminosity. Before even considering inferred molecular gas masses, it is worth emphasizing the relationship between far-IR luminosity, $L_{FIR}$, and CO luminosity, $L_{CO}$. As shown in Figure \[fig:ISM-gas-genzel2010\], @genzel2010a have assembled measurements of local star-forming galaxies (both quiescent and merging systems), as well as $z\sim 1.5-2.0$ UV-selected (“BX") and $BzK$ systems and (more IR-luminous) SMGs. Most strikingly (and also pointed out by @daddi2010), at a fixed $L_{CO}$, high-redshift SMGs appear to produce $4-10$ times more far-IR luminosity than their UV-selected and $BzK$ counterparts, which cannot simply be attributed to enhanced AGN activity in SMGs. This trend mirrors the one observed between local mergers and more quiescently star-forming galaxies.
In terms of star-formation rate and molecular gas surface densities, SMGs are characterized by significantly higher $\Sigma_{SFR}$ at a given $\Sigma_{gas}$. The UV-selected and $BzK$ galaxies follow relations between $\Sigma_{SFR}$ and $\Sigma_{gas}$ that are similar to the one observed among quiescently star-forming galaxies in the local Universe, with a slope of $N=1.1-1.2$ [@genzel2010a]. SMGs follow an analogous relation but with a higher overall normalization, perhaps reflective of their shorter dynamical timescales [@genzel2010a], and also of processes that increase the efficiency of star formation in the turbulent, merger-driven environment that may be common in SMGs [@engel2010]. More speculatively, a top-heavy IMF in SMGs may produce more far-IR luminosity for a given mass of stars formed. However, the evidence for IMF variations at high-redshift is only indirect at this point, and not specific to SMGs [@vandokkum2008a; @dave2008]. At the same time, @ivison2011 offer a potential caveat regarding the inferred bimodality of star-formation efficiencies among SMGs and other, more quiescent $z\sim 2$ systems. In so far as the $L_{CO}$ values are based on a range of CO transitions, the conversion of these to CO(1-0) relies on a proper characterization of the gas excitation conditions. Larger samples of $z\sim 2$ star-forming galaxies with uniform CO(1-0) measurements will be crucial for characterizing the diversity (and actual bimodality) among star-formation efficiencies in high-redshift star-forming systems.
The inferred molecular gas reservoirs in UV-selected and $BzK$ galaxies are extended on scales of several kpc, with typical masses of $0.5-1.0\times 10^{11} M_{\odot}$ [@tacconi2010; @daddi2010]. These correspond to median gas fractions of 0.44 and 0.57, respectively, for the 10 UV-selected and 6 $BzK$ galaxies, and typical gas depletion timescales of $\sim 0.5$ Gyr [@genzel2010a]. The typical size of the molecular gas distribution is smaller (half-light radii of $\sim 2-3$ kpc) in SMGs, and, while the molecular gas masses and fractions are similar on average, the apparent gas depletion scales are a factor of several shorter than for the UV-selected and $BzK$ galaxies, due to the higher star-formation rates. With the Atacama Large Millimeter Array (ALMA) it will be possible to extend these studies down to fainter luminosities and characterize the gas reservoirs of more typical systems at $z>2$ that make up the bulk of the star formation at those early epochs.
Metal Content {#subsec:ISM-metals}
-------------
While molecular gas provides the material out of which stars form, heavy elements constitute an important product of star formation, returned to the ISM by supernova explosions and stellar winds. As such, the metal content of galaxies reflects the past integral of star formation, modified by the effects of gas inflow (i.e. gas accretion) and outflow (i.e. feedback from star formation or black-hole accretion). The relative abundances of different chemical elements also provides clues about the past history of star formation, in so far as so-called $\alpha$ elements (e.g., O, S, Si, Mg) are produced primarily in Type II supernovae events, on short timescales ($\sim 10$ Myr), while Fe-peak elements (e.g., Fe, Mn, Ni) are produced primarily in Type Ia supernovae over longer timescales ($\sim 1$ Gyr). The metal content of galaxies is especially meaningful when considered in concert with their stellar and gas masses, since the relationships among these quantities – and deviations from “closed-box" expectations provide constraints on the nature of large-scale gas flows (in both directions).
There are many different methods for measuring the metallicities of galaxies at high redshift, probing both their stellar and gaseous components using rest-frame UV and optical spectroscopic features. Stellar metallicity is measured from absorption lines, while the metal content of the interstellar gas can be gauged from either absorption lines arising in the neutral and ionized ISM, or emission features originating in H II regions.
In general, the continuum S/N and spectral resolution obtained for the rest-frame UV and optical spectra of typical $z\geq 2$ galaxies are not sufficient for robust absorption-line metallicity measurements in individual objects (even using $8-10$-meter class telescopes). The rest-frame UV spectra of star-forming galaxies include a host of interstellar features arising from neutral hydrogen and both neutral and ionized metal species, but only the strongest, highly-saturated interstellar absorption lines are detected on an individual-object basis within high-redshift samples [@shapley2003]. These saturated features are not useful for metallicity estimates. In exceptional cases of the spectra of strongly gravitationally-lensed objects, for which both the continuum S/N and resolution are at least an order of magnitude better than average, weak, unsaturated interstellar metal absorption features are detected and can be used for interstellar metallicity estimates. @pettini2002 measure weak features from $\alpha$, Fe-peak, and intermediate elements (i.e. nitrogen) in the spectrum of the gravitationally-lensed $z=2.73$ galaxy, cB58 (see Section \[subsec:ISM-dustext\]), to infer an actual interstellar abundance pattern. Based on the relative enhancement of $\alpha$ to both Fe-peak elements and nitrogen, @pettini2002 estimate a “young" age for cB58, of less than $\sim 300$ Myr, the timescale for nitrogen enrichment – an unusual case of a [*chemical*]{} constraint on the past history of star formation.
Rest-frame UV stellar absorption features from both the photospheric and wind features of hot stars can in principle be used to infer stellar metallicity. @rix2004 develop calibrations for metal-sensitive stellar photospheric absorption indices at 1370, 1425, and 1978 Å. These have been applied to a small number of individual lensed and unlensed star-forming galaxies at $z\sim 2-3$ [@steidel2004; @quider2009; @dessauges2010], yielding results consistent for the most part with other metallicity indicators. On the other hand, the 1978 Å Fe III index was measured in a composite spectrum of 75 star-forming galaxies drawn from the GMASS survey [@halliday2008], in fact suggesting a systematic enhancement of $\alpha$ relative to Fe, when compared with the expected oxygen abundance for objects of the same stellar mass (see below). The shape of the C IV$\lambda 1549$ P-Cygni wind feature from O and B stars is also sensitive to metallicity (as well as the form of the IMF), and has also been used to estimate stellar metallicity in a few (mainly gravitationally-lensed) $z\sim 2-3$ galaxies with adequate S/N and spectral resolution [@pettini2000; @quider2009; @quider2010]. These interstellar and stellar absorption metallicities offer intriguing and detailed probes of small numbers of special high-redshift galaxies, but await the power of future 30-meter-class ground-based telescopes for application to large samples of individual, unlensed objects.
Most results about the metal content of high-redshift galaxies are based on measurements of rest-frame optical emission lines from H II regions. These include combinations of hydrogen recombination lines (H$\alpha$, H$\beta$), and collisionally excited forbidden lines from heavy elements such as oxygen (\[OIII\], \[OII\]), nitrogen (\[NII\]), and neon (\[NeIII\]). At $z\geq 2$, rest-frame optical features shift out of the observed optical range and require near-IR spectroscopic observations. The study of rest-frame optical emission from H II regions at high redshift to date has relied on both long-slit and IFU spectroscopy, using instruments such as NIRSPEC and OSIRIS at the Keck Observatory, ISAAC and SINFONI at the VLT, MOIRCS on Subaru, and GNIRS on Gemini-South. Emission lines are primarily used to infer the gas-phase abundance of oxygen [expressed as $12+\log(\mbox{O/H})$, where the solar value in these units is 8.66; @asplund2004], and are based on the relations between particular sets of emission-line ratios and metallicity. These relations have been both empirically calibrated in the local Universe [e.g., @pettinipagel2004; @nagao2006] and theoretically modelled using photoionization codes [e.g., @kewley2002; @tremonti2004]. Two commonly-used indicators of $12+\log(\mbox{O/H})$ at high redshift are the so-called $N2$ index, defined as $\log(\mbox{[NII]}\lambda
6584 / \mbox{H}\alpha)$, and $R_{23} \equiv
(\mbox{[OIII]}+\mbox{[OII]})/\mbox{H}\beta$. From the ground, $R_{23}$ can be measured within near-IR windows of atmospheric transmision for various redshift intervals between $z\sim 2$ and $z\sim 4$, while $N2$ is only measurable up to $z\sim
2.6$, at which point the thermal background becomes preventively high.
In addition to the significant scatter among different emission-line indicators (differences as large as 0.7 dex in metallicity for the same galaxy) [@kewley2008], another potential source of bias when using locally-calibrated metallicity indicators to interpret the emission-line ratios of high-redshift galaxies is the assumption that the physical conditions in high-redshift galaxy H II regions are similar to those in local galaxies. Small samples of UV-selected galaxies at $z\sim 2$ with measurements of both \[OIII\]/H$\beta$ and \[NII\]/H$\alpha$, and/or \[OIII\]/\[OII\] and $R_{23}$, indicate systematic offsets from the excitation sequence of low-redshift galaxies [@erb2006a; @hainline2009]. These differences may be indicative of a systematically higher ionization parameter and/or electron density, or harder ionizing radiation spectrum, with correspondingly different translations between empirical line ratios and physical metallicities. Larger samples of objects at $z\sim 2$ with such measurements will be required to determine the origin of these apparent offsets, relative to local star-forming galaxies.
At this point, $N2$ measurements have been obtained for a wide variety of $z\sim 2$ sources, numbering $\sim 200$ and ranging from UV- and near-IR selected galaxies [e.g., @shapley2004; @erb2006a; @hayashi2009; @yoshikawa2010; @kriek2007] to SMGs [@swinbank2004]. The sample of objects with emission-line metallicity measurements at $z\geq 3$ is much smaller [@pettini2001; @maiolino2008; @mannucci2009], with metallicities based on $R_{23}$ or other combinations of oxygen, \[NeIII\], and Balmer lines for $\sim 30$ objects.
The measurement of galaxy metallicities along with stellar masses allows for the construction of the galaxy mass-metallicity (or $M_{star}$-Z) relation. This relationship has been studied in the local Universe for $>50,000$ galaxies with SDSS [@tremonti2004], and used as a tool to constrain the importance of inflows and outflows as a function of galaxy mass. Another necessary component for this type of analysis is an estimate of the gas mass, which enables a calculation of the metallicity as a function of gas fraction, or, equivalently, the effective yield, $y_{eff}$. Given the small sample of galaxies with direct atomic and/or molecular gas measurements both locally and at high redshift, gas masses have typically been estimated indirectly on the basis of $\Sigma_{SFR}$ values and galaxy sizes, and an assumption that the locally-calibrated relation between $\Sigma_{SFR}$ and $\Sigma_{gas}$ applies [@tremonti2004; @erb2006a]. We note here that the average gas fraction of $\sim 50$% inferred by @erb2006b for UV-selected star-forming galaxies at $z\sim 2$ agrees well with the CO-based estimate of $44$% from @tacconi2010.
The largest survey of rest-frame optical emission lines at $z\geq 2$ is presented in @erb2006a, where spectra covering the H$\alpha$ region were obtained for 87 UV-selected galaxies. As shown in Figure \[fig:ISM-metals-gas-erb2006a\], the empirically-measured $N2$ indicator increases monotonically in bins of increasing stellar mass, corresponding to an increase in $12+\log(\mbox{O/H})$. At a given stellar mass, @erb2006a find that $z\sim 2$ galaxies are $0.3$ dex lower in metallicity than the local galaxies studied in @tremonti2004. Based on indirect estimates of gas masses and gas fractions, these authors compare their observations with simple chemical evolution models. The shallow slope of the $z\sim 2$ $M_{star}-Z$ and gas-fraction$-Z$ relations are then used to demonstrate the importance of galaxy-scale winds with mass-outflow rates roughly equal to the star-formation rate [@erb2008]. The observed $M_{star}-Z$ slope at $z\sim 2$ has been viewed as evidence in favor of a “momentum-driven" feedback recipe in the simulations of @finlator2008, however we caution against placing too much significance on the precise observed value of this slope, given the known limitations of the $N2$ indicator (which saturates at roughly solar metallicity).
@maiolino2008 construct the $M_{star}-Z$ relation for 9 galaxies at $z\sim 3.5$, showing that, at fixed stellar mass, galaxies at $z\sim 3.5$ are 0.4 dex lower in $12+\log(\mbox{O/H})$ than at $z\sim 2$. @mannucci2010 explain the observed evolution of the $M_{star}-Z$ relation up to $z\sim 2.5$ in terms of galaxies populating a more general $M_{star}-Z-SFR$ relation, with no redshift evolution. According to this “Fundamental Metallicity Relation" [FMR; @mannucci2010], galaxies at fixed stellar mass but higher star-formation rate will have lower metallicity, resulting from the interplay between infalling IGM gas and outflowing enriched material. The rapid apparent evolution in the $M_{star}-Z$ relation beyond $z\sim 2.5$ requires some evolution in the FMR, yet the sample of galaxies at these redshifts with metallicity measurements is too small to draw definitive conclusions.
In order to use the $M_{star}-Z$ relation to constrain models of gas inflow and outflow in star-forming galaxies at $z\sim 2-4$, significantly larger (i.e. at least an order of magnitude) samples of galaxies with metallicity measurements are required, which are complete down to a given stellar mass or star-formation rate. Furthermore, constructing the relations between metal and baryonic (stellar and gas) content for galaxies with direct estimates of molecular gas content will provide more robust estimates of the metallicity as a function of gas fraction. Spatially-resolved estimates of chemical abundance gradients will offer additional constraints on the inflow and outflow of gas, and the build-up of the galaxy stellar population. These have been now reported for a handful of objects, with conflicting results about the sign of the radial gradient [@cresci2010; @jones2010]. Deeper data for larger samples will be required to settle this discrepancy. The next generation of near-IR multi-slit spectrographs and IFUs should easily enable these observations.
STRUCTURAL AND DYNAMICAL PROPERTIES {#sec:structure}
===================================
The diversity among galaxy structural and dynamical parameters in the local Universe reflects a range of mass assembly histories, and the relative importance of mergers and smoother mass accretion. Furthermore, properties such as size, concentration, mass-surface density, and velocity dispersion are correlated with indicators of the nature of stellar populations, such as luminosity, stellar mass, and specific star-formation rate [@kauffmann2003b; @shen2003]. Reproducing the observed connections among these galaxy structural and stellar properties and their redshift evolution within a unified framework constitutes a crucial test for models of galaxy formation. In this section we review recent results about the structural and dynamical properties of high-redshift galaxies. Due to the small apparent sizes of most $z\geq 2$ galaxies ($\lesssim 1$"), [*HST*]{} imaging and ground-based IFU maps assisted by adaptive optics (AO) are both critical for resolving overall shapes and $\sim$kpc-scale fine structure. Interferometric observations of CO line emission have also provided valuable insight into the dynamical properties of the most extreme sources at high-redshift, such as SMGs. While high-resolution imaging and dynamical maps are each powerful probes individually, combining these two methods of observations [@forsterschreiber2011] will provide the deepest insights into the nature of the galaxy assembly process at high redshift.
Structural Properties {#subsec:structure-structure}
---------------------
In general, the morphologies of galaxies at $z\geq 2$ are characterized by two important differences with respect to those of local galaxies. First, the traditional Hubble sequence of regular spirals and elliptical galaxies has not settled into place by $z\sim 2$, and a much higher frequency of clumpy, irregular morphologies is observed among star-forming systems [@ravindranath2006; @lotz2006]. Second, both star-forming and quiescent galaxies are more compact at fixed stellar mass or rest-frame optical luminosity [@buitrago2008]. In order to characterize the structural properties of high-redshift galaxies, both parametric and non-parametric methods have been employed. In the former, model Sérsic profiles are fit to the data, yielding best-fit values for $n$, the Sérsic power-law index, and $r_{e}$, the effective radius, within which half the galaxy luminosity is emitted. Sérsic profile fits work best for regular, axially-symmetric morphologies with well-defined centers, so, in order to address the clumpy, irregular structures observed among high-redshift systems non-parametric statistics such as the $Gini$, $M_{20}$, $Multiplicity$, and $CAS$ coefficients have been developed [@abraham2003; @lotz2004; @lotz2006; @law2007a; @conselice2003] to characterize their concentrations, asymmetries, and clumpiness (i.e. deviations from maximally compact configurations).
The first high-redshift galaxies to be observed with [*HST*]{} were UV-selected galaxies at $z\sim 3$ [@giavalisco1996; @lowenthal1997]. The irregular morphologies of these systems, often composed of multiple compact components and/or irregular nebulosity, were first attributed to the fact that the Wide Field Planetary Camera 2 (WFPC2) and ACS are optical imagers, probing rest-frame UV wavelengths at $z\sim 3$ – a so-called “morphological $k$-correction." In principle, measurements at these wavelengths might be sensitive to only the most active regions of star formation rather than the bulk of the stellar mass, and potentially affected by patchy dust extinction. However, in practice, rest-frame optical imaging with the Near Infrared Camera and Multi-Object Spectrometer (NICMOS), using both NIC2 and NIC3 cameras [@dickinson2000; @papovich2005; @forsterschreiber2011; @kriek2009] and the Wide Field Camera 3 (WFC3) [@overzier2010] has demonstrated that actively star-forming galaxies at $z\sim 2-3$ have very similar rest-frame UV and optical morphologies. On the other hand, among the most massive, evolved systems at $z\sim 2-3$, there is evidence for more centrally-concentrated, regular morphologies at rest-frame optical wavelengths, compared with the structure observed in the rest-frame UV [@toft2005; @cameron2010]. With the recent installation of WFC3, it will now be possible to measure rest-frame optical morphologies for statistical samples of $z\sim 2-3$ objects at $\sim$kpc-scale resolution. The performance of the IR channel of WFC3 is superior to the NICMOS NIC3 camera in terms of resolution, sensitivity, and area.
While the traditional Hubble sequence of disk and elliptical is not in place by $z\sim 2$, [*HST*]{} imaging has revealed a large diversity of structural parameters among galaxies at $z\sim 2-3$, which correlate with their stellar populations in analogy with the trends observed in the local Universe. Based on NICMOS/NIC3 F160W imaging, @zirm2007 and @toft2007 show that quiescent, high-redshift $z\sim 2$ galaxies have systematically smaller sizes and higher stellar-mass surface densities than actively star-forming systems of the same stellar mass. @franx2008, @toft2009, and @williams2010 use deep, ground-based $K$-band images to demonstrate the same trend. Based on higher-resolution NICMOS/NIC2 F160W imaging for a sample of spectroscopically-confirmed $K$-selected massive galaxies at $z\sim 2$, @kriek2009 presents a comparison of the Sérsic profile fits for the quiescent and emission-line objects. As shown in Figure \[fig:structure-structure-kriek2009\], galaxies with redder rest-frame $U-B$ colors, lower specific star-formation rates, and SEDs indicative of more evolved stellar populations, are characterized by smaller $r_e$ and more concentrated surface-brightness profiles (larger $n$). This figure also shows the clumpy rest-frame optical morphologies of the massive, blue galaxies. Objects whose emission-line ratios actually suggest AGN activity show structural properties and SEDs that are similar to those of the quiescent galaxies. @forsterschreiber2011 find even larger sizes and shallower profiles in NIC2 images of the six clumpy UV-selected galaxies in their sample, which also have higher specific star-formation rates. Also similar to the trends observed in the local Universe, @franx2008 and @forsterschreiber2011 find a correlation between stellar mass surface density and specific star-formation rate, such that galaxies with higher stellar mass densities have lower specific star-formation rates [@kauffmann2003b].
We now highlight two important observed morphological phenomena, which have sparked much interest from the theoretical community. One is the nature of the clumpy morphology and “clumps" in actively star-forming galaxies at $z\geq 2$. The other is the incredibly compact nature of massive, quiescent galaxies at high redshift, and their connection to today’s massive, early-type galaxies.
### CLUMPY MORPHOLOGIES IN STAR-FORMING SYSTEMS {#subsubsec:structure-structure-clump}
While it is possible to model the rest-frame UV and optical morphologies of $z\geq 2$ galaxies using smooth Sérsic profiles, significant residuals result due to the presence of non-axisymmetric surface-brightness fluctuations. These “clumps" have been fairly ubiquitously observed among the rest-frame UV and optical morphologies of distant galaxies [@cowie1995; @elmegreen2005; @law2007a; @lotz2004], and are visible among the actively star-forming galaxies in Figure \[fig:structure-structure-kriek2009\]. Clumps are characterized by typical sizes of $\sim 1$ kpc, and make up as much as $\sim 40$% of the rest-frame UV light distribution [@elmegreen2009]. While clumpy, irregular structure was first explained as the natural outcome of the increased prevalence of major merger events at high redshift [e.g., @conselice2003], various lines of evidence suggest that other factors may contribute to this phenomenon.
First, @law2007a demonstrated a decoupling between the $Gini$ coefficient (and any other rest-frame UV non-parametric morphological statistic) and star-formation rate. If clumpy morphology was connected with the incidence of a major merger, the $Gini$ coefficient, sensitive to non-uniformities in the light distribution, should display some correlation with the parameters describing galaxy stellar populations. @swinbank2010 also highlight the fact that the distribution of rest-frame UV and optical morphologies of SMGs [plausibly the sites of major merger events; @engel2010; @swinbank2006] were statistically indistinguishable from those of UV-selected star-forming galaxies, which are significantly more quiescent in terms of bolometric luminosity and star-formation rate. Furthermore, the clumpy phenomenon is far too common among star-forming galaxies at $z\sim 2-3$ to be accounted for quantitatively by the predicted rate of major merger events in, e.g., the Millennium Simulation [@genel2008; @conroy2008]. Finally, as described below in Section \[subsubsec:structure-dynamics-ifu\], clumpy structures are commonly observed in galaxies with ordered velocity fields indicative of rotating disks – demonstrating that a major merger is not necessarily the cause for clumps. Minor-merger interactions and instabilities in gas-rich, turbulent disks are more likely alternatives for the formation of clumps [@genzel2008; @bournaud2009; @dekel2009].
### THE COMPACTNESS OF QUIESCENT GALAXIES {#subsubsec:structure-structure-compact}
At fixed stellar mass, both star-forming and quiescent galaxies have smaller radii at higher redshift. The evolution in the $M_{star}-r_e$ relationship is especially dramatic for quiescent systems. As noted by many authors [e.g., @daddi2005; @trujillo2007; @toft2007; @cimatti2008; @vandokkum2008b; @buitrago2008; @damjanov2009] using rest-frame UV and optical [*HST*]{} imaging and ground-based data, $z\sim 1.5-2$ quiescent galaxies have radii that are $\sim 2-5$ times smaller than those of local early-type galaxies of the same stellar mass. For example, $z\sim 2$ quiescent galaxies with $M_{star}\sim 10^{11} M_{\odot}$ are measured to have $r_e\sim 1$ kpc [@vandokkum2008b], compared with $\sim 3$ kpc in local early-type galaxies of the same mass [@damjanov2009]. The observed differences in size correspond to factors of $\sim 10-100$ in physical density! Furthermore, @taylor2010 demonstrate the extreme scarcity of such compact, early-type systems in the local Universe. The resulting challenge from these discoveries consists of relating high-redshift massive, compact quiescent galaxies to their low-redshift counterparts – since structural evolution is required for the distant objects to resemble the current ones. Scenarios considered to explain the observed evolution in size include the effects of major and minor dissipationless mergers [@naab2009; @hopkins2009; @hopkins2010] as well as energetic feedback from an accreting black hole [@fan2008].
Many important caveats have been raised about the observations of apparent compactness among quiescent galaxies. These include $(a)$ the possibility of not accounting for extended low-surface brightness emission which would tend to increase the inferred radius; $(b)$ using in some cases rest-frame UV rather than rest-frame optical imaging; $(c)$ radially-dependent stellar $M/L$ ratios that increase outwards, causing an underestimate of the mass at larger radii; $(d)$ a reliance on photometric rather than spectroscopic redshifts; and $(e)$ uncertainties in stellar population models used to estimate stellar masses [@vandokkum2008b; @vandokkum2010]. Most of these concerns have been addressed, using samples with spectroscopic redshifts and deeper surface-brightness limits. Additionally, crude dynamical information has been obtained from stacked and individual spectra of compact, quiescent galaxies, based on an estimate of velocity dispersion from stellar absorption lines [@vandokkum2009_nature; @cenarro2009; @cappellari2009]. The estimated velocity dispersions appear to confirm the large masses inferred from stellar population modeling. Furthermore, results at $z\sim 1$ based on a comparison of radii and [*dynamical*]{} mass estimates for 50 galaxies suggest a factor of $2$ evolution in size at fixed dynamical mass, consistent with the evolutionary trend observed to $z\sim 2$ based on stellar masses [@vanderwel2008].
Employing a complementary technique, @vandokkum2010 stack the ground-based near-IR surface-brightness profiles for rare, massive galaxies of constant number density in five redshift bins between $z=2.0$ and $z=0.1$, such that the higher-redshift objects can plausibly represent the progenitors of the objects at lower redshift. The observed evolution in these stacked surface brightness profiles shows that the mass within an inner region of $r=5$ kpc remains roughly constant, while the mass at larger radii builds up smoothly as a function of decreasing redshift. This result suggests the importance of minor merger events in the growth of massive, elliptical galaxies [@naab2009; @hopkins2009], and that the structure of elliptical galaxies is not self-similar as a function of redshift. Based on deep near-IR spectroscopy with upcoming instruments on the ground and in space, statistical samples of dynamical mass estimates for passive galaxies at $z\geq 2$ will be crucial for even more robustly distinguishing among evolutionary scenarios.
Dynamical Properties {#subsec:structure-dynamics}
--------------------
Dynamical studies of high-redshift galaxies have mainly been limited to star-forming objects, using rest-frame optical emission lines from ionized gas or millimeter-wave emission from CO tracing molecular gas. While the sample of existing rest-frame UV spectra for high-redshift galaxies is much larger than either of these other types of dataset [@steidel2003; @steidel2004], the emission and absorption features typically detected in the rest-frame UV range are broadened by non-virial motions from outflowing gas and radiative transfer effects, and unfortunately do not permit useful dynamical measurements [@pettini2001; @shapley2003; @steidel2010]. The first dynamical probes of high-redshift star-forming galaxies consisted of long-slit near-IR spectroscopy of $\sim 100$ UV-selected galaxies [@pettini2001; @erb2003; @erb2004; @erb2006b] and CO maps of very small samples of SMGs [@genzel2003; @neri2003]. Indeed, CO observations can be used not only for estimating molecular gas masses but also for probing the dynamical properties of a galaxy. These early observations revealed a typical H$\alpha$ or \[OIII\] emission-line velocity dispersion of $\sigma\sim 100\mbox{ km s}^{-1}$ for UV-selected galaxies. Measured linewidths were $\sim 2-3$ times larger in the SMG CO maps, which also showed evidence for complex, multi-component emission-line morphology. A particularly striking result from the long-slit H$\alpha$ studies of @erb2006b consists of the measurement of spatially-resolved, tilted emission-lines, indicative of velocity shear, and, perhaps, rotation – i.e., preliminary evidence for disks at high redshift. Below we summarize some of the latest results from both near-IR IFU and CO dynamical studies of high-redshift galaxies, highlighting many open questions in this rapidly evolving field.
### INTEGRAL-FIELD UNIT OBSERVATIONS OF TURBULENT STAR-FORMING GALAXIES {#subsubsec:structure-dynamics-ifu}
With the advent of both seeing-limited and AO-assisted near-IR IFU spectrographs on Keck, the VLT and Gemini telescopes several years ago, the study of high-redshift galaxy dynamics has advanced considerably. Some striking trends have emerged from a total sample of $\sim 100$ objects with IFU kinematic observations, which are no longer subject to the spatial-sampling limitations of long-slit spectroscopy taken at a fixed position angle. We focus here on the VLT/SINFONI survey of @forsterschreiber2009 and the Keck/OSIRIS survey of @law2009, which represent two of the largest campaigns at $z\geq 2$ using this new observational capability. The “SINS H$\alpha$ Survey“ presented in @forsterschreiber2009 includes 62 star-forming galaxies at $1.3 \leq z \leq 2.6$ selected in the rest-frame UV, optical, near-IR, and submillimeter. The sample is representative of massive, $M_{star}>10^{10} M_{\odot}$, star-forming galaxies in this redshift range, with a median stellar mass and star-formation rate of $M_{star}=3\times 10^{10} M_{\odot}$ and $SFR=70 \; M_{\odot}\mbox{yr}^{-1}$, respectively. The sample of @law2009 includes 12 UV-selected star-forming galaxies at $2.0\leq z \leq 2.5$ with a median stellar mass and star-formation rate of $M_{star}=10^{10} M_{\odot}$ and $SFR=40\; M_{\odot}\mbox{yr}^{-1}$, respectively. In addition to having lower stellar masses on average, the objects in the [@law2009] sample are also characterized by a smaller median H$\alpha$ half-light radius than the SINS H$\alpha$ sample (1.3 versus 3.1 kpc). All OSIRIS observations in @law2009 are AO-assisted, with an effective PSF of $\sim 0.15$”, corresponding to $\sim 1.2$ kpc. The majority ($\sim 85$%) of the SINFONI observations discussed by @forsterschreiber2009 are seeing-limited, with a typical resolution of $\sim 0.6$", corresponding to $\sim 4.9$ kpc; the remainder were observed using either laser- or natural-guide-star AO, with resolutions ranging from $0.17"-0.41"$, corresponding to $\sim 1.6-3.4$ kpc.
These IFU surveys have revealed a wealth of diversity among the dynamical properties of distant star-forming galaxies. The key observables are maps of velocity, velocity dispersion, and line-intensity, which can then be used as inputs to detailed dynamical models, or analyzed in a more empirical sense. Figure \[fig:structure-dynamics-disks-forsterschreiber2009\] demonstrates the range of velocity fields observed in the SINS H$\alpha$ survey. Roughly 1/3 of these velocity maps show clear gradients and are classified as “rotation-dominated," with corresponding maps of velocity dispersion showing central maxima; roughly another 1/3 qualify as “mergers" as evidenced by kinemetric modeling [@shapiro2009] or distinct multiple components; finally, 1/3 of the sample are “dispersion-dominated", with no evidence for rotation or shear, but characterized by a significant velocity dispersion. Half of the @law2009 sample belong to this last category, with no evidence for rotation.
Another basic result consists of the high degree of turbulence observed in the ISM of high-redshift galaxies, along with a small ratio of rotational to random motions. The spatially-resolved velocity dispersion maps indicate local velocity dispersion values ranging from $\sigma_{local}=30-90 \mbox{ km s}^{-1}$ in the SINS H$\alpha$ sample and $\sigma_{local}=60-100 \mbox{ km s}^{-1}$ in the sample of @law2009. Even in rotation-dominated systems, the velocity dispersion is significant. The median value of $v_{rot}/\sigma$ is $4.5$, which is considerably lower than the values of $\sim 10-20$ observed in local spiral galaxies [@forsterschreiber2009]. Of course, in the dispersion-dominated systems, $v_{rot}/\sigma$ is even smaller (i.e., $\leq 1$). The prevalence of rotation and rotation-dominated systems appears to be correlated with stellar mass, with a higher fraction of rotating systems at higher stellar masses. At the same time, galaxies with small stellar masses and large gas fractions tend to have negligible rotation [@law2009; @forsterschreiber2009].
The star-forming galaxies with IFU observations also appear to follow scaling relations between their basic dynamical and stellar properties. Both @bouche2007b and @forsterschreiber2009 present a strong correlation between galaxy circular velocity, $v_{rot}$, and size for galaxies selected in the rest-frame UV, optical, and near-IR. This correlation is indistinguishable from the one describing local disk galaxies (despite the significant differences in $v_{rot}/\sigma$), and both UV-selected and $BzK$ galaxies follow it. On the other hand, SMGs occupy a distinct region of $v_{rot}-$size parameter space, with significantly larger velocities and smaller sizes. This difference potentially reflects the lower angular momenta and higher matter densities present in SMGs relative to more quiescently star-forming UV-selected and $BzK$ systems [@bouche2007b]. Limiting their analysis to 18 rotation-dominated systems from the SINS H$\alpha$ sample, @cresci2009 discover a “Tully-Fisher" correlation between stellar mass and $v_{rot}$, which has the same slope as the correlation among local disk galaxies, but offset by $\sim 0.4$ dex towards lower stellar mass at fixed $v_{rot}$. Furthermore, a comparison of dynamical and stellar masses indicates a strong correlation, but suggests that galaxies must contain significant gas fractions to explain the differences between the two mass estimates. In particular, galaxies with young stellar ages appear to contain larger gas fractions, consistent with the previous, long-slit results of @erb2006b.
The spatially-resolved H$\alpha$ maps indicate that clumpy morphologies are a feature of not only continuum surface brightness [@elmegreen2005], but also the emission from ionized gas [@forsterschreiber2009; @genzel2008]. Given that star-forming clumps are present even in galaxies with ordered rotation fields and no evidence of major merging, and that the clumps appear to follow the general velocity field of the larger systems in which they are embedded, it is unlikely that clumps constitute distinct, accreted systems. An alternative explanation for observed clump properties is based on predicting the size and mass-scale on which gas should fragment in a turbulent, high-surface-density disk. Specifically, the high turbulent velocities observed (as described above) lead to an expected Jeans length for fragmentation of $\sim 2.5$ kpc [@genzel2008]. Folding in the typical observed disk surface densities ($\sim 10^2 M_{\odot}\mbox{ pc}^{-2}$), @genzel2008 estimate clump masses of $\sim 10^9 M_{\odot}$, similar to what is observed. Numerical simulations of turbulent gas-rich disks also appear to produce $\sim$kpc-scale clumps [@bournaud2009]. Theoretical arguments (both analytical and numerical) suggest that these clumps may migrate inwards and coalesce to form a bulge on $\sim0.5-1.0$ Gyr timescales [@dekel2009; @elmegreen2008]. On the other hand, feedback from star formation may disrupt the clumps before they reach the center, and they may simply contribute to the overall growth of the disk. The role of clumpy structures in the growth of massive galaxies is an extremely active area of research; new observational results will help to elucidate the origin and fate of the clumps.
Local velocity dispersions of $\sim 50-100 \mbox{ km s}^{-1}$ appear to be a generic property of high-redshift star-forming galaxies at $z\sim 2$. Determining the cause of this large apparent turbulence is an important goal for models of galaxy formation. Proposed scenarios include the conversion of the gravitational potential energy from infalling accreting matter into random kinetic energy or collisions between large clumps as they migrate inwards [see, e.g., @genzel2010b for a review of these possibilities]. On the other hand, @lehnert2009 argue that mechanical energy feedback associated with star formation – in the form of stellar winds and supernovae explosions – is the driving force behind the large observed interstellar turbulence. Various authors have compared the surface-density of star-formation $\Sigma_{SFR}$ with the local velocity dispersion (presumably an indication of the connection between star-formation feedback and interstellar turbulence) and have arrived at different conclusions about what the weak observed correlation (if any) signifies [@lehnert2009; @forsterschreiber2009; @genzel2010b]. The cause of large turbulent velocities in high-redshift star-forming galaxies clearly remains an open question.
### THE NATURE OF SMGS {#subsubsec:structure-dynamics-smgs}
Dynamical information can also be potentially used to investigate the origin of the extreme luminosities among SMGs. Both near-IR IFU maps and CO observations of SMGs have been used in this endeavor. The observed CO profiles are broad (FWHM typically several hundred $\mbox{km s}^{-1}$) and often double-peaked [@greve2005]. A broad, double-peaked profile can be indicative of either a massive, rotating disk or, alternatively, a merger event. @swinbank2006 present near-IR IFU observations of 8 SMGs at $1.3 \leq z\leq 2.6$, probing rest-frame optical line emission. At least five of these systems show evidence for two or more distinct dynamical components, suggestive of merging. Based on subarcsecond-resolution CO maps for 12 SMGs at $1.2\leq z \leq 3.4$, @engel2010 find evidence in 5 systems for two distinct spatial components, with mass ratios (when possible to determine) closer than $1:3$ – the standard threshold for being considered a major merger. In the remaining systems, morphologies are either disturbed or compact, and @engel2010 argue that these represent later-stage, coalesced merger events. A significant fraction of SMGs clearly show dynamical evidence for major merging. Whether an alternative scenario of smooth gas infall and minor mergers [e.g., @dave2010] can explain the luminosities of the remaining sources will require more robust observations of the space densities, and stellar and dynamical masses of SMGs. The extreme matter densities inferred for SMGs [@bouche2007b], their high apparent star-formation efficiencies [@daddi2010; @genzel2010a], and the deviation of these systems from global scaling relations between star-formation rate and stellar mass [@daddi2007], all suggest that they constitute very unusual events. It remains an open challenge to relate SMGs to other ULIRGs at the same epochs [e.g., the $24\mu$m-selected objects of @yan2007], as well plausible descendants among the lower-redshift massive-galaxy population.
CONCLUDING REMARKS {#sec:future}
==================
Our path has traveled through many different wavelength ranges and observational techniques along the way to characterizing the stars, dust, gas, heavy elements, structure and dynamics of high-redshift galaxies at $2\leq z \leq 4$. By design, this review has focused primarily on the translation of observed quantities to physical ones, instead of making systematic comparisons with particular theories (although at times, when appropriate, a connection was made). At the same time, the rapid development of observations and their physical interpretation over the previous decade has yielded an important body of data for input into models of galaxy formation (i.e., the diversity of star-formation histories and structures among the highest-mass systems; the connection between stellar mass and star-formation rate; the nature of dust extinction as a function of luminosity; the estimate of star-formation efficiency for LIRGs and ULIRGs; the mass-metallicity relation as a function of redshift; the compactness of massive, quiescent systems; the high degree of turbulence in the ISM of star-forming galaxies; and so on).
There are some definite limitations in the nature of this review. Indeed, the description of physical properties was often still couched in terms of how the results applied to a specific galaxy sample, assembled using a specific selection technique from among the ones described in Section \[sec:technique\]. In order for measurements of galaxy properties to have true discriminating power among galaxy formation models, it is crucial to use samples that are complete with respect to a well-defined property, such as rest-frame optical or near-IR luminosity, stellar mass, dynamical mass, or star-formation rate. The majority of the samples described in this review slice the underlying high-redshift galaxy population in particular ways that add an extra layer of complexity if a comparison with a simulated galaxy population is desired. We advocate the design of future surveys with the type of physical completeness described above in mind. The NEWFIRM Medium-Band Survey [@vandokkum2009_pasp] represents an important step in this direction, but an even deeper survey would be desirable, with both rest-UV and optical spectroscopic follow-up.
Furthermore, we point out that many of the results reported here were (by necessity) based on small samples – these include mid-IR spectra of the most luminous sources, measurements of molecular gas content and dynamics, individual emission and absorption-line metallicity measurements at $z\geq 2$, high-resolution rest-frame optical morphological measurements, and AO-assisted emission-line maps. While these results highlight truly compelling questions, they also await confirmation from samples an order of magnitude larger for a robust comparison with models.
Also, by necessity, some of the most intriguing results regarding gas, dust, and structural properties are limited to the luminous extreme of the galaxy population. With the steep luminosity functions described in Section \[subsec:empirical-LF\], galaxies with $L\leq 10^{12} L_{\odot}$ comprise the bulk ($\sim 75$%) of the luminosity and star-formation rate density of the Universe at $z\sim 2$. Even galaxies with $L\leq 10^{11} L_{\odot}$ may contribute $\sim 30-35$% of the bolometric luminosity density at this redshift [@reddy2008]. We must find ways of extending our physical studies towards fainter luminosities – either by using gravitationally-lensed objects, or the next generation of instruments and telescopes. These faint objects are important, and perhaps more analogous to the galaxies playing a crucial role in the reionization process at $z\geq 6$.
We close by highlighting two important observational challenges, and looking towards the future. First, the direct measurement of gas inflow (accretion) and outflow (star-formation and AGN feedback) received only passing reference during the course of this review. The ubiquity of galaxy-scale outflows among $z\geq 2$ UV-selected galaxies has long been known on the basis of rest-frame UV and optical spectra [@pettini2001; @adelberger2003; @shapley2003; @steidel2004; @adelberger2005; @steidel2010], yet obtaining robust constraints on the physical properties associated with these outflows (e.g., mass outflow rates) remains a challenge [but see, e.g., @steidel2010]. At the same time, there is no obvious connection between these observations and the extremely popular theoretical models of cold gas accretion [e.g., @keres2005; @keres2009; @dekel2009_nature]. An open challenge is to observe a “smoking gun" of gas accretion in high-redshift galaxies. Second, the study of galaxy environments at $z\geq 2$ is a field in its infancy. While overdensities of LAEs have been identified near luminous radio galaxies [@venemans2007], and a small number of apparent protoclusters have been discovered serendipitously during the course of high-redshift galaxy spectroscopic surveys [@steidel1998; @steidel2005], the study of the dependence of galaxy properties on environment at $z\geq 2$ remains largely untapped. Environmental studies will require extensive spectroscopy of mass-complete samples in both overdense and “field" environments, and will have potentially fundamental implications for understanding the origin of local galaxy environmental trends. Finally, we look forward to the truly exciting insights that will be made possible by upcoming facilities, including sensitive multi-object near-IR spectrographs on $8-10$-meter class telescopes, ALMA, JWST, and the extremely large ground-based telescopes of the next decade.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Katherine Kornei and Gwen Rudie for careful readings of the manuscript, and Naveen Reddy for many stimulating discussions and helpful comments. I would also like to thank the numerous authors who so generously shared their figures for this review, and note that figures previously appearing in [*The Astrophysical Journal*]{} have been reproduced by permission of the AAS. Very importantly, my thanks go to Sandy Faber, for giving me the opportunity to review the field of high-redshift galaxy properties, and for her patience and constructive feedback as editor. Finally, I would like to thank my collaborators for useful dialogs over the past decade and more, which have shaped my understanding of this rapidly developing field. AES is a Packard Fellow; financial support from the Packard Foundation is gratefully acknowledged.
Acronyms/Definitions {#acronymsdefinitions .unnumbered}
====================
1. LBG: “Lyman Break Galaxy." Star-forming $z\geq 3$ galaxy selected on the basis of its rest-frame UV colors, which are indicative of a Lyman Break – i.e. significant absorption at wavelengths below 912 Å.
2. UV selection: More general use of rest-frame UV photometry to select star-forming galaxies not only at $z\geq 3$ using the Lyman Break, but also at $1.4 \leq z \leq 2.5$ using different rest-frame UV color criteria.
3. DRG: “Distant Red Galaxy." Galaxy selected on the basis of a red observed-frame $J-K$ color, which corresponds to a Balmer or 4000 Å break, or significant dust extinction, for objects at $z\sim 2-4$. The majority of DRGs are actively forming stars, but typically with higher $M/L$ and stellar masses than UV-selected galaxies and sBzK galaxies.
4. sBzK: “Star-forming BzK Galaxy." Galaxy selected on the basis of its $B-z$ and $z-K$ colors to lie at $1.4\leq z \leq 2.5$ and be actively star-forming. The sBzK criteria are tuned to find galaxies with Balmer breaks, ongoing star formation, and a wide range of dust extinction properties.
5. pBzK: “Passive BzK Galaxy." Galaxy selected on the basis of its $B-z$ and $z-K$ colors to lie at $1.4\leq z \leq 2.5$ and be devoid of star formation. The pBzK criteria are tuned to find galaxies with prominent 4000 Å breaks and a lack of current star formation.
6. SMG: “Submillimeter Galaxy." Galaxy selected on the basis of its powerful luminosity in the submillimeter range of the spectrum. SCUBA 850 $\mu$m selection has commonly been used to find such systems. The prolific IR luminosity is due to reprocessed emission from dust (powered either by stars, an active nucleus, or both).
7. LAE: “Ly$\alpha$ Emitter." Galaxy selected on the basis of having strong emission in the Ly$\alpha$ feature at rest-frame 1216 Å. As a result of the selection based on line strength, LAEs tend to be significantly fainter in the rest-frame UV continuum than other star-forming sources at the same redshift.
8. LIRG: “Luminous Infrared Galaxy." Galaxy characterized by $10^{11} L_{\odot} < L_{IR} \leq 10^{12} L_{\odot}$. Such systems, along with ULIRGs, are much more common at $z\sim 2$ than in the local Universe.
9. ULIRG: “Ultra-luminous Infrared Galaxy." Galaxy characterized by $L_{IR} > 10^{12} L_{\odot}$. Such systems, along with LIRGs, are much more common at $z\sim 2$ than in the local Universe.
|
---
address: 'School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad, Pakistan.'
author:
- 'Bushra Majeed$^1$, Mubasher Jamil$^2$ and Azad A. Siddiqui$^3$'
title: 'Holographic dark energy with time varying ${n^2}$ parameter in non-flat universe'
---
.We consider a holographic dark energy model, with a varying parameter, $n$, which evolves slowly with time. We obtain the differential equation describing evolution of the dark energy density parameter, $\Omega_d$, for the flat and non-flat FRW universes. The equation of state parameter in this generalized version of holographic dark energy depends on $n$.\
\
PACS [98.80.-k, 95.36.+x]{}
Introduction
============
The current observations [@c1]-[@c4] strongly support that our universe is in the accelerated expansion phase. In the standard cosmological structure, the existence of a component with antigravity effect is necessary for explaining this accelerated expansion. Using different approaches, many models have been suggested to explain the dark energy. One of them is the use of the Einstein cosmological constant, but it suffers from two problems (the “fine-tuning” and the “coincidence”) [@c7]. The dynamical dark energy models with a variable equation of state have been investigated, scalar field models are one of these [@essence]. Another interesting approach for exploring the behavior of dark energy is through the use of the principles of quantum gravitation [@Witten:2000zk]. Proposal of holographic dark energy (HDE) is an example of such models [@Cohen:1998zx]-[@Li:2004rb]. The expression for energy density of HDE is: $$\label{de}
\rho_d=\frac{3n^2}{8\pi G L^2},$$ where $L$ is the infrared (IR) cut-off, $n$ is a constant, and $G$ is the Newton gravitational constant. The holographic dark energy scenario is one of the most widely studied model and there are many versions of this model in the literature [@j12]-[@obs3].
If we take $L$ as a Hubble horizon, it results in a wrong equation of state (EoS) and the accelerated expansion of the universe can not be obtained. However, this issue can be resolved in the case of interacting HDE. This model does not work with the particle horizon as well, but when $L$ is chosen as future event horizon, the results favor the accelerated expansion.
There is no strong evidence for $n$ to be taken a constant, so the HDE parameter, $n^2$, has a vital role in characterizing the properties of the model. For example in future, the model could be like a phantom or quintessence dark energy model depends on whether the value of $n^2$ is larger or smaller than $1$ respectively. Model of HDE with variable $n^2$ parameter for a flat universe has been studied in [@CPL]. There are many models of HDE in literature with a variable gravitational constant, $G$. Following the approach adopted by Jamil et al. [@G], here we investigate the HDE model with a variable $n^2$ parameter, allowing the consequent modifications to the EoS parameter, $w$, of the dark energy. Plan of our work is as follows: In section \[model\] we build the HDE model with a time varying $n^2(z)$ and extract the evolution equations for the dark energy density parameter. In section \[discussion\] we calculate the corrections to the parameter, $w$, at low redshifts. In section \[Numerical Results\] we demonstrate the numerical results and in section \[Conclusions\] we summarize our results.
Holographic Dark Energy (HDE) with variable ${n^2}$ parameter {#model}
=============================================================
Flat FRW geometry
-----------------
To construct the HDE model with a variable $n^2$, we consider the flat Robertson-Walker geometry given as $$\label{met}
ds^{2}=-dt^{2}+a^{2}(t)\left(dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right),$$ where $a(t)$ is the scale factor and $t$ is the comoving time. The first Friedmann equation is given by $$\label{FR1}
H^2=\frac{8\pi G}{3}(\rho_m+\rho_d),$$ where $H$ is the Hubble parameter, $\rho_m=\frac{\rho_{m0}}{a^3}$ denotes the matter density, and $\rho_d$ is the dark energy density. The present value of a quantity is represented by index $``0''$. Using the density parameter, $ \Omega_d\equiv\frac{8\pi
G}{3H^2}\rho_d$, with Eq. (\[de\]) we get $$\label{OmegaL2}
\ol=\frac{n^2(z)}{H^2L^2}.$$ As discussed before, for flat universe, defining $L$ as the future event horizon is the best option [@Hsu:2004ri; @Li:2004rb; @Guberina; @Guberina0], i.e. taking $L\equiv R_ h(a)$ as $$R_ h(a)=a\int_t^\infty{dt\over
a(t)}=a\int_a^\infty{da\over Ha^2}~.\label{eh}$$ To denote the time derivative we use a dot, and prime is used for the differentiation with respect to the independent variable $\ln
a$, i.e. we acquire $\dot{J}=J'H$, for every quantity $J$. Differentiating Eq. (\[OmegaL2\]), using Eq. (\[eh\]), and $\dot{R}_h=HR_h-1$, we attain $$\label{OmegaLdif}
\frac{\Omega'_d}{\Omega^2_d}=\frac{2}{\Omega_d}\Big[\frac{n^\prime}{n}-1-\frac{\dot{H}}{H^2}+\frac{\sqrt{\Omega_d}}{n}\Big].$$ We can see that the varying behavior of $n^2$ has become apparent. To eliminate $\dot{H}$ we differentiate the Friedman equation (\[FR1\]) and use the expression $$\label{rhoodot}
\rho'_d=\rho_d\left(\frac{n'}{n}-2+\frac{2\sqrt{\Omega_d}}{n}\right),$$ to get $$\label{Hdot}
2\frac{\dot{H}}{H^2}=-3+\Omega_d\left(1+\frac{2\sqrt{\Omega_d}}{n}\right)+\frac{2n'}{n}\Omega_d,$$ where $n$ is considered to be time dependent. Finally, using Eq. (\[Hdot\]) in Eq. (\[OmegaLdif\]) we have $$\label{OmegaLdif3}
\Omega_d'=\Omega_d(1-\Omega_d)\Big[1+\frac{2\sqrt{\Omega_d}}{n}\Big]+2\Omega_d(1-\Omega_d)\frac{n'}{n}.$$ Note that the second term is the correction term appearing because of variable $n$, here $n'/n$ is a dimensionless number.
Non-flat FRW geometry
---------------------
Now we extend the work presented in the previous subsection for the FRW universe with metric $$\label{metr}
ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^2}{1-kr^2}+r^2(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right) ,$$ where $(t,r,\theta,\varphi)$ are comoving coordinates and $k$ represents the spacial curvature with $k = -1, 0, 1$ respectively corresponding to the open, flat and the closed universes. In this geometry the first Friedmann equation becomes $$\label{FR1nf}
H^2+\frac{k}{a^2}=\frac{8\pi G}{3}(\rho_m+\rho_d).$$ In non-flat metric, the cosmological length, $L$, for the HDE model takes the following form [@nonflat] $$\label{Lnonflat}
L\equiv\frac{a(t)}{\sqrt{|k|}}\,\mbox{sinn}\left(\frac{\sqrt{|k|}R_h}{a(t)}\right),$$ with $$\frac{1}{\sqrt{|k|}}\mbox{sinn}(\sqrt{|k|}y)=\cases{
\sin y & \, \, $k=+1$,\\
y & \, \, $k=0$,\\
\sinh y & \, \, $k=-1$.\\
}$$ It is easy to find that $$\label{Ldot}
\dot{L}=H L-\mbox{cosn}\left(\frac{\sqrt{|k|}R_h}{a}\right),$$ where $$\mbox{cosn}(\sqrt{|k|}y)=\cases{
\cos y & \, \, $k=+1$,\\
1 & \, \, $k=0$,\\
\cosh y & \, \, $k=-1$.\\
}$$ Following the same steps as adopted in the previous subsection, differentiating Eq. (\[OmegaL2\]) and using Eqs. (\[Lnonflat\]) and (\[Ldot\]) we obtain $$\label{Hdotnf0}
\frac{\Omega^\prime_d}{\Omega^2_d}=\frac{2}{\Omega_d} \left(
-1+\frac{n'}{n}-\frac{\dot H}{H^2}+\frac{\sqrt{\Omega_d}}{n}
\,\mbox{cosn}(\sqrt{|k|}y) \right).$$ From Friedmann equation (\[FR1nf\]) we get $$\label{Hdotnf}
2\frac{\dot H}{H^2}=-3-\Omega_k+\Omega_d+2\frac{\Omega_d^{3/2}}{n}\,
\mbox{cosn}\left(\frac{\sqrt{|k|}R_h}{a}\right)
+2\Omega_d\frac{n^\prime}{n},$$ where $\Omega_k\equiv\frac{k}{(aH)^2}$ is the curvature density parameter. Using Eq. (\[Hdotnf\]) into Eq. (\[Hdotnf0\]) we have [$$\label{Omprimenf}
\Omega_d^\prime=\Omega_d\left[1+\Omega_k-\Omega_d+\frac{2\sqrt{\Omega_d}}{n}\,
\mbox{cosn}\left(\frac{\sqrt{|k|}R_h}{a}\right)(1-\Omega_d)\right]
+2\Omega_d(1-\Omega_d)\frac{n^\prime}{n}.$$]{} Here the correction made to the HDE differential equation in non-flat universe because of the variable $n$ can be observed. Clearly, when $k=0$ (and thus $\Omega_k=0$) we get Eq. (\[OmegaLdif3\]).
Equation of State Parameter ($w(z)$) {#discussion}
====================================
We find $w(z)$ for small values of redshifts $z$. Since $\rho_d\sim
a^{-3(1+w)}$, taking the derivatives at the present time $a_0=1$ (so $\Omega_d=\Omega_d^0$) we get $$\ln\rho_d =\ln \rho^0_d+{d\ln\rho_d \over d\ln a} \ln a +\frac{1}{2}
{d^2\ln\rho_d \over d(\ln a)^2}(\ln a)^2+\dots....$$ Then, $w(\ln a)$ up to second order is given by $$w(\ln a)=-1-{1\over 3}\left[{d\ln\rho_d \over d\ln a} +\frac{1}{2}
{d^2\ln\rho_d \over d(\ln a)^2}\ln a\right].$$ Using $\ln a=-\ln(1+z)\simeq -z$, which is applicable for small redshifts, one can easily compute $w(z)$, as $$w(z)=-1-{1\over 3}\left({d\ln\rho_d \over d\ln a}\right)+
\frac{1}{6} \left[{d^2\ln\rho_d \over d(\ln a)^2}\right]\,z\equiv
w_0+w_1z.$$
Flat FRW geometry
-----------------
Using the expression for $\Omega_d'$, given in Eq. (\[OmegaLdif3\]) and aforementioned procedure leads to $$\begin{aligned}
&&w_0=-{1\over 3}-{2\over 3n}\sqrt{\Omega^0_d}
-\frac{2\Delta_n}{3}\label{w0fl},\\
\label{w1fl}
&&w_1={1\over
6n}\sqrt{\Omega^0_d}(1-\Omega^0_d)\left(1+{2\over
n}\sqrt{\Omega^0_d}\right)+2\frac{(1-\Omega^0_d)\sqrt{\Omega^0_d}}{6n}\Delta_n.
\ \ \ \ \ \ \ \ \ \ \ \\end{aligned}$$ These $w_0$ and $w_1$ are for the HDE with varying $n^2$, in a flat universe. It is clear that when $n$-variation $\Delta_n=0$, we obtain the results which are consistent with those of [@Li:2004rb].
The best fit value for $n$ obtained from supernovae type Ia observational data, within $1-\sigma$ error range [@obs3a], is $n=0.21$ and from the analysis of $X-$ray gas mass fraction of galaxy clusters it comes out as $n=0.61$ [@obs2]. While combing the results from different sources we have: the data obtained by the observations of type Ia supernovae, Cosmic Microwave Background (CMB) radiation and large scale structure gives $n=0.91$ [@obs1], whereas combining the observations of Baryon Acoustic Oscillation, $X-$ray gas and type Ia supernovae lead to $n=0.73$ [@Wu:2007fs].
Non-flat FRW geometry
---------------------
Using the expression of $\Omega_d'$ for non-flat case, given in Eq. (\[Omprimenf\]), we have $$\begin{aligned}
\label{w0nonfl}
&&w_0=-{1\over 3}-{2\over
3n}\sqrt{\Omega^0_d}\,\mbox{cosn}\frac{\sqrt{|k|}R_{h0}}{a_0}
-\frac{2 \Delta_n}{3}\\
\label{w1nonfl}
&&w_1=\frac{\sqrt{\Omega^0_d}}{6n}\left[1+\Omega_k^0-\Omega^0_d+\frac{2\sqrt{\Omega^0_d}}{n}\,
\mbox{cosn}\left(\frac{\sqrt{|k|}R_{h0}}{a_0}\right)(1-\Omega^0_d)
\right]\mbox{cosn}\left(\frac{\sqrt{|k|}R_{h0}}{a_0}\right)\nonumber\\&&
+\frac{\Omega^0_d}{3n^2}\,q\left(\frac{\sqrt{|k|}R_{h0}}{a_0}\right)
+2\frac{\sqrt{\Omega^0_d}}{6n}(1-\Omega^0_d)\mbox{cosn}\left(\frac{\sqrt{|k|}R_{h0}}{a_0}\right)\Delta_n,\end{aligned}$$ where $$q(\sqrt{|k|}y)=\cases{
\sin^2 y & \, \, $k=+1$,\\
0 & \, \, $k=0$,\\
-\sinh^2 y & \, \, $k=-1$.\\
}$$ Clearly, for $k=0$, Eqs. (\[w0nonfl\]) and (\[w1nonfl\]) reduce to Eqs. (\[w0fl\]) and (\[w1fl\]) respectively.
The expressions given by Eqs. (\[w0nonfl\]) and (\[w1nonfl\]) involve present values of the parameters $\Omega_d^0$, $\Omega_k^0$, $a_0$, and $R_{h0}$. From Eq. (\[OmegaL2\]) we obtain $L_0=n/(H_0\sqrt{\Omega_\Lambda^0})$. Also from Eq. (\[Lnonflat\]), we get $R_{h0}/a_0=\frac{1}{\sqrt{|k|}}\mbox{sinn}^{-1}(\sqrt{|k|}L_0/a_0)$. Hence, $$\begin{aligned}
\frac{R_{h0}}{a_0}&=&\frac{1}{\sqrt{|k|}}\mbox{sinn}^{-1}\left(\frac{n\sqrt{|k|}}{a_0H_0\sqrt{\Omega_\Lambda^0}}\right),
\nonumber\\
&=&
\frac{1}{\sqrt{|k|}}\mbox{sinn}^{-1}\left(\frac{n\sqrt{|\Omega_k^0|}}{\sqrt{\Omega_\Lambda^0}}\right).\label{eqn27}\end{aligned}$$ Substituting Eq. (\[eqn27\]) in Eqs. (\[w0nonfl\]) and (\[w1nonfl\]), we finally obtain $$\begin{aligned}
\label{w0nonflb}
w_0&=&-{1\over 3}-{2\over 3n}\sqrt{\Omega^0_d-n^2\Omega^0_k}
-2\frac{\Delta_n}{3},\\ \label{w1nonflb}
w_1&=&\frac{\Omega_k^0}{3}+\frac{1}{6n}\sqrt{\Omega^0_d-n^2\Omega_k^0}
\left[1+\Omega_k^0-\Omega^0_d +
\frac{2}{n}(1-\Omega^0_d)\sqrt{\Omega^0_d-n^2\Omega_k^0}\right]\nonumber\\&&
+\frac{1}{6n}\sqrt{\Omega^0_d-n^2\Omega^0_k}
\left(1-\Omega^0_d\right)\Delta_n.\end{aligned}$$ These $w_0$ and $w_1$ are for non-flat universe, depending only on $\Omega_d^0$, $\Omega_k^0$, $n$, and $\Delta_n$.
Numerical Results {#Numerical Results}
=================
By solving the equations for EoS parameters we can give a numerical description of the evolution of GHDE model. For the choice of model parameter, $n(z)$, we use the parameterization known as Chavallier-Polarski-Linder (CPL) [@CPL] given as $$n(z)=n_0+n_1\frac{z}{1+z}.\label{CPL}$$ When $z\rightarrow\infty$ (in the early universe), we see that $n\rightarrow n_0+n_1$ and as $z\rightarrow0$ (at the present time), $n\rightarrow n_0$. Therefore, the value of $n$ varies from $n_0+n_1$ to $n_0$ with passage of time. Also the positive energy condition of GHDE model requires that $$n_0>0,~~~~n_0+n_1>0\label{CPL1}.$$ Since $w(\ln a)\equiv w_0+w_1z$, using Eq. (\[CPL\]) and solving $w_0$ and $w_1$ given in Eqs. (\[w0fl\]) and (\[w1fl\]) we plot the evolutionary behavior of EoS parameter of GHDE model versus redshift variable (Fig. (\[flat\])). We choose the values for $n_0$ and $n_1$ such that they satisfy Eq. (\[CPL1\]).
![Evolution of EoS parameter for flat universe versus redshift parameter $z$. Model parameters $n_0$ and $n_1$ are $0.75$ and $0.1$ respectively. The values for $\Omega_d$ and $\Omega_m$ are taken as $0.7$ and $0.3$ respectively.[]{data-label="flat"}](flat.eps){width="8cm"}
Similarly we can draw the EoS parameter for non-flat case by solving Eqs. (\[w0nonflb\]) and (\[w1nonflb\]) with Eq. (\[CPL\]). The behavior of curve is shown in Fig. (\[nonflatc00\]). We see that the curve for EoS parameter enters from $w_d>-1$ to $w_d<-1$. So it crosses the phantom line, $w_d=-1$, without taking into account the interaction of dark energy and dark matter.
![Evolution of EoS parameter for non-flat universe versus redshift parameter $z$. Model parameters $n_0$ and $n_1$ are $0.5$ and $0.1$. The values for $\Omega_d$, $\Omega_m$ and $\Omega_k$ are taken as $0.7$, $0.285$, and $0.015$ respectively.[]{data-label="nonflatc00"}](nonflatc00.eps){width="8cm"}
Conclusions {#Conclusions}
===========
In this paper, we have considered the generalized holographic dark energy model for spatially flat and non-flat universes with a future event horizon. Since the holographic parameter, $n$, is generally not constant and can be assumed as a function of cosmic redshift, we have provided the complete expressions for cosmological parameters, introducing the correction terms due to varying $n$. For GHDE with future event horizon, we have obtained the EoS parameter at small redshifts by performing the Taylor series expansion up to the first order i.e. $w(z) \equiv w_0+w_1z$. We have obtained $w_0$ and $w_1$ in terms of $\Omega_d^0$, $\Omega_k^0$ and $n$-variation $\Delta n$. The expressions for evolution of the dark energy density have an additional term depending on $\Delta n$.
For further exploration we have also considered the CPL parametrization in which $ n(z)=n_0+n_1\frac{z}{1+z}$ . An investigation has been made for the effect of correction term on EoS parameters for flat and non-flat geometries. Obtaining numerical values of the cosmological parameters, we have plotted them against $z(t)$. Graphical behavior of EoS parameters for our model shows that with a varying $n$ a shifting from quintessence regime ($w_d>-1$) to phantom regime ($w_d<-1$) is possible, without taking into account the interaction of dark matter and dark energy. Hence the GHDE model has a correspondence with the observational data.
References {#references .unnumbered}
==========
[10]{} Riess A G [*et al.*]{} 1998 *Astron. J.* **116** 1009; Perlmutter S [*et al.*]{} 1999 *Astrophys. J.* **517** 565.
Bennett C L [*et al.*]{} 2003 *Astrophys. J. Suppl.* **148** 1. Tegmark M [*et al.*]{} 2004 *Phys. Rev. D* **69** 103501. Allen S W *et al.* 2004 *Mon. Not. Roy. Astron. Soc.* **353** 457.
Sahni V and Starobinsky A 2000 *Int. J. Mod. Phy. D* **9** 373; Peebles P J and Ratra B 2003 *Rev. Mod. Phys.* **75** 559.
Jamil M, Momeni D, Serikbayev N S and Myrzakulov R 2012 *Astrophys. Space Sci.* **339** 37.
Witten E hep-ph/0002297.
Cohen A G, Kaplan D B and Nelson A E 1999 *Phys. Rev. Lett.* **82** 4971.
Hooft G ’t gr-qc/9310026; Susskind L 1995 *J. Math. Phys.* **36** 6377.
Hsu S D H 2004 *Phys. Lett. B* **594** 13.
Li M 2004 *Phys. Lett. B* **603** 1.
Setare W R and Jamil M 2010 *J. Cosmol. Astropart. Phys.* **02** 010; Karami K, Khaledian M S and Jamil M 2011 *Phys. Scr.* **83** 025901; Sheykhi A and Jamil M 2011 *Gen. Rel. Grav.* **43** 2661; Pasqua A, Jamil M, Myrzakulov R and Majeed B 2012 *Phys. Scr.* **86** 045004.
Huang Q G and Gong Y G 2004 *JCAP* 0408 006.
Chang Z, Zhang X and Wu F Q 2006 *Phys. Lett. B* **633** 14.
Zhang X and Wu F Q 2005 *Phys. Rev. D* **72** 043524.
Ma Y Z and Gong Y 2009 *Eur. Phys. J. C* **60** 303.
Enqvist K, Hannestad S and Sloth M S 2005 *JCAP* 0502 004.
Jamil M, Saridakis E N and Setare M R 2009 *Phys.Lett.B* **679** 172.
Guberina B, Horvat R and Nikolic H 2005 *Phys. Rev. D* **72** 125011.
Setare M R 2007 *JCAP* 701 23.
Ratra B and Peebles P J E 1988 *Phys. Rev. D* **37** 3406.
Huang Q G and Li M 2004 *JCAP* 408 13.
Malekjani M, Zarei R and Honari-Jafarpour M 2013 *Astrophys. Space Sci.* **343** 799.
|
---
abstract: 'We consider the effect of spin-orbit coupling on the low energy excitation spectrum of an Andreev billiard (a quantum dot weakly coupled to a superconductor), using a dynamical numerical model (the spin Andreev map). Three effects of spin-orbit coupling are obtained in our simulations: In zero magnetic field: (1) the narrowing of the distribution of the excitation gap; (2) the appearance of oscillations in the average density of states. In strong magnetic field: (3) the appearance of a peak in the average density of states at zero energy. All three effects have been predicted by random-matrix theory.'
author:
- 'B. B[é]{}ri'
- 'J. H. Bardarson'
- 'C. W. J. Beenakker'
date: 'December, 2006'
title: 'Effect of spin-orbit coupling on the excitation spectrum of Andreev billiards'
---
Introduction
============
A quantum dot in a two-dimensional electron gas has a mean level spacing which is independent of energy and depends only on geometrical factors (area) and material properties (effective mass). The nature of the electron dynamics (chaotic versus integrable) and the presence or absence of symmetries, such as time-reversal and spin-rotation symmetry, have no effect on the mean density of states. The situation changes if the quantum dot is coupled to a superconductor (see Fig. \[fig:setup\]). The presence of the superconductor strongly affects the excitation spectrum of such an Andreev billiard. The density of states at the Fermi level is suppressed in a way which is sensitive to the nature of the dynamics and existing symmetries.[@Bee05] While the effect of broken time-reversal symmetry on the density of states has been studied extensively,[@Alt96; @Fra96; @Mel97; @Goo05] the effect of broken spin-rotation symmetry due to spin-orbit coupling has only been partially investigated.[@Alt97; @Cht03; @Dim06]
![Sketch of an Andreev Billiard: A quantum dot (N) connected to a superconductor (S) by a point contact. Spin-orbit coupling is present in the quantum dot.[]{data-label="fig:setup"}](setup.eps){width="0.6\columnwidth"}
In this paper we study in computer simulations three effects of the spin-orbit coupling. All of these effects have been predicted by random-matrix theory (RMT),[@Tra96; @Alt97; @Vav01] but have so far not been confirmed in a dynamical model. The first two of these effects,[@Tra96; @Vav01] present in the absence of a magnetic field, are the reduction of the sample-to-sample fluctuations of the excitation gap and the appearance of oscillations as a function of energy in the average density of states. The third effect appears in a magnetic field strong enough to close the excitation gap. While in the absence of spin-orbit coupling the average density of states vanishes at the Fermi level, in the presence of spin-orbit coupling it peaks at the Fermi level at twice the value in the normal state.[@Alt97]
Predictions of random-matrix theory {#sec:RMT}
===================================
We begin by briefly summarizing the RMT of the Andreev billiard.[@Bee05] In perturbation theory the density of states of non-degenerate levels (in zero or weak magnetic field) has a square root dependence on energy near the gap,[@Mel96; @Mel97] $$\label{eq:rhoMF}
\rho_\text{pert}(E) = \frac{1}{\pi}\sqrt{\frac{E-E_\text{gap}}{\Delta_\text{gap}^3}},\quad E\rightarrow E_\text{gap}.$$ The parameters $E_\text{gap}$ and $\Delta_\text{gap}$ are given by $$\label{eq:Egap}
E_\text{gap}=c E_T,\quad \Delta_\text{gap}
= \left(\frac{s\delta}{2}\right)^{2/3}\left(\frac{dE_T}{4\pi^2}\right)^{1/3}\!\!\!.$$ Here $\delta = 2\pi\hbar^2/mA$ is the mean level spacing in the isolated quantum dot (area $A$, effective mass $m$), $N =
{\rm Int}[k_FW/\pi]$ is the number of modes in the ballistic point contact (width $W$, Fermi wave vector $k_F$) connecting it to the superconductor, and $E_T =
N\delta/4\pi$ is the Thouless energy. These parameters refer to two-fold degenerate levels and modes, corresponding to $s=2$. If both spin-rotation and time-reversal symmetries are broken, the two-fold degeneracy is lifted and one should take $s = 1$. The numerical coefficients $c$ and $d$ are magnetic field dependent. For $B=0$ one has $$\label{eq:cdparams}
c=2\gamma^{5/2}, \quad d=(5-2\sqrt{5})\gamma^{5/2},$$ with $\gamma =(\sqrt{5}-1)/2$ the reciprocal of the golden ratio. For $B \neq 0$ they should be calculated from the RMT solution given in Ref. .
The perturbation theory has $\delta/E_T$ as a small parameter and gives the density of states with an energy resolution of order $E_T$, which is a macroscopic energy scale. Since spin-orbit effects typically appear as quantum corrections, no sign of spin-orbit coupling can be seen in such a calculation. In particular, the magnetic field dependence of the perturbative density of states is the same with and without spin-orbit coupling — the only difference being that in a magnetic field, in the absence of spin-rotation symmetry, there is no level degeneracy thus $\Delta_{\text{gap}}$ is $2^{2/3}$ times smaller than in the spin-rotation symmetric case.
To capture the spectral properties on the mesoscopic energy scale of order $\Delta_\text{gap}$, one needs to go beyond perturbation theory. According to the universality hypothesis of Vavilov [*et al. *]{}[@Vav01] the probability distribution of the lowest level $E_1$ in properly scaled units is universal and identical to the distribution of the smallest eigenvalue of random Gaussian matrices from the three symmetry classes of RMT. The appropriate scaling is in terms of the dimensionless variable $x_1=(E_1-E_\text{gap})/\Delta_\text{gap}$ and the universal distributions are given by[@Tra96; @Ede05] $$\label{eq:probdist}
P_\beta(x_1)=-\frac{d}{dx_1}F_\beta(x_1),$$ where
$$\begin{aligned}
F_1(x)&=\sqrt{F_2(x)}\exp \left(-\frac{1}{2}\int_{-\infty}^x q(x')\,dx'\right),\\
F_2(x)&=\exp \left( -\int_{-\infty}^x (x-x')q(x')^2\,dx' \right), \\
F_4\left(\frac{x}{2^{2/3}}\right) &= \sqrt{F_2(x)}\cosh
\left(\frac{1}{2}\int_{-\infty}^x q(x')\,dx'\right).\end{aligned}$$
The function $q(x)$ is the solution of the differential equation $$q''(x)=-xq(x)+2q^3(x).$$ The boundary condition is $q(x)\rightarrow {\rm Ai}(-x)$ as , with ${\rm Ai}(x)$ being the Airy function. The three distributions are plotted in Fig. \[fig:P(x)\] (top panel). The symmetry index $\beta$ takes values $\beta=1$ for time-reversal and spin-rotation invariant systems, $\beta=4$ when time-reversal symmetry is present but spin-rotation symmetry is broken, and $\beta=2$ for systems with broken time-reversal symmetry.
Near the gap, the average density of states in terms of the variable $x=(E-E_\text{gap})/\Delta_\text{gap}$ is given by[@FNH; @Tra05]
$$\begin{aligned}
\rho_1(x)=\rho_2(x)&+ \tfrac{1}{2}{\rm Ai}(-x)\left[1-\int_{-x}^{\infty} {\rm Ai}(y)dy \right],\\
\rho_2(x)&=x{\rm Ai}^2(-x)+[{\rm Ai}'(-x)]^2,\\
2^{1/3}\rho_4\left(\frac{x}{2^{2/3}}\right)&=\rho_2(x)-\frac{1}{2}{\rm Ai}(-x)\int_{-x}^{\infty} {\rm Ai}(y)dy.\end{aligned}$$
The distribution $P_2$ and the density of states $\rho_2$ are applicable in an intermediate magnetic field range, which exists because the flux needed to close the gap is much larger than the flux needed to break the time-reversal symmetry. For intermediate fluxes ( with $\tau_\text{erg}=\sqrt{A}/v_{\rm F}$ the ergodic time and $v_{\rm F}$ the Fermi velocity), there will still be a gap, but its fluctuations are governed by the $\beta = 2$ symmetry class. In this case the presence or absence of spin-orbit coupling only affects the parameter $\Delta_\text{gap}$ (which is reduced by a factor $2^{2/3}$ in the absence of spin-orbit coupling, because the level degeneracy parameter goes from $s=2$ to $s=1$); the gap distribution $P_2$ and the density of states $\rho_2$ in rescaled variables do not depend on the presence or absence of spin-rotation symmetry.
If the flux is made much larger ($\Phi \gg (h/e) \sqrt{N\tau_\text{erg}\delta/\hbar}$), such that the gap closes, spin-orbit coupling starts to play a role again. The reason is that an energy level $E$ and its mirror level at $-E$ can repel each other, and this repulsion depends on the presence or absence of spin-orbit coupling. When there is still a gap these levels are widely separated and this repulsion is not effective.[@Alt97]
According to Altland and Zirnbauer,[@Alt97] the RMT of an Andreev billiard in strong magnetic field is in a new symmetry class called C (D) in the absence (presence) of spin-orbit coupling. The average density of states in these symmetry classes is $$\label{eq:rhoCD}
\rho_{\pm}(E) = \frac{4}{s\delta}\left[1\pm\frac{\sin(8\pi E/s\delta)}{8\pi E/s\delta}\right].$$ The minus (plus) sign should be taken for symmetry class C (D). This result expresses the fact that in the absence of spin-orbit coupling, a level and its mirror level repel each other leading to a vanishing density of states at $E=0$. In the presence of spin-orbit coupling the repulsion disappears and levels pile up at the Fermi level, leading to a peak in the density of states at $E=0$.
Spin Andreev map
================
To verify these predictions of RMT in a dynamical model we combine the general construction of an Andreev map[@Jac03] with the spin kicked rotator.[@Sch89; @Bar05] The starting point of our discussion is the spin generalized Bogoliubov-De Gennes Hamiltonian [@DeGennes] $$\label{eq:BdG}
\mathcal{H}_{\text{BdG}} =
\begin{pmatrix}
H-E_{\rm F} & \Delta \\
\Delta^* & E_{\rm F}-\mathcal{T}H\mathcal{T}^{-1}
\end{pmatrix}.$$ Here $H$ is the single particle Hamiltonian, $E_{\rm F}$ is the Fermi energy, and $\Delta$ is the superconducting pair-potential. The operator $\mathcal{T}$ stands for time-reversal, and will be specified later.
With the Bogoliubov-De Gennes Hamiltonian as a guide we construct the spin Andreev map. First we note that if an electron in the normal metal evolves with time-evolution operator $F(t)$, the hole evolves with the transformed time-evolution operator $\mathcal{T}F(t)\mathcal{T}^{-1}$. Second, since we are interested in low energy phenomena, only the dynamics on long time scales is important. On time scales much larger than $\tau_{\rm erg}$, the dynamics can be described as a mapping on a two-dimensional Poincaré surface of section. This amounts to a stroboscopic description where we are only concerned with the state of the electron when it bounces off the boundary.
The quantum map we use is the computationally efficient spin kicked rotator, given in terms of a Floquet matrix,[@Sch89; @Bar05] $$\label{eq:Floquet}
F_{ll'}=e^{i\varepsilon_0}(\Pi U X U^\dagger \Pi)_{ll'}, \quad l,l' = 0,1,\ldots, M-1.$$ The integer $M$ sets the level spacing $\delta = 2\pi/M$. The $M \times M$ matrices appearing in Eq. have quaternion matrix elements, and are given by
$$\begin{aligned}
\Pi_{ll'}&=\delta_{ll'}e^{-i\pi (l+l_0)^2/M }\sigma_0, \\
U_{ll'}&=M^{-1/2}e^{-i2\pi ll'/M}\sigma_0,\\
X_{ll'}&= \delta_{ll'}e^{-i(M/4\pi)V(2\pi l/M)},\end{aligned}$$
with $$V(\theta) = K\cos(\theta+\theta_0)\,\sigma_0 + K_\text{so}(\sigma_1\sin2\theta + \sigma_3\sin\theta).$$ The quaternions are represented using the Pauli matrices $\sigma_i$ with $\sigma_0$ the $2 \times 2$ unit matrix. The matrix $X$ corresponds to the spin-orbit coupled free motion inside the dot and $\Pi$ gives scattering off the boundaries of the dot. This map is classically chaotic for kicking strength $K \gtrsim 7.5$. The parameter $K_\text{so}$ breaks spin-rotation symmetry, $\theta_0$ breaks time-reversal symmetry and $l_0$ breaks other symmetries of the map. The spin-orbit coupling time $\tau_\text{so}$ (in units of the stroboscopic period $\tau_0 \approx \tau_{\rm erg}$) is related to $K_\text{so}$ through $\tau_\text{so} = 32\pi^2/(K_\text{so}M)^2$. The parameter $\varepsilon_0$ corresponds to the Fermi energy. In the above representation of the Floquet matrix, the time-reversal operator is given by ${\cal T}=i \sigma_2 {\cal K}$ where ${\cal K}$ is the operator of complex conjugation.[@Bar05] Therefore, the hole Floquet matrix is given by $\sigma_2 F^* \sigma_2 \equiv \bar{F}$, where the overbar denotes quaternionic complex conjugation.
The spin Andreev map is constructed from the electron and hole Floquet matrices in the same way as in the absence of spin-orbit coupling,[@Jac03]
$$\begin{aligned}
\label{eq:aKR}
\mathcal{F} &= \mathcal{P}
\begin{pmatrix}
F & 0 \\ 0 & \bar{F}\\
\end{pmatrix},\\
\mathcal{P} &=
\begin{pmatrix}
1-P^TP & -iP^TP \\
-iP^TP & 1-P^TP
\end{pmatrix}.\end{aligned}$$
The projection matrix $P$ projects onto the contact with the superconductor. Its matrix elements are $P_{kl} =
\delta_{kl}\sigma_0\sum_{i=1}^N\delta_{l,n_i}$ where the set of indices $\{ n_i \}$ corresponds to the modes coupled to the superconductor. The dwell time is therefore $\tau_{\rm dwell}=M/N$. The corresponding Thouless energy is $E_T = N\delta/4\pi = (2\tau_\text{dwell})^{-1}$. As shown in Ref. , the magnetic field scale at which the gap closes is given by $\theta_c = 4\pi\sqrt{N}/(KM^{3/2})$. From the definitions in Eq. the scaling parameters in the spin Andreev map become $$\label{eq:scalingpar}
E_\text{gap} = \frac{cN}{2M}, \quad
\Delta_\text{gap} = \frac{(s^2dN)^{1/3}}{2M}.$$
![Probability distribution of the rescaled excitation gap $x=(E_{1}-E_\text{gap})/\Delta_\text{gap}$. Smooth curves (upper panel) are the predictions of random-matrix theory, for the three symmetry classes $\beta=1,2,4$. Histograms are the results of the numerical simulation of the spin Andreev map, without spin-orbit coupling (zero magnetic field, $\beta=1$) and with spin-orbit coupling (zero magnetic field, $\beta=4$; weak magnetic field, $\beta=2$). The histograms in the lower panel are plotted without any fitting, while in the upper panel $E_\text{gap}$ and $\Delta_\text{gap}$ are treated as fit parameters. []{data-label="fig:P(x)"}](datascale.eps "fig:"){width="0.95\columnwidth"} ![Probability distribution of the rescaled excitation gap $x=(E_{1}-E_\text{gap})/\Delta_\text{gap}$. Smooth curves (upper panel) are the predictions of random-matrix theory, for the three symmetry classes $\beta=1,2,4$. Histograms are the results of the numerical simulation of the spin Andreev map, without spin-orbit coupling (zero magnetic field, $\beta=1$) and with spin-orbit coupling (zero magnetic field, $\beta=4$; weak magnetic field, $\beta=2$). The histograms in the lower panel are plotted without any fitting, while in the upper panel $E_\text{gap}$ and $\Delta_\text{gap}$ are treated as fit parameters. []{data-label="fig:P(x)"}](dataraw.eps "fig:"){width="0.95\columnwidth"}
The Floquet matrix has the symmetry $$\label{eq:Symmetry}
\mathcal{F} = {\cal CT} \mathcal{F}({\cal CT})^{-1}, \quad
{\cal C} = \begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix},$$ corresponding to the ${\cal CT}$-antisymmetry of $\mathcal{H}_{\text{BdG}}$, the fundamental discrete symmetry of normal-superconducting systems.[@Alt97] The eigenphases of the Floquet matrix $\mathcal{F}$, defined as the solutions of $$\label{eq:eqseq}
\det(\mathcal{F} - e^{-i\varepsilon})=0,$$ play the role of the discrete excitation energies in the Andreev billiard. From the symmetry it follows that they come in pairs, $\pm\varepsilon$, as required.
Numerical results and comparison with RMT
=========================================
![Average density of states in rescaled energy units, $x=(E-E_\text{gap})/\Delta_\text{gap}$, in zero magnetic field (upper panel, $\beta=1,4$) and in a weak magnetic field (lower panel, $\beta=2$). The smooth curves are the RMT predictions, the histograms are the results of the simulation (using the same values of the fit parameters $E_\text{gap},\Delta_\text{gap}$ as in the upper panel of Fig. \[fig:P(x)\]).[]{data-label="fig:rhoscale(x)"}](scaledos14bw.eps "fig:"){width="0.95\columnwidth"} ![Average density of states in rescaled energy units, $x=(E-E_\text{gap})/\Delta_\text{gap}$, in zero magnetic field (upper panel, $\beta=1,4$) and in a weak magnetic field (lower panel, $\beta=2$). The smooth curves are the RMT predictions, the histograms are the results of the simulation (using the same values of the fit parameters $E_\text{gap},\Delta_\text{gap}$ as in the upper panel of Fig. \[fig:P(x)\]).[]{data-label="fig:rhoscale(x)"}](scaledos2.eps "fig:"){width="0.95\columnwidth"}
![Average density of states in a magnetic field sufficiently strong to close the excitation gap. Upper panel: without spin-orbit coupling; Lower panel: with spin-orbit coupling. The smooth curves are the RMT predictions from Eq. and the histograms are the results of the simulation (now without any fit parameters).[]{data-label="fig:rhoCD"}](dos_b2nosp.eps "fig:"){width="0.6\columnwidth"} ![Average density of states in a magnetic field sufficiently strong to close the excitation gap. Upper panel: without spin-orbit coupling; Lower panel: with spin-orbit coupling. The smooth curves are the RMT predictions from Eq. and the histograms are the results of the simulation (now without any fit parameters).[]{data-label="fig:rhoCD"}](dos_b2wsp.eps "fig:"){width="0.6\columnwidth"}
In Fig. \[fig:P(x)\] we plot the excitation gap distribution (histograms) from our numerical simulation with parameters $K=41.123$, $M=4096$, $N=205$. The smallest $\varepsilon$ solving Eq. was calculated for some 6000 different Fermi energies and positions of the contact to the superconductor. To generate the three symmetry classes we took: $\theta_{0}/\theta_{c}=0$, $\tau_{\rm
dwell}/\tau_{\rm so}=0$ ($\beta=1$); $\theta_{0}/\theta_{c}=0$, $\tau_{\rm dwell}/\tau_{\rm so}=625$ ($\beta=4$); $\theta_{0}/\theta_{c}=0.4$, $\tau_{\rm dwell}/\tau_{\rm so}=625$ ($\beta=2$). The values of the parameters $c$ and $d$ for $\theta_0/\theta_c = 0$ are given by Eq. . For $\theta_0/\theta_c = 0.4$ we calculate $c = 0.427$, $d =
0.339$ from Ref. . The data is shown without any fit parameter in the lower panel and with $E_\text{gap}$, $\Delta_\text{gap}$ as fit parameters in the upper panel. (A similar fitting procedure was used in Ref. .) The fitted values of $E_\text{gap}$ and $\Delta_\text{gap}$ do not vary much (by about 5% and 10%, respectively) from their nominal values \[given by Eq. \], but the agreement with the RMT predictions improves considerably if we allow for this variation.[@FOOTNOTE]
The characteristics of the gap distribution are clearly obtained in our simulation. In zero magnetic field, the gap distribution becomes narrower as the strength of spin-orbit coupling is increased ($\beta=1 \rightarrow 4$). The RMT prediction for the standard deviation of the scaled distributions $P_\beta(x)$ is $\sigma_1 = 1.27$ and $\sigma_4 = 0.64$. The corresponding values obtained in our numerical simulation [*without any fitting*]{} are $\sigma_1 = 1.34$ and $\sigma_4 = 0.72$. If a weak magnetic field is present ($\beta=2$), the width of the distribution is predicted to be intermediate between the cases with $\beta=1,4$. As seen, the dynamical model follows the prediction. The theoretical value of the standard deviation is $\sigma_2 = 0.90$, the numerical result (without fitting) is $\sigma_2 = 0.99$.
Using the same fit parameters as in Fig. \[fig:P(x)\] we plot the average density of states close to the gap in Fig. \[fig:rhoscale(x)\]. As seen, the numerical data follow closely the analytical predictions, the deviations becoming significant only outside the universal regime $|E-E_\text{gap}|\ll E_T$, i.e. $|x|\ll N^{2/3}$. In the absence of magnetic field the spin-orbit coupling induced oscillations are clearly obtained.
The density of states in a strong magnetic field is given in Fig. \[fig:rhoCD\]. The upper panel shows the data without spin-orbit coupling,[@Goo05] and the lower panel shows what happens if spin-rotation symmetry is broken. The numerical data follows closely the analytical prediction of RMT. In particular the enhanced density of states in the presence of spin-orbit coupling is clearly seen. The first oscillations in the density of states are also captured in the dynamical model. The frequency doubling due to the reduced degeneracy ($s=2 \rightarrow s=1$) is apparent.
Conclusion
==========
In conclusion, we have introduced a quantum map for the dynamics of a chaotic quantum dot with spin-orbit coupling connected to a superconductor. We have demonstrated three effects of spin-orbit coupling on the excitation spectrum of this Andreev billiard: The narrowing of the distribution of the excitation gap and the appearance of oscillations in the density of states in the absence of a magnetic field; and the peak in the density of states at the Fermi level in strong magnetic field. Our numerical simulations confirm the predictions of random-matrix theory. The third effect is particularly interesting from an experimental point of view. In view of the possibility to tune the strength of spin-orbit coupling in quantum dots,[@Zum02; @Mil03] one can imagine tuning the density of states at the Fermi level from zero to a value of twice the normal density of states.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
Discussions and correspondence with J. Cserti, M. C. Goorden, P. Jacquod, and J. Tworzyd[ł]{}o are gratefully acknowledged. This work was supported by the Dutch Science Foundation NWO/FOM. We acknowledge support by the European Community’s Marie Curie Research Training Network under contract MRTN-CT-2003-504574, Fundamentals of Nanoelectronics.
[22]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ****, ().
, ****, ().
, , , , ****, ().
, , , , ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, , , , ****, ().
, .
, , , ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, , , ****, ().
, ** (, ).
, , , , ****, ().
, , , , , ****, ().
, , , , , , , ****, ().
|
---
abstract: 'The correlated motion of electrons in the presence of strong orbital fluctuations and correlations is investigated with respect to magnetic couplings and excitations in an orbitally degenerate ferromagnet within the framework of a non-perturbative Goldstone-mode-preserving approach based on a systematic inverse-degeneracy expansion scheme. Introduction of the orbital degree of freedom results in a class of diagrams representing spin-orbital coupling which become particularly important near the orbital ordering instability. Low-energy staggered orbital fluctuation modes, particularly with momentum near $(\pi/2,\pi/2,0)$ (corresponding to period $4a$ orbital correlations as in CE phase of manganites involving staggered arrangement of nominally $\rm Mn^{3+} / \rm Mn^{4+}$ ions, and staggered ordering of occupied $3x^2 - r^2 / 3y^2 - r^2$ orbitals on alternating $\rm Mn^{3+}$ sites), are shown to generically yield strong intrinsically non-Heisenberg $(1-\cos q)^2$ magnon self energy correction, resulting in no spin stiffness reduction, but strongly suppressed zone-boundary magnon energies in the $\Gamma$-X direction. The zone-boundary magnon softening is found to be strongly enhanced with increasing hole doping and for narrow-band materials, which provides insight into the origin of zone-boundary anomalies observed in ferromagnetic manganites.'
author:
- 'Dheeraj Kumar Singh, Bhaskar Kamble, and Avinash Singh'
title: |
Orbital fluctuations, spin-orbital coupling, and anomalous\
magnon softening in an orbitally degenerate ferromagnet
---
Introduction
============
The orbital degree of freedom of the electron has attracted considerable attention in recent years due to the rich variety of electronic, magnetic, and transport properties exhibited by orbitally degenerate systems such as the ferromagnetic manganites, which have highlighted the interplay between spin and orbital degrees of freedom in these correlated electron systems.[@tokura_2003; @khaliullin_2005] Orbital fluctuations, correlations, and orderings have been observed in Raman spectroscopic studies[@saitoh_2001] of orbiton modes in $\rm LaMnO_3$, polarization-contrast-microscopy studies[@ogasawara_2001] of $\rm La_{0.5}Sr_{1.5}MnO_4$, magnetic susceptibility and inelastic neutron scattering studies[@khalifah_2002] of $\rm La_4 Ru_2 O_{10}$, and resonant inelastic soft X-ray scattering studies[@ulrich_2008] of $\rm YTiO_3$ and $\rm LaTiO_3$. A new detection method for orbital structures and ordering based on spectroscopic imaging scanning tunneling microscopy is of strong current interest[@lee_2009] in orbitally active metallic systems such as strontium ruthenates and iron pnictide superconductors.
The role of orbital fluctuations on magnetic couplings and excitations is of strong current interest in view of the several zone-boundary anomalies observed in spin-wave excitation measurements in the metallic ferromagnetic phase of colossal magnetoresistive (CMR) manganites.[@hwang_98; @dai_2000; @tapan_2002; @ye_2006; @ye_2007; @zhang_2007; @moussa_2007] The presence of short-range dynamical orbital fluctuations has been suggested in neutron scattering studies of ferromagnetic metallic manganite $\rm La_{1-x}(Ca_{1-y}Sr_y)_{x}MnO_3$.[@moussa_2007] These observations are of the crucial importance for a quantitative understanding of the carrier-induced spin-spin interactions, magnon excitations, and magnon damping, and have highlighted possible limitations of existing theoretical approaches.
For example, the observed magnon dispersion in the $\Gamma$-X direction shows significant softening near the zone boundary, indicating non-Heisenberg behaviour usually modeled by including a fourth neighbour interaction term $J_4$, and highlighting the limitation of the double-exchange model. Similarly, the prediction of magnon-phonon coupling as the origin of magnon damping[@dai_2000] and of disorder as the origin of zone-boundary anomalous softening[@furukawa_2005] have been questioned in recent experiments.[@ye_2006; @ye_2007; @zhang_2007; @moussa_2007] Furthermore, the dramatic difference in the sensitivity of long-wavelength and zone-boundary magnon modes on the density of mobile charge carriers has emerged as one of the most puzzling feature. Observed for a finite range of carrier concentrations, while the spin stiffness remains almost constant, the anomalous softening and broadening of the zone-boundary modes show substantial enhancement with increasing hole concentration.[@ye_2006; @ye_2007]
Theoretically, the role of orbital-lattice fluctuations and correlations on magnetic couplings and excitations has been investigated within an orbitally degenerate double exchange model with an inter-orbital interaction $V$ and the Jahn-Teller coupling.[@khaliullin_2000] Based on a strong coupling expansion, this approach is restricted to the strong coupling limit $V \gg t$. The final calculations for the magnon self energy, carried out in terms of a phenomenological parameter, show significant zone-boundary magnon softening only for ferromagnetic orbital correlations, and extremely close to the orbital ordering instability.
If orbital fluctuations have signature effects on magnetic excitations in a ferromagnet with orbital degree of freedom, they can be probed indirectly through neutron scattering studies. A detailed investigation of the orbital fluctuation magnon self energy is therefore of strong current interest, especially with respect to dependence on inter-orbital interaction strength, band filling, and different orbital fluctuation modes. In this paper we will present a theory for spin-orbital coupling and magnon self energy, and examine how the correlated motion of electrons in the presence of strong orbital correlations near the orbital ordering instability influences magnetic couplings and excitations.
We will employ a diagrammatic approach which allows interpolation in the full range of interaction strength from weak to strong coupling. In this approach correlation effects in the form of self energy and vertex corrections are incorporated systematically so that the Goldstone mode is preserved order by order. Based on a systematic inverse-degeneracy expansion scheme,[@singh] the approach has been used recently to study spin-charge coupling effects, which give rise to strong magnon energy softening, damping, and non-Heisenberg behaviour in metallic ferromagnets.[@spch3; @qfklm]
The present work will also extend the recent investigation into role of orbital degeneracy and Hund’s coupling on magnetic couplings and excitations in a band ferromagnet.[@hunds] Orbital degeneracy and Hund’s coupling were shown to enhance ferromagnetism by strongly suppressing correlation-induced quantum corrections to spin stiffness and magnon energies. An effective quantum parameter was obtained for determining the magnitude of quantum corrections, and the theory was applied to calculate the spin stiffness for a realistic multi-orbital system such as iron. We will show here that the spin stiffness remains essentially unaffected by the interaction $V$ due to the non-Heisenberg $(1-\cos q)^2$ behaviour of the magnon self energy resulting from orbital fluctuations and correlations.
In manganites, an important role is also played by the cooperative Jahn-Teller distortion of O$^{2-}$ ions which lifts the two-fold degeneracy of $\rm e_g$ electronic levels of Mn due to a combination of orbital geometry and electrostatic repulsion, leading to staggered orbital correlations. This is qualitatively similar to the local orbital moment and staggered orbital correlations introduced by the inter-orbital density interaction $V n_{i\alpha} n_{i\beta}$ which relatively pushes up the $\beta$ orbital energy if the $\alpha$ orbital density $\langle n_{i\alpha} \rangle$ is more than average, thus self consistently lifting the orbital degeneracy. Therefore, orbital correlations and fluctuations due to dynamical Jahn-Teller distortion can be qualitatively treated in terms of an effective inter-orbital interaction.
The structure of the paper is as follows. Starting with a degenerate two-orbital Hubbard model including an inter-orbital interaction $V$, the first order quantum correction diagrams for the irreducible particle-hole propagator are obtained in Section II. As basic ingredients in the diagrammatics, spin and orbital fluctuations are briefly discussed in section III. The magnon self energy contributions due to orbital fluctuations and spin-orbital coupling are obtained in sections IV and V. The interplay between magnetic and charge contributions to the spin-orbital interaction vertex is discussed in section VI, and orbital fluctuations near $(\pi/2,\pi/2,0)$ are shown to yield strong zone-boundary magnon softening. Extension to finite Hund’s coupling $J$ and the ferromagnetic Kondo lattice model are discussed in sections VII and VIII, and conclusions are presented in Section IX.
Two-orbital Hubbard model
=========================
We will consider a two-orbital Hubbard model $$\begin{aligned}
H &=& -t \sum_{\langle ij\rangle \sigma} (a^\dagger _{i\alpha\sigma} a_{j\alpha\sigma} + a^\dagger _{i\beta\sigma} a_{j\beta\sigma} ) + U \sum_i (n_{i\alpha\uparrow} n_{i\alpha\downarrow} + n_{i\beta\uparrow} n_{i\beta\downarrow} ) \nonumber \\ &+& \sum_{i\sigma\sigma'} (V-\delta_{\sigma\sigma'}J) n_{i\alpha\sigma} n_{i\beta\sigma'}
- J \sum_{i,\sigma \ne \sigma'} a^\dagger _{i\alpha\sigma} a_{i\alpha\sigma'} a^\dagger _{i\beta\sigma'} a_{i\beta\sigma}\end{aligned}$$ on a simple cubic lattice with two orbitals (labeled by $\alpha,\beta$) per site and intra-orbital nearest-neighbor hopping $t$. The Coulomb interaction matrix elements included here are the intra-orbital interaction $U$, the inter-orbital density interaction $V$, and the inter-orbital exchange interaction (Hund’s coupling) $J$. The last term represents the transverse part ($S_{i\alpha}^- S_{i\beta}^+ + S_{i\alpha}^+ S_{i\beta}^-$) of the Hund’s coupling, and the density interaction term yields the longitudinal part $S_{i\alpha}^z S_{i\beta}^z$, so that altogether the Hund’s coupling term has the form $-J {\bf S}_{i\alpha} . {\bf S}_{i\beta}$. The Hamiltonian therefore possesses continuous spin rotation symmetry, and hence the Goldstone mode must exist in the spontaneously broken symmetry state.
Hund’s coupling has been shown to strongly enhance ferromagnetism in an orbitally degenerate system by strongly suppressing the correlation-induced quantum corrections.[@hunds] An effective quantum parameter $[U^2 + ({\cal N}-1)J^2]/[U+({\cal N}-1)J]^2$ was obtained for determining the magnitude of quantum corrections for an ${\cal N}$-orbital system, in analogy with $1/S$ for quantum spin systems. The rapid decrease of this quantum factor from 1 to $1/{\cal N}$ as $J/U$ increases from 0 to 1 results in strong suppression of quantum corrections and hence significant stabilization of ferromagnetism by Hund’s coupling, particularly for large ${\cal N}$.
In order to highlight the role of inter-orbital Coulomb interaction $V$ and orbital fluctuations on magnetic couplings and excitations in this paper, we will first set $J=0$. The case of finite Hund’s coupling will be treated later in section VII.
In a band ferromagnet, all information regarding carrier-induced spin interactions $J_{ij} = U^2 \phi_{ij}$ and excitations are contained in the irreducible particle-hole propagator $\phi({\bf q},\omega)$, which then yields the exact transverse spin fluctuation (magnon) propagator:[@hunds] $$\chi^{-+}({\bf q},\omega) = \frac {\phi({\bf q},\omega)}{1- U\phi({\bf q},\omega)} \; .$$ Our approach is to incorporate correlation effects in $\phi({\bf q},\omega)$ in the form of self-energy and vertex corrections using a systematic expansion $\phi = \phi^{(0)} + \phi^{(1)} + \phi^{(2)} + ... $ which preserves the Goldstone mode order by order. Rooted within an inverse-degeneracy expansion scheme, this systematic approach is non-perturbative with respect to the interaction terms and therefore yields a controlled approximation which remains valid in the strong coupling limit. Contributions to the first order quantum correction $\phi^{(1)}$ due to the Hubbard interaction $U$ and Hund’s coupling $J$ have been discussed earlier.[@hunds]
The additional first order diagrams for $\phi$ arising from the inter-orbital interaction $V$ are shown in Fig. 1. The diagrams shown here are for a saturated ferromagnet in which minority ($\downarrow$) spin particle-hole fluctuations are absent. Here Fig. 1(a) and (b) represent quantum corrections due to electronic self energy renormalization by orbital fluctuations, Fig. 1(c) represents the corresponding vertex correction, and Fig. 1(d) represents vertex corrections involving coupling between transverse spin and orbital fluctuations. The vertex correction diagrams as in Fig. 1(d) (nine such diagrams) can be represented in terms of an effective spin-orbital interaction vertex $\Gamma_{\rm sp-orb}$ as shown in Fig. 2(a). The spin-orbital interaction vertex has three contributions involving three-fermion vertices, as shown in Fig. 2(b). The missing fourth diagram vanishes because of the assumption of complete polarization.
As the Goldstone-mode condition $U\phi=1$ at $q=0$ is already exhausted by the zeroth-order (classical) term $\phi^{(0)}$, the sum of the higher order (quantum) terms $\phi^{(1)}+\phi^{(2)}+ \; ... $ must exactly vanish at $q=0$. For this cancellation to hold for arbitrary $U$, $J$, and $V$, each higher order term $\phi^{(n)}$ in the expansion must individually vanish, implying that the Goldstone mode is preserved order by order. We will demonstrate this exact cancellation explicitly for the new contributions due to $V$ in the first-order quantum correction $\phi^{(1)}$.
Systematics in our two-orbital model can be formally introduced, in analogy with the inverse-degeneracy ($1/{\cal N}$) expansion for the Hubbard model, by: i) treating the two physical orbitals $\alpha,\beta$ as pseudo spins, ii) introducing ${\cal N}$ pseudo orbitals ($\mu$) for each pseudo spin, and iii) generalizing the inter-orbital density interaction to $(V/{\cal N})\sum_{i\mu\nu} n_{i\alpha\mu} n_{i\beta\nu}$. Now, each interaction line $V$ yields a factor $1/{\cal N}$ and each bubble yields a factor ${\cal N}$ from the summation over pseudo orbitals, resulting in an overall $1/{\cal N}$ factor for the bubble series, and an overall $(1/{\cal N})^n$ factor in the $n^{\rm th}$-order quantum correction $\phi^{(n)}$.
Spin and Orbital fluctuations
=============================
The diagrammatic expansion above involves spin and orbital fluctuation propagators, the characteristic energy and momentum distribution of which are important in view of the spin-orbital coupling investigated in this work. The ladder series in Fig. 1(d) yields the effective intra-orbital transverse spin interaction: $$U_{\rm eff}^{\alpha\alpha}({\bf Q},\Omega) = \frac{U}{1-U\chi_0 ({\bf Q},\Omega)} \approx U^2 \frac{\chi_0 ({\bf Q},\Omega)}{1-U\chi_0 ({\bf Q},\Omega)} \equiv U^2 \chi_{\rm sp}({\bf Q},\Omega)
\approx U^2 \frac{m_{\bf Q}}{\Omega + \omega_{\bf Q}^0 - i \eta}$$ in terms of the RPA-level magnon propagator $\chi_{\rm sp}$, having an advanced pole in the saturated ferromagnetic state ($n_\uparrow = m,\; n_\downarrow = 0$). Here $\chi_0 ({\bf Q},\Omega)$ is the bare antiparallel-spin particle-hole propagator, $m_{\bf Q} \approx m$ and $\omega_{\bf Q}^0$ are the magnon-mode amplitude and energy, and the small weight of gapped Stoner excitations has been neglected for simplicity.
Similarly, the bubble series in Fig. 1 involving odd number of bubbles yields, in terms of the RPA-level orbital fluctuation propagator, the effective intra-orbital density interaction: $$V_{\rm eff} ^{\alpha\alpha} ({\bf Q},\Omega) = -\frac{V^2 \chi_{0\uparrow} ({\bf Q},\Omega)}
{1 - V^2 \chi_{0\uparrow}^2 ({\bf Q},\Omega)} \equiv -V^2 \chi_{\rm orb} ({\bf Q},\Omega) \; \;
\approx \frac{-V^2}{2} \frac{\chi_{0\uparrow} ({\bf Q},\Omega)}{1 - V \chi_{0\uparrow} ({\bf Q},\Omega)}$$ near the orbital ordering instability where $V\chi_{0\uparrow} \sim 1$. Here $\chi_{0\uparrow}({\bf Q},\Omega)$ is the bare spin-$\uparrow$ particle-hole propagator. The orbital fluctuation propagator is symmetric $\chi_{\rm orb} (-{\bf Q},-\Omega) = \chi_{\rm orb} ({\bf Q},\Omega)$, with a spectral representation: $$\chi_{\rm orb} ({\bf Q},\Omega) = - \int_0^\infty \frac{d\Omega'}{\pi} \frac{{\rm Im} [\chi_{\rm orb} ({\bf Q},\Omega')]}{\Omega - \Omega' + i \eta}$$ for its retarded part, with a continuum distribution over the orbital fluctuation energy $\Omega'$.
Exactly at quarter filling $(m=n=0.5$ per orbital), the orbital fluctuation propagator diverges at ${\bf Q} = (\pi,\pi,\pi)$ in the absence of any NNN hopping $t'$ terms which destroy Fermi surface nesting, indicating instability towards staggered orbital ordering. With increasing hole doping, the spectral function peak shifts below $(\pi,\pi,\pi)$.
Fig. 3(a) shows the momentum integrated orbital fluctuation (para-orbiton) spectral function $\sum_{\bf Q} (1/\pi) {\rm Im} \chi_{\rm orb} ({\bf Q},\Omega)$ with increasing interaction strength $V$. Here, and in the following, we have set the hopping term $t=1=W/12$ as the unit of energy, where $W$ is the bandwidth. In analogy with the well-known para-magnon response with approaching magnetic instability, the para-orbiton energy scale is strongly suppressed from order bandwidth in the weak-coupling regime to relatively very low energies near the orbital ordering instability.
Fig. 3(b) shows the momentum dependence of the low-energy $(\Omega<2)$ integrated part of the orbital fluctuation spectral function near the R point $(\pi,\pi,\pi)$, which shows that low-energy orbital fluctuations are concentrated near the wavevector $(\pi,\pi,\pi)$ corresponding to staggered orbital fluctuations. With increasing doping away from quarter filling, the peak shifts below $(\pi,\pi,\pi)$, indicating incommensurate fluctuations.
Orbital fluctuation magnon self energy
======================================
We will first consider diagrams for $\phi$ in Fig. 1 (a,b,c) involving electronic self energy corrections due to orbital fluctuations and the corresponding vertex correction. Absent in the single-orbital case, these diagrams are characteristic of orbital degeneracy, inter-orbital interaction, and orbital fluctuations, and strongly influence magnetic couplings and excitations through electronic band renormalization, particularly in vicinity of the orbital ordering instability. The vertex correction diagrams (d) involving both spin and orbital fluctuations will be discussed in the next section.
Quantum corrections to the irreducible particle-hole propagator $\phi$ in Eq. (2) yield the magnon self energy: $$\Sigma ({\bf q},\omega) = mU^2[\phi^{(1)}({\bf q},\omega) + \phi^{(2)}({\bf q},\omega) + ...] \; ,$$ in terms of which the magnon propagator $\chi^{-+}({\bf q},\omega) = m/[\omega + \omega_{\bf q} ^{(0)} - \Sigma ({\bf q},\omega)]$. The first-order magnon self energy corresponding to diagrams in Fig. 1(a,b,c) involving only orbital fluctuations is then obtained by summing over the bosonic degrees of freedom of the orbital fluctuations: $$\Sigma_{\rm orb} ^{(1)} ({\bf q},\omega) = mU^2 [\phi^{(a)} + \phi^{(b)} + \phi^{(c)}]
= m V^2 \sum_{\bf Q} \int_{-\infty} ^{\infty} \frac{d\Omega}{2\pi i}
\; \chi_{\rm orb} ({\bf Q-q},\Omega-\omega) \; \Gamma_4 ({\bf Q,q},\Omega,\omega)$$ where $\Gamma_4$ is the four-fermion vertex obtained by integrating out the fermionic degrees of freedom in the diagrams for $\phi$ shown in Fig. 1(a,b,c). As will be further discussed in the following three subsections, this orbital fluctuation magnon self energy physically represents contributions due to (i) coupling between Stoner excitations and orbital fluctuations, and (ii) self-energy corrections involving band-energy renormalization and spectral-weight transfer.
Using the spectral representation for the orbital fluctuation propagator (retarded part, since $\Gamma_4$ has only advanced poles with respect to $\Omega$), we obtain: $$\begin{aligned}
\Sigma_{\rm orb} ^{(1)} ({\bf q},\omega) &=& - m V^2 \sum_{\bf Q} \int_{-\infty}^{\infty} \frac{d\Omega}{2\pi i}
\int_0 ^\infty \frac{d\Omega'}{\pi} \frac{{\rm Im} \chi_{\rm orb} ({\bf Q-q},\Omega')}{\Omega-\omega-\Omega' + i\eta}
\Gamma_4 ({\bf Q,q},\Omega,\omega) \nonumber \\
&=& m V^2 \sum_{\bf Q'} \int_0^\infty \frac{d\Omega'}{\pi}
{\rm Im} \chi_{\rm orb} ({\bf Q'},\Omega') \; \Gamma_4 ({\bf Q',q},\Omega',\omega) \nonumber \\
&=& m V^2 \langle \Gamma_4 ({\bf Q',q},\Omega',\omega) \rangle_{{\bf Q'},\Omega'} \; ,\end{aligned}$$ where the average four-fermion vertex $\langle \Gamma_4 \rangle_{{\bf Q'},\Omega'}$ denotes averaging over orbital fluctuation modes ${\bf Q'}\equiv {\bf Q-q}$. Evaluation of the four-fermion vertex $\Gamma_4$, resolved into different contributions corresponding to distinct physical mechanisms, is discussed below. Term by term, the four-fermion vertex $\Gamma_4$ vanishes identically for $q=0$, in accord with the Goldstone mode.
Stoner-orbital coupling
-----------------------
In the single-orbital case, the magnon self energy due to spin-charge coupling included a Stoner contribution representing coupling of charge excitations with the gapped part of spin excitations.[@spch3] In analogy, diagrams Fig. 1(a,b,c) yield contributions which represent a Stoner-orbital coupling: $$\Gamma_4 ^{\rm Stoner} = U^2 \sum_{\bf k}
\left ( \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \Omega - i \eta} \right )
\left ( \frac{1}{\epsilon_{\bf k-q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \omega - i \eta}
- \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k-Q+q}^{\uparrow} + \omega - i \eta} \right )^2$$ in which the first term represents the Stoner excitation mode $({\bf Q},\Omega)$ and the quadratic term is the Stoner-orbital interaction vertex, which involves only magnetic energy denominators. Here $\epsilon_{\bf k}^\sigma = \epsilon_{\bf k} - \sigma \Delta$ are the ferromagnetic state band energies for the two spins in terms of the free-particle energy $\epsilon_{\bf k} = -2t(\cos k_x + \cos k_y + \cos k_z)$ for the sc lattice and the exchange splitting $2\Delta = mU$. The band energy superscripts $+ (-)$ refer to particle (hole) states. There is no restriction on the energy $\epsilon_{\bf k-Q+q}^{\uparrow}$ in Eq. (9) as both particle and hole states contribute. Of the four terms in this quadratic interaction vertex, the two square terms arise from diagrams Fig. 1(a) and (b), while the cross terms are from diagram Fig. 1(c); the characteristic quadratic structure therefore stems from orbital fluctuations renormalizing electrons of both spins, and is clearly absent in the single-orbital case involving spin-down renormalization only. This intrinsic quadratic structure resulting from orbital degeneracy yields a characteristic non-Heisenberg $(1-\cos q)^2$ magnon self energy, resulting in no spin stiffness reduction but strong zone-boundary magnon energy reduction.
Electronic band renormalization
-------------------------------
Due to exchange of inter-orbital fluctuations in the diagram Fig. 1(a) involving intermediate spin-$\uparrow$ states, the spin-$\uparrow$ hole (particle) energies are pulled down (pushed up), increasing the particle-hole energy gap, and thereby suppressing the particle-hole propagator $\phi$. Including the corresponding contributions from the vertex correction diagram Fig. 1(c), we obtain the electronic band renormalization contribution: $$\Gamma_4 ^{\rm band} = - U^2 \sum_{\bf k}
\left ( \frac{1}{\epsilon_{\bf k-Q+q}^{\uparrow +} - \epsilon_{\bf k}^{\uparrow -} + \Omega - \omega} \right )
\left ( \frac{1}{\epsilon_{\bf k-q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \omega}
- \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k-Q+q}^{\uparrow +} + \omega} \right )^2$$ involving one charge and two magnetic energy denominators. The finite infinitesimal term $i\eta$ as in Eq. (9) has been dropped for compactness.
Spectral weight transfer
------------------------
The electronic self energy correction in diagram Fig. 1(a) also results in spectral-weight transfer and redistribution between occupied and unoccupied spin-$\uparrow$ states. However, there is no net change in occupancy and magnetization. The corresponding spectral weight transfer contribution: $$\Gamma_4 ^{\rm spectral} = - U^2 \sum_{\bf k}
\left ( \frac{1}{\epsilon_{\bf k-Q+q}^{\uparrow +} - \epsilon_{\bf k}^{\uparrow -} + \Omega - \omega} \right )^2
\left ( \frac{1}{\epsilon_{\bf k-q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \omega}
- \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k-Q+q}^{\uparrow +} + \omega} \right )$$ involves one magnetic and two charge energy denominators. The first (negative) contribution corresponds to loss of spin-$\uparrow$ hole spectral weight due to transfer to particle states, and the second (positive) contribution corresponds to the reverse process.
### Cancellation of most singular contributions {#cancellation-of-most-singular-contributions .unnumbered}
Singular contributions in Eqs. (9-11) for $\Gamma_4$ exactly cancel out. For example, the most singular contribution in Eq. (9) involving two powers of the vanishing energy denominator ($\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k-Q+q}^{\uparrow +} + \omega$) exactly cancels with the corresponding contribution from Eq. (10). Similarly, the next most singular contributions in Eqs. (9-11) also exactly cancel out.
### Average over orbital fluctuation modes {#average-over-orbital-fluctuation-modes .unnumbered}
The average $\langle \Gamma_4 \rangle_{{\bf Q'},\Omega'}$ of the four-fermion vertex over orbital fluctuation modes directly yields the magnon self energy from Eq. (8). Since orbital fluctuations peak below $(\pi,\pi,\pi)$ for finite doping \[Fig. 3\], the vertex $\langle \Gamma_4 \rangle$ was estimated by averaging over a selected ${\bf Q'}$ region ($-0.8 < \cos Q'_\mu < -0.6$) near $(\pi,\pi,\pi)$, with ${\rm Im} \chi_{\rm orb}({\bf Q'},\Omega')$ assumed flat inside and zero outside.
The $q$ dependence of the averaged four-fermion vertex $\langle \Gamma_4 (q)\rangle $ in the $\Gamma$-X direction of the Brillouin zone is shown in Fig. 4. The band contribution is negative due to renormalization of band energies by orbital fluctuations, as discussed above. The band contribution rapidly diminishes in the strong coupling limit \[Fig. 4(b)\], as does the Stoner contribution. The spectral contribution survives in the strong coupling limit. All contributions have strongly non-Heisenberg character, with negligible magnitude at small $q$ but rising sharply near the zone boundary, implying no spin stiffness correction but appreciable zone boundary magnon softening. Orbital fluctuation modes near $(\pi,0,\pi)$ etc. also yield significant non-Heisenberg character to the interaction vertex $\Gamma_4 (q)$, and the three contributions exhibit similar behaviour.
How does the orbital fluctuation magnon self energy compare with the bare magnon energy? Taking the orbital fluctuation energy $\Omega'$ to be negligible in comparison to the bandwidth near the orbital ordering instability, and the estimated average $\langle \Gamma_4 ({\bf q}) \rangle_{\bf Q'}\approx 0.3$ near the zone boundary from Fig. 4(c), we obtain (for $m=0.35$ and $V=3$): $$\Sigma_{\rm orb} ({\bf q}) \sim m (V^2 /2) \langle \Gamma_4 ({\bf q}) \rangle_{\bf Q'}
= 0.35 \times (9/2) \times 0.3 \approx 0.5$$ which is comparable to the bare zone-boundary magnon energy $\omega_{\bf q}^0 \approx 0.35 (1-\cos q) \approx 0.7$ for realistic strength of the inter-orbital interaction.
Spin-orbital coupling magnon self energy
========================================
In the previous section, we considered the diagrams of Fig. 1 (a,b,c) involving electronic self-energy corrections due to orbital fluctuations. We will now examine the vertex correction diagrams of Fig. 1(d) representing spin-orbital coupling, which are particularly important near the orbital ordering instability where orbital fluctuations are soft. In contrast to the single-orbital case where self energy and vertex correction diagrams were of qualitatively similar order,[@singh] introduction of the orbital degree of freedom lifts this constraint and allows qualitatively independent self energy and vertex correction contributions. The corresponding first-order magnon self energy: $$\begin{aligned}
\Sigma_{\rm sp-orb}^{(1)} ({\bf q},\omega) &=& mU^2 \phi ^{(d)} ({\bf q},\omega) \nonumber \\
&=& -m U^2 \sum_{\bf Q}\int\frac{d\Omega}{2\pi i} [U_{\rm eff} ^{\alpha\alpha} ({\bf Q},\Omega) ]
[\Gamma_3 ({\bf Q,q},\Omega,\omega)]^2 [V_{\rm eff} ^{\alpha\alpha} ({\bf q-Q},\omega-\Omega) ] \nonumber \\
&=& m \sum_{\bf Q}\int\frac{d\Omega}{2\pi i} [\chi_{\rm sp} ({\bf Q},\Omega) ]
[\Gamma_{\rm sp-orb} ({\bf Q,q},\Omega,\omega)]^2 [\chi_{\rm orb} ({\bf q-Q},\omega-\Omega) ]\end{aligned}$$ where $\chi_{\rm sp}$ and $\chi_{\rm orb}$ are the spin and orbital fluctuation propagators (section III), and $\Gamma_{\rm sp-orb} \equiv U^2 V \Gamma_3$ represents the spin-orbital interaction vertex in terms of the three-fermion vertex $\Gamma_3$, evaluation of which is discussed in the Appendix.
In analogy with the spin-charge coupling process,[@spch3] this correlation-induced spin-orbital coupling is analogous to a second-order Raman scattering process in which the magnon $({\bf q},\omega)$ scatters into an intermediate-state magnon $({\bf Q},\Omega)$ along with an internal orbital excitation $({\bf q-Q},\omega-\Omega)$, leading to significant magnon energy renormalization and magnon damping.
The ${\bf Q},\Omega$ integrals in the above equation represent integrating out the bosonic (both spin and orbital) degrees of freedom. As the magnon propagator $[\chi_{\rm sp} ({\bf Q},\Omega)]$ is purely advanced in nature, only the retarded part of the product $[\Gamma_{\rm sp-orb}]^2 [\chi_{\rm orb}]$ contributes in the $\Omega$ integral. For simplicity, considering the dominant contribution to the spectral representation of this product coming from the orbital fluctuation propagator, from Eqs. (13), (3), (5), and the symmetry property given above Eq. (5), we obtain: $$\begin{aligned}
\Sigma_{\rm sp-orb}^{(1)} ({\bf q},\omega)
&=& - m \sum_{\bf Q}\int\frac{d\Omega}{2\pi i} \left ( \frac{m}{\Omega + \omega_{\bf Q}^0 - i \eta} \right )
[\Gamma_{\rm sp-orb} ({\bf Q,q},\Omega,\omega)]^2
\int_0^\infty \frac{d\Omega'}{\pi} \;
\frac{{\rm Im} \chi_{\rm orb}({\bf Q-q},\Omega')}{\Omega - \omega - \Omega' + i \eta} \nonumber \\
&=& m^2 \sum_{\bf Q}
\int_0^\infty \frac{d\Omega'}{\pi} \;
\frac {[\Gamma_{\rm sp-orb} ({\bf Q,q},\Omega,\omega)]^2 }
{\omega_{\bf Q}^0 + \omega + \Omega' - i \eta}
{\rm Im}\chi_{\rm orb}({\bf Q-q},\Omega')\end{aligned}$$
Due to magnon decay into internal spin and orbital excitations ($-\omega=\omega_{\bf q}= \omega_{\bf Q}^0 + \Omega'$), the above magnon self energy yields a finite imaginary part representing finite magnon damping and linewidth, as discussed earlier for spin-charge coupling.[@spch3]
An approximate evaluation of the resulting spin-orbital magnon self energy illustrates the importance of the orbital fluctuation softening near the orbital-ordering instability. With $\Omega_{\rm orb}=\Omega'$ and $\Omega_{\rm spin}=\omega_{\bf Q}^0 $ representing characteristic orbital and spin fluctuation energy scales, we obtain (for $\omega=0$): $$\Sigma_{\rm sp-orb}^{(1)} ({\bf q}) \approx
m^2 \frac{\langle [\Gamma_{\rm sp-orb} ({\bf q})]^2 \rangle_{{\bf Q'},\Omega'} }
{\Omega_{\rm spin} + \Omega_{\rm orb} - i \eta }$$ where the angular brackets $\langle \; \rangle$ again refer to averaging over the orbital fluctuation modes ${\bf Q' = Q-q}$, as in Eq. (8). Far from the orbital ordering instability, the orbital fluctuation energy $\Omega_{\rm orb}$ is of order bandwidth $W$, which strongly suppresses the magnon self energy. However, near the orbital-ordering instability, spin-orbital coupling becomes important due to the relatively much smaller energy denominator $\Omega_{\rm spin} + \Omega_{\rm orb} \sim t$.
The spin-orbital interaction vertex $\Gamma_{\rm sp-orb}$ is obtained by integrating out the fermion degrees of freedom in the three-fermion interaction vertices. This interaction vertex explicitly vanishes at momentum $q=0$ in accordance with the Goldstone mode requirement, and yields the dominant $q$ dependence of the magnon self energy. In order to illustrate the characteristic non-Heisenberg character of the interaction vertex, we consider its magnetic part with energy denominators involving the Stoner gap. This term qualitatively differs from the charge part of the vertex with energy denominators involving excitation energies of order bandwidth. Evaluation of the three-fermion vertices contributing to $\Gamma_{\rm sp-orb}$ is discussed in the Appendix.
For the purely magnetic part, we obtain: $$\Gamma_{\rm sp-orb} ^{\rm mag} = U^2 V
\sum_{\bf k}\left ( \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \Omega - i \eta} \right )
\left [ \frac{1}{\epsilon_{\bf k-q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \omega - i \eta} -
\frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k+q-Q}^{\uparrow} + \omega - i \eta} \right ]$$ where there is no restriction on the $\epsilon_{\bf k+q-Q}^{\uparrow}$ energies. The leading order $q$ dependence of the above term is approximately $(1-\cos q)$ as shown below. For a fixed orbital fluctuation momentum ${\bf Q'=Q-q}$, with ${\bf Q'}$ close to $(\pi,\pi,\pi)$ corresponding to staggered orbital fluctuations, expansion in powers of $\epsilon/\Delta$ of the square-bracket terms in Eq. (16) yields a contribution $(\epsilon_{\bf k-q} - \epsilon_{\bf k})/2\Delta \sim (1-\cos q)$ to leading order. The formally second-order structure of the spin-orbital coupling magnon self energy (Eq. 15) involving $[\Gamma_{\rm sp-orb}]^2$ therefore directly yields an intrinsically non-Heisenberg $(1-\cos q)^2$ contribution to the magnon self energy, which yields no spin stiffness quantum correction, but significant magnon energy reduction and damping near the zone boundary, and therefore accounts for the observed zone-boundary magnon anomalies.
The $q$ dependence of the magnetic part $[\Gamma_{\rm sp-orb} ^{\rm mag}]^2$ of the spin-orbital interaction term is shown in Fig. 5(a). Being small in comparison to the Stoner gap, the boson energies $\omega$ and $\Omega$ in Eq. (16) were set to zero, and the momentum ${\bf Q'}$ was selected in a range near $(\pi,\pi,\pi)$, as in Fig. 4. While $[\Gamma_{\rm sp-orb} ^{\rm mag}]^2$ shows strong anomalous momentum dependence for orbital fluctuation modes near $(\pi,\pi,\pi)$, when both magnetic and charge terms (see Appendix) are included, the net contribution from these modes is small due to a cancellation, as shown in Fig. 5(b).
Orbital fluctuations near $(\pi/2,\pi/2,0)$
===========================================
In contrast, for orbital fluctuation modes with momentum ${\bf Q'}$ near $(\pi/2,\pi/2,0)$, the above cancellation is avoided as the magnetic contribution to $\Gamma_{\rm sp-orb}$ is small. We find that the total spin-orbital interaction term $[\Gamma_{\rm sp-orb}]^2$ exhibits a strong anomalous momentum dependence \[Fig. 5(b)\], and that it is strongly enhanced with increasing hole doping, as the band filling changes from 0.5 (quarter filling) to 0.25 (one-eighth filling).
In order to quantitatively examine the effect of this anomalous momentum behaviour of $\Gamma_{\rm sp-orb}^2$ on the magnon dispersion, the magnon self energy was evaluated approximately using Eq. (15). On averaging $[\Gamma_{\rm sp-orb}]^2$ over ${\bf Q'}$, we find that the anomalous momentum behaviour remains qualitatively similar for $|Q_z '| \lesssim 1$ and drops sharply for $Q_z' \gtrsim 1$, whereas in the $Q_x ' - Q_y '$ plane the dominant and qualitatively similar contribution comes from diagonal strips along $|Q_x ' + Q_y '| = \pi$, yielding a factor of $\sim 1/2$ on planar averaging, resulting in an overall phase-space factor of $\sim (1/3)(1/2)$.
The magnon self energy was therefore estimated using Eq. (15) with $\Gamma_{\rm sp-orb}^2$ as obtained in Fig. 5(b) for the orbital fluctuation mode $(\pi/2,\pi/2,0)$, and including the above phase-space factor (taken as 1/10 in our calculations) to account for momentum averaging over orbital fluctuation modes. Also, the bare magnon energy $\omega_Q ^0$ and the orbital fluctuation energy $\Omega_{\rm orb}$ were taken to be of order $t$. The renormalized magnon energy: $$\omega_{\bf q} = \omega_{\bf q} ^0 - \Sigma_{\rm sp-orb} ^{(1)} ({\bf q})$$ is shown in Fig. 6 for different band fillings $n=(1-x)/2$. It is evident that while the spin stiffness remains essentially unchanged, the magnon self energy at the zone-boundary becomes comparable to the bare magnon energy for realistic values of the inter-orbital interaction $V$, resulting in a zone-boundary softening which is strongly enhanced with hole doping. In the presence of staggered orbital correlations such as near $(\pi/2,\pi/2,0)$, weighted averaging over ${\bf Q'}$ with a peaked orbital spectral function will further enhance $\langle [\Gamma_{\rm sp-orb}]^2 \rangle$ and therefore the anomalous magnon self energy.
The orbital fluctuation modes near $(\pi/2,\pi/2,0)$ correspond to period $4a$ planar staggered correlations, as shown in Fig. 7. Such orbital correlations have been observed in the CE-type charge-ordered phase of the half-doped $(x=0.5)$ manganites such as the narrow-band compounds like $\rm Pr_{1-x} Ca_x Mn O_3$ and $\rm La_{1-x} Ca_x Mn O_3$,[@kajimoto_2002] and the layered material $\rm La_{1/2}Sr_{3/2}MnO_4$ in which magnetic excitations were found to be dominated by ferromagnetic couplings.[@senff_2006] Therefore, as $x$ approaches 0.5, orbital fluctuation modes near $(\pi/2,\pi/2,0)$ may form the dominant contribution to the low-energy part of the orbital fluctuation spectral function.
The different layers in $\rm La_{1/2}Sr_{3/2}MnO_4$ are magnetically decoupled due to negligible interlayer couplings. Extension of the present investigation of spin-orbital coupling to the two-dimensional case is therefore of interest with respect to renormalization of magnetic couplings. In the case of spin-charge coupling, the renormalized magnon dispersion for a square lattice does show strong zone-boundary softening near $(\pi,0)$ and $(0,\pi)$, while the magnon energy near $(\pi,\pi)$ remains undiminished, indicating softening of the nearest-neighbour ferromagnetic bonds but strong ferromagnetic correlations along the diagonal directions.[@spch3]
Orbital fluctuation modes near ${\bf Q'}=(\pm\pi,0,0)$ and $(0,\pm\pi,0)$ were also found to yield significant contribution to the spin-orbital interaction vertex. Involving similar density $\langle n_{i\alpha} + n_{i\beta} \rangle$ on all sites, such configurations should, however, be relatively suppressed by an intersite density interaction $V'n_i n_j$. In contrast, allowing for reduced density on alternating “empty” sites with vanishing orbital “magnetization”, configurations corresponding to modes near $(\pi/2,\pi/2,0)$ with period $4a$ orbital correlations minimize the intersite interaction energy $V' n_i n_j$, and would therefore be relatively more important.
Finite Hund’s coupling
======================
So far we had set the Hund’s coupling $J=0$ in order to highlight the role of inter-orbital interaction and fluctuation. For finite $J$, it is convenient to proceed in two steps. The part $V=J$ of the inter-orbital interaction $V$ together with Hund’s coupling effectively amounts to a purely magnetic interaction $-J{\bf S}_{i\alpha}.{\bf S}_{i\beta}$, and has been investigated earlier.[@hunds] This is because for $V=J$, the inter-orbital interaction $(J-J\delta_{\sigma\sigma'})n_{i\alpha\sigma} n_{i\beta\sigma'}$ acts only between opposite-spin electrons, and so the resulting diagrammatics is similar to the Hubbard interaction case. The remaining part $(V-J)$ is purely non-magnetic, and yields diagrammatic contributions essentially as in section II. Thus, corresponding to Fig. 1 (a,b,c) diagrams involving the bubble series for the orbital fluctuation propagator, the magnon self energy is obtained by simply replacing $V$ by $(V-J)$ in Eq. (7).
For the spin-orbital coupling magnon energy, however, the interaction ladders in diagrams Fig. 1(d) and Fig. 2 now involve Hund’s coupling $J$ as well, and since the transverse part of $J$ flips the orbital index in the ladder series, there are now two contributions to the irreducible particle-hole propagator: $$\phi_{\alpha\mu}({\bf q},\omega) = - \sum_{\bf Q} \int \frac{d\Omega}{2\pi i}
U_{\rm eff} ^{\alpha\mu} ({\bf Q},\Omega)
[\Gamma_3 ({\bf Q,q},\Omega,\omega)]^2
V_{\rm eff} ^{\alpha\mu} ({\bf q-Q},\omega-\Omega)$$ involving intra $(\mu=\alpha)$ and inter $(\mu=\beta)$ orbital spin and orbital fluctuations. Involving ladders of $U$ and $J$, the effective intra and inter-orbital transverse spin interactions: $$U_{\rm eff} ^{\alpha\mu} = \frac{1}{2} \left [\frac{U^+}{1-U^+ \chi_0} \pm \frac{U^-}{1-U^- \chi_0} \right ]
\approx \frac{(U+J)^2}{2} \left [\frac{\chi_0}{1-U^+ \chi_0} \pm \frac{\chi_0}{1-U^- \chi_0} \right ]$$ can be expressed as in-phase $(\mu=\alpha)$ and out-of-phase $(\mu=\beta)$ combinations of the acoustic and optical branches,[@hunds] with $U^{\pm} \equiv U \pm J$. Similarly, the effective intra and inter-orbital density interactions: $$\begin{aligned}
V_{\rm eff} ^{\alpha\alpha} &=& -\frac{(V-J)^2\chi_{0\uparrow}}{1 - (V-J)^2\chi_{0\uparrow} ^2} \nonumber \\
V_{\rm eff} ^{\alpha\beta} &=& \frac{(V-J)}{1 - (V-J)^2\chi_{0\uparrow} ^2}\end{aligned}$$ involve odd and even number of bubbles.
Now, in the investigation of role of Hund’s coupling on quantum corrections,[@hunds] it was shown that the inter-orbital component $U_{\rm eff} ^{\alpha\beta} ({\bf Q},\Omega)$ yields a small $\Omega$-integrated contribution due to partial cancellation from the out-of-phase combination of the acoustic and optical modes, essentially reflecting an inter-orbital incoherence. Therefore, the inter-orbital component $\phi_{\alpha\beta}$ should be relatively small, and the intra-orbital component $\phi_{\alpha\alpha}$ is approximately given by Eq. (15), with $V$ replaced by $(V-J)$ and $U$ replaced by $(U+J)$.
Due to the purely opposite-spin density interaction $(J-J\delta_{\sigma\sigma'})n_{i\alpha\sigma} n_{i\beta\sigma'}$, the interaction line $U$ connected to the bubble in the third diagram in Fig. 2(b) for the three-fermion vertex $\Gamma_3$ is also replaced by $(U+J)$. This is consistent with the enhanced exchange splitting to $(U+J)m$ in the $\chi^0$ energy denominator, which changes the magnon pole condition to $(U+J)\chi^0 = 1$, and ensures that the three-fermion vertex $\Gamma_3$ exactly vanishes at $q=0$ in accordance with the Goldstone-mode condition (see Appendix).
Extension to the ferromagnetic Kondo lattice model
==================================================
Magnetic couplings and excitations in ferromagnetic manganites have been theoretically investigated using the ferromagnetic Kondo lattice model (FKLM) and its strong coupling limit, the double exchange ferromagnet. In this model, the $S=3/2$ core spins due to localized $\rm t_{2g}$ electrons of the magnetic Mn$^{++}$ ions are exchange coupled to the mobile $\rm e_g$ band electrons, represented by an interaction term $-{\cal J} \sum_i {\bf S}_i . {\mbox{\boldmath $\sigma$}}_i$, with $\cal J$ of order bandwidth $W$. The fermionic representation approach for evaluating magnon self energy corrections in the FKLM, which allows conventional diagrammatic tools to be employed for evaluating quantum corrections beyond the leading order,[@qfklm] can be readily extended to include effects of orbital fluctuations.
With an inter-orbital interaction $V$ included between the two degenerate $\rm e_g$ orbitals, the FKLM magnon self energy due to spin-orbital coupling can be directly obtained from our magnon renormalization analysis of Fig. 6. The required correspondence is: $U\rightarrow {\cal J}$ and $m\rightarrow 2S$, so that the exchange splitting $Um \rightarrow 2{\cal J}S$. The FKLM magnon self energy is thus obtained from our calculated Hubbard model result using a multiplicative factor $f={\cal J}^4(2S)^2/U^4 m^2$. With $U/t=20$, $m=0.35$, ${\cal J}/t=4$, and $2S=3$, we obtain $f\approx 1/6$. As the FKLM bare magnon energy $\sim (t/18)(1-\cos q)$ is smaller than the Hubbard model bare magnon energy $\sim (t/3)(1-\cos q)$ by roughly the same factor, the FKLM renormalized magnon energy $\omega_{\bf q}=\omega_{\bf q}^0 - \Sigma_{\rm sp-orb} ({\bf q})$ will also be as in Fig. 6, only scaled down by a factor (1/6). Taking the hopping energy scale $t\sim 200$meV corresponding to a realistic bandwidth $\sim 2$eV, the magnon energy scale in Fig. 6 is $\sim 30$meV for ferromagnetic manganites, in agreement with the measured magnon energies.[@zhang_2007]
Since the FKLM magnon self energy goes as the fourth power of ${\cal J}/t$ explicitly, the anomalous zone-boundary softening effect should be especially pronounced in narrow-band systems. Indeed, zone-boundary magnon softening is clearly seen to occur in the relatively low-$T_{\rm C}$ or narrow-band materials, and broad-band materials such as $\rm La_{0.7} Pb_{0.3} Mn O_3$ show nearly Heisenberg behavior, in agreement with this prediction. Systematic studies of doping dependence of the zone-boundary magnon softening indicates that the higher the doping level, the larger the zone-boundary softening.[@zhang_2007] Furthermore, doping dependence of spin dynamics indicates that the measure spin stiffness $D \sim 160 \pm 15$ meV Å$^2$ remains essentially unchanged, while the zone-boundary magnon softening (denoted by the ratio $J_4/J_1$) is found to be enhanced linearly with increasing doping.
Conclusions
===========
The correlated motion of electrons in the presence of strong orbital fluctuations and correlations was investigated in an orbitally degenerate ferromagnet with two orbitals per site with respect to magnetic couplings and excitations. A systematic Goldstone-mode-preserving approach was employed to incorporate correlation effects in the form of self energy and vertex corrections, so that both long-wavelength and zone-boundary magnon modes could be studied on an equal footing. Our investigation focussed on the anomalous momentum dependence of the three- and four-fermion interaction vertices which determine the magnon self energy and the role of different orbital fluctuation modes, particularly near the orbital ordering instability where orbital fluctuations are relatively soft.
Orbital fluctuations were generically shown to impart an intrinsically non-Heisenberg $(1- \cos q)^2$ character to the magnon self energy in the $\Gamma$-X direction of interest. This generic behaviour was ascribed to a quadratic structure of the spin-orbital interaction vertex resulting from a new class of diagrammatic contributions associated with the orbital degree of freedom. These diagrams are absent in the single-orbital case, the essential difference being that orbital fluctuations couple to electrons of both spin.
The absence of any $q^2$ contribution in this non-Heisenberg magnon self energy for small $q$ implies that the spin stiffness is not renormalized by orbital fluctuations generated by the inter-orbital density interaction $V$. In a multi-orbital ferromagnet, with intra-orbital Coulomb interaction $U$, inter-orbital interaction $V$, Hund’s coupling $J$, and orbital degeneracy ${\cal N}$, the spin stiffness therefore continues to be essentially determined by the intra-atomic factors $U$, $J$, ${\cal N}$, through the effective quantum parameter $[U^2 + ({\cal N}-1)J^2]/[U + ({\cal N}-1)J]^2$ as obtained earlier,[@hunds] and the interaction $V$ does not play an important role.
However, the strong enhancement of magnon self energy near the zone boundary resulted in a strong anomalous magnon softening in the $\Gamma$-X direction, which increases significantly with hole doping away from quarter filling. Our investigation thus clarifies the completely different roles of interactions $J$ and $V$, representing magnetic and charge parts of the inter-orbital Coulomb interaction. While Hund’s coupling enhances ferromagnetism by strongly suppressing the effective quantum parameter, orbital fluctuations and correlations due to $V$ destabilize ferromagnetism by strongly suppressing zone-boundary magnon energies near the orbital ordering instability.
With regard to relative importance of different orbital fluctuation modes, staggered fluctuations with ${\bf Q}$ near $(\pi,\pi,\pi)$ and $(\pi,0,\pi)$ were found to be most important for the orbital fluctuation magnon self energy. The spin-orbital coupling magnon self energy was found to be strongly sensitive to orbital fluctuation modes due to an interference between magnetic and charge terms in the interaction vertex. Thus, while the magnetic part showed strong anomalous momentum dependence for orbital modes near $(\pi,\pi,\pi)$, the net contribution to the total vertex was found to be small due to cancellation with the charge term. Rather, fluctuation modes near $(\pi/2,\pi/2,0)$ were found to be important for the total vertex including the charge part. The strong zone-boundary magnon softening near $(0,0,\pi)$, arising from staggered orbital fluctuations with ${\bf Q}$ near $(\pi/2,\pi/2,0)$, suggests an instability towards a composite structure of spin-orbital correlations involving period $4a$ orbital ordering in ferromagnetic planes and intra-orbital AF spin correlations in the perpendicular direction.
These results provide a plausible explanation of the observed anomalies in neutron scattering studies of spin-wave excitations in ferromagnetic manganites, where spin stiffness is seen to remain essentially unchanged whereas the zone-boundary magnon softening is enhanced with increasing hole doping and the approach towards CE-type charge-orbital ordered states near $x=0.5$. Our results of strong anomalous magnon self energy contribution from different orbital fluctuation modes such as $(\pi,\pi,\pi)$ and $(\pi/2,\pi/2,0)$ show the zone-boundary softening to be a more generic feature of spin-orbital coupling. Only ferromagnetic orbital correlations extremely close to the orbital ordering instability were found to yield significant magnon softening in earlier studies.[@khaliullin_2000]
The observed zone-boundary magnon softening has been usually modeled by including a fourth neighbour interaction term $J_4$.[@zhang_2007] As $J_4$ yields no contribution to the zone-boundary magnon energy, but contributes significantly to spin stiffness, it must be accompanied by a corresponding reduction $\Delta J_1 = 4J_4$ in the NN coupling so that the spin stiffness remains unchanged, as observed experimentally; the net magnon energy reduction then has the non-Heisenberg form $2J_4 (1-\cos q)^2$. Our anomalous magnon self energy result of this form thus provides fundamental insight into the role of orbital fluctuations on magnetic couplings and excitations.
Instead of the inter-orbital interaction $Vn_{i\alpha} n_{i\beta}$ considered here, inter-site interactions $V' n_i n_j$ would generate similar diagrammatic contributions to the magnon self energy, which become important near the charge-ordering instability where charge excitations become relatively soft, resulting in similar magnon self energy and anomalous zone-boundary magnon softening.
Appendix {#appendix .unnumbered}
========
The spin-orbital interaction vertex is obtained as: $$\Gamma_{\rm sp-orb} = U^2 V \left [ \Gamma_3^{(a)} + \Gamma_3^{(b)} + \Gamma_3^{(c)} \right ]$$ in terms of the three types of fermion vertices shown in Fig. 1(e), which are evaluated by integrating out the fermion degrees of freedom as discussed below.
For the first fermion vertex with interaction line $V$ attaching to spin-$\downarrow$ fermion, we obtain: $$\Gamma_3^{(a)} = -\sum_{\bf k} \left ( \frac{1}{\epsilon_{\bf k-q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \omega - i \eta}
\right ) \left ( \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \Omega - i \eta} \right ) \; .$$ The second vertex $\Gamma_3^{(b)} = \Gamma_3^{(b1)} + \Gamma_3^{(b2)}$ consists of a similar all magnetic term: $$\Gamma_3^{(b1)} = \sum_{\bf k} \left ( \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k-Q+q}^{\uparrow } + \omega - i \eta} \right ) \left ( \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \Omega - i \eta} \right )$$ and a magnetic-charge term: $$\begin{aligned}
\Gamma_3^{(b2)} &=& - \sum_{\bf k} \left [\left ( \frac{1}{\epsilon_{\bf k-q}^{\downarrow +} - \epsilon_{\bf k}^{\uparrow -} + \omega - i \eta} \right )\left ( \frac{1}{\epsilon_{\bf k-q+Q}^{\uparrow +} - \epsilon_{\bf k}^{\uparrow -} + \omega - \Omega - i \eta} \right ) \right . \nonumber \\
& + & \left . \left ( \frac{1}{\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k-Q+q}^{\uparrow +} + \omega - i \eta} \right ) \left ( \frac{1}{\epsilon_{\bf k-Q+q}^{\uparrow +} - \epsilon_{\bf k}^{\uparrow -} - \omega + \Omega - i \eta} \right ) \right ] \; .\end{aligned}$$ In Eq. (23), there is no restriction on the fermion energy $\epsilon_{\bf k-Q+q}^{\uparrow }$ as contributions with both particle (+) and hole (-) energies are included. Finally, the third vertex $\Gamma_3^{(c)}$ involves a $U$ interaction line and a spin-$\uparrow$ bubble attached to the spin-$\downarrow$ fermion lines, and is given by: $$\Gamma_3^{(c)} = - \Gamma_3^{(a)} \; U \; \chi_{0\uparrow}({\bf q-Q},\omega-\Omega)$$
For $q=0$, the spin-orbital interaction vertex $\Gamma_{\rm sp-orb}$ vanishes identically, ensuring that the Goldstone mode is explicitly preserved. Both $-\Gamma_3^{(a)}$ and $\Gamma_3^{(b1)}$ reduce to $\chi_0({\bf Q},\Omega)/(2\Delta +\omega)$ for $q=0$, whereas both $-\Gamma_3^{(b2)}$ and $\Gamma_3^{(c)}$ reduce to $\chi_{0\uparrow}({\bf Q},\omega-\Omega)$ on setting $U\chi_0({\bf Q},\Omega)=1$ at the magnon pole, so that from Eq. (21) $\Gamma_{\rm sp-orb}=0$ for $q=0$.
Singular contributions in the three-fermion vertex $\Gamma_3$ exactly cancel out, as for the four-fermion vertex $\Gamma_4$ discussed below Eq. (11). Thus, the singular contribution of Eq. (23) due to vanishing energy denominator $\epsilon_{\bf k-Q}^{\downarrow +} - \epsilon_{\bf k-Q+q}^{\uparrow +} + \omega$ exactly cancels the corresponding contribution from the second term of Eq. (24).
[05]{}
Y. Tokura, Physics Today, 50, (2003).
G. Khaliullin, Prog. Theor. Phys. Suppl. [**160**]{}, 155 (2005).
E. Saitoh, S. Okamoto, K. T. Takahashi, K. Tobe, K. Yamamoto, T. Kimura, S. Ishihara, S. Maekawa and Y. Tokura, Nature [**410**]{}, 180 (2001).
T. Ogasawara, T. Kimura, T. Ishikawa, M. Kuwata-Gonokami, and Y. Tokura, Phys. Rev. B [**63**]{}, 113105 (2001).
P. Khalifah, R. Osborn, Q. Huang, H. W. Zandbergen, R. Jin, Y. Liu, D. Mandrus, R. J. Cava, Science, [**297**]{}, 2237 (2002).
C. Ulrich, G. Ghiringhelli, A. Piazzalunga, L. Braicovich, N. B. Brookes, H. Roth, T. Lorenz, and B. Keimer, Phys. Rev. B [**77**]{}, 113102 (2008).
W.-C. Lee and C. Wu, arXiv:0906.1973 (2009).
H. Y. Hwang, P. Dai, S-W. Cheong, G. Aeppli, D. A. Tennant, and H. A. Mook, Phys. Rev. Lett. [**80**]{}, 1316 (1998).
P. Dai, H. Y. Hwang, J. Zhang, J. A. Fernandez-Baca, S.-W. Cheong, C. Kloc, Y. Tomioka, and Y. Tokura, Phys. Rev. B [**61**]{}, 9553 (2000).
T. Chatterji, L. P. Regnault, and W. Schmidt, Phys. Rev. B [**66**]{}, 214408 (2002).
F. Ye, Pengcheng Dai, J. A. Fernandez-Baca, Hao Sha, J. W. Lynn, H. Kawano-Furukawa, Y. Tomioka, Y. Tokura, and Jiandi Zhang, Phys. Rev. Lett. [**96**]{}, 047204 (2006).
F. Ye, P. Dai, J. A. Fernandez-Baca, D. T. Adroja, T. G. Perring, Y. Tomioka, and Y. Tokura, Phys. Rev. B [**75**]{}, 144408 (2007).
J. Zhang, F. Ye, H. Sha, P. Dai, J. A. Fernandez-Baca, and E. W. Plummer, J. Phys.: Condens. Matter [**19**]{}, 315204 (2007).
F. Moussa, M. Hennion, P. Kober Lehouelleur, D. Reznik, S. Petit, H. Moudden, A. Ivanov, Ya. M. Mukovskii, R. Privezentsev, and F. Albenque-Rullier, Phys. Rev. B [**76**]{}, 064403 (2007).
Y. Motome and N. Furukawa, Phys. Rev. B [**71**]{}, 014446 (2005).
G. Khaliullin and R. Kilian, Phys. Rev. B [**61**]{}, 3494 (2000).
A. Singh, Phys. Rev. B 74, 224437 (2006).
S. Pandey and A. Singh, Phys. Rev. B [**78**]{}, 014414 (2008).
S. Pandey, S. K. Das, B. Kamble, S. Ghosh, D. K. Singh, R. Ray, and A. Singh, Phys. Rev. B 77, 134447 (2008).
B. Kamble and A. Singh, Phys. Rev. B [**79**]{}, 064410 (2009).
S. Ghosh and A. Singh, Phys. Rev. B [**77**]{}, 094430 (2008).
R. Kajimoto, H. Yoshizawa, Y. Tomioka, and Y. Tokura, Phys. Rev. B [**66**]{}, 180402 (2002).
D. Senff, F. Krüger, S. Scheidl, M. Benomar, Y. Sidis, F. Demmel, and M. Braden, Phys. Rev. Lett. [**96**]{}, 257201 (2006).
|
---
abstract: |
We study a class of measures on the real line with a kind of self-similar structure, which we call *dynamically driven self-similar measures*, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$ dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg’s long-standing conjecture on the dimension of the intersections of $\times p$ and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions.
The proof of the main result is inspired by M. Hochman’s approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman’s theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
address: 'Department of Mathematics and Statistics, Torcuato Di Tella University, and CONICET, Buenos Aires, Argentina'
author:
- Pablo Shmerkin
title: 'On Furstenberg’s intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions'
---
[^1]
Introduction and main results
=============================
Transversality of times p, times q
----------------------------------
In the 1960s, H. Furstenberg proposed a series of conjectures which, in different ways, aim to capture the heuristic principle that “expansions in multiplicatively independent bases (such as $2$ and $3$) should have no common structure”. Recall that $p,q\in {\mathbb{N}}$ are called multiplicatively independent if they are not powers of a common integer or, equivalently, $\log p/\log q$ is irrational. For $p\in{\mathbb{N}}_{\ge 2}$, let $T_p:[0,1)\to [0,1)$, $x\mapsto p x\bmod 1$ denote multiplication by $p$ on the circle. In [@Furstenberg67], Furstenberg proved a pioneering result of this type: if $p,q\in{\mathbb{N}}_{\ge 2}$ are multiplicatively independent, then no infinite proper closed subset of $[0,1]$ can be simultaneously invariant under $T_p$ and $T_q$. This gave rise to the famous $\times 2,\times 3$ conjecture, which remains open today: if $\mu$ is a Borel probability measure on the circle invariant under $T_2$ and $T_3$, then $\mu$ is a linear combination of Lebesgue measure and a purely atomic measure.
Furstenberg proposed other conjectures with a more geometric flavor. Let $A,B$ be closed subsets of the circle $[0,1)$ invariant under $T_p, T_q$ respectively, with $p,q$ again multiplicatively independent. Furstenberg conjectured that $$\operatorname{dim_{\mathsf{H}}}(A+B) = \min(\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B),1),$$ where $\operatorname{dim_{\mathsf{H}}}$ stands for Hausdorff dimension, and $A+B=\{a+b:a\in A, b\in B\}$ is the arithmetic sum. This fits into the general heuristic principle mentioned above, since the inequality $\operatorname{dim_{\mathsf{H}}}(A+B) \le \min(\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B),1)$ always holds, and a strict inequality should only occur if $A$ and $B$ have some shared structure at many scales. This conjecture was proved in [@PeresShmerkin09] in the special case that $A,B$ are defined by restricting the digits in their base $p,q$ expansion to a fixed digit set, and in [@HochmanShmerkin12] in the general case. Moreover, in [@HochmanShmerkin12] a corresponding result for invariant measures was obtained: if $\mu,\nu$ are Borel probability measures invariant under $\times p,\times q$ respectively, then $$\label{eq:dim-conv-invariant-measures}
\operatorname{dim_{\mathsf{H}}}(\mu*\nu) = \min(\operatorname{dim_{\mathsf{H}}}(\mu)+\operatorname{dim_{\mathsf{H}}}(\nu),1).$$ Here $\operatorname{dim_{\mathsf{H}}}$ denotes the lower Hausdorff dimension of a measure, defined as $$\label{eq:measure-dissonance}
\operatorname{dim_{\mathsf{H}}}(\eta) = \inf\{\operatorname{dim_{\mathsf{H}}}(A):\eta(A)>0\}.$$
We note that this result is trivial if either $\mu$ or $\nu$ have zero entropy (since zero entropy implies zero dimension), but in the positive entropy case it is stronger than the $\times 2,\times 3$ conjecture. We recall the Rudolph-Johnson theorem, asserting that if $\mu,\nu$ are ergodic and invariant under $\times p,\times q$ (with $\log p/\log q$ irrational) and $\mu$ has positive but not full entropy with respect to $\times p$, then $\mu$ and $\nu$ are singular. We showed in [@HochmanShmerkin12] that the Rudolph-Johnson Theorem can be obtained as an easy corollary of .
There is an obvious heuristic relationship between the size of the sumset $A+B$ and the size of the fibers $\ell_z=\{ (x,y):x\in A,y\in B, x+y=z\}$. Namely, if the sumset is “large” then “many fibers” should be small, and conversely. Another conjecture of Furstenberg, and one of the few to be stated explicitly in print [@Furstenberg70 Conjecture 1], asserts that for sets invariant under $\times 2,\times 3$, *all* fibers should be small:
\[conj:Furstenberg\] If $A,B$ are closed subsets of the circle $[0,1)$, invariant under $T_p, T_q$ respectively, with $p$ and $q$ multiplicatively independent, then $$\label{eq:upper-bound-dim-fibers}
\operatorname{dim_{\mathsf{H}}}(A\cap B) \le \max(\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)-1,0).$$
In Furstenberg’s terminology, the dynamics of $T_p$ and $T_q$ should be *transverse*. Again, this fits into the general heuristics of “lack of common structure” since a fiber of larger than expected size can be seen as some shared structure between $A$ and $B$ (and hence between expansions in bases $p$ and $q$). To see why the right-hand side in is the natural bound, one can think of the analogous formula for the dimension of the intersection of transversal linear subspaces, or Marstrand’s intersection theorem asserting that for any Borel set $E\subset {\mathbb{R}}^2$, $$\operatorname{dim_{\mathsf{H}}}(E\cap \ell)\le \max(\operatorname{dim_{\mathsf{H}}}(E)-1,0)$$ for *almost all* lines $\ell$, and this fails for any smaller value on the right-hand side (we note that $E=A\times B$ has dimension $\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)$). See for example [@Mattila95 Chapter 10].
Also in [@Furstenberg70], Furstenberg showed that if $$\operatorname{dim_{\mathsf{H}}}(A\cap g(B)) \ge c$$ for some invertible affine map $g:{\mathbb{R}}\to{\mathbb{R}}$, then for almost all slopes $a$ there is an affine map $g_a(x)=a x+ b(a)$ such that $$\operatorname{dim_{\mathsf{H}}}(A\cap g_a(B)) \ge c.$$ Using this, it is not hard to show that Conjecture \[conj:Furstenberg\] holds when $\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)\le 1/2$; see [@Hochman14b Theorem 7.9] for an exposition of the argument. More generally, combining Furstenberg’s result with estimates of Wolff [@Wolff99] on the dimension of sets that contain, for almost every $v\in S^1$, a subset of a line in direction $v$ with Hausdorff dimension at least $c$, one gets $$\operatorname{dim_{\mathsf{H}}}(A\cap g(B)) \le \max(\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)-1/2,0).$$ Note that this is vacuous if $\operatorname{dim_{\mathsf{H}}}(A)\ge 1/2$.
In a different direction, in [@FHR14] it was shown that $A$ cannot be affinely embedded into $B$ if $0<\operatorname{dim_{\mathsf{H}}}(A)<\operatorname{dim_{\mathsf{H}}}(B)<1$. More precisely, it follows from [@FHR14 Theorem 1.6] that in this case there is some (non-effective) $\delta=\delta(A,B)>0$ such that $$\operatorname{dim_{\mathsf{H}}}(A\cap g(B)) \le \operatorname{dim_{\mathsf{H}}}(A)-\delta$$ for all $C^1$ diffeomorphisms $g$ of ${\mathbb{R}}$ (here, and whenever clear from context, we think of $A,B$ as subsets of $[0,1)\subset {\mathbb{R}}$ rather than the circle). Although in [@FHR14] this is stated only for the case in which $A$ is a $p$-Cantor set and $B$ is a $q$-Cantor set, the case of general invariant sets follows by a standard upper approximation. De-Jun Feng (Private Communication) developed an algorithm that yields effective values of $\delta$ in specific cases, for example if $A$ is the middle-one quarter Cantor set and $B$ is the middle-thirds Cantor set; the computed values are still far from those predicted by Furstenberg’s conjecture.
D-J. Feng (Private Communication) also constructed, for any multiplicatively independent $p,q$ and for any $0<s,t<1$ and ${\varepsilon}>0$, closed $T_p,T_q$-invariant sets $A,B\subset [0,1)$ of dimension $s,t$ respectively, for which $$\operatorname{dim_{\mathsf{H}}}(A\cap g(B)) \le \max(\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)-1,0)+{\varepsilon},$$ for all affine maps $g$. Although this comes close, we note that not a single example of sets $A,B$ (for some multiplicatively independent $p,q$) for which the conjecture holds, was known, apart from the trivial cases in which one of the sets has dimension $0$ or $1$, and the case in which $\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)\le 1/2$, as explained above.
In this article, we prove the following strong version of Furstenberg’s conjecture which, in his terminology, says that the maps $T_p$ and $T_q$ on the circle are *strongly transverse*:
\[thm:Furstenberg\] Let $p,q\in{\mathbb{N}}_{\ge 2}$ be multiplicatively independent. Then for any closed sets $A,B$ of the circle $[0,1)$ invariant under $T_p, T_q$ respectively, and for any invertible affine map $g:{\mathbb{R}}\to {\mathbb{R}}$, $$\operatorname{{\overline{dim}_{\mathsf{B}}}}(A\cap g(B)) \le \max(\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)-1,0).$$
Here $\operatorname{{\overline{dim}_{\mathsf{B}}}}$ denotes upper box-counting dimension, which is always at least as large as Hausdorff dimension.
The method we use to establish Theorem \[thm:Furstenberg\] yields several other new results on classical problems in fractal geometry and dynamics. Before discussing our general approach, we present some of these results.
Dimension and densities of Bernoulli convolutions
-------------------------------------------------
Given ${\lambda}\in(0,1)$, let $\nu_{\lambda}$ be the distribution of the random series $\sum_{n=0}^\infty \pm{\lambda}^n$, with the signs chosen independently with equal probabilities. This is the family of *Bernoulli convolutions*, whose study goes back to the 1930s. For ${\lambda}\in (0,1/2)$, it is well known that $\nu_{\lambda}$ is (up to an affine bijection) a constant multiple of Hausdorff measure (of the appropriate dimension) on the central Cantor set constructed by removing a central interval of length $1-2{\lambda}$ from $[0,1]$ and iterating. The properties of $\nu_{\lambda}$ for ${\lambda}\in [1/2,1)$ have been studied for some 80 years but are far from being properly understood. We prove new properties of the densities and dimension of $\nu_{\lambda}$ outside of a small set of parameters.
Perhaps the most significant open problem on Bernoulli convolutions is to determine for which values of ${\lambda}$ the measure $\nu_{\lambda}$ turns out to be absolutely continuous. Erdős already in 1939 [@Erdos39] showed that if ${\lambda}^{-1}$ is a Pisot number (an algebraic unit $>1$ such that all its algebraic conjugates are $<1$ in modulus), then $\nu_{\lambda}$ is singular. It is still not known if there is any ${\lambda}\in (1/2,1)$ such that $\nu_{\lambda}$ is singular and ${\lambda}^{-1}$ is not Pisot.
In light of this open problem, a fruitful strand of research developed to prove results of the form: $\nu_{\lambda}$ is absolutely continuous, with certain regularity of the density, outside of some “small” set. This line was also initiated by Erdős [@Erdos40], who proved that for every $k\in{\mathbb{N}}$ there is ${\varepsilon}_k>0$ such that $\nu_{\lambda}$ has a $k$-times continuously differentiable density for almost all ${\lambda}\in (1-{\varepsilon}_k,1)$. Several decades later, Kahane [@Kahane71] noted that Erdős’ argument yields a stronger statement, namely, that for every $k\in{\mathbb{N}}$, $$\lim_{{\varepsilon}\downarrow 0} \operatorname{dim_{\mathsf{H}}}\{{\lambda}\in (1-{\varepsilon},1): \nu_{\lambda}\text{ does not have a $C^k$ density }\} = 0.$$ The proof of Erdős-Kahane is based on a combinatorial study of the Fourier transform of $\nu_{\lambda}$, and no other proof of the statement is known.
The Erdős-Kahane argument only gives non-trivial information very close to $1$. In a landmark paper from 1995, Solomyak [@Solomyak95] showed that $\nu_{\lambda}$ is absolutely continuous with an $L^2$ density for almost all ${\lambda}\in (1/2,1)$. A simpler proof was obtained by Peres and Solomyak [@PeresSolomyak96]. The $L^2$ part of the result is a by-product of the transversality technique used by Solomyak, and a natural question is whether $L^2$ can be replaced by a better space. In [@PeresSchlag00], Peres and Schlag proved that for any ${\varepsilon}>0$ there is some (explicit) $\delta>0$ such hat $\nu_{\lambda}$ has fractional derivatives of order $\delta$ in $L^2$, for almost all ${\lambda}\in (1/2+{\varepsilon},1)$. By the Sobolev embedding theorem, in particular this implies that $\nu_{\lambda}$ has a density in $L^q$ for some $q=q({\varepsilon})>2$ for almost all ${\lambda}\in (1/2+{\varepsilon},1)$. Their result still relies on transversality techniques, which cannot go beyond $L^2$ for ${\lambda}$ close to $1/2$.
Besides improving on the smoothness of the density, another natural line to pursue is to make the exceptional set of ${\lambda}$ smaller. In the same article [@PeresSchlag00], Peres and Schlag proved that for every ${\varepsilon}>0$, there is an explicit $\delta>0$ such that $$\operatorname{dim_{\mathsf{H}}}\{{\lambda}\in (1/2+{\varepsilon}, 1):\nu_{\lambda}\text{ does not have an $L^2$ density } \} \le 1-\delta.$$
Much more recently, the author [@Shmerkin14] (relying on deep work of Hochman [@Hochman14] that will be discussed in some detail below) proved that $\nu_{\lambda}$ is absolutely continuous for all ${\lambda}$ outside of a set of zero Hausdorff dimension. Moreover, in [@ShmerkinSolomyak16] it was shown that, again outside of a set of zero Hausdorff dimension of parameters, $\nu_{\lambda}$ has a density in $L^q$ for some $q>1$ that is not explicit and depends on ${\lambda}$.
These three lines of work yield somewhat complementary results: the stronger the information about the densities, the weaker the information about the exceptional set. They also leave open the question of what is the smallest natural function space that contains the density of $\nu_{\lambda}$ for almost all ${\lambda}$. In this article, we prove:
\[thm:abc-cont-BCs\]
1. There exists a set $\mathcal{E}\subset (1/2,1)$ of zero Hausdorff dimension such that if ${\lambda}\in (1/2,1)\setminus \mathcal{E}$, then $\nu_{\lambda}$ has a density in $L^q$ for all finite $q>1$.
2. There exists a set $\mathcal{E}'\subset (1/\sqrt{2},1)$ of zero Hausdorff dimension such that if ${\lambda}\in (1/\sqrt{2},1)\setminus \mathcal{E}'$, then $\nu_{\lambda}$ has a continuous density.
The new contribution is part (i); part (ii) then follows by a standard argument. In turn, part (i) follows from a new result about *dimensions* of Bernoulli convolutions, together with a result from [@ShmerkinSolomyak16]. To state the dimensional result, we define the following set (which appears already in [@Hochman14]).
\[def:ssc-BCs\] Let $\mathcal{P}_n$ be the family of all non-zero polynomials of degree at most $n$ and coefficients in $\{-1,0,1\}$. Let $$\label{eq:def-super-exp-set-BCs}
\mathcal{E} = \left\{ {\lambda}\in (1/2,1): \frac1n \log\left(\min_{P\in\mathcal{P}_n} |P({\lambda})|\right) \to -\infty \right\} .$$
It is shown in [@Hochman14] that $\mathcal{E}$ has zero packing dimension (in particular, zero Hausdorff dimension) and does not contain any algebraic number which is not a root of a polynomial in $\mathcal{P}_n$ for some $n$. In particular, no rational number in $(1/2,1)$ is in $\mathcal{E}$.
\[thm:infinity-dim-BCs\] Let ${\lambda}\in (1/2,1)\setminus \mathcal{E}$. Then for every ${\varepsilon}>0$ there is $C=C({\varepsilon},{\lambda})>0$ such that $$\nu_{\lambda}(B(x,r)) \le C \, r^{1-{\varepsilon}} \quad\text{for all } x\in {\mathbb{R}}, r\in (0,1].$$
It is known (see [@FengHu09]) that for any ${\lambda}$, the limit $$\lim_{r\downarrow 0} \frac{\log\nu_{\lambda}(B(x,r))}{\log r}$$ exists and is constant $\nu_{\lambda}$-almost everywhere; this constant value is denoted $\dim(\nu_{\lambda})$ and equals the Hausdorff, packing and entropy dimensions of $\nu_{\lambda}$. In [@Hochman14], it is proved that if ${\lambda}\in (1/2,1)\setminus\mathcal{E}$, then $\dim(\nu_{\lambda})=1$. Theorem \[thm:infinity-dim-BCs\] strengthens this, since it implies in particular that $$\liminf_{r\downarrow 0} \frac{\log\nu_{\lambda}(B(x,r))}{\log r} \ge 1$$ for *all* (rather than almost all) $x$. On the other hand, for any locally finite measure $\mu$ on the real line it holds that $$\limsup_{r\downarrow 0} \frac{\log\mu(B(x,r))}{\log r} \le 1$$ for $\mu$ almost all $x$. Nevertheless, for any ${\lambda}\in (1/2,1)$ there are two points $x$ (the boundary points of the support of $\nu_{\lambda}$) for which $$\lim_{r\downarrow 0} \frac{\log\nu_{\lambda}(B(x,r))}{\log r} = \frac{\log 2}{\log(1/{\lambda})}>1,$$ and if ${\lambda}$ is close to $1/2$ there is a positive dimensional set of such points, see [@JSS11 Theorem 1.5]. These remarks indicate that Theorem \[thm:infinity-dim-BCs\] is optimal in a number of ways.
Very recently, some striking progress on the dimensions and absolute continuity of Bernoulli convolutions for algebraic parameters was achieved by P. Varjú [@Varju16] and E. Breuillard and P. Varjú [@BreuillardVarju15]. The latter article also uncovers some deep connections between Bernoulli convolutions, the famous Lehmer’s conjecture from number theory, and the growth of subgroups of linear groups. This line of work goes in a transversal direction to ours: while they obtain new information for many algebraic (and not only) parameters, which our work is far from being able to replicate, their methods do not seem to be able to give information about Frostman exponents or $L^q$ densities for any $q>1$.
Lq dimensions, Frostman exponents, and the size of fibers
---------------------------------------------------------
At first sight, Theorems \[thm:Furstenberg\] and \[thm:infinity-dim-BCs\] may appear to have little in common. However, we will obtain both as rather direct consequences of a single general result. Our common approach is based on *$L^q$ dimensions*. Let $\mu$ be a Borel probability measure on $[0,1]$. We denote the family of $2^{-m}$-intervals $\{ [j 2^{-m}, (j+1) 2^{-m})\}$, $j\in{\mathbb{Z}}$ by ${\mathcal{D}}_m$. If $q>1$, then $$\frac{\log \sum_{I\in{\mathcal{D}}_m} \mu(I)^q}{(1-q)m} \in [0,1],$$ for any $m\in{\mathbb{N}}$, as can be easily seen from Hölder’s inequality. Here and throughout the article, the logarithms are to base $2$. Moreover, a small value indicates that $\mu$ is nearly concentrated on few intervals in ${\mathcal{D}}_m$, while a value close to $1$ implies that $\mu(I), I\in{\mathcal{D}}_m$ is a fairly uniform probability vector. Thus, it makes sense to consider the limit as $m\to\infty$ of the left-hand side as a notion of dimension of $\mu$.
\[def:Lq-dim\] Let $q\in (1,\infty)$. If $\mu$ is a probability measure on ${\mathbb{R}}$ with bounded support, then $$\tau(\mu,q) = \tau_\mu(q) = \liminf_{m\to\infty} -\frac{\log \sum_{I\in{\mathcal{D}}_m} \mu(I)^q}{m}$$ is the *$L^q$ spectrum* of $\mu$, and $$D(\mu,q) = D_\mu(q) = \frac{\tau_\mu(q)}{q-1}$$ is the *$L^q$ dimension* of $\mu$.
It is also possible to define $L^q$ dimensions for other values of $q$, but we will not need to do so here. It is well-known that, for a fixed measure $\mu$, the map $q\mapsto D(\mu,q)$ is continuous and decreasing on $(1,\infty)$. Moreover, $$\label{eq:hausdorff-dim-larger-than-Lq-dim}
\operatorname{dim_{\mathsf{H}}}\mu \ge \lim_{q\downarrow 1} D(\mu,q).$$ See [@FLR02] for proofs of these standard facts.
If $\mu$ is a finite measure on a metric space $X$, we say that $\mu$ has *Frostman exponent* $s$ if $\mu(B(x,r)) \le C\,r^s$ for some $C>0$ and all $x\in X,r>0$. It is easy to see that $L^q$ dimensions for large $q$ provide information about Frostman exponents:
\[lem:Lq-dim-to-Frostman-exp\] Let $\mu$ be a probability measure on a compact interval of ${\mathbb{R}}$. If $D(\mu,q)> s$ for some $q\in (1,\infty)$, then there is $r_0>0$ such that $$\mu(B(x,r)) \le \, r^{(1-1/q)s} \text{ for all } x\in{\mathbb{R}}, r\in (0,r_0].$$
If $D(\mu,q)>s$, then there is $s'>s$ such that for all large enough $m$ and each $J\in{\mathcal{D}}_m$, $$\mu(J)^q \le \sum_{I\in{\mathcal{D}}_m} \mu(I)^q \le 2^{-m (q-1) s'}.$$ Since any ball can be covered by $O(1)$ dyadic intervals of size smaller than the radius, we get that if $r$ is sufficiently small then $$\mu(B(x,r)) \le C\, r^{(1-1/q)s'},$$ where $C$ is independent of $x$ and $r$. This gives the claim.
Hence, in order to establish Theorem \[thm:infinity-dim-BCs\] it is enough to show that, under the hypotheses of the theorem, $D(\nu_{\lambda},q)=1$ for arbitrarily large $q$; and this is what we will do.
Next, we show how Frostman exponents (and therefore, also $L^q$ dimensions) of projected measures give information about the size of fibers. We recall the definition of upper box-counting (or Minkowski) dimension in a totally bounded metric space $(X,d)$. Given $A\subset X$, let $N_{\varepsilon}(A)$ denote the maximal cardinality of an ${\varepsilon}$-separated subset of $A$. The upper box-counting dimension of $A$ is then defined as $$\operatorname{{\overline{dim}_{\mathsf{B}}}}(A) = \limsup_{{\varepsilon}\downarrow 0} \frac{\log(N_{\varepsilon}(A))}{\log(1/{\varepsilon})}.$$
\[lem:Frostman-exp-to-small-fiber\] Let $X$ be a compact metric space, and suppose $\pi:X\to {\mathbb{R}}$ is a Lipschitz map. Let $\mu$ be a probability measure on $X$ such that $\mu(B(x,r)) \ge r^s$ for all $x\in X$ and all sufficiently small $r$. If $\pi\mu$ has Frostman exponent $\alpha$, then there exists $C>0$ such that for all balls $B_{\varepsilon}$ of radius ${\varepsilon}$ in ${\mathbb{R}}$, any ${\varepsilon}$-separated subset of $\pi^{-1}(B_{\varepsilon})$ has size at most $C{\varepsilon}^{-(s-\alpha)}$.
In particular, for any $y\in{\mathbb{R}}$, $$\operatorname{{\overline{dim}_{\mathsf{B}}}}(\pi^{-1}(y)) \le s-\alpha$$
Let $(x_j)_{j=1}^M$ be an ${\varepsilon}$-separated subset of $\pi^{-1}(B_{\varepsilon})$ with ${\varepsilon}$ small. Then $$\mu\left(\bigcup_{j=1}^M B(x_j,{\varepsilon}/2)\right) \ge M ({\varepsilon}/2)^s,$$ while the set in question projects onto an interval of size at most $O({\varepsilon})$. Hence $M=O({\varepsilon}^{\alpha-s})$, giving the claim.
A class of dynamically-driven self-similar measures {#subsec:DDSSM}
---------------------------------------------------
It is easy to see that in order to prove Theorem \[thm:Furstenberg\], it is enough to consider the case in which $A$ is a $p$-Cantor set and $B$ is a $q$-Cantor set, that is, $A$ is the set of points whose base $p$-expansion digits lie in some set $D_1\subset\{0,1,\ldots,p-1\}$, and likewise for $B$ and a set $D_2\subset \{0,1,\ldots,q-1\}$. Let $\Delta_i = \frac1{|D_i|} \sum_{d\in D_i} \delta_d$, and let $\eta_1$, $\eta_2$ be the distributions of the random sums $\sum_{i=1}^\infty X_i p^{-i}$, $\sum_{i=1}^\infty Y_i q^{-i}$, respectively, where $X_i$ are i.i.d. random variables with distribution $\Delta_1$, and $Y_i$ are i.i.d. random variables, also independent of the $X_i$, with distribution $\Delta_2$. Finally, set $\mu=\eta_1\times\eta_2$.
It is easy to see that $\mu(B(x,r)) = \Theta(r^{\operatorname{dim_{\mathsf{H}}}A+\operatorname{dim_{\mathsf{H}}}B})$ for $x\in{\text{supp}}(\mu)=A\times B$. Our goal is to apply Lemma \[lem:Frostman-exp-to-small-fiber\] to $\mu$ and, in light of Lemma \[lem:Lq-dim-to-Frostman-exp\], we will do this by investigating the $L^q$ dimension of projections of $\mu$. Up to a smooth change of coordinates in the parametrization, and an affine change of coordinates in the projections, the family of linear projections of $\mu$ in directions with strictly positive slope is given by $$\{ \mu_x := \eta_1 * S_{e^x} \eta_2: x\in {\mathbb{R}}\} ,$$ where $S_a(x)=ax$ scales by $a$. Note that $\mu_x$ is an infinite convolution of Bernoulli random variables, since $\eta_1,\eta_2$ are. Unlike $\eta_1,\eta_2$, the measures $\mu_x$ are not self-similar because $\eta_1,\eta_2$ are constructed with different contraction ratios. However, it is still possible to express $\mu_x$ in a way that resembles self-similarity, but with the geometry at different scales driven by a dynamical system. Namely, suppose $p<q$ and let $X=[0,\log q)$, $\mathbf{T}:X\to X$, $x\mapsto x+\log p\bmod(\log q)$. Moreover, for each $x\in X$, let $\Delta(x)$ be the finitely supported measure given by $$\Delta(x) = \left\{
\begin{array}{ll}
\Delta_1* S_{e^x}\Delta_2 & \text{if } x\in [0,\log p) \\
\Delta_1 & \text{if } x\in [\log p,\log q)
\end{array}
\right..$$ It is then easy to see that $\mu_x$ is the distribution of the random sum $\sum_{i=1}^\infty Z_i p^{-i}$, where the $Z_i$ are independent and have distribution $\Delta(\mathbf{T}^i x)$. Indeed, let $$n'(x)=|\{j\in\{1,\ldots,n\}: \mathbf{T}^j(x)\in [0,\log p)\}|.$$ Note that $$\mathbf{T}^n(x)= x+ n \log p - n'(x) \log q,$$ so that $$e^{\mathbf{T}^n(x)}p^{-n} = e^x q^{-n'(x)}.$$ Hence the distribution $\mu_{x,n}$ of $\sum_{i=1}^n Z_i p^{-i}$ is equal to the distribution of $$\sum_{i=1}^n X_i p^{-i}+ \sum_{i=1}^{n'(x)} e^x Y_i q^{-i},$$ where $X_i, Y_i$ are independent and have distribution $\Delta_1,\Delta_2$ respectively. This shows that $\mu_{x,n}\to\mu_x$ weakly.
Although in different language, this decomposition of $\mu_x$ can be traced back to Furstenberg [@Furstenberg70], and was also used more explicitly in [@NPS12] to study the $L^2$ dimensions of $\mu_x$.
Based on the above discussion, we introduce the following setup. Let $\mathcal{A}$ be the collection of all probability measures supported on a finite set, i.e. $$\mathcal{A} = \left\{ \sum_{i=1}^{N} p_i \delta(t_i) : N\in{\mathbb{N}}, p_i> 0, \sum_i p_i =1 , t_i\in{\mathbb{R}}\right\}.$$ (We denote a delta mass at $t$ either by $\delta_t$ or $\delta(t)$.) We topologize $\mathcal{A}$ in the natural way: it consists of countably many connected components, corresponding to the number of atoms $N$, and for each $N$ it inherits the topology from ${\mathbb{R}}^{2N}$.
If $\mu$ is a measure on a metric space $X$ and $f:X\to Y$ is a Borel map, then we denote by $f\mu$ the push-forward measure: $f\mu(A)=\mu(f^{-1}A))$. Fix ${\lambda}\in (0,1)$. If $\Delta_i$ is a sequence of measures in $\mathcal{A}$, all supported on a fixed compact interval, then we can form the infinite Bernoulli convolution $$\mu = *_{i=0}^\infty S_{{\lambda}^i} \Delta_i.$$ (Equivalently, $\mu$ is the distribution of the random sum $\sum_{i=0}^\infty {\lambda}^i Z_i$, where the $Z_i$ are independent and have distribution $\Delta_i$.) We are interested in the situation in which the $\Delta_i$ are generated dynamically. Let $(X,\mathbf{T})$ be a dynamical system, and suppose $\Delta:X\to \mathcal{A}$ is a map such that, for some compact interval $I_0$, ${\text{supp}}(\Delta(x))\subset I_0$ for all $x\in X$. Then we can consider the family of measures $$\label{eq:def-mu-x}
\mu_x = *_{i=0}^\infty S_{{\lambda}^i} \Delta(\mathbf{T}^i x), \quad x\in X.$$ These measures enjoy a dynamical version of self-similarity. Write $$\label{eq:def-mu-x-n}
\mu_{x,n} = *_{i=0}^{n-1} S_{{\lambda}^i} \Delta(\mathbf{T}^i x).$$ Then, clearly, $$\label{eq:mu-x-self-similarity}
\mu_x = \mu_{x,n} * S_{{\lambda}^n} \mu_{\mathbf{T}^n x}.$$
We will call the tuple $\mathcal{X}=(X,\mathbf{T},\Delta,{\lambda})$ a *model* generating the measures $\mu_x$. We will also refer to the measures $\mu_x$ themselves as *dynamically driven self-similar measures*.
Trivially, Bernoulli convolutions also fall into this setting, with $X$ the one-point space.
Lq dimensions of dynamically driven self-similar measures
---------------------------------------------------------
In order to prove Theorem \[thm:Furstenberg\] along the lines we have been describing, we need to derive estimates on the $L^q$ dimensions of $\eta_1 * S_{e^x} \eta_2$ for *all* values of $x$. As a matter of fact, by self-similarity, it is enough to deal with all $x$ in some nonempty open set, but it is not enough to gain information for almost all values of $x$. Note that the underlying dynamical system $(X,\mathbf{T})$ is an irrational rotation on the circle (thanks to $p$ and $q$ being multiplicatively independent) while, in the case of Bernoulli convolutions, $(X,\mathbf{T})$ is the trivial one-point system. In the general case of dynamically driven self-similar measures generated by a model $(X,\mathbf{T},\Delta,{\lambda})$, if one hopes to gain any information for all $x\in X$, it is reasonable to impose strong rigidity and continuity assumptions on the dynamics. The next definition, clearly satisfied by our two main classes of examples, introduces the kind of regularity that will be needed in the abstract setting. Recall that a Borel transformation $T:X\to X$ is called *uniquely ergodic* if there exists exactly one Borel probability measure ${\mathbb{P}}$ on $X$ such that $T{\mathbb{P}}={\mathbb{P}}$.
\[def:pleasant\] We say that a model $(X,\mathbf{T},\Delta,{\lambda})$ is *pleasant* if $X$ is a compact metric space, $\mathbf{T}$ is a uniquely ergodic transformation on $X$, the measures $\mu_x$ are all non-atomic and supported on some fixed bounded interval, and the map $x\mapsto \mu_{x}$ is continuous (in the weak topology), outside of a null set (with respect to the unique invariant measure).
In most of our applications, $X$ will equal either the trivial group $\{0\}$ or the circle, and in all applications $X$ will be a torus or the product of a torus and a cyclic group. In all cases, $\mathbf{T}$ will be a translation on $X$. We recall that if $X$ is a compact Abelian group, and $\mathbf{T}(x)=x+y$ is translation by $y\in X$, then $\mathbf{T}$ is uniquely ergodic if and only if the orbit $\{ n y:n\in{\mathbb{N}}\}$ is dense in $X$. See e.g. [@EinsiedlerWard11 Theorem 4.14].
We will also need to impose a separation condition, albeit an extremely weak one.
\[def:exponential-separation\] Let $\mathcal{X}=(X,\mathbf{T},\Delta,{\lambda})$ be a pleasant model with unique invariant measure $\mathbb{P}$. We say that $\mathcal{X}$ has *exponential separation* if for $\mathbb{P}$-almost all $x$ there is $R>0$ such that the following holds for infinitely many $n$: all the atoms of $\mu_{x,n}$ are distinct and ${\lambda}^{R n}$-separated. By the atoms of $\mu_{x,n}$ being distinct we mean that $$|{\text{supp}}(\mu_{x,n})| = \prod_{i=0}^{n-1} |{\text{supp}}(\Delta(\mathbf{T}^i x))|,$$ i.e. there are no exact coincidences among the atoms that make up ${\text{supp}}(\mu_{x,n})$.
This definition coincides with the notion of (lack of) super-exponential separation introduced in [@Hochman14] in the case of self-similar measures (i.e. when $X$ is a one-point set). As will become clear later, if $X$ is infinite, then under very mild non-degeneracy assumptions on the map $\Delta$, exponential separation holds almost automatically.
The following is the main result of the paper, from which more general versions of Theorems \[thm:Furstenberg\], \[thm:abc-cont-BCs\] and \[thm:infinity-dim-BCs\], as well as other applications, will follow.
\[thm:L-q-dim-dyn-ssm\] Let $(X,\mathbf{T},\Delta,{\lambda})$ be a pleasant model with exponential separation, and denote the unique invariant measure by $\mathbb{P}$. Assume further that the map $x\mapsto \Delta(x)$ is continuous ${\mathbb{P}}$-almost everywhere, and the number of atoms of $\Delta(x)$ is uniformly bounded. Then for all $q\in (1,+\infty)$ $$\label{eq:main-thm}
\lim_{m\to\infty} -\frac{\log\sum_{I\in{\mathcal{D}}_m} \mu_x(I)^q}{(q-1)m} = \min\left( \frac{\int_X \log\|\Delta(x)\|_q^q \,d\mathbb{P}(x)}{(q-1)\log{\lambda}},1\right),$$ uniformly in $x\in X$. That is, the limit in the definition of $L^q$ dimension of $\mu_x$ exists and equals the constant value on the right-hand side, for all $x\in X$.
In the above statement, and throughout the paper, the $L^q$ norm of a finitely supported measure $\Delta$ is given by $$\|\Delta\|_q^q = \sum_{y\in{\text{supp}}(\Delta)} \Delta(y)^q.$$
Outline of proof
----------------
We conclude this introduction by presenting an outline of the main steps of the proof of Theorem \[thm:L-q-dim-dyn-ssm\]. The overall strategy is inspired by the ideas of [@Hochman14]. Additional complications are caused by the fact that our model allows measures which are not strictly self-similar; this will be dealt with the help of a cocycle introduced in [@NPS12]. The key difference, however, is that Hochman’s method is based on entropy, while we need to deal with $L^q$ norms. As we will see, this forces substantial changes in the implementation of the general strategy.
At the heart of [@Hochman14] is an inverse theorem for the growth of entropy under convolutions, see [@Hochman14 Theorem 2.7]. We prove an inverse theorem for the decay of $L^q$ norms under convolutions, which may be of independent interest. This theorem is stated in Section \[sec:inverse-thm\] and proved in Section \[sec:inverse-thm-proof\]. Here we give a heuristic description. Let $\mu,\nu$ be two probability measures supported on $2^{-m}{\mathbb{Z}}\cap [0,1]$. By Young’s inequality, $\|\mu*\nu\|_q \le \|\mu\|_q$. The question the inverse theorem aims to answer is: what can be said if we are close to an equality? Here, and in the rest of the paper, “close” is meant in a very weak sense: up to some small exponential loss. More concretely, the inverse theorem asserts that if $\|\mu*\nu\|_q \ge 2^{-{\varepsilon}m} \|\mu\|_q$ for some small ${\varepsilon}>0$, then $\mu$ and $\nu$ are forced to have a multi-scale structure of a certain kind. We note that equality in Young’s theorem happens if either $\mu$ is the uniform measure on $2^{-m}{\mathbb{Z}}\cap [0,1]$, or if $\nu$ is a single atom. The inverse theorem asserts that, after restricting $\mu,\nu$ to suitable subsets $A,B$ which are “large” and “regular” in a certain sense, there is a multi-scale decomposition such that, at each scale, either $\mu|_A$ is “almost uniform” or $\nu|_B$ is “almost discrete”. In spirit this is not unlike [@Hochman14 Theorem 2.7], although the details differ substantially; see Section \[sec:inverse-thm\] below for further discussion. The two main tools in the proof of the inverse theorem come from additive combinatorics: an asymmetric version of the Balog-Szemerédi-Gowers Theorem, due to Tao and Vu, and a structure result on sets with “small” sumset, due to Bourgain. These results are recalled in Section \[sec:inverse-thm-proof\].
We note that the inverse theorem is a statement about arbitrary measures; no self-similarity is involved. Now let us consider a pleasant model $(X,\mathbf{T},\Delta,{\lambda})$ generating measures $\mu_x,x\in X$. The right-hand side in is easily seen to be an upper bound for the left-hand side (for all $x$), so the task is to show the reverse inequality. The self-similarity expressed by , in conjunction with the pleasantness of the model, can be used to show that there is a function $T:(1,\infty)\to [0,1]$, such that $\tau_{\mu_x}=T$ for ${\mathbb{P}}$-almost all $x$, and $\tau_{\mu_x}\ge T$ for *all* $x\in X$ - see Proposition \[prop:cocycle\] and Corollary \[cor:unif-continuity\]. Thus, in order to complete the proof, one needs to show that $T(q)/(q-1)$ equals the right-hand side of .
We point out that the strategy of studying $L^q$ dimensions via the function $T(q)$ is borrowed from [@NPS12]. The innovation of this work consists in being able to calculate $T(q)$ for a wider range of models and, crucially, for all finite $q\ge 1$ (while the method of [@NPS12], based on Marstrand’s projection theorem, is restricted to $q\in (1,2]$).
It is known from general considerations that $T(q)$ is concave, so in particular it is continuous and differentiable outside of at most a countable set. The rest of the proof focuses on the study of $T(q)$ for a fixed differentiability point $q$. The “multifractal structure” of a measure $\mu$ is known to behave in a regular way for points $q$ of differentiability of the spectrum $\tau_\mu$. Extending some elementary results in this direction to the function $T(q)$, we show that if $\alpha=T'(q)$ exists and $\tau_{\mu_x}(q)=T(q)$ (which we have seen happens for almost all $x$) then, for large enough $m$, “almost all” of the contribution to the sum $\sum_{I\in{\mathcal{D}}_m}\mu_x(I)^q$ comes from $\approx 2^{T^*(\alpha)m}$ intervals $I$ such that $\mu_x(I)\approx 2^{\alpha m}$; here $T^*$ is the Legendre transform of $T$. Moreover, using the self-similarity of $\mu_x$, we establish also a multi-scale version of this fact, see Proposition \[prop:Lq-over-small-set-is-small\].
Let $\mu_x^{(m)}$ supported on $2^{-m}{\mathbb{Z}}$ be given by $$\label{eq:def-mu-m}
\mu_x^{(m)}(j 2^{-m}) = \mu_x([j2^{-m},(j+1)2^{-m})).$$ Then $\mu_x^{(m)}$ is a discretization of $\mu_x$ at scale $2^{-m}$, and $\|\mu_x^{(m)}\|_q^q = \sum_{I\in{\mathcal{D}}_m} \mu_x(I)^q \lesssim 2^{-m T(q)}$. The inverse theorem, together with the study of the multifractal structure of $\mu_x$, is used to show that either $T(q)=q-1$ (in which case we are done) or, otherwise, the following holds: if $\rho$ is an arbitrary measure supported on $2^{-m}{\mathbb{Z}}\cap [0,1]$ such that $\|\rho\|_q \le 2^{-\sigma m}$, then $$\label{eq:outline-conv-flattens-mu-x}
\|\rho*\mu_x^{(m)}\|_q^q \le 2^{-{\varepsilon}m} 2^{-T(q)m} \quad\text{for all }x\in X,$$ where ${\varepsilon}={\varepsilon}(\sigma,q)>0$. That is, convolving $\mu_x$ with $\rho$ results in an exponential flattening of the $L^q$ norm (a priori this is not necessarily true for all $x$, since $\|\mu_x^{(m)}\|_q^q$ can be far smaller than $2^{-T(q)m}$ for some $x$, but all that is needed later is an exponential gain over $2^{-T(q)m}$). The heuristic reason for this is the following: suppose the opposite is true. The inverse theorem then asserts that there is a regular subset $A$ of ${\text{supp}}(\mu_x^{(m)})$ which captures much of the $L^q$ norm. By the inverse theorem, and since $\rho$ is assumed to have exponentially small $L^q$ norm, $A$ must have almost full growth (or branching) on a positive density set of scales in a multi-scale decomposition. But $A$ itself does not have full growth (this follows from the assumption $T(q)<q-1$, which rules out $\mu_x^{(m)}$ having too small $L^q$ norm). So there must also be a positive density set of scales on which $A$ has smaller than average growth. The regularity of the multifractal spectrum discussed above rules this out, since it forces $A$ to have an almost constant growth on almost all scales.
The conclusion of the proof of Theorem \[thm:L-q-dim-dyn-ssm\] from goes along the same lines of [@Hochman14]. By the exponential separation assumption, there is $x\in X$ such that $\tau_{\mu_x}(q)=T(q)$ and, for some $R=R(x)\in {\mathbb{N}}$, $$\frac{\log \|\mu_{x,n}^{(R m)}\|_q^q}{(q-1)n\log(1/{\lambda})} = \frac{\|\mu_{x,n}\|_q^q}{(q-1)n\log(1/{\lambda})} = \frac{\sum_{i=0}^{n-1} \log\|\Delta(\mathbf{T}^i x)\|_q^q}{(q-1)n\log(1/{\lambda})},$$ where $m=m(n)$ is chosen so that $2^{-m}\sim {\lambda}^n$. Under our running assumption that $T(q)<q-1$, the ergodic theorem for uniquely ergodic systems implies that the right-hand side above tends to the right-hand side of as $n\to\infty$. Hence, it remains to show that $$\label{eq:outline-Lq-norm-scale-Rm}
\lim_{n\to\infty}\frac{\log \|\mu_{x,n}^{(R m)}\|_q^q}{n\log(1/{\lambda})} = T(q).$$ In other words, we need to show that the $L^q$ norm of $\mu_{x,n}$ at scale $2^{-m}\approx {\lambda}^n$ (which is easily seen to be comparable to the $L^q$ norm of $\mu_x$ at scale $2^{-m}$) nearly exhausts the $L^q$ norm of $\mu_{x,n}$ at the much finer scale $2^{-R m}$ which, in turn, equals the full $L^q$ norm of $\mu_{x,n}$, by the exponential separation assumption.
To show , we recall that $\mu_x = \mu_{x,n}* S_{{\lambda}^n}\mu_{\mathbf{T}^n x}$, and use this to decompose $$\mu_{x}^{((R+1) m)} = \sum_{I\in{\mathcal{D}}_m} \mu(I) {\widetilde}{\rho}_I * S_{{\lambda}^n}\mu_{\mathbf{T}^n x},$$ where ${\widetilde}{\rho}_I$ is the normalized restriction of $\mu_{x,n}$ to $I$. Since the supports of ${\widetilde}{\rho}_I * S_{{\lambda}^n}\mu_{\mathbf{T}^n x}$ have bounded overlap, it is not hard to deduce that $$\|\mu_{x}^{((R+1) m)}\|_q^q \approx \sum_{I\in{\mathcal{D}}_m} \mu_x(I)^q \| \rho_I * \mu_{\mathbf{T}^n x}^{(R m)}\|_q^q,$$ where $\rho_I = S_{{\lambda}^{-n}}{\widetilde}{\rho}_I$. This is the point where we apply , to conclude that if on the right-hand side above we only add over those $I$ such that $\|\rho_I\|_q \ge 2^{-\sigma q}$, where $\sigma>0$ is arbitrary, then, provided $n$ is large enough depending on $\sigma$, we still capture almost all of the left-hand side. This follows since can be shown to imply that the contribution of the remaining $I$ is exponentially smaller than the left-hand side. A similar calculation, now with $\mu_{x,n}^{((R+1)m)}$ in place of $\mu_{x}^{((R+1) m)}$ in the left-hand side, then shows that holds, finishing the proof.
We point out that, simultaneously and independently of this work, Meng Wu [@Wu16] obtained an elegant alternative proof of Theorem \[thm:Furstenberg\]. Wu’s proof is purely ergodic-theoretical and completely different from ours. His methods do not seem to yield any analogs of Theorem \[thm:L-q-dim-dyn-ssm\] and, in particular, are unable to reproduce our results on the dimensions and densities of Bernoulli convolutions. Nevertheless, some of our concrete applications (besides Furstenberg’s conjecture) also follow from Wu’s approach: this is the case for Corollaries \[cor:dim-intersection-ssm\] and \[cor:dim-intersection-sss-lines\].
Organization of the paper and summary of applications
-----------------------------------------------------
We outline the organization of the rest of the paper. Sections \[sec:inverse-thm\]–\[sec:proof-of-main-thm\] are devoted to the proof of Theorem \[thm:L-q-dim-dyn-ssm\], while the remaining Sections \[sec:dim-ssm-and-applications\]–\[sec:abs-cont\] contain the applications of Theorem \[thm:L-q-dim-dyn-ssm\]. More precisely:
- In Section \[sec:inverse-thm\] we state and discuss the inverse theorem for the $L^q$ norms of convolutions of discrete measures. The inverse Theorem is proved in Section \[sec:inverse-thm-proof\].
- Section \[sec:properties-dyn-ssm\] develops some properties of dynamically driven self-similar measures. In Section \[sec:proof-of-main-thm\], these are combined with the inverse theorem to conclude the proof of Theorem \[thm:L-q-dim-dyn-ssm\].
- In Section \[sec:dim-ssm-and-applications\] we apply Theorem \[thm:L-q-dim-dyn-ssm\] to study $L^q$ dimensions and Frostman exponents of self-similar measures on the line. In particular, we prove Theorem \[thm:dim-ssm\], which generalizes Theorem \[thm:infinity-dim-BCs\] to homogeneous self-similar measures on ${\mathbb{R}}$, and Theorem \[thm:dim-general-ssm\], which extends this to arbitrary self-similar measures on the line (not necessarily homogeneous).
- In Section \[sec:convolutions-and-Furst-conj\], we conclude the proof of Theorem \[thm:Furstenberg\]. We also study the $L^q$ dimensions of convolutions of self-similar measures (Theorems \[thm:dim-conv-ssm\] and \[thm:dim-conv-many-ssm\]), and deduce a variant of Furstenberg’s conjecture for self-similar sets, Corollary \[cor:dim-intersection-ssm\].
- Section \[sec:planar-sss\] contains further applications of Theorem \[thm:L-q-dim-dyn-ssm\] to projections and sections of planar self-similar sets and measures. In particular, we prove an upper bound for the dimensions of arbitrary linear sections of some self-similar sets on the plane, see Corollary \[cor:dim-intersection-sss-lines\].
- Finally, in Section \[sec:abs-cont\] we turn our focus to the densities of the measures studied in the previous sections. We present a general result in the framework of dynamically defined measures, Theorem \[thm:abs-continuity\], and deduce Theorem \[thm:abc-cont-BCs\], as well as several other applications, as corollaries.
Notation
--------
We use Landau’s $O(\cdot)$ and related notation: if $X,Y$ are two positive quantities, then $Y=O(X)$ means that $Y\le C X$ for some constant $C>0$, while $Y=\Omega(X)$ means that $X=O(Y)$, and $Y=\Theta(X)$ that $Y=O(X)$ and $X=O(Y)$. If the constant $C$ is allowed to depend on some parameters, these are often denoted by subscripts. For example, $Y=O_q(X)$ means that $Y\le C(q) X$, where $C(q)$ is a function depending on the parameter $q$.
The following table summarizes some of the notational conventions to be used throughout the paper.
------------------------------------------------ -- -------------------------------------------------------
${\mathbb{N}}$ Natural numbers $\{1,2,\ldots\}$
$B(x,r)$ Open ball of center $x$, radius $r$.
$\operatorname{dim_{\mathsf{H}}}$ Hausdorff dimension
$\operatorname{{\overline{dim}_{\mathsf{B}}}}$ Upper box-counting dimension
$[n]$ $\{0,1,\ldots,n-1\}$
$\delta,{\varepsilon},\eta,\kappa,\sigma$ Small positive numbers
$\mu,\nu,\eta,\rho$ Measures (always positive and finite, often discrete)
$\mu^{(m)}$ Discretization of $\mu$ at scale $2^{-m}$
$\|\cdot\|_q$ Discrete $L^q$ norm
$q'$ Dual exponent to $q$
$\delta_t,\delta(t)$ Delta mass at $t$
$\mathcal{A}$ Space of finitely supported measures
$\Delta_i, {\widetilde}{\Delta}$ Elements of $\mathcal{A}$
$\Delta(x)$ $\mathcal{A}$-valued functions
$(X,\mathbf{T},\Delta,{\lambda})$ A model generating DDSSMs
$\mu_x$ The DDSSM corresponding to $x\in X$
$\mu_{x,n}$ Discrete approximations to $\mu_x$
$S_{\lambda}$ Map that scales by ${\lambda}$
$\tau(\mu,q)$ or $\tau_\mu(q)$ $L^q$ spectrum
$D(\mu,q)$ or $D_\mu(q)$ $L^q$ dimension
$\mathcal{E},\mathcal{E}_i$ Small exceptional sets
$\mathcal{D}_s$ Dyadic intervals of length $2^{-s}$
$\mathcal{D}_s(A)$ Elements of $\mathcal{D}_s$ hitting $A$
$\mathcal{N}_s(A)$ or $\mathcal{N}(A,s)$ $|\mathcal{D}_s(A)|$
$D$ $2^D$=base for tree representation of sets
$\ell$ Height of tree representing a set
$\mathcal{S},\mathcal{S}',\mathcal{S}_i$ Subsets of $[\ell]$ (representing sets of scales)
$R_s, R'_s, R''_s$ Branching numbers of trees representing regular sets
$T(q)$ The function from Proposition \[prop:cocycle\]
$f,g,h$ Maps ${\mathbb{R}}\to{\mathbb{R}}$, often affine
$(f_i)_{i\in\mathcal{I}}$ Iterated function system of similarities
------------------------------------------------ -- -------------------------------------------------------
**Acknowledgments**. I am grateful to Mike Hochman and Izabella [Ł]{}aba for inspiring discussion related to the themes in this paper, and to Eino Rossi and Julien Barral for a careful reading and for suggesting numerous small corrections.
An inverse theorem for the decay of Lq norms under convolution {#sec:inverse-thm}
==============================================================
Let $\mu,\nu$ be probability measures on ${\mathbb{R}}$ (or the circle ${\mathbb{R}}/{\mathbb{Z}}$). For any reasonable notion of smoothness, the convolution $\mu*\nu$ is at least as smooth as $\nu$. A natural question is then: if $\mu*\nu$ is not “much smoother” than $\mu$, can we deduce any information about the measures $\mu$ and $\nu$? Of course, this depends on the notion of smoothness under consideration, and on the precise meaning of “much smoother”.
We are interested in general, possibly fractal, measures, and their discrete approximations. A general method for defining notions of dimension (or smoothness) of a measure is to discretize it at a certain scale ${\varepsilon}$, measure smoothness at that scale in some standard way (for example, by means of entropy or $L^q$ norms) and then study the growth/decay of this quantity as ${\varepsilon}\downarrow 0$. Indeed, we have seen that $L^q$ dimensions are defined precisely in this way, and there is a parallel notion for entropy.
Let $\mu$ be a probability measure on ${\mathbb{R}}/{\mathbb{Z}}$. Its normalized level $m$ entropy is $$H_m(\mu) = \frac{1}{m}\sum_{I\in\mathcal{D}_m} -\mu(I)\log(\mu(I)),$$ with the usual convention $0\log 0=0$. In [@Hochman14 Theorem 2.7], Hochman showed that if $$H_m(\nu*\mu) \le H_m(\mu)+{\varepsilon},$$ where ${\varepsilon}>0$ is small, then $\nu$ and $\mu$ have a certain structure which, very roughly, is of this form: the set of dyadic scales $0\le s<m$ can be split into three sets $\mathcal{A}\cup \mathcal{B}\cup \mathcal{C}$. At scales in $\mathcal{A}$, the measure $\nu$ looks “roughly atomic”, at scales in $\mathcal{B}$ the measure $\mu$ looks “roughly uniform”, and the set $\mathcal{C}$ is small. This theorem was motivated in part by its applications to the dimension theory of self-similar measures, as discussed above.
Our goal is to develop a corresponding theory for $L^q$ norms. Given $m\in{\mathbb{N}}$, we will say that $\mu$ is a $2^{-m}$-measure if $\mu$ is a probability measure supported on $2^{-m}{\mathbb{Z}}\cap [0,1)$ (and we sometimes identify $[0,1)$ with the circle). Recall from that if $\mu$ is a probability measure on $[0,1)$, we denote by $\mu^{(m)}$ the associated $2^{-m}$-measure, that is, $\mu^{(m)}(j 2^{-m}) = \mu([j 2^{-m},(j+1)2^{-m}))$. We also recall that, given a purely atomic measure $\mu$, we define the $L^q$ norms $$\|\mu\|_q = \left(\sum \mu(y)^q\right)^{1/q},$$ and $\|\mu\|_\infty= \max_y \mu(y)$.
From now on, the convolutions are always assumed to take place on the circle unless otherwise indicated; however, all results immediately transfer to the real line, using the fact that the map $(x,y)\mapsto x+y$ is two-to-one on the circle so, for example, if $\mu,\nu$ are $2^{-m}$-measures, then the $L^q$ norms of $\mu*\nu$ as convolutions on the circle or the real line are comparable up to a multiplicative constant.
By Young’s inequality (which in this context is a direct consequence the convexity of $t\mapsto t^q$), we know that $\|\mu*\nu\|_q\le \|\mu\|_q$, for any $q\ge 1$. We aim to understand under what circumstances $\|\mu*\nu\|_q \approx \|\mu\|_q$, where the closeness is in a weak, exponential sense. More precisely, we are interested in what structural properties of the measures $\mu,\nu$ ensure an exponential flattening of the $L^q$ norm of the form $$\label{eq:intro-L2-norm-flattens}
\|(\mu*\nu)^{(m)}\|_q \le 2^{-{\varepsilon}m} \|\mu^{(m)}\|_q.$$ The Balog-Szemerédi-Gowers Theorem (particularly, its asymmetric formulation, see Theorem \[thm:BSG\] below) can be seen as providing a partial answer in a special case, i.e. when $\mu^{(m)},\nu^{(m)}$ are indicator functions.
While we are not aware of any general results in this direction, we note that a special case has received considerable attention: if $A\subset 2^{-m}{\mathbb{Z}}$, then $\|\mathbf{1}_A*\mathbf{1}_A\|_2^2$ is nothing but the additive energy of $A$ (see below), and estimates of the form $$\|\mathbf{1}_A*\mathbf{1}_A\|_2^2 \le |A|^{3-{\varepsilon}}$$ arise repeatedly in dynamics, combinatorics and analysis: see e.g. [@DyatlovZahl16; @ALL16] for some recent examples. In particular, S. Dyatlov and J. Zahl [@DyatlovZahl16 Theorem 6] showed that if $\mu$ is an Ahlfors-regular measure, that is, if there are $C,s>0$ such that $$C^{-1} r^s \le \mu(B(x,r)) \le C r^s \quad\text{for all }x\in{\text{supp}}(\mu), r\in (0,1],$$ then $$\|(\mu*\mu)^{(m)}\| \le 2^{-{\varepsilon}m}\|\mu^{(m)}\|_2,$$ where ${\varepsilon}>0$ depends only on the parameters $C,s$. Their proof does not appear to readily extend to the convolution of two different measures, or beyond the Ahlfors-regular case. Outside of the Euclidean setting, the $L^2$ norm of self-convolutions has been studied in many groups as part of the Bourgain-Gamburd expansion machine developed to prove that Cayley graphs are expanders, see e.g. [@BourgainGamburd08].
Here we go in a different direction, by investigating general geometric conditions on the measures $\mu,\nu$ that ensure flattening in the sense of . We make the trivial observation that if $\nu=\delta_{k 2^{-m}}$ or $\mu={\lambda}=$Lebesgue measure on ${\mathbb{R}}/{\mathbb{Z}}$, then $\|(\mu*\nu)^{(m)}\|_q = \|\mu^{(m)}\|_q$. Furthermore, if $\nu=2^{-{\varepsilon}m}\delta_x + (1- 2^{{\varepsilon}m})\lambda$ and $\mu$ is an arbitrary measure, then we still have $\|(\mu*\nu)^{(m)}\|_q \ge 2^{-{\varepsilon}m}\|\mu^{(m)}\|_q$. This shows that a subset of measure $2^{-{\varepsilon}m}$ is able to prevent smoothening in the sense of , so that (unlike the case of entropy) in order to guarantee exponential smoothing we need to impose conditions on the structure of the measures inside sets of exponentially small measure.
There are also less trivial situations in which $\|\mu*\mu\|_q\approx \|\mu\|_q$. Let $D\gg 1$ be a large integer, fix $\ell\gg D$, and for given subset $\mathcal{S}$ of $\{0,\ldots,\ell-1\}$ define $A$ as the set of all $x\in 2^{-\ell D}{\mathbb{Z}}\cap [0,1)$, such that the $s$-th digit in the $2^{-D}$-base expansion of $x$ is $0$ for all $s\in\mathcal{S}$ (and is arbitrary otherwise). Then it is not hard to check that $\|\mathbf{1}_A*\mathbf{1}_A\|_q\approx \|\mathbf{1}_A\|_1\|\mathbf{1}_A\|_q$. In more combinatorial terms, $A$ looks like an arithmetic progression at all scales. In similar ways one constructs probability measures $\mu,\nu$ supported on sets of widely different sizes, such that $\|\mu*\nu\|_q\approx \|\mu\|_q$.
Our inverse theorem asserts that if fails to hold then one can find subsets $A\subset {\text{supp}}(\mu)$ and $B\subset{\text{supp}}(\nu)$, such that $A$ captures a “large” proportion of the $L^q$ norm of $\mu$ and $B$ a “large” proportion of the mass of $\nu$, and moreover $\mu|_A,\nu|_B$ are fairly regular (for example, they are constant up to a factor of $2$). The main conclusion, however, is that $A$ and $B$ have a structure resembling the example above, and also the conclusion of Hochman’s inverse theorem for entropy: if $D$ is a large enough integer, then for each $s$, either $B$ has no branching between scales $2^{sD}$ and $2^{(s+1)D}$ (in other words, once the first $s$ digits in the $2^D$-adic expansion of $y\in B$ are fixed, the next digit is uniquely determined), or $A$ has nearly full branching between scales $2^{sD}$ and $2^{(s+1)D}$ (whatever the first $s$ digits of $x\in A$ in the $2^D$-adic expansion, the next digit can take almost any value).
Before stating the theorem, we summarize our notation for dyadic intervals to be used throughout the paper (some of it was introduced before):
- ${\mathcal{D}}_s$ is the family of dyadic intervals $[j 2^{-s},(j+1) 2^{-s})$. We also refer to elements of ${\mathcal{D}}_s$ as $2^{-s}$-intervals.
- Given a set $A\subset{\mathbb{R}}$ or ${\mathbb{R}}/{\mathbb{Z}}$, we write ${\mathcal{D}}_s(A)$ for the family of $2^{-s}$-intervals that hit $A$. We also write $\mathcal{N}(A,s)$ or ${\mathcal{N}}_s(A)$ for $|{\mathcal{D}}_s(A)|$, i.e. the number of $2^{-s}$ intervals that hit $A$.
- Given $x\in{\mathbb{R}}$ or ${\mathbb{R}}/{\mathbb{Z}}$, we write ${\mathcal{D}}_s(x)$ for the only $2^{-s}$-interval that contains $x$.
- We write $a I$ for the interval of the same center as $I$ and length $a$ times the length of $I$.
We also write $[\ell]=\{0,1,\ldots,\ell-1\}$.
\[thm:inverse-thm\] Given $q\in (1,\infty)$, $\delta>0$ and $D_0\in{\mathbb{N}}$, there are ${\varepsilon}>0$, $D\ge D_0$, such that the following holds for all large enough $\ell$.
Let $m=\ell D$, and let $\mu,\nu$ be $2^{-m}$-measures such that $$\|\mu*\nu\|_q \ge 2^{-{\varepsilon}m}\|\mu\|_q.$$ After translating the measures $\mu,\nu$ by appropriate numbers of the form $k 2^{-m}$, there exist sets $A\subset {\text{supp}}(\mu), B\subset {\text{supp}}(\nu)$, such that:
1. $\|\mu|_A\|_q \ge 2^{-\delta m}\|\mu\|_q$.
2. $\mu(y)\le 2\mu(x)$ for all $x,y\in A$.
3. There is a sequence $R'_s$, $s\in [\ell]$, such that ${\mathcal{N}}_{(s+1)D}(A\cap I)=R'_s$ for all $I\in {\mathcal{D}}_{s D}(A)$.
4. $x\in \frac12 {\mathcal{D}}_{s D}(x)$ for every $x\in A, s\in [\ell]$.
5. $\nu(B)\ge 2^{-\delta m}$.
6. $\nu(y)\le 2\nu(x)$ for all $x,y\in B$.
7. There is a sequence $R''_s$, $s\in [\ell]$, such that ${\mathcal{N}}_{(s+1)D}(B\cap I)=R''_s$ for all $I\in{\mathcal{D}}_{s D}(B)$.
8. $y\in \frac12 {\mathcal{D}}_{s D}(y)$ for every $y\in B, s\in [\ell]$.
Moreover,
1. For each $s$, either $R''_s=1$, or $$\label{eq:scales-with-full-branching}
R'_s \ge 2^{(1-\delta)D}.$$
2. Let $\mathcal{S}$ be the set of $s$ such that holds. Then $$\log(\|\nu\|_q^{-q'}) - \delta m \le D |\mathcal{S}| \le \log(\|\mu\|_q^{-q'})+\delta m.$$
Here, and throughout the paper, $q'=q/(q-1)$ denotes the dual exponent. We make some remarks on the statement.
1. The initial translation of the measures, as well as their convolution, take place on the circle. However, by decomposing the measures into finitely many pieces it is easy to deduce the same statement with both the translation and the convolution taking place on the real line.
2. The translation is only needed for (A-iv) and (B-iv), which are technical claims that we include in the theorem as they are often useful in applications.
3. The main claim in the theorem is part (v). Obtaining sets $A,B$ satisfying (A-i)–(A-iv) and (B-i)–(B-iv) is not hard, and (vi) is a straightforward calculation using (v).
4. The theorem fails for $q=1$ and $q=\infty$. In the first case there is an equality $\|\mu*\nu\|_1=\|\mu\|_1$ for any $2^{-m}$-measures, and in the second case there is always an equality $\|{\mathbf{1}}_A*{\mathbf{1}}_{-A}\|_\infty= \|{\mathbf{1}}_A\|_1$. On the other hand, the case of arbitrary $1<q<\infty$ is easily reduced to the case $q=2$: see Lemma \[lem:q-to-2\] below.
We emphasize that the proof of Theorem \[thm:inverse-thm\] (including the proofs of the results it relies on) is elementary, in particular avoiding any use of the Fourier transform or quantitative probabilistic estimates such as the Berry-Esseen Theorem, which is crucial in the approach of [@Hochman14]. The value of ${\varepsilon}$ is effective in principle, although it is certainly very poor; the worst loss occurs in the application of the asymmetric Balog-Szemerédi-Gowers Theorem (Theorem \[thm:BSG\] below).
Proof of the inverse theorem {#sec:inverse-thm-proof}
============================
Preliminaries
-------------
In this section we prove Theorem \[thm:inverse-thm\]. We begin by describing the two main tools involved in the proof: a version of the Balog-Szemerédi-Gowers Theorem that is effective even when the sets have very different sizes, due to Tao and Vu, and the additive part of Bourgain’s discretized sum-product theorem. We begin with the latter.
We say that $A\subset [0,1]$ or ${\mathbb{R}}/{\mathbb{Z}}$ is a $2^{-m}$-set if each element of $A$ is an integer multiple of $2^{-m}$. For a finite set $A\subset{\mathbb{R}}$, we define its doubling constant as $\sigma[A]=|A+A|/|A|$. We will call a $2^{-m}$ set $A$ such that $\sigma[A]\le 2^{\delta m}$ an **$(m,\delta)$-small doubling set**.
The structure of sets $A$ such that $\sigma[A]\le K$ (where $K$ is independent of $|A|$) is characterized by Freiman’s Theorem (see e.g. [@TaoVu10 Theorem 5.32]): such sets can be densely embedded in a generalized arithmetic progression. However Freiman’s Theorem gives no information when the doubling constant grows exponentially with the size of the set. The following structural property of sets with small exponential doubling is proved by Bourgain [@Bourgain10]. Although it is not explicitly stated in [@Bourgain10], this theorem emerges from the constructions in Sections 2 and 3, in particular see [@Bourgain10 Equations (3.15), (3.20), (3.21), (3.22)].
\[thm:Bourgain\] Given a large $T\in{\mathbb{N}}$, the following holds for sufficiently large $m_1\in {\mathbb{N}}$ (depending on $T$).
Let $m=m_1 T$, and suppose $H$ is a $(m,2^{-2T-1})$-small doubling set. Then $H$ contains a subset $H'$ such that the following holds:
1. $|H'|\ge 2^{-(2 \log T/\sqrt{T} )m} |H|$.
2. There are a set $\mathcal{S}\subset \{0, \ldots,m_1-1\}$ and integers $R_s, n_s$, $s\in\mathcal{S}$, with $n_s\in [s T, (s+1)T)$, such that:
1. If $s\notin\mathcal{S}$, then $\mathcal{N}(H'\cap I,(s+1)T)= 1$ for each $I\in\mathcal{D}_{sT}(H')$.
2. If $s\in\mathcal{S}$, then $\mathcal{N}(H'\cap I,n_s)=R_s$ for each $I\in\mathcal{D}_{sT}(H')$, and $\mathcal{N}(H'\cap J,(s+1)T)= 1$ for each $J\in\mathcal{D}_{n_s}(H')$.
3. $2^{(1-T^{-1/2})(n_s-sT)} < R_s \le 2^{n_s - sT}$ for all $s\in\mathcal{S}$.
In particular, $|H'|=\prod_{s\in\mathcal{S}} R_s$.
Thus, the theorem says that a set with small exponential doubling contains a fairly dense subset which has no branching between the scales $2^{-Ts}$ and $2^{-(T+1)s}$ for $s\notin\mathcal{S}$ and between the scales $2^{-n_s}$ and $2^{-(T+1)s}$ for $s\in\mathcal{S}$; and has “uniform and nearly full branching” between the scales $2^{-Ts}$ and $2^{n_s}$, $s\in\mathcal{S}$.
We remark that the proof of Theorem \[thm:Bourgain\] is ingenious but elementary, only relying on the Plünnecke-Ruzsa inequalities, for which a short elementary proof was recently found by Petridis [@Petridis11].
Another crucial ingredient in the proof of Theorem \[thm:inverse-thm\] is the following version of the celebrated Balog-Szemerédi-Gowers Theorem, due to Tao and Vu [@TaoVu10], which allows the sets to have widely different sizes. Recall that the additive energy between two finite sets $A,B$ in a common ambient group is $$\label{eq:def-additive-energy}
E(A,B) = |\{ (a_1,a_2,b_1,b_2)\in A^2\times B^2: a_1+b_1=a_2+b_2 \}| = \|{\mathbf{1}}_A*{\mathbf{1}}_B\|_2^2.$$
\[thm:BSG\] Given $\kappa>0$, there is $\tau>0$ such that the following holds for $m\in{\mathbb{N}}$ large enough. Let $A,B\subset[0,1]$ or ${\mathbb{R}}/{\mathbb{Z}}$ be $2^{-m}$-sets such that $$E(A,B) \ge 2^{-\tau m} |A| |B|^2 =2^{-\tau m} \|\mathbf{1}_B\|_1^2 \|\mathbf{1}_A\|_2^2.$$ Then there are a $(m,\kappa)$-small doubling set $H$ and a $2^{-m}$-set $X$ such that:
1. $|A\cap (X+H)| \ge 2^{-\kappa m} |A|\ge 2^{-2\kappa m}|X||H|$,
2. $|B\cap H| \ge 2^{-\kappa m} |B|$.
This follows from [@TaoVu10 Theorem 3.5]. Indeed, take $L=2^m$, $\alpha=2^{-\tau m}/2$, ${\varepsilon}=\kappa/4$. Then by making $\tau>0$ small enough in terms of $\kappa$, we can ensure that $$\Omega_\kappa \left(\alpha^{O_\kappa(1)} L^{-\kappa/4}\right)\ge \Omega_\kappa(2^{-(\kappa/2)m}) \ge 2^{-\kappa m}$$ if $\delta$ is small enough and $m$ large enough in terms of $\kappa$.
The proof of Theorem \[thm:BSG\] is also elementary, although it is rather lengthy.
Overview
--------
We give a rough sketch of the proof of the inverse theorem. Our goal is to apply the asymmetric Balog-Szemerédi-Gowers Theorem, Theorem \[thm:BSG\]. In §\[subsec:analytical-lemmas\] we present two lemmas involving $L^q$ norms. Recall that our assumption is that $\|\mu*\nu\|_q\ge 2^{-{\varepsilon}m}\|\mu\|_q$. In Lemma \[lem:non-flattening-to-subset\] we extract two sets $A,B$, which already satisfy properties (A-i), (A-ii), (B-i), (B-ii), and such that similar bounds hold for their indicator functions. Lemma \[lem:q-to-2\] (a simple application of Hölder’s inequality) shows that one can pass from the $L^q$ norm to the $L^2$ norm, enabling the application of Theorem \[thm:BSG\].
In §\[subsec:combinatorial-lemmas\], we present some combinatorial regularization lemmas, inspired in [@Bourgain10]. Theorem \[thm:BSG\] produces a set $H$ of small exponential doubling such that $B+H$ is not much larger than $H$ and $A+H$ is not much larger than $A$. Together with the information on the structure of $H$ provided by Theorem \[thm:Bourgain\] (or, rather, the version given by Corollary \[cor:Bourgain\] below), and with the lemmas in §\[subsec:combinatorial-lemmas\], this allows us to deduce the remaining properties of $A$ and $B$ (after passing to suitable dense subsets).
Finally, (vi) is a straightforward consequence of the previous claims.
A point of notation: throughout this section, $\ell$ and $m$ will denote sufficiently large integers (given any other relevant data); any inequalities involving them are understood to hold if they are larger than a constant that is allowed to depend on any other parameters involved.
Analytical lemmas {#subsec:analytical-lemmas}
-----------------
We begin with a lemma, based on Young’s inequality and dyadic pigeonholing, that enables the use of the Balog-Szemerédi-Gowers Theorem. It is an $L^q$ asymmetric version of (the proof of) [@BourgainGamburd08 Proposition 2].
\[lem:non-flattening-to-subset\] Given ${\varepsilon}>0$ and $q\in (1,\infty)$, the following holds for large enough $m\in{\mathbb{N}}$. Suppose $\mu,\nu$ are $2^m$-measures satisfying $\|\mu*\nu\|_q \ge 2^{-{\varepsilon}m}\|\mu\|_q$. Then there exist $j, j'\le 2{\varepsilon}q' m$ such that, setting $$\begin{aligned}
A &= \{ x: 2^{-j-1}\|\mu\|_q^{q'} < \mu(x) \le 2^{-j}\|\mu\|_q^{q'}\},\\
B &= \{ y: 2^{-j'-1}2^{-m} < \nu(y) \le 2^{-j'}2^{-m}\},\end{aligned}$$ the following holds:
1. $\|\mathbf{1}_A*\mathbf{1}_B\|_q \ge 2^{-2{\varepsilon}m}\|\mathbf{1}_A\|_q\|\mathbf{1}_B\|_1$,
2. $\|\mu|_A\|_q \ge 2^{-2{\varepsilon}m} \|\mu\|_q$,
3. $\|\nu|_B\|_1 =\nu(B) \ge 2^{-2{\varepsilon}m}$.
We use the notation $X\gtrsim Y$ to mean $X \ge C^{-1}m^{-C} Y$, where $C>0$ depends on $q$ only. For $j\in{\mathbb{Z}}$, let $$\begin{aligned}
A_j &= \{ x: 2^{-j-1}\|\mu\|_q^{q'} < \mu(x) \le 2^{-j}\|\mu\|_q^{q'}\},\\
B_j &= \{ y: 2^{-j-1}2^{-m} < \nu(y) \le 2^{-j}2^{-m}\}.\end{aligned}$$ Firstly, note that $A_j=\varnothing, B_j=\varnothing$ if $j\le -(m+1)$ since, by Hölder’s inequality, $$1= \sum_x \mu(x) \le \|\mu\|_q 2^{m/q'} \,\Longrightarrow \, \|\mu\|_q^{q'} \ge 2^{-m}.$$ Write $\ell =\lceil 2{\varepsilon}q' m\rceil$, and let $E=\cup_{j\ge\ell} A_j, F=\cup_{j\ge\ell} B_j$. Note that $$\begin{aligned}
\|\mu|_E\|_q^q &\le \left(\max_{x\in E}\mu(x)^{q-1}\right) \sum_{x\in E}\mu(x) \le 2^{-\ell(q-1)}\|\mu\|_q^q,\\
\|\nu|_F\|_1 &= \nu(F) \le 2^m 2^{-\ell-m} = 2^{-\ell}.\end{aligned}$$ By Young’s inequality, $$\max(\|\mu|_E *\nu\|_q,\|\mu*\nu|_F\|_q) \le 2^{-\ell/q'} \|\mu\|_q \le 2^{-{\varepsilon}m} \|\mu*\nu\|_q.$$ It follows from the bilinearity of convolution and the triangle inequality that, if $m\gg_{\varepsilon}1$, $$\sum_{- m\le j,j'<\ell} \|\mu|_{A_j}*\nu|_{B_{j'}}\|_q \ge \frac12 \|\mu*\nu\|_q.$$ Pigeonholing and applying Young’s inequality once again, we can pick $j,j'<\ell$ such that, setting $A=A_j, B=B_{j'}$, we have $$\|\mu|_A\|_q \|\nu|_B\|_1 \ge \|\mu|_A *\nu|_B\|_q \gtrsim \|\mu*\nu\|_q \ge 2^{-{\varepsilon}m}\|\mu\|_q $$ From here it follows that $\|\nu|_B\|_1 \gtrsim 2^{-{\varepsilon}m}$ and $\|\mu|_A\|_q \gtrsim 2^{-{\varepsilon}m}\|\mu\|_q$. Note that $2^{j'+m}\gtrsim |B|$. We conclude that $$\begin{aligned}
\|{\mathbf{1}}_A*{\mathbf{1}}_B\|_q &\gtrsim (2^{j}\|\mu\|_q^{-q'} 2^{j'+m}) \|\mu|_A*\nu|_B\|_q \\
&\gtrsim (2^{j}\|\mu\|_q^{-q'} 2^{j'+m}) 2^{-{\varepsilon}m}\|\mu|_A\|_q \\
&\gtrsim 2^{-{\varepsilon}m} \|{\mathbf{1}}_A\|_q \|{\mathbf{1}}_B\|_1.\end{aligned}$$
The following simple consequence of Hölder’s inequality will allow us to apply the Balog-Szmerédi-Gowers also in the context of $L^q$ norms, $q\in (1,+\infty)$.
\[lem:q-to-2\] Let $A,B$ be two $2^{-m}$-sets and let $q\in (1,\infty)$. If $\|{\mathbf{1}}_A *{\mathbf{1}}_B\|_q\ge 2^{-\kappa m}|A|^{1/q}|B|$, then $$\|{\mathbf{1}}_A *{\mathbf{1}}_B\|_2^2 \ge 2^{-(\max(q,q'))\kappa m} |A||B|^2.$$
Consider first the case $q\in (1,2)$. Applying Hölder’s inequality in the form $$\sum_x f(x)^q = \sum_x f(x)^{2-q} f(x)^{2(q-1)} \le \left(\sum_x f(x)\right)^{2-q} \left(\sum_x f(x)^2\right)^{q-1}$$ to $f={\mathbf{1}}_A*{\mathbf{1}}_B$ yields $$\|{\mathbf{1}}_A*{\mathbf{1}}_B\|_q^q \le |A|^{2-q} |B|^{2-q} \|{\mathbf{1}}_A*{\mathbf{1}}_B\|_2^{2(q-1)}.$$ Hence, using the assumption, $$\|{\mathbf{1}}_A*{\mathbf{1}}_B\|_2^{2(q-1)} \ge 2^{-q\kappa m}\frac{|A| |B|^q}{|A|^{2-q} |B|^{2-q}} = 2^{-q\kappa m}|A|^{q-1} |B|^{2(q-1)},$$ which gives the claim when $q\in (1,2)$.
Suppose now $q\in (2,+\infty)$. Then $$2^{-q\kappa m} |A||B|^q \le \|{\mathbf{1}}_A*{\mathbf{1}}_B\|_q^q \le \|{\mathbf{1}}_A*{\mathbf{1}}_B\|_2^2 \|{\mathbf{1}}_A*{\mathbf{1}}_B\|_\infty^{q-2} \le \|{\mathbf{1}}_A*{\mathbf{1}}_B\|_2^2 |B|^{q-2},$$ and this completes the proof.
Combinatorial lemmas {#subsec:combinatorial-lemmas}
--------------------
In this section we establish several elementary combinatorial lemmas. In both the statement and the proof of Theorem \[thm:inverse-thm\] an important rôle is played by sets with a “regular tree structure”. We begin by formalizing this concept. Recall that $[\ell]=\{0,1,\ldots,\ell-1\}$.
Let $D,\ell\in{\mathbb{N}}$ and set $m=\ell D$. Given a sequence $(R_s)_{s\in [\ell]}$ taking values in $[1,2^D]$, we say that a $2^{-m}$-set $A$ is **$(D,\ell,R)$-uniform** if ${\mathcal{N}}(A\cap I,(s+1)D)=R_s$ for each $s\in[\ell]$ and $I\in{\mathcal{D}}_{s\ell}(A)$.
Further, we say that $A$ is **$(D,\ell)$-uniform** if there is a sequence $R$ such that $A$ is $(D,\ell,R)$-uniform.
Given an arbitrary $2^{-m}$-set $A$ and $D|m$, one may associate to it the tree whose vertices of level $s$ are the $2^{-s D}$-intervals intersecting $A$. Then $A$ is $(D,\ell)$-uniform if and only if the associated tree is spherically symmetric, i.e. the number of offspring of a vertex is constant over all vertices at the same distance to the root (but may vary between vertices of different levels). We will often informally refer to the tree description of sets, for example by speaking of branching at certain levels.
In our first lemma we show that any $2^{-m}$ set contains a fairly large uniform subset. This fact goes back at least to [@Bourgain10]; we provide details for completeness.
\[lem:regular-subset\] Let $D,\ell\in {\mathbb{N}}$, and let $A$ be a $2^{-m}$-set, where $m=\ell D$. Then there exists a $(D,\ell)$-uniform subset $A'\subset A$ such that $$|A'|\ge (2D)^{-\ell}|A| = 2^{(-\log(2D)/D) m}|A|.$$
The construction is similar to that in [@Bourgain10 Section 2]. We begin from the bottom of the tree, setting $A^{(\ell)}:=A$. Once $A^{(s+1)}$ is constructed, we let $$A^{(s,j)} =\bigcup \{ A^{(s+1)}\cap J : J\in{\mathcal{D}}_{sD}(A^{(s+1)}), {\mathcal{N}}(J\cap A^{(s+1)},(s+1)D) \in [2^j+1,2^{j+1}] \}.$$ Since $j$ takes at most $D$ values, we can pick $j=j_s$ such that $|A^{(s,j)}|\ge |A^{(s+1)}|/D$. By removing at most half of the intervals in $A^{(s+1)}$ from each interval $J$ making up $A^{(s,j)}$, we obtain a set $A^{(s)}$ such that $|A^{(s)}|\ge |A^{(s+1)}|/(2D)$ and ${\mathcal{N}}(J\cap A^{(s)},(s+1)D)=2^j$ for all $J\in {\mathcal{D}}_{sD}(A^{(s)})$. We see inductively that ${\mathcal{N}}(J\cap A^{(s)}, (s'+1) D)$ is constant over all $J\in{\mathcal{D}}_{s' D}(A^{(s)})$, for all $s'=s,s+1,\ldots,\ell-1$.
The lemma follows by taking $A'=A^{(0)}$.
The next simple lemma (which is also implicit in [@Bourgain10]) asserts that, given a $(D,\ell,R)$-uniform set, it is possible to reduce some of the numbers $R_s$ to $1$ without decreasing the size of the set too much.
\[lem:collapsing\] Given $D,\ell\in{\mathbb{N}}$, the following holds. Suppose $A$ is $(D,\ell,R)$-uniform. Then, if $\mathcal{S}\subset[\ell]$ is any set, there exists a subset $A'\subset A$ which is $(D,\ell,R')$ uniform, where $R'_s=1$ for $s\in\mathcal{S}$ and $R'_s=R_s$ for $s\in [\ell]\setminus \mathcal{S}$, and $$|A'| \ge \left(\prod_{s\in\mathcal{S}} \frac{1}{R_s}\right)|A| \ge 2^{-|\mathcal{S}|D}|A|.$$
We inductively construct a sequence of sets $A^{(s)}$, $s\in [\ell]$. Set $A^{(0)}=A$. Once $A^{(s)}$ is defined, if $s\notin\mathcal{S}$ set $A^{(s+1)}=A^{(s)}$. Otherwise, for each $I\in{\mathcal{D}}_{sD}(A^{(s)})$, let $J_I$ be any interval in ${\mathcal{D}}_{(s+1)D}(A\cap I)$, and let $A^{(s+1)}$ be the union of all such intervals $J_I$. Since $R_s\le 2^{D}$, it is clear that $A'=A^{(\ell-1)}$ has the desired properties.
Given a set $A$, the next lemma extracts a large subset $A'$ of a suitable translation of $A$, such that points in $A'$ are “not too close to the boundary” of $2^D$-adic intervals.
\[lem:centering\] Let $D\in{\mathbb{N}}_{\ge 2}$, $\ell\in{\mathbb{N}}$, and let $A$ be a $2^{-m}$-set in ${\mathbb{R}}/{\mathbb{Z}}$, where $m=\ell D$. Then there are a point $x=k 2^{-m}$, and a subset $A'\subset A$ such that:
1. $|A'|\ge 2^{-(\log 3/D) m}|A|$.
2. For all $y\in A'$ and all $s\in [\ell]$, $y+x\in\frac12 {\mathcal{D}}_{sD}(y+x)$.
We note the following simple fact: for any $y\in [0,1)$ and any $j\le m-2$, there is $t\in\{-2^{-(j+2)},0,2^{-(j+2)}\}$ such that $y+t\subset \tfrac12 {\mathcal{D}}_j(y)$. With this in mind, we prune the tree in a similar way to Lemma \[lem:regular-subset\] to construct sets $A^{(s)}$, starting from $A^{(\ell)}=A$ and moving up to $A^{(0)}$, such that for each $s\in [\ell]$,
1. There is $t_s\in\{ -2^{-(s D+2)},0,2^{-(s D+2)}\}$ such that $y+x_s:=y+\sum_{s'=s}^{\ell-1} t_{s'}\in\frac12{\mathcal{D}}_{s D}(y+x_s)$, for all $y\in A^{(s)}$.
2. Moreover, $|A^{(s)}|\ge |A^{(s+1)}|/3$.
Set $x=x_0=\sum_{s=0}^{\ell-1} t_s$ and $A'=A^{(0)}$. It is clear that $|A'|\ge 3^{-\ell}|A|= 2^{-(\log 3/D) m}|A|$. Also, since $\sum_{s'=0}^{s-1} t_{s'}$ is a multiple of $2^{-sD}$, we have $y+x\in{\mathcal{D}}_{s D}(y+x)$ for all $y\in A'$ and $s\in [\ell]$, as claimed.
The next lemma will allow us to show that if $H$ has small doubling and $A+H$ is “not too large”, then $A$ and $H$ have a certain shared structure.
\[lem:large-sumset-regular-sets\] Let $D,\ell\in{\mathbb{N}}$, and write $m=\ell D$. Suppose $H,A$ are $2^{-m}$-sets such $H$ is $(D,\ell,R)$-uniform and $A$ is $(D,\ell,R')$-uniform. Then $$|A+H| \ge 2^{-(1/D)m} |H| \prod_{s:R_s=1} R'_s.$$
Write $\mathcal{S}=\{ s:R_s=1\}$. By replacing $A$ with the subset given by Lemma \[lem:collapsing\], we may assume that $R'_s=1$ for all $s\notin\mathcal{S}$. This makes the problem symmetric: for each $s$, either $R_s=1$ or $R'_s=1$. With this in mind, we inductively show that for each $s=0,1,\ldots,\ell$, there are families $\mathcal{I}_s\subset {\mathcal{D}}_{s D}(A)$, $\mathcal{J}_s\subset {\mathcal{D}}_{s D}(H)$, such that:
1. $|\mathcal{I}_s||\mathcal{J}_s| \ge 2^{-s} {\mathcal{N}}_{s D}(A){\mathcal{N}}_{s D}(H)$
2. The intervals $\{ I+J: I\in\mathcal{I}_s, J\in\mathcal{J}_s\}$ are pairwise disjoint.
The base case $s=0$ is trivial. Suppose this holds for some $s<\ell$. Without loss of generality, $R_{s+1}=1$. Hence, for each $J\in\mathcal{J}_s$ we pick the single $J'\in{\mathcal{D}}_{(s+1)D}(J\cap H)$ and let $\mathcal{J}_{s+1}$ be the union of all such $J'$. Next, for each $I\in\mathcal{I}_s$, let $(I'_j)_{j=1}^{N_I}$ be a subcollection of ${\mathcal{D}}_{(s+1)D}(I\cap A)$ such that no two of the $I'_j$ are adjacent, and $N_I\ge \lceil R'_{s+1}/2\rceil$. We let $\mathcal{I}_{s+1}$ be the union of all $I'_j$ over all $I\in\mathcal{I}_s$. It is clear from this construction that (1)–(2) hold.
The claim follows from (1)–(2) applied with $s=\ell$.
We conclude this section with a version of Theorem \[thm:Bourgain\] in which the lengths of the intervals over which there is either no or close to full branching is kept constant (at the price of worsening the quantitative estimates). This reduction is a matter of simplicity; a version of Theorem \[thm:inverse-thm\] in which the intervals of almost full/no branching have varying lengths could be deduced directly from Theorem \[thm:Bourgain\].
\[cor:Bourgain\] Given a large $D\in{\mathbb{N}}$, the following holds for sufficiently large $\ell\in {\mathbb{N}}$ (depending on $D$).
Let $m=\ell D$. Suppose $H$ is an $(m,2^{-2D^2-1})$-small doubling set. Then there are is a subset $H_1\subset H$ such that the following holds:
1. $|H_1|\ge 2^{-(4 (\log D)D^{-1/4} )m} |H|$.
2. $H_1$ is $(D,\ell,R)$-uniform, where for each $u$ either $R_u=1$, or $\log R_u\ge (1-D^{-1/4})D$.
In particular, $|H_1|=\prod_{u:R_u>1} R_u$.
Let $H',\mathcal{S}$, $n_s$, $\widetilde{R}_s$ (in place of $R_s$) be as given by Theorem \[thm:Bourgain\] with $T=D^2$. We assume that $m$ is a multiple of $T$; the general case can be deduced by applying this special case to $\max\{m_1 T: m_1 T\le m\}$.
Let $S=\sum_{s\in\mathcal{S}} n_s - s T$. If $S<T^{-1/4}m$, then $$|H| \le 2^{(2\log T/\sqrt{T}) m} |H'| \le 2^{(2\log T/\sqrt{T}) m} 2^{T^{-1/4}m} \le 2^{2T^{-1/4} m}.$$ so that a singleton satisfies the conditions in the statement. We therefore assume that $S\ge T^{-1/4}m$.
We apply Lemma \[lem:regular-subset\] to $H'$ and $D$, to obtain a $(D,\ell,R)$-uniform set $H''$ such that $$\label{eq:H''-lower-bound}
|H''| \ge 2^{-(\log(2D)/D) m}|H'|,$$ It is clear that $R_u=1$ for all $u$ of the form $sD+j, j\in[D]$, with $s\notin\mathcal{S}$, and also with $s\in\mathcal{S}$ and $jD \ge n_s-s T$, since over those scales already $H'$ had no branching. Therefore, there is a set $\mathcal{U}$ such that $R_u=1$ for $u\notin\mathcal{U}$, and $$\label{eq:U-lower-bound}
D|\mathcal{U}| \le S + (m/T)D \le S(1+D^{-1/2}),$$ using that $S\ge D^{-1/2}m$. Using Theorem \[thm:Bourgain\], and $S\ge D^{-1/2}m$ again, we get $$ (1-1/D)S \le \log|H'| \le \log|H''|+ \frac{\log(2D)}{D} D^{1/2} S,$$ so that, recalling , $$\log|H''| \ge S(1-1/D-\log(2D) D^{-1/2}) \ge \frac{1-(2\log D) D^{-1/2}}{1+D^{-1/2}} D|\mathcal{U}|.$$ Hence, $$\frac{1}{|\mathcal{U}|}\sum_{u\in\mathcal{U}} \frac{\log R_u}{D} = \frac{\log |H''|}{D|\mathcal{U}|} \ge 1-3(\log D)D^{-1/2}.$$ Since $\log R_u/D\in [0,1]$ for all $u$, Markov’s inequality yields that $\log(R_u) \ge (1- D^{-1/4})|D|$ for $u$ outside of a set $\mathcal{U}'$ with $$|\mathcal{U}'|\le 3(\log D)D^{-1/4}|\mathcal{U}| \le 3(\log D)D^{-1/4}m/D,$$ provided $D$ is larger than an absolute constant. To obtain our final set $H_1$, we apply Lemma \[lem:collapsing\] to $H''$ and the set $\mathcal{U}'$ (that is, we collapse all $R_u$ intervals to a single one for $u\in\mathcal{U}'$). Recalling Theorem \[thm:Bourgain\](i) and , the resulting set satisfies $$|H_1| \ge 2^{-D|\mathcal{U}'|}|H''| \ge 2^{-4(\log D)D^{-1/4} m}|H|,$$ while the claim on the branching structure is clear from the construction.
Proof of Theorem \[thm:inverse-thm\]
------------------------------------
Let $D\in{\mathbb{N}},{\varepsilon}>0$. In the course of the proof, we will impose several lower bounds to $D$ (depending on $D_0,\delta,q$ only) and upper bounds on ${\varepsilon}$ (depending on $D,\delta,q$ only), resulting in the verification of all the claims in the theorem. To begin, we assume $D\ge D_0$. In the course of the proof, we write $m=D\ell$, and understand $\ell$ and $m$ to be sufficiently large that any claims involving them hold.
Let $\tau>0$ be the value given by Theorem \[thm:BSG\] for $\kappa:=2^{-2D^2-1}$. We take $${\varepsilon}\le \frac{\tau}{2\max(q,q')}.$$ (Later we will impose further conditions on ${\varepsilon}$.)
Apply Lemma \[lem:non-flattening-to-subset\] to obtain sets $A_1$, $B_1$ and $j,j'\le 2{\varepsilon}q' m$ satisfying (i)–(iii) in the lemma (with $A_1, B_1$ in place of $A,B$). By our choice of ${\varepsilon}$ and Lemma \[lem:q-to-2\], $$\|\mathbf{1}_{A_1} * \mathbf{1}_{B_1}\|_2^2 \ge 2^{-\tau m}|A_1| |B_1|^2,$$ so that we can apply Theorem \[thm:BSG\] to $A_1,B_1$ to obtain an $(m,\kappa)$-small doubling set $H$ and a $2^{-m}$-set $X$ such that $$\begin{aligned}
|A_1\cap (X+H)| & \ge 2^{-\kappa m}|A_1|, \label{eq:BSG-i} \\
|A_1| &\ge 2^{-\kappa m}|X||H|, \label{eq:BSG-ii} \\
|B_1\cap H| & \ge 2^{-\kappa m}|B_1|. \label{eq:BSG-iii}\end{aligned}$$ Thanks to Lemma \[lem:non-flattening-to-subset\], the sets $A_1, B_1$ already satisfy (A-ii), (B-ii). As the final sets $A,B$ will be subsets of $A_1, B_1$, these properties are established.
Our next step is to pass to suitable regular subsets of (a translation of) $A_1, B_1, H$:
1. By our choice $\kappa=2^{-2 D^2-1}$, we can apply Corollary \[cor:Bourgain\] to $H$. Let $H'\subset H$ be the resulting set, with branching numbers $R_s, s\in [\ell]$.
2. We first apply Lemma \[lem:centering\] (this is the point where we need to translate the original measure), and then Lemma \[lem:regular-subset\] and , to the set $A_1\cap (X+H)$, to obtain a set $A\subset A_1\cap (X+H)$ such that:
1. $|A|\ge 2^{- (2\log D/D) m}|A_1\cap (X+H)|\ge 2^{-(3\log D/D)m}|A_1|$. Hence, in light of (A-ii), property (A-i) holds if $D$ is taken large enough in terms of $\delta$.
2. The set $A$ is $(\ell,D,R')$-uniform for some sequence $(R'_s)_{s\in[\ell]}$. This shows that (A-iii) holds.
3. $x\in \frac12 {\mathcal{D}}_{s D}(x)$ for all $x\in A$ and $s\in [\ell]$. That is, (A-iv) holds.
3. Similarly, we apply Lemma \[lem:centering\], and then Lemma \[lem:regular-subset\] and , to $B_1\cap H$ to obtain a set $B_2\subset B_1\cap H$ (not yet our final set $B$) such that:
1. $|B_2|\ge 2^{- (2\log D/D) m}|B_1\cap H|\ge 2^{-(3\log D/D)m}|B_1|$.
2. The set $B_2$ is $(\ell,D,{\widetilde}{R})$-uniform for some sequence $({\widetilde}{R}_s)_{s\in[\ell]}$. This shows that (B-iii) holds for $B_2$.
3. $y\in \frac12 {\mathcal{D}}_{s D}(y)$ for all $y\in B_2$. As the final set $B$ will be a subset of $B_2$, this establishes (B-iv).
Next, we note that as $A+H'\subset X+H+H$, we can use and (2)(a) above to estimate $$\label{eq:A+H'-small}
|A+H' | \le |X||H+H| \le 2^{\kappa m}|X||H| \le 2^{2\kappa m}|A_1| \le 2^{(4\log D/D)m}|A|.$$ Let $\mathcal{S}_0 = \{ s\in[\ell]: R_s=1\}$, $\mathcal{S}_1=[\ell]\setminus\mathcal{S}_0$, so that $\mathcal{S}_1$ indexes the scales over which $H'$ has almost full branching. We will see that $A$ has almost full branching for a large subset of scales $\mathcal{S}\subset\mathcal{S}_1$; eventually $B$ will be obtained from $B_2$ by collapsing all the branching at the scales in $[\ell]\setminus\mathcal{S}$ using Lemma \[lem:collapsing\].
According to Lemma \[lem:large-sumset-regular-sets\] applied to $A$ and $H'$ (which we have seen meet the hypotheses), $$\label{eq:A+H'-large}
|A+H'| \ge 2^{-(1/D) m} |H'| \prod_{s\in\mathcal{S}_0} R'_s.$$ Since $|A|=\prod_s R'_s$, $|H'|=\prod_s R_s$ and $R_s\ge 2^{(1-D^{-1/4})D}$ for $s\in\mathcal{S}_1$, we may combine and to deduce that $$\prod_{s\in\mathcal{S}_1} R'_s = \frac{|A|}{\prod_{s\in\mathcal{S}_0} R'_s} \ge 2^{-(5\log D/D)m}|H'| \ge 2^{-(5\log D/D)m} 2^{(1-D^{-1/4})|\mathcal{S}_1|D}.$$ Consider two cases.
1. If $|\mathcal{S}_1|< D^{-1/2}\ell$ (which we note implies $H'$, hence $H$ and $B$, are very small) we set $\mathcal{S}' = \mathcal{S}_1$ and $\mathcal{S}=\varnothing$.
2. If $|\mathcal{S}_1|\ge D^{-1/2}\ell$, then we further deduce from the above that $$\prod_{s\in\mathcal{S}_1} R'_s \ge 2^{(1-2 D^{-1/4})D|\mathcal{S}_1|}.$$ Let $$\begin{aligned}
\mathcal{S}&=\{ s\in\mathcal{S}_1: R'_s \ge 2^{(1-D^{-1/8})D}\},\\
\mathcal{S}'&=\{ s\in\mathcal{S}_1: R'_s < 2^{(1-D^{-1/8})D}\}.\end{aligned}$$ Since $R'_s\le 2^D$ for all $s$, we have $$(1-2 D^{-1/4})|\mathcal{S}_1| \le \sum_{s\in\mathcal{S}_1} \frac{\log R'_s}{D} \le (1-D^{-1/8})|\mathcal{S}'| + |\mathcal{S}_1|-|\mathcal{S}'|,$$ so that $$|\mathcal{S}'| \le 2D^{-1/8} |\mathcal{S}_1| \le 2 D^{-1/8} \ell.$$
We note for later reference that, in either case $$\label{eq:scales-prunned}
|\mathcal{S}'| \le \max(2 D^{-1/8}\ell,D^{-1/2}\ell) = 2 D^{-1/8}\ell.$$
We move on to the construction of $B$. By Theorem \[thm:BSG\] and Corollary \[cor:Bourgain\], $$|B_2+H'| \le |(B_1\cap H)+H| \le 2^{\kappa m}|H| \le 2^{(5(\log D) D^{-1/4}) m}|H'|.$$ Applying Lemma \[lem:large-sumset-regular-sets\] to $B_2$ and $H'$, we deduce that $$\prod_{s\in\mathcal{S}_0} {\widetilde}{R}_s \le 2^{(1/D)m} 2^{(5(\log D) D^{-1/4}) m} \le 2^{(6(\log D) D^{-1/4})m}.$$ We apply Lemma \[lem:collapsing\] to $B_2$ and the set $\mathcal{S}_0$, to obtain a new set $B_3\subset B_2$ such that for all $s\in\mathcal{S}_0$ and $I\in{\mathcal{D}}_{s D}(B_3)$, there is a single $J\in{\mathcal{D}}_{(s+1)D}(B_3\cap I)$, while if $s\notin\mathcal{S}_0$, then ${\mathcal{N}}(B_3\cap I,(s+1)D)={\widetilde}{R}_s$ for all $I\in{\mathcal{D}}_{s D}(B_3)$. By Lemma \[lem:collapsing\] and (3)(a) above, $$|B_3| \ge 2^{-(6(\log D)D^{-1/4})m}|B_2| \ge 2^{-(7(\log D)D^{-1/4})m} |B_1|.$$ Finally, recall that we defined a set $\mathcal{S}'$, satisfying . We obtain our final set $B$ by applying Lemma \[lem:collapsing\] to $B_3$ and $\mathcal{S}'$. Then $$|B| \ge 2^{-2 D^{-1/8} m} |B_3| \ge 2^{- 3 D^{-1/8} m} |B_1|,$$ and ${\mathcal{N}}_{(s+1)D}(I\cap B)=1$ for all $I={\mathcal{D}}_{s D}(B)$ for each $s\in\mathcal{S}_0\cup \mathcal{S}'=[\ell]\setminus\mathcal{S}$. We had already established (B-ii) and (B-iv). The set $B$ satisfies (B-i) if $D$ is large enough (thanks to (B-ii)); and it still satisfies (B-iii), with $R''_s=1$ for $s\notin\mathcal{S}$ and $R''_s={\widetilde}{R}_s$ for $s\in\mathcal{S}$.
The claim (v) follows from the construction if $D$ is large enough: either $s\in\mathcal{S}$, in which case $R'_s\ge 2^{(1-o_{D\to\infty}(1))(D)}$ or $s\notin \mathcal{S}$, in which case $R''_s=1$ as we have just observed.
It remains to establish (vi). It follows from (B-i)–(B-ii) that $\nu(x) \ge \frac12 2^{-\delta m}|B|^{-1}$ for all $x\in B$. On the other hand, we know from (B-iii) and (v) that $|B|\le 2^{D|\mathcal{S}|}$. We get $$\|\nu\|_q^{-q'} \le O_q(1) 2^{\delta m q'} |B| \le O_q(1) 2^{\delta m q'} 2^{D|\mathcal{S}|},$$ which gives the left-hand inequality in (vi), with $O_q(\delta)$ in place of $\delta$.
By Lemma \[lem:non-flattening-to-subset\], $\mu(x) \ge \tfrac12 2^{-2{\varepsilon}q' m}\|\mu\|_q^{q'}$ for all $x\in A\subset A_1$, whence $$2^{-q} 2^{-(2q{\varepsilon}) m}\|\mu\|_q^{q q'} |A| \le \|\mu\|_q^q,$$ so that $|A|\le 2^{3 q{\varepsilon}m} \|\mu\|_q^{-q'}$. Since $|A|\ge 2^{(1-\delta)D|\mathcal{S}|}$ by (A-iii) and (v), the right-hand side inequality in (vi) also follows (with $2\delta$ in place of $\delta$, say), concluding the proof.
Properties of dynamically driven self-similar measures {#sec:properties-dyn-ssm}
======================================================
Preliminary lemmas
------------------
In this section we initiate the study of measures generated by pleasant models (recall Definition \[def:pleasant\]). We start by collecting some standard lemmas for later reference. The short proofs are included for completeness.
\[lem:Holder\] Let $(Y,\mu,\mathcal{B})$ be a probability space. Suppose $\mathcal{P},\mathcal{Q}$ are finite families of measurable subsets of $Y$ such that each element of $\mathcal{P}$ can be covered by at most $M$ elements of $\mathcal{Q}$ and each element of $\mathcal{Q}$ intersects at most $M$ elements of $\mathcal{P}$. Then, for every $q\ge 1$, $$\sum_{P\in\mathcal{P}} \mu(P)^q \le M^q \sum_{Q\in\mathcal{Q}} \mu(Q)^{q}$$
Let $Q_{P,1},\ldots, Q_{P,M_P}$, $M_P\le M$, be a minimal sub-collection of $\mathcal{Q}$ that covers $P\in\mathcal{P}$. Using Hölder’s inequality in the form $(\sum_{i=1}^m a_i)^q \le m^{q-1} \sum_{i=1}^m a_i^q$, we get $$\sum_{P\in\mathcal{P}} \mu(P)^q \le M^{q-1} \sum_{P\in\mathcal{P}} \sum_{i=1}^{M_P}\mu(Q_{P_i})^q \le M^q \sum_{Q\in\mathcal{Q}} \mu(Q)^q.$$
\[lem:L-q-norm-almost-disj-supports\] Let $\mu=\sum_{i=1}^\ell \mu_i$, where $\mu_i$ are finitely supported measures on a space $Y$, such that each point is in the support of at most $M$ of the $\mu_i$. Then $$\|\mu\|_q^q \le M^{q-1} \sum_{i=1}^\ell \|\mu_i\|_q^q.$$
For each $x$, Hölder’s inequality, together with the assumption that $\mu_i(x)>0$ for at most $M$ values of $i$, gives $(\sum_i \mu_i(x))^q \le M^{q-1} \sum_i \mu_i(x)^q$. The claim follows.
\[lem:discr-norm-conv-equivalence\] For any probability measures $\mu,\nu$ on ${\mathbb{R}}/{\mathbb{Z}}$, and any $q\in (1,\infty)$, $$\|(\mu*\nu)^{(m)}\|_q^q = \Theta_q(1) \|\mu^{(m)}*\nu^{(m)}\|_q^q.$$
Given $I=[k 2^{-m},(k+1)2^{-m})\in{\mathcal{D}}_m$, let $P_I=\{ (x,y)\in({\mathbb{R}}/{\mathbb{Z}})^2: x+y\in I\}$ and $$Q_I = \bigcup_{ i\in{\mathbb{Z}}/2^m{\mathbb{Z}}} [i 2^{-m},(i+1) 2^{-m}) \times [(k-i) 2^{-m},(k-i+1) 2^{-m}).$$ Then $\|(\mu*\nu)^{(m)}\|_q^q=\sum_I (\mu\times\nu)(P_I)^q$ and $\|\mu^{(m)}*\nu^{(m)}\|_q^q=\sum_I (\mu\times\nu)(Q_I)^q$, so the claim follows from Lemma \[lem:Holder\].
A sub-multiplicative cocycle, and consequences
----------------------------------------------
Throughout the rest of this section, we use the following notation. We work with a measure-preserving system $(X,\mathbf{T},\mathbb{P})$, i.e. $\mathbf{T}:X\to X$ is a measurable map, and $\mathbf{T}\mathbb{P}=\mathbb{P}$. A model $\mathcal{X}=(X,\mathbf{T},\Delta,{\lambda})$ is fixed, and $\mu_x, \mu_{x,n}$ are as defined in , . Moreover, $m=m(n)$ will denote the smallest integer such that $2^{-m}\le {\lambda}^n$ (the dependence is omitted when it is clear from context). We assume that $$\label{eq:assumption-support-mu-x}
{\text{supp}}(\mu_x)\subset [0,1] \text{ for all }x\in X,$$ which can always be achieved by a change of coordinates, i.e. by replacing the map $\Delta$ by $g\circ\Delta$ for an appropriate affine map $g$.
For each $x\in X$ we define a *code space* $\Omega_x = \prod_{n=0}^\infty {\text{supp}}(\Delta(\mathbf{T}^n x))$ and a *coding map* $\pi_x:\Omega_x \to {\mathbb{R}}$, via $\omega\mapsto \sum_{n=0}^\infty \omega_n {\lambda}^n$. Then, by definition, $\mu_x$ is the push-down of the product measure $\prod_n \Delta(\mathbf{T}^n x)$ under this coding map. We also define the *truncated coding maps* $\pi_{x,n}: \Omega_x \to {\mathbb{R}}$, $\omega\mapsto \sum_{i=0}^{n-1} \omega_i {\lambda}^i$. Then $\mu_{x,n}$ is the image of $\prod_n \Delta(\mathbf{T}^n x)$ under the truncated coding map.
\[lem:comparison-mu-m-mu\] For every $x\in X$, $\|\mu_x^{(m)}\|_q^q = \Theta_{{\lambda},q}(1) \|\mu_{x,n}^{(m)}\|_q^q$.
Let $\eta=\prod_{n=0}^\infty \Delta(\mathbf{T}^n x)$, so that $\mu_n=\pi_n\eta$ and $\mu_{n,x}=\pi_{n,x}\eta$. Then $$\begin{aligned}
\|\mu_x^{(m)}\|_q^q &= \sum_{I\in{\mathcal{D}}_m} \eta(\pi_x^{-1}I)^q,\\
\|\mu_{x,n}^{(m)}\|_q^q &= \sum_{I\in{\mathcal{D}}_m} \eta(\pi_{x,n}^{-1}I)^q.\end{aligned}$$ Since $\|\pi_x-\pi_{x,n}\|_\infty \le O({\lambda}^n) = O_{\lambda}(2^m)$, the lemma follows easily from Lemma \[lem:Holder\].
We recall some well-known properties of the $L^q$ spectrum $\tau_\mu$. See e.g. [@LauNgai99] for the proofs.
\[lem:basic-tau-properties\] For any probability measure on ${\mathbb{R}}$ of bounded support, the function $\tau=\tau_\mu:[0,\infty)\to {\mathbb{R}}$ is increasing, concave, and satisfies $\tau(1)=0$.
The next proposition introduces a sub-multiplicative cocycle (which was first used in [@NPS12], in a special case) that will play a crucial rôle in the proof of Theorem \[thm:L-q-dim-dyn-ssm\]. Let us define the following sequence of functions, parametrized by $q\in [1,\infty)$: $$\phi_n^q(x) = \| \mu_x^{(m(n))} \|_q^q.$$
\[prop:cocycle\] For any $n,n'\in{\mathbb{N}}$, $$\phi_{n+n'}^q(x) \le O_{q,{\lambda}}(1) \phi_n^q(x) \phi_{n'}^q(\mathbf{T}^n x).$$ In particular, for each $q\in [1,\infty)$ there exists a number $T(q)$ such that $$\label{eq:T-q-for-almost-all-x}
\lim_{n\to\infty} -\frac{1}{m} \log\|\mu_{x}^{(m)}\|_q^q = T(q)$$ for $\mathbb{P}$-a.e. $x$. Moreover, for $\mathbb{P}$-a.e. $x$ it holds that $T(q) = \tau_{\mu_x}(q)$ for all $q\in [1,+\infty)$. In particular, $T:[1,\infty)\to{\mathbb{R}}$ is increasing and concave, and $T(1)=0$.
We estimate: $$\begin{aligned}
\|\mu_{x}^{(m(n+n'))}\|_q^q &\le O_{{\lambda},q}(1) \| \mu_{x,n}^{(m(n+n'))} * \left(S_{{\lambda}^n} \mu_{\mathbf{T}^n x}\right)^{(m(n+n'))}\|_q^q \\
&\le O_{{\lambda},q}(1) \sum_{I\in{\mathcal{D}}_{m(n)}} \|\mu_{x,n}^{(m(n+n'))}|_I * \left(S_{{\lambda}^n} \mu_{\mathbf{T}^n x}\right)^{(m(n+n'))}\|_q^q\\
&\le O_{{\lambda},q}(1) \sum_{I\in{\mathcal{D}}_{m(n)}} \mu_{x,n}(I)^q \sum_{J\in{\mathcal{D}}_{m(n+n')}} \mu_{\mathbf{T}^n x}(S_{{\lambda}^{-n}}J)^q \\
&\le O_{{\lambda},q}(1) \|\mu_{x}^{(m(n))}\|_q^q \|\mu_{\mathbf{T}^n x}^{(m(n'))}\|_q^q.\end{aligned}$$ We have used the self-similarity relation and Lemma \[lem:discr-norm-conv-equivalence\] in the first line, Lemma \[lem:L-q-norm-almost-disj-supports\] in the second line (which is justified since the support of $S_{{\lambda}^n} \mu_{\mathbf{T}^n x}$ has diameter $O_{\lambda}(2^{-m(n)})$), Young’s inequality in the third line, and Lemmas \[lem:Holder\], \[lem:comparison-mu-m-mu\] in the last line.
The subadditive ergodic theorem applied to the sequence of (bounded and measurable) functions $x\mapsto \log \phi_n^q(x) - C_{{\lambda},q}$ for a sufficiently large constant $C_{{\lambda},q}$ yields . More precisely, we know the convergence for the subsequence $m(n)$, $n\in\mathbb{N}$, but since this sequence has positive density, follows from the monotonicity of $m\mapsto \|\nu^{(m)}\|_q^q$.
Finally, if $(q_j)$ is a dense subset of $(1,\infty)$, then we know from the previous claim that $\tau_{\mu_x}(q_j)=T(q_j)$ for all $j$, for $\mathbb{P}$-almost all $x$. Since $\tau_\mu$ is concave and increasing, and $T(q)$ is clearly increasing, we deduce that the equality extends to all $q\in (1,\infty)$.
The last claim is immediate from Lemma \[lem:basic-tau-properties\]
In order to prove Theorem \[thm:L-q-dim-dyn-ssm\], we would like to draw conclusions for *all* $x$ rather than almost all. Indeed, the strategy will be to prove that the convergence in holds for all $x$, and $T(q)$ has the “expected” value. It is well known that for uniquely ergodic systems, the ergodic averages of sufficiently regular (a.e. continuous) observables converge uniformly. The next known lemma asserts that a one-sided version of this remains valid for subadditive cocycles.
\[lem:uniform-convergence\] Let $(X,\mathbf{T},\mathbb{P})$ be a uniquely ergodic measure-preserving system, with $X$ a compact manifold, and $\mathbf{T}$ continuous. Suppose $f_n:X\to{\mathbb{R}}$ are continuous ${\mathbb{P}}$-almost everywhere and bounded, and $$f_{n+n'}(x) \le f_n(x) + f_{n'}(\mathbf{T}^n x)$$ for all $n,n'\in{\mathbb{N}}$, $x\in X$. Then, denoting by $L$ the almost sure limit of $f_n(x)/n$, we have $$\limsup_{n\to\infty} \frac1n f_n(x) \le L \quad\text{uniformly in }x\in X.$$
For continuous $f_n$, the claim was observed by Furman [@Furman97 Theorem 1]. In the case the $f_n$ are only a.e. continuous and bounded, a classical exercise in measure theory yields that for each $n$ and ${\varepsilon}>0$ there exists a continuous function $g_{n,{\varepsilon}}$ such that $f_n\le g_{n,{\varepsilon}}$ pointwise, and $\int g_{n,{\varepsilon}}-f_n \,d{\mathbb{P}}\le {\varepsilon}$. The claim is then a special case of [@GSSY16 Theorem 3.5 and Corollary 3.6], which in turn is established by inspecting the proof of the subadditive ergodic theorem given by Katznelson and Weiss [@KatznelsonWeiss82].
Furman [@Furman97 Theorem 1] also showed that, even in the continuous case, the set of $x$ such that $f_n(x)/n\not\to L$ may be nonempty and, indeed, can equal any $F_\sigma$, $\mathbb{P}$-null set.
From Proposition \[prop:cocycle\] and Lemma \[lem:uniform-convergence\] we obtain the following crucial corollary; this is the main place where the pleasantness of the model gets used.
\[cor:unif-continuity\] Suppose $(X,\mathbf{T},\Delta,{\lambda})$ is a pleasant model. Then $$\liminf_{m\to\infty} -\frac1m \log\|\mu_x^{(m)}\|_q^q \ge T(q) \quad\text{uniformly in }x\in X,$$ where $T(q)$ is the function from Proposition \[prop:cocycle\].
Let $\psi_m:{\mathbb{R}}/{\mathbb{Z}}\to [0,1]$ be a continuous bump function supported on the interval $[-2^{-m},2^{-m}]$ such that $\psi_m\equiv 1$ on $[-2^{-m}/2,2^{-m}/2]$. It follows easily from Lemma \[lem:Holder\] that $$\Psi_m(x):= \sum_{k=0}^{2^m-1} \left(\int \psi_m(t+k 2^{-m}) d \mu_x(t)\right)^q = \Theta_q(\|\mu_x^{(m)}\|_q^q).$$ Since the model is pleasant, $\Psi_m$ is bounded and continuous ${\mathbb{P}}$-a.e. The corollary is now immediate from (the proof of) Proposition \[prop:cocycle\] and Lemma \[lem:uniform-convergence\].
We point out that, in the special case given by Lemma \[lem:mu-x-conv-ssm\] below, this corollary was first obtained in [@NPS12].
Multifractal structure {#subsec:Lq-spectrum}
----------------------
Next, we investigate the scaling (or multifractal) properties of measures generated by pleasant models. Throughout the rest of this section, we always assume the following:
**Standing assumption**. $(X,\mathbf{T},\Delta,{\lambda})$ is a pleasant model, $T(q)$ is the function given by Proposition \[prop:cocycle\] for this model. Any constants or parameters are allowed to depend on the model (in particular, on the function $T$).
We will establish some regularity of the multifractal structure for those values of $q$ such that $T$ is differentiable at $q$. The Legendre transform plays a key role in multifractal analysis. Given a concave function $\tau:{\mathbb{R}}\to{\mathbb{R}}$, its Legendre transform $\tau^*:{\mathbb{R}}\to [-\infty,\infty)$ is defined as $$\tau^*(\alpha) = \inf_{q\in{\mathbb{R}}} \alpha q -\tau(q).$$ It is easy to check that if $\tau$ is concave and is differentiable at $q$, then $$\tau^*(\alpha)=\alpha q-\tau(q) \text{ for } \alpha=\tau'(q).$$
The next lemma is also well known; the short proof is included for completeness.
\[lem:f-alpha-smaller-than-one\] If $T$ is differentiable at $q>1$, $T(q)<q-1$, and $\alpha=T'(q)$, then $T^*(\alpha) \le \alpha < 1$
Since $T(1)=0$ and $T(q)<q-1$, we have $(T(q)-T(1))/(q-1)<1$. On the other hand, as $T$ is concave and differentiable at $q$, we must have $\alpha\le (T(q)-T(1))/(q-1)<1$. Furthermore, $T^*(\alpha)\le \alpha\cdot 1-T(1)=\alpha$, so the lemma follows.
It is known that the multifractal structure of general measures displays some regularity for values of $q$ such that $\tau_\mu$ is differentiable at $q$ (or, dually, values of $\alpha$ such that $\tau^*$ is strictly concave at $\alpha$); see for example [@LauNgai99 Theorem 5.1]. The following lemmas, which are proved with similar ideas, are a further illustration of this. For a single measure $\mu$, the heuristic to keep in mind is that, whenever $\alpha=\tau_\mu'(q)$ exists, almost all of the contribution to $\|\mu^{(m)}\|_q^q$ comes from $\approx 2^{\tau^*(\alpha) m}$ intervals, each of mass $\approx 2^{-\alpha m}$. In our case, we are dealing with a family $(\mu_x)_{x\in X}$; with the help of Corollary \[cor:unif-continuity\] we will establish results which are uniform in $x$, at the price of dealing with $T(q)$ in place of $\tau_{\mu_x}(q)$.
\[lem:size-set-A-in-terms-of-f-alpha\] Suppose that $\alpha_0=T'(q_0)$ exists for some $q_0\in (1,\infty)$.
Given ${\varepsilon}>0$, the following holds if $\delta$ is small enough in terms of ${\varepsilon}, q_0$ and $m$ is large enough in terms of ${\varepsilon}, q_0$ and $\delta$.
Suppose $\mathcal{D}'\subset\mathcal{D}_m$ is such that, for some $x\in X$:
1. $2^{-\alpha m}\le\mu_x(I)\le 2\cdot 2^{-\alpha m}$ for all $I\in\mathcal{D}'$ and some $\alpha\ge 0$.
2. $\sum_{I\in\mathcal{D}'} \mu_x(I)^{q_0} \ge 2^{-(T(q_0)+\delta)m}$.
Then $|\mathcal{D}'|\le 2^{m(T^*(\alpha_0)+{\varepsilon})}$.
Set $\eta:={\varepsilon}/(3q_0)$, and pick $\delta\le \eta^2/9$, and also small enough that, if $q_1=q_0-\delta^{1/2}$, then $$\label{eq:using-tau-diff-1}
T(q_0)-T(q_1) \le \delta^{1/2}\alpha_0 + \delta^{1/2}\eta .$$ On one hand, using (1) and Corollary \[cor:unif-continuity\], we get $$\label{eq:bound-norm-mu-m-1}
2^{-(T(q_1)-\delta)m} \ge \|\mu_x^{(m)}\|_{q_1}^{q_1} \ge |\mathcal{D}'| 2^{- \alpha q_1 m},$$ if $m$ is large enough (depending on $q_0, T$, but not on $x$). On the other hand, by the assumptions (1)–(2), $$|\mathcal{D}'| 2^{-\alpha q_0 m} \ge 2^{-q_0} 2^{(-T(q_0)-\delta)m} \ge 2^{(-T(q_0)-2\delta)m}$$ if $m\gg_{\delta,q_0} 1$. Eliminating $|\mathcal{D}'|$ from the last two displayed equations yields $$\alpha q_0 -T(q_0) -2\delta \le \alpha (q_0-\delta^{1/2})-T(q_0-\delta^{1/2})+\delta,$$ so that, recalling , $$\delta^{1/2}\alpha \le T(q_0)-T(q_0-\delta^{1/2}) +3\delta \le \delta^{1/2}\alpha_0 +\delta^{1/2}\eta+3\delta.$$ Hence $\alpha-\alpha_0< 2\eta$, since we assumed $\delta\le (\eta/3)^2$. Using this, a further application of Corollary \[cor:unif-continuity\] guarantees that if $m\gg_{\varepsilon}1$, then $$2^{(-T(q_0)+{\varepsilon}/3)m} \ge \|\mu_x^{(m)}\|_{q_0}^{q_0} \ge 2^{-q_0\alpha m} |\mathcal{D}'| \ge 2^{-q_0\alpha_0 m} 2^{-(q_0 2 \eta) m} |\mathcal{D}'|.$$ The conclusion follows from the formula $T^*(\alpha_0)=q_0 \alpha_0-T(q_0)$ and our choice $\eta={\varepsilon}/(3q_0)$.
\[lem:Lq-sum-large-mass\]
Let $q_0>0$ be such that $\alpha_0=T'(q_0)$ exists. Given $\sigma>0$, there is ${\varepsilon}={\varepsilon}(\sigma,q_0)>0$ such that the following holds for large enough $m$ (in terms of $\sigma, q_0$): for all $x\in X$, $$\label{eq:Lq-sum-large-mass}
\sum \{ \mu_x(I)^{q_0}: I\in{\mathcal{D}}_m, \mu_x(I) \ge 2^{-m(\alpha_0-\sigma)} \} \le 2^{-m(T(q_0)+{\varepsilon})}.$$
Let $\eta\in (0,1)$ be small enough that $$\label{eq:using-tau-diff-2}
T(q_0+\eta) \ge T(q_0) + \eta \alpha_0 - \delta,$$ where $\delta= \eta\sigma/(4+2 q_0)$.
Let $\alpha_j = \alpha_0 -\delta j$, and write $N_x(\alpha_j,m)$ for the number of intervals $I$ in ${\mathcal{D}}_m$ such that $2^{-m\alpha_j}\le\mu_x(I)< 2^{-m\alpha_{j+1}}$. By Corollary \[cor:unif-continuity\], for any fixed value of $q$, if $m\gg_q 1$ then, $$\label{eq:bound-norm-mu-m-2}
N_x(\alpha_j,m) 2^{-m q\alpha_j} \le \|\mu_x^{(m)}\|_q^q \le 2^{-m(T(q)-\delta)}.$$ Applying this to $q=q_0+\eta$, and using , we estimate $$\begin{aligned}
N_x(\alpha_j,m) 2^{-m q_0\alpha_j} &\le 2^{m \eta \alpha_j} 2^{-m(T(q_0+\eta)-\delta)} \\
&\le 2^{2\delta m} 2^{-j\delta\eta m} 2^{-T(q_0)m}.\end{aligned}$$ Let $S_x$ be the sum in the left-hand side of that we want to estimate. Using that $\delta=\eta\sigma/(4+2 q_0)$, we conclude that $$\begin{aligned}
S_x &\le \sum_{j: \delta (j+1)\ge \sigma} N_x(\alpha_j,m) 2^{-m q_0\alpha_{j+1}} \\
&\le \sum_{j: \delta (j+1)\ge \sigma} 2^{\delta q_0 m} 2^{2\delta m} 2^{-j\delta\eta m} 2^{-T(q_0)m}\\
&\le \sum_{j\ge 0} 2^{-j\delta\eta m} 2^{(2+q_0)\delta m} 2^{-\eta\sigma m} 2^{-T(q_0)m} \\
&\le O_{\delta\eta}(1) 2^{(\eta\sigma/2-\eta\sigma)m} 2^{-T(q_0)m},\end{aligned}$$ as claimed.
\[lem:Lq-sum-over-small-set\] Let $q_0>1$ be such that $\alpha_0=T'(q_0)$ exists. Given $\kappa>0$, there is ${\varepsilon}={\varepsilon}(\kappa,q_0)>0$ such that the following holds for large enough $m$ (in terms of $q_0,{\varepsilon}$) and all $x\in X$.
If $\mathcal{D}'\subset{\mathcal{D}}_m$ has $\le 2^{(T^*(\alpha_0)-\kappa)m}$ elements, then $$\sum_{I\in\mathcal{D}'} \mu_x(I)^{q_0} \le 2^{-(T(q_0)+{\varepsilon})m}$$ for all $x\in X$.
Let $\sigma=\kappa/(2 q_0)$ and fix $x\in X$. In light of Lemma \[lem:Lq-sum-large-mass\], we only need to worry about those $I$ with $\mu_x(I)\le 2^{-m(\alpha_0-\sigma)}$. But $$\begin{aligned}
\sum\{ \mu_x(I)^{q_0}: I\in\mathcal{D}',\mu_x(I) \le 2^{-m(\alpha_0-\sigma)} \} &\le 2^{(T^*(\alpha_0)-\kappa)m} 2^{-(q_0\alpha_0-q_0\sigma)m} \\
&= 2^{-(\kappa-q_0\sigma)m}2^{-T(q_0)m}.\end{aligned}$$ By our choice of $\sigma$, $\kappa-q_0\sigma=\kappa/2>0$, so this gives the claim.
The second part of the following proposition can be used to give another (though closely related) proof of Proposition \[prop:cocycle\], and was obtained in [@PeresSolomyak00; @NPS12] in special cases. The first part is proved in a similar way, relying on Lemma \[lem:Lq-sum-over-small-set\].
\[prop:Lq-over-small-set-is-small\] Let $q>1$ be such that $\alpha=T'(q)$ exists.
1. Given $\kappa>0$, there is $\eta=\eta(\kappa,q)>0$ such that the following holds for all large enough $m$: for any $s\in{\mathbb{N}}$, $I\in{\mathcal{D}}_s$ and $x\in X$, if $\mathcal{D}'$ is a collection of intervals in ${\mathcal{D}}_{s+m}(I)$ with $|\mathcal{D}'|\le 2^{(T^*(\alpha)-\kappa)m}$, then $$\sum_{J\in\mathcal{D}'} \mu_x(J)^q \le 2^{-(T(q)+\eta)m} \mu_{x}(2 I)^q.$$
2. Given $\delta>0$, the following holds for all large enough $m$: for any $I\in{\mathcal{D}}_s$, $s\in{\mathbb{N}}$, and $x\in X$, $$\sum_{J\in{\mathcal{D}}_{s+m}(I)} \mu_x(J)^q \le 2^{-(T(q)-\delta)m} \mu_{x}(2 I)^q.$$
We prove (i) first. Let $n$ be the smallest integer such that ${\lambda}^n < 2^{-s-2}$. Let $y_j$ be the atoms of $\mu_{x,n}$ such that $[y_j,y_j+{\lambda}^{n}]\cap I\neq \varnothing$, let $p_j$ be their respective masses, and write $$\mu_{x,n,I} = \sum_j p_j \delta_{y_j}.$$ Then the support of $\mu_{x,n,I}$ is contained in the ${\lambda}^n$-neighborhood of $I$. Moreover, since $\delta_z * S_{{\lambda}^n}\mu_{T^n x}$ is supported on $[z,z+{\lambda}^n]$, thanks to , it follows from the self-similarity relation $\mu_x = \mu_{x,n}* S_{{\lambda}^n}\mu_{\mathbf{T}^n x}$ and the definition of $\mu_{x,n,I}$ that $\mu_x|_I =(\mu_{x,n,I}*S_{{\lambda}^n}\mu_{\mathbf{T}^n x})|_I$. Write $$p = \|\mu_{x,n,I}\|_1 = \sum_j p_j \le \mu_x(2 I),$$ using that, again by , the support of $\mu_{x,n}$ is contained in the ${\lambda}^n$ neighborhood of the support of $\mu_x$, and that $4 {\lambda}^n \le 2^{-s}$.
We can then estimate $$\begin{aligned}
\sum_{J\in{\mathcal{D}}'} \mu_x(J)^q &= \sum_{J\in{\mathcal{D}}'} \left( \sum_j p_j \delta_{y_j} * S_{{\lambda}^n}\mu_{\mathbf{T}^n x}(J) \right)^q\\
&= \sum_{J\in{\mathcal{D}}'} \left( \sum_j p_j \mu_{\mathbf{T}^n x}({\lambda}^{-n}(J-y_j)) \right)^q\\
&\le \sum_{J\in{\mathcal{D}}'} p^{q-1} \sum_j p_j \, \mu_{\mathbf{T}^n x}({\lambda}^{-n}(J-y_j))^q\\
&= p^{q-1}\sum_j p_j \sum_{J\in{\mathcal{D}}'} \mu_{\mathbf{T}^n x}({\lambda}^{-n}(J-y_j))^q,\end{aligned}$$ where we used the convexity of $t^q$ in the third line. Now for each fixed $j$, each interval ${\lambda}^{-n}(J-y_j)$ with $J\in{\mathcal{D}}'$ can be covered by $O_{{\lambda}}(1)$ intervals in ${\mathcal{D}}_m$, and reciprocally each interval in ${\mathcal{D}}_m$ hits at most $2$ intervals among the ${\lambda}^{-n}(J-y_j)$. We deduce from Lemmas \[lem:Holder\] and \[lem:Lq-sum-over-small-set\] that, still for a fixed $j$, $$\sum_{J\in{\mathcal{D}}'} \mu_{\mathbf{T}^n x}({\lambda}^{-n}(J-y_j))^q \le O_{{\lambda},q}(1) 2^{-(T(q)+{\varepsilon})m},$$ provided $m$ is taken large enough, where ${\varepsilon}={\varepsilon}(\kappa,q)>0$ is given by Lemma \[lem:Lq-sum-over-small-set\]. Combining the last three displayed equations yields the first claim with $\eta={\varepsilon}/2$.
The second claim follows in the same way, adding over ${\mathcal{D}}_{s+m}(I)$ instead of ${\mathcal{D}}'$, and using Corollary \[cor:unif-continuity\] instead of Lemma \[lem:Lq-sum-over-small-set\].
Proof of Theorem \[thm:L-q-dim-dyn-ssm\] {#sec:proof-of-main-thm}
========================================
Flattening of Lq norm for dynamically driven self-similar measures
------------------------------------------------------------------
As noted in the introduction, we aim to prove a generalization of [@Hochman14 Theorem 1.1], by following the same broad outline. One of the key steps in the proof of [@Hochman14 Theorem 1.1] consists in showing that convolving a self-similar measure with an arbitrary measure, on which only a lower bound on the entropy is assumed, results in an entropy increment: see [@Hochman14 Corollary 5.5]. In turn, this is derived from the inverse theorem of [@Hochman14] by proving that the entropy of self-similar measures is roughly constant at most scales and locations, a property that Hochman termed *uniform entropy dimension*, see [@Hochman14 Definition 5.1 and Proposition 5.2] for precise details. Once again, we will follow a different path to obtain a statement for $L^q$ norms which is similar in spirit.
We continue to work with a fixed pleasant model $(X,\mathbf{T},\Delta,{\lambda})$, and the function $T$ from Proposition \[prop:cocycle\].
\[thm:conv-with-ssm-flattens\] Given $\sigma>0$ and $q>1$ such that $T$ is differentiable at $q$ and $T(q)<q-1$, there is ${\varepsilon}={\varepsilon}(\sigma,q)>0$ such that the following holds for $m$ large enough in terms of all previous parameters:
If $\nu$ is a $2^{-m}$-measure with $\|\nu\|_q^{q'}\le 2^{-\sigma m}$, and $x\in X$, then $$\| \nu * \mu_x^{(m)}\|_q^q \le 2^{-(T(q)+{\varepsilon}) m}.$$
The analogy with [@Hochman14 Corollary 5.5] is clear. However, there is no useful analog of the notion of uniform entropy dimension for $L^q$ norms. One of the key differences is that nearly all of the $L^q$ norm may be (and often is) captured by sets of extremely small measure; while sets of small measure also have small entropy. Instead, we will use the regularity of the multifractal spectrum established in the previous section in the following manner: if the flattening claimed in the theorem does not hold, then the inverse theorem provides a regular set $A$ which captures much of the $L^q$ norm of $\mu_x$. The upper bound on $\|\nu\|_q^{q'}$, together with (v)–(vi) in the inverse theorem imply that $A$ has nearly full branching for a positive proportion of $2^{D}$-scales, so it must have substantially less than average branching also on a positive proportion of scales. On the other hand, we will call upon the lemmas from the previous section to show that, in fact, $A$ must have nearly constant branching on nearly all scales (this is the part that uses the differentiability of $T$ at $q$), obtaining the desired contradiction.
Suppose $\nu$ is a $2^{-m}$-measure with $\|\nu\|_q^{q'} \le 2^{-\sigma m}$. In the course of the proof, we will choose $\delta'>0$ small enough in terms of $\sigma, q$; $\delta>0$ small enough in terms of $\delta',\sigma,q$; and $D_0\in{\mathbb{N}}$ large enough in terms of $\delta,\sigma,q$. We will also introduce other small quantities, all of which depend on $\sigma,q$ only.
Write $\alpha=T'(q)$, and define $\kappa$ as $$\label{eq:def-kappa}
\kappa = (1-T^*(\alpha))\sigma/4.$$ (The reason for this choice will become clear later.) Then $\kappa>0$ thanks to Lemma \[lem:f-alpha-smaller-than-one\], and the assumption $T(q)<q-1$.
We fix $x\in X$ for the rest of the proof, and observe that all estimates will in fact be independent of $x$. Let $\xi>0$ be a small enough number to be chosen later in terms of $\sigma$ and $q$. If $\|\mu_x^{(m)}\|_q^q \le 2^{-(T(q)+\xi)m}$ then there is nothing to do, so from now on we assume that $$\label{eq:lower-bound-assumption}
\|\mu_x^{(m)}\|_q^q \ge 2^{-(T(q)+\xi)m}.$$
We apply Proposition \[prop:Lq-over-small-set-is-small\] to obtain a sufficiently large $D_0$ (in terms of $\delta,\sigma,q$, with $\delta$ yet to be specified) such that
1. For any $D'\ge D_0-2$, any $I\in{\mathcal{D}}_{s'}$, $s'\in{\mathbb{N}}$, and any subset ${\mathcal{D}}'\subset {\mathcal{D}}_{s'+D'}(I)$ with $|{\mathcal{D}}'|\le 2^{(T^*(\alpha)-\kappa)D'}$, $$\sum_{J\in{\mathcal{D}}'} \mu_x(J)^q \le 2^{-(T(q)+\eta)D'}\mu_{x}(2I)^q,$$ where $\eta$ depends on $\kappa$, hence on $\sigma,q$ only.
2. For any $D'\ge D_0-2$ and any $I\in{\mathcal{D}}_{s'}$, $s'\in{\mathbb{N}}$, $$\sum_{J\in{\mathcal{D}}_{s'+D'}(I)} \mu_x(J)^q \le 2^{-(T(q)-\delta)D'}\mu_{x}(2I)^q.$$
3. $1/D_0 <\delta$.
Let ${\varepsilon}>0,D\in{\mathbb{N}}$ be the numbers given by Theorem \[thm:inverse-thm\] applied to $\delta, D_0$ and $q$. Suppose $$\label{eq:non-flattening-assumption}
\| \nu*\mu_x^{(m)}\|_q^q \ge 2^{-(T(q)+{\varepsilon}q/2) m}$$ We will derive a contradiction from this provided $m=\ell D$ is large enough, proving the theorem with ${\varepsilon}q/2$ in place of ${\varepsilon}$ (if $m$ is not of the form $\ell D$, we apply the argument to $\lfloor m/D\rfloor D$ instead).
By Corollary \[cor:unif-continuity\], if $m$ is large enough (depending only on ${\varepsilon},q$) and holds, then $$\| \nu*\mu_x^{(m)}\|_q \ge 2^{-{\varepsilon}m} \|\mu_x^{(m)}\|_q.$$ We apply Theorem \[thm:inverse-thm\] to obtain (assuming $m$ is large enough) a set $A\subset{\text{supp}}(\mu_x^{(m)})$ as in the theorem, with corresponding branching numbers $R'_s$.
The key to the proof is to show, using the structure of $A$ provided by Theorem \[thm:inverse-thm\], that $$\label{eq:small-branching-at-many-scales}
|\{ s\in [\ell]: R'_s \le 2^{(T^*(\alpha)-\kappa)D} \}| \ge \gamma \ell,$$ where $\gamma>0$ depends on $\sigma,q$ only (and $\kappa$ is given by ). We first show how to complete the proof assuming this. Consider the sequence $$L_s = -\log\sum_{I\in{\mathcal{D}}_{s D}(A)} \mu_x(I)^q.$$ By (2) applied with $s'=sD+2$ and $D'=D-2$, $$L_{s+1} \ge (T(q)-\delta)(D-2) -\log \sum_{I\in {\mathcal{D}}_{sD+2}(A)} \mu_x(2I)^q.$$ But if $I\in {\mathcal{D}}_{sD+2}(A)$, then $2I$ is contained in a single interval in ${\mathcal{D}}_{sD}(A)$ by property (A-iv) from Theorem \[thm:inverse-thm\], and conversely $J\in{\mathcal{D}}_{sD}(A)$ hits at most two intervals $2I$, $I\in{\mathcal{D}}_{sD+2}(A)$. We deduce that $$L_{s+1} \ge L_s +(T(q)-\delta)(D-2)-1$$ for all $s\in[\ell]$. Likewise, by (1), $$L_{s+1} \ge L_s + (T(q)+\eta)(D-2)-1,$$ whenever $R'_s \le 2^{(T^*(\alpha)-\kappa)D}$. Recall that $\eta$ depends on $\kappa$, and hence on $\sigma,q$ only. In light of , and using also (3), we have $$\begin{aligned}
L_{\ell} &\ge (T(q)+\eta)\gamma \ell (D-2) + (T(q)-\delta)(1-\gamma)\ell (D-2)-\ell\\
&\ge (T(q)+\eta\gamma-\delta(1-\gamma))m - 2\delta (T(q)+\eta) m -\delta m.\end{aligned}$$
Hence, by choosing $\delta$ small enough in terms of $T(q),\gamma$ and $\eta$ (hence only in terms of $\sigma,q$) we can ensure that, for $m$ large enough, $$L_{\ell} = -\log\|\mu_x^{(m)}|_A\|_q^q \ge (T(q)+\eta\gamma/2)m.$$ On the other hand, by (A-i) in Theorem \[thm:inverse-thm\] and our assumption , $$\|\mu_x^{(m)}|_A\|_q^q \ge 2^{-q\delta m}\|\mu_x^{(m)}\|_q^q \ge 2^{-q\delta m} 2^{-(T(q)+\xi)m}.$$ From the last two displayed equations, $$\eta\gamma/2 \le q\delta +\xi.$$ Recall that $\eta,\gamma$ are given in terms of $\sigma,q$ only, while $\delta,\xi$ have not yet been specified. By ensuring $q\delta <\eta\gamma/8$ and $\xi\le \eta\gamma/8$ we reach a contradiction. Hence cannot hold, which is what we wanted to show.
It remains to establish . The idea is very simple: Theorem \[thm:inverse-thm\] (together with the assumption that $\|\nu\|_q^{q'}\le 2^{-\sigma m}$) imply that $A$ has “nearly full branching” on a positive proportion of scales. On the other hand, Lemma \[lem:size-set-A-in-terms-of-f-alpha\] says the size of $A$ is at most roughly $2^{T^*(\alpha)m}\ll 2^m$ (by Lemma \[lem:f-alpha-smaller-than-one\]), so there must be a positive proportion of scales on which the average $2^D$-adic branching is far smaller than $2^{T^*(\alpha)D}$, which is what says.
We proceed to the details. Using (A-i), (A-ii) in Theorem \[thm:inverse-thm\] and , we get that (for $m\gg_\delta 1$) there is ${\widetilde}{\alpha}>0$ such that $\mu_x(a)\in [2^{-{\widetilde}{\alpha} m},2^{1-{\widetilde}{\alpha} m}]$ for all $a\in A$, and $$\label{eq:sum-Lq-in-A-large}
\sum_{I\in{\mathcal{D}}_m(A)} \mu_x(I)^q \ge 2^{-q\delta m} \sum_{I\in{\mathcal{D}}_m}\mu_x(I)^q \ge 2^{-(T(q)+ q\delta+\xi)m}.$$ We let $\delta\le \delta'$ and $\xi$ be small enough in terms of $\delta'$ (where $\delta'$ is yet to be specified) that, invoking Lemma \[lem:size-set-A-in-terms-of-f-alpha\], $$\label{eq:size-A-upper-bound}
|A| \le 2^{(T^*(\alpha)+\delta')m}.$$
Let $\mathcal{S}'=[\ell]\setminus \mathcal{S}$, where $\mathcal{S} = \{ s: R'_s \ge 2^{(1-\delta)D}\}$. Using (A-iii) in Theorem \[thm:inverse-thm\], we see that $$\label{eq:size-A-lower-bound}
|A| = \prod_{s=0}^{\ell-1} R'_s \ge 2^{(1-\delta)D|\mathcal{S}|} \prod_{s\in\mathcal{S}'} R'_s.$$ Let $m_1=D|\mathcal{S}|$, $m_2=D|\mathcal{S}'|=m-m_1$. Combining and , and using that $\delta\le \delta'$, we deduce $$\label{eq:upper-bound-Rs-S-prime}
\prod_{s\in\mathcal{S}'} R'_s \le 2^{-(1-\delta)m_1} 2^{(T^*(\alpha)+\delta')m} \le 2^{-(1-T^*(\alpha)-2\delta')m_1} 2^{(T^*(\alpha)+\delta')m_2}.$$ Note that $1-T^*(\alpha)>0$ by Lemma \[lem:f-alpha-smaller-than-one\]; we assume that $\delta'$ is small enough that $1-T^*(\alpha)-2\delta'>0$. Using (vi) in Theorem \[thm:inverse-thm\], and the assumptions and $\|\nu\|_q^{q'}\le 2^{-\sigma m}$, we further estimate $$\label{eq:bound-total-branching-scales}
(\sigma - \delta)m \le m_1 \le \left( (T(q)+\xi)/(q-1)+\delta\right)m.$$ We can plug in the left inequality (together with $m_2\le m$) into , to obtain the key estimate $$\log\prod_{s\in\mathcal{S}'} R'_s \le \left(T^*(\alpha)+ \delta' -(1-T^*(\alpha)-2\delta')(\sigma-\delta)\right)m_2.$$ Recalling , this shows that by making $\delta'$ (hence also $\delta$) small enough in terms of $\sigma,q$ only, we have $$\log\prod_{s\in\mathcal{S}'} R'_s \le (T^*(\alpha)-2\kappa) m_2.$$ Let $\mathcal{S}_1 = \{ s\in\mathcal{S}': \log R'_s \le (T^*(\alpha)-\kappa)D\}$. Recall that our goal is to show , i.e. $|\mathcal{S}_1|\ge \gamma(q,\sigma)\ell$. We have $$D|\mathcal{S}'\setminus\mathcal{S}_1| \le \frac{1}{T^*(\alpha)-\kappa}\sum_{s\in\mathcal{S}'\setminus\mathcal{S}_1} \log R'_s \le \frac{T^*(\alpha)-2\kappa}{T^*(\alpha)-\kappa} D|\mathcal{S}'|,$$ so that, using the right-most inequality in , and recalling that $D|\mathcal{S}'|=m-m_1$, $$D|\mathcal{S}_1| \ge \frac{\kappa (m-m_1)}{T^*(\alpha)-\kappa} \ge \left(\frac{\kappa(1-(T(q)+\xi)/(q-1)-\delta)}{T^*(\alpha)-\kappa}\right) m.$$ By ensuring that $\delta,\xi$ are small enough in terms of $q$, the right-hand side above can be bounded below by $$\left(\frac{\kappa(1-T(q)/(q-1))/2}{T^*(\alpha)-\kappa}\right) m,$$ confirming that holds.
Lq norms of discrete approximations at finer scales
---------------------------------------------------
Theorem \[thm:L-q-dim-dyn-ssm\] will be an easy consequence of the following proposition, which relies on Theorem \[thm:conv-with-ssm-flattens\]. It is an analog of [@Hochman14 Theorem 1.4], and we follow a similar outline.
\[prop:ssm-scale-Rm-norm\] Let $(X,\mathbf{T},\Delta,{\lambda})$ be a pleasant model, and let $T$ be the function from Proposition \[prop:cocycle\]. Let $q\in (1,\infty)$ be such that $T$ is differentiable at $q$ and $T(q)<q-1$, and let $x\in X$ be such that $$\label{eq:Lq-dim-exists-equal-Tq}
\lim_{m\to\infty} \frac1m \log\|\mu_x^{(m)}\|_q^q = -T(q).$$
Fix $R\in{\mathbb{N}}$. Then $$\lim_{n\to\infty} \frac{\log \|\mu_{x,n}^{(Rm(n))}\|_q^q}{n\log{\lambda}} = T(q),$$ where $m(n)$ is the smallest integer with $2^{-m(n)}\le {\lambda}^n$.
Fix $n\in{\mathbb{N}}$. We write $m=m(n)$ for simplicity, and allow all implicit constants to depend on $q$ and the model only. Using the self-similarity relation and Lemma \[lem:discr-norm-conv-equivalence\], we get $$\begin{aligned}
\|\mu_x^{((R+1)m)}\|_q^q &\le O(1) \| \mu_{x,n}^{((R+1)m)} * (S_{{\lambda}^n} \mu_{\mathbf{T}^n x})^{((R+1)m)} \|_q^q \\
&= O(1) \big\| \sum_{I\in{\mathcal{D}}_m} \mu_{x,n}(I) (\mu_{x,n})_I^{((R+1)m)} * (S_{{\lambda}^n} \mu_{\mathbf{T}^n x})^{((R+1)m)} \big\|_q^q.\end{aligned}$$ Here $(\mu_{x,n})_I = \mu_{x,n}|_I / \mu_{x,n}(I)$ is the normalized restriction of $\mu_{x,n}$ to $I$ (note that we are only summing over $I$ such that $\mu_{x,n}(I)>0$). Since the measures $(\mu_{x,n})_I^{((R+1)m)} * (S_{{\lambda}^n} \mu_{\mathbf{T}^n x})^{((R+1)m)}$ are supported on $I+[0,{\lambda}^n]$, the support of each of them hits the supports of $O(1)$ others. We can then apply Lemma \[lem:L-q-norm-almost-disj-supports\] to obtain $$\|\mu_x^{((R+1)m)}\|_q^q \le O(1)\sum_{I\in{\mathcal{D}}_m} \mu_{x,n}(I)^q \|(\mu_{x,n})_I^{((R+1)m)} * (S_{{\lambda}^n} \mu_{\mathbf{T}^n x})^{((R+1)m)} \|_q^q$$ Let $\rho_{x,I} = S_{{\lambda}^{-n}}(\mu_{x,n})_I$ (we suppress the dependence on $n$ from the notation, but keep it in mind). Note that $S_a(\eta)*S_a(\eta')=S_a(\eta*\eta')$ for any $a>0$ and measures $\eta,\eta'$. It follows from Lemmas \[lem:Holder\] and \[lem:discr-norm-conv-equivalence\] that $$\|(\mu_{x,n})_I^{((R+1)m)} * (S_{{\lambda}^n} \mu_{\mathbf{T}^n x})^{((R+1)m)} \|_q^q \le O(1) \| \rho_{x,I}^{(Rm)} * \mu_{\mathbf{T}^n x}^{(Rm)} \|_q^q,$$ so that, combining the last two displayed formulas, $$\label{eq:estimate-mu-Rp1}
\|\mu_x^{((R+1)m)}\|_q^q \le O(1) \sum_{I\in{\mathcal{D}}_m} \mu_{x,n}(I)^q \| \rho_{x,I}^{(Rm)} * \mu_{\mathbf{T}^n x}^{(Rm)} \|_q^q.$$ On the other hand, using Lemma \[lem:Holder\] again, $$\label{eq:estimate-mu-m-Rp1}
\|\mu_{x,n}^{((R+1)m)}\|_q^q = \sum_{I\in{\mathcal{D}}_m} \mu_{x,n}(I)^q \|(\mu_{x,n})_I^{((R+1)m)}\|_q^q \ge \Omega(1) \sum_{I\in{\mathcal{D}}_m} \mu_{x,n}(I)^q \| \rho_{x,I}^{(Rm)} \|_q^q.$$
Fix $\sigma>0$, and let $\mathcal{D}' = \{ I\in{\mathcal{D}}_{m}: \| \rho_{x,I}^{(Rm)}\|_q^q \le 2^{-\sigma m}\}$. According to Theorem \[thm:conv-with-ssm-flattens\], there is ${\varepsilon}={\varepsilon}(\sigma,q)>0$ such that, if $n$ is taken large enough, then $$I\in\mathcal{D}' \quad\Longrightarrow\quad\|\rho_{x,I}^{(Rm)} * \mu_{\mathbf{T}^{n}(x)}^{(Rm)}\|_q^q \le 2^{-(T(q)+{\varepsilon})Rm}.$$ Applying this to , we get $$\begin{aligned}
\|\mu_x^{((R+1)m)}\|_q^q &\le O(1) 2^{-(T(q)+{\varepsilon})Rm} \sum_{I\in{\mathcal{D}}'} \mu_{x,n}(I)^q + O(1) \sum_{I\notin {\mathcal{D}}'} \mu_{x,n}(I)^q \| \mu_{\mathbf{T}^n x}^{(Rm)} \|_q^q \\
&\le O(1) 2^{-(T(q)+{\varepsilon})Rm} \|\mu^{(m)}_{x}\|_q^q + O(1)\| \mu_{\mathbf{T}^n x}^{(Rm)} \|_q^q \sum_{I\notin{\mathcal{D}}'} \mu_{x,n}(I)^q\\\end{aligned}$$ using Young’s inequality in the first line, and Lemma \[lem:comparison-mu-m-mu\] in the second. On the other hand, our assumption implies that $$2^{-(T(q)+{\varepsilon})Rm} \|\mu^{(m)}_{x}\|_q^q \le 2^{-{\varepsilon}m/2} \|\mu_x^{((R+1)m)}\|_q^q$$ if $n$ is large enough (depending on $x$ and $R$). Inspecting the last two displayed equations, we deduce that if $n\gg_{x,\sigma} 1$, then $$\sum_{I\notin {\mathcal{D}}'} \mu_{x,n}(I)^q \ge \Omega(1) \frac{\|\mu_x^{((R+1)m)}\|_q^q}{\|\mu_{\mathbf{T}^n x}^{(Rm)} \|_q^q } \ge 2^{-m(T(q)+\sigma)},$$ where for the right-most inequality we used the assumption and Corollary \[cor:unif-continuity\]. Recalling , we conclude that $$\begin{aligned}
\|\mu_{x,n}^{((R+1)m)}\|_q^q &\ge \Omega(1) \sum_{I\notin{\mathcal{D}}'} \mu_{x,n}(I)^q \| \rho_{x,I}^{(Rm)} \|_q^q \\
&\ge \Omega(1) 2^{-\sigma m}\sum_{I\notin{\mathcal{D}}'} \mu_{x,n}(I)^q \ge \Omega(1) 2^{-2\sigma m} 2^{-m T(q)}.\end{aligned}$$ The inequality $\|\mu_{x,n}^{((R+1)m)}\|_q^q \le \|\mu_{x,n}^{(m)}\|_q^q$ holds trivially, so that by Lemma \[lem:comparison-mu-m-mu\] $$\|\mu_{x,n}^{((R+1)m)}\|_q^q \le \|\mu_{x,n}^{(m)}\|_q^q \le 2^{\sigma m} 2^{-m T(q)},$$ provided $n\gg_\sigma 1$. Since $\sigma>0$ was arbitrary and $2^{-m}=\Theta({\lambda}^n)$, this concludes the proof.
Proof of Theorem \[thm:L-q-dim-dyn-ssm\] {#proof-of-theorem-thml-q-dim-dyn-ssm}
----------------------------------------
We can now conclude the proof of the theorem.
We continue to write $m=m(n)=\lceil n\log(1/{\lambda})\rceil$. To begin, we note that, without any assumptions on the model, for any $q\in (1,\infty)$, $$\label{eq:main-thm-upper-bound-1}
\|\mu_{x,n}^{(m)}\|_q^q \ge \|\mu_{x,n}\|_q^q \ge \prod_{i=0}^{n-1} \|\Delta(\mathbf{T}^i x)\|_q^q.$$ (The latter inequality is an equality if and only if there are no overlaps among the atoms of $\mu_{x,n}$.) By our assumptions on the map $\Delta(\cdot)$, the function $x\mapsto \|\Delta(x)\|_q^q$ is bounded away from zero and continuous ${\mathbb{P}}$-a.e. Then, by unique ergodicity, $$\label{eq:main-thm-upper-bound-2}
\lim_{n\to\infty} \frac1n\log\prod_{i=0}^{n-1} \|\Delta(\mathbf{T}^i x)\|_q^q = \int_X \log\|\Delta(x)\|_q^q \,d\mathbb{P}(x)\quad\text{uniformly in }x\in X.$$ This property of uniquely ergodic systems is well known, or one can apply Lemma \[lem:uniform-convergence\] to the additive sequence $\log\prod_{i=0}^{n-1} \|\Delta(\mathbf{T}^i x)\|_q^q$. Since $\|\nu^{(m)}\|_q^{q'} \ge 2^{-m}$ for any probability measure $\nu$, from , and Lemma \[lem:comparison-mu-m-mu\], we deduce that $$\limsup_{m\to\infty} -\frac{\log\|\mu_x^{(m)}\|_q^q}{(q-1)m} \le \min\left( \frac{\int_X \log\|\Delta(x)\|_q^q \,d\mathbb{P}(x)}{(q-1)\log{\lambda}},1\right),$$ uniformly in $x\in X$. In light of this and Corollary \[cor:unif-continuity\], the proof will be completed if we can show that for each $q\in (1,\infty)$, either $T(q)\ge q-1$ (so that in fact $T(q)=q-1$) or $$\label{eq:formula-T-q-want-to-prove}
T(q) = \frac{\int_X \log\|\Delta(x)\|_q^q \,d\mathbb{P}(x)}{\log{\lambda}}.$$ Since $T(q)$ is concave, it is enough to prove this for all $q$ such that $T$ is differentiable at $q$. Hence, we fix $q$ such that $T(q)<q-1$ and $T$ is differentiable at $q$, and we set out to prove .
By Proposition \[prop:cocycle\] and the exponential separation assumption, there is $x\in X$ such that holds, and the atoms of $\mu_{x,n}$ are ${\lambda}^{R n}$-separated for infinitely many $n$ and some $R\in{\mathbb{N}}$ (indeed, this holds for ${\mathbb{P}}$-almost all $x$). We known from Proposition \[prop:ssm-scale-Rm-norm\] that $$\label{eq:tau-mu-x-equals-T}
\lim_{n\to\infty} \frac{\log \|\mu_{x,n}^{(Rm(n))}\|_q^q}{n\log{\lambda}} = T(q).$$ On the other hand, if $n$ is such that the atoms of $\mu_{x,n}$ are ${\lambda}^{R n}$-separated then (since ${\lambda}^{Rn}\ge 2^{-R m(n)}$) $$\label{eq:tau-mu-x-exp-separation}
\|\mu_{x,n}^{(Rm(n))}\|_q^q = \|\mu_{x,n}\|_q^q = \prod_{i=0}^{n-1} \| \Delta(\mathbf{T}^i x)\|_q^q.$$ Combining Equations , and , we conclude that holds, finishing the proof.
Lq dimensions of self-similar measures, and applications {#sec:dim-ssm-and-applications}
========================================================
In this section we apply Theorem \[thm:L-q-dim-dyn-ssm\] to prove Theorem \[thm:infinity-dim-BCs\]; in fact, we will obtain a far more general result for self-similar measures on the line. We also derive some geometric applications.
Background on self-similar sets and measures {#subsec-sssm}
--------------------------------------------
We begin by recalling some basic facts about self-similar sets and measures, fixing notation along the way. For further background, see e.g. [@Falconer97].
Let $\mathcal{I}$ be a finite set with at least two elements. Let $(f_i)_{i\in\mathcal{I}}$ be a collection of strictly contracting similarities on ${\mathbb{R}}^d$ (usually referred to as an *iterated function system* or IFS). That is, $f_i(x)=\lambda_i O_i(x)+t_i$, where ${\lambda}_i\in (0,1)$, $O_i$ is an orthogonal map on ${\mathbb{R}}^d$, and $t_i\in{\mathbb{R}}^d$. Then there exists a unique nonempty compact set $A\subset{\mathbb{R}}^d$ such that $$A = \bigcup_{i\in\mathcal{I}} f_i(A).$$ If a probability vector $(p_i)_{i\in\mathcal{I}}$ is also given, then there is a unique Borel probability measure $\mu$ such that $$\mu = \sum_{i\in\mathcal{I}} p_i\, f_i\mu.$$ Moreover, ${\text{supp}}(\mu)\subset A$, with equality if $p_i>0$ for all $i$.
If one replaces $\mathcal{I}$ by $\mathcal{I}^n$, $(f_i)$ by $(f_{i_1}\circ\cdots\circ f_{i_n})$, and $(p_i)$ by $(p_{i_1}\cdots p_{i_n})$, then the invariant set $A$ and the invariant measure $\mu$ do not change.
The Hausdorff and box counting dimensions agree for any self-similar set. The *open set condition* holds if there is a nonempty open set $U$ such that $f_i(U)\subset U$ and $f_i(U)\cap f_j(U)=\varnothing$ for all $i\neq j\in\mathcal{I}$. In this case, the Hausdorff dimension of $A$ is the only positive number $s$ such that $\sum_{i\in\mathcal{I}} {\lambda}_i^s=1$. Moreover, the *uniform self-similar measure* $\mu$ given by the weights ${\lambda}_i^s$ satisfies $\mu(B(x,r))=\Theta(r^s)$ for $x\in A$ and $r\in (0,1]$, with the implicit constants depending only on $(f_i)$.
In this article we will be mostly concerned with *homogeneous* iterated function systems: those for which $\lambda_i\equiv{\lambda}$ and $O_i\equiv O$ are constant for all $i\in\mathcal{I}$. In this case, the self-similar set $A$ can be explicitly written as an infinite arithmetic sum: $$A = \sum_{i=0}^\infty S_{{\lambda}^i}(O^i E),$$ where $E=\{t_i:i\in\mathcal{I}\}$ is the set of translations, and the self-similar measure $\mu$ can be expressed as an infinite convolution: $$\mu = *_{i=0}^\infty S_{{\lambda}^i}(O^i\Delta),$$ where $\Delta= \sum_{i\in\mathcal{I}} p_i \delta(t_i)$. Note that in dimension $1$ (where most of the focus will be), $O$ is either the identity or minus the identity, and the latter case can always be reduced to the first by iterating the IFS, as above.
If the system is homogeneous and the open set condition holds, then there is $c>0$ such that for all $n\in{\mathbb{N}}$, the points in the finite approximation $$A_n = \sum_{i=0}^{n-1} S_{{\lambda}^i}(O^i E)$$ are all distinct (i.e. there are $|E|^n$ of them) and $c{\lambda}^n$ separated. See e.g. [@LauNgai99 Example 1 in Section 6]. Moreover, in this case the $L^q$ dimensions of $\mu$ are given by $$D(\mu,q) = \frac{\log \|\Delta\|_q^q}{(q-1)\log{\lambda}}.$$ The right-hand side majorizes the $L^q$ dimension without any separation assumption (always assuming homogeneity).
Finally, we point out that the limit in the definition of $L^q$ dimension exists for arbitrary self-similar measures, see [@PeresSolomyak00].
Lq dimensions and Frostman exponents of self-similar measures
-------------------------------------------------------------
Next, we obtain Theorem \[thm:infinity-dim-BCs\] as a special case of a result valid for more general self-similar measures on ${\mathbb{R}}$. Fix $\Delta=\sum_{i\in\mathcal{I}} p_i \delta_{t_i}\in\mathcal{A}$ and ${\lambda}\in (0,1)$, and let $$\label{eq:def-ssm-hom}
\mu = \mu_{\Delta,{\lambda}} = *_{i=0}^\infty S_{{\lambda}^i}\Delta$$ be the associated self-similar measure. Bernoulli convolutions correspond to the special case $\Delta=\frac12(\delta_{-1}+\delta_1)$.
\[def:exponential-separation-hom-ssm\] Given a set $E\subset{\mathbb{R}}$ and $n\in{\mathbb{N}}$, we let $\mathcal{P}_{E,n}$ be the family of non-zero polynomials of degree at most $n$ and coefficients in $E-E$. Slightly abusing notation, we write $\mathcal{P}_{\Delta,n}=\mathcal{P}_{{\text{supp}}(\Delta),n}$.
We say that a measure $\mu$ as in has *exponential separation* if there exists $R>0$ such that, for infinitely many $n$, $$\label{eq:ssm-exponential-separation}
\min_{P\in\mathcal{P}_{\Delta,n}} |P({\lambda})| \ge {\lambda}^{R n}.$$
Note that this is a property of ${\text{supp}}(\Delta)$ and ${\lambda}$, and not of the particular distribution of mass on ${\text{supp}}(\Delta)$. Recall that if the open set condition holds, then there is $c>0$ such that $$|P({\lambda})| \ge c{\lambda}^n \quad\text{for all }n\in{\mathbb{N}}, P\in\mathcal{P}_{\Delta,n}.$$ Hence, exponential separation is a weaker property than the open set condition.
\[thm:dim-ssm\] Let $\mu=\mu_{\Delta,{\lambda}}$ be a self-similar measure as in with exponential separation. Then for all $q\in (1,+\infty)$, $$D(\mu,q) = \min\left(\frac{\log\|\Delta\|_q^q}{(q-1)\log{\lambda}},1\right).$$ In particular, for every $$\alpha < \min\left(\frac{\log \|\Delta\|_\infty}{\log{\lambda}},1\right),$$ it holds that $\mu(B(x,r)) \le r^\alpha$ for all $r\in (0,r_0(\alpha))$ and all $x\in{\mathbb{R}}$.
Before presenting the short deduction from Theorem \[thm:L-q-dim-dyn-ssm\], we make some remarks on this statement:
1. Theorem \[thm:infinity-dim-BCs\] is an immediate consequence of the last claim in the theorem.
2. Recall from §\[subsec-sssm\] that the claim in the theorem is well-known under the open set condition. The point is that the separation assumption is far weaker than the open set condition. This notion of “exponential separation” was introduced in [@Hochman14] and, as explained there, it is a quantitative version of the “no exact overlaps” condition which is conjectured to already imply the claims in Theorem \[thm:dim-ssm\].
3. As shown by Hochman [@Hochman14], if ${\lambda}$ and all points in ${\text{supp}}(\Delta)$ are algebraic, then either holds for all $n$, or ${\lambda}$ is a root of some $P\in\mathcal{P}_{\Delta,n}$, $n\in{\mathbb{N}}$ (which corresponds to an exact overlap).
4. Hochman ([@Hochman14 Theorem 1.8], [@Hochman15 Theorem 1.10]) has also shown that in quite general parametrized families of self-similar measures, the exponential separation assumption in Theorem \[thm:dim-ssm\] holds outside of a set of parameters of packing and Hausdorff co-dimension at least $1$.
5. The analog of Theorem \[thm:dim-ssm\] for exact (or Hausdorff) dimension was established by Hochman [@Hochman14 Theorem 1.1]. We recover his result in the homogeneous case by letting $q\to 1^{+}$ in Theorem \[thm:dim-ssm\].
6. Note that $q\mapsto-\log\|\Delta\|_q^q$ is linear if and only if $\Delta$ is uniform on its support; otherwise, it is a strictly concave real-analytic function. It follows from the theorem that, under the separation assumption , the map $q\mapsto \tau_\mu(q)$ is differentiable except, perhaps, at a single point $q>1$ such that $\|\Delta\|_q^q = {\lambda}^{q-1}$. It follows from a result of D-J. Feng [@Feng07] that the multifractal formalism holds for $\mu$ and all $q\in (1,\infty)$ outside, possibly, of this point. See [@Feng07] for details.
We apply Theorem \[thm:L-q-dim-dyn-ssm\] with a constant function $\Delta$ (corresponding to a one-point set $X=\{0\}$). Such a trivial model is clearly pleasant and satisfies the continuity assumption in Theorem \[thm:L-q-dim-dyn-ssm\]. The support of $\mu_n:=\mu_{0,n}$ is $$\sum_{i=0}^{n-1} S_{{\lambda}^i}(\Delta) = \left\{ \sum_{i=0}^{n-1} {\lambda}^i y_i : y_i\in{\text{supp}}(\Delta)\right\},$$ so the model has exponential separation if and only if $\mu$ has exponential separation. The application of Theorem \[thm:L-q-dim-dyn-ssm\] is therefore justified, and yields the claimed formula for $D_\mu(q)$. The latter claim for the Frostman exponent then follows from Lemma \[lem:Lq-dim-to-Frostman-exp\] by letting $q\uparrow\infty$.
Some applications
-----------------
We present some consequences of Theorem \[thm:dim-ssm\]. Recall that the one-dimensional Sierpiński Gasket $S$ is the set of all points in $[0,1]^2$ of the form $$\left\{ \sum_{n=1}^\infty X_n 3^{-n} : X_n\in \{(0,0),(0,1),(1,0)\} \right\}.$$ The gasket $S$ is a self-similar set, with open set condition, of Hausdorff dimension $1$. Furstenberg conjectured that all orthogonal projections of $S$ in directions with irrational slope also have Hausdorff dimension $1$; this was proved in [@Hochman14 Theorem 1.6]. We can deduce a stronger statement from Theorem \[thm:dim-ssm\]:
\[cor:sierpinski-gasket\] Let $\Pi_t(x,y)=x+t y$. For every Borel subset $A\subset S$ and for every $t\in{\mathbb{R}}\setminus\mathbb{Q}$, $$\operatorname{dim_{\mathsf{H}}}(\Pi_t A) = \operatorname{dim_{\mathsf{H}}}(A).$$
Let $\mu$ be the uniform self-similar measure on $S$, so that $\mu(B(x,r))=\Theta(r)$ for $x\in S$ and $r\le 1$. For each $t\in{\mathbb{R}}$, the projection $\Pi_t\mu$ is the uniform self-similar measure for the iterated function system $\{ x/3, x/3+1, x/3+t\}$. As shown in the proof of [@Hochman14 Theorem 1.6], this IFS satisfies the exponential separation hypothesis for all irrational $t$. From now on let $t$ be a fixed irrational number. We deduce from Theorem \[thm:dim-ssm\] that $$\Pi_t\mu(B(y,{\varepsilon})) = O_{t,\delta}({\varepsilon}^{1-\delta})\quad\text{for all }\delta>0.$$ In turn, Lemma \[lem:Frostman-exp-to-small-fiber\] says that $\Pi_t^{-1}(B(y,{\varepsilon}))$ can be covered by $O_{t,\delta}({\varepsilon}^{-\delta})$ balls of radius ${\varepsilon}$ for any $y\in{\mathbb{R}}$. Indeed, if $(x_j)_j$ is a maximal ${\varepsilon}$-separated subset of some set, then $(B(x_j,{\varepsilon}))_j$ covers the set.
Now fix a Borel subset $A\subset S$ of Hausdorff dimension $s$, and $\delta>0$. By Frostman’s Lemma (see e.g. [@Mattila95 Theorem 8.8]) there is a Borel probability measure $\nu$ supported on $A$ such that $\nu(B(x,{\varepsilon})) = O_{A,\delta}({\varepsilon}^{s-\delta})$ for all $x\in{\mathbb{R}}^2, {\varepsilon}>0$. It follows that $$\Pi_t\nu(B(y,{\varepsilon})) = O_{A,\delta,t}({\varepsilon}^{s-2\delta}).$$ Since $\delta>0$ was arbitrary, the conclusion follows from the mass distribution principle (see e.g. [@Falconer97 Proposition 2.1]).
The gasket $S$ could be replaced by the attractor of any iterated function system in the plane, satisfying the open set condition and of Hausdorff dimension at most $1$, of the form $(\lambda x+a_i,\lambda y+b_i)_{i\in\mathcal{I}}$ with $\lambda,a_i,b_i$ all rational. If ${\lambda},a_i,b_i$ are only assumed to be algebraic, then the same holds assuming that $t$ is transcendental, instead of irrational. The proof works verbatim since in this more general situation $\Pi_t S$ continues to be a self-similar set satisfying , see the proof of [@Shmerkin15 Theorem 5.3].
When the Hausdorff dimension of the self-similar set is larger than $1$ we cannot reach the same conclusion, but Lemma \[lem:Frostman-exp-to-small-fiber\] still provides an upper bound for the size of the fibers. We conclude this section by discussing some concrete classes of examples.
\[cor:intersection-Cantor-set\] Let $A\subset [0,1)$ be a $p$-Cantor set, $p\ge 2$. Then for every irrational number $t\in{\mathbb{R}}$ and any $u\in{\mathbb{R}}$, $$\operatorname{{\overline{dim}_{\mathsf{B}}}}(A\cap (t A+ u)) \le \min(2\operatorname{dim_{\mathsf{H}}}(A)-1,0).$$
The product set $S=A\times A$ is the attractor of an iterated function system with rational coefficients satisfying the open set condition, and $\operatorname{dim_{\mathsf{H}}}(S)=2\operatorname{dim_{\mathsf{H}}}A$. As pointed out above, it is shown in the proof of [@Shmerkin15 Theorem 5.3] that $\Pi_t S$ is a self-similar set satisfying whenever $t$ is irrational (the argument for this holds regardless of the dimension of the self-similar set). Since the fiber $\Pi_t^{-1}(u)\cap S$ is, up to an affine change of coordinates, equal to $A\cap (t A+ u)$, the conclusion follows from Theorem \[thm:dim-ssm\] and Lemma \[lem:Frostman-exp-to-small-fiber\].
The corollary generalizes to $T_p$-invariant sets, by embedding them in $p$-Cantor sets of arbitrarily close dimension, see the proof of Theorem \[thm:Furstenberg\] in §\[subsec:proof-of-Furst-conjecture\] below. The dimension of the intersections of the middle-thirds Cantor set with translates of itself (without scaling) was investigated by Hawkes [@Hawkes75], and this was greatly generalized to $T_p$-invariant sets by Kenyon and Peres [@KenyonPeres91]. Without scaling, the situation is very different; in particular, $\operatorname{dim_{\mathsf{H}}}(A\cap A+u) > 2\operatorname{dim_{\mathsf{H}}}(A)-1$ for many values of $u$. We mention also a related result of M. Hochman [@Hochman12] for invariant *measures*: if $\mu$ is $T_p$-invariant, $\operatorname{dim_{\mathsf{H}}}\mu\in (0,1)$ and $f(x)=t x+u$ with $\log t/\log p\notin {\mathbb{Q}}$, then $\mu$ and $f\mu$ are mutually singular.
Likewise, if $S$ is the standard Sierpiński gasket or the Sierpiński carpet, or more generally if $S$ is the attractor of an IFS of the form $\left(N^{-1}(x+a_i,y+b_i)\right)_{i\in I}$ with $(a_i,b_i)\in [N]^2$, and $|\mathcal{I}|>N$ (so that $\operatorname{dim_{\mathsf{H}}}S>1$), then $$\operatorname{{\overline{dim}_{\mathsf{B}}}}(S\cap\ell) \le \operatorname{dim_{\mathsf{H}}}(S)-1$$ for all lines $\ell$ with irrational slope. The intersections of these carpets with lines of *rational* slope was investigated in several papers; see [@BFS12; @BaranyRams14] and references there. In particular, in those two papers it is shown that for the gasket and many other carpets $S$, there are many lines with a given rational slope that intersect $S$ in a set of dimension $> \operatorname{dim_{\mathsf{H}}}(S)-1$. More precisely, given a rational slope, a typical slice (with respect to the uniform self-similar measure) has a constant dimension strictly larger than $\operatorname{dim_{\mathsf{H}}}(S)-1$.
General self-similar measures on the line {#subsec:general-ssm}
-----------------------------------------
We conclude this section by extending Theorem \[thm:dim-ssm\] to general (not necessarily homogeneous) self-similar measures on ${\mathbb{R}}$. Although we are no longer in a setting in which Theorem \[thm:L-q-dim-dyn-ssm\] can be applied, we will see that the same approach, with minor changes, can be used to directly establish the desired result.
We begin by defining a notion of exponential separation, which again agrees with that in [@Hochman14], and extends the one given here in the homogeneous case. We define a distance between two affine maps $g_i(x)={\lambda}_i x +t_i$ on ${\mathbb{R}}$ as $$d(g_1,g_2) = \left\{ \begin{array}{ccc}
|t_1-t_2| & \text{ if } & {\lambda}_1={\lambda}_2 \\
1 & \text{ if } & {\lambda}_1\neq{\lambda}_2
\end{array}
\right..$$
Let $(f_i)_{i\in\mathcal{I}}$ be strictly contractive, invertible affine maps on ${\mathbb{R}}$, i.e. $f_i(x)={\lambda}_i(x)+t_i$, where $|{\lambda}_i|\in (0,1)$ and $t_i\in{\mathbb{R}}$. Given a finite word $u\in\mathcal{I}^k$, we write $f_u = f_{u_1}\cdots f_{u_k}$, $f_u(x)={\lambda}_u x+t_u$, and $p_u=p_{u_1}\cdots p_{u_k}$. If $k\ge 1$, we also write $u^{-}$ for the word obtained from $u$ by deleting the last symbol.
Given $m\in{\mathbb{N}}$, let $\Omega_m$ be the family of all words $u$ such that ${\lambda}_u\le 2^{-m}$ but ${\lambda}_{u^{-}}>2^{-m}$. We can now define:
We say that the IFS $(f_i)_{i\in\mathcal{I}}$ has *exponential separation* if there are $R>0$ and a sequence $m_j\to\infty$ such that $$d(f_u,f_v) \ge 2^{-R m_j} \quad\text{for all }u\neq v\in \Omega_{m_j}.$$
\[thm:dim-general-ssm\] Let $(f_i)_{i\in\mathcal{I}}$ be an IFS with exponential separation, and consider a self-similar measure $$\mu = \sum_{i\in\mathcal{I}} p_i\, f_i\mu.$$ Then $D(\mu,q)=\min({\widetilde}{\tau}(q)/(q-1),1)$, where ${\widetilde}{\tau}(q)$ is the only solution to $\sum_{i\in\mathcal{I}} p_i^q |{\lambda}_i|^{-{\widetilde}{\tau}(q)}=1$.
As many of the steps in the proof of Theorem \[thm:dim-general-ssm\] are small variants of corresponding steps in the proof of Theorem \[thm:L-q-dim-dyn-ssm\], we will present an outline emphasizing the main differences, and leave the verification of the details to the interested reader. For simplicity we will assume that ${\lambda}_i>0$ for all $i$; the general case can be deduced with minor notational changes.
Let $\tau(q)=\tau(\mu,q)$. We have to show that either $\tau(q)=q-1$ or $\tau(q)={\widetilde}{\tau}(q)$. Hence, in order to prove Theorem \[thm:dim-general-ssm\] it is enough to establish:
\[prop:dim-general-ssm\] Under the assumptions of Theorem \[thm:dim-general-ssm\], if $q\in (1,\infty)$ is such that $\tau(q)<q-1$, then $\tau(q)= {\widetilde}{\tau}(q)$.
To prove the proposition, we begin by observing that Lemmas \[lem:f-alpha-smaller-than-one\]–\[lem:Lq-sum-over-small-set\] hold if $T(q)$ if replaced by $\tau(q)$ and $\mu_x$ by $\mu$. Indeed, the proofs only use concavity of $T$, and Corollary \[cor:unif-continuity\], both of which remain true for $\tau$ and $\mu$ by the definition and basic properties of $\tau$ (since we are dealing with just one measure, one needs not worry about uniform convergence in this context).
As a consequence, Proposition \[prop:Lq-over-small-set-is-small\] also remains valid with $\tau$ in place of $T$ and $\mu$ in place of $\mu_x$. Indeed, given $m\in{\mathbb{N}}$, we define $$\mu_m = \sum_{u\in\Omega_m} p_u \delta(t_u).$$ We note that this does not fully agree with our earlier notation in the homogeneous case. Given $s\in{\mathbb{N}}$ and $I\in{\mathcal{D}}_s$, we let $y_j$ be the atoms of $\mu_{s+2}$ such that $[y_j,y_j+2^{-s-2}]\cap I\neq\varnothing$, let $p_j$ be their masses, and define $$\mu_{I} = \sum_j p_j \delta_{y_j}.$$ The proof of Proposition \[prop:Lq-over-small-set-is-small\] then goes through using the measures $\mu_{I}$ instead of $\mu_{x,n,I}$.
In turn, Theorem \[thm:conv-with-ssm-flattens\] remains valid if, once again, we replace $T(q)$ by $\tau(q)$ and $\mu_x$ by the fixed self-similar measure $\mu$. This is because the proof of Theorem \[thm:conv-with-ssm-flattens\] relies only on Corollary \[cor:unif-continuity\], Lemmas \[lem:f-alpha-smaller-than-one\] and \[lem:size-set-A-in-terms-of-f-alpha\], and Proposition \[prop:Lq-over-small-set-is-small\], all of which we have seen continue to hold in our context.
The main change comes in the proof of the analog of Proposition \[prop:ssm-scale-Rm-norm\], which nevertheless remains valid:
\[prop:general-ssm-smaller-scale\] Using the notation above, fix $q\in (1,\infty)$ such that $\tau$ is differentiable at $q$ and $\tau(q)<q-1$. Then, for any $R\in{\mathbb{N}}$ $$\lim_{m\to\infty} \frac{\log\|\mu_{m}^{(R m)}\|_q^q }{m} = -\tau(q)$$
The key difference with the setting of Proposition \[prop:ssm-scale-Rm-norm\] is that $\mu$ is no longer a convolution of a scaled down version of itself and a discrete approximation. However, $\mu$ is still a convex combination of a “small” number of measures which do have this structure. Indeed, given $m\in{\mathbb{N}}$, let $\Lambda_m$ be the set of contraction ratios $\{ {\lambda}_u: u\in\Omega_m\}$. For ${\lambda}\in\Lambda_m$, define $$\mu_{m,{\lambda}} = \sum\{ p_u \delta(t_u) : u\in\Omega_m,{\lambda}_u = {\lambda}\}.$$ The elements of $\Lambda_m$ are of the form $\prod_{i\in\mathcal{I}} {\lambda}_i^{n_i}$, where ${\lambda}_i^{n_i} \ge (\min_{i\in\mathcal{I}}{\lambda}_i)2^{-m}$. It follows that $$\label{eq:Lambda-size-small}
|\Lambda_m| \le O(m^{|\mathcal{I}|}),$$ with the implicit constant depending only on $|\mathcal{I}|$ and $({\lambda}_i)_{i\in\mathcal{I}}$. By self-similarity we have $$\label{eq:decomposition-ssm-mu}
\mu = \sum_{u\in\Omega_m} p_u \,f_u\mu = \sum_{{\lambda}\in \Lambda_m} \mu_{m,{\lambda}} * S_{{\lambda}}\mu,$$ The idea is to apply the argument of the proof of Proposition \[prop:ssm-scale-Rm-norm\] to the convolutions $\mu_{m,{\lambda}} * S_{{\lambda}}\mu$. Since, thanks to and , $\mu$ is the sum of a sub-exponential number of such measures, the proof will go through with minor changes.
Recall that $\mu_m = \sum_{u\in\Omega_m} p_u\delta(t_u)$, so that $\mu_m= \sum_{{\lambda}\in\Lambda_m} \mu_{m,{\lambda}}$. An argument similar to the proof of Lemma \[lem:comparison-mu-m-mu\] shows that $$\|\mu_m^{(m)}\|_q^q =\Theta_q(1) \|\mu^{(m)}\|_q^q.$$ Using and the Hölder bound $\|\sum_{j\in\Lambda}\nu_j\|_q^q \le |\Lambda|^{q-1}\sum _{j\in\Lambda}\|\nu_j\|_q^q$, and arguing as in the proof of Proposition \[prop:ssm-scale-Rm-norm\] for each measure $\mu_{m,{\lambda}} * S_{{\lambda}}\mu$, we estimate $$\|\mu^{((R+1)m)}\|_q^q \le O(1) |\Lambda_m|^{q-1} \sum_{{\lambda}\in\Lambda_m} \sum_{I\in{\mathcal{D}}_m} \mu_{m,{\lambda}}(I)^q \|\rho_{m,{\lambda},I}^{(Rm)} * \mu^{(Rm)}\|_q^q,$$ where $\rho_{m,{\lambda},I}=S_{1/{\lambda}}(\mu_{m,{\lambda}}|_I)/\mu_{m,{\lambda}}(I)$. Hence, we can fix ${\lambda}^*={\lambda}^*(m)\in\Lambda_m$ such that $$\|\mu^{((R+1)m)}\|_q^q \le O(1) |\Lambda_m|^q \sum_{I\in{\mathcal{D}}_m} \mu_{m,{\lambda}^*}(I)^q \|\rho_{m,{\lambda}^*,I}^{(Rm)} * \mu^{(Rm)}\|_q^q,$$ On the other hand, similarly to , and using that $\mu_m\ge \mu_{m,{\lambda}^*}$ pointwise, we also get $$\|\mu_{m}^{((R+1)m)}\|_q^q \ge \Omega(1) \sum_{I\in{\mathcal{D}}_m} \mu_{m,{\lambda}^*}(I)^q \| \rho_{m,{\lambda}^*,I}^{(Rm)} \|_q^q.$$ We recall that $-\tfrac 1m \log\|\mu^{(m)}\|_q^q$ converges for all self-similar measures. Combining this with the last two displayed equations, the proof is concluded as in Proposition \[prop:ssm-scale-Rm-norm\], with $\rho_{m,{\lambda}^*,I}$ playing the rôle of $\rho_{x,I}$. The new factor $|\Lambda_m|^q$ is of no consequence, as it has subexponential growth thanks to .
It is enough to prove the statement for $q$ such that $\tau$ is differentiable at $q$. Iterating the definition of ${\widetilde}{\tau}(q)$, we see that $$\sum_{u\in\Omega_m} p_u^q {\lambda}_u^{-{\widetilde}{\tau}(q)} =1.$$ Since ${\lambda}_u \in (c 2^{-m},2^{-m})$ for $u\in\Omega_m$ and a constant $c>0$ depending only on the IFS, we deduce that $$\label{eq:symbolic-Lq-spectrum-general-ssm}
{\widetilde}{\tau}(q) = \lim_{m\to\infty} \frac{-\log\left(\sum_{u\in\Omega_m} p_u^q\right) }{m}.$$
By the exponential separation assumption, there exist $R\in{\mathbb{N}}$ and a sequence $m_j\to\infty$ such that, for fixed ${\lambda}\in\Lambda_{m_j}$, the distance between any two distinct atoms of $\mu_{m,{\lambda}}$ is at least $2^{-Rm_j}$. Hence, by and Hölder’s inequality, $$\label{eq:norm-at-smaller-scale-equals-symbolic-norm}
\|\mu_{m_j}^{(Rm_j)}\|_q^q \le |\Lambda_{m_j}|^{q-1} \sum_{{\lambda}\in\Lambda_{m_j}} \|\mu_{m_j,{\lambda}}^{(Rm)}\|_q^q
= |\Lambda_{m_j}|^{q-1} \sum_{u\in\Omega_{m_j}} p_u^q.$$ On the other hand, one always has $$\label{eq:norm-at-smaller-scale-equals-symbolic-norm-2}
\|\mu_m^{(Rm)}\|_q^q \ge \|\mu_m\|_q^q \ge \sum_{u\in\Omega_m} p_u^q.$$
Combining Proposition \[prop:general-ssm-smaller-scale\] and Equations , , and yields the claimed equality $\tau(q) = {\widetilde}{\tau}(q)$.
This concludes the proof of Theorem \[thm:dim-general-ssm\].
Convolutions of self-similar measures and the proof of Theorem \[thm:Furstenberg\] {#sec:convolutions-and-Furst-conj}
==================================================================================
Convolutions of two self-similar measures and Furstenberg’s conjectures {#subsec:proof-of-Furst-conjecture}
-----------------------------------------------------------------------
We turn to convolutions of homogeneous self-similar measures, and deduce Theorem \[thm:Furstenberg\] as a corollary. As we observed in §\[subsec:DDSSM\], the convolutions of the natural measures on a $p$-Cantor set and a $q$-Cantor set fit naturally into the setting of dynamically driven self-similar measures. The same argument works in greater generality:
\[lem:mu-x-conv-ssm\] Let $0<{\lambda}_2<{\lambda}_1<1$ and $\Delta_1,\Delta_2\in\mathcal{A}$, and consider the self-similar measures $$\label{eq:def-ssm-for-convolution}
\eta_i = \eta_i(\Delta_i,{\lambda}_i)= *_{n=0}^\infty S_{{\lambda}_i^n}\Delta_i.$$
Write $a_i=|\log({\lambda}_i)|$. On $X=[0,a_2)$, define the map $$\mathbf{T}(x) = x+a_1 \bmod(a_2).$$ Moreover, let $\Delta:X\to \mathcal{A}$ be given by $$\Delta(x) = \left\{
\begin{array}{ll}
\Delta_1* S_{e^x}\Delta_2 & \text{if } x\in [0,a_1)\\
\Delta_1 & \text{if } x\in [a_1,a_2)
\end{array}
\right..$$ Then if $\mu_x$ is given by with ${\lambda}={\lambda}_1$, we have $$\mu_x = \left\{
\begin{array}{ll}
\eta_1 * S_{e^x}\eta_2 & \text{ if } x\in [0,a_1)\\
\eta_1 * S_{e^{x-a_2}}\eta_2 & \text{ if } x\in [a_1,a_2)
\end{array}
\right..$$ for all $x\in X$.
Let $n'(x)=|\{i\in[1,n]: \mathbf{T}^i(x)\in [0,a_1)\}|$. Then $\mathbf{T}^n(x)=x+n a_1-n'(x)a_2$, so that $e^{\mathbf{T}^n(x)}{\lambda}_1^n = e^x {\lambda}_2^{n'(x)}$, and therefore $$\label{eq:model-for-conv-ssm}
*_{i=1}^n \Delta(\mathbf{T}^i x) = \left(*_{i=1}^n S_{{\lambda}_1^i}\Delta_1\right) * S_{e^x} \left( *_{i=1}^{n'(x)} S_{{\lambda}_2}^i\Delta_2 \right).$$ The claim follows by convolving with $\Delta(x)$ to get $\mu_{n+1,x}$, and then letting $n\to\infty$.
\[thm:dim-conv-ssm\] Let $\eta_1,\eta_2$ be as in . Assume $\log{\lambda}_2/\log{\lambda}_1\notin\mathbb{Q}$. Moreover, suppose that there is $R>0$ such that for infinitely many $n$ and all $P_j\in\mathcal{P}_{\Delta_j,n}$ (recall Definition \[def:exponential-separation-hom-ssm\]), $j=1,2$ it holds that $$|P_1({\lambda}_1)|,|P_2({\lambda}_2)| \ge {\lambda}_1^{Rn}.$$ Then $$\label{eq:Lq-dim-conv-ssm}
D(\eta_1*\eta_2,q) = \min\left(D(\eta_1,q)+D(\eta_2,q),1\right)$$ for all $q\in (1,\infty)$.
Let $(X,\mathbf{T},\Delta,{\lambda}_1)$ be the model given by Lemma \[lem:mu-x-conv-ssm\]. We identify $X$ with the circle (i.e. we identify $0$ and $\log{\lambda}_1$), so that the $X$ becomes compact, and $\mathbf{T}$ is rotation by $\log {\lambda}_1/\log{\lambda}_2$ (which is irrational by assumption) on the circle. Hence $\mathbf{T}$ is uniquely ergodic (with the unique invariant measure $\mathbb{P}$ being normalized Lebesgue measure on $X$). If $\Delta_1$ and $\Delta_2$ are supported on a single point, then $\mu_x$ is an atom for all $x$ and there is nothing to do; otherwise, $\mu_x$ is non-atomic for all $x$. Finally, the map $x\mapsto \mu_x$ has a single discontinuity at $a_1$, as is evident from Lemma \[lem:mu-x-conv-ssm\]. We have then checked that the model is pleasant. The assumptions on $x\mapsto \Delta(x)$ in Theorem \[thm:L-q-dim-dyn-ssm\] also hold trivially.
We claim that our assumption on the separation of $\eta_1,\eta_2$ implies that our model has exponential separation. Let $$\Delta_{n,j} = \sum_{i=0}^{n-1} S_{{\lambda}_j^i}(\Delta_j) = \left\{ \sum_{i=0}^{n-1} y_i {\lambda}_j^i : y_i\in\Delta_j \right\}.$$ Recall from that all atoms of $\mu_{x,n}$ have the form $$\{ u_1 + e^x u_2 : u_1\in\Delta_{n,1}, u_2\in\Delta_{n,2}\}.$$ Thus, for given $x\in X$, the smallest distance between atoms of $\mu_{x,n}$ is bounded above by $$\Phi_n(x) = \min\{ |P_1({\lambda}_1)|, |e^x P_2({\lambda}_2)|, |P_1({\lambda}_1) - e^x P_2({\lambda}_2)| : P_1\in \mathcal{P}_{\Delta_1,n}, P_2\in\mathcal{P}_{\Delta_2,n} \}.$$ Here $|P_1({\lambda}_1)|$ corresponds to differences between pairs of atoms for which $u_2$ coincide, $|e^x P_2({\lambda}_2)|$ to pairs of atoms for which $u_1$ coincide, and $|P_1({\lambda}_1) - e^x P_2({\lambda}_2)|$ to pairs of atoms for which neither $u_1$ nor $u_2$ coincide. By assumption, $|P_j({\lambda}_j)|\ge {\lambda}_1^{Rn}$ for infinitely many $n$, so we only have to deal with the third type of differences. Fix, then, $n$ such that $|P_j({\lambda}_j)|\ge {\lambda}_1^{Rn}$ for all $P_j\in \mathcal{P}_{\Delta_j,n}$.
Let $R'\gg R$. For fixed $P_j\in \mathcal{P}_{\Delta_j,n}$, $$|\{x: |P_1({\lambda}_1) - e^x P_2({\lambda}_2)| \le {\lambda}_1^{R' n}\}| \le O_{\Delta_1,\Delta_2}(1) {\lambda}_1^{(R'-R)n}.$$ Since $| \mathcal{P}_{\Delta_j,n}|\le O_{|\Delta_j|}(1)^n$, we deduce that $$|\{x: |P_1({\lambda}_1) - e^x P_2({\lambda}_2)| \le {\lambda}_1^{R' n} \text{ for some } P_j\in \mathcal{P}_{\Delta_j,n} \} | \le O_{\Delta_1,\Delta_2}(1)^n {\lambda}_1^{(R'-R)n}.$$ Hence, if $R'$ is taken large enough (in terms of $R, \Delta_1, \Delta_2$ only), then there are infinitely many $n\in{\mathbb{N}}$ such that for almost all $x\in X$ it holds that $|P_1({\lambda}_1) - e^x P_2({\lambda}_2)|\ge {\lambda}_1^{R' n}$ for any choice of $P_j\in\mathcal{P}_{\Delta_j,n}$. This establishes exponential separation.
We have verified that the application of Theorem \[thm:L-q-dim-dyn-ssm\] is justified. In light of this theorem, we only need to check that the right-hand side in equals the right-hand side in . Note that $$\|\Delta_1 * S_{e^x}\Delta_2\|_q^q = \|\Delta_1\|_q^q \|\Delta_2\|_q^q$$ outside of a finite set of $x$. Hence, keeping in mind the definition of the map $\Delta$ from Lemma \[lem:mu-x-conv-ssm\], $$\int_X \log\|\Delta(x)\|_q^q \,d\mathbb{P}(x) = \log\|\Delta_1\|_q^q + \frac{\log({\lambda}_1)}{\log({\lambda}_2)}\log\|\Delta_2\|_q^q.$$ Dividing by $(q-1)\log({\lambda}_1)$ we get that $$D(\eta_1*\eta_2,q) = \min\left(\frac{\log\|\Delta_1\|_q^q}{(q-1)\log({\lambda}_1)}+\frac{\log\|\Delta_2\|_q^q}{(q-1)\log({\lambda}_2)}1\right).$$ Theorem \[thm:dim-ssm\] applied to $\eta_1$ and $\eta_2$ concludes the proof.
We point out that in the range $q\in (1,2]$, the above result was proved in [@NPS12] in some special cases and then, extending the same ideas, in [@GSSY16 Corollary 6.2], in even greater generality. For example, in [@GSSY16] no separation assumptions are made on $\eta_1,\eta_2$. However, the methods of [@NPS12; @GSSY16] ultimately rely on Marstrand’s projection theorem, which is known to fail in general if $q>2$.
As a corollary, we obtain a Furstenberg-like bound on the intersections of self-similar sets, which also answers affirmatively a question of De-Jun Feng.
\[cor:dim-intersection-ssm\] Let ${\lambda}_1,{\lambda}_2\in (0,1)$ with $\log{\lambda}_1/\log{\lambda}_2\notin\mathbb{Q}$. Suppose $E_1, E_2$ are finite sets such that $\{ {\lambda}_j x + t: t\in E_j\}$ satisfies the open set condition for $j=1,2$. Let $A_1, A_2$ denote the corresponding self-similar sets.
Then for all invertible affine maps $g:{\mathbb{R}}\to{\mathbb{R}}$, $$\operatorname{{\overline{dim}_{\mathsf{B}}}}(A_1 \cap g(A_2)) \le \max(\operatorname{dim_{\mathsf{H}}}(A_1)+\operatorname{dim_{\mathsf{H}}}(A_2)-1,0).$$
Let $\eta_i$ be the uniform self-similar measure on $A_i$, and write $\mu=\eta_1\times\eta_2$ and $s=\operatorname{dim_{\mathsf{H}}}(A_1)+\operatorname{dim_{\mathsf{H}}}(A_2)$. Then $\mu(B(x,r)) = \Theta(r^s)$ for $x\in{\text{supp}}(\mu)$, since the corresponding fact holds for $\eta_1,\eta_2$ thanks to the open set condition.
As rescaling $A_2$ does not change the assumptions, it is enough to prove the claim when $g$ is a translation. Let $\Delta_j$ be the uniform probability measure on $E_j$, and $\eta_j=\eta_j({\lambda}_j,\Delta_j)$ the associated self-similar measure. The hypotheses of Theorem \[thm:dim-conv-ssm\] are met, so we know that $$D(\eta_1 * \eta_2,q) = \min\left(\frac{\log|E_1|}{\log(1/{\lambda}_1)}+\frac{\log|E_2|}{\log(1/{\lambda}_2)}, 1\right) = \min(s,1)$$ for all $q>1$. The claim now follows from Lemmas \[lem:Lq-dim-to-Frostman-exp\] and \[lem:Frostman-exp-to-small-fiber\] applied to the function $(x,y)\mapsto x-y$ restricted to $A_1\times A_2$.
We can now finish the proof of Theorem \[thm:Furstenberg\].
Let $A, B$ be $T_p$-invariant and $T_q$-invariant respectively, with $p$ and $q$ multiplicatively independent, and fix $\delta>0$. Given $N\in{\mathbb{N}}$, let $$E_{A,N} = \{ j p^{-N}: A\cap [j p^{-N},(j+1) p^{-N})\neq\varnothing \},$$ and define $E_{B,N}$ likewise. It is well known that Hausdorff and box-counting dimensions coincide for $T_p, T_q$-invariant sets, see e.g. [@Furstenberg08 Theorem 5.1] for a more general fact. Hence by taking $N$ large enough we can ensure that $$|E_{A,N}| \le p^{N(\operatorname{dim_{\mathsf{H}}}(A)+\delta)}, \quad |E_{B,N}| \le q^{N(\operatorname{dim_{\mathsf{H}}}(B)+\delta)}.$$ Let $A'$ be the homogeneous self-similar set with contraction $p^{-N}$ and translation set $E_{A,N}$, and define $B'$ analogously. The open set condition holds for $A',B'$ with open set $(0,1)$. Then $$\operatorname{dim_{\mathsf{H}}}(A') = \frac{\log|E_{A,N}|}{\log p^N} < \operatorname{dim_{\mathsf{H}}}(A)+\delta,$$ and likewise for $B'$. Also, by invariance of $A,B$ under $T_{p^N}, T_{q^N}$ respectively, $A\subset A', B\subset B'$. (Symbolically, $E_{A,N}$ corresponds to all initial words of length $N$ in $A$, and $A'$ to all concatenations of such words).
Since $\delta>0$ was arbitrary, the theorem follows from Corollary \[cor:dim-intersection-ssm\] applied to $A',B'$.
Corollary \[cor:dim-intersection-ssm\] and Theorem \[thm:Furstenberg\] remain valid for $C^1$ maps $g$. It is not hard to deduce this from the affine case and Furstenberg’s theory of CP-processes [@Furstenberg08], but since it would take us too far in a different direction, we defer a detailed proof of these and related results to a forthcoming article.
Recall from the introduction that another conjecture of Furstenberg, settled in [@HochmanShmerkin12], concerns the dimension of the arithmetic sum of a $\times p$ and a $\times q$ invariant set. As a corollary, we are able to sharpen this when the sum of the dimensions is at most $1$:
\[cor:sums-of-invariant-sets\] Let $p,q$ be multiplicatively independent, and suppose that $A,B\subset [0,1)$ are closed and $T_p,T_q$-invariant, respectively. Assume $\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)\le 1$. Then for any subsets $A'\subset A, B'\subset B$, $$\operatorname{dim_{\mathsf{H}}}(A'+B') = \operatorname{dim_{\mathsf{H}}}(A'\times B')$$
We note that in general $\operatorname{dim_{\mathsf{H}}}(A'\times B') \ge \operatorname{dim_{\mathsf{H}}}(A')+\operatorname{dim_{\mathsf{H}}}(B')$ and the inequality can be strict, but there is an equality if either $A'$ or $B'$ have equal Hausdorff and upper box-counting dimensions.
Suppose first that $\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)<1$. By embedding $A,B$ in $p^N, q^N$-Cantor sets of almost the same dimension as in the proof of Theorem \[thm:Furstenberg\], we may assume that $A,B$ are already a $p, q$-Cantor set respectively. The proof is now nearly identical to that of Corollary \[cor:sierpinski-gasket\], using Theorem \[thm:dim-conv-ssm\] in place of Theorem \[thm:dim-ssm\].
If $\operatorname{dim_{\mathsf{H}}}(A)+\operatorname{dim_{\mathsf{H}}}(B)=1$, then we proceed in the same way but now the sums of the dimensions of the $p,q$-Cantor sets containing $A,B$ is $1+\delta$, where $\delta$ is arbitrarily small. The argument of Corollary \[cor:sierpinski-gasket\] still goes through with very minor modifications; details are left to the interested reader.
A minor variant of the same argument recovers the full conjecture of Furstenberg on sums of $T_p$ and $T_q$ invariant sets. However, apart from some special cases, the methods from this paper do not appear to yield a different proof of the corresponding statement for convolutions of invariant measures, recall .
Convolutions of several self-similar measures
---------------------------------------------
Theorem \[thm:dim-conv-ssm\] generalizes easily to convolutions of an arbitrary number of self-similar measures. This provides an example of application of Theorem \[thm:L-q-dim-dyn-ssm\] in which $X$ is a torus of arbitrary dimension.
\[thm:dim-conv-many-ssm\] Let $0<{\lambda}_1<\ldots<{\lambda}_k<1$, $k\ge 2$, be numbers such that $(1/\log{\lambda}_j)_{j=1}^k$ is linearly independent over ${\mathbb{Q}}$. Fix $\Delta_1,\ldots,\Delta_k\in\mathcal{A}$, and write $$\eta_j =\eta_j(\Delta_j,{\lambda}_j) = *_{n=0}^\infty S_{{\lambda}_j^n} \Delta_j$$ for the corresponding self-similar measures. Moreover, suppose that there is $R>0$ such that for infinitely many $n$ it holds that $$\label{eq:exponential-sep-component-measures}
|P_j({\lambda}_j)| \ge {\lambda}_k^{Rn}\quad \text{for all } P_j\in\mathcal{P}_{\Delta_j,n}, j=1,\ldots,k.$$ Then $$\label{eq:Lq-dim-conv-many-ssm}
D(\eta_1*\cdots*\eta_k,q) = \min\left(\sum_{j=1}^k D(\eta_j,q),1\right).$$ for all $q\in (1,\infty)$.
The proof is similar to that of Theorem \[thm:dim-conv-ssm\], so we will skip some details. We write $a_j=|\log({\lambda}_j)|$. Let $X=[0,a_k)^{k-1}$, thought of as the $(k-1)$-dimensional torus, and let $\mathbf{T}:X\to X$ be given by $$\mathbf{T}(x_1,\ldots,x_{k-1}) = (x_1+a_k \bmod a_1,\ldots, x_{k-1}+a_k \bmod a_{k-1}).$$ Up to re-parametrization, this is translation by $(a_k/a_1,\ldots,a_k/a_{k-1})$ on the torus, which is uniquely ergodic if (and only if) $(1,a_k/a_1,\ldots,a_k/a_{k-1})$ is linearly independent over ${\mathbb{Q}}$; see e.g. [@EinsiedlerWard11 Corollary 4.15]. An easy calculation using the linear independence of $1/\log {\lambda}_j$ shows that this is indeed the case.
Given $x\in X$, we let $J(x)=\{ j\in\{1,\ldots,k-1\}: x_j \in [0,a_k)\}$, and define $\Delta:X\to \mathcal{A}$ as $$\Delta(x) = \left( *_{j\in J(x)} \Delta_j \right) * \Delta_k .$$ We have already remarked that $(X,\mathbf{T})$ is uniquely ergodic. The same argument from Lemma \[lem:mu-x-conv-ssm\] shows that the measures generated by this model are $$\mu_x = S_{\exp(x_1-\sigma_1(x)a_1)}\eta_1 * \cdots * S_{\exp(x_{k-1}-\sigma_{k-1}(x)a_{k-1})}\eta_{k-1}* \eta_k,$$ with $\sigma_j(x)=0$ if $x_j\in [0,a_j)$ and $\sigma_j(x)=1$ otherwise. The model $(X,\mathbf{T},\Delta,{\lambda}_k)$ is now readily checked to be pleasant, while the map $\Delta(\cdot)$ also meets the hypotheses in Theorem \[thm:L-q-dim-dyn-ssm\].
To establish exponential separation, we notice that the difference between two atoms of $\mu_{x,n}$ has the form $$\sum_{j=1}^k s_j e^{x_j} P_j({\lambda}_j),$$ where $s_j\in\{0,1\}$, not all $s_j$ are zero, $P_j\in \mathcal{P}_{\Delta_j,n}$, and we set $x_k=0$. For $n$ such that holds, the same argument in the proof of Theorem \[thm:dim-conv-ssm\], together with Fubini and an induction on the number of non-zero $s_j$, shows that the distance between atoms of $\mu_{n,x}$ is at least ${\lambda}_k^{R' n}$ for a.e. $x$, where $R'$ depends on $R$, the $\Delta_i$ and $k$ only.
We have checked that Theorem \[thm:L-q-dim-dyn-ssm\] can be applied. A calculation like the one in the proof of Theorem \[thm:dim-conv-ssm\] yields $$\int_X \log\|\Delta(x)\|_q^q \,d{\mathbb{P}}(x) = \log\|\Delta_k\|_q^q + \sum_{j=1}^{k-1} \frac{\log {\lambda}_k}{\log{\lambda}_j} \|\Delta_j\|_q^q,$$ so that Theorems \[thm:L-q-dim-dyn-ssm\] and \[thm:dim-ssm\] yield the desired conclusion.
Embeddings of self-similar sets
-------------------------------
Let us denote by $A_{\lambda}$ any self-similar set arising from a homogeneous IFS with contraction ratio ${\lambda}$, satisfying the open set condition and of dimension strictly smaller than $1$. A special case of a conjecture of D-J. Feng, W. Huang and H. Rao [@FHR14 Conjecture 1.2] asserts that $A_{\lambda}$ cannot be affinely embedded into $A_{{\lambda}'}$ unless $\log{\lambda}/\log{\lambda}'\in{\mathbb{Q}}$. In [@FHR14] this is proved in some special cases, and some further new cases were recently established by A. Algom [@Algom16]. However the general case was not known even for central Cantor sets (i.e. self-similar sets generated by two maps). It follows immediately from Theorem \[cor:dim-intersection-ssm\] that if $\log{\lambda}/\log{\lambda}'\notin{\mathbb{Q}}$, then for every affine map $h:{\mathbb{R}}\to{\mathbb{R}}$, $$\operatorname{dim_{\mathsf{H}}}(A_{\lambda}\cap h(A_{{\lambda}'})) \le \operatorname{dim_{\mathsf{H}}}(A_{\lambda})+\operatorname{dim_{\mathsf{H}}}(A_{{\lambda}'})-1 < \min(\operatorname{dim_{\mathsf{H}}}(A_{\lambda}),\operatorname{dim_{\mathsf{H}}}(A_{{\lambda}'})),$$ so that no affine immersion is possible. We can easily extend this to the case in which the set we want to embed is an arbitrary non-trivial self-similar set:
Suppose $A=\bigcup_{i\in\mathcal{I}} \lambda_i A+t_i$, $B=\bigcup_{j\in\mathcal{J}} {\lambda}' B+t'_j$ are self-similar sets, with $A$ not a singleton, and $B$ homogeneous, satisfying the open set condition, and of dimension strictly smaller than $1$. If there is a $C^1$ map $h:{\mathbb{R}}\to{\mathbb{R}}$ such that $h(A)\subset B$, then $\log {\lambda}_i/\log{\lambda}'$ is rational for all $i$.
Suppose that, on the contrary, $\log {\lambda}_i/\log{\lambda}'$ is irrational for some $i$, and yet $h(A)\subset B$ for some $C^1$ map $h$. Without loss of generality, assume that $\log{\lambda}_1/\log{\lambda}'$ is irrational. We may also assume that, writing $f_j(x)={\lambda}_j x+t_j$, the fixed points of $f_1$ and $f_2$ are different (if all the $f_j$ had the same fixed point, then $A$ would equal this point). If $N$ is sufficiently large, then $(f_{2}f_{1}^N, f_{1}^N f_{2})$ is a homogeneous IFS satisfying the open set condition, and its attractor $A_N$ is contained in $A$, so that $h(A_N)\subset A$. On the other hand, if $\log({\lambda}_2{\lambda}_1^N)/\log({\lambda}')$ is rational then, by our assumption, $\log({\lambda}_2{\lambda}_1^{N+1})/\log({\lambda}')$ is irrational.
We have thus reduced the problem to the case of $A$ homogeneous and satisfying the open set condition. Under these assumptions, [@FHR14 Theorem 1.1] implies that there is an *affine* embedding of $A$ into $B$. But, as we have seen, this is ruled out by Corollary \[cor:dim-intersection-ssm\].
Sections and projections of planar self-similar sets {#sec:planar-sss}
====================================================
Our next geometric application involves homogeneous self-similar sets and measures on the plane. It was observed in several previous works, going back at least to [@PeresShmerkin09], that methods devised to study geometric properties of cartesian products of linear self-similar sets and measures often can also be applied to the study of self-similar sets and measures on the plane. The next lemma may help clarify the reason behind this; compare with Lemma \[lem:mu-x-conv-ssm\].
\[lem:model-proj-hom-ssm\] Fix $\alpha\in [0,2\pi)$, ${\lambda}\in (0,1)$ and a finitely supported probability measure ${\widetilde}{\Delta}=\sum_{i\in\mathcal{I}} p_i \delta(t_i)$ on ${\mathbb{R}}^2$. Denote rotation by $\alpha$ by $\mathbf{R}_\alpha$, and let $$\mu = *_{n=0}^\infty S_{{\lambda}^n}\mathbf{R}_\alpha^n({\widetilde}{\Delta})$$ be the associated homogeneous self-similar measure. Given $x\in S^1$, let $P_x(y) = \langle x,y\rangle$ be the orthogonal projection onto a line in direction $x$. Furthermore, let $\Delta(x)=P_x{\widetilde}{\Delta}$.
Then the measures $\mu_x$ generated by the model $(S^1,\mathbf{R}_{-\alpha}, \Delta,{\lambda})$ are the projections $P_x\mu$. Moreover, the model is pleasant if and only if $\alpha/\pi\notin{\mathbb{Q}}$.
Immediate, since $$\left\langle x, S_{{\lambda}^n} \mathbf{R}_\alpha^n(y) \right\rangle = \left\langle \mathbf{R}_{-\alpha}^n x, S_{{\lambda}^n} (y) \right\rangle,$$ and rotation by $\beta\in [0,2\pi)$ is uniquely ergodic if and only if $\beta/\pi$ is irrational.
\[thm:dim-proj-planar-ssm\] Let $\mu$ and $P_x$ be as in Lemma \[lem:model-proj-hom-ssm\]. Assume further that $\alpha/\pi\notin{\mathbb{Q}}$, and that the open set condition holds. Then for every $x\in S^1$ and every $q\in (1,\infty)$, $$D(P_x\mu,q) = \min\left( \frac{\log\|{\widetilde}{\Delta}\|_q^q}{(q-1)\log{\lambda}},1 \right).$$
Let $\mu_n = *_{i=0}^{n-1} S_{{\lambda}^n}\mathbf{R}_\alpha^n({\widetilde}{\Delta})$. By the open set condition, $\mu_n$ has $|\mathcal{I}|^n$ atoms, which are $c {\lambda}^n$-separated for some $c>0$. Note that $P_x(\mu_n)=\mu_{x,n}$ (the measures generated by the model from Lemma \[lem:model-proj-hom-ssm\]). In particular, the atoms of $\mu_{x,n}$ are the projections of the atoms of $\mu_n$.
Let $R$ be a large enough integer to be chosen later. By elementary geometry, for a given pair $a,b$ of distinct atoms of $\mu_n$, the set of $x\in S^1$ such that $|P_x a - P_x b|\le {\lambda}^{Rn}$ has measure $O_c({\lambda}^{(R-1)n})$. Hence, the set of $x\in S^1$ such that the atoms of $\mu_{n,x}$ are all distinct and ${\lambda}^{Rn}$-separated has measure $1-O_c(|\mathcal{I}|^{2n}) {\lambda}^{(R-1)n}$. This implies that if $R$ is taken large enough in terms of $|\mathcal{I}|$, then for almost all $x\in S^1$ there is $n_0=n_0(x)$ such that the atoms of $\mu_{n,x}$ are distinct and ${\lambda}^{Rn}$ separated for all $n\ge n_0$. Hence the model from Lemma \[lem:model-proj-hom-ssm\] has exponential separation.
Since the hypothesis on $\Delta$ is trivially satisfied, we can apply Theorem \[thm:L-q-dim-dyn-ssm\] to conclude that $$D(P_x\mu,q) = \min\left( \frac{\int \log \|P_x{\widetilde}{\Delta}\|_q^q\,dx}{(q-1)\log {\lambda}},1\right),$$ which gives the claim since $P_x$ is injective on ${\widetilde}{\Delta}$ for all but a finite set of $x$.
We obtain the following corollary on linear sections of planar self-similar sets; compare with Corollary \[cor:dim-intersection-ssm\].
\[cor:dim-intersection-sss-lines\] Fix ${\lambda}\in (0,1)$, $\alpha\in [0,2\pi)$ such that $\alpha/\pi$ is irrational, and a finite set $(t_i)_{i\in\mathcal{I}}$ of translations in ${\mathbb{R}}^2$. Assume that the IFS $\{ {\lambda}\mathbf{R}_\alpha(x)+t_i\}_{i\in\mathcal{I}}$ satisfies the open set condition, and denote its invariant set by $E$.
Then $$\operatorname{{\overline{dim}_{\mathsf{B}}}}(E\cap \ell) \le \max(\operatorname{dim_{\mathsf{H}}}(E)-1,0).$$ for all lines $\ell\subset\mathbb{R}^2$.
Immediate from Lemmas \[lem:Lq-dim-to-Frostman-exp\] and \[lem:Frostman-exp-to-small-fiber\] applied to $P_x$, and Theorem \[thm:dim-proj-planar-ssm\] applied to the uniform self-similar measure on $E$.
We make some remarks about this corollary.
1. Let $E$ be any Borel set with $\operatorname{dim_{\mathsf{H}}}(E)\ge 1$. It follows from Marstrand’s intersection theorem (see e.g. [@Mattila95 Theorem 10.10]) that, given a direction $x\in S^1$, *almost all* lines $\ell$ in direction $x$ satisfy $\operatorname{dim_{\mathsf{H}}}(E\cap\ell) \le \operatorname{dim_{\mathsf{H}}}(E)-1$. There has been great interest in improving almost all-type of results for classes of natural sets, but most of the progress achieved concerns projections rather than the more subtle problem of intersections. For some classes of *random* stochastically self-similar sets, even stronger bounds on intersections were obtained in [@ShmerkinSuomala15 Section 11]. D-J. Feng has some unpublished results for deterministic sets, using ad-hoc constructions. To the best of our knowledge, Corollary \[cor:dim-intersection-sss-lines\] is the first result of this kind for a natural class of deterministic sets.
2. It is also natural to consider the dual question of obtaining *lower* bounds on the dimension of $E\cap\ell$ for lines $\ell\subset{\mathbb{R}}^2$ when $\operatorname{dim_{\mathsf{H}}}(E)>1$. Of course, many such intersections are empty, but one would like to know that the intersections are large (of dimension equal or close to $\operatorname{dim_{\mathsf{H}}}(E)-1$) for *many* lines $\ell$ in a given direction (measured, for example, in terms of Hausdorff dimension). Progress on this problem was achieved recently by K. Falconer and X. Jin [@FalconerJin15].
3. Using Furstenberg’s theory of CP-processes and galleries [@Furstenberg08], it is possible to obtain a version of Corollary \[cor:dim-intersection-sss-lines\] where lines are replaced by $C^1$ or even differentiable curves; we hope to address this at detail in a forthcoming paper. On the other hand, no such result can hold for Lipschitz curves since any set of upper box-counting dimension less than $1$ can be embedded in a Lipschitz curve.
4. The hypothesis that $\alpha/\pi$ is irrational is necessary: if $C$ is the middle-thirds Cantor set, then the diagonal of $C\times C$ is an affine copy of $C$. However, the homogeneity assumption is likely an artifact of the proof.
Absolute continuity and Lq densities {#sec:abs-cont}
====================================
We turn to the problem of absolute continuity, and smoothness of the densities, of self-similar and related measures. Compared to Sections \[sec:dim-ssm-and-applications\]–\[sec:planar-sss\], our results here will be less explicit: we show that in many parametrized families, the measures have a density in $L^q$ for all parameters outside of some very small set. In particular, we will establish Theorem \[thm:abc-cont-BCs\]. Unfortunately, however, either for Bernoulli convolutions or the other parametrized families we consider, we do not know how to find even one explicit parameter which is not exceptional.
The main ideas in this section are borrowed from [@Shmerkin14; @ShmerkinSolomyak16]; the reason we improve upon existing results is that Theorem \[thm:L-q-dim-dyn-ssm\] provides stronger information about $L^q$ dimensions to begin with.
Recall that the Fourier transform of a Borel probability measure $\mu$ on ${\mathbb{R}}$ is defined as $$\widehat{\mu}(\xi) = \int \exp(2\pi i x\xi)\,d\mu(x).$$ Given a model $(X,\mathbf{T},\Delta,{\lambda})$ and $k\in{\mathbb{N}}$, let us consider the measures $$\mu_x^{(k)} = *_{i=0}^\infty \Delta(\mathbf{T}^{ki}(x)).$$ These are precisely the measures generated by the model $(X,\mathbf{T}^k,\Delta,{\lambda})$, which is pleasant whenever the original model is; however we will not need to use this.
The next theorem presents our general result on densities of $\mu_{x}$. We will deduce several applications afterwards.
\[thm:abs-continuity\] Let $(X,\mathbf{T},\Delta,{\lambda})$ be a model satisfying the assumptions of Theorem \[thm:L-q-dim-dyn-ssm\], and assume furthermore that either $X$ is trivial, or no point in $X$ is $\mathbf{T}$-periodic. Fix $q\in (1,+\infty)$ and assume also that $$\frac{\int_X \log\|\Delta(x)\|_q^q \,d\mathbb{P}(x)}{(q-1)\log{\lambda}} > 1.$$ Suppose $y\in X$ is such that for infinitely many $k\in{\mathbb{N}}$ there exist constants $C(k),\delta(k)>0$ such that the Fourier transform of $\mu_y^{(k)}$ satisfies $$\left|\widehat{\mu_y^{(k)}}(\xi)\right| \le C(k)\, |\xi|^{-\delta(k)} \quad\text{for all }\xi\neq 0.$$
Then $\mu_y$ is absolutely continuous and has a density in $L^q$.
Using the convolution structure of $\mu_y$, we decompose $$\mu_y = \left(*_{k\nmid i} \Delta(\mathbf{T}^{i}y)\right)*\left(*_{k\mid i} \Delta(\mathbf{T}^{i}y)\right)=:\nu_y^{(k)}*\mu_y^{(k)}.$$ If we can show that $$\label{eq:dim-skip-meas-1}
D(\nu_y^{(k)},q)=1$$ for all large enough $k$, then [@ShmerkinSolomyak16 Theorem 4.4], together with our assumption on the Fourier decay $\mu_y^{(k)}$, will allow us to conclude that $\mu_y$ has a density in $L^q$.
For fixed $k$, consider the model $(X',\mathbf{T}',\Delta',{\lambda})$, where $X'=X\times [k]$, $\mathbf{T}'(x,j)=(\mathbf{T}x,j+1 \bmod k)$ and $$\Delta'(x,j) = \left\{
\begin{array}{ll}
\Delta(x) & \text{ if } j\neq 0\\
\delta_0 & \text{ if } j=0
\end{array}
\right..$$ The measures $\mu'_x$ generated by this model are precisely $\nu_x^{(k)}$, as is immediate from the definition of $\Delta'$. This model satisfies all the assumptions in Theorem \[thm:L-q-dim-dyn-ssm\]. Indeed, exponential separation is inherited from the base model, since the atoms of $\mu'_{x,n}$ are a subset of the atoms of $\mu_{x,n}$. Unique ergodicity (with invariant measure $\mathbb{P}'=\mathbb{P}\times (\tfrac1k \sum_j \delta_j)$) follows from the unique ergodicity of $(X,\mathbf{T})$ and the fact that there are no periodic points, and the rest of the assumptions in Theorem \[thm:L-q-dim-dyn-ssm\] are immediate.
Applying Theorem \[thm:L-q-dim-dyn-ssm\] and recalling the form of $\mathbb{P}'$, we conclude that for any $y\in X$, $$D(\nu_y^{(k)},q) = \min\left( \frac{\frac{k-1}{k}\int_X \log\|\Delta(x)\|_q^q \,d\mathbb{P}(x)}{(q-1)\log{\lambda}},1\right) =1,$$ provided $k$ is large enough. This establishes and concludes the proof.
We remark that the theorem provides the correct range for the possibility of having an $L^q$ density (other than perhaps the endpoint), since measures $\mu$ with an $L^q$ density satisfy $D(\mu,q)=1$; this can be seen from the inequality $(\int_I f)^q \le |I|^{q-1} \int_I f^q$ for all intervals $I$, where $f$ is the $L^q$ density of $\mu$.
As a first application, we can now conclude the proof of Theorem \[thm:abc-cont-BCs\].
Erdős [@Erdos40] and Kahane [@Kahane71] proved that there is a set $\mathcal{E}'\subset (0,1)$ of zero Hausdorff dimension, such that if ${\lambda}\in (0,1)\setminus\mathcal{E}'$, then $|\widehat{\nu_{\lambda}}(\xi)| \le C({\lambda}) |\xi|^{-\delta({\lambda})}$ for some $C({\lambda}),\delta({\lambda})>0$. See also [@PSS00] for an exposition of the argument.
Let $\mathcal{E}_1 = \{ {\lambda}\in (0,1/2):{\lambda}^k\in\mathcal{E}' \text{ for some } k\}$, which still has zero Hausdorff dimension. Consider the model $\mathcal{X}_{\lambda}$ with trivial dynamics as in the proof of Theorem \[thm:dim-ssm\], and recall from Definition \[def:ssc-BCs\] and the discussion afterwards that there is another zero-dimensional set $\mathcal{E}_2$ such $\mathcal{X}_{\lambda}$ has exponential separation for ${\lambda}\in (1/2,1)\setminus\mathcal{E}_2$. The measure $\mu_x^{(k)}$ for the model $\mathcal{X}_{\lambda}$ is just $\nu_{{\lambda}^k}$. Part (i) of the theorem then follows from Theorem \[thm:abs-continuity\] with exceptional set $\mathcal{E}=\mathcal{E}_1\cup \mathcal{E}_2$.
The second part follows from the first, the identity $\nu_{\lambda}= \nu_{{\lambda}^2} * S_{\lambda}\nu_{{\lambda}^2}$, and the fact that the convolution of two $L^2$ functions is continuous.
The method of Erdős-Kahane has been applied to many other parametrized families of fractal measures, see [@ShmerkinSolomyak16 Section 3] for some examples related to the measures investigated in this paper. Using this, one can extend Theorem \[thm:abc-cont-BCs\] to more general families of self-similar measures, and in particular to the families satisfying the assumptions of [@ShmerkinSolomyak16 Theorem A]. Moreover, the exceptional set depends only on the iterated function system (that is, the contraction ratio and translations), and not on the probability weights.
Note, however, that just as in [@ShmerkinSolomyak16] here we are limited to homogeneous iterated function systems, as the argument to pass from full $L^q$ dimension to $L^q$ density depends strongly on the structure of the measures as infinite Bernoulli convolutions.
As another application of Theorem \[thm:abs-continuity\], we obtain the following result on projections of planar self-similar measures:
\[cor:abs-cont-proj-ssm\] Let $\mu$ be as in Lemma \[lem:model-proj-hom-ssm\]. Assume that the open set condition holds. Then there is a set $\mathcal{E}\subset [0,2\pi)$ of zero Hausdorff dimension (depending only ${\lambda},\alpha,{\text{supp}}({\widetilde}{\Delta})$), such that $P_x\mu$ is absolutely continuous with an $L^q$ density for all $q$ such that $\|{\widetilde}{\Delta}\|_q^q<{\lambda}^{(q-1)}$.
If $\alpha/\pi\in{\mathbb{Q}}$, then we can assume that $\alpha=0$ by iterating the original IFS. In this case $(P_x\mu)_{x\in S^1}$ is a family of self-similar measures satisfying the assumptions of [@ShmerkinSolomyak16 Theorem A], so the claim holds as explained in the above discussion.
If $\alpha/\pi\notin {\mathbb{Q}}$, consider the model $(S^1,\mathbf{R}_{-\alpha},\Delta,{\lambda})$ from Lemma \[lem:model-proj-hom-ssm\]. The measures $\mu_x^{(k)}$ are projections of the self-similar measure $*_{n=0}^\infty S_{{\lambda}^n} \mathbf{R}_{k\alpha}^n {\widetilde}{\Delta}$. It follows from [@ShmerkinSolomyak16 Proposition 3.3] that there exists a set $\mathcal{E}_k\subset [0,2\pi)$ of zero Hausdorff dimension, depending only on ${\lambda},\alpha,k$ and ${\text{supp}}({\widetilde}{\Delta})$, such that the projection $\mu_x^{(k)}$ has a power Fourier decay for all $x\in [0,2\pi)\setminus \mathcal{E}_k$. The claim then follows from Theorem \[thm:abs-continuity\] with exceptional set $\mathcal{E}=\cup_{k\in{\mathbb{N}}} \mathcal{E}_k$.
Recall that the Fourier transform of a measure $\mu$ on ${\mathbb{R}}^2$ is $$\widehat{\mu}(\xi) = \int \exp(2\pi i \langle y,\xi \rangle)\,d\mu(y),$$ and that if $v\in S^1$, then $\widehat{P_v\mu}(\xi)=\widehat{\mu}(\xi v)$. In particular, if $\mu$ has power Fourier decay (in the sense that $|\widehat{\mu}(\xi)|=O(|\xi|^{-\delta})$ for $\xi\in{\mathbb{R}}^2\setminus\{0\}$ and some $\delta>0$), then so do all its projections $P_x\mu$.
If the planar self-similar measure $\mu$ has power Fourier decay and $\alpha/\pi$ is irrational, then the proof of the Corollary \[cor:abs-cont-proj-ssm\] together with the above observations show that $P_x\mu$ has an $L^q$ density for *all* $x\in S^1$, whenever $\|{\widetilde}{\Delta}\|_q^q<{\lambda}^{(q-1)}$. Although we know of no explicit example of such measure $\mu$, in parameter space power Fourier decay occurs outside of very small exceptional sets; see [@ShmerkinSolomyak16b Theorem D].
We obtain a further corollary for convolutions of two self-similar measures, with the parameter coming in the scaling. A direct application of Theorem \[thm:abs-continuity\] is somewhat awkward because the corresponding measures $\mu_k^{(k)}$ do not have a particularly nice structure. However, the proof of [@ShmerkinSolomyak16 Theorem D], using Theorem \[thm:dim-conv-ssm\] to calculate the $L^q$ dimensions of self-similar measures and their convolutions, yields our final result; the verification of the details is left to the reader.
Let $$\eta_j=\eta_j(\Delta_j,{\lambda}_j) = *_{n=0}^\infty S_{{\lambda}_j^n} \Delta_j$$ be two homogeneous self-similar measures satisfying the open set condition on the real line.
Then there is a set $\mathcal{E}\subset{\mathbb{R}}$ of zero Hausdorff dimension, such that if $t\in{\mathbb{R}}\setminus\mathcal{E}$ and $q>1$ is such that $D(\eta_1,q)+D(\eta_2,q)>1$, then the convolution $\eta_1 * S_t \eta_2$ is absolutely continuous with a density in $L^q$.
[10]{}
Christoph Aistleitner, Gerhard Larcher, and Mark Lewko. Additive [E]{}nergy and the [H]{}ausdorff dimension of the exceptional set in metric pair correlation problems. Preprint, arXiv:1606.03591, 2016.
Amir Algom. Affine embeddings of [C]{}antor sets on the line. Preprint, arXiv:1607.02849, 2016.
Bal[á]{}zs B[á]{}r[á]{}ny, Andrew Ferguson, and K[á]{}roly Simon. Slicing the [S]{}ierpiński gasket. , 25(6):1753–1770, 2012.
Bal[á]{}zs B[á]{}r[á]{}ny and Micha[ł]{} Rams. Dimension of slices of [S]{}ierpiński-like carpets. , 1(3):273–294, 2014.
Jean Bourgain. The discretized sum-product and projection theorems. , 112:193–236, 2010.
Jean Bourgain and Alex Gamburd. Uniform expansion bounds for [C]{}ayley graphs of [${\rm SL}\sb 2(\Bbb
F\sb p)$]{}. , 167(2):625–642, 2008.
Emmanuel Breuillard and Péter Varjú. Entropy of bernoulli convolutions and uniform exponential growth for linear groups. Preprint, arXiv:1510.04043, 2015.
Semyon Dyatlov and Joshua Zahl. Spectral gaps, additive energy, and a fractal uncertainty principle. , 26(4):1011–1094, 2016.
Manfred Einsiedler and Thomas Ward. , volume 259 of [*Graduate Texts in Mathematics*]{}. Springer-Verlag London, Ltd., London, 2011.
Paul Erdős. On a family of symmetric [B]{}ernoulli convolutions. , 61:974–976, 1939.
Paul Erdős. On the smoothness properties of a family of [B]{}ernoulli convolutions. , 62:180–186, 1940.
Kenneth Falconer. . John Wiley & Sons, Ltd., Chichester, 1997.
Kenneth Falconer and Xiong Jin. Dimension conservation for self-similar sets and fractal percolation. , (24):13260–13289, 2015.
Ai-Hua Fan, Ka-Sing Lau, and Hui Rao. Relationships between different dimensions of a measure. , 135(3):191–201, 2002.
De-Jun Feng. Gibbs properties of self-conformal measures and the multifractal formalism. , 27(3):787–812, 2007.
De-Jun Feng and Huyi Hu. Dimension theory of iterated function systems. , 62(11):1435–1500, 2009.
De-Jun Feng, Wen Huang, and Hui Rao. Affine embeddings and intersections of [C]{}antor sets. , 102(6):1062–1079, 2014.
Alex Furman. On the multiplicative ergodic theorem for uniquely ergodic systems. , 33(6):797–815, 1997.
Harry Furstenberg. Disjointness in ergodic theory, minimal sets, and a problem in [D]{}iophantine approximation. , 1:1–49, 1967.
Harry Furstenberg. Intersections of [C]{}antor sets and transversality of semigroups. In [*Problems in analysis ([S]{}ympos. [S]{}alomon [B]{}ochner, [P]{}rinceton [U]{}niv., [P]{}rinceton, [N]{}.[J]{}., 1969)*]{}, pages 41–59. Princeton Univ. Press, Princeton, N.J., 1970.
Hillel Furstenberg. Ergodic fractal measures and dimension conservation. , 28(2):405–422, 2008.
Daniel Galicer, Santiago Saglietti, Pablo Shmerkin, and Alexia Yavicoli. dimensions and projections of random measures. , 29(9):2609–2640, 2016.
John Hawkes. Some algebraic properties of small sets. , 26(102):195–201, 1975.
Michael Hochman. Geometric rigidity of [$\times m$]{} invariant measures. , 14(5):1539–1563, 2012.
Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. , 8(3-4):437–497, 2014.
Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy. , 180(2):773–822, 2014.
Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$. arXiv:1503.09043v1, 2015.
Michael Hochman and Pablo Shmerkin. Local entropy averages and projections of fractal measures. , 175(3):1001–1059, 2012.
Thomas Jordan, Pablo Shmerkin, and Boris Solomyak. Multifractal structure of [B]{}ernoulli convolutions. , 151(3):521–539, 2011.
J.-P. Kahane. Sur la distribution de certaines séries aléatoires. In [*Colloque de [T]{}héorie des [N]{}ombres ([U]{}niv. [B]{}ordeaux, [B]{}ordeaux, 1969)*]{}, pages 119–122. Bull. Soc. Math. France, Mém. No. 25, Soc. Math. France Paris, 1971.
Y. Katznelson and B. Weiss. A simple proof of some ergodic theorems. , 42(4):291–296, 1982.
Richard Kenyon and Yuval Peres. Intersecting random translates of invariant [C]{}antor sets. , 104(3):601–629, 1991.
Ka-Sing Lau and Sze-Man Ngai. Multifractal measures and a weak separation condition. , 141(1):45–96, 1999.
Pertti Mattila. , volume 44 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability.
Fedor Nazarov, Yuval Peres, and Pablo Shmerkin. Convolutions of [C]{}antor measures without resonance. , 187:93–116, 2012.
Yuval Peres and Wilhelm Schlag. Smoothness of projections, [B]{}ernoulli convolutions, and the dimension of exceptions. , 102(2):193–251, 2000.
Yuval Peres, Wilhelm Schlag, and Boris Solomyak. Sixty years of [B]{}ernoulli convolutions. In [*Fractal geometry and stochastics, [II]{} ([G]{}reifswald/[K]{}oserow, 1998)*]{}, volume 46 of [*Progr. Probab.*]{}, pages 39–65. Birkhäuser, Basel, 2000.
Yuval Peres and Pablo Shmerkin. Resonance between [C]{}antor sets. , 29(1):201–221, 2009.
Yuval Peres and Boris Solomyak. Absolute continuity of [B]{}ernoulli convolutions, a simple proof. , 3(2):231–239, 1996.
Yuval Peres and Boris Solomyak. Existence of [$L\sp q$]{} dimensions and entropy dimension for self-conformal measures. , 49(4):1603–1621, 2000.
Giorgis Petridis. Plünnecke’s inequality. , 20(6):921–938, 2011.
Pablo Shmerkin. On the exceptional set for absolute continuity of [B]{}ernoulli convolutions. , 24(3):946–958, 2014.
Pablo Shmerkin. Projections of self-similar and related fractals: a survey of recent developments. In Christoph Bandt, Kenneth J. Falconer, and Martina Zähle, editors, [*Fractal Geometry and Stochastics V*]{}, pages 53–74. Springer, 2015.
Pablo Shmerkin and Boris Solomyak. Absolute continuity of complex [B]{}ernoulli convolutions. , FirstView, 2016.
Pablo Shmerkin and Boris Solomyak. Absolute continuity of self-similar measures, their projections and convolutions. , 368(7):5125–5151, 2016.
Pablo Shmerkin and Ville Suomala. Spatially independent martingales, intersections and applications. , to appear, 2015. arXiv:1409.6707v4.
Boris Solomyak. On the random series [$\sum\pm\lambda\sp n$]{} (an [E]{}rd[ő]{}s problem). , 142(3):611–625, 1995.
Terence Tao and Van H. Vu. , volume 105 of [*Cambridge Studies in Advanced Mathematics*]{}. Cambridge University Press, Cambridge, 2010. Paperback edition \[of MR2289012\].
Péter Varjú. Absolute continuity of [B]{}ernoulli convolutions for algebraic parameters. Preprint, arXiv:1602.00261, 2016.
Thomas Wolff. Recent work connected with the [K]{}akeya problem. In [*Prospects in mathematics ([P]{}rinceton, [NJ]{}, 1996)*]{}, pages 129–162. Amer. Math. Soc., Providence, RI, 1999.
Meng Wu. A proof of [F]{}urstenberg’s conjecture on the intersections of $\times
p$ and $\times q$-invariant sets. Preprint, 2016.
[^1]: P.S. was partially supported by Projects PICT 2013-1393 and PICT 2014-1480 (ANPCyT)
|
---
abstract: 'The Mini-EUSO telescope is designed by the JEM-EUSO Collaboration to observe the UV emission of the Earth from the vantage point of the International Space Station (ISS) in low Earth orbit. The main goal of the mission is to map the Earth in the UV, thus increasing the technological readiness level of future EUSO experiments and to lay the groundwork for the detection of Extreme Energy Cosmic Rays (EECRs) from space [@Ebisuzaki:2014wka]. Due to its high time resolution of , Mini-EUSO is capable of detecting a wide range of UV phenomena in the Earth’s atmosphere. In order to maximise the scientific return of the mission, it is necessary to implement a multi-level trigger logic for data selection over different timescales. This logic is key to the success of the mission and thus must be thoroughly tested and carefully integrated into the data processing system prior to the launch. This article introduces the motivation behind the trigger design and details the integration and testing of the logic.'
address:
- 'D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University, 1(2), Leninskie Gory, Moscow, 119991, Russia'
- 'Faculty of Physics, M.V. Lomonosov Moscow State University, 1(2), Leninskie Gory, Moscow, 119991, Russia'
- |
Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino,\
Corso Duca degli Abruzzi, 24, 10129 Torino, Italy
- 'Istituto Nazionale di Fisica Nucleare, sez. di Torino, via P. Giuria,1, 10125 Torino, Italy'
- 'Department of Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden'
- 'The Oskar Klein Centre for Cosmoparticle Physics, SE-106 91 Stockholm, Sweden'
author:
- Alexander Belov
- Mario Bertaina
- Francesca Capel
- Federico Fausti
- Francesco Fenu
- Pavel Klimov
- Marco Mignone
- Hiroko Miyamoto
- 'for the JEM-EUSO Collaboration'
bibliography:
- 'mybibfile.bib'
title: 'The integration and testing of the Mini-EUSO multi-level trigger system'
---
=1
front-end,readout electronics,trigger ,DAQ ,data management ,EUSO ,EECRs
Introduction {#sec:INTRO}
============
The Mini-EUSO instrument is designed for the measurement and mapping of the UV night-time emissions from the Earth and is being developed by the JEM-EUSO Collaboration as a pathfinder for the detection of EECRs from space. The cosmic ray energy spectrum extends over many orders of magnitude in both flux and energy. Notably, at the high energies of EECRs, defined as cosmic rays with energy E $>$ , the flux drops to less than 1 particle//century and the exposure of a proposed detector becomes a critical consideration. Existing ground-based experiments make use of large detector arrays and fluorescence-detecting telescopes in order to capture the particle showers and UV light emitted by the interaction of EECRs with the Earth’s atmosphere (for example the Telescope Array Project [@Kawai:2008gz] and the Pierre Auger Observatory [@Abraham:2004dt]). By taking the fluorescence detection concept into Earth orbit, the exposure of such an experiment to EECRs could be dramatically increased, whilst also allowing for coverage of both hemispheres. Increasing the statistics of these measurements is key to improving our understanding of the astrophysical sources and mechanisms which are capable of producing such high energy particles.
The JEM-EUSO collaboration proposes such space-based experiments including the KLYPVE/K-EUSO [@Panasyuk15] on the Russian Segment of the International Space Station (ISS) and JEM-EUSO (Extreme Universe Space Observatory on board the Japanese Experiment Module) [@Collaboration15; @Olinto15]. Mini-EUSO is currently approved by both the Russian (Roscosmos) and Italian (ASI) space agencies and is set to be launched to the Zvezda module of the ISS in early 2018, where it will look down on the Earth from a nadir-facing, UV-transparent window, in order to verify this detection concept.
Mini-EUSO is made up of three main sub-systems: the Fresnel-based optical system, the Photo-Detector Module (PDM) and the readout electronics. There are also two ancillary cameras installed at the level of the front lens, to provide complementary information in the visible and near infra-red range. The design of Mini-EUSO is shown in Figure \[fig:minieuso\_highlevel\]. The optical system consists of 2 double-sided Fresnel lenses with a diameter of allowing for a compact system with a large aperture, ideal for space application [@Hachisu:2011vx]. The lenses focus the light onto the focal surface, where it is detected by the PDM. 36 multi-anode photomultiplier tubes (MAPMTs) with UV filters make up the PDM, each with 64 pixels, resulting in a readout of 2304 pixels. Signals are pre-amplified and converted to digital by the SPACIROC3 ASIC [@BlinBondil:2014ve], before being passed to the data processing unit (PDM-DP) for data handling and storage.
[![The design of the Mini-EUSO instrument. The main sub-systems are shown: the two double-sided Fresnel lenses, the PDM and the readout electronics. The near infra-red and visible cameras are mounted at the level of the first lens, outside of the optical system. The dimensions of the instrument are 37 $\times$ 37 $\times$ 62 .[]{data-label="fig:minieuso_highlevel"}](figures/minieusomechanicalparts.png "fig:"){width="90.00000%"}]{}
The main scientific goal of Mini-EUSO is to produce a high-resolution map of the Earth in the UV range (300 - 400 nm). With a spatial resolution of $\sim$ and a temporal resolution of , Mini-EUSO will present results of unprecedented detail in this range. Such observations are key to the understanding of the detection threshold of EECRs from space, in addition to estimating the duty cycle of future experiments. Although the energy threshold of Mini-EUSO is likely too high to detect EECRs, the instrument’s capability to detect such events will be tested by triggering on EECR-like laser tracks produced by ground-based laser systems. Due to its high resolution, Mini-EUSO is also capable of capturing a variety of both atmospheric and terrestrial phenomena, such as transient luminous events (TLEs), meteors, space debris, bioluminescence and anthropogenic lights. The duration of such events varies by 6 orders of magnitude, motivating a multi-level trigger system to maximise the scientific return, given constraints on the duty cycle and data storage. For more details on the Mini-EUSO instrument, see @Mini-EUSO (in submission).
TLEs are upper-atmospheric events which occur above areas of thunderstorm activity [@Pasko:2011ev]. Imaged for the first time in 1989, as described in @Franz:1990cu, the term TLE is used to describe a wide range of phenomena which are broadly classified into 3 main groups: blue jets, sprites and elves. Blue jets and sprites are more localised, column-like events which can occur in groups and have a spatial extent of a few when viewed from above. Elves are very fast flashes which form large halos, extending over . Generally speaking, the timescales associated with TLE events are between $\sim$ and $\sim$ , with rise times on the scale of . TLEs are luminous in the UV and occur with a high frequency, making it important to study such events due to their contribution to the EECR-signal background. In addition, Mini-EUSO will be able to provide high-time-resolution imaging of these events which could contribute to a better understanding of their structure and development in relation to the associated thunderstorms. The Atmosphere-Space Interactions Monitor (ASIM, @Neubert:2009ih) is designed for the study of atmospheric phenomena, including TLEs, and will operate on-board the ISS during the mission of Mini-EUSO. A specific trigger level for the study of TLEs has been implemented in Mini-EUSO to capture these events and will allow for the joint study of TLEs by both instruments.
The trigger algorithm
=====================
The digitised data path {#sec:signal path}
-----------------------
Photons focused onto the Mini-EUSO focal surface are detected by the 36 MAPMTs, each with 64 pixels (Hamamatsu R11265-M64). Each PMT is read out by a 64 channel SPACIROC3 ASIC (AMS 0.35 $ \mu$m SiGe) operating in single photon counting mode. This data is digitised for each acquisition window of which is referred to hereafter as a gate time unit (GTU), for a data sampling rate of . The output of the SPACIROC3 ASIC is then passed to the PDM data processing, or PDM-DP. The PDM-DP consists of 3 boards, the cross board, the Zynq board and the power board, as shown in Figure \[fig:pdm-dp\].
The cross board contains 3 synchronised Xilinx Artix7 FPGAs which perform data gathering from the ASICs, pixel mapping and data multiplexing. Data is output from the cross board in a 48 $\times$ 48 pixel format transferred at , using a double data rate. The Zynq board interfaces to the cross board and contains a Zynq XC7Z030 system of programmable logic (PL) Xilinx Kintex7 FPGA, with an embedded dual core ARM9 CPU processing system (PS). The Zynq board does the majority of the data handling including data buffering, configuration of the SPACIROC3 ASICs, triggering, synchronisation, and interfacing with the separate CPU system for data storage. In addition to these tasks, the high-voltage control to the PMTs is also taken care of by the Zynq board. The power board provides the necessary voltages to the system. Figure \[fig:dataflow\] summarises the digitised data path.
[![The PDM-DP is shown with dimensions in mm. The 3 separate boards can be seen with the mechanical support for the SPACIROC3 ASICs on their left. []{data-label="fig:pdm-dp"}](figures/pdm-dp.png "fig:"){width="\textwidth"}]{}
The multi-level trigger {#sec:Trigger}
-----------------------
The Mini-EUSO trigger logic is implemented in VHDL inside the PL of the Zynq Board and consists of two levels, level 1 (L1) and level 2 (L2), that work with different time resolutions. Each level is dedicated to a specific category of events that will be observed by Mini-EUSO. The motivation behind the trigger algorithm is to capture different events of interest on short timescales, but also to provide continuous imaging on slower timescales as Mini-EUSO orbits around the Earth. In order to achieve this efficiently, 3 different types of data are stored, each with different time resolution.
[![A block diagram summarising the trigger logic. Top: L1, bottom: L2. The trigger outputs 3 separate types of data with time resolutions of , and . []{data-label="fig:L1andL2"}](figures/whole_trigger.png "fig:"){width="90.00000%"}]{}
The L1 trigger gives data with a time resolution of and looks for signal excess on a timescale of , as this corresponds to the timescale of EECR-like events. Each pixel is considered as independent, motivated by the fact that its field of view at ground is $\sim$ , so light takes at least $\sim$ to cross one pixel. Pixel signal is integrated over 8 consecutive GTUs and compared with the background level, determined by averaging over 128 GTU, to look for an excess. If the signal is $8\sigma$ above background, the event is triggered, the whole focal surface is read out and a packet of 128 GTU is stored, centred on the trigger. In addition to this, the data integrated over 128 GTU () in order to set the background level is then passed to the L2 trigger.
The L2 trigger receives the integration of 128 GTU (=1 L2 GTU, or GTU$_{L2}$) as input from the L1. It operates with a similar logic, but with a time resolution of , well-suited to capturing fast atmospheric events, such as TLEs and lightning, which have timescales from $\sim$ up to $\sim$ . Background is set by integrating 128 GTU$_{L2}$, which is also stored as the level 3 (L3) data, or 1 GTU$_{L3}$. An L2 trigger occurs when the signal in P GTU$_{L2}$ is greater than N times the background level, and the event is stored. These key parameters can be altered in-flight to optimise the trigger performance. Following offline optimisation of the trigger using simulations (as described in Section \[l2\_test\_esaf\]), the default values are N = 4 and P = 8. The background is determined as a sum of the pixel counts over 128 GTUs, which is then rescaled on P GTUs as shown in Figure \[fig:L1andL2\].
After the accumulation of 128 GTU$_{L3}$, or every , all stored events from L1, L2 and L3 data are transferred to the CPU for formatting and storage on the disk. If no L1 or L2 events are triggered, instead the last 128 GTU or 128 GTU$_{L2}$ present on the overwritten buffer are read out. In this way, a continuous and controlled readout is achieved with a resolution of whilst also capturing interesting events at faster timescales. This “movie” will be used to search for meteors, space debris and strange quark matter using offline trigger algorithms, as well as for the mapping of the Earth in UV. The L1 and L2 trigger algorithm is summarised in Figure \[fig:L1andL2\].
L1 and L2 thresholds are set to trigger, on average, at a rate lower than 1 event per . Assuming that 3 bytes/pixel are recorded, the presented trigger algorithm gives a data readout of 507 kB/s. Assuming an optimistic duty cycle of 50%, this results in a data storage requirement of 660 GB/month. Assuming some ancillary data from the camera and housekeeping systems, it is still reasonable to estimate a maximum data output of 1 TB/month.
The L1 trigger defined for Mini-EUSO is based on the same concept of observing a signal excess relative to the UV background level as that of the trigger for future large-scale EUSO instruments, such as JEM-EUSO. However, the main difference in the implementation is that 1 single pixel of Mini-EUSO has a field of view (FoV) slightly larger than the FoV of a whole MAPMT in JEM-EUSO. Therefore, in Mini-EUSO, each pixel is considered independent and the signal is integrated over 8 GTUs, which is the minimum time that the signal will stay within 1 pixel. In contrast, the JEM-EUSO trigger system will utilise a 3 $\times$ 3 pixel array to search for signal excess on timescales of 5 GTU, optimised for the observation of typical EECR signal development by a larger scale instrument. Notably, Mini-EUSO is the first implementation of a specific trigger level dedicated to “slower” events (i.e. $>$ scale) in a pathfinder mission for JEM-EUSO. The second level trigger of the future JEM-EUSO system will be dedicated to further filtering in the EECR signal time domain.
Verification of the trigger algorithm {#sec:testalg}
=====================================
Prior to the implementation of the trigger algorithm in hardware, the logic has been tested extensively using both simulated data and data taken at the TurLab facility.
L1 trigger tests at TurLab
--------------------------
The EUSO@TurLab project is an ongoing activity aimed to reproduce atmospheric and luminous phenomena that the JEM-EUSO and EUSO style telescopes will observe from Earth orbit [@Bertaina:2015hha]. TurLab is a laboratory equipped with a diameter rotating tank and located below ground level. Therefore, without artificial illumination, the room is darker than the night sky by several orders of magnitude. The EUSO@TurLab project makes use of the TurLab rotating tank with a series of different light configurations to reproduce the UV emission of the Earth. The Mini-EUSO detector is represented by one elementary cell (EC) unit of 4 MAPMTs and the necessary readout electronics. The detector is suspended from the ceiling and looks down on the rotating tank to mimic the observation from orbit (see Figure \[fig:vasca\]).
![The TurLab rotating tank. The black tube on the ceiling shows the collimator of the experimental setup used to mimic the Mini-EUSO telescope. Light sources and materials used to mimic other UV sources are also shown.[]{data-label="fig:vasca"}](figures/vasca.png){width="70.00000%"}
The capability of controlling the tank rotation speed ( - 20 minutes per turn) allows for the reproduction of events of different duration and spatial extent, as seen from ISS, with the same configuration.
Vital to the testing of the trigger algorithm in this setup is the choice and variety of light sources. There are two types of light source: 1) direct light emitting sources; 2) materials reflecting ambient light. A range of different light sources are used, with the intent of reproducing different kinds of phenomena: a) LEDs inside tubes of different dimensions, in order to reproduce extended intense light directly pointing towards the MAPMT, thus representing urban areas; b) an oscilloscope generating Lissajous curves for events such as meteors; c) LEDs driven by a pulse generator for fast luminous events such as lightning; d) LEDs or optical fibers driven by Arduino for light pulses with duration. A more detailed discussion of the setup is reported in @Bertaina:2015hha.
For these measurements in EUSO@TurLab, the apparatus consists of one fully-equipped EC unit, similar to those used in Mini-EUSO, with a 1 inch focusing lens ( focal length) placed directly in front of the MAPMTs. A test board is used to retrieve data from the EC ASICs and a LabView program is used for data acquisition. The main differences between the TurLab setup and Mini-EUSO are that data is acquired in packets of 100 GTU instead of the nominal 128 GTU, and the system has a $\sim$ delay between two consecutive acquisitions of 100 GTU. This condition slows down the measurements and introduces artificial discontinuities in the recorded light between two acquisitions. 200 simulated data packets were added between two experimental packets in order to smooth out such discontinuities. In this way, it was possible to extend the 8.2$\times$10$^5$ GTUs collected in around of rotation, to a total number of 1.6$\times$10$^8$ GTUs used to test the L1 trigger offline.
Figure \[fig:L1trigger-TurLab\] shows an example of the performance of the trigger logic described in Section \[sec:Trigger\] for one EC unit. The figure is divided into 4 different blocks. In each block the top plot shows the average number of counts per pixel, normalised at the PMT and packet level, as a function of time for one PMT, while the bottom plot indicates the time when the L1 trigger was activated. The different letters (from A to I) in the plot of PMT 2 indicate different types of light surface or reflective source present in the tank, which are responsible for a different signal seen by the PMTs. The same pattern is apparent in all 4 PMTs, but with different intensities and slightly shifted in time due to the movement of the tank and the size of the light source.
Pictures of these sources are displayed in Figure \[fig:vasca\]. A represents clouds; B and D represent the response to ground glass in which D looks brighter because it is glass illuminated by an LED; C, E and F are the reflections from sand, brick and moss, respectively; G is due to meteor-like signals; H is due to an Arduino-emulated cosmic ray and I to the reflection of clear water. The Arduino event looks quite dim compared to other signals because the track is limited to a few pixels, therefore, it is almost overwhelmed by the total number of counts in the PMT.
Despite the presence of several light sources of different intensity, duration and extension, most of the triggers occur in coincidence with the Arduino EECR-like signal transit in the field of view of the telescope for all four PMTs. The rate of spurious triggers re-scaled to one full PDM is $\sim$ , which is compatible with the acquisition logic. The rate of the Arduino events was not controlled as the main purpose of these initial tests was to verify the ability of the trigger logic to avoid light signals which are not EECR-like, and this was indeed achieved. Regarding the ability of the trigger to capture EECR-like events, the trigger efficiency has been tested both via ESAF simulations and through hardware tests as detailed in Sections \[l1\_test\_esaf\] and \[l1\_test\_hw\].
This data from TurLab was also used for online testing of the VHDL code implemented in FPGA, and the same results were obtained. These results demonstrate that the L1 trigger is sensitive to the presence of EECR-like light signals.
L1 trigger tests with ESAF {#l1_test_esaf}
--------------------------
The main objectives of the TurLab tests were the verification of the capability of the L1 trigger logic and the optimisation of the trigger thresholds with variations of light intensity. This is important in order to keep the rate of false triggers at an acceptable level. The logic demonstrated the capability of recognising and triggering on EECR-like signals. Events of longer duration such as meteors, city lights, clouds, etc. do not generate triggers, as required.
In order to evaluate the trigger performance for EECR observation, simulations using the ESAF code were performed. The EUSO Simulation and Analysis Framework (ESAF) [@Berat10] is currently used as the simulation and analysis software for the JEM-EUSO and its pathfinder missions. ESAF performs the simulation of the shower development, photon production and transport in the atmosphere, and detector simulations for optics and electronics. Furthermore, algorithms and tools for the reconstruction of the shower properties are included in the ESAF package [@Bertaina:2014fk]. Recently, the Mini-EUSO mission configuration was implemented in ESAF, including the L1 trigger logic, in order to assess its performance.
Figure \[fig:eecr\] shows the expected track (left) and light curve (right) of a EECR with energy $E$ = . Figure \[fig:eecr\_eff\] shows the trigger efficiency curve for Mini-EUSO when adopting the L1 trigger logic described in section \[sec:Trigger\].
![The trigger efficiency of Mini-EUSO as a function of the energy of the air shower, E, is shown in black. The threshold is around $E_{th} = 10^{21}$ . The effective aperture as a function of energy is also shown in red. []{data-label="fig:eecr_eff"}](figures/efficiencyExposure.pdf){width="\textwidth"}
Despite its energy threshold being too high for cosmic ray detection\
($E_{thr}\sim$), with its annual exposure of $\sim$ y , Mini-EUSO will provide a significant contribution in estimating an absolute limit on the cosmic ray flux above such energies for a null detection.
As ESAF allows the simulation of phenomena of longer durations such as TLEs, meteors, cities, etc., a few examples of these classes of events were generated in ESAF. These simulations confirmed the capability of the L1 trigger logic to avoid triggers on meteors and cities, whilst in the case of TLEs it was verified that the L1 would trigger if the rising phase of the light curve is so steep that the adaptation of the trigger thresholds at steps of is still too slow to follow the light increase. Even though the detection of TLEs and lightning is one of the main objectives of the L2 trigger logic, the L1 will allow the recording of the raising phase of the brightest and fastest signals with much higher time resolution.
An estimation of the rate of false triggers due to lower energy galactic cosmic rays hitting the PMTs directly has not yet been taken into account. The effects are currently being investigated using both dedicated simulations of the MAPMT structure and data from the recent EUSO-SPB flight [@Wiencke:2016uf], where such events were detected. From the experience of the TUS instrument [@Zotov:2017vh], these events are indeed expected, and whilst they can occur on similar timescales to those of EECR-like signals, their resulting light profile is sufficiently different such that false triggering can be blocked by a simple trigger veto.
L2 trigger tests {#l2_test_esaf}
----------------
As described in section \[sec:Trigger\], the L2 trigger operates on integrated packets of 128 GTU generated by the L1 trigger. Triggering is performed on the timescale of $\sim$ with a time resolution of , designed to capture the range of transient luminous events (TLEs) in the Earth’s atmosphere that will be visible to Mini-EUSO. TLEs are important to study, as they are part of the UV background that will be encountered by future instruments looking to study EECRs from space. However, the high temporal and spatial resolution of Mini-EUSO means that it will also be possible to make unique observations of these atmospheric events, complementing those of other dedicated instruments scheduled to fly in Earth orbit during the same period (e.g. TUS [@Panasyuk:2010gk], ASIM [@Neubert:2009ih]).
In order to test the algorithm, the ESAF simulation software was used to generate a range of typical TLE events (namely blue jets, elves and sprites), as would be seen by the Mini-EUSO focal surface. Background was superimposed onto the simulated data packets, assuming a Poisson distribution of background events centred on 1 photon/pixel/GTU [@AdamsJr:2013bd]. Examples of the TLE events considered are shown in Figure \[fig:l2test\_event\]. The L2 trigger was then run on this simulated data to test its performance. Two key parameters, the threshold level and the persistence, were varied to investigate their effect on the trigger efficiency. The threshold is simply the level at which the signal is triggered and the persistence is the time frame used to compare the instantaneous signal to background.
![The result of running the L2 trigger algorithm on the simulated TLE. The included events (from left to right) are a blue jet, a sprite, another blue jet, an elf and a final sprite. The events were spread over different areas of the focal surface, and the triggered light curves shown are for a single pixel. The grey dashed line marks an L2 trigger. This result was achieved following the optimisation of the trigger parameters and a trigger efficiency of 100% is seen.[]{data-label="fig:l2test_result"}](figures/trig_vis_MixEBG_14L2.png){width="\textwidth"}
The testing of the trigger algorithm confirmed its ability to distinguish events of interest from typical background levels and also allowed approximate lower limits to be set on the magnitude of the TLEs that Mini-EUSO will be able to detect (for typical sprites and blue jets, an absolute magnitude of $\sim$ 3, and for elves an absolute magnitude of $\sim$ 1). Figure \[fig:l2test\_result\] shows the trigger response to five different simulated TLEs. The trigger performs well for a threshold of greater than 4 $\times$ the background level and a persistence of 8 frames. It should be noted that a longer persistence increases the sensitivity of the algorithm to the more diffuse elves, but at the expense of the detection of the more localised blue jets and sprites. The final implementation of the L2 trigger should allow for some compromise here and ideally have parameters which are adjustable in-flight.
Trigger implementation {#sec: Zoom IN}
======================
First tests {#l1_test_hw}
-----------
The Mini-EUSO trigger algorithm is coded in VHDL. Following testbench and synthesis simulations, the code was implemented in the PL of the Zynq board and subsequently tested using a pulse generator to induce L1 trigger events. This allowed to test the data acquisition chain from the EC ASIC board to the L1 trigger in the Zynq board of the PDM-DP system.
In order to do this, a pulse generator was connected via a kapton cable to the MAPMT interface on the EC ASIC board, set to generate an input pulse of ($\approx$ 2 photo-electron charge equivalent, for a $5\times10^6$ PMT gain) with a duration of . This was then connected directly to the PDM-DP system, made up of the cross board, the Zynq board and the power board. The trigger algorithm was programmed to give a L1 event signal to an output pin of the Zynq board upon triggering. This L1 event signal was measured using an oscilloscope. The input pulse amplitude and shape were chosen to simply verify the correct operation of the trigger algorithm in terms of signal over threshold, and not to correspond to a typical signal expected to be detected during the Mini-EUSO mission. As shown in Table \[tab:pulsetestres\], the time between pulses was fixed at .
The number of pulses was varied for a burst rate of and a constant threshold value on the first version of the EC ASIC board (SPACIROC), prior to the upgrade to the SPACIROC 3 ASICs. The main improvements present in the SPACIROC3 are reduced power consumption, improved double pulse separation and a larger charge dynamic range [@BlinBondil:2014ve], thus the results presented here are also valid for SPACIROC3, which will be used in Mini-EUSO. As the electrical noise was set to $\sim$ 1 count/pixel/GTU, the L1 logic presented in Section \[sec:Trigger\] is expected to start triggering with high efficiency above $\sim$ 30 pulses. This is indeed confirmed by the results shown in Table \[tab:pulsetestres\].
Following these initial verification tests, more sophisticated tests of the trigger performance in hardware are currently under way making use of both simulated data and data from the recent flight of EUSO-SPB [@Wiencke:2016uf] passed directly into the front-end electronics.
**No. of pulses** **No. of trigger/min** **Trigger efficiency \[%\]** **Burst \[**\]
------------------- ------------------------ ------------------------------ ----------------
40 61 102 4
38 60 100 3.8
36 60 100 3.6
34 42 70 3.4
32 37 62 3.2
30 37 62 3
20 10.3 17 2
: Table showing the results of the pulse generator tests of the L1 trigger. Measurement was taken at a constant DAC level in the SPACIROC1 ASIC, with fixed intervals of between pulses and a burst rate of . The efficiency of 102% is due to statistical fluctuations in the background for which the average value was 1 count/pixel/GTU.
\[tab:pulsetestres\]
Ancillary trigger elements
--------------------------
In addition to the main trigger logic, the artificial data generator, pixel masking module and time stamp generator were also developed and implemented in the PL of the Zynq board, as shown in Figure \[fig:surround\]. The artificial data generator allows the generation of realistic trigger stimuli within the Zynq board, providing a useful stand-alone testing system for the trigger logic that can easily be used without the main instrument subsystems. It provides both L1 and L2 modes in order to fully test the trigger logic. Pixel masking is implemented in order to mask pixels showing unexpected behaviour. This is important in order to control fake triggers an maximise the scientific return of the instrument. The pixel masking interfaces to the PS of the Zynq and the Mini-EUSO CPU, to allow pixels to be masked in-flight via the uploading of a configuration file.
The time stamp generator is needed in order to tag triggered events precisely. Upon boot, the Zynq board is synchronised with the Mini-EUSO CPU and counts the incoming data at the GTU level. When a trigger occurs, the corresponding GTU number is read out with the event and this information is passed back to the CPU.
[![Schematic representation of the trigger logic surroundings. ADG: artificial data generator, PM: pixel masking, FC: format converter, L1: level 1 trigger, L2: level 2 trigger, TSG: time stamp generator.[]{data-label="fig:surround"}](figures/Picturef.png "fig:"){width="75.00000%"}]{}
Conclusion {#sec:conclusion}
==========
The Mini-EUSO trigger algorithm has been integrated in the Zynq Board FPGA. Prior to this, the trigger algorithm was tested successfully using simulated data and data generated as part of the EUSO@TurLab project. Once integrated in the hardware, the trigger was then tested using a pulse generator and the complete data acquisition chain. The artificial data generator implemented in the Zynq board will allow for stand alone testing of the trigger logic. Following the trigger implementation in the PDM-DP system will now be integrated with the Mini-EUSO instrument, allowing for end-to-end testing of the data acquisition system.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the Italian Ministry of Foreign Affairs and International Cooperation, Italian Space Agency (ASI) contract 2016-1-U.0, the Russian Foundation for Basic Research, grants \#15-35-21038 and \# 16-29-13065, and the Olle Engkvist Byggm[ä]{}stare Foundation. The technical support of G. Cotto, R. Forza, and M. Manfrin during validation tests at TurLab is deeply acknowledged.
References {#references .unnumbered}
==========
|
---
abstract: 'This work presents a spectroscopic study of 163 Herbig Ae/Be stars. Amongst these, we present new data for 30 objects. Stellar parameters such as temperature, reddening, mass, luminosity and age are homogeneously determined. Mass accretion rates are determined from $\rm H\alpha$ emission line measurements. Our data is complemented with the X-Shooter sample from previous studies and we update results using [*Gaia*]{} DR2 parallaxes giving a total of 78 objects with homogeneously determined stellar parameters and mass accretion rates. In addition, mass accretion rates of an additional 85 HAeBes are determined. We confirm previous findings that the mass accretion rate increases as a function of stellar mass, and the existence of a different slope for lower and higher mass stars respectively. The mass where the slope changes is determined to be $3.98^{+1.37}_{-0.94}\,\rm M_{\sun}$. We discuss this break in the context of different modes of disk accretion for low- and high mass stars. Because of their similarities with T Tauri stars, we identify the accretion mechanism for the late-type Herbig stars with the Magnetospheric Accretion. The possibilities for the earlier-type stars are still open, we suggest the Boundary Layer accretion model may be a viable alternative. Finally, we investigated the mass accretion - age relationship. Even using the superior [*Gaia*]{} based data, it proved hard to select a large enough sub-sample to remove the mass dependency in this relationship. Yet, it would appear that the mass accretion does decline with age as expected from basic theoretical considerations.'
author:
- |
C. Wichittanakom,$^{1,2}$[^1] R. D. Oudmaijer,$^{2}$ J. R. Fairlamb,$^{3}$ I. Mendigutía,$^{4}$ M. Vioque,$^{2}$ and K. M. Ababakr$^{5}$\
$^{1}$Department of Physics, Faculty of Science and Technology, Thammasat University, Rangsit Campus, Pathum Thani 12120, Thailand\
$^{2}$School of Physics and Astronomy, EC Stoner Building, University of Leeds, Leeds LS2 9JT, UK\
$^{3}$Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI, 96822, USA\
$^{4}$Centro de Astrobiología (CSIC-INTA), Departamento de Astrofísica, ESA-ESAC Campus, PO Box 78,\
28691 Villanueva de la Cañada, Madrid, Spain\
$^{5}$Erbil Polytechnic University, Kirkuk Road, Erbil, Iraq
bibliography:
- 'references.bib'
date: 'Accepted 2020 January 07. Received 2020 January 01; in original form 2019 October 09'
title: 'The accretion rates and mechanisms of Herbig Ae/Be stars'
---
\[firstpage\]
accretion, accretion discs – stars: formation – stars: fundamental parameters – stars: pre-main-sequence – stars: variables: T Tauri, Herbig Ae/Be – techniques: spectroscopic
Introduction
============
Herbig Ae/Be stars (HAeBes) are optically visible intermediate pre-main sequence (PMS) stars whose masses range from about 2 to 10$\rm M_{\sun}$. These PMS stars were first identified by [@Herbig1960], using three criteria: “Stars with spectral type A and B with emission lines; lie in an obscured region; and illuminate fairly bright nebulosity in its immediate vicinity”. HAeBes play an important role in understanding massive star formation, because they bridge the gap between low-mass stars whose formation is relatively well understood, and high-mass stars whose formation is still unclear. The study of their formation is not well understood as massive stars are very rare and as a consequence on average far away, and often optically invisible [@Lumsden2013]. A long standing problem has also been that their brightness is very high, such that radiation pressure can in principle stop accretion onto the stellar surface [@Kahn1974]. Moreover, they form very quickly and reach the main sequence before their surrounding cloud disperses [@Palla1993]. This suggests that their evolutionary processes are very different from T Tauri stars.
The accretion onto classical T Tauri low-mass stars is magnetically controlled. The magnetic field from the star truncates material in the disc and from this point, material falls onto the star along the field line. After matter hits the photosphere, it produces X-ray radiation that is absorbed by surrounding particles. Then, these particles heat up and re-radiate at ultraviolet wavelengths, producing an UV-excess that can be observed and from which the accretion luminosity can be calculated [@Bouvier2007; @Hartmann2016]. It was found later, that a correlation between the line luminosity and the accretion luminosity exists, allowing accretion rates of classical T Tauri stars to be determined from the UV-excess and absorption line veiling [@Calvet2004; @Ingleby2013].
HAeBes have similar properties as classical T Tauri stars, for instance having emission lines, UV-excess, a lower surface gravity than main-sequence stars [@Hamann1992; @Vink2005] and they are usually identified by an infrared (IR) excess from circumstellar discs [@van_den_Ancker2000; @Meeus2001]. Because their envelopes are radiative, no magnetic field is expected to be generated in HAeBe stars as this usually happens in stars by convection. Indeed, magnetic fields have rarely been detected towards them [@Catala2007; @Alecian2013]. As a result magnetically controlled accretion is not necessarily expected to apply in the case of the most massive objects, requiring another accretion mechanism. However, how the material arrives at the stellar surface in the absence of magnetic fields is still unclear.
Several lines of evidence have indicated a difference between the lower mass Herbig Ae and higher mass Herbig Be stars. Spectropolarimetric studies suggest that Herbig Ae stars and T Tauri stars may form in the same process [@Vink2005; @Ababakr2017]. Accretion rates of HAeBes are determined from the measurement of UV-excess [@Mendigutia2011b] who have found that the relationship between the line luminosity and the accretion luminosity is similar to the classical T Tauri stars. A spectroscopic variability study suggests that a Herbig Ae star is undergoing magnetospheric accretion in the same manner as classical T Tauri stars [@Scholler2016] while @Mendigutia2011a find the Herbig Be stars to have different H$\alpha$ variability properties than the Herbig Ae stars. Therefore, magnetically influenced accretion in HAeBes would still be possible. A study employing X-shooter spectra for 91 HAeBes was carried out by @Fairlamb2015 [@Fairlamb2017 hereafter F15 and F17 respectively]. They determined the stellar parameters in a homogenous fashion, derived mass accretion rates from the UV-excess and found the relationship between accretion luminosity and line luminosity for 32 emission lines in the range 0.4–2.4 $\mu$m. 47 objects in their sample were taken from @The1994 [table 1], which to this date contains the strongest (candidate) members of the group. The Fairlamb papers were published before the [*Gaia*]{} parallaxes became available, and the distances used may need revision as these can affect parameters such as the radius, luminosity, stellar mass and mass accretion rate.
A [*Gaia*]{} astrometric study of 252 HAeBes was carried out by [@Vioque2018] hereafter V18. They presented parallaxes for all known HAeBes from the [*Gaia*]{} Data Release 2 ([*Gaia*]{} DR2) Catalogue [@Gaia2016; @Gaia2018]. Also, they collected effective temperatures, optical and infrared photometry, visual extinctions, $\rm H\alpha$ equivalent widths, emission line profiles and binarities from the literature. They derived distances, luminosities, masses, ages, infrared excesses, photometric variabilities for most of their sample. This is the largest astrometric study of HAeBes to date. These [*Gaia*]{} DR2 parallaxes are useful for improving the stellar parameters of the previous studies.
The main aim of this paper is to determine stellar parameters and accretion rates of 30 Northern HAeBes using a homogeneous approach, extending the mostly Southern F15 sample. In parallel, the results of F15 will be updated using the distances determined from the [ *Gaia*]{} DR2 parallaxes and a redetermined extinction towards the objects. As such, this becomes the largest homogeneous spectroscopic analysis of HAeBes to date. In addition, the accretion rates of HAeBes in the V18 study are determined for the 104 objects for which H$\alpha$ emission line equivalent widths are collected from the literature or determined from archival spectra.
The structure of this paper is as follows. Section 2 presents the spectroscopic data observation of all targets. Details of observations, instrumental setups and reduction procedures are given in this section. Section 3 and 4 detail the determination of stellar parameters and mass accretion rates. Sections 5 and 6 focus on the analysis and a discussion respectively. Section 7 provides a summary of the main conclusions of this paper.
Observations and Data Reduction
===============================
IDS and sample selection
------------------------
The data were collected in June 2013 using the Intermediate Dispersion Spectrograph (IDS) instrument and the RED+2 CCD detector with $2048
\times 4096$ pixels (pixel size 24), which is attached to the Cassegrain focus of the 2.54-m Isaac Newton Telescope (INT) at the Observatorio del Roque de los Muchachos, La Palma, Spain. The observations spanned six nights between the $\rm 20^{th}$ and the $\rm
25^{th}$ June 2013. Bias frames, flat field frames, object frames and arc frames of Cu-Ar and Cu-Ne comparison lamp were taken each night to prepare for the data reduction.
In the first two nights the spectrograph was set up with a 1200linesmm$^{-1}$ R1200B diffraction grating and a 0.9 or 1.0 arcsecond wide slit in order to obtain spectra across the Balmer discontinuity. The wavelength coverage was 3600–4600Å, centred at 4000.5Å. In this range the hydrogen lines of $\rm
H\gamma, H\delta$, H(7-2), H(8-2), H(9-2) and H(10-2) can be analysed. This combination provides a reciprocal dispersion of 0.53Åpixel$^{-1}$ with a spectral resolution of $\sim$ 1Å and a resolving power of $R\sim4000$.
For the next two nights the R1200Y grating was used to observe spectra in the visible. Its spectral range covers 5700 to 6700Å, centred at 6050.4Å. This range includes the line at 5876Å, the \[$_{\lambda6300}$\] line and the $\rm H\alpha$ line at 6562Å. This setup results in a reciprocal dispersion of 0.52Åpixel$^{-1}$ and a resolving power of $R\sim6500$. The final 2 nights used the R1200R grating to observe the spectral range of 8200–9200Å, centred at 8597.2 and 8594.4Å. This range covers the line at 8446Å, all lines of the triplet and many lines of the Paschen series. With this setup the reciprocal dispersion becomes 0.51Åpixel$^{-1}$, giving a resolving power of $R\sim9000$.
The sample consists of 45 targets, with 30 HAeBes, and 15 standard stars. 26 Herbig stars were chosen from the catalogue of @The1994 [table 1] and 4 from [@Vieira2003]. About 67 per cent of HAeBes in the final sample are in the northern hemisphere. There are 7 HAeBes which are also in the sample of F15. Spectra of standard stars were observed each night for spectral comparisons. A log of the observations of the 30 HAeBes is shown in Table \[tab:log\] while a log of the observations of the standard stars is presented in Table \[tab:A1\_standard\_stars\] (See Appendix A in the online version of this paper).
The data reduction was performed using the Image Reduction and Analysis Facility ([IRAF]{}[^2]). Standard procedures were used in order to process all frames. Generally, there are three steps of data reduction; bias subtraction, flat field division and wavelength calibration. First, many bias frames were taken and averaged to reduce some noise and the bias level was removed from the CCD data. Next, flat field division was used to remove the variations of CCD signal in each pixel. All flat frames were averaged before subtracting the bias. Then, all of the object frames were divided by the normalised flat-field frame in order to correct the pixel-to-pixel variation of the detector sensitivity. The object frame was extracted to a one-dimensional spectrum by defining the extraction aperture on the centre of the profile and subtracting the sky background. Finally, the data were wavelength calibrated. The accuracy of the wavelength calibration is measured from an RMS of the fit and is less than 10 per cent of the reciprocal dispersion ($<0.05$Åpixel$^{-1}$). For illustration, the $\rm H\alpha$ profiles of all 30 objects are displayed in Figure \[fig:B1\_EW\] (See Appendix B in the online version of this paper).
Stellar Parameters and Mass Accretion Rate Determinations
=========================================================
Measurement of accretion rates requires accurate stellar parameters of the star in question. Therefore, this section aims to determine accretion rates by first determining the stellar parameters, such as the effective temperature, surface gravity, distance, radius, reddening, luminosity, mass and age. These will be determined by combining the IDS spectra, stellar model atmosphere grids, photometry from the literature, the [*Gaia*]{} DR2 parallaxes and stellar isochrones.
Effective temperature and surface gravity
-----------------------------------------
To estimate the effective temperature of the target, a comparison of the known standard star spectra with the unknown target spectrum is performed. The list of standard stars including spectral types is shown in Table \[tab:A1\_standard\_stars\] (See Appendix A in the online version of this paper). The conversion from the spectral type to effective temperature can be found from [@Straizys1981]. Then, the range of effective temperatures was explored in more detail with spectra computed from model atmospheres.
The model atmospheres used in this work are grids of BOSZ-Kurucz model atmospheres computed by [@Bohlin2017]. The BOSZ models are calculated from ATLAS-APOGEE ATLAS9 [@Meszaros2012] which came from the original ATLAS code version 9 [@Kurucz1993]. The metallicity $[\rm M/H]=0$, carbon abundance $\rm [C/H]=0$, alpha-element abundance $[\alpha/ \rm H]=0$, microturbulent velocity $\xi=2.0$kms$^{-1}$, rotational broadening velocity $v\sin i=0.0$kms$^{-1}$, and instrumental broadening $R=5000$ are adopted. This instrumental broadening is chosen for matching the resolution of the blue spectra. The range of effective temperature $T_{\rm eff}$ is from 3500K to 30000K with steps of 250K (from 3500K to 12000K), 500K (from 12000K to 20000K) and 1000K (from 20000K to 30000K). The range of surface gravity $\log(g)$ is from 0.0dex to 5.0dex with steps of 0.5dex. By using linear interpolation, $\log(g)$ with steps of 0.1dex were calculated.
The procedure to obtain the effective temperature and surface gravity in this work follows the same method by F15. The effective temperature and surface gravity determination is carried out by primarily comparing the wings of the observed hydrogen Balmer lines $\rm H\gamma$, $\rm H\delta$ and $\rm H\epsilon$ with synthetic profiles produced with solar metallicity from BOSZ models. The shapes of the hydrogen profiles are not very sensitive to metallicity [@Bohlin2017] while their widths are dominated by pressure broadening, while rotation hardly affects the shapes. $\rm
H\alpha$ is not used for spectral typing because it is frequently strongly in emission for HAeBes. This phenomenon can affect the shape of the wings and cause difficulty of fitting the BOSZ models to target spectra. $\rm H\beta$ is also not used for spectral typing because it appears around the edge of the blue spectrum which makes a proper characterization of $\rm H\beta$’s line profile troublesome.
In order to carry out the fitting, both the observed profiles and the synthetic profiles are normalised based on the continuum on both the blue and red side of the profile. Next, the observed wavelengths of the target profiles were corrected to the vacuum wavelength of the synthetic profile using the [IRAF DOPCOR]{} task. The effective temperature and surface gravity are obtained from the fit of the normalised synthetic spectra to the normalised observed spectra by considering continuum features and the wings of the profile above the normalised intensity of 0.8. This intensity was chosen because this part of the wings is sensitive to variation of the $\log(g)$ while the central part of the profile of a Herbig Ae/Be star can be contaminated by emission.
The width of the Balmer lines depends upon both effective temperature and surface gravity. Different combinations of $T_{\rm eff}$ and $\log(g)$, can create the same width. This degeneracy can be solved by visual inspection of absorption features in the wings and continuum either side of the lines. The final value for the effective temperature and surface gravity are the average value of the best fit for each profile ($\rm H\gamma$, $\rm H\delta$ and $\rm H\epsilon$). The uncertainties of $T_{\rm eff}$ and $\log(g)$ were chosen to be the typical difference in the values determined for the three lines respectively. If the standard error becomes zero or less than that, the step size will be adopted. An example of spectral typing by fitting the Balmer line profile of HD 141569 (black) and the BOSZ model (blue) is demonstrated in Figure \[fig:HD141569\_Hgamma\].
------------------- ------------ ------------- ---------------- --------- ---------- --------------- ----- ----- ----- ------------------- --------- --------- ------------- -------------- --------------
Name RA DEC Obs Date Spectral Type
\[3pt\] (J2000) (J2000) (June 2013) R1200B R1200Y R1200R B Y R [*B*]{} [*V*]{} [*R$_c$*]{} [*R*]{} [*I$_c$*]{}
\[3pt\] V594 Cas 00:43:18.3 +61:54:40.1 21; 23; 24 600 600 600 152 160 50 B8$^{1}$ 11.08 10.51 10.08 - 9.55$^{13}$
\[3pt\] PDS 144 15:49:15.3 -26:00:54.8 22; 24 - 900 900 - 97 83 A5 V$^{2}$ 13.28 12.79 12.49 - 12.16$^{2}$
\[3pt\] HD 141569 15:49:57.8 -03:55:16.3 20; 22; 24 60 60 60 302 385 194 A0 Ve$^{3}$ 7.23 7.13 7.08 - 7.02$^{2}$
\[3pt\] HD 142666 15:56:40.0 -22:01:40.0 20; 22; 24 90; 180 90; 150 150 19 147 47 A8 Ve$^{3}$ 9.17 8.67 8.35 - 8.01$^{2}$
\[3pt\] HD 145718 16:13:11.6 -22:29:06.7 21; 23; 25 90 180 180 30 160 74 A8 IV$^{2}$ 9.62 9.10 8.79 - 8.45$^{2}$
\[3pt\] HD 150193 16:40:17.9 -23:53:45.2 21; 23; 24; 25 90 300 120; 300 54 258 96 A2 IVe$^{3}$ 9.33 8.80 8.41 - 7.93$^{14}$
\[3pt\] PDS 469 17:50:58.1 -14:16:11.8 23; 25 - 1200 1200 - 127 62 A0$^{2}$ 13.33 12.77 12.39 - 11.95$^{2}$
\[3pt\] HD 163296 17:56:21.3 -21:57:21.9 20; 23; 24; 25 60 60 60 60 270 125 A1 Vep$^{3}$ 6.94 6.83 6.77 - 6.73$^{14}$
\[3pt\] MWC 297 18:27:39.5 -03:49:52.1 21; 22; 24; 25 900 10; 60 1200; 600 22 16 102 B0$^{2}$ 14.28 12.03 10.19 - 8.80$^{14}$
\[3pt\] VV Ser 18:28:47.9 +00:08:39.8 21; 23; 25 900 600 600 91 180 113 B6$^{4}$ 12.82 11.81 11.10 - 10.31$^{13}$
\[3pt\] MWC 300 18:29:25.7 -06:04:37.3 21; 23; 25 900 600; 60 600 48 29 19 B1 Ia+\[e\]$^{5}$ 12.81 11.78 11.09 - 10.58$^{14}$
\[3pt\] AS 310 18:33:21.3 -04:58:04.8 20; 22; 24 40; 900 1200 1200 35 144 128 B1e$^{4}$ 13.56 12.59 11.93 - 11.20$^{14}$
\[3pt\] PDS 543 18:48:00.7 +02:54:17.1 21; 22; 24 900 1200 1200 39 153 149 B1$^{2}$ 14.56 12.52 11.25 - 9.96$^{2}$
\[3pt\] HD 179218 19:11:11.3 +15:47:15.6 21; 23; 25 60 120 120 140 311 166 A0 IVe$^{3}$ 7.47 7.39 7.34 - 7.29$^{2}$
\[3pt\] HD 190073 20:03:02.5 +05:44:16.7 21; 23; 25 60 120 120; 300 65 205 90 A0 IVp+sh$^{2}$ 7.86 7.73 - 7.66$^{15}$ -
\[3pt\] V1685 Cyg 20:20:28.2 +41:21:51.6 20; 22; 24; 25 600 600; 60 600; 60 120 114 82 B2 Ve$^{3}$ 11.48 10.69 - 9.63$^{16}$ -
\[3pt\] LkHA 134 20:48:04.8 +43:47:25.8 20; 22; 24 600 600 600 79 192 95 B8$^{6}$ 12.02 11.35 - 10.56$^{16}$ -
\[3pt\] HD 200775 21:01:36.9 +68:09:47.8 20; 22; 24; 25 60 60; 30 30; 60 161 192 50 B3$^{4}$ 7.77 7.37 - 6.84$^{16}$ -
\[3pt\] LkHA 324 21:03:54.2 +50:15:10.0 21; 23; 25 900 1200 1200 52 166 123 B8$^{4}$ 13.71 12.56 11.85 - 11.09$^{13}$
\[3pt\] HD 203024 21:16:03.0 +68:54:52.1 21; 23; 24; 25 120 180 180 56 187 65 A5 V$^{3}$ 9.24 9.01 9.02 - 8.62$^{17}$
\[3pt\] V645 Cyg 21:39:58.3 +50:14:20.9 21; 23; 25 900 1200 1200 20 86 152 O8.5$^{7}$ 14.55 13.47 - 12.28$^{16}$ -
\[3pt\] V361 Cep 21:42:50.2 +66:06:35.1 20; 22; 24; 25 300 300 300 146 219 84 B4$^{4}$ 10.65 10.21 9.89 - 9.50$^{13}$
\[3pt\] V373 Cep 21:43:06.8 +66:06:54.2 20; 22; 24 900 1200 1200 96 143 103 B8$^{8}$ 13.22 12.33 11.7 - 11.01$^{13}$
\[3pt\] V1578 Cyg 21:52:34.1 +47:13:43.6 21; 23; 24; 25 300 300; 600 300 86 225 191 A0$^{4}$ 10.53 10.13 - 9.67$^{16}$ -
\[3pt\] LkHA 257 21:54:18.8 +47:12:09.7 20; 22; 23; 24 900 1200 1200 54 179 129 A2e$^{9}$ 14.00 13.29 12.86 - 12.39$^{17}$
\[3pt\] SV Cep 22:21:33.2 +73:40:27.1 20; 22; 24 240 300 300 87 188 112 A2 IVe$^{3}$ 11.37 10.98 - 10.58$^{16}$ -
\[3pt\] V375 Lac 22:34:41.0 +40:40:04.5 23; 25 - 1200 1200 - 109 71 A7 Ve$^{10}$ 14.22 13.38 12.84 - 12.25$^{13}$
\[3pt\] HD 216629 22:53:15.6 +62:08:45.0 20; 22; 24 90 180 180 91 251 110 B3 IVe+A3$^{11}$ 10.01 9.28 - 8.53$^{16}$ -
\[3pt\] V374 Cep 23:05:07.5 +62:15:36.5 20; 22; 24 600 450 450 121 173 138 B5 Vep$^{12}$ 11.14 10.28 9.71 - 9.12$^{17}$
\[3pt\] V628 Cas 23:17:25.6 +60:50:43.4 21; 23; 24; 25 900 600; 60 600 26 130 149 B0eq$^{1}$ 12.63 11.37 10.38 - 9.37$^{13}$
\[3pt\]
------------------- ------------ ------------- ---------------- --------- ---------- --------------- ----- ----- ----- ------------------- --------- --------- ------------- -------------- --------------
: Log of observations of 30 HAeBes. Column 1 gives the object name. Columns 2 and 3 are right ascension (RA) in the units of time ( ) and declination (DEC) in the units of angle ( ) respectively. Column 4 lists the observation dates. Columns 5–7 present the exposure times for each grating. The signal-to-noise ratios for each grating are given in columns 8–10. These were determined using a 20Å wide line free spectral regions centred at the wavelength 4200, 6050 and 8800Å for R1200B, R1200Y and R1200R grating respectively. Spectral types and photometry are listed along with references in columns 11–16.[]{data-label="tab:log"}
\
[**References.**]{} $^{(1)}$ @Hillenbrand1992; $^{(2)}$ @Vieira2003; $^{(3)}$ @Mora2001; $^{(4)}$ @Hernandez2004; $^{(5)}$ @Wolf1985; $^{(6)}$ @Herbig1958; $^{(7)}$ @Clarke2006; $^{(8)}$ @Dahm2015; $^{(9)}$ @Walker1959; $^{(10)}$ @Calvet1978; $^{(11)}$ @Skiff2014; $^{(12)}$ @Garrison1970; $^{(13)}$ @Fernandez1995; $^{(14)}$ @deWinter2001; $^{(15)}$ @Oudmaijer2001; $^{(16)}$ @Herbst1999; $^{(17)}$ @Zacharias2012.
![The normalised spectrum of HD 141569 fits with BOSZ model of $T_{\rm eff}=9500$K and $\log(g)=4.2$.[]{data-label="fig:HD141569_Hgamma"}](HD141569_HgdeD_Teff_logg_s15_20190610.png){width="\columnwidth"}
For a given $T_{\rm eff}$, normally, the higher the $\log(g)$, the broader the Balmer profile. Therefore, the region of the wings of the hydrogen profile near the continuum level can be used to obtain both effective temperature and surface gravity as was mentioned earlier. Unfortunately, there is a non-linear relationship between the surface gravity and width of the Balmer profile for objects that have $T_{\rm
eff} < 8000$K. For this reason, a spectroscopic $\log(g)$ can not be determined and the surface gravity is calculated instead using the stellar mass and radius (see later). For 3 objects (PDS 144S, PDS 469 and V375 Lac) no near UV blue spectra were taken during the observations. Fortunately, a FEROS spectrum of PDS 469[^3] was found in the ESO Science Archive Facility and was used to determine temperature and surface gravity. In the case of PDS 144S and V375 Lac, visible spectra including $\rm
H\alpha$ profiles in combination with spectral types from the literature were investigated in order to adopt their effective temperatures.
7 objects show extremely strong emission lines. These strong emissions have an influence on the wings on even the whole Balmer profiles. Both $T_{\rm eff}$ and $\log(g)$ could therefore not be determined by this method. Instead, for these objects, estimated temperatures from the literature are adopted. The stars for which this process is performed on are noted in the last column of Table \[tab:all\_stellar\_parameters\].
There are 7 objects in this work that overlap with F15. The difference in temperatures of these objects is on average 180K, which is smaller than the step-size used by both studies. Figure \[fig:logT\] compares the effective temperature derived in this work (Table \[tab:all\_stellar\_parameters\]) with estimated values from the literature. The good correlation would suggest that our method is reliable, the fact that it is applied to the entire sample ensures a homogeneous study.
![The effective temperature derived in this work compared to temperatures derived from spectral types in the literature listed in Table \[tab:log\] and Table \[tab:A1\_standard\_stars\]. The spectral type was converted to temperature using the values provided in [@Straizys1981], and an uncertainty of a subclass of spectral type was assigned for the temperature. HAeBes and standard stars are denoted in circle and square respectively. Circles with larger circle around them indicate the objects that temperature from [@Fairlamb2015] is used. The standard deviation ($\sigma$) between both $\log(T_{\rm eff})$ is only 0.02. The solid line is the expected line of correlation and the dashed lines are $3\,\sigma$ deviation from the solid line.[]{data-label="fig:logT"}](logT_20190929.png){width="\columnwidth"}
Visual extinction, distance and radius
--------------------------------------
The second step is to use the synthetic BOSZ spectral energy distribution and previous photometry results to determine the visual extinction or reddening ($A_V$) by fitting the synthetic surface flux density to observed photometry. In the case of zero extinction or for extinction corrected photometry, the flux density $f$ of the BOSZ model and the observed fluxes differ by the ratio of distance to the star and its radius ($D/R_*$). The spectral energy distribution grid of the BOSZ models are set up for the effective temperatures from the previous step with a range of scaling factors $D/R_*$. The value of $\log(g)$ does not have a significant effect on the spectral energy distribution shape. Therefore, $\log(g)=4.0$ was adopted at this stage.
The observed photometry is dereddened using the extinction values $A_\lambda /A_V$ from [@Cardelli1989] with the standard ratio of total to selective extinction parameter $R_V=3.1$ and zero-magnitude fluxes from [@Bessell1979]. By varying $A_V$ in steps of 0.01magnitude, the best fit of the photometry and BOSZ models will then yield the best fitting reddening values. Only the [*BVRI*]{} magnitudes are used, as the [*U*]{}-band and [*JHKLM*]{} photometry are often affected by the Balmer continuum excess and IR-excess emission respectively, i.e. they can not automatically be used in the fitting. All of the observed [*BVR$_c$I$_c$*]{} or [ *BVR*]{} photometry from the literature are shown in Table \[tab:log\]. For 3 objects (HD 203024, LkHA 257 and V374 Cep) their Sloan photometry needed to be converted to Johnson-Cousins photometry. This was done using the transformation equations provided by [@Smith2002]. The uncertainties in the resulting $A_V$ and $D/R_*$ are assigned to be at values that resulted twice of the minimum chi-squared value. Figure \[fig:HD141569\_photometry\] demonstrates an example of photometry fitting.
![ The synthetic spectral energy distribution BOSZ model (black line) is fitted to the dereddened photometry (red point) of HD 141569. The observed [*BVR$_c$I$_c$*]{} photometry of HD 141569 were taken from [@Vieira2003]. A synthetic spectral energy distribution BOSZ model with $T_{\rm eff}=9500$K and $\log(g)=4.0$ is fitted to the dereddened photometry. This provides the reddening $A_V=0.38^{+0.02}_{-0.03}$mag and the scaling factor $D/R_* =
63.7^{+1.1}_{-0.7}$pc/$\rm R_{\sun}$. []{data-label="fig:HD141569_photometry"}](HD141569_photometry3pt3_BVRcIc_SED_20190923.png){width="\columnwidth"}
The scaling factor $D/R_*$ allows us to calculate stellar radius $R_*$, provided the stellar distance $D$ is known. The [*Gaia*]{} DR2 catalogue provides astrometric parameters, such as positions, proper motions and parallaxes for more than 1.3 billion targets including most of the known HAeBes. The distances to most of our targets were determined by V18 using [*Gaia*]{} DR2 parallaxes. Re-normalised unit weight error (RUWE) is used to select sources with good astrometry. We adopted RUWE $<1.4$ as a criterion for good parallaxes (see [*Gaia*]{} Data Release 2 document). 7 stars have low quality parallaxes, and for 2 stars no parallaxes are presented in the [*Gaia*]{} archive. These are noted in the final column of Table \[tab:all\_stellar\_parameters\]. In total, stellar radii could be determined for 26 out of the 30 targets.
Stellar luminosity, mass and age
--------------------------------
Using the stellar radius $R_*$ and the effective temperature $T_{\rm
eff}$, the luminosity $L_*$ can be determined from the Stefan-Boltzmann law. The next step is to estimate the mass and age of the HAeBes using isochrones. Stellar isochrones of [@Marigo2017] from 0.01 to 100Myr are used in order to extract a mass and an age of the target from the luminosity-temperature Hertzsprung-Russell diagram. A metallicity $Z=0.01$ and helium mass fraction $Y=0.267$ are chosen, because these values are close to solar values.
After each star is placed on the HR diagram, the two closest points on an isochrone are used to obtain the mass of the star by interpolating between those points. Uncertainties of mass and age are derived from the error bars of the effective temperature $T_{\rm eff}$ and the luminosity $L_*$ on the HR diagram. All determined parameters of all targets are presented in Table \[tab:all\_stellar\_parameters\]. The positions of the 21 HAeBes on the HR diagram are represented with red symbols in Figure \[fig:HR\_all\] As mentioned above, the 9 targets that are not included in the plot have low quality parallaxes or do not have parallaxes at all. Since the 3 objects (HD 142666, HD 145718 and SV Cep) have had $T_{\rm
eff}<8000$K, their surface gravity cannot be determined from fitting the spectra with stellar atmospheric models. Instead, their surface gravities were calculated from the stellar mass and radius derived from the parallexes instead. These are also noted in the $\log(g)$ column of Table \[tab:all\_stellar\_parameters\].
------------------- -------------------------------- ------------------------------- ------------------------ -------------------------- ------------------------- ------------------------ --------------------------- --------------------------
Name $T_{\rm eff}$ $\log(g)$ $A_V$ $D$ $R_*$ $\log(L_*)$ $M_*$ Age
\[3pt\] (K) \[cms$^{-2}$\] (mag) (pc) ($\rm R_{\sun}$) \[$\rm L_{\sun}$\] ($\rm M_{\sun}$) (Myr)
V594 Cas $11500^{+250}_{-250}$ $3.70^{+0.10}_{-0.10}$ $2.27^{+0.18}_{-0.23}$ $569^{+16}_{-14}$ $3.86^{+0.52}_{-0.53}$ $2.37^{+0.15}_{-0.17}$ $3.51^{+0.45}_{-0.42}$ $1.29^{+0.53}_{-0.38}$
\[3pt\] PDS 144S $7750^{+500}_{-500}$ $^{a}$ $4.00^{+0.30}_{-0.30}$ $1.02^{+0.07}_{-0.07}$ -$^{d}$ - - - -
\[3pt\] HD 141569 $9500^{+250}_{-250}$ $4.20^{+0.10}_{-0.10}$ $0.38^{+0.02}_{-0.03}$ $110.63^{+0.91}_{-0.88}$ $1.74^{+0.03}_{-0.04}$ $1.34^{+0.06}_{-0.07}$ $2.06^{+0.02}_{-0.15}$ $5.89^{+1.87}_{-0.64}$
\[3pt\] HD 142666 $7250^{+250}_{-250}$ $4.00^{+0.10}_{-0.10}$ $^{c}$ $0.82^{+0.07}_{-0.08}$ $148.3^{+2}_{-1.9}$ $2.21^{+0.11}_{-0.12}$ $1.08^{+0.10}_{-0.11}$ $1.64^{+0.12}_{-0.11}$ $7.76^{+1.79}_{-1.30}$
\[3pt\] HD 145718 $7750^{+250}_{-250}$ $4.20^{+0.10}_{-0.10}$ $^{c}$ $1.10^{+0.06}_{-0.06}$ $152.5^{+3.2}_{-3}$ $1.85^{+0.10}_{-0.10}$ $1.05^{+0.10}_{-0.10}$ $1.62^{+0.07}_{-0.03}$ $8.71^{+0.84}_{-1.12}$
\[3pt\] HD 150193 $9250^{+250}_{-250}$ $4.10^{+0.10}_{-0.10}$ $1.88^{+0.16}_{-0.20}$ $150.8^{+2.7}_{-2.5}$ $2.34^{+0.26}_{-0.27}$ $1.56^{+0.14}_{-0.15}$ $2.12^{+0.21}_{-0.12}$ $4.57^{+0.93}_{-1.02}$
\[3pt\] PDS 469 $9500^{+750}_{-750}$ $^{a}$ $3.80^{+0.30}_{-0.30}$ $1.94^{+0.12}_{-0.13}$ -$^{d}$ - - - -
\[3pt\] HD 163296 $9000^{+250}_{-250}$ $4.10^{+0.10}_{-0.10}$ $0.29^{+0.01}_{-0.02}$ $101.5^{+2}_{-1.9}$ $1.87^{+0.05}_{-0.05}$ $1.31^{+0.07}_{-0.07}$ $1.95^{+0.07}_{-0.07}$ $6.03^{+0.28}_{-0.27}$
\[3pt\] MWC 297 $24000^{+2000}_{-2000}$ $^{b}$ $4.00^{+0.10}_{-0.10}$ $7.87^{+0.41}_{-0.64}$ $375^{+22}_{-18}$ $9.28^{+3.12}_{-3.04}$ $4.41^{+0.39}_{-0.50}$ $14.53^{+6.11}_{-4.84}$ $0.04^{+0.07}_{-0.02}$
\[3pt\] VV Ser $14000^{+1000}_{-1000}$ $4.30^{+0.30}_{-0.30}$ $3.74^{+0.22}_{-0.27}$ -$^{d}$ - - - -
\[3pt\] MWC 300 $23000^{+2000}_{-2000}$ $^{b}$ $3.00^{+0.20}_{-0.20}$ $3.85^{+0.21}_{-0.28}$ $1400^{+250}_{-160}$ $6.02^{+1.96}_{-1.44}$ $3.96^{+0.39}_{-0.39}$ $10.09^{+3.76}_{-1.89}$ $0.09^{+0.09}_{-0.05}$
\[3pt\] AS 310 $26000^{+2000}_{-2000}$ $4.40^{+0.35}_{-0.35}$ $3.86^{+0.20}_{-0.24}$ $2110^{+350}_{-240}$ $5.70^{+1.71}_{-1.27}$ $4.13^{+0.36}_{-0.36}$ $11.60^{+3.92}_{-2.18}$ $0.07^{+0.08}_{-0.04}$
\[3pt\] PDS 543 $28500^{+2500}_{-2500}$ $^{b}$ $4.00^{+0.10}_{-0.10}$ $7.11^{+0.28}_{-0.40}$ $1410^{+240}_{-160}$ $16.24^{+5.99}_{-4.57}$ $5.19^{+0.42}_{-0.45}$ $30.02^{+18.28}_{-10.85}$ $0.01^{+0.01}_{-0.01}$
\[3pt\] HD 179218 $9500^{+250}_{-250}$ $3.95^{+0.10}_{-0.10}$ $0.33^{+0.02}_{-0.02}$ $266^{+5.6}_{-5.2}$ $3.62^{+0.12}_{-0.11}$ $1.98^{+0.07}_{-0.07}$ $2.86^{+0.16}_{-0.20}$ $2.04^{+0.47}_{-0.26}$
\[3pt\] HD 190073 $9750^{+250}_{-250}$ $3.50^{+0.10}_{-0.10}$ $0.20^{+0.04}_{-0.04}$ $870^{+100}_{-70}$ $9.23^{+1.28}_{-0.94}$ $2.84^{+0.16}_{-0.14}$ $5.62^{+0.78}_{-0.65}$ $0.28^{+0.14}_{-0.10}$
\[3pt\] V1685 Cyg $23000^{+4000}_{-4000}$ $^{b}$ $4.06^{+0.10}_{-0.10}$ $3.33^{+0.34}_{-0.51}$ $910^{+46}_{-39}$ $5.48^{+1.51}_{-1.51}$ $3.88^{+0.49}_{-0.61}$ $9.53^{+4.57}_{-2.55}$ $0.11^{+0.27}_{-0.07}$
\[3pt\] LkHA 134 $11000^{+250}_{-250}$ $4.00^{+0.10}_{-0.10}$ $2.44^{+0.19}_{-0.24}$ $843^{+36}_{-31}$ $4.53^{+0.71}_{-0.70}$ $2.43^{+0.17}_{-0.19}$ $3.77^{+0.56}_{-0.56}$ $1.02^{+0.60}_{-0.34}$
\[3pt\] HD 200775 $19000^{+3000}_{-3000}$ $4.27^{+0.25}_{-0.25}$ $1.85^{+0.15}_{-0.17}$ -$^{d}$ - - - -
\[3pt\] LkHA 324 $12500^{+500}_{-500}$ $4.00^{+0.10}_{-0.10}$ $3.94^{+0.14}_{-0.16}$ $605^{+16}_{-14}$ $3.15^{+0.34}_{-0.33}$ $2.34^{+0.16}_{-0.17}$ $3.36^{+0.41}_{-0.30}$ $1.51^{+0.49}_{-0.41}$
\[3pt\] HD 203024 $8500^{+500}_{-500}$ $3.83^{+0.29}_{-0.29}$ $0.52^{+0.24}_{-0.31}$ -$^{e}$ - - - -
\[3pt\] V645 Cyg $30000^{+7000}_{-7000}$ $^{b}$ $3.75^{+0.35}_{-0.35}$ $4.21^{+0.29}_{-0.40}$ -$^{d}$ - - - -
\[3pt\] V361 Cep $16750^{+500}_{-500}$ $4.00^{+0.10}_{-0.10}$ $1.97^{+0.14}_{-0.15}$ $893^{+35}_{-31}$ $4.34^{+0.52}_{-0.48}$ $3.12^{+0.15}_{-0.15}$ $5.56^{+0.65}_{-0.57}$ $0.40^{+0.16}_{-0.11}$
\[3pt\] V373 Cep $11500^{+1250}_{-1250}$ $^{b}$ $3.50^{+0.50}_{-0.50}$ $3.24^{+0.20}_{-0.23}$ -$^{d}$ - - - -
\[3pt\] V1578 Cyg $10500^{+500}_{-500}$ $3.80^{+0.20}_{-0.20}$ $1.46^{+0.08}_{-0.08}$ $773^{+30}_{-27}$ $4.77^{+0.41}_{-0.37}$ $2.39^{+0.15}_{-0.15}$ $3.74^{+0.45}_{-0.41}$ $1.02^{+0.39}_{-0.28}$
\[3pt\] LkHA 257 $9250^{+250}_{-250}$ $4.05^{+0.10}_{-0.10}$ $2.28^{+0.07}_{-0.08}$ $794^{+18}_{-16}$ $1.84^{+0.11}_{-0.11}$ $1.35^{+0.10}_{-0.10}$ $1.98^{+0.06}_{-0.04}$ $5.76^{+1.32}_{-0.51}$
\[3pt\] SV Cep $8000^{+500}_{-500}$ $4.37^{+0.11}_{-0.11}$ $^{c}$ $0.78^{+0.04}_{-0.05}$ $344.3^{+4}_{-3.8}$ $1.48^{+0.05}_{-0.06}$ $0.91^{+0.14}_{-0.15}$ $1.63^{+0.04}_{-0.14}$ $11.00^{+10.40}_{-2.49}$
\[3pt\] V375 Lac $8000^{+750}_{-750}$ $^{a}$ $4.30^{+0.10}_{-0.10}$ $2.31^{+0.14}_{-0.16}$ -$^{e}$ - - - -
\[3pt\] HD 216629 $21500^{+1000}_{-1000}$ $4.00^{+0.10}_{-0.10}$ $2.86^{+0.13}_{-0.15}$ $805^{+31}_{-27}$ $7.59^{+0.94}_{-0.83}$ $4.04^{+0.18}_{-0.18}$ $10.83^{+1.71}_{-1.54}$ $0.07^{+0.04}_{-0.02}$
\[3pt\] V374 Cep $15500^{+1000}_{-1000}$ $3.50^{+0.10}_{-0.10}$ $3.20^{+0.15}_{-0.18}$ $872^{+40}_{-35}$ $7.72^{+1.05}_{-0.98}$ $3.49^{+0.22}_{-0.23}$ $7.50^{+1.46}_{-1.28}$ $0.16^{+0.12}_{-0.06}$
\[3pt\] V628 Cas $31000^{+5000}_{-5000}$ $^{b}$ $4.00^{+0.10}_{-0.10}$ $5.05^{+0.36}_{-0.55}$ -$^{d}$ - - - -
\[3pt\]
------------------- -------------------------------- ------------------------------- ------------------------ -------------------------- ------------------------- ------------------------ --------------------------- --------------------------
\
[**Notes.**]{} $^{(a)}$ Stars for which NUV-B spectra were not obtained. $^{(b)}$ Stars which display extremely strong emission lines. $^{(c)}$ Stars for which parallactic $\log(g)$ is used. $^{(d)}$ Stars which have low quality parallaxes in the [*Gaia*]{} DR2 Catalogue (see the text for discussion). $^{(e)}$ Stars which do not have parallaxes in the [*Gaia*]{} DR2 Catalogue.
Extending the sample with HAeBes in the southern hemisphere
-----------------------------------------------------------
The original driver for the INT observations presented here was to extend the spectroscopic analysis of the, mostly southern, sample of F15 to the northern hemisphere. However, F15 determined spectroscopic distances using their spectral derived values of the surface gravity or literature values for the distances to their 91 HAeBes. The availability of [*Gaia*]{}-derived distances warrants a re-determination of the stellar parameters. To this end we use the distances from the [*Gaia*]{} DR2 parallax (V18). In parallel, we re-assessed the extinction values using the same photometric bands [*BVR$_c$I$_c$*]{} as above to ensure consistency in the determination of the extinction. Most of the photometry used for the photometry fitting can be found in @Fairlamb2015 [table A1]. We collated Sloan photometry from @Zacharias2012 for 3 objects (HD 290500, HT CMa and HD 142527) and one object, HD 95881, from [*APASS*]{}[^4] DR10 and converted these to the Johnson-Cousins system. Moreover, we also used the brightest [*V*]{}-band magnitude of Johnson [*BVR*]{} photometry for 2 objects, HD 250550 and KK Oph, and Johnson [*BVRI*]{} photometry for Z CMa from [@Herbst1999]. The redetermined stellar parameters for the 91 HAeBes from F15 are listed in Table \[tab:C1\_Fairlamb\_para\] (See Appendix C in the online version of this paper). For the sample as a whole, the luminosities resulting from the revised distances and extinctions are on average similar to the original F15 values, however the contribution to the scatter around the mean differences is dominated by the new distances. We thus conclude that the improvement in distance values dominates that of the extinction when arriving at the final luminosities.
We now have the largest spectroscopic sample of HAeBes and the homogeneously determined stellar parameters including mass accretion rates, which will be determined in the next section. 7 targets in F15’s sample which are also in this work’s sample were left out, leaving 84 HAeBes in their sample. Unfortunately, 22 out of the 84 objects (not including PDS 144S) have low quality or do not have a [*Gaia*]{} DR2 parallax. These objects cannot be placed on the HR diagram at this stage. The final sample contains 83 Herbig Ae/Be objects. Figure \[fig:HR\_all\] demonstrates all these HAeBes ($21+62$) placed in the HR diagram. Many targets are gathered around $2\,\rm M_{\sun}$ and few targets are located at high-mass tracks. This can be understood by the initial mass function IMF [@Salpeter1955]. Low-mass stars are more common than high-mass stars. Moreover, low-mass stars evolve slower across the HR diagram.
{width="12cm"}
Combining with additional HAeBes with [*Gaia*]{} DR2 data from V18
------------------------------------------------------------------
In order to expand the sample further, the rest of 144/252 objects in the sample of V18 were investigated. There are 101 objects which satisfy the condition RUWE $<1.4$. They are also included in Figure \[fig:HR\_all\]. H$\alpha$ equivalent widths are provided for most objects in @Vioque2018 [their table 1 and table 2] with references. Spectra for 14 objects without H$\alpha$ data in the V18 sample were found in the ESO Science Archive Facility. These were downloaded and their H$\alpha$ EWs were measured. The stellar radii and surface gravities were calculated using the temperatures, luminosities and masses provided in V18. The intrinsic equivalent widths were measured from BOSZ models for the temperature provided in @Vioque2018 [table 1 and table 2] and the calculated surface gravity. This will enable us to compute mass accretion rates for a further 85 objects below. Table \[tab:D1\_Vioque\_Mdot\] presents all of the emission lines measurements and determined accretion rates of the V18 sample (See Appendix D in the online version of this paper).
Accretion rate determination
============================
We now will determine the mass accretion rates of the combined sample of Herbig Ae/Be stars. The underlying assumption of the methodology is that the stars accrete material according to the MA paradigm, which, as demonstrated by [@Muzerolle2004], can explain the observed excess fluxes in Herbig Ae/Be stars. The infalling material shocking the photosphere gives rise to UV-excess emission, whose measurement can be converted into an accretion luminosity. Knowledge of the stellar parameters can then return a value of the mass accretion rate.
F15 derived the accretion rates for their sample from the UV excess following the methodology of [@Muzerolle2004] and extending the work of @Mendigutia2011b. Unfortunately, as mentioned before, our current observational set-up did not allow for such accurate measurements. However, as for example pointed out by [@Mendigutia2011b], the accretion luminosity correlates with the line strengths of various types of emission lines. They will therefore also correlate with the mass accretion rate (see also @Fairlamb2017). As such, the line strengths do provide an observationally cheap manner to derive the accretion rate of an object without having to resort to the rather delicate and time-consuming process of measuring the UV excess. We should note that despite this, it is not clear whether the observed correlation is intrinsically due to accretion or some other effect [@Mendigutia2015a].
We have chosen the H$\alpha$ line for the accretion luminosity determination, as these lines are strongest lines present in the spectra. To arrive at a measurement of the total line emission, we need to take account of the fact that the underlying line absorption is filled in with emission. Therefore, the intrinsic absorption equivalent width $EW_{\rm int}$ needs to be subtracted from the observed equivalent width $EW_{\rm obs}$. The intrinsic EW is measured from the synthetic spectra corresponding to the effective temperature and surface gravity determined earlier.
The line luminosity $L_{\rm line}$ was calculated using the unreddened stellar flux at the wavelength of H$\alpha$ and distance to the star. The relationship between accretion luminosity and line luminosity goes as (cf. e.g. @Mendigutia2011b).
$$\label{eq:Lacc_Lline}
\log\left( \frac{L_{\rm
acc}}{\rm L_{\sun}} \right)=A+B \times \log\left( \frac{L_{\rm line}}{\rm L_{\sun}} \right),$$
where $A$ and $B$ are constants corresponding to the intercept and the gradient of the relation between $\log(L_{\rm acc}/\rm L_{\sun})$ and $\log(L_{\rm line}/\rm L_{\sun})$ respectively. This relationship has, most recently, been determined for 32 accretion diagnostic emission lines by F17. For the case of $\rm H\alpha$, the constants are $A=2.09\pm0.06$ and $B=1.00\pm0.05$. The mass accretion rate is then determined from the accretion luminosity, stellar radius and stellar mass by Equation \[eq:Macc\].
$$\label{eq:Macc}
\dot M_{\rm acc}=\frac{L_{\rm acc}R_*}{GM_*}$$
Table \[tab:EW\_Mdot\] summarises the EW measurements for the $\rm
H\alpha$ line and mass accretion rates in all HAeBes.
The final set of 78 mass accretion rates (21+57) based on the INT and X-Shooter data is the largest, and arguably the best such collection to date as they were determined in a homogeneous and consistent manner. To arrive at the largest sample possible however, we expand the sample using data provided in the comprehensive study by V18. Adding 85 objects from V18 results in a large sample of 163 Herbig Ae/Be stars with mass accretion rates[^5]
Analysis
========
We have obtained new spectroscopic data of 30 northern Herbig Ae/Be stars which are used to determine their stellar parameters, extinction and H$\alpha$ emission line fluxes. Combined with [*Gaia*]{} DR2 parallaxes, this led to the determination of the mass accretion rates for 21 of these objects. This dataset complements the large southern sample of F15, for which we re-determined stellar parameters using the [*Gaia*]{} parallaxes and extinctions in the same manner as for the northern sample. This full sample contains 78 objects with homogeneously determined values. To this we add 85 objects from the V18 study for which literature values have been adapted and new H$\alpha$ line measurements have been added using data from archives. This led to a total of 163 objects for which we have stellar parameters, distances, H$\alpha$ emission line fluxes and mass accretion rates derived from these using the MA paradigm available. In the following we will investigate the dependence of the MA-derived accretion rate on stellar mass, and find that there is a break in properties around 4M$_{\odot}$.
Accretion luminosity as a function of stellar luminosity
--------------------------------------------------------
Before addressing the mass accretion rate, let us first discuss the accretion luminosity, as that is less dependent on the stellar parameters. In Figure \[fig:LLacc\_wfv\] we show the accretion luminosity versus the stellar luminosity for the entire sample. The accretion luminosity increases monotonically with stellar luminosity, which confirms earlier reports (F15, @Mendigutia2011b). However, the sample under consideration is much larger than previously.
It would appear that the slope decreases, while the scatter in the relationship increases, with mass. To investigate whether there is a significant difference in accretion luminosities between high- and low mass objects, and if so, to determine the mass where there is a turn-over, we split the data into a low mass and a high mass sample, with a varying turnover mass. We then fitted a straight line to the data from the lowest mass to this intermediate mass, and a straight line from the intermediate mass to the largest mass. The intermediate mass was varied from the second smallest to the second largest mass.
This resulted in values of the slopes and their statistical uncertainty for a range of masses. When taking the difference in slopes and expressing this in terms of the respective uncertainties in the fit, we arrive at a statistical assessment whether the low- and high mass samples have a different slope. This approach takes into account the issue that the absolute value of the difference-in-slopes may not be a reliable indicator of the turn-over point. This is because each slope can vary depending on both the number and spread of data points used. For example, at both the high and low mass ends, the slopes will have a large uncertainty simply because of small number statistics.
We quantify the difference in slopes by combining the uncertainties on both low and high mass gradients, $\sigma$, and compare this to the difference in slopes, $\Delta$(slope). This is shown in the top panel of Figure \[fig:LLacc\_wfv\]. The $\Delta$(slope) reaches its maximum significance of 4$\sigma$ for a luminosity of 194 L$_{\odot}$, which was determined by a triple-Gaussian fit to the curve. For the lower mass HAeBes, the linear best fit provides the empirical calibration of $\log\left( \frac{L_{\rm acc}}{\rm L_{\sun}} \right)=(-0.87 \pm 0.11)+(1.03 \pm 0.08) \times \log\left( \frac{L_{*}}{\rm L_{\sun}} \right)$. This is in agreement with the best fit for low-mass stars in the work of [@Mendigutia2011b] ($L_{\rm acc}
\propto L^{1.2}_*$). For the higher mass HAeBes, the best fit provides the expression of $\log\left( \frac{L_{\rm acc}}{\rm L_{\sun}} \right)=(0.19 \pm 0.27)+(0.60 \pm 0.08) \times \log\left( \frac{L_{*}}{\rm L_{\sun}} \right)$.
Mass accretion rate as a function of stellar mass
-------------------------------------------------
One of the main questions regarding the formation of massive stars is at which mass the mass accretion mechanism changes from magnetically controlled accretion to another mechanism, which could be direct accretion from the disk onto the star. Above we saw that there is a difference in the accretion luminosity to stellar luminosity relationships for different masses. So, we now move to look at the mass accretion rates and investigate various subsamples individually.
------------------- --------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
Name $\rm H\alpha$ $EW_{\rm obs}$ $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] profile (Å) (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
V594 Cas IV B $-81.79\pm1.23$ $8.21\pm0.19$ $-90.00\pm1.24$ $1.11\times10^{-15}$ $(9.96\pm0.14)\times10^{-14}$ $0.00^{+0.03}_{-0.03}$ $2.09^{+0.09}_{-0.09}$ $-5.36^{+0.09}_{-0.10}$
\[3pt\] PDS 144S I $-14.55\pm0.53$ $12.82\pm0.14$ $-27.37\pm0.55$ $4.70\times10^{-17}$ $(1.29\pm0.03)\times10^{-15}$ - - -
\[3pt\] HD 141569 II R $6.38\pm0.01$ $14.80\pm0.26$ $-8.42\pm0.26$ $4.17\times10^{-15}$ $(3.51\pm0.11)\times10^{-14}$ $-1.87^{+0.02}_{-0.02}$ $0.22^{+0.17}_{-0.18}$ $-7.35^{+0.18}_{-0.15}$
\[3pt\] HD 142666 IV R $2.46\pm0.08$ $11.01\pm0.17$ $-8.54\pm0.19$ $1.84\times10^{-15}$ $(1.57\pm0.04)\times10^{-14}$ $-1.97^{+0.02}_{-0.02}$ $0.12^{+0.18}_{-0.18}$ $-7.24^{+0.17}_{-0.17}$
\[3pt\] HD 145718 I $10.83\pm0.42$ $14.18\pm0.25$ $-3.35\pm0.49$ $1.50\times10^{-15}$ $(5.03\pm0.73)\times10^{-15}$ $-2.44^{+0.08}_{-0.09}$ $-0.35^{+0.26}_{-0.27}$ $-7.79^{+0.26}_{-0.29}$
\[3pt\] HD 150193 III B $-0.29\pm0.05$ $14.86\pm0.18$ $-15.15\pm0.19$ $3.85\times10^{-15}$ $(5.83\pm0.07)\times10^{-14}$ $-1.38^{+0.02}_{-0.02}$ $0.71^{+0.15}_{-0.15}$ $-6.75^{+0.15}_{-0.18}$
\[3pt\] PDS 469 I $4.65\pm0.17$ $13.50\pm0.17$ $-8.85\pm0.24$ $1.02\times10^{-16}$ $(9.04\pm0.24)\times10^{-16}$ - - -
\[3pt\] HD 163296 I $-11.04\pm0.87$ $15.48\pm0.13$ $-26.51\pm0.88$ $5.11\times10^{-15}$ $(1.35\pm0.05)\times10^{-13}$ $-1.36^{+0.03}_{-0.03}$ $0.73^{+0.16}_{-0.16}$ $-6.79^{+0.15}_{-0.16}$
\[3pt\] MWC 297 I $-455.84\pm1.41$ $3.50\pm0.02$ $-459.34\pm1.41$ $4.81\times10^{-14}$ $(2.21\pm0.01)\times10^{-11}$ $1.99^{+0.05}_{-0.04}$ $4.08^{+0.21}_{-0.20}$ $-3.61^{+0.19}_{-0.20}$
\[3pt\] VV Ser II R $-39.07\pm0.10$ $8.04\pm0.07$ $-47.11\pm0.12$ $1.31\times10^{-15}$ $(6.15\pm0.02)\times10^{-14}$ - - -
\[3pt\] MWC 300 III B $-140.04\pm0.40$ $1.95\pm0.06$ $-141.99\pm0.40$ $1.35\times10^{-15}$ $(1.91\pm0.01)\times10^{-13}$ $1.07^{+0.14}_{-0.11}$ $3.16^{+0.26}_{-0.21}$ $-4.56^{+0.25}_{-0.24}$
\[3pt\] AS 310 I $0.83\pm0.04$ $4.13\pm0.03$ $-3.29\pm0.05$ $6.60\times10^{-16}$ $(2.17\pm0.03)\times10^{-15}$ $-0.52^{+0.14}_{-0.11}$ $1.57^{+0.22}_{-0.20}$ $-6.23^{+0.21}_{-0.22}$
\[3pt\] PDS 543 I $0.16\pm0.01$ $2.66\pm0.02$ $-2.50\pm0.02$ $1.42\times10^{-14}$ $(3.55\pm0.02)\times10^{-14}$ $0.34^{+0.14}_{-0.11}$ $2.43^{+0.22}_{-0.18}$ $-5.33^{+0.15}_{-0.13}$
\[3pt\] HD 179218 I $-1.70\pm0.05$ $12.43\pm0.13$ $-14.14\pm0.14$ $3.12\times10^{-15}$ $(4.41\pm0.05)\times10^{-14}$ $-1.01^{+0.02}_{-0.02}$ $1.08^{+0.13}_{-0.13}$ $-6.31^{+0.12}_{-0.12}$
\[3pt\] HD 190073 IV B $-24.79\pm0.50$ $10.60\pm0.19$ $-35.39\pm0.53$ $1.99\times10^{-15}$ $(7.05\pm0.11)\times10^{-14}$ $0.22^{+0.10}_{-0.08}$ $2.31^{+0.18}_{-0.15}$ $-4.97^{+0.18}_{-0.14}$
\[3pt\] V1685 Cyg II B $-107.70\pm0.67$ $3.63\pm0.03$ $-111.33\pm0.67$ $2.63\times10^{-15}$ $(2.93\pm0.02)\times10^{-13}$ $0.88^{+0.05}_{-0.04}$ $2.97^{+0.15}_{-0.14}$ $-4.77^{+0.09}_{-0.15}$
\[3pt\] LkHA 134 I $-68.47\pm0.36$ $9.64\pm0.08$ $-78.11\pm0.37$ $6.45\times10^{-16}$ $(5.04\pm0.02)\times10^{-14}$ $0.05^{+0.04}_{-0.03}$ $2.14^{+0.10}_{-0.10}$ $-5.28^{+0.11}_{-0.10}$
\[3pt\] HD 200775 II R $-66.71\pm0.23$ $5.21\pm0.06$ $-71.92\pm0.24$ $1.36\times10^{-14}$ $(9.76\pm0.04)\times10^{-13}$ - - -
\[3pt\] LkHA 324 II R $-2.54\pm0.03$ $7.74\pm0.13$ $-10.28\pm0.13$ $7.60\times10^{-16}$ $(7.81\pm0.10)\times10^{-15}$ $-1.05^{+0.03}_{-0.03}$ $1.04^{+0.14}_{-0.14}$ $-6.48^{+0.13}_{-0.15}$
\[3pt\] HD 203024 I $8.88\pm0.04$ $14.42\pm0.20$ $-5.54\pm0.20$ $8.72\times10^{-16}$ $(4.83\pm0.18)\times10^{-15}$ - - -
\[3pt\] V645 Cyg I $-99.65\pm1.32$ $2.15\pm0.05$ $-101.80\pm1.33$ $4.39\times10^{-16}$ $(4.47\pm0.06)\times10^{-14}$ - - -
\[3pt\] V361 Cep I $-30.74\pm0.24$ $5.11\pm0.06$ $-35.85\pm0.25$ $1.05\times10^{-15}$ $(3.75\pm0.03)\times10^{-14}$ $-0.03^{+0.04}_{-0.03}$ $2.06^{+0.10}_{-0.10}$ $-5.54^{+0.10}_{-0.10}$
\[3pt\] V373 Cep III B $-55.39\pm0.39$ $7.40\pm0.15$ $-62.80\pm0.42$ $5.10\times10^{-16}$ $(3.21\pm0.02)\times10^{-14}$ - - -
\[3pt\] V1578 Cyg III B $-19.29\pm0.06$ $10.30\pm0.09$ $-29.59\pm0.11$ $7.77\times10^{-16}$ $(2.30\pm0.01)\times10^{-14}$ $-0.37^{+0.04}_{-0.03}$ $1.72^{+0.11}_{-0.11}$ $-5.67^{+0.10}_{-0.10}$
\[3pt\] LkHA 257 I $-5.40\pm0.00$ $12.92\pm0.25$ $-18.32\pm0.25$ $8.56\times10^{-17}$ $(1.57\pm0.02)\times10^{-15}$ $-1.51^{+0.03}_{-0.02}$ $0.58^{+0.16}_{-0.16}$ $-6.95^{+0.17}_{-0.18}$
\[3pt\] SV Cep III R $2.18\pm0.16$ $14.89\pm0.45$ $-12.71\pm0.48$ $2.08\times10^{-16}$ $(2.65\pm0.10)\times10^{-15}$ $-2.01^{+0.03}_{-0.03}$ $0.08^{+0.19}_{-0.19}$ $-7.46^{+0.19}_{-0.17}$
\[3pt\] V375 Lac IV B $-18.88\pm0.21$ $12.37\pm0.02$ $-31.24\pm0.21$ $9.02\times10^{-17}$ $(2.82\pm0.02)\times10^{-15}$ - - -
\[3pt\] HD 216629 II B $-17.02\pm0.09$ $3.68\pm0.04$ $-20.70\pm0.10$ $5.84\times10^{-15}$ $(1.21\pm0.01)\times10^{-13}$ $0.39^{+0.03}_{-0.03}$ $2.48^{+0.12}_{-0.11}$ $-5.17^{+0.10}_{-0.09}$
\[3pt\] V374 Cep II B $-32.95\pm0.31$ $4.62\pm0.05$ $-37.56\pm0.31$ $3.09\times10^{-15}$ $(1.16\pm0.01)\times10^{-13}$ $0.44^{+0.04}_{-0.04}$ $2.53^{+0.13}_{-0.12}$ $-4.95^{+0.11}_{-0.10}$
\[3pt\] V628 Cas IV B $-124.89\pm2.44$ $2.10\pm0.01$ $-126.99\pm2.44$ $6.52\times10^{-15}$ $(8.28\pm0.16)\times10^{-13}$ - - -
\[3pt\]
------------------- --------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
: The equivalent width measurements and accretion rates. Column 2 provides the $\rm H\alpha$ emission line profile classification scheme according to [@Reipurth1996]: single-peaked (I); double-peaked and the secondary peak rises above half strength of the primary peak (II); double-peaked and the secondary peak rises below half strength of the primary peak (III), when the secondary peak is located blueward or redward of the primary peak, class II and III are label with B or R respectively; regular P-Cygni profile (IV B); and inverse P-Cygni profile (IV R). The classifications are based on the emission lines in Figure \[fig:B1\_EW\] corrected for absorption. Columns 3–10 present observed equivalent width, intrinsic equivalent width, corrected equivalent width, continuum flux density at central wavelength of the $\rm
H\alpha$ profile, line flux, line luminosity, accretion luminosity and mass accretion rate respectively.[]{data-label="tab:EW_Mdot"}
{width="17.5cm"}
![The logarithmic accretion luminosities versus stellar luminosities for the full, 163 stars, sample. Also shown are linear fits to the full sample and to the low- and high-mass stars respectively. As can be seen in the top panel, the difference in slopes between the low- and high-mass objects is most significant at at $L_*>194\,\rm L_{\sun}$ (see text for details). The gradients of best fits for the whole sample, low-mass and high-mass HAeBes are $0.85\pm0.03$, $1.03\pm0.08$ and $0.60\pm0.08$ respectively. []{data-label="fig:LLacc_wfv"}](LLacc_20190929_wfv.png){width="8cm"}
Let us first consider the sample with homogeneously derived mass accretion rates, before investigating the full sample. The relationship between mass accretion rate and stellar mass of the combined sample from this work and [@Fairlamb2015; @Fairlamb2017] that are present in @The1994 [table 1], is shown in the left hand panel of Figure \[fig:MMdot\_wft\_wf\_wfv\]. The sample of [@The1994], contains the best established HAeBes, and is thus least contaminated by possible misclassified objects. It can be seen that the mass accretion rate increases with stellar mass, and that there is a different behaviour for objects with masses below and above the mass range with $\log(M_*)=0.4$–0.6. This break is consistent with the major finding in F15 that the relationship between mass accretion rate and stellar mass shows a break around the boundary between Herbig Ae and Herbig Be stars. In particular, they found that the relationship between mass accretion rate and mass has a different slope for the lower and higher mass objects respectively. With our improved sample in hand, we can now revisit this finding and we determine the mass at which the break occurs with higher precision in the same way as we found the turnover luminosity earlier.
We found that the maximum slope difference is at the $4.4\,\sigma$ level for the Thé et al. sample at $\log(M_*)=0.56^{+0.14}_{-0.14}$ or $M_*=3.61^{+1.38}_{-0.98}\,\rm M_{\sun}$. The uncertainty in the mass is decided where the difference in slope is 1$\sigma$ smaller than at maximum. If we focus on the total combined sample of 78 objects from this work and F15, the break is established at $\log(M_*)=0.58^{+0.14}_{-0.14}$ or $M_*=3.81^{+1.46}_{-1.05}\,\rm
M_{\sun}$ with the maximum gradient difference about $6.4\,\sigma$. as shown in the middle panel of Figure \[fig:MMdot\_wft\_wf\_wfv\]. It can be seen that this plot shows the same relationship as the one in the left-hand panel. When considering the full, combined sample of 163 objects from this work, F15 and V18 whose mass accretion rates are shown in the right-hand panel of Figure \[fig:MMdot\_wft\_wf\_wfv\], the break is now found at the $\log(M_*)=0.60^{+0.13}_{-0.12}$ or $M_*=3.98^{+1.37}_{-0.94}\,\rm M_{\sun}$ with the maximum difference of gradient about $6.6\,\sigma$.
We note that both the significance of the difference in slopes as well as the mass at which the break occurs increases with the number of objects. This could be due to the number of objects; especially as the [@The1994] sample is sparsely populated at the high mass end, which could increase the uncertainty on the resulting slope and thus lower the significance of the difference in slopes between high and low mass objects. Alternatively, it could be affected by a larger contamination of non-Herbig Be stars in the full sample. It is notoriously hard to differentiate between a regular Be star and a Herbig Be star. Given the low number statistics of the [@The1994] sample and the similarity of the break in mass of the other two samples, we will proceed with a break in slope at 4 M$_{\odot}$.
{width="5.8cm"} {width="5.8cm"} {width="5.8cm"}
### Literature comparisons with T Tauri stars
It is interesting to see how the accretion luminosities compare to those of T Tauri stars at the low mass range. With the large caveat that no [*Gaia*]{} DR2 study of T Tauri stars and their stellar parameters and accretion rates exists at the moment, we show in Figure \[fig:LLacc\_CTTs\] the $L_{\rm acc}$ against $L_*$ for the whole sample of 163 HAeBes in this work compared to classical T Tauri stars of which all luminosity values are taken from [@Hartmann1998], [@White2003], [@Calvet2004] and [@Natta2006]. On one hand, the gradient of best fit for the classical T Tauri stars is $1.17\pm0.09$ which is close, and well within the errorbars, to $1.03\pm0.08$ for low-mass HAeBes. On the other hand, high-mass HAeBes shows a flatter relationship of $L_{\rm acc} \propto
L^{0.60\pm0.08}_*$. It would seem that the Herbig Ae and Be stars with masses up until 4M$_{\odot}$ behave similarly to the T Tauri stars. Again, stellar parameters of these classical T Tauri stars were not derived in the same manner as HAeBes in this work. This may have an effect on our conclusion.
![Accretion luminosity versus stellar luminosity for 163 HAeBes in this work and classical T Tauri stars from [@Hartmann1998], [@White2003], [@Calvet2004] and [@Natta2006]. Red, green and blue dashed lines are the best fit for T Tauri stars ($L_{\rm
acc}\propto L^{1.17\pm0.09}_*$), low-mass HAeBes ($L_{\rm
acc}\propto L^{1.03\pm0.08}_*$) and high-mass HAeBes ($L_{\rm
acc}\propto L^{0.60\pm0.08}_*$) respectively.[]{data-label="fig:LLacc_CTTs"}](LLacc_20190929_wfv_CTTs_all_NoMuzerolle2005.png){width="\columnwidth"}
Mass accretion rate as a function of stellar age
------------------------------------------------
In Figure \[fig:HR\_Mdot\] we illustrate the mass accretion rates across the HR-diagram. As expected, it can be seen that the accretion rates are largest for the brightest and most massive objects. However, the sample may allow an investigation into the evolution of the mass accretion rate as a function of stellar age as provided by the evolutionary models. In general it can be stated that, in terms of evolutionary age, the younger objects have higher accretion rates than older objects. As illustration, we show the mass accretion rate as a function of age in the top panel of Figure \[fig:Mdot\_age\]. It is clear that the accretion rate decreases with age (as also shown in F15); the Pearson correlation coefficient is -0.87.
However, when considering any properties as a function of age, there is always the issue that higher mass stars evolve faster than lower mass stars. Hence, the observed fact that higher mass objects have higher accretion rates could be the main underlying reason for a trend of decreasing accretion rate with time. Indeed, in the top panel of Figure \[fig:Mdot\_age\] the range in accretion rates covers the entire accretion range of the accretion vs. mass relation, so it is hard to disentangle from this graph whether there is also an accretion rate - age relation.
This can in principle be circumvented when selecting a sample of objects in a small mass range, so that the spread in mass accretion rates is minimized. This needs to be offset against the number of objects under consideration. In the lower panels of Figure \[fig:Mdot\_age\] we show how the accretion rate change with the age of stars in 3 different mass bins (2.0-$2.5\,\rm M_{\sun}$, 2.5-$3.0\,\rm M_{\sun}$ and 3.0-$3.5\,\rm M_{\sun}$. A best fit taken into account the uncertainties in both the mass accretion rate and age to the 30 objects where $2.0\,\rm M_{\sun}<M_*<2.5\,\rm M_{\sun}$ shows a $\dot M_{\rm acc}\propto Age^{-1.95\pm0.49}$ with a Pearson’s correlation coefficient -0.55. The 9 stars with masses $2.5\,\rm
M_{\sun}<M_*<3.0\,\rm M_{\sun}$ have a gradient of a best fit is $-1.40\pm1.47$ (correlation coefficient 0.40). The error of the best fit gradient becomes slightly larger than its own value, which is due to the smaller number of objects involved. The 8 objects with $3.0\,\rm M_{\sun}<M_*<3.5\,\rm M_{\sun}$ in the bottom panel of Figure \[fig:Mdot\_age\] have a gradient of a best fit is $-0.37\pm1.25$ with a linear correlation -0.36. It can be seen that the error of the best fit gets larger than 3 times of its own absolute value. This is more likely to be due to the lack of HAeBes when the stellar mass increases.
The 2.0-2.5M$_{\odot}$ sample is best suited to study any trend in accretion rate with age in the sense that the accretion rates change by 0.4dex across this mass range (cf. the slope of 4.05 found for the low mass objects in Sec. 5.2). The spread in accretion rates in the graph is twice that, and if the additional decrease in mass accretion rate is due to evolution, we find $\dot M_{\rm acc}\propto
Age^{-1.95\pm0.49}$. This value is the only such determination for Herbig Ae/Be stars in a narrow mass range, other determinations were based on full samples of HAeBe stars which contain many different masses and suffer from a mass-age degeneracy (e.g. F15, [@Mendigutia2012]). Despite the relatively large errorbars, we note that our determination is remarkably close to the observed values for T Tauri stars by [@Hartmann1998], who find an $\eta$ between 1.5-2.8, while theoretically these authors predict $\eta$ to be larger than 1.5 for a single $\alpha$ viscous disk. We also refer the reader to the discussion in [@Mendigutia2012], who, remarkably, find a similar value of the exponent to ours: 1.8$^{+1.4}_{-0.7}$ as determined for their full sample.
![ Ages versus mass accretion rates. From top to bottom all sample, the mass range 2.0-$2.5\,\rm M_{\sun}$, 2.5-$3.0\,\rm M_{\sun}$ and 3.0-$3.5\,\rm M_{\sun}$. The best fit is shown in the dashed line where $\dot M_{\rm acc}\propto Age^{-1.11\pm0.05}$, $\dot M_{\rm acc}\propto Age^{-1.95\pm0.49}$, $\dot M_{\rm acc}\propto Age^{1.40\pm1.47}$ and $\dot M_{\rm acc}\propto Age^{-0.37\pm1.25}$ respectively. The colour map indicates stellar mass of each mass bin. []{data-label="fig:Mdot_age"}](Mdot_Age_20190929_wfv_pearson.png){width="\columnwidth"}
Discussion
==========
In the above we have worked out the mass accretion rates for a large sample of 163 Herbig Ae/Be stars. To this end new and spectroscopic archival data were employed. A large fraction of the sample now has homogeneously determined astrophysical parameters, while [*Gaia*]{} parallaxes were used to arrive at updated luminosities for the sample objects. We compared the mass accretion rates derived using the MA paradigm for Herbig Ae/Be stars with various properties. We find that:
- The mass accretion rate increases with stellar mass, but the sample can be split into two subsets depending on their masses.
- The low mass Herbig Ae stars’ accretion rates have a steeper dependence on mass than the higher mass Herbig Be stars.
- The above findings corroborate previous reports in the literature. The larger sample and improved data allows us to determine the mass where the largest difference in slopes between high and low mass objects occurs. We find it is 4 M$_{\odot}$.
- Bearing in mind the caveat that T Tauri stars do not yet have [*Gaia*]{} based accretion rates, it appears that the Herbig Ae stars’ accretion properties display a similar dependence on luminosity as the T Tauri stars.
- In general, younger objects have larger mass accretion rates, but it proves not trivial to disentangle a mass dependence from an age dependence. We present the first attempt to do so. A small subset in a narrow mass range leads us to suggest that the accretion rate decreases with time.
In the following we aim to put these results into context, but we start by discussing the various assumptions that had to be made to arrive at these results.
On the viability of the MA model to determine accretion rates of Herbig Ae/Be stars
-----------------------------------------------------------------------------------
In the above we have determined accretion luminosities and accretion rates to young stars. However, the question remains whether the magnetic accretion shock modelling that underpins these computations should be applicable. After all, A and B-type stars typically do not have magnetic field detections, and indeed, are not expected to harbour magnetic fields as their radiative envelopes would not create a dynamo that in turn can create a magnetic field. This was verified in a study of Intermediate Mass T Tauri stars (IMTTS) by @Villebrun2019 who found that the fraction of pre-Main Sequence objects with a magnetic field detection drops markedly as they become hotter (for similar masses). These authors suspect that any B-field detections in Herbig Ae/Be stars could be fossil fields from the evolutionary stage immediately preceding them. The fields would not only be weaker, but also more complex, hampering their direct detection.
@Muzerolle2004 showed that the MA models provided qualitative agreement with the observed emission line profiles of the Herbig Ae star UX Ori. In the process, they also found that the B-field geometry should be more complex than the usual dipole field in T Tauri stars and the expected field strength would be much below the usual T Tauri detections, in line with the @Villebrun2019 findings.
Since, based on statistical studies and targeted individual investigations evidence has emerged that Herbig Ae stars are similar to the T Tauri stars. @Garcia2006 showed that the accretion rate properties of the Herbig Ae stars constitute a natural extension to the T Tauri stars, while F15 demonstrated that the MA model can reproduce the observed UV excesses towards most of their Herbig Ae/Be stars with realistic parameters for the shocked regions. A small number of objects, all Herbig Be stars, could not be explained with the MA model. They exhibit such a large UV-excess that they would require shock covering factors larger than 100%, which is clearly unphysical.
@Vink2002 and @Vink2005 were the first to point out the remarkable similarity in the observed linear spectropolarimetric properties of the H$\alpha$ line in Herbig Ae stars and T Tauri stars. The polarimetric line effects in these types of objects can be explained with geometries consistent with light scattering off (magnetically) truncated disks. In contrast, the very different effects observed towards the Herbig Be stars were more consistent with disks reaching onto the central star - hinting at a different accretion mechanism in those. The large sample by @Ababakr2017 allowed them to identify the transition region to be around spectral type B7/8.
@Cauley2014 studied the He [i]{} 1.083 $\mu$m line profiles of a large sample of Herbig stars and found that both T Tauri and Herbig Ae stars could be explained with MA, while the Herbig Be stars could not. @Reiter2018 did not detect a difference in line morphology between the few (5) magnetic Herbig Ae/Be stars and the rest of the sample. They argued that Herbig Ae stars may therefore not accrete similarly to T Tauri stars. However, based on Poisson statistics alone, even if all magnetic objects showed either a P Cygni or Inverse P Cygni profile, a sample of 5 would not be sufficient to conclusively demonstrate that the magnetic objects are, or are not, different from the non-magnetic objects. Clearly, more work needs to be done in this area. @Costigan2014 investigated the H$\alpha$ line variability properties of Herbig Ae/Be stars and T Tauri stars and found that both the timescales and amplitude of the variability were similar for the Herbig Ae and T Tauri, which also led these authors to suggest that the mode of accretion of these objects are similar. A notion also implied by our result in Figure \[fig:LLacc\_CTTs\] that the accretion luminosities of both Herbig Ae and T Tauri stars appear to have the same dependence on the stellar luminosity. Finally, @Mendigutia2011a found that the H$\alpha$ line width variability of Herbig Be stars is considerably smaller than for Herbig Ae stars, which is in turn smaller than for TTs (see Figure 37 in @Fang2013). This may suggest smaller line emitting regions/magnetospheres as the stellar mass increases, and thus a eventual transition from MA to some other mechanism responsible for accretion.
To summarize this section, various studies of different observational properties have confirmed the many similarities between T Tauri and Herbig Ae stars, which in turn hint at a similar accretion mechanism, the magnetically controlled accretion. In turn, this would validate our use of the MA shock modelling to derive mass accretion rates for at least the lower mass end of the Herbig Ae/Be star range. Before we discuss the accretion mechanisms for low- and high mass stars, we address the use of line luminosities to arrive at accretion luminosities and rates below.
On the use of line emission as accretion rate diagnostic
--------------------------------------------------------
The only “direct” manner to determine the mass accretion rate is to measure the accretion luminosity, which is essentially the amount of gravitational potential energy that is converted into radiation at ultra-violet wavelengths. This can then be turned into a mass accretion rate once the stellar mass and radius are known. The determination of the contribution of the accretion shock to the total UV emission for HAeBe objects is not straightforward due to the fact that these stars intrinsically emit many UV photons by virtue of their higher temperatures.
@Calvet2004 showed that the Br$\gamma$ emission strengths observed towards a sample of IMTTS correlated with the accretion luminosity as determined by the UV excess for a large mass range extending to 3.7 M$_{\odot}$ (the mass of their most massive target, GW Ori, with spectral type G0 - cool enough to unambiguously determine the UV excess emission). This was already known for lower mass objects, but, significantly, these authors extended it to higher masses. In addition, these authors also showed that the mass accretion rate correlated with the stellar mass over this mass range. Therefore they also demonstrated that line emission, in this case the hydrogen recombination Br$\gamma$, can be used to measure accretion rates to masses of at least up to $\sim$4 M$_{\odot}$. @Mendigutia2011b measured the UV-excess of Herbig Ae/Be stars using UV-blue spectroscopy and photometry respectively and noted the correlation with emission line strengths (see also @Donehew2011). F15 and F17 took this further and demonstrated the strong correlation between emission line strength and accretion luminosity for a much larger sample and a mass range going up to 10–15 M$_{\odot}$. It extended to an accretion luminosity of 10$^4$ L$_{\odot}$ and was shown to hold for a large number of different emission lines.
One should keep in mind the caveat, also pointed out by F17, that high spatial resolution optical and near-infrared studies of the hydrogen line emission do not necessarily identify the line emitting regions of Herbig Ae/Be stars with the magnetospheric accretion channels. Already in 2008, @Kraus2008 reported that their milli-arcsec resolution AMBER interferometric data indicated different Br$\gamma$ line forming regions which were consistent with the MA scenario for some objects but with disk-winds for others. Further studies by e.g. @Mendigutia2015b find the line emission region consistent with a rotating disk. Similarly, @Tam2016 and @Kreplin2018 find the disk-wind a more likely explanation for the emitting region than the compact accretion channels. Even higher resolution H$\alpha$ CHARA data discussed by @Mendigutia2017 present a similar diversity.
There thus remains the question whether we can use emission lines to probe accretion if they are apparently not related to the accretion process itself, or indeed, why the line luminosities correlate with the accretion luminosity at all. It is well known that accretion onto young stars drives jets and outflows, where the mass ejection rates are of order 10% of the accretion rates (see e.g. @Purser2016). If this is also the case for Herbig Ae/Be stars, then one could expect the emission lines to be correlated with the accretion rates. On the other hand, @Mendigutia2015a pointed out that while the physical origin of the lines may not be related to the accretion process [ *per se*]{}, it is the underlying correlation between accretion luminosity and stellar luminosity that gives rise to a correlation between the line strengths and accretion. Hence, although the lines may not necessarily be directly accretion-related, the empirical correlations between emission line luminosities and accretion luminosities are strong enough to validate their use as accretion tracers.
When we revisit the various studies mentioned above, it is notable that a distinction between Herbig Ae and Herbig Be stars is found, spectroscopically [@Cauley2014], spectropolarimetrically [@Vink2005], due to spectral variability [@Costigan2014], and even when considering the accretion rates from UV-excesses (F15).
Based on the break in accretion rates, we can move this boundary to a critical mass of 4M$_{\odot}$, remarkably close, but not perfectly so, to the boundary of around B7/8 put forward by @Ababakr2017. It is thus implied that there is a transition from magnetically controlled accretion in low-mass HAeBes to another accretion mechanism in high-mass HAeBes at around $4\,\rm M_{\sun}$. The remaining question is what this mechanism should be.
If MA does not operate in massive objects, what then?
-----------------------------------------------------
The spectropolarimetric finding of a disk reaching onto the star is reminiscent of the Boundary Layer (BL) accretion mechanism that was found to be a natural consequence of a viscous circumstellar disk around a stellar object (@Lynden1974). The BL is a thin annulus close to the star in which the material reduces its (Keplerian) velocity to the slow rotation of the star when it reaches the stellar surface. It is here that kinetic energy and angular momentum will be dissipated. The BL mechanism was originally used to explain the observed UV excesses of low mass pre-Main Sequence stars (@Bertout1988) until observations led to strong support for the magnetic accretion scenario instead (@Bouvier2007). One of the difficulties the BL mechanism had was the smaller redshifted absorption linewidths predicted from the Keplerian widths expected from the BL than from freefall in the case of MA (@Bertout1988). It had been suggested in the past to operate in Herbig Ae/Be objects (eg @Blondel2006 who studied Herbig Ae stars; @Mendigutia2011b; @Cauley2014) but has never been adapted and tested for masses of Herbig Be stars and greater.
It is very much beyond the scope of this paper to address the BL theoretically, but let us suffice with a basic consideration to see whether we would be able to expect a different slope in the derived accretion rate or luminosity for an object undergoing MA or BL accretion. In both situations infalling material converts energy into radiation. In MA the energy released by a mass $dM$ falling onto a star with mass and radius $M_*$ and $R_*$ respectively will be the gravitational potential energy of the infalling material, $\frac{GM_*dM}{R_*}$, times a factor close to 1 accounting for the fact that material is not falling from infinity.
How would this amount of released energy compare to that in the BL scenario for infalling material of the same mass $dM$? For the BL case, the accretion energy will be at most the kinetic energy of the rotating material prior to being decelerated in the very thin layer close to the stellar surface. As the material rotates Keplerian, we know that the centripetal force equals the force of gravity, $\frac{dMv^2}{R_*} = \frac{GM_*dM}{R_*^2}$. We thus find that the kinetic energy $\frac{1}{2}dMv^2 = \frac{GM_*dM}{2R_*}$, which is half the gravitational potential energy for the same mass.
In other words, the energy released in the BL scenario will be [ *less*]{} than that released by MA for the same mass. Therefore if the mass accretion rate is the same, we obtain a lower accretion luminosity for objects accreting material through a Boundary Layer than through Magnetospheric Accretion. This also means that the accretion rate has to be larger in BL to arrive at the same accretion luminosity. In turn, this has as implication that if we derive the mass accretion rates using the MA paradigm, while the accretion is due to BL, then the resulting accretion luminosities and rates would have been underestimated.
Earlier, we found that the (MA derived) accretion luminosities have the same dependence of the accretion luminosity on the stellar luminosity for T Tauri stars and Herbig Ae stars, while we find a smaller gradient for the more massive stars. If we would assume that this dependence would hold for more massive stars too, then it could be concluded that the accretion rates have been underestimated for massive Herbig Be stars. If the BL scenario was the acting mechanism for the Herbig Be stars then the accretion luminosities would be larger - possibly resulting in the same relationship between accretion and stellar luminosity as the lower mass stars, and no break would be visible.
Could that be the case here? The accretion onto the stars likely depends on the rates the accretion disks are fed and mass needs to be transported through the disk, so it may be reasonable to assume that this process is less dependent on the stellar parameters and a simple correlation between accretion onto the star and stellar luminosity (or mass) would be expected. It is intriguing that the BL scenario for more massive stars might explain why we obtain lower accretion rates when we assume MA for the more massive objects resulting in a break at around $4\,\rm M_{\sun}$. An in-depth investigation into the BL is certainly warranted.
Final Remarks
=============
In this paper we presented an analysis of a new set of optical spectroscopy of 30 northern Herbig Ae/Be stars. This was combined with our data and analysis of southern objects in F15 resulting in a set of temperatures, gravities, extinctions and luminosities that were derived in a consistent manner. As such this constitutes the largest homogeneously analysed sample of 78 Herbig Ae/Be stars to date. To these we added 85 objects from the [*Gaia*]{} DR2 study of V18. The total sample of 163 objects allowed us to derive the accretion luminosities and mass accretion rates using the empirical power-law relationship between accretion luminosity and line luminosity as derived under the MA paradigm.
We identified a subset in the total sample as being the strongest Herbig Ae/Be star candidates known. The set contains 60 per cent of the objects in Table 1 from the [@The1994] catalogue. All trends found in the large sample are also present in this subsample. This implies that the large sample likely has a low contamination and is therefore a good representation of the Herbig Ae/Be class.
- We find that the mass accretion rate increases with stellar mass, and that the lower mass Herbig Ae stars’ accretion rates have a steeper dependence on mass than the higher mass Herbig Be stars. This confirms previous findings, but the large sample allows us to determine the mass where this break occurs. This is found to be $4\,\rm M_{\sun}$.
- A comparison of accretion luminosities of the Herbig Ae/Be stars with those of T Tauri stars from the literature indicates that the Herbig Ae stars’ accretion rates display a similar dependence on luminosity as the T Tauri stars. This provides further evidence that Herbig Ae stars may accrete in a similar fashion as the T Tauri stars. We do caution however that T Tauri stars do not yet have [*Gaia*]{}-based accretion rates.
- We also find that in general, younger objects have larger mass accretion rates. However, it is not trivial to disentangle a mass and age dependence from each other: More massive stars have larger accretion rates, but have much smaller ages as well. A small subset selected in a narrow mass range leads us to suggest that the accretion rate does indeed decrease with time. The best value could be determined for the mass range 2.0-2.5M$_{\odot}$. We find $\dot M_{\rm
acc}\propto Age^{-1.95\pm0.49}$.
Finally, we discussed the similarities and differences between the accretion properties of lower mass and higher mass Herbig pre-Main Sequence stars. In particular we discuss the various lines of evidence that suggest they accrete in different fashions. In addition, from linear spectropolarimetric studies, the Herbig Be stars are found to have disks reaching onto the stellar surface while the Herbig Ae stars (with the break around 4M$_{\sun}$) have disks with inner holes, similar to the T Tauri stars.
We therefore put forward the Boundary Layer mechanism as a viable manner for the accretion onto the stellar surface of massive pre-Main Sequence stars. More work needs to be done, but an initial, crude, estimate of the accretion luminosity dependency on mass – assuming a global correlation between mass accretion rate and stellar mass for all masses – can explain the observed break in accretion properties.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank to the anonymous reviewer who provided constructive comments which helped improve the manuscript. CW thanks Thammasat University for financial support in the form of a PhD scholarship at the University of Leeds. IM acknowledges the funds from a “Talento” Fellowship (2016-T1/TIC-1890, Government of Comunidad Autónoma de Madrid, Spain). M. Vioque was funded through the STARRY project which received funding from the European Union’s Horizon 2020 research and innovation programme under MSCA ITN-EID grant agreement No 676036. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} (<https://www.cosmos.esa.int/gaia>), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement. This research has made use of [IRAF]{} which is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. This publication has made use of SIMBAD data base, operated at CDS, Strasbourg, France. It has also made use of NASA’s Astrophysics Data System and the services of the ESO Science Archive Facility. Based on observations collected at the European Southern Observatory under ESO programmes 60.A-9022(C), 073.D-0609(A), 075.D-0177(A), 076.B-0055(A), 082.A-9011(A), 082.C-0831(A), 082.D-0061(A), 083.A-9013(A), 084.A-9016(A) and 085.A-9027(B). This research was made possible through the use of the AAVSO Photometric All-Sky Survey (APASS), funded by the Robert Martin Ayers Sciences Fund and NSF AST-1412587.
Supporting Information
======================
Supplementary appendices are available for download alongside this article on the MNRAS website (https://academic.oup.com/mnras).
\[lastpage\]
------------------- ------------ ------------- ------------- -------- -------- -------- -------- -------- -------- ------------- ------
Name RA DEC Obs Date
\[3pt\] (J2000) (J2000) (June 2013) R1200B R1200Y R1200R R1200B R1200Y R1200R Ref.
Feige 98 14:38:15.8 +27:29:33.0 23 - 300 - - 94 - sdB9III:He1 1
\[3pt\] HD 130109 14:46:14.9 +01:53:34.4 21; 23 60; 20 20 - 179 420 - A0 IIInn 2
\[3pt\] BD+262606 14:49:02.4 +25:42:09.2 23 - 150 - - 124 - A5 3
\[3pt\] HD 147394 16:19:44.4 +46:18:48.1 21; 23 20 20; 10 - 319 376 - B5 IV 4
\[3pt\] HD 152614 16:54:00.5 +10:09:55.3 20; 22; 24 30 15 30; 15 257 339 179 B8 V 5
\[3pt\] HD 153653 17:00:29.4 +06:35:01.3 20; 22; 24 60 40 15; 30 36 247 69 A7 V 6
\[3pt\] HD 157741 17:24:33.8 +15:36:21.8 22; 23; 24 - 40; 30 40 - 534 157 B9 V 6
\[3pt\] HD 158485 17:26:04.8 +58:39:06.8 21; 23 60 60 - 52 439 - A4 V 6
\[3pt\] HD 160762 17:39:27.9 +46:00:22.8 21; 23 20 10; 5 - 247 302 - B3 IV 7
\[3pt\] HD 161056 17:43:47.0 -07:04:46.6 23 - 30 - - 464 - B3 Vn 8
\[3pt\] HD 175426 18:53:43.6 +36:58:18.2 20; 22; 24 60 40 40 245 390 204 B2 V 9
\[3pt\] HD 177724 19:05:24.6 +13:51:48.5 21; 23; 25 20; 15 10 10 184 224 395 A0 IV-Vnn 10
\[3pt\] HD 186307 19:41:57.6 +40:15:14.9 25 - - 60 - - 125 A6 V 6
\[3pt\] HD 196504 20:37:04.7 +26:27:43.0 25 - - 40 - - 235 B8 V 11
\[3pt\] HD 199081 20:53:14.8 +44:23:14.1 21; 23; 25 20 20 20; 60 234 492 118 B5 V 7
\[3pt\]
------------------- ------------ ------------- ------------- -------- -------- -------- -------- -------- -------- ------------- ------
: Log of observations of standard stars. Column 1 gives the object name. Columns 2 and 3 are right ascension (RA) in the units of time ( ) and declination (DEC) in the units of angle ( ) respectively, column 4 lists the observation dates. Columns 5–7 present the exposure times for each grating. The signal-to-noise ratios for each grating are given in columns 8–10. Spectral type is listed along with references in columns 11–12.[]{data-label="tab:A1_standard_stars"}
\
References: (1) @Drilling2013; (2) @Abt1995; (3) @Fulbright2000; (4) @Morgan1973; (5) @Cowley1972; (6) @Cowley1969; (7) @Lesh1968; (8) @deVaucouleurs1957; (9) @Hill1977; (10) @Gray2003; (11) @Hube1970.
{width="17"}
{width="17"} \[fig:B1\_EW2\]
------------------- ------------ ------------- ------------------------- ------------------------ ------------------------------- ----------------------- ------------------------- ------------------------ ------------------------ ------------------------
Name RA DEC $T_{\rm eff}$ $\log(g)$ $A_V$ D $R_*$ $\log(L_*)$ $M_*$ Age
\[3pt\] (J2000) (J2000) (K) \[cms$^{-2}$\] (mag) (pc) ($\rm R_{\sun}$) \[$\rm L_{\sun}$\] ($\rm M_{\sun}$) (Myr)
UX Ori 05:04:29.9 -03:47:14.2 $8500^{+250}_{-250}$ $3.90^{+0.25}_{-0.25}$ $0.99^{+0.04}_{-0.03}$ $324.9^{+9.3}_{-8.4}$ $1.80^{+0.09}_{-0.08}$ $1.18^{+0.09}_{-0.09}$ $1.82^{+0.07}_{-0.08}$ $7.08^{+0.51}_{-0.47}$
\[3pt\] PDS 174 05:06:55.5 -03:21:13.3 $17000^{+2000}_{-2000}$ $4.10^{+0.40}_{-0.40}$ $3.36^{+0.29}_{-0.39}$ $398^{+10}_{-9}$ $1.12^{+0.23}_{-0.23}$ $1.98^{+0.35}_{-0.42}$ $3.18^{+0.57}_{-0.87}$ $2.75^{+3.01}_{-1.13}$
\[3pt\] V1012 Ori 05:11:36.5 -02:22:48.4 $8500^{+250}_{-250}$ $4.38^{+0.15}_{-0.15}$ $1.15^{+0.05}_{-0.06}$ -$^{d}$ - - - -
\[3pt\] HD 34282 05:16:00.4 -09:48:35.3 $9500^{+250}_{-250}$ $4.40^{+0.15}_{-0.15}$ $0.55^{+0.03}_{-0.02}$ $311.5^{+7.9}_{-7.2}$ $1.48^{+0.06}_{-0.05}$ $1.21^{+0.08}_{-0.08}$ $1.87^{+0.11}_{-0.04}$ $8.91^{+2.89}_{-2.15}$
\[3pt\] HD 287823 05:24:08.0 +02:27:46.8 $8375^{+125}_{-125}$ $4.23^{+0.11}_{-0.15}$ $0.39^{+0.03}_{-0.03}$ $359^{+12}_{-11}$ $2.06^{+0.11}_{-0.10}$ $1.27^{+0.07}_{-0.07}$ $1.80^{+0.07}_{-0.04}$ $6.61^{+0.64}_{-0.58}$
\[3pt\] HD 287841 05:24:42.8 +01:43:48.2 $7750^{+250}_{-250}$ $4.27^{+0.12}_{-0.12}$ $0.33^{+0.03}_{-0.03}$ $366^{+10}_{-9}$ $1.85^{+0.08}_{-0.07}$ $1.04^{+0.09}_{-0.09}$ $1.60^{+0.07}_{-0.02}$ $8.91^{+0.64}_{-1.15}$
\[3pt\] HD 290409 05:27:05.4 +00:25:07.6 $9750^{+500}_{-500}$ $4.25^{+0.25}_{-0.25}$ $0.41^{+0.03}_{-0.03}$ $455^{+31}_{-25}$ $1.87^{+0.16}_{-0.13}$ $1.45^{+0.16}_{-0.16}$ $2.18^{+0.01}_{-0.24}$ $4.79^{+2.97}_{-0.42}$
\[3pt\] HD 35929 05:27:42.7 -08:19:38.4 $7000^{+250}_{-250}$ $3.47^{+0.11}_{-0.11}$ $0.35^{+0.05}_{-0.04}$ $387^{+13}_{-12}$ $6.39^{+0.38}_{-0.34}$ $1.94^{+0.11}_{-0.11}$ $3.40^{+0.31}_{-0.33}$ $1.05^{+0.36}_{-0.24}$
\[3pt\] HD 290500 05:29:48.0 -00:23:43.5 $9500^{+500}_{-500}$ $3.80^{+0.40}_{-0.40}$ $1.03^{+0.06}_{-0.07}$ $^{b}$ $438^{+24}_{-20}$ $1.56^{+0.14}_{-0.12}$ $1.25^{+0.16}_{-0.17}$ $1.89^{+0.21}_{-0.11}$ $8.13^{+4.77}_{-3.00}$
\[3pt\] HD 244314 05:30:19.0 +11:20:20.0 $8500^{+250}_{-250}$ $4.15^{+0.11}_{-0.15}$ $0.51^{+0.06}_{-0.06}$ $432^{+19}_{-17}$ $2.14^{+0.17}_{-0.15}$ $1.33^{+0.12}_{-0.12}$ $1.86^{+0.16}_{-0.07}$ $6.17^{+0.91}_{-1.27}$
\[3pt\] HK Ori 05:31:28.0 +12:09:10.1 $8500^{+500}_{-500}$ $4.22^{+0.13}_{-0.13}$ $1.31^{+0.20}_{-0.25}$ -$^{e}$ - - - -
\[3pt\] HD 244604 05:31:57.2 +11:17:41.3 $9000^{+250}_{-250}$ $3.99^{+0.15}_{-0.13}$ $0.59^{+0.04}_{-0.05}$ $421^{+19}_{-17}$ $2.77^{+0.19}_{-0.18}$ $1.65^{+0.11}_{-0.11}$ $2.26^{+0.22}_{-0.16}$ $3.80^{+0.77}_{-0.85}$
\[3pt\] UY Ori 05:32:00.3 -04:55:53.9 $9750^{+250}_{-250}$ $4.30^{+0.20}_{-0.20}$ $1.37^{+0.13}_{-0.14}$ -$^{d}$ - - - -
\[3pt\] HD 245185 05:35:09.6 +10:01:51.4 $10000^{+500}_{-500}$ $4.25^{+0.25}_{-0.25}$ $0.40^{+0.03}_{-0.02}$ $429^{+37}_{-29}$ $1.78^{+0.19}_{-0.14}$ $1.46^{+0.17}_{-0.16}$ $2.21^{+0.04}_{-0.24}$ $5.01^{+2.93}_{-0.84}$
\[3pt\] T Ori 05:35:50.4 -05:28:34.9 $9000^{+500}_{-500}$ $3.60^{+0.30}_{-0.30}$ $1.78^{+0.12}_{-0.13}$ $408^{+13}_{-11}$ $2.91^{+0.30}_{-0.27}$ $1.70^{+0.18}_{-0.18}$ $2.34^{+0.38}_{-0.27}$ $3.47^{+1.32}_{-1.18}$
\[3pt\] V380 Ori 05:36:25.4 -06:42:57.6 $9750^{+750}_{-750}$ $4.00^{+0.35}_{-0.35}$ $2.05^{+0.26}_{-0.35}$ -$^{d}$ - - - -
\[3pt\] HD 37258 05:36:59.2 -06:09:16.3 $9750^{+500}_{-500}$ $4.25^{+0.25}_{-0.25}$ $0.54^{+0.03}_{-0.04}$ -$^{d}$ - - - -
\[3pt\] HD 290770 05:37:02.4 -01:37:21.3 $10500^{+250}_{-250}$ $4.20^{+0.30}_{-0.30}$ $0.34^{+0.05}_{-0.05}$ $399^{+21}_{-18}$ $2.08^{+0.17}_{-0.15}$ $1.67^{+0.11}_{-0.11}$ $2.44^{+0.05}_{-0.14}$ $3.63^{+1.05}_{-0.16}$
\[3pt\] BF Ori 05:37:13.2 -06:35:00.5 $9000^{+250}_{-250}$ $3.97^{+0.15}_{-0.13}$ $0.74^{+0.06}_{-0.07}$ $389^{+14}_{-12}$ $2.25^{+0.16}_{-0.15}$ $1.47^{+0.11}_{-0.11}$ $2.01^{+0.15}_{-0.08}$ $5.13^{+0.76}_{-0.86}$
\[3pt\] HD 37357 05:37:47.0 -06:42:30.2 $9500^{+250}_{-250}$ $4.10^{+0.10}_{-0.10}$ $0.40^{+0.03}_{-0.03}$ -$^{d}$ - - - -
\[3pt\] HD 290764 05:38:05.2 -01:15:21.6 $7875^{+375}_{-375}$ $3.90^{+0.17}_{-0.15}$ $0.58^{+0.07}_{-0.08}$ $398^{+18}_{-15}$ $2.58^{+0.22}_{-0.20}$ $1.36^{+0.15}_{-0.16}$ $1.94^{+0.24}_{-0.20}$ $5.25^{+1.83}_{-1.36}$
\[3pt\] HD 37411 05:38:14.5 -05:25:13.3 $9750^{+250}_{-250}$ $4.35^{+0.15}_{-0.15}$ $0.59^{+0.07}_{-0.07}$ -$^{e}$ - - - -
\[3pt\] V599 Ori 05:38:58.6 -07:16:45.6 $8000^{+250}_{-250}$ $3.72^{+0.13}_{-0.12}$ $4.92^{+0.25}_{-0.35}$ $410^{+12}_{-11}$ $3.34^{+0.60}_{-0.62}$ $1.61^{+0.20}_{-0.23}$ $2.34^{+0.41}_{-0.39}$ $3.24^{+2.01}_{-1.15}$
\[3pt\] V350 Ori 05:40:11.7 -09:42:11.0 $9000^{+250}_{-250}$ $4.18^{+0.11}_{-0.16}$ $1.06^{+0.08}_{-0.08}$ -$^{d}$ - - - -
\[3pt\] HD 250550 06:01:59.9 +16:30:56.7 $11000^{+500}_{-500}$ $3.80^{+0.40}_{-0.40}$ $1.55^{+0.10}_{-0.13}$ $^{b}$ $697^{+94}_{-64}$ $15.35^{+3.09}_{-2.38}$ $3.49^{+0.24}_{-0.23}$ $8.81^{+1.83}_{-1.51}$ $0.08^{+0.06}_{-0.03}$
\[3pt\] V791 Mon 06:02:14.8 -10:00:59.5 $15000^{+1500}_{-1500}$ $4.30^{+0.16}_{-0.16}$ $1.48^{+0.11}_{-0.13}$ $887^{+53}_{-44}$ $3.47^{+0.44}_{-0.39}$ $2.74^{+0.27}_{-0.29}$ $4.31^{+0.92}_{-0.40}$ $0.81^{+0.42}_{-0.36}$
\[3pt\] PDS 124 06:06:58.4 -05:55:06.7 $10250^{+250}_{-250}$ $4.30^{+0.20}_{-0.20}$ $1.79^{+0.02}_{-0.01}$ $853^{+64}_{-51}$ $2.06^{+0.18}_{-0.14}$ $1.62^{+0.11}_{-0.10}$ $2.33^{+0.02}_{-0.04}$ $3.98^{+0.59}_{-0.26}$
\[3pt\] LkHa 339 06:10:57.8 -06:14:39.6 $10500^{+250}_{-250}$ $4.20^{+0.20}_{-0.20}$ $3.11^{+0.36}_{-0.54}$ $857^{+33}_{-29}$ $2.50^{+0.67}_{-0.67}$ $1.83^{+0.25}_{-0.31}$ $2.47^{+0.46}_{-0.27}$ $3.24^{+2.13}_{-1.20}$
\[3pt\] VY Mon 06:31:06.9 +10:26:04.9 $12000^{+4000}_{-4000}$ $3.75^{+0.50}_{-0.50}$ $5.62^{+0.34}_{-0.51}$ -$^{d}$ - - - -
\[3pt\] R Mon 06:39:09.9 +08:44:09.5 $12000^{+2000}_{-2000}$ $4.00^{+0.11}_{-0.24}$ $2.47^{+0.23}_{-0.29}$ -$^{e}$ - - - -
\[3pt\] V590 Mon 06:40:44.6 +09:48:02.1 $12500^{+1000}_{-1000}$ $4.20^{+0.30}_{-0.30}$ $1.03^{+0.19}_{-0.24}$ -$^{d}$ - - - -
\[3pt\] PDS 24 06:48:41.6 -16:48:05.6 $10500^{+500}_{-500}$ $4.20^{+0.30}_{-0.30}$ $1.45^{+0.10}_{-0.12}$ $1130^{+42}_{-37}$ $1.59^{+0.15}_{-0.15}$ $1.44^{+0.16}_{-0.17}$ $2.10^{+0.32}_{-0.12}$ $6.31^{+2.60}_{-2.42}$
\[3pt\]
------------------- ------------ ------------- ------------------------- ------------------------ ------------------------------- ----------------------- ------------------------- ------------------------ ------------------------ ------------------------
: Stellar parameters of 91 Herbig Ae/Be stars in the [@Fairlamb2015] sample. Column 1 gives the object name. Columns 2 and 3 are right ascension (RA) in the units of time ( ) and declination (DEC) in the units of angle ( ) respectively. Columns 4–11 are effective temperature, surface gravity, visual extinction, distance, radius, luminosity, mass and age respectively. Temperature and surface gravity are taken from [@Fairlamb2015]. Distance is obtained from [@Vioque2018]. The rest of parameters are redetermined in this work.[]{data-label="tab:C1_Fairlamb_para"}
\[tab:C1\_Fairlamb\_para2\]
------------------------------- ------------ ------------- ------------------------- ------------------------ ------------------------------- ------------------------ ------------------------- ------------------------ --------------------------- -------------------------
Name RA DEC $T_{\rm eff}$ $\log(g)$ $A_V$ D $R_*$ $\log(L_*)$ $M_*$ Age
\[3pt\] (J2000) (J2000) (K) \[cms$^{-2}$\] (mag) (pc) ($\rm R_{\sun}$) \[$\rm L_{\sun}$\] ($\rm M_{\sun}$) (Myr)
PDS 130 06:49:58.5 -07:38:52.2 $10500^{+250}_{-250}$ $3.90^{+0.20}_{-0.20}$ $2.38^{+0.13}_{-0.15}$ $1316^{+62}_{-53}$ $2.68^{+0.34}_{-0.31}$ $1.89^{+0.14}_{-0.15}$ $2.56^{+0.28}_{-0.17}$ $2.95^{+0.68}_{-0.76}$
\[3pt\] PDS 229 06:55:40.0 -03:09:50.5 $12500^{+250}_{-250}$ $4.20^{+0.20}_{-0.20}$ $2.24^{+0.15}_{-0.17}$ -$^{d}$ - - - -
\[3pt\] GU CMa 07:01:49.5 -11:18:03.3 $22500^{+1500}_{-1500}$ $3.90^{+0.40}_{-0.40}$ $0.80^{+0.10}_{-0.12}$ -$^{e}$ - - - -
\[3pt\] HT CMa 07:02:42.5 -11:26:11.8 $10500^{+500}_{-500}$ $4.00^{+0.20}_{-0.20}$ $1.61^{+0.09}_{-0.09}$ $^{b}$ $1121^{+83}_{-66}$ $3.19^{+0.41}_{-0.33}$ $2.05^{+0.19}_{-0.18}$ $2.87^{+0.44}_{-0.36}$ $2.14^{+0.95}_{-0.69}$
\[3pt\] Z CMa 07:03:43.1 -11:33:06.2 $8500^{+500}_{-500}$ $2.53^{+0.17}_{-0.17}$ $3.09^{+0.23}_{-0.28}$ $^{b}$ -$^{d}$ - - - -
\[3pt\] HU CMa 07:04:06.7 -11:26:08.5 $13000^{+250}_{-250}$ $4.20^{+0.20}_{-0.20}$ $1.29^{+0.07}_{-0.07}$ $1170^{+100}_{-80}$ $2.72^{+0.35}_{-0.28}$ $2.28^{+0.14}_{-0.13}$ $3.26^{+0.26}_{-0.02}$ $1.70^{+0.12}_{-0.35}$
\[3pt\] HD 53367 07:04:25.5 -10:27:15.7 $29500^{+1000}_{-1000}$ $4.25^{+0.25}_{-0.25}$ $2.12^{+0.13}_{-0.14}$ -$^{d}$ - - - -
\[3pt\] PDS 241 07:08:38.7 -04:19:04.8 $26000^{+1500}_{-1500}$ $4.00^{+0.30}_{-0.30}$ $2.92^{+0.27}_{-0.37}$ $2890^{+600}_{-400}$ $6.51^{+2.63}_{-1.90}$ $4.24^{+0.39}_{-0.40}$ $12.60^{+5.09}_{-2.58}$ $0.06^{+0.06}_{-0.03}$
\[3pt\] NX Pup 07:19:28.2 -44:35:11.2 $7000^{+250}_{-250}$ $3.78^{+0.13}_{-0.13}$ $0.26^{+0.14}_{-0.17}$ -$^{d}$ - - - -
\[3pt\] PDS 27 07:19:35.9 -17:39:17.9 $17500^{+3500}_{-3500}$ $3.16^{+0.27}_{-0.27}$ $4.96^{+0.33}_{-0.49}$ $2550^{+460}_{-310}$ $13.84^{+5.77}_{-4.42}$ $4.21^{+0.62}_{-0.72}$ $12.89^{+9.62}_{-5.67}$ $0.04^{+0.15}_{-0.03}$
\[3pt\] PDS 133 07:25:04.9 -25:45:49.6 $14000^{+2000}_{-2000}$ $4.08^{+0.12}_{-0.11}$ $1.92^{+0.09}_{-0.10}$ $1475^{+92}_{-76}$ $2.09^{+-0.11}_{-0.01}$ $2.18^{+0.19}_{-0.27}$ $3.26^{+0.40}_{-0.39}$ $2.00^{+1.31}_{-0.65}$
\[3pt\] HD 59319$^{c}$ 07:28:36.7 -21:57:49.2 $12500^{+500}_{-500}$ $3.50^{+0.20}_{-0.20}$ $0.06^{+0.02}_{-0.01}$ $668^{+39}_{-33}$ $4.00^{+0.28}_{-0.22}$ $2.54^{+0.13}_{-0.12}$ $3.89^{+0.41}_{-0.36}$ $1.00^{+0.32}_{-0.26}$
\[3pt\] PDS 134 07:32:26.6 -21:55:35.7 $14000^{+500}_{-500}$ $3.40^{+0.30}_{-0.30}$ $1.70^{+0.06}_{-0.07}$ $2550^{+370}_{-260}$ $5.00^{+0.92}_{-0.69}$ $2.94^{+0.21}_{-0.19}$ $5.05^{+0.94}_{-0.67}$ $0.49^{+0.25}_{-0.20}$
\[3pt\] HD 68695 08:11:44.5 -44:05:08.7 $9250^{+250}_{-250}$ $4.40^{+0.15}_{-0.15}$ $0.37^{+0.05}_{-0.05}$ $396^{+10}_{-9}$ $1.86^{+0.10}_{-0.09}$ $1.35^{+0.09}_{-0.09}$ $1.98^{+0.11}_{-0.03}$ $5.76^{+0.70}_{-0.39}$
\[3pt\] HD 72106 08:29:34.8 -38:36:21.1 $8750^{+250}_{-250}$ $3.89^{+0.13}_{-0.12}$ $0.00^{+0.03}_{-0.00}$ -$^{d}$ - - - -
\[3pt\] TYC 8581-2002-1$^{c}$ 08:44:23.6 -59:56:57.8 $9750^{+250}_{-250}$ $4.00^{+0.10}_{-0.10}$ $1.54^{+0.05}_{-0.07}$ $558^{+12}_{-11}$ $1.98^{+0.10}_{-0.11}$ $1.50^{+0.09}_{-0.09}$ $2.14^{+0.11}_{-0.03}$ $4.79^{+0.22}_{-0.32}$
\[3pt\] PDS 33 08:48:45.6 -40:48:21.0 $9750^{+250}_{-250}$ $4.40^{+0.15}_{-0.15}$ $1.03^{+0.03}_{-0.02}$ $951^{+43}_{-37}$ $1.79^{+0.11}_{-0.09}$ $1.41^{+0.10}_{-0.09}$ $2.16^{+0.04}_{-0.19}$ $5.01^{+2.24}_{-0.33}$
\[3pt\] HD 76534 08:55:08.7 -43:27:59.8 $19000^{+500}_{-500}$ $4.10^{+0.20}_{-0.20}$ $0.95^{+0.07}_{-0.08}$ $911^{+55}_{-46}$ $6.61^{+0.69}_{-0.60}$ $3.71^{+0.13}_{-0.13}$ $8.49^{+0.90}_{-0.88}$ $0.13^{+0.05}_{-0.03}$
\[3pt\] PDS 281$^{c}$ 08:55:45.9 -44:25:14.1 $16000^{+1500}_{-1500}$ $3.50^{+0.30}_{-0.30}$ $2.32^{+0.09}_{-0.09}$ $932^{+47}_{-40}$ $10.14^{+1.03}_{-0.90}$ $3.78^{+0.24}_{-0.25}$ $9.31^{+2.03}_{-1.70}$ $0.09^{+0.07}_{-0.04}$
\[3pt\] PDS 286 09:06:00.0 -47:18:58.1 $30000^{+3000}_{-3000}$ $4.25^{+0.16}_{-0.16}$ $6.44^{+0.35}_{-0.56}$ $1820^{+210}_{-150}$ $17.72^{+6.33}_{-5.58}$ $5.36^{+0.43}_{-0.51}$ $36.13^{+24.39}_{-14.82}$ $0.01^{+0.01}_{-0.00}$
\[3pt\] PDS 297 09:42:40.3 -56:15:34.1 $10750^{+250}_{-250}$ $4.00^{+0.20}_{-0.20}$ $1.23^{+0.06}_{-0.06}$ $1590^{+140}_{-110}$ $3.44^{+0.43}_{-0.34}$ $2.15^{+0.14}_{-0.13}$ $3.06^{+0.35}_{-0.29}$ $1.82^{+0.58}_{-0.47}$
\[3pt\] HD 85567 09:50:28.5 -60:58:02.9 $13000^{+500}_{-500}$ $3.50^{+0.30}_{-0.30}$ $1.00^{+0.16}_{-0.20}$ $1023^{+53}_{-45}$ $8.60^{+1.29}_{-1.22}$ $3.28^{+0.19}_{-0.20}$ $6.79^{+1.16}_{-1.04}$ $0.19^{+0.13}_{-0.07}$
\[3pt\] HD 87403 10:02:51.4 -59:16:54.6 $10000^{+250}_{-250}$ $3.30^{+0.10}_{-0.10}$ $0.34^{+0.04}_{-0.05}$ $1910^{+280}_{-190}$ $10.49^{+1.81}_{-1.30}$ $2.99^{+0.18}_{-0.16}$ $6.22^{+0.98}_{-0.77}$ $0.21^{+0.11}_{-0.08}$
\[3pt\] PDS 37 10:10:00.3 -57:02:07.3 $17500^{+3500}_{-3500}$ $2.94^{+0.35}_{-0.35}$ $5.64^{+0.38}_{-0.60}$ $1930^{+360}_{-230}$ $11.33^{+5.23}_{-4.04}$ $4.03^{+0.65}_{-0.77}$ $11.19^{+8.37}_{-5.14}$ $0.06^{+0.27}_{--0.10}$
\[3pt\] HD 305298 10:33:04.9 -60:19:51.3 $34000^{+1000}_{-1000}$ $4.31^{+0.16}_{-0.16}$ $1.51^{+0.12}_{-0.14}$ $4040^{+630}_{-440}$ $6.19^{+1.46}_{-1.07}$ $4.66^{+0.23}_{-0.22}$ $19.41^{+3.25}_{-2.58}$ $0.03^{+0.03}_{-0.01}$
\[3pt\] HD 94509 10:53:27.2 -58:25:24.5 $11500^{+1000}_{-1000}$ $2.90^{+0.40}_{-0.40}$ $0.35^{+0.01}_{-0.02}$ $1830^{+210}_{-150}$ $9.32^{+1.14}_{-0.87}$ $3.13^{+0.25}_{-0.24}$ $6.45^{+1.40}_{-1.17}$ $0.20^{+0.18}_{-0.09}$
\[3pt\] HD 95881 11:01:57.6 -71:30:48.3 $10000^{+250}_{-250}$ $3.20^{+0.10}_{-0.10}$ $0.72^{+0.03}_{-0.04}$ $^{b}$ $1168^{+82}_{-66}$ $12.29^{+1.10}_{-0.95}$ $3.13^{+0.12}_{-0.11}$ $7.02^{+0.62}_{-0.67}$ $0.14^{+0.05}_{-0.03}$
\[3pt\] HD 96042$^{c}$ 11:03:40.5 -59:25:59.0 $25500^{+1500}_{-1500}$ $3.80^{+0.20}_{-0.20}$ $1.13^{+0.08}_{-0.08}$ $3100^{+510}_{-350}$ $15.68^{+3.38}_{-2.37}$ $4.97^{+0.27}_{-0.25}$ $24.10^{+7.71}_{-5.24}$ $0.01^{+0.01}_{-0.01}$
\[3pt\] HD 97048 11:08:03.3 -77:39:17.4 $10500^{+500}_{-500}$ $4.30^{+0.20}_{-0.20}$ $1.37^{+0.05}_{-0.06}$ $184.8^{+2.2}_{-2.1}$ $2.28^{+0.09}_{-0.10}$ $1.76^{+0.12}_{-0.12}$ $2.40^{+0.15}_{-0.03}$ $3.55^{+0.25}_{-0.60}$
\[3pt\] HD 98922 11:22:31.6 -53:22:11.4 $10500^{+250}_{-250}$ $3.60^{+0.10}_{-0.10}$ $0.45^{+0.07}_{-0.07}$ $689^{+28}_{-25}$ $12.02^{+0.99}_{-0.86}$ $3.20^{+0.11}_{-0.11}$ $7.17^{+0.67}_{-0.69}$ $0.14^{+0.05}_{-0.03}$
\[3pt\] HD 100453 11:33:05.5 -54:19:28.5 $7250^{+250}_{-250}$ $4.08^{+0.15}_{-0.13}$ $0.12^{+0.04}_{-0.04}$ $104.2^{+0.7}_{-0.69}$ $1.68^{+0.04}_{-0.05}$ $0.85^{+0.08}_{-0.09}$ $1.48^{+0.03}_{-0.02}$ $11.00^{+0.80}_{-1.00}$
\[3pt\] HD 100546 11:33:25.4 -70:11:41.2 $9750^{+500}_{-500}$ $4.34^{+0.06}_{-0.06}$ $0.17^{+0.02}_{-0.03}$ $110^{+1}_{-1}$ $1.87^{+0.04}_{-0.05}$ $1.45^{+0.11}_{-0.11}$ $2.18^{+0.02}_{-0.17}$ $4.79^{+1.82}_{-0.22}$
\[3pt\] HD 101412 11:39:44.4 -60:10:27.7 $9750^{+250}_{-250}$ $4.30^{+0.20}_{-0.20}$ $0.69^{+0.03}_{-0.03}$ $411.3^{+8.1}_{-7.6}$ $2.75^{+0.10}_{-0.10}$ $1.79^{+0.08}_{-0.08}$ $2.44^{+0.16}_{-0.13}$ $3.24^{+0.48}_{-0.55}$
\[3pt\] PDS 344 11:40:32.8 -64:32:05.7 $15250^{+500}_{-500}$ $4.30^{+0.20}_{-0.20}$ $1.27^{+0.07}_{-0.07}$ $2440^{+170}_{-140}$ $2.36^{+0.26}_{-0.22}$ $2.43^{+0.15}_{-0.14}$ $3.85^{+0.33}_{-0.20}$ $1.29^{+0.53}_{-0.34}$
\[3pt\]
------------------------------- ------------ ------------- ------------------------- ------------------------ ------------------------------- ------------------------ ------------------------- ------------------------ --------------------------- -------------------------
\[tab:C1\_Fairlamb\_para3\]
------------------------- ------------ ------------- ------------------------- ------------------------ ------------------------------- -------------------------- ------------------------- ------------------------ ------------------------- --------------------------
Name RA DEC $T_{\rm eff}$ $\log(g)$ $A_V$ D $R_*$ $\log(L_*)$ $M_*$ Age
\[3pt\] (J2000) (J2000) (K) \[cms$^{-2}$\] (mag) (pc) ($\rm R_{\sun}$) \[$\rm L_{\sun}$\] ($\rm M_{\sun}$) (Myr)
HD 104237 12:00:05.0 -78:11:34.5 $8000^{+250}_{-250}$ $3.89^{+0.12}_{-0.12}$ $0.29^{+0.05}_{-0.06}$ -$^{d}$ - - - -
\[3pt\] V1028 Cen 13:01:17.8 -48:53:18.7 $14000^{+500}_{-500}$ $3.80^{+0.30}_{-0.30}$ $0.81^{+0.11}_{-0.13}$ -$^{d}$ - - - -
\[3pt\] PDS 361 13:03:21.4 -62:13:26.2 $18500^{+1000}_{-1000}$ $3.80^{+0.30}_{-0.30}$ $2.20^{+0.13}_{-0.15}$ $2950^{+430}_{-300}$ $4.37^{+1.01}_{-0.75}$ $3.30^{+0.27}_{-0.26}$ $6.26^{+1.37}_{-0.56}$ $0.30^{+0.14}_{-0.13}$
\[3pt\] HD 114981 13:14:40.6 -38:39:05.6 $16000^{+500}_{-500}$ $3.60^{+0.20}_{-0.20}$ $0.15^{+0.02}_{-0.03}$ $705^{+57}_{-44}$ $5.91^{+0.55}_{-0.46}$ $3.31^{+0.13}_{-0.12}$ $6.46^{+0.68}_{-0.62}$ $0.25^{+0.09}_{-0.07}$
\[3pt\] PDS 364 13:20:03.5 -62:23:54.0 $12500^{+1000}_{-1000}$ $4.20^{+0.20}_{-0.20}$ $2.02^{+0.18}_{-0.22}$ -$^{d}$ - - - -
\[3pt\] PDS 69 13:57:44.1 -39:58:44.2 $15000^{+2000}_{-2000}$ $4.00^{+0.35}_{-0.35}$ $1.65^{+0.19}_{-0.23}$ $643^{+33}_{-28}$ $3.60^{+0.60}_{-0.56}$ $2.77^{+0.35}_{-0.39}$ $4.39^{+1.29}_{-0.74}$ $0.78^{+0.73}_{-0.42}$
\[3pt\] DG Cir 15:03:23.7 -63:22:58.8 $11000^{+3000}_{-3000}$ $4.41^{+0.18}_{-0.18}$ $4.00^{+0.25}_{-0.32}$ $833^{+52}_{-43}$ $1.89^{+0.41}_{-0.38}$ $1.67^{+0.59}_{-0.75}$ $2.45^{+0.84}_{-0.83}$ $3.98^{+12.22}_{-2.47}$
\[3pt\] HD 132947$^{c}$ 15:04:56.0 -63:07:52.6 $10250^{+250}_{-250}$ $3.90^{+0.10}_{-0.10}$ $0.29^{+0.03}_{-0.02}$ $382^{+15}_{-13}$ $2.33^{+0.13}_{-0.11}$ $1.73^{+0.09}_{-0.08}$ $2.35^{+0.11}_{-0.04}$ $3.72^{+0.26}_{-0.48}$
\[3pt\] HD 135344B 15:15:48.4 -37:09:16.0 $6375^{+125}_{-125}$ $3.94^{+0.12}_{-0.12}$ $0.53^{+0.18}_{-0.22}$ $135.8^{+2.4}_{-2.3}$ $2.42^{+0.28}_{-0.28}$ $0.94^{+0.13}_{-0.14}$ $1.67^{+0.18}_{-0.16}$ $6.46^{+1.86}_{-1.56}$
\[3pt\] HD 139614 15:40:46.3 -42:29:53.5 $7750^{+250}_{-250}$ $4.31^{+0.12}_{-0.12}$ $0.21^{+0.04}_{-0.04}$ $134.7^{+1.6}_{-1.6}$ $1.49^{+0.05}_{-0.05}$ $0.86^{+0.08}_{-0.09}$ $1.61^{+0.02}_{-0.11}$ $10.20^{+11.20}_{-0.65}$
\[3pt\] PDS 144S$^{a}$ 15:49:15.3 -26:00:54.7 $7750^{+250}_{-250}$ $4.13^{+0.14}_{-0.16}$ $1.02^{+0.07}_{-0.07}$ -$^{d}$ - - - -
\[3pt\] HD 141569$^{a}$ 15:49:57.7 -03:55:16.3 $9750^{+250}_{-250}$ $4.35^{+0.15}_{-0.15}$ $0.45^{+0.03}_{-0.03}$ $110.63^{+0.91}_{-0.88}$ $1.78^{+0.05}_{-0.04}$ $1.41^{+0.07}_{-0.07}$ $2.16^{+0.04}_{-0.15}$ $5.01^{+1.60}_{-0.33}$
\[3pt\] HD 141926 15:54:21.7 -55:19:44.3 $28000^{+1500}_{-1500}$ $3.75^{+0.25}_{-0.25}$ $2.61^{+0.17}_{-0.20}$ $1340^{+150}_{-110}$ $11.61^{+2.59}_{-2.05}$ $4.87^{+0.27}_{-0.26}$ $21.92^{+6.34}_{-4.51}$ $0.02^{+0.01}_{-0.01}$
\[3pt\] HD 142666$^{a}$ 15:56:40.0 -22:01:40.0 $7500^{+250}_{-250}$ $4.13^{+0.11}_{-0.16}$ $0.95^{+0.08}_{-0.08}$ $148.3^{+2}_{-1.9}$ $2.19^{+0.12}_{-0.12}$ $1.14^{+0.10}_{-0.11}$ $1.69^{+0.13}_{-0.11}$ $7.41^{+1.50}_{-1.38}$
\[3pt\] HD 142527 15:56:41.8 -42:19:23.2 $6500^{+250}_{-250}$ $3.93^{+0.08}_{-0.08}$ $1.20^{+0.04}_{-0.05}$ $^{b}$ $157.3^{+2}_{-1.9}$ $3.92^{+0.13}_{-0.15}$ $1.39^{+0.09}_{-0.10}$ $2.26^{+0.20}_{-0.13}$ $3.09^{+0.46}_{-0.63}$
\[3pt\] HD 144432 16:06:57.9 -27:43:09.7 $7500^{+250}_{-250}$ $4.05^{+0.17}_{-0.14}$ $0.48^{+0.04}_{-0.05}$ $155.4^{+2.4}_{-2.2}$ $2.32^{+0.09}_{-0.09}$ $1.18^{+0.09}_{-0.09}$ $1.74^{+0.13}_{-0.10}$ $6.76^{+1.18}_{-1.14}$
\[3pt\] HD 144668 16:08:34.2 -39:06:18.3 $8500^{+250}_{-250}$ $3.75^{+0.13}_{-0.12}$ $0.87^{+0.03}_{-0.03}$ $161.1^{+3.1}_{-2.9}$ $4.34^{+0.16}_{-0.15}$ $1.95^{+0.08}_{-0.08}$ $2.97^{+0.17}_{-0.21}$ $1.74^{+0.40}_{-0.23}$
\[3pt\] HD 145718$^{a}$ 16:13:11.5 -22:29:06.6 $8000^{+250}_{-250}$ $4.37^{+0.15}_{-0.15}$ $1.23^{+0.06}_{-0.06}$ $152.5^{+3.2}_{-3}$ $1.85^{+0.10}_{-0.09}$ $1.10^{+0.10}_{-0.10}$ $1.67^{+0.06}_{-0.02}$ $8.13^{+0.58}_{-0.88}$
\[3pt\] PDS 415 16:18:37.2 -24:05:18.1 $6250^{+250}_{-250}$ $4.47^{+0.15}_{-0.15}$ $1.54^{+0.12}_{-0.13}$ -$^{d}$ - - - -
\[3pt\] HD 150193$^{a}$ 16:40:17.9 -23:53:45.1 $9000^{+250}_{-250}$ $4.27^{+0.17}_{-0.17}$ $1.80^{+0.15}_{-0.17}$ $150.8^{+2.7}_{-2.5}$ $2.32^{+0.24}_{-0.23}$ $1.50^{+0.13}_{-0.14}$ $2.05^{+0.19}_{-0.12}$ $4.90^{+0.99}_{-1.01}$
\[3pt\] AK Sco 16:54:44.8 -36:53:18.5 $6250^{+250}_{-250}$ $4.26^{+0.10}_{-0.10}$ $0.51^{+0.03}_{-0.02}$ $140.6^{+2.1}_{-2}$ $2.21^{+0.07}_{-0.06}$ $0.82^{+0.09}_{-0.09}$ $1.57^{+0.11}_{-0.12}$ $7.25^{+1.87}_{-1.08}$
\[3pt\] PDS 431 16:54:59.1 -43:21:49.7 $10500^{+500}_{-500}$ $3.70^{+0.20}_{-0.20}$ $2.10^{+0.12}_{-0.14}$ $1810^{+160}_{-120}$ $3.20^{+0.52}_{-0.43}$ $2.05^{+0.21}_{-0.21}$ $2.87^{+0.52}_{-0.40}$ $2.14^{+1.10}_{-0.79}$
\[3pt\] KK Oph 17:10:08.1 -27:15:18.8 $8500^{+500}_{-500}$ $4.38^{+0.15}_{-0.15}$ $1.57^{+0.10}_{-0.12}$ $^{b}$ -$^{d}$ - - - -
\[3pt\] HD 163296$^{a}$ 17:56:21.2 -21:57:21.8 $9250^{+250}_{-250}$ $4.30^{+0.20}_{-0.20}$ $0.37^{+0.04}_{-0.05}$ $101.5^{+2}_{-1.9}$ $1.86^{+0.08}_{-0.08}$ $1.36^{+0.08}_{-0.09}$ $2.04^{+0.07}_{-0.10}$ $5.50^{+0.67}_{-0.13}$
\[3pt\] MWC 297$^{a}$ 18:27:39.5 -03:49:52.1 $24500^{+1500}_{-1500}$ $4.00^{+0.30}_{-0.30}$ $7.87^{+0.41}_{-0.64}$ $375^{+22}_{-18}$ $9.12^{+3.09}_{-3.00}$ $4.43^{+0.36}_{-0.46}$ $14.79^{+5.59}_{-4.61}$ $0.04^{+0.06}_{-0.02}$
\[3pt\]
------------------------- ------------ ------------- ------------------------- ------------------------ ------------------------------- -------------------------- ------------------------- ------------------------ ------------------------- --------------------------
\
[**Notes.**]{} $^{(a)}$ Stars which are also in this work’s sample. $^{(b)}$ Stars of which photometry were used for the photometry fitting different from values listed in @Fairlamb2015 [table A1] (see text for details). $^{(c)}$ Stars which have small emissions. $^{(d)}$ Stars which have low quality parallaxes in the [*Gaia*]{} DR2 Catalogue (see the text for discussion). $^{(e)}$ Stars which do not have parallaxes in the [*Gaia*]{} DR2 Catalogue.
------------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
Name $EW_{\rm obs}$ $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (Å) (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
UX Ori $2.39\pm0.34$ $14.65\pm1.47$ $-12.26\pm1.51$ $4.03\times10^{-16}$ $(4.94\pm0.61)\times10^{-15}$ $-1.79^{+0.07}_{-0.08}$ $0.30^{+0.22}_{-0.23}$ $-7.20^{+0.23}_{-0.23}$
\[3pt\] PDS 174 $-54.19\pm1.34$ $6.46\pm0.65$ $-60.65\pm1.49$ $3.59\times10^{-16}$ $(2.18\pm0.05)\times10^{-14}$ $-0.97^{+0.03}_{-0.03}$ $1.12^{+0.14}_{-0.14}$ $-6.82^{+0.15}_{-0.10}$
\[3pt\] V1012 Ori $4.74\pm0.55$ $16.36\pm1.64$ $-11.62\pm1.73$ $1.06\times10^{-16}$ $(1.23\pm0.18)\times10^{-15}$ - - -
\[3pt\] HD 34282 $4.18\pm0.40$ $16.23\pm1.62$ $-12.05\pm1.67$ $3.83\times10^{-16}$ $(4.61\pm0.64)\times10^{-15}$ $-1.86^{+0.08}_{-0.09}$ $0.23^{+0.23}_{-0.24}$ $-7.36^{+0.22}_{-0.25}$
\[3pt\] HD 287823 $10.18\pm0.50$ $15.58\pm1.56$ $-5.40\pm1.64$ $4.18\times10^{-16}$ $(2.26\pm0.69)\times10^{-15}$ $-2.04^{+0.14}_{-0.18}$ $0.05^{+0.30}_{-0.36}$ $-7.39^{+0.30}_{-0.37}$
\[3pt\] HD 287841 $8.36\pm0.43$ $13.22\pm1.32$ $-4.86\pm1.39$ $2.61\times10^{-16}$ $(1.27\pm0.36)\times10^{-15}$ $-2.28^{+0.13}_{-0.17}$ $-0.19^{+0.30}_{-0.35}$ $-7.62^{+0.30}_{-0.36}$
\[3pt\] HD 290409 $0.35\pm0.55$ $15.00\pm1.50$ $-14.65\pm1.60$ $2.97\times10^{-16}$ $(4.36\pm0.48)\times10^{-15}$ $-1.55^{+0.10}_{-0.10}$ $0.54^{+0.23}_{-0.24}$ $-7.02^{+0.27}_{-0.22}$
\[3pt\] HD 35929 $1.54\pm0.69$ $8.69\pm0.87$ $-7.15\pm1.11$ $2.00\times10^{-15}$ $(1.43\pm0.22)\times10^{-14}$ $-1.18^{+0.09}_{-0.10}$ $0.91^{+0.21}_{-0.22}$ $-6.31^{+0.19}_{-0.20}$
\[3pt\] HD 290500 $-1.46\pm0.25$ $13.14\pm1.31$ $-14.60\pm1.33$ $2.14\times10^{-16}$ $(3.12\pm0.28)\times10^{-15}$ $-1.73^{+0.08}_{-0.08}$ $0.36^{+0.23}_{-0.23}$ $-7.22^{+0.22}_{-0.24}$
\[3pt\] HD 244314 $-14.04\pm0.61$ $15.54\pm1.55$ $-29.58\pm1.67$ $3.23\times10^{-16}$ $(9.56\pm0.54)\times10^{-15}$ $-1.26^{+0.06}_{-0.06}$ $0.83^{+0.18}_{-0.19}$ $-6.60^{+0.18}_{-0.20}$
\[3pt\] HK Ori $-61.86\pm0.72$ $15.79\pm1.58$ $-77.65\pm1.74$ $2.17\times10^{-16}$ $(1.68\pm0.04)\times10^{-14}$ - - -
\[3pt\] HD 244604 $2.32\pm0.51$ $14.79\pm1.48$ $-12.47\pm1.57$ $6.52\times10^{-16}$ $(8.14\pm1.02)\times10^{-15}$ $-1.35^{+0.09}_{-0.09}$ $0.74^{+0.21}_{-0.23}$ $-6.66^{+0.20}_{-0.22}$
\[3pt\] UY Ori $4.98\pm0.26$ $15.28\pm1.53$ $-10.30\pm1.55$ $5.81\times10^{-17}$ $(5.99\pm0.90)\times10^{-16}$ - - -
\[3pt\] HD 245185 $-13.56\pm0.50$ $14.48\pm1.45$ $-28.04\pm1.53$ $3.22\times10^{-16}$ $(9.03\pm0.49)\times10^{-15}$ $-1.29^{+0.10}_{-0.09}$ $0.80^{+0.21}_{-0.21}$ $-6.78^{+0.25}_{-0.20}$
\[3pt\] T Ori $-4.15\pm0.43$ $13.07\pm1.31$ $-17.22\pm1.38$ $7.65\times10^{-16}$ $(1.32\pm0.11)\times10^{-14}$ $-1.17^{+0.06}_{-0.06}$ $0.92^{+0.18}_{-0.18}$ $-6.48^{+0.15}_{-0.17}$
\[3pt\] V380 Ori $-81.88\pm0.48$ $13.61\pm1.36$ $-95.49\pm1.44$ $9.35\times10^{-16}$ $(8.93\pm0.14)\times10^{-14}$ - - -
\[3pt\] HD 37258 $-0.33\pm0.39$ $15.00\pm1.50$ $-15.33\pm1.55$ $4.66\times10^{-16}$ $(7.14\pm0.72)\times10^{-15}$ - - -
\[3pt\] HD 290770 $-24.08\pm0.35$ $12.98\pm1.30$ $-37.06\pm1.35$ $5.55\times10^{-16}$ $(2.06\pm0.07)\times10^{-14}$ $-0.99^{+0.06}_{-0.06}$ $1.10^{+0.17}_{-0.17}$ $-6.47^{+0.19}_{-0.17}$
\[3pt\] BF Ori $-0.02\pm0.46$ $14.70\pm1.47$ $-14.72\pm1.54$ $5.03\times10^{-16}$ $(7.41\pm0.78)\times10^{-15}$ $-1.46^{+0.07}_{-0.08}$ $0.63^{+0.20}_{-0.21}$ $-6.81^{+0.20}_{-0.22}$
\[3pt\] HD 37357 $4.76\pm0.41$ $14.64\pm1.46$ $-9.88\pm1.52$ $8.69\times10^{-16}$ $(8.58\pm1.32)\times10^{-15}$ - - -
\[3pt\] HD 290764 $-2.38\pm0.43$ $13.31\pm1.33$ $-15.69\pm1.40$ $4.48\times10^{-16}$ $(7.02\pm0.63)\times10^{-15}$ $-1.46^{+0.08}_{-0.07}$ $0.63^{+0.20}_{-0.21}$ $-6.74^{+0.19}_{-0.20}$
\[3pt\] HD 37411 $-1.07\pm0.47$ $15.55\pm1.56$ $-16.62\pm1.63$ $4.29\times10^{-16}$ $(7.13\pm0.70)\times10^{-15}$ - - -
\[3pt\] V599 Ori $1.69\pm0.57$ $13.35\pm1.34$ $-11.66\pm1.46$ $7.44\times10^{-16}$ $(8.68\pm1.08)\times10^{-15}$ $-1.34^{+0.08}_{-0.08}$ $0.75^{+0.20}_{-0.21}$ $-6.59^{+0.20}_{-0.22}$
\[3pt\] V350 Ori $3.09\pm0.33$ $15.66\pm1.57$ $-12.57\pm1.60$ $2.70\times10^{-16}$ $(3.39\pm0.43)\times10^{-15}$ - - -
\[3pt\] HD 250550 $-48.83\pm0.39$ $10.03\pm1.00$ $-58.86\pm1.07$ $1.08\times10^{-14}$ $(6.35\pm0.12)\times10^{-13}$ $0.98^{+0.12}_{-0.09}$ $3.07^{+0.23}_{-0.20}$ $-4.18^{+0.23}_{-0.19}$
\[3pt\] V791 Mon $-88.08\pm0.25$ $7.94\pm0.79$ $-96.02\pm0.83$ $5.71\times10^{-16}$ $(5.49\pm0.05)\times10^{-14}$ $0.13^{+0.05}_{-0.05}$ $2.22^{+0.12}_{-0.11}$ $-5.37^{+0.09}_{-0.12}$
\[3pt\] PDS 124 $-13.83\pm1.37$ $14.13\pm1.41$ $-27.96\pm1.97$ $1.13\times10^{-16}$ $(3.16\pm0.22)\times10^{-15}$ $-1.14^{+0.09}_{-0.09}$ $0.95^{+0.20}_{-0.21}$ $-6.60^{+0.24}_{-0.23}$
\[3pt\] LkHa 339 $-5.39\pm1.37$ $12.98\pm1.30$ $-18.37\pm1.89$ $1.74\times10^{-16}$ $(3.20\pm0.33)\times10^{-15}$ $-1.14^{+0.08}_{-0.08}$ $0.95^{+0.19}_{-0.20}$ $-6.54^{+0.22}_{-0.28}$
\[3pt\] VY Mon $-17.81\pm1.76$ $8.45\pm0.84$ $-26.26\pm1.95$ $2.75\times10^{-15}$ $(7.21\pm0.54)\times10^{-14}$ - - -
\[3pt\] R Mon $-114.51\pm1.76$ $9.35\pm0.93$ $-123.86\pm1.99$ $3.69\times10^{-16}$ $(4.57\pm0.07)\times10^{-14}$ - - -
\[3pt\] V590 Mon $-60.10\pm0.52$ $9.63\pm0.96$ $-69.73\pm1.09$ $5.06\times10^{-17}$ $(3.53\pm0.06)\times10^{-15}$ - - -
\[3pt\] PDS 24 $-25.47\pm1.79$ $12.98\pm1.30$ $-38.45\pm2.21$ $4.03\times10^{-17}$ $(1.55\pm0.09)\times10^{-15}$ $-1.21^{+0.06}_{-0.05}$ $0.88^{+0.17}_{-0.18}$ $-6.74^{+0.15}_{-0.19}$
\[3pt\]
------------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
: The equivalent width measurements and accretion rates of 91 Herbig Ae/Be stars in the [@Fairlamb2015] sample. Columns 2–9 present observed equivalent width, intrinsic equivalent width, corrected equivalent width, continuum flux density at central wavelength of the $\rm H\alpha$ profile, line flux, line luminosity, accretion luminosity and mass accretion rate respectively. Equivalent widths are taken from [@Fairlamb2017]. The rest of parameters are redetermined in this work.[]{data-label="tab:C2_Fairlamb_Mdot"}
\[tab:C2\_Fairlamb\_Mdot2\]
------------------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
Name $EW_{\rm obs}$ $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (Å) (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
PDS 130 $-31.17\pm0.70$ $11.41\pm1.14$ $-42.58\pm1.34$ $8.43\times10^{-17}$ $(3.59\pm0.11)\times10^{-15}$ $-0.71^{+0.05}_{-0.05}$ $1.38^{+0.15}_{-0.15}$ $-6.10^{+0.15}_{-0.17}$
\[3pt\] PDS 229 $7.41\pm0.82$ $9.63\pm0.96$ $-2.22\pm1.26$ $9.37\times10^{-17}$ $(2.08\pm1.18)\times10^{-16}$ - - -
\[3pt\] GU CMa $-14.86\pm0.48$ $4.75\pm0.47$ $-19.61\pm0.67$ $9.95\times10^{-15}$ $(1.95\pm0.07)\times10^{-13}$ - - -
\[3pt\] HT CMa $-20.94\pm0.35$ $11.87\pm1.19$ $-32.81\pm1.24$ $1.65\times10^{-16}$ $(5.40\pm0.20)\times10^{-15}$ $-0.68^{+0.08}_{-0.07}$ $1.41^{+0.17}_{-0.17}$ $-6.03^{+0.16}_{-0.16}$
\[3pt\] Z CMa $-63.55\pm0.99$ $9.97\pm1.00$ $-73.52\pm1.41$ $1.65\times10^{-14}$ $(1.21\pm0.02)\times10^{-12}$ - - -
\[3pt\] HU CMa $-52.00\pm0.40$ $9.09\pm0.91$ $-61.09\pm0.99$ $1.62\times10^{-16}$ $(9.87\pm0.16)\times10^{-15}$ $-0.38^{+0.08}_{-0.07}$ $1.71^{+0.15}_{-0.15}$ $-5.86^{+0.17}_{-0.20}$
\[3pt\] HD 53367 $-7.62\pm0.52$ $4.02\pm0.40$ $-11.64\pm0.66$ $2.27\times10^{-14}$ $(2.64\pm0.15)\times10^{-13}$ - - -
\[3pt\] PDS 241 $-8.36\pm0.29$ $4.21\pm0.42$ $-12.57\pm0.51$ $4.57\times10^{-16}$ $(5.75\pm0.23)\times10^{-15}$ $0.17^{+0.18}_{-0.15}$ $2.26^{+0.26}_{-0.21}$ $-5.52^{+0.26}_{-0.26}$
\[3pt\] NX Pup $-37.01\pm0.45$ $8.74\pm0.87$ $-45.75\pm0.98$ $4.73\times10^{-16}$ $(2.16\pm0.05)\times10^{-14}$ - - -
\[3pt\] PDS 27 $-73.20\pm0.73$ $4.40\pm0.44$ $-77.60\pm0.85$ $1.39\times10^{-15}$ $(1.08\pm0.01)\times10^{-13}$ $1.34^{+0.15}_{-0.12}$ $3.43^{+0.28}_{-0.24}$ $-4.04^{+0.19}_{-0.15}$
\[3pt\] PDS 133 $-103.11\pm3.91$ $7.84\pm0.78$ $-110.95\pm3.99$ $6.73\times10^{-17}$ $(7.47\pm0.27)\times10^{-15}$ $-0.30^{+0.07}_{-0.06}$ $1.79^{+0.14}_{-0.14}$ $-5.89^{+0.07}_{-0.09}$
\[3pt\] HD 59319 $5.84\pm0.07$ $7.13\pm0.71$ $-1.29\pm0.71$ $1.00\times10^{-15}$ - - - -
\[3pt\] PDS 134 $-12.22\pm0.41$ $5.98\pm0.60$ $-18.20\pm0.73$ $1.30\times10^{-16}$ $(2.36\pm0.09)\times10^{-15}$ $-0.32^{+0.13}_{-0.11}$ $1.77^{+0.20}_{-0.19}$ $-5.73^{+0.20}_{-0.20}$
\[3pt\] HD 68695 $0.48\pm0.48$ $16.51\pm1.65$ $-16.03\pm1.72$ $3.51\times10^{-16}$ $(5.63\pm0.60)\times10^{-15}$ $-1.56^{+0.07}_{-0.07}$ $0.53^{+0.20}_{-0.21}$ $-6.99^{+0.20}_{-0.23}$
\[3pt\] HD 72106 $8.78\pm0.63$ $14.60\pm1.46$ $-5.82\pm1.59$ $8.58\times10^{-16}$ $(4.99\pm1.36)\times10^{-15}$ - - -
\[3pt\] TYC 8581-2002-1 $11.05\pm0.42$ $13.61\pm1.36$ $-2.56\pm1.42$ $2.22\times10^{-16}$ - - - -
\[3pt\] PDS 33 $-3.10\pm0.48$ $15.83\pm1.58$ $-18.93\pm1.65$ $6.26\times10^{-17}$ $(1.18\pm0.10)\times10^{-15}$ $-1.48^{+0.07}_{-0.07}$ $0.61^{+0.20}_{-0.21}$ $-6.96^{+0.22}_{-0.19}$
\[3pt\] HD 76534 $-11.00\pm0.34$ $5.84\pm0.58$ $-16.84\pm0.67$ $2.80\times10^{-15}$ $(4.72\pm0.19)\times10^{-14}$ $0.09^{+0.07}_{-0.06}$ $2.18^{+0.14}_{-0.12}$ $-5.43^{+0.13}_{-0.12}$
\[3pt\] PDS 281 $4.30\pm0.49$ $5.42\pm0.54$ $-1.12\pm0.73$ $4.89\times10^{-15}$ - - - -
\[3pt\] PDS 286 $-26.84\pm0.42$ $3.93\pm0.39$ $-30.77\pm0.57$ $1.12\times10^{-14}$ $(3.45\pm0.06)\times10^{-13}$ $1.55^{+0.10}_{-0.08}$ $3.64^{+0.25}_{-0.22}$ $-4.16^{+0.15}_{-0.15}$
\[3pt\] PDS 297 $7.42\pm0.41$ $11.36\pm1.14$ $-3.94\pm1.21$ $9.95\times10^{-17}$ $(3.92\pm1.21)\times10^{-16}$ $-1.51^{+0.19}_{-0.22}$ $0.58^{+0.32}_{-0.37}$ $-6.86^{+0.32}_{-0.37}$
\[3pt\] HD 85567 $-50.25\pm0.47$ $6.76\pm0.68$ $-57.01\pm0.83$ $2.11\times10^{-15}$ $(1.20\pm0.02)\times10^{-13}$ $0.59^{+0.05}_{-0.05}$ $2.68^{+0.14}_{-0.13}$ $-4.71^{+0.13}_{-0.13}$
\[3pt\] HD 87403 $6.79\pm0.63$ $9.76\pm0.98$ $-2.97\pm1.17$ $5.60\times10^{-16}$ $(1.66\pm0.65)\times10^{-15}$ $-0.72^{+0.26}_{-0.31}$ $1.37^{+0.35}_{-0.42}$ $-5.90^{+0.35}_{-0.42}$
\[3pt\] PDS 37 $-119.89\pm0.47$ $3.87\pm0.39$ $-123.76\pm0.61$ $1.63\times10^{-15}$ $(2.01\pm0.01)\times10^{-13}$ $1.37^{+0.15}_{-0.11}$ $3.46^{+0.29}_{-0.24}$ $-4.03^{+0.21}_{-0.16}$
\[3pt\] HD 305298 $0.01\pm0.41$ $3.25\pm0.32$ $-3.24\pm0.52$ $3.53\times10^{-16}$ $(1.14\pm0.18)\times10^{-15}$ $-0.24^{+0.19}_{-0.18}$ $1.85^{+0.25}_{-0.26}$ $-6.14^{+0.28}_{-0.28}$
\[3pt\] HD 94509 $-16.84\pm0.53$ $6.33\pm0.63$ $-23.17\pm0.82$ $6.27\times10^{-16}$ $(1.45\pm0.05)\times10^{-14}$ $0.18^{+0.11}_{-0.09}$ $2.27^{+0.18}_{-0.15}$ $-5.07^{+0.15}_{-0.11}$
\[3pt\] HD 95881 $-12.53\pm0.39$ $9.35\pm0.94$ $-21.88\pm1.02$ $2.06\times10^{-15}$ $(4.50\pm0.21)\times10^{-14}$ $0.28^{+0.08}_{-0.07}$ $2.37^{+0.16}_{-0.14}$ $-4.88^{+0.16}_{-0.13}$
\[3pt\] HD 96042 $3.20\pm0.80$ $3.95\pm0.39$ $-0.75\pm0.89$ $2.23\times10^{-15}$ - - - -
\[3pt\] HD 97048 $-24.43\pm0.34$ $13.54\pm1.35$ $-37.97\pm1.39$ $3.11\times10^{-15}$ $(1.18\pm0.04)\times10^{-13}$ $-0.90^{+0.03}_{-0.03}$ $1.19^{+0.13}_{-0.13}$ $-6.33^{+0.12}_{-0.15}$
\[3pt\] HD 98922 $-14.64\pm0.44$ $10.04\pm1.00$ $-24.68\pm1.09$ $6.20\times10^{-15}$ $(1.53\pm0.07)\times10^{-13}$ $0.35^{+0.05}_{-0.05}$ $2.44^{+0.13}_{-0.13}$ $-4.83^{+0.13}_{-0.12}$
\[3pt\] HD 100453 $5.27\pm0.77$ $10.23\pm1.02$ $-4.96\pm1.28$ $2.14\times10^{-15}$ $(1.06\pm0.27)\times10^{-14}$ $-2.44^{+0.11}_{-0.14}$ $-0.35^{+0.28}_{-0.32}$ $-7.79^{+0.29}_{-0.33}$
\[3pt\] HD 100546 $-23.96\pm0.42$ $15.50\pm1.55$ $-39.46\pm1.61$ $5.12\times10^{-15}$ $(2.02\pm0.08)\times10^{-13}$ $-1.12^{+0.03}_{-0.03}$ $0.97^{+0.14}_{-0.14}$ $-6.59^{+0.15}_{-0.12}$
\[3pt\] HD 101412 $-0.15\pm0.69$ $15.28\pm1.53$ $-15.43\pm1.68$ $7.90\times10^{-16}$ $(1.22\pm0.13)\times10^{-14}$ $-1.19^{+0.06}_{-0.07}$ $0.90^{+0.18}_{-0.19}$ $-6.55^{+0.17}_{-0.18}$
\[3pt\] PDS 344 $-22.23\pm0.45$ $7.80\pm0.78$ $-30.03\pm0.90$ $3.59\times10^{-17}$ $(1.08\pm0.03)\times10^{-15}$ $-0.70^{+0.07}_{-0.06}$ $1.39^{+0.16}_{-0.16}$ $-6.32^{+0.17}_{-0.18}$
\[3pt\]
------------------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
\[tab:C2\_Fairlamb\_Mdot3\]
-------------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
Name $EW_{\rm obs}$ $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (Å) (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
HD 104237 $-13.74\pm0.98$ $13.75\pm1.38$ $-27.49\pm1.69$ $7.42\times10^{-15}$ $(2.04\pm0.13)\times10^{-13}$ - - -
\[3pt\] V1028 Cen $-101.61\pm0.52$ $7.01\pm0.70$ $-108.62\pm0.87$ $2.50\times10^{-16}$ $(2.71\pm0.02)\times10^{-14}$ - - -
\[3pt\] PDS 361 $-3.97\pm0.45$ $5.35\pm0.53$ $-9.32\pm0.70$ $1.12\times10^{-16}$ $(1.05\pm0.08)\times10^{-15}$ $-0.55^{+0.15}_{-0.13}$ $1.54^{+0.23}_{-0.22}$ $-6.11^{+0.23}_{-0.26}$
\[3pt\] HD 114981 $-7.60\pm0.46$ $5.64\pm0.56$ $-13.24\pm0.72$ $2.90\times10^{-15}$ $(3.84\pm0.21)\times10^{-14}$ $-0.23^{+0.09}_{-0.08}$ $1.86^{+0.16}_{-0.16}$ $-5.67^{+0.15}_{-0.15}$
\[3pt\] PDS 364 $-78.38\pm0.74$ $9.63\pm0.96$ $-88.01\pm1.21$ $5.77\times10^{-17}$ $(5.07\pm0.07)\times10^{-15}$ - - -
\[3pt\] PDS 69 $-69.42\pm1.30$ $7.00\pm0.70$ $-76.42\pm1.48$ $1.17\times10^{-15}$ $(8.96\pm0.17)\times10^{-14}$ $0.06^{+0.05}_{-0.05}$ $2.15^{+0.12}_{-0.11}$ $-5.43^{+0.07}_{-0.10}$
\[3pt\] DG Cir $-47.98\pm0.83$ $13.10\pm1.31$ $-61.08\pm1.55$ $1.14\times10^{-16}$ $(6.98\pm0.18)\times10^{-15}$ $-0.82^{+0.06}_{-0.06}$ $1.27^{+0.16}_{-0.16}$ $-6.34^{+0.12}_{-0.08}$
\[3pt\] HD 132947 $10.38\pm0.33$ $11.96\pm1.20$ $-1.58\pm1.24$ $7.22\times10^{-16}$ - - - -
\[3pt\] HD 135344B $-4.60\pm0.75$ $5.66\pm0.57$ $-10.26\pm0.94$ $1.69\times10^{-15}$ $(1.73\pm0.16)\times10^{-14}$ $-2.00^{+0.05}_{-0.06}$ $0.09^{+0.21}_{-0.22}$ $-7.25^{+0.21}_{-0.23}$
\[3pt\] HD 139614 $0.48\pm0.61$ $13.26\pm1.33$ $-12.78\pm1.46$ $1.24\times10^{-15}$ $(1.59\pm0.18)\times10^{-14}$ $-2.05^{+0.06}_{-0.06}$ $0.04^{+0.22}_{-0.23}$ $-7.49^{+0.23}_{-0.21}$
\[3pt\] PDS 144S $-16.12\pm0.68$ $13.07\pm1.31$ $-29.19\pm1.48$ $4.68\times10^{-17}$ $(1.36\pm0.07)\times10^{-15}$ - - -
\[3pt\] HD 141569 $5.13\pm0.57$ $15.55\pm1.56$ $-10.42\pm1.66$ $4.57\times10^{-15}$ $(4.76\pm0.76)\times10^{-14}$ $-1.74^{+0.07}_{-0.08}$ $0.35^{+0.21}_{-0.23}$ $-7.23^{+0.22}_{-0.21}$
\[3pt\] HD 141926 $-43.48\pm0.43$ $3.40\pm0.34$ $-46.88\pm0.55$ $7.76\times10^{-15}$ $(3.64\pm0.04)\times10^{-13}$ $1.31^{+0.10}_{-0.08}$ $3.40^{+0.23}_{-0.20}$ $-4.37^{+0.20}_{-0.19}$
\[3pt\] HD 142666 $5.14\pm0.68$ $11.70\pm1.17$ $-6.56\pm1.35$ $2.02\times10^{-15}$ $(1.32\pm0.27)\times10^{-14}$ $-2.04^{+0.09}_{-0.11}$ $0.05^{+0.25}_{-0.28}$ $-7.33^{+0.24}_{-0.27}$
\[3pt\] HD 142527 $-6.95\pm0.98$ $6.18\pm0.62$ $-13.13\pm1.16$ $3.56\times10^{-15}$ $(4.67\pm0.41)\times10^{-14}$ $-1.44^{+0.05}_{-0.05}$ $0.65^{+0.18}_{-0.19}$ $-6.61^{+0.15}_{-0.18}$
\[3pt\] HD 144432 $-0.98\pm0.68$ $11.65\pm1.17$ $-12.63\pm1.35$ $2.06\times10^{-15}$ $(2.60\pm0.28)\times10^{-14}$ $-1.71^{+0.06}_{-0.06}$ $0.38^{+0.20}_{-0.21}$ $-6.99^{+0.18}_{-0.20}$
\[3pt\] HD 144668 $-8.41\pm0.45$ $14.12\pm1.41$ $-22.53\pm1.48$ $9.59\times10^{-15}$ $(2.16\pm0.14)\times10^{-13}$ $-0.76^{+0.04}_{-0.05}$ $1.33^{+0.14}_{-0.15}$ $-6.00^{+0.13}_{-0.13}$
\[3pt\] HD 145718 $8.34\pm0.52$ $14.53\pm1.45$ $-6.19\pm1.54$ $1.66\times10^{-15}$ $(1.02\pm0.26)\times10^{-14}$ $-2.13^{+0.11}_{-0.14}$ $-0.04^{+0.28}_{-0.32}$ $-7.49^{+0.28}_{-0.33}$
\[3pt\] PDS 415 $3.10\pm1.30$ $5.04\pm0.50$ $-1.94\pm1.39$ $1.83\times10^{-16}$ $(3.55\pm2.55)\times10^{-16}$ - - -
\[3pt\] HD 150193 $-5.59\pm0.57$ $16.07\pm1.61$ $-21.66\pm1.71$ $3.57\times10^{-15}$ $(7.74\pm0.61)\times10^{-14}$ $-1.26^{+0.05}_{-0.05}$ $0.83^{+0.17}_{-0.18}$ $-6.61^{+0.17}_{-0.20}$
\[3pt\] AK Sco $-0.82\pm0.95$ $5.06\pm0.51$ $-5.88\pm1.08$ $1.23\times10^{-15}$ $(7.22\pm1.32)\times10^{-15}$ $-2.35^{+0.09}_{-0.10}$ $-0.26^{+0.26}_{-0.28}$ $-7.61^{+0.24}_{-0.26}$
\[3pt\] PDS 431 $1.27\pm0.50$ $10.49\pm1.05$ $-9.22\pm1.16$ $6.40\times10^{-17}$ $(5.90\pm0.74)\times10^{-16}$ $-1.22^{+0.13}_{-0.12}$ $0.87^{+0.24}_{-0.25}$ $-6.58^{+0.23}_{-0.24}$
\[3pt\] KK Oph $-25.15\pm0.31$ $16.36\pm1.64$ $-41.51\pm1.67$ $1.84\times10^{-15}$ $(7.64\pm0.31)\times10^{-14}$ - - -
\[3pt\] HD 163296 $-3.63\pm0.36$ $16.02\pm1.60$ $-19.65\pm1.64$ $5.38\times10^{-15}$ $(1.06\pm0.09)\times10^{-13}$ $-1.47^{+0.05}_{-0.05}$ $0.62^{+0.18}_{-0.19}$ $-6.91^{+0.19}_{-0.19}$
\[3pt\] MWC 297 $-590.00\pm0.90$ $4.52\pm0.45$ $-594.52\pm1.01$ $4.81\times10^{-14}$ $(2.86\pm0.01)\times10^{-11}$ $2.10^{+0.05}_{-0.04}$ $4.19^{+0.22}_{-0.21}$ $-3.52^{+0.21}_{-0.22}$
\[3pt\]
-------------------- ------------------ ---------------- ------------------ ---------------------- ------------------------------- ------------------------- ------------------------- ---------------------------------
[width=1.33]{}
------------------- ------------ ------------- ----------------- ---------------- ---------------- ----------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
Name RA DEC $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (J2000) (J2000) (Å) Ref. (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
HBC 1 00:07:02.6 +65:38:38.3 $-31.00\pm3.10$ 1 $14.43\pm0.05$ $-45.43\pm3.10$ $1.15\times10^{-18}$ $(5.24\pm0.36)\times10^{-17}$ -$^{a}$ - -
\[3pt\] HBC 324 00:07:30.6 +65:39:52.6 $-16.90\pm0.85$ 2 $13.99\pm0.11$ $-30.89\pm0.85$ $2.42\times10^{-17}$ $(7.48\pm0.21)\times10^{-16}$ -$^{a}$ - -
\[3pt\] MQ Cas 00:09:37.5 +58:13:10.7 - - $15.82\pm0.12$ - $4.25\times10^{-17}$ - -$^{a}$ - -
\[3pt\] VX Cas 00:31:30.6 +61:58:50.9 $-22.10\pm0.66$ 3 $14.94\pm0.11$ $-37.04\pm0.67$ $1.16\times10^{-16}$ $(4.31\pm0.08)\times10^{-15}$ $-1.41^{+0.04}_{-0.03}$ $0.68^{+0.17}_{-0.17}$ $-6.96^{+0.22}_{-0.21}$
\[3pt\] HBC 7 00:43:25.3 +61:38:23.3 $-41.60\pm2.08$ 2 $4.06\pm0.04$ $-45.66\pm2.08$ $4.17\times10^{-16}$ $(1.91\pm0.09)\times10^{-14}$ $0.65^{+0.09}_{-0.08}$ $2.74^{+0.19}_{-0.17}$ $-4.85^{+0.39}_{-0.42}$
\[3pt\] PDS 2 01:17:43.4 -52:33:30.7 $-4.93\pm0.20$ 4 $8.37\pm0.34$ $-13.30\pm0.39$ $1.25\times10^{-16}$ $(1.66\pm0.05)\times10^{-15}$ $-2.05^{+0.03}_{-0.03}$ $0.04^{+0.19}_{-0.19}$ $-7.34^{+0.24}_{-0.21}$
\[3pt\] HD 9672 01:34:37.7 -15:40:34.8 $12.28\pm0.04$ 5 $15.97\pm0.19$ $-3.69\pm0.19$ $1.19\times10^{-14}$ $(4.41\pm0.23)\times10^{-14}$ $-2.35^{+0.03}_{-0.03}$ $-0.26^{+0.21}_{-0.21}$ $-7.80^{+0.25}_{-0.22}$
\[3pt\] HBC 334 02:16:30.1 +55:22:57 $-0.20\pm0.01$ 2 $7.65\pm0.08$ $-7.85\pm0.08$ $3.13\times10^{-17}$ $(2.45\pm0.02)\times10^{-16}$ $-1.62^{+0.09}_{-0.08}$ $0.47^{+0.23}_{-0.22}$ $-7.42^{+0.39}_{-0.44}$
\[3pt\] HD 17081 02:44:07.3 -13:51:31.3 $5.95\pm0.05$ 6 $7.19\pm0.00$ $-1.24\pm0.05$ $3.89\times10^{-14}$ $(4.83\pm0.21)\times10^{-14}$ $-1.77^{+0.08}_{-0.07}$ $0.32^{+0.22}_{-0.22}$ $-7.18^{+0.34}_{-0.32}$
\[3pt\] BD+30 549 03:29:19.7 +31:24:57.0 - - $11.64\pm0.16$ - $6.12\times10^{-16}$ - - - -
\[3pt\] PDS 4 03:39:00.5 +29:41:45.7 $-9.00\pm0.45$ 7 $14.75\pm0.13$ $-23.75\pm0.47$ $2.59\times10^{-16}$ $(6.15\pm0.12)\times10^{-15}$ $-1.52^{+0.04}_{-0.04}$ $0.57^{+0.17}_{-0.17}$ $-7.02^{+0.25}_{-0.28}$
\[3pt\] IP Per 03:40:46.9 +32:31:53.7 $-21.40\pm1.07$ 2 $15.18\pm0.32$ $-36.58\pm1.12$ $2.48\times10^{-16}$ $(9.06\pm0.28)\times10^{-15}$ $-1.57^{+0.05}_{-0.05}$ $0.52^{+0.19}_{-0.19}$ $-7.01^{+0.26}_{-0.28}$
\[3pt\] XY Per A 03:49:36.3 +38:58:55.4 $-9.80\pm0.29$ 3 $12.13\pm0.05$ $-21.93\pm0.30$ $1.04\times10^{-15}$ $(2.28\pm0.03)\times10^{-14}$ -$^{a}$ - -
\[3pt\] V892 Tau 04:18:40.6 +28:19:15.6 $-17.80\pm0.89$ 2 $9.43\pm0.14$ $-27.23\pm0.90$ $1.51\times10^{-16}$ $(4.10\pm0.14)\times10^{-15}$ -$^{a}$ - -
\[3pt\] AB Aur 04:55:45.8 +30:33:04.2 $-45.00\pm4.50$ 8 $13.66\pm0.14$ $-58.66\pm4.50$ $3.51\times10^{-15}$ $(2.06\pm0.16)\times10^{-13}$ $-0.77^{+0.05}_{-0.05}$ $1.32^{+0.14}_{-0.15}$ $-6.13^{+0.25}_{-0.27}$
\[3pt\] HD 31648 04:58:46.2 +29:50:36.9 $-19.40\pm0.58$ 3 $15.10\pm0.22$ $-34.50\pm0.62$ $2.07\times10^{-15}$ $(7.13\pm0.13)\times10^{-14}$ $-1.24^{+0.03}_{-0.03}$ $0.85^{+0.15}_{-0.15}$ $-6.57^{+0.21}_{-0.17}$
\[3pt\] HD 34700 05:19:41.4 +05:38:42.7 $-2.40\pm0.07$ 3 $5.20\pm0.25$ $-7.60\pm0.26$ $6.69\times10^{-16}$ $(5.08\pm0.18)\times10^{-15}$ $-1.70^{+0.04}_{-0.04}$ $0.39^{+0.18}_{-0.19}$ $-6.87^{+0.20}_{-0.19}$
\[3pt\] HD 290380 05:23:31.0 -01:04:23.6 $-7.00\pm0.70$ 9 $6.66\pm0.06$ $-13.66\pm0.70$ $1.98\times10^{-16}$ $(2.71\pm0.14)\times10^{-15}$ $-1.98^{+0.05}_{-0.05}$ $0.11^{+0.20}_{-0.21}$ $-7.23^{+0.24}_{-0.22}$
\[3pt\] HD 35187 05:24:01.1 +24:57:37.5 $-3.50\pm0.20$ 10 $14.14\pm0.14$ $-17.64\pm0.25$ $2.29\times10^{-15}$ $(4.03\pm0.06)\times10^{-14}$ $-1.48^{+0.03}_{-0.03}$ $0.61^{+0.16}_{-0.16}$ $-6.94^{+0.29}_{-0.31}$
\[3pt\] CO Ori 05:27:38.3 +11:25:38.9 $-21.10\pm0.63$ 3 $6.28\pm0.08$ $-27.38\pm0.64$ $7.01\times10^{-16}$ $(1.92\pm0.05)\times10^{-14}$ -$^{a}$ - -
\[3pt\] HD 36112 05:30:27.5 +25:19:57.0 $-6.30\pm0.63$ 11 $12.85\pm0.19$ $-19.15\pm0.66$ $1.36\times10^{-15}$ $(2.61\pm0.09)\times10^{-14}$ $-1.68^{+0.03}_{-0.03}$ $0.41^{+0.17}_{-0.18}$ $-7.00^{+0.24}_{-0.22}$
\[3pt\] RY Ori 05:32:09.9 -02:49:46.7 $-15.80\pm0.47$ 3 $6.19\pm0.05$ $-21.99\pm0.48$ $1.94\times10^{-16}$ $(4.26\pm0.09)\times10^{-15}$ $-1.75^{+0.03}_{-0.03}$ $0.34^{+0.18}_{-0.18}$ $-6.98^{+0.20}_{-0.22}$
\[3pt\] HD 36408 05:32:14.1 +17:03:29.2 $4.00\pm0.20$ 7 $6.80\pm0.03$ $-2.80\pm0.20$ $1.01\times10^{-14}$ $(2.82\pm0.20)\times10^{-14}$ $-0.78^{+0.10}_{-0.09}$ $1.31^{+0.19}_{-0.19}$ $-6.04^{+0.27}_{-0.28}$
\[3pt\] HD 288012 05:33:04.7 +02:28:09.7 - - $13.09\pm0.09$ - $6.26\times10^{-16}$ - - - -
\[3pt\] HBC 442 05:34:14.1 -05:36:54.1 $-1.30\pm0.07$ 2 $6.00\pm0.02$ $-7.30\pm0.07$ $2.35\times10^{-16}$ $(1.72\pm0.02)\times10^{-15}$ $-2.10^{+0.03}_{-0.02}$ $-0.01^{+0.19}_{-0.19}$ $-7.32^{+0.20}_{-0.20}$
\[3pt\] HD 36917 05:34:46.9 -05:34:14.5 $-2.50\pm0.10$ 10 $8.82\pm0.03$ $-11.32\pm0.11$ $1.94\times10^{-15}$ $(2.20\pm0.02)\times10^{-14}$ $-0.81^{+0.06}_{-0.05}$ $1.28^{+0.15}_{-0.15}$ $-6.15^{+0.28}_{-0.28}$
\[3pt\] HD 36982 05:35:09.8 -05:27:53.2 - - $5.58\pm0.02$ - $1.64\times10^{-15}$ - - - -
\[3pt\] NV Ori 05:35:31.3 -05:33:08.8 $-4.00\pm0.12$ 3 $8.45\pm0.35$ $-12.45\pm0.37$ $3.59\times10^{-16}$ $(4.47\pm0.13)\times10^{-15}$ $-1.68^{+0.05}_{-0.04}$ $0.41^{+0.19}_{-0.19}$ $-6.89^{+0.23}_{-0.23}$
\[3pt\] CQ Tau 05:35:58.4 +24:44:54.0 $-4.80\pm0.14$ 3 $8.28\pm0.34$ $-13.08\pm0.37$ $9.67\times10^{-16}$ $(1.26\pm0.04)\times10^{-14}$ -$^{a}$ - -
\[3pt\] V1787 Ori 05:38:09.3 -06:49:16.5 $-12.00\pm0.60$ 12 $15.29\pm0.27$ $-27.29\pm0.66$ $2.74\times10^{-16}$ $(7.49\pm0.18)\times10^{-15}$ $-1.45^{+0.04}_{-0.04}$ $0.64^{+0.17}_{-0.17}$ $-6.80^{+0.22}_{-0.22}$
\[3pt\] HD 37371 05:38:09.9 -00:11:01.1 $-4.30\pm0.22$ 13 $9.51\pm0.16$ $-13.81\pm0.27$ $2.99\times10^{-15}$ $(4.12\pm0.08)\times10^{-14}$ $-0.66^{+0.05}_{-0.04}$ $1.43^{+0.14}_{-0.14}$ $-5.96^{+0.22}_{-0.24}$
\[3pt\]
------------------- ------------ ------------- ----------------- ---------------- ---------------- ----------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
: The equivalent width measurements and accretion rates of 144 Herbig Ae/Be stars in the [@Vioque2018] sample. Column 1 gives the object name. Columns 2 and 3 are right ascension (RA) in the units of time ( ) and declination (DEC) in the units of angle ( ) respectively. Observed equivalent width is listed along with references in columns 4–5. The rest of parameters are derived in this work. Columns 6–12 present intrinsic equivalent width, corrected equivalent width, continuum flux density at central wavelength of the $\rm H\alpha$ profile, line flux, line luminosity, accretion luminosity and mass accretion rate respectively.[]{data-label="tab:D1_Vioque_Mdot"}
\[tab:D1\_Vioque\_Mdot2\]
[width=1.33]{}
----------------------- ------------ ------------- ------------------- ---------------- ---------------- ------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
Name RA DEC $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (J2000) (J2000) (Å) Ref. (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
HD 37490 05:39:11.1 +04:07:17.2 $-3.50\pm0.18$ 14 $4.60\pm0.02$ $-8.10\pm0.18$ $3.27\times10^{-14}$ $(2.64\pm0.06)\times10^{-13}$ $-0.10^{+0.21}_{-0.14}$ $1.99^{+0.27}_{-0.21}$ $-5.50^{+0.38}_{-0.39}$
\[3pt\] RR Tau 05:39:30.5 +26:22:26.9 $-25.60\pm0.77$ 3 $11.70\pm0.02$ $-37.30\pm0.77$ $3.55\times10^{-16}$ $(1.33\pm0.03)\times10^{-14}$ $-0.61^{+0.06}_{-0.06}$ $1.48^{+0.15}_{-0.15}$ $-5.93^{+0.20}_{-0.20}$
\[3pt\] HD 245906 05:39:30.4 +26:19:55.1 $-3.60\pm0.18$ 2 $13.23\pm0.20$ $-16.83\pm0.27$ $2.94\times10^{-16}$ $(4.94\pm0.08)\times10^{-15}$ -$^{a}$ - -
\[3pt\] HD 37806 05:41:02.2 -02:43:00.7 $-17.00\pm0.85$ 14 $10.40\pm0.08$ $-27.40\pm0.85$ $1.51\times10^{-15}$ $(4.13\pm0.13)\times10^{-14}$ $-0.63^{+0.05}_{-0.05}$ $1.46^{+0.14}_{-0.14}$ $-5.96^{+0.22}_{-0.24}$
\[3pt\] HD 38087 05:43:00.5 -02:18:45.3 - - $8.06\pm0.05$ - $1.42\times10^{-15}$ - - - -
\[3pt\] HD 38120 05:43:11.8 -04:59:49.8 $-37.00\pm1.85$ 7 $11.58\pm0.11$ $-48.58\pm1.85$ $5.74\times10^{-16}$ $(2.79\pm0.11)\times10^{-14}$ $-0.85^{+0.07}_{-0.06}$ $1.24^{+0.17}_{-0.17}$ $-6.30^{+0.32}_{-0.28}$
\[3pt\] V351 Ori 05:44:18.7 +00:08:40.4 $-0.90\pm0.05$ 13 $13.64\pm0.03$ $-14.54\pm0.06$ $6.70\times10^{-16}$ $(9.74\pm0.04)\times10^{-15}$ $-1.45^{+0.02}_{-0.02}$ $0.64^{+0.16}_{-0.16}$ $-6.72^{+0.20}_{-0.17}$
\[3pt\] HD 39014 05:44:46.3 -65:44:07.9 $8.52\pm0.34$ 15 $13.58\pm0.18$ $-5.05\pm0.39$ $4.19\times10^{-14}$ $(2.12\pm0.16)\times10^{-13}$ $-1.89^{+0.05}_{-0.05}$ $0.20^{+0.20}_{-0.21}$ $-7.15^{+0.24}_{-0.22}$
\[3pt\] PDS 123 05:50:54.2 +20:14:50.0 - - $5.61\pm0.03$ - $1.17\times10^{-16}$ - - - -
\[3pt\] V1818 Ori 05:53:42.5 -10:24:00.7 $-40.00\pm2.00$ 12 $6.56\pm0.06$ $-46.56\pm2.00$ $2.20\times10^{-15}$ $(1.02\pm0.04)\times10^{-13}$ -$^{a}$ - -
\[3pt\] GSC 5360-1033 05:57:49.4 -14:05:33.6 $-4.00\pm0.20$ 7 $6.24\pm0.04$ $-10.24\pm0.20$ $2.30\times10^{-17}$ $(2.36\pm0.05)\times10^{-16}$ -$^{a}$ - -
\[3pt\] HD 249879 05:58:55.7 +16:39:57.3 $-41.00\pm2.05$ 7 $11.58\pm0.07$ $-52.58\pm2.05$ $1.24\times10^{-16}$ $(6.54\pm0.26)\times10^{-15}$ $-1.04^{+0.10}_{-0.08}$ $1.05^{+0.21}_{-0.20}$ $-6.62^{+0.33}_{-0.36}$
\[3pt\] PDS 22 06:03:37.0 -14:53:03.1 $-25.00\pm1.25$ 7 $12.66\pm0.13$ $-37.66\pm1.26$ $1.97\times10^{-16}$ $(7.43\pm0.25)\times10^{-15}$ -$^{a}$ - -
\[3pt\] HD 41511 06:04:59.1 -16:29:03.9 $0.24\pm0.31$ 16 $11.22\pm0.10$ $-10.98\pm0.33$ $3.57\times10^{-14}$ $(3.92\pm0.12)\times10^{-13}$ $-0.22^{+0.07}_{-0.07}$ $1.87^{+0.14}_{-0.14}$ $-5.35^{+0.21}_{-0.20}$
\[3pt\] GSC 1876-0892 06:07:15.3 +29:57:55.0 $-5.00\pm0.25$ 12 $3.94\pm0.04$ $-8.94\pm0.25$ $3.36\times10^{-16}$ $(3.01\pm0.09)\times10^{-15}$ $-0.07^{+0.17}_{-0.13}$ $2.02^{+0.23}_{-0.20}$ $-5.58^{+0.43}_{-0.46}$
\[3pt\] LkHA 208 06:07:49.5 +18:39:26.4 $-4.90\pm0.25$ 2 $13.80\pm0.13$ $-18.70\pm0.28$ $8.62\times10^{-17}$ $(1.61\pm0.02)\times10^{-15}$ -$^{a}$ - -
\[3pt\] PDS 211 06:10:17.3 +29:25:16.6 $-35.00\pm1.75$ 7 $11.53\pm0.01$ $-46.53\pm1.75$ $9.60\times10^{-17}$ $(4.47\pm0.17)\times10^{-15}$ $-0.80^{+0.07}_{-0.06}$ $1.29^{+0.16}_{-0.16}$ $-6.22^{+0.27}_{-0.30}$
\[3pt\] LkHA 338 06:10:47.1 -06:12:50.6 $-51.00\pm2.55$ 2 $14.72\pm0.11$ $-65.72\pm2.55$ $3.07\times10^{-17}$ $(2.02\pm0.08)\times10^{-15}$ $-1.31^{+0.08}_{-0.07}$ $0.78^{+0.20}_{-0.20}$ $-6.96^{+0.34}_{-0.37}$
\[3pt\] PDS 126 06:13:37.2 -06:25:01.6 $-5.00\pm0.25$ 7 $13.61\pm0.12$ $-18.61\pm0.28$ $1.01\times10^{-16}$ $(1.89\pm0.03)\times10^{-15}$ $-1.38^{+0.06}_{-0.05}$ $0.71^{+0.18}_{-0.18}$ $-6.65^{+0.23}_{-0.23}$
\[3pt\] MWC 137 06:18:45.5 +15:16:52.2 $-370.00\pm37.00$ 8 $2.30\pm0.04$ $-372.30\pm37.00$ $1.78\times10^{-15}$ $(6.64\pm0.66)\times10^{-13}$ $2.24^{+0.20}_{-0.17}$ $4.33^{+0.39}_{-0.34}$ $-3.45^{+0.55}_{-0.48}$
\[3pt\] CPM 25 06:23:56.3 +14:30:28.0 $-200.00\pm10.00$ 12 $5.50\pm0.06$ $-205.50\pm10.00$ $6.66\times10^{-17}$ $(1.37\pm0.07)\times10^{-14}$ -$^{a}$ - -
\[3pt\] NSV 2968 06:26:53.9 -10:15:34.9 $-40.00\pm2.00$ 7 $3.97\pm0.07$ $-43.97\pm2.00$ $6.85\times10^{-16}$ $(3.01\pm0.14)\times10^{-14}$ -$^{a}$ - -
\[3pt\] HD 45677 06:28:17.4 -13:03:11.1 $-57.00\pm5.70$ 8 $5.80\pm0.03$ $-62.80\pm5.70$ $1.28\times10^{-15}$ $(8.02\pm0.73)\times10^{-14}$ $-0.02^{+0.09}_{-0.09}$ $2.07^{+0.16}_{-0.15}$ $-5.57^{+0.26}_{-0.35}$
\[3pt\] HD 46060 06:30:49.8 -09:39:14.7 $3.61\pm0.01$ 17 $4.00\pm0.02$ $-0.40\pm0.02$ $3.22\times10^{-15}$ $(1.27\pm0.06)\times10^{-15}$ $-1.47^{+0.11}_{-0.09}$ $0.62^{+0.24}_{-0.23}$ $-7.03^{+0.46}_{-0.42}$
\[3pt\] PDS 129 06:31:03.6 +10:01:13.5 - - $7.13\pm0.12$ - $6.91\times10^{-17}$ - - - -
\[3pt\] LKHA 215 06:32:41.7 +10:09:34.2 $-25.70\pm1.29$ 2 $6.82\pm0.07$ $-32.52\pm1.29$ $7.11\times10^{-16}$ $(2.31\pm0.09)\times10^{-14}$ $-0.44^{+0.07}_{-0.06}$ $1.65^{+0.14}_{-0.14}$ $-5.91^{+0.24}_{-0.26}$
\[3pt\] HD 259431 06:33:05.1 +10:19:19.9 $-71.00\pm2.13$ 11 $6.24\pm0.08$ $-77.24\pm2.13$ $1.75\times10^{-15}$ $(1.35\pm0.04)\times10^{-13}$ $0.34^{+0.06}_{-0.06}$ $2.43^{+0.14}_{-0.13}$ $-5.07^{+0.35}_{-0.33}$
\[3pt\] HBC 217 06:40:42.1 +09:33:37.4 $-10.30\pm0.52$ 2 $6.12\pm0.08$ $-16.42\pm0.52$ $4.60\times10^{-17}$ $(7.56\pm0.24)\times10^{-16}$ $-1.94^{+0.06}_{-0.05}$ $0.15^{+0.21}_{-0.21}$ $-7.20^{+0.24}_{-0.24}$
\[3pt\] HBC 222 06:40:51.1 +09:44:46.1 $-1.20\pm0.06$ 2 $6.18\pm0.02$ $-7.38\pm0.06$ $4.79\times10^{-17}$ $(3.54\pm0.03)\times10^{-16}$ $-2.26^{+0.04}_{-0.04}$ $-0.17^{+0.21}_{-0.21}$ $-7.51^{+0.25}_{-0.25}$
\[3pt\] HD 50138 06:51:33.3 -06:57:59.4 $-71.00\pm7.10$ 8 $11.09\pm0.06$ $-82.09\pm7.10$ $4.65\times10^{-15}$ $(3.82\pm0.33)\times10^{-13}$ $0.23^{+0.07}_{-0.07}$ $2.32^{+0.15}_{-0.14}$ $-4.99^{+0.21}_{-0.19}$
\[3pt\] HD 50083 06:51:45.7 +05:05:03.8 $-43.00\pm1.29$ 11 $4.16\pm0.03$ $-47.16\pm1.29$ $5.99\times10^{-15}$ $(2.83\pm0.08)\times10^{-13}$ $1.02^{+0.09}_{-0.08}$ $3.11^{+0.20}_{-0.19}$ $-4.33^{+0.29}_{-0.35}$
\[3pt\] PDS 25 06:54:27.8 -25:02:15.8 $-13.00\pm0.65$ 7 $16.11\pm0.28$ $-29.11\pm0.71$ $2.03\times10^{-17}$ $(5.91\pm0.14)\times10^{-16}$ -$^{a}$ - -
\[3pt\] HD 56895B 07:18:30.5 -11:11:53.8 - - $9.15\pm0.27$ - $1.16\times10^{-15}$ - - - -
\[3pt\] GSC 6546-3156 07:24:17.5 -26:16:05.2 $-9.00\pm0.45$ 7 $14.31\pm0.12$ $-23.31\pm0.47$ $2.61\times10^{-17}$ $(6.08\pm0.12)\times10^{-16}$ $-1.43^{+0.06}_{-0.05}$ $0.66^{+0.19}_{-0.19}$ $-6.91^{+0.29}_{-0.29}$
\[3pt\]
----------------------- ------------ ------------- ------------------- ---------------- ---------------- ------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
\[tab:D1\_Vioque\_Mdot3\]
[width=1.33]{}
----------------------- ------------ ------------- -------------------- ---------------- ---------------- -------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
Name RA DEC $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (J2000) (J2000) (Å) Ref. (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
GSC 6542-2339 07:24:36.9 -24:34:47.4 $-25.00\pm1.25$ 7 $14.43\pm0.05$ $-39.43\pm1.25$ $2.00\times10^{-16}$ $(7.90\pm0.25)\times10^{-15}$ -$^{a}$ - -
\[3pt\] HD 58647 07:25:56.0 -14:10:43.5 $-11.40\pm0.30$ 3 $9.68\pm0.01$ $-21.08\pm0.30$ $5.05\times10^{-15}$ $(1.06\pm0.02)\times10^{-13}$ $-0.47^{+0.03}_{-0.02}$ $1.62^{+0.11}_{-0.11}$ $-5.76^{+0.14}_{-0.15}$
\[3pt\] GSC 5988-2257 07:41:41.0 -20:00:13.4 $-15.00\pm0.75$ 12 $5.60\pm0.03$ $-20.60\pm0.75$ $2.24\times10^{-17}$ $(4.62\pm0.17)\times10^{-16}$ -$^{a}$ - -
\[3pt\] GSC 8143-1225 07:59:11.5 -50:22:46.8 $1.71\pm0.05$ 17 $8.07\pm0.29$ $-6.36\pm0.29$ $6.89\times10^{-17}$ $(4.38\pm0.20)\times10^{-16}$ $-2.69^{+0.03}_{-0.03}$ $-0.60^{+0.22}_{-0.23}$ $-8.10^{+0.28}_{-0.28}$
\[3pt\] PDS 277 08:23:11.8 -39:07:01.6 $-3.00\pm0.15$ 7 $8.34\pm0.39$ $-11.34\pm0.42$ $2.58\times10^{-16}$ $(2.93\pm0.11)\times10^{-15}$ $-1.96^{+0.03}_{-0.03}$ $0.13^{+0.19}_{-0.19}$ $-7.22^{+0.22}_{-0.21}$
\[3pt\] V388 Vel 08:42:16.5 -40:44:09.9 - - $11.09\pm0.05$ - $1.06\times10^{-16}$ - -$^{a}$ - -
\[3pt\] PDS 34 08:49:58.5 -45:53:05.6 $-50.00\pm2.50$ 7 $5.52\pm0.08$ $-55.52\pm2.50$ $7.01\times10^{-17}$ $(3.89\pm0.18)\times10^{-15}$ $-0.26^{+0.07}_{-0.07}$ $1.83^{+0.14}_{-0.14}$ $-6.01^{+0.37}_{-0.38}$
\[3pt\] PDS 290 09:26:11.0 -52:42:26.9 $-7.00\pm0.35$ 7 $10.26\pm0.06$ $-17.26\pm0.36$ $6.14\times10^{-17}$ $(1.06\pm0.02)\times10^{-15}$ $-1.60^{+0.04}_{-0.04}$ $0.49^{+0.18}_{-0.18}$ $-7.28^{+0.33}_{-0.34}$
\[3pt\] HD 87643 10:04:30.2 -58:39:52.1 $-145.00\pm14.50$ 8 $3.25\pm0.06$ $-148.25\pm14.50$ $4.22\times10^{-15}$ $(6.26\pm0.61)\times10^{-13}$ -$^{a}$ - -
\[3pt\] PDS 322 10:52:08.6 -56:12:06.8 $4.00\pm0.02$ 7 $5.52\pm0.08$ $-1.52\pm0.08$ $1.09\times10^{-16}$ $(1.66\pm0.09)\times10^{-16}$ -$^{a}$ - -
\[3pt\] PDS 324 10:57:24.2 -62:53:13.2 $-4.00\pm0.20$ 7 $4.42\pm0.05$ $-8.42\pm0.21$ $8.15\times10^{-17}$ $(6.86\pm0.17)\times10^{-16}$ $-0.75^{+0.10}_{-0.09}$ $1.34^{+0.20}_{-0.19}$ $-6.58^{+0.46}_{-0.40}$
\[3pt\] PDS 138 11:53:13.2 -62:05:20.9 $-1.35\pm0.15$ 18 $2.18\pm0.04$ $-3.52\pm0.16$ $1.12\times10^{-15}$ $(3.96\pm0.18)\times10^{-15}$ $0.42^{+0.15}_{-0.13}$ $2.51^{+0.24}_{-0.20}$ $-5.27^{+0.44}_{-0.34}$
\[3pt\] GSC 8645-1401 12:17:47.5 -59:43:59.0 $-9.00\pm0.45$ 7 $9.27\pm0.33$ $-18.27\pm0.56$ $1.58\times10^{-16}$ $(2.89\pm0.09)\times10^{-15}$ $-0.55^{+0.09}_{-0.08}$ $1.54^{+0.18}_{-0.17}$ $-5.64^{+0.23}_{-0.23}$
\[3pt\] Hen 2-80 12:22:23.1 -63:17:16.8 $-150.00\pm7.50$ 19 $8.07\pm0.07$ $-158.07\pm7.50$ $2.29\times10^{-16}$ $(3.61\pm0.17)\times10^{-14}$ -$^{a}$ - -
\[3pt\] Hen 3-823 12:48:42.3 -59:54:34.9 $-25.00\pm1.25$ 19 $5.63\pm0.06$ $-30.63\pm1.25$ $4.05\times10^{-16}$ $(1.24\pm0.05)\times10^{-14}$ -$^{a}$ - -
\[3pt\] DK Cha 12:53:17.2 -77:07:10.7 $-88.00\pm4.40$ 20 $10.64\pm0.33$ $-98.64\pm4.41$ $1.66\times10^{-16}$ $(1.63\pm0.07)\times10^{-14}$ -$^{a}$ - -
\[3pt\] GSC 8994-3902 13:19:03.9 -62:34:10.1 $3.39\pm0.07$ 18 $4.21\pm0.06$ $-0.82\pm0.09$ $3.75\times10^{-16}$ $(3.09\pm0.33)\times10^{-16}$ $-1.26^{+0.17}_{-0.15}$ $0.83^{+0.28}_{-0.28}$ $-6.80^{+0.49}_{-0.53}$
\[3pt\] PDS 371 13:47:31.4 -36:39:49.6 $-40.00\pm2.00$ 12 $2.18\pm0.01$ $-42.18\pm2.00$ $1.26\times10^{-16}$ $(5.30\pm0.25)\times10^{-15}$ -$^{a}$ - -
\[3pt\] Hen 3-938 13:52:42.8 -63:32:49.1 $-90.00\pm4.50$ 12 $1.98\pm0.03$ $-91.98\pm4.50$ $9.75\times10^{-16}$ $(8.97\pm0.44)\times10^{-14}$ $1.62^{+0.15}_{-0.13}$ $3.71^{+0.30}_{-0.26}$ $-4.20^{+0.44}_{-0.35}$
\[3pt\] HD 130437 14:50:50.2 -60:17:10.3 $-53.01\pm0.25$ 19 $3.28\pm0.06$ $-56.29\pm0.26$ $1.89\times10^{-15}$ $(1.06\pm0.01)\times10^{-13}$ $0.95^{+0.09}_{-0.07}$ $3.04^{+0.20}_{-0.18}$ $-4.68^{+0.44}_{-0.38}$
\[3pt\] PDS 389 15:14:47.0 -62:16:59.7 $-9.00\pm0.45$ 7 $13.64\pm0.09$ $-22.64\pm0.46$ $3.83\times10^{-16}$ $(8.68\pm0.18)\times10^{-15}$ $-0.76^{+0.05}_{-0.04}$ $1.33^{+0.14}_{-0.14}$ $-5.99^{+0.18}_{-0.20}$
\[3pt\] HD 135344 15:15:48.9 -37:08:55.7 $-6.50\pm0.65$ 8 $8.47\pm0.41$ $-14.97\pm0.77$ $2.26\times10^{-15}$ $(3.38\pm0.17)\times10^{-14}$ $-1.67^{+0.04}_{-0.04}$ $0.42^{+0.18}_{-0.19}$ $-6.89^{+0.20}_{-0.20}$
\[3pt\] Hen 3-1121 15:58:09.6 -53:51:18.3 $-0.40\pm0.05$ 19 $3.63\pm0.02$ $-4.03\pm0.05$ $5.33\times10^{-16}$ $(2.14\pm0.03)\times10^{-15}$ -$^{a}$ - -
\[3pt\] CPD-53 6867 15:58:09.6 -53:51:35.0 $-0.10\pm0.01$ 19 $3.32\pm0.05$ $-3.42\pm0.05$ $1.73\times10^{-15}$ $(5.90\pm0.09)\times10^{-15}$ -$^{a}$ - -
\[3pt\] HD 143006 15:58:36.9 -22:57:15.2 $-8.35\pm0.15$ 21 $3.17\pm0.01$ $-11.52\pm0.15$ $3.91\times10^{-16}$ $(4.51\pm0.06)\times10^{-15}$ $-2.41^{+0.04}_{-0.04}$ $-0.32^{+0.22}_{-0.22}$ $-7.73^{+0.25}_{-0.25}$
\[3pt\] WRAY 15-1435 16:13:06.7 -50:23:20.0 $-20.00\pm1.00$ 19 $4.07\pm0.07$ $-24.07\pm1.00$ $3.48\times10^{-16}$ $(8.37\pm0.35)\times10^{-15}$ $-0.05^{+0.15}_{-0.12}$ $2.04^{+0.21}_{-0.19}$ $-5.80^{+0.48}_{-0.40}$
\[3pt\] Hen 3-1191 16:27:15.1 -48:39:26.8 $-1093.17\pm23.58$ 22 $4.16\pm0.11$ $-1097.33\pm23.58$ $1.95\times10^{-16}$ $(2.14\pm0.05)\times10^{-13}$ -$^{a}$ - -
\[3pt\] HD 149914 16:38:28.6 -18:13:13.7 $9.33\pm0.06$ 23 $11.08\pm0.03$ $-1.75\pm0.07$ $9.56\times10^{-15}$ $(1.68\pm0.07)\times10^{-14}$ $-1.88^{+0.04}_{-0.04}$ $0.21^{+0.19}_{-0.19}$ $-7.21^{+0.26}_{-0.28}$
\[3pt\] V921 Sco 16:59:06.7 -42:42:08.4 $-194.30\pm19.43$ 24 $2.56\pm0.05$ $-196.86\pm19.43$ $4.16\times10^{-15}$ $(8.20\pm0.81)\times10^{-13}$ $1.79^{+0.17}_{-0.14}$ $3.88^{+0.32}_{-0.28}$ $-3.94^{+0.50}_{-0.44}$
\[3pt\] HD 155448 17:12:58.7 -32:14:33.5 $4.06\pm0.02$ 25 $8.62\pm0.07$ $-4.55\pm0.07$ $1.09\times10^{-15}$ $(4.95\pm0.08)\times10^{-15}$ $-0.86^{+0.09}_{-0.07}$ $1.23^{+0.19}_{-0.18}$ $-6.11^{+0.29}_{-0.28}$
\[3pt\] PDS 453 17:20:56.1 -26:03:30.5 $4.00\pm0.20$ 7 $9.41\pm0.35$ $-5.41\pm0.40$ $4.10\times10^{-17}$ $(2.22\pm0.16)\times10^{-16}$ -$^{a}$ - -
\[3pt\] MWC 878 17:24:44.7 -38:43:51.4 $-54.00\pm2.70$ 19 $3.28\pm0.06$ $-57.28\pm2.70$ $1.67\times10^{-15}$ $(9.58\pm0.45)\times10^{-14}$ $0.97^{+0.16}_{-0.13}$ $3.06^{+0.27}_{-0.23}$ $-4.66^{+0.51}_{-0.45}$
\[3pt\] HD 319896 17:31:05.8 -35:08:29.2 $-27.00\pm1.35$ 7 $5.48\pm0.07$ $-32.48\pm1.35$ $6.68\times10^{-16}$ $(2.17\pm0.09)\times10^{-14}$ $0.06^{+0.14}_{-0.11}$ $2.15^{+0.21}_{-0.17}$ $-5.39^{+0.29}_{-0.25}$
\[3pt\] HD 158643 17:31:24.9 -23:57:45.5 $-3.30\pm0.10$ 3 $11.32\pm0.07$ $-14.62\pm0.12$ $2.37\times10^{-14}$ $(3.47\pm0.03)\times10^{-13}$ $-0.79^{+0.06}_{-0.05}$ $1.30^{+0.16}_{-0.15}$ $-6.07^{+0.22}_{-0.24}$
\[3pt\]
----------------------- ------------ ------------- -------------------- ---------------- ---------------- -------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
\[tab:D1\_Vioque\_Mdot4\]
[width=1.33]{}
-------------------- ------------ ------------- ------------------- ---------------- ---------------- ------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
Name RA DEC $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (J2000) (J2000) (Å) Ref. (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
HD 323771 17:34:04.6 -39:23:41.3 $-54.00\pm2.70$ 7 $6.83\pm0.03$ $-60.83\pm2.70$ $2.32\times10^{-16}$ $(1.41\pm0.06)\times10^{-14}$ $-0.30^{+0.11}_{-0.10}$ $1.79^{+0.18}_{-0.18}$ $-5.86^{+0.30}_{-0.28}$
\[3pt\] SAO 185668 17:43:55.6 -22:05:44.6 - - $4.42\pm0.04$ - $1.87\times10^{-15}$ - - - -
\[3pt\] MWC 593 17:49:10.1 -24:14:21.2 $-35.00\pm1.75$ 19 $4.81\pm0.06$ $-39.81\pm1.75$ $1.54\times10^{-15}$ $(6.14\pm0.27)\times10^{-14}$ $0.54^{+0.15}_{-0.12}$ $2.63^{+0.25}_{-0.20}$ $-4.85^{+0.32}_{-0.26}$
\[3pt\] PDS 477 18:00:30.3 -16:47:25.8 $-120.00\pm6.00$ 7 $4.13\pm0.06$ $-124.13\pm6.00$ $1.68\times10^{-16}$ $(2.09\pm0.10)\times10^{-14}$ -$^{a}$ - -
\[3pt\] HD 313571 18:01:07.1 -22:15:04.0 $-34.00\pm1.70$ 19 $4.87\pm0.12$ $-38.87\pm1.70$ $9.73\times10^{-16}$ $(3.78\pm0.17)\times10^{-14}$ $0.36^{+0.18}_{-0.13}$ $2.45^{+0.26}_{-0.20}$ $-5.08^{+0.37}_{-0.39}$
\[3pt\] LkHA 260 18:19:09.3 -13:50:41.2 - - $8.22\pm0.05$ - $7.93\times10^{-17}$ - - - -
\[3pt\] HD 169142 18:24:29.7 -29:46:49.3 $-13.97\pm0.15$ 21 $13.76\pm0.10$ $-27.73\pm0.18$ $2.82\times10^{-15}$ $(7.83\pm0.05)\times10^{-14}$ $-1.50^{+0.01}_{-0.01}$ $0.59^{+0.15}_{-0.15}$ $-7.09^{+0.26}_{-0.29}$
\[3pt\] MWC 930 18:26:25.2 -07:13:17.8 - - $8.76\pm0.09$ - $1.50\times10^{-13}$ - - - -
\[3pt\] PDS 520 18:30:06.1 +00:42:33.5 $-33.00\pm1.65$ 7 $8.32\pm0.28$ $-41.32\pm1.67$ $9.42\times10^{-17}$ $(3.89\pm0.16)\times10^{-15}$ -$^{a}$ - -
\[3pt\] PDS 530 18:41:34.3 +08:08:20.7 $-28.00\pm1.40$ 7 $14.71\pm0.22$ $-42.71\pm1.42$ $3.44\times10^{-17}$ $(1.47\pm0.05)\times10^{-15}$ -$^{a}$ - -
\[3pt\] MWC 953 18:43:28.4 -03:46:17.0 $-32.00\pm1.60$ 19 $3.66\pm0.05$ $-35.66\pm1.60$ $2.35\times10^{-15}$ $(8.39\pm0.38)\times10^{-14}$ $0.89^{+0.16}_{-0.13}$ $2.98^{+0.28}_{-0.23}$ $-4.57^{+0.48}_{-0.48}$
\[3pt\] HD 174571 18:50:47.1 +08:42:10.0 $2.00\pm0.20$ 26 $3.64\pm0.04$ $-1.64\pm0.20$ $6.00\times10^{-15}$ $(9.83\pm1.22)\times10^{-15}$ $-0.43^{+0.15}_{-0.13}$ $1.66^{+0.22}_{-0.22}$ $-5.89^{+0.41}_{-0.46}$
\[3pt\] PDS 551 18:55:22.9 +04:04:35.2 $-50.00\pm2.50$ 7 $2.87\pm0.06$ $-52.87\pm2.50$ $6.02\times10^{-18}$ $(3.18\pm0.15)\times10^{-16}$ -$^{a}$ - -
\[3pt\] HD 176386 19:01:38.9 -36:53:26.5 $-0.17\pm0.05$ 4 $12.73\pm0.12$ $-12.89\pm0.13$ $2.68\times10^{-15}$ $(3.46\pm0.03)\times10^{-14}$ $-1.56^{+0.02}_{-0.02}$ $0.53^{+0.16}_{-0.16}$ $-7.08^{+0.27}_{-0.27}$
\[3pt\] TY CrA 19:01:40.8 -36:52:33.8 $-5.82\pm0.37$ 15 $13.14\pm0.04$ $-18.96\pm0.37$ $2.48\times10^{-15}$ $(4.71\pm0.09)\times10^{-14}$ -$^{a}$ - -
\[3pt\] R CrA 19:01:53.6 -36:57:08.1 $-85.50\pm4.28$ 27 $14.43\pm0.05$ $-99.93\pm4.28$ $2.87\times10^{-16}$ $(2.87\pm0.12)\times10^{-14}$ -$^{a}$ - -
\[3pt\] MWC 314 19:21:33.9 +14:52:56.9 $-125.00\pm15.00$ 28 $3.62\pm0.04$ $-128.62\pm15.00$ $1.43\times10^{-14}$ $(1.84\pm0.21)\times10^{-12}$ $2.70^{+0.20}_{-0.17}$ $4.79^{+0.40}_{-0.36}$ $-2.36^{+0.68}_{-0.66}$
\[3pt\] HD 344261 19:21:53.5 +21:31:50.5 - - $9.23\pm0.23$ - $1.97\times10^{-16}$ - - - -
\[3pt\] WW Vul 19:25:58.7 +21:12:31.3 $-19.10\pm0.57$ 3 $15.06\pm0.10$ $-34.16\pm0.58$ $2.67\times10^{-16}$ $(9.11\pm0.16)\times10^{-15}$ $-1.14^{+0.03}_{-0.03}$ $0.95^{+0.15}_{-0.15}$ $-6.51^{+0.21}_{-0.19}$
\[3pt\] PX Vul 19:26:40.2 +23:53:50.7 $-14.40\pm0.43$ 3 $8.37\pm0.32$ $-22.77\pm0.54$ $2.03\times10^{-16}$ $(4.63\pm0.11)\times10^{-15}$ -$^{a}$ - -
\[3pt\] PDS 581 19:36:18.8 +29:32:50.8 $-200.00\pm10.00$ 7 $5.16\pm0.07$ $-205.16\pm10.00$ $4.10\times10^{-16}$ $(8.42\pm0.41)\times10^{-14}$ -$^{a}$ - -
\[3pt\] MWC 623 19:56:31.5 +31:06:20.1 $-129.00\pm14.00$ 29 $3.85\pm0.04$ $-132.85\pm14.00$ $2.56\times10^{-15}$ $(3.40\pm0.36)\times10^{-13}$ $2.06^{+0.18}_{-0.16}$ $4.15^{+0.35}_{-0.31}$ $-3.19^{+0.45}_{-0.41}$
\[3pt\] V1686 Cyg 20:20:29.3 +41:21:28.4 $-22.70\pm0.68$ 3 $5.59\pm0.03$ $-28.29\pm0.68$ $1.07\times10^{-16}$ $(3.02\pm0.07)\times10^{-15}$ -$^{a}$ - -
\[3pt\] MWC 342 20:23:03.6 +39:29:49.9 $-290.00\pm80.00$ 30 $2.58\pm0.06$ $-292.58\pm80.00$ $5.20\times10^{-15}$ $(1.52\pm0.42)\times10^{-12}$ $2.19^{+0.17}_{-0.19}$ $4.28^{+0.35}_{-0.35}$ $-3.43^{+0.52}_{-0.57}$
\[3pt\] BD+41 3731 20:24:15.7 +42:18:01.3 - - $5.56\pm0.01$ - $5.45\times10^{-16}$ - - - -
\[3pt\] HBC 694 20:24:29.5 +42:14:02 - - $14.43\pm0.05$ - $2.10\times10^{-18}$ - -$^{a}$ - -
\[3pt\] MWC 1021 20:29:26.9 +41:40:43.8 - - $2.87\pm0.02$ - $4.74\times10^{-14}$ - - - -
\[3pt\] V1478 Cyg 20:32:45.6 +40:39:36.1 - - $2.87\pm0.02$ - $3.61\times10^{-14}$ - -$^{a}$ - -
\[3pt\] PV Cep 20:45:53.9 +67:57:38.6 $-49.90\pm2.50$ 31 $14.43\pm0.05$ $-64.33\pm2.50$ $2.54\times10^{-17}$ $(1.63\pm0.06)\times10^{-15}$ $-2.22^{+0.05}_{-0.05}$ $-0.13^{+0.22}_{-0.22}$ -
\[3pt\] V1977 Cyg 20:47:37.4 +43:47:24.9 $-32.70\pm0.98$ 3 $8.89\pm0.03$ $-41.59\pm0.98$ $6.92\times10^{-16}$ $(2.88\pm0.07)\times10^{-14}$ $-0.18^{+0.05}_{-0.05}$ $1.91^{+0.12}_{-0.12}$ $-5.49^{+0.15}_{-0.15}$
\[3pt\] HBC 705 20:51:02.7 +43:49:31.9 $-26.90\pm1.35$ 2 $4.21\pm0.06$ $-31.11\pm1.35$ $5.13\times10^{-16}$ $(1.59\pm0.07)\times10^{-14}$ $0.33^{+0.10}_{-0.09}$ $2.42^{+0.18}_{-0.16}$ $-5.21^{+0.39}_{-0.42}$
\[3pt\] V1493 Cyg 20:52:04.6 +44:37:30.4 $-9.50\pm0.48$ 2 $12.76\pm0.09$ $-22.26\pm0.48$ $3.03\times10^{-16}$ $(6.75\pm0.15)\times10^{-15}$ -$^{a}$ - -
\[3pt\] HBC 717 20:52:06.0 +44:17:16.0 $-19.00\pm0.95$ 2 $6.56\pm0.13$ $-25.56\pm0.96$ $1.41\times10^{-16}$ $(3.61\pm0.14)\times10^{-15}$ -$^{a}$ - -
\[3pt\] HD 199603 20:58:41.8 -14:28:59.2 $9.44\pm0.27$ 15 $10.96\pm0.39$ $-1.52\pm0.48$ $1.01\times10^{-14}$ $(1.54\pm0.48)\times10^{-14}$ $-2.42^{+0.13}_{-0.18}$ $-0.33^{+0.31}_{-0.37}$ $-7.65^{+0.32}_{-0.37}$
\[3pt\]
-------------------- ------------ ------------- ------------------- ---------------- ---------------- ------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
\[tab:D1\_Vioque\_Mdot5\]
[width=1.33]{}
----------------------- ------------ ------------- ------------------- ---------------- ---------------- ------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
Name RA DEC $EW_{\rm int}$ $EW_{\rm cor}$ $F_{\lambda}$ $F_{\rm line}$ $\log(L_{\rm line})$ $\log(L_{\rm acc})$ $\log(\dot M_{\rm acc})$
\[3pt\] (J2000) (J2000) (Å) Ref. (Å) (Å) (Wm$^{-2}$Å$^{-1}$) (Wm$^{-2}$) \[$\rm L_{\sun}$\] \[$\rm L_{\sun}$\] \[$\rm M_{\sun}$$\rm yr^{-1}$\]
V594 Cyg 21:20:23.3 +43:18:10.2 - - $6.39\pm0.08$ - $3.04\times10^{-15}$ - - - -
\[3pt\] HD 235495 21:21:27.4 +50:59:47.6 - - $14.00\pm0.09$ - $2.53\times10^{-16}$ - - - -
\[3pt\] AS 470 21:36:14.2 +57:21:30.8 $-40.00\pm4.00$ 32 $11.44\pm0.04$ $-51.44\pm4.00$ $1.84\times10^{-16}$ $(9.48\pm0.74)\times10^{-15}$ $0.68^{+0.16}_{-0.14}$ $2.77^{+0.26}_{-0.22}$ $-4.37^{+0.44}_{-0.38}$
\[3pt\] GSC 3975-0579 21:38:08.4 +57:26:47.6 $-6.90\pm0.35$ 13 $14.83\pm0.21$ $-21.73\pm0.40$ $9.95\times10^{-17}$ $(2.16\pm0.04)\times10^{-15}$ $-1.22^{+0.05}_{-0.04}$ $0.87^{+0.17}_{-0.17}$ $-6.55^{+0.25}_{-0.26}$
\[3pt\] BH Cep 22:01:42.8 +69:44:36.4 $-6.20\pm0.19$ 3 $7.70\pm0.44$ $-13.90\pm0.48$ $1.80\times10^{-16}$ $(2.50\pm0.09)\times10^{-15}$ $-2.06^{+0.02}_{-0.02}$ $0.03^{+0.19}_{-0.19}$ $-7.34^{+0.23}_{-0.25}$
\[3pt\] BO Cep 22:16:54.0 +70:03:44.9 $-7.50\pm0.23$ 3 $7.71\pm0.41$ $-15.21\pm0.47$ $7.33\times10^{-17}$ $(1.11\pm0.03)\times10^{-15}$ $-2.31^{+0.02}_{-0.02}$ $-0.22^{+0.20}_{-0.20}$ $-7.69^{+0.23}_{-0.22}$
\[3pt\] V669 Cep 22:26:38.7 +61:13:31.5 $-127.00\pm60.00$ 33 $6.60\pm0.02$ $-133.60\pm60.00$ $3.41\times10^{-16}$ $(4.56\pm2.05)\times10^{-14}$ -$^{a}$ - -
\[3pt\] MWC 655 22:38:31.8 +55:50:05.3 $-14.30\pm0.72$ 26 $3.47\pm0.07$ $-17.77\pm0.72$ $7.05\times10^{-16}$ $(1.25\pm0.05)\times10^{-14}$ $0.26^{+0.11}_{-0.09}$ $2.35^{+0.19}_{-0.16}$ $-5.40^{+0.43}_{-0.38}$
\[3pt\] MWC 657 22:42:41.8 +60:24:00.6 $-185.35\pm13.00$ 34 $3.20\pm0.03$ $-188.55\pm13.00$ $1.73\times10^{-15}$ $(3.26\pm0.22)\times10^{-13}$ $2.00^{+0.13}_{-0.11}$ $4.09^{+0.30}_{-0.27}$ $-3.41^{+0.94}_{-0.71}$
\[3pt\] BP Psc 23:22:24.6 -02:13:41.3 $-13.17\pm0.35$ 35 $3.15\pm0.45$ $-16.32\pm0.58$ $1.64\times10^{-16}$ $(2.68\pm0.09)\times10^{-15}$ -$^{a}$ - -
\[3pt\] LkHA 259 23:58:41.6 +66:26:12.7 $-24.00\pm1.20$ 2 $12.03\pm0.18$ $-36.03\pm1.21$ $8.27\times10^{-17}$ $(2.98\pm0.10)\times10^{-15}$ $-1.28^{+0.05}_{-0.05}$ $0.81^{+0.17}_{-0.18}$ $-6.54^{+0.21}_{-0.23}$
\[3pt\]
----------------------- ------------ ------------- ------------------- ---------------- ---------------- ------------------- ---------------------- ------------------------------- ------------------------- -------------------------- ---------------------------------
\
[**Notes.**]{} $^{(a)}$ Stars which have low quality parallaxes in the [*Gaia*]{} DR2 Catalogue (see the text for discussion). References: (1) @Herbig1988; (2) @Hernandez2004; (3) @Mendigutia2011a; (4) @Pogodin2012; (5) ESO Programme 082.A-9011(A); (6) ESO Programme 076.B-0055(A); (7) @Sartori2010; (8) @Baines2006; (9) @Miroshnichenko1999; (10) @Boehm1995; (11) @Wheelwright2010; (12) @Vieira2011; (13) @Hernandez2005; (14) @Oudmaijer1999; (15) ESO Programme 085.A-9027(B); (16) ESO Programme 082.D-0061(A); (17) ESO Programme 084.A-9016(A); (18) ESO Programme 083.A-9013(A); (19) @Carmona2010; (20) @Spezzi2008; (21) @Dunkin1997; (22) ESO Programme 075.D-0177(A); (23) ESO Programme 60.A-9022(C); (24) @BorgesFernandes2007; (25) ESO Programme 073.D-0609(A); (26) @Ababakr2017; (27) @Manoj2006; (28) @Frasca2016; (29) @Polster2012; (30) @Kucerova2013; (31) @Acke2005; (32) @Nakano2012; (33) @Miroshnichenko2002; (34) @Miroshnichenko2000; (35) @Zuckerman2008.
[^1]: E-mail: pycw@leeds.ac.uk (CW)
[^2]: [IRAF]{} is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.
[^3]: Based on observations collected at the European Southern Observatory under ESO programmes 084.A-9016(A).
[^4]: https://www.aavso.org/apass-dr10-download
[^5]: When drafting this manuscript, a paper by [@Arun2019] was published reporting on the accretion rates using [*Gaia*]{} data of a somewhat smaller sample than here. Notable differences are that we use homogeneously determined stellar parameters for a large fraction of the sample and the fact that their sample is not selected for parallax quality and therefore includes faulty [*Gaia*]{} parallax measurements affecting the derived distances. It would also appear that their determination of the H$\alpha$ line luminosity does not account for underlying absorption, which will affect especially the weakly emitting sources. As the authors do not provide all relevant datatables, it proved hard to investigate and assess further differences.
|
---
author:
- 'Yosuke <span style="font-variant:small-caps;">Imamura</span>[^1]andKeisuke <span style="font-variant:small-caps;">Kimura</span>[^2]'
title: 'On the Moduli Space of Elliptic Maxwell-Chern-Simons Theories'
---
Introduction
============
Recently, there has been great interest in 3-dimensional superconformal field theories as theories for describing multiple M2-branes in various backgrounds. This was triggered by the proposal of a new class of 3-dimensional theories by Bagger and Lambert[@Bagger:2006sk; @Bagger:2007jr; @Bagger:2007vi], and Gusstavson[@Gustavsson:2007vu; @Gustavsson:2008dy]. The model (BLG model) possesses ${\cal N}_{(d=3)}=8$ superconformal symmetry and is based on Lie $3$-algebra. The action of the BLG model includes the structure constant $f^{abc}{}_d$ of a Lie $3$-algebra, which determines the form of the interactions, and a metric $h^{ab}$, which appears in the coefficients of the kinetic terms. These tensors must satisfy certain conditions required by the supersymmetry invariance of the action. If these tensors satisfy the conditions, we can write down the action of a BLG model. The constraint imposed on the structure constant is called a fundamental identity. It was soon realized that the identity is very restrictive[@Ho:2008bn], and it was proved that if we assume that the metric is positive definite and the algebra is finite dimensional, there is only one nontrivial Lie $3$-algebra[@Papadopoulos:2008sk; @Gauntlett:2008uf], which is called an $A_4$ algebra. The BLG model based on the $A_4$ algebra is a $SU(2)\times SU(2)$ Chern-Simons theory with levels $k$ and $-k$ for each $SU(2)$ factor. Analysis of this model showed that it describes a pair of M2-branes in certain orbifold backgrounds[@VanRaamsdonk:2008ft; @Lambert:2008et; @Distler:2008mk]. As a theory for an arbitrary number of M2-branes, a model based on an algebra with a Lorenzian metric was proposed in Refs. . Because of the indefinite metric, the model includes unwanted ghost modes. Although the ghost modes can be removed by treating them as background fields satisfying classical equations of motion[@Ho:2008ei; @Honma:2008un], or by gauging certain symmetries and fixing them[@Bandres:2008kj; @Gomis:2008be], this procedure breaks the conformal invariance, and the theory becomes D2-brane theory[@Ho:2008ei; @Bandres:2008kj; @Ezhuthachan:2008ch] by the mechanism proposed in Ref. unless the parameter corresponding to the Yang-Mills coupling is sent to infinity or integrated over all values as a dynamical parameter[@Gomis:2008be].
There has also been some progress in 3-dimensional Chern-Simons theories with supersymmetries of less than $8$, which are closely related to M2-branes. Gaiotto and Witten[@Gaiotto:2008sd] proposed ${\cal N}_{(\rm d=3)}=4$ superconformal Chern-Simons theories, and Hosomichi et al.[@Hosomichi:2008jd] extended the theories by introducing twisted hypermultiplets. They derived the relation between their models and the BLG model, and showed that the $A_4$ BLG model is included as a special case of their ${\cal N}_{(d=3)}=4$ Chern-Simons theories. They also studied the M-crystal model [@Lee:2006hw; @Lee:2007kv; @Kim:2007ic], which is described by a circular quiver diagram with $2n$ vertices. The vertices represent Chern-Simons fields at level $\pm k$ with alternate signatures, and by analyzing the moduli space of the model they showed that it can be regarded as a theory describing M2-branes in the orbifold $({\mathbb C}^2/{\mathbb Z}_n)^2$.[^3] They also presented a realization of this model by using D3-, D5-, and NS5-branes, which give the model at level $\pm1$, and reproduce the orbifold as the M-theory dual of the brane system.
Aharony et al. also proposed a similar model[@Aharony:2008ug] based on $U(N)\times U(N)$ Chern-Simons theory with levels $k$ and $-k$ for each $U(N)$ factor. They showed that the action possesses ${\cal N}_{(d=3)}=6$ superconformal symmetry, and describes $N$ M2-branes in the orbifold ${\mathbb C}^4/{\mathbb Z}_k$. Although ${\cal N}_{(d=3)}=8$ supersymmetry, which is expected when $k=1$ or $2$ is not manifest, the action does not have dimensionful parameters and the scale invariance is manifest. In Ref. it is also shown that the theory can be realized as a theory on a brane system in type IIB string theory. The brane system consists of $N$ D3-, one NS5-, and one $(k,1)$5-branes. They showed that by T-duality and M-theory lift, M2-branes in the orbifold ${\mathbb C}^4/{\mathbb Z}_k$ are obtained.
The purpose of this paper is to extend the models proposed in Refs. and by generalizing the brane configurations in these references. In §\[brane.sec\] we consider a brane system with $n_A$ NS5-branes and $n_B$ $(k,1)$5-branes, and analyze the moduli space of the theory realized by the brane system. The theory is a $U(N)^{n_A+n_B}$ quiver gauge theory with nonvanishing Chern-Simons terms for some of the $U(N)$ factors. Some of the $U(N)$ fields are Yang-Mills fields without Chern-Simons coupling. The supersymmetry of this theory is ${\cal N}_{(d=3)}=3$, which is expected to be enhanced to ${\cal N}_{(d=3)}=4$ in the strong gauge-coupling limit. The reason for this is as follows. This theory can be obtained from the $U(N)\times U(N)$ theory proposed in Ref. by combining two extensions. One is the inclusion of twisted hypermultiplets, as mentioned above, and the other is the inclusion of gauge groups with vanishing Chern-Simons couplings. The latter extension is discussed in Ref. to describe general nonlinear sigma models of hypermultiplets. Both extensions are known to give ${\cal N}_{(d=3)}=4$ supersymmetric Chern-Simons theory, and it is plausible that the theory we discuss in this paper possesses ${\cal N}_{(d=3)}=4$ supersymmetry.
In §\[moduli.sec\] we determine the moduli space of the theory. We focus only on the Higgs branch, which describes a mobile M2-brane. Under a certain assumption for flux quantization, we obtain a 4-dimensional orbifold ${\mathbb C}^4/\Gamma$, where $\Gamma$ is a discrete subgroup depending on $k$, $n_A$, and $n_B$. We reproduce the same orbifold in §\[dual.sec\] as an M-theory dual of the brane configuration. In §\[more.sec\] we consider models with more than two kinds of fivebranes. The moduli space is also a 4-dimensional manifold, but it is nontoric. The last section is devoted to discussion.
Brane configuration and action {#brane.sec}
==============================
The model proposed in Ref. is a Chern-Simons theory with a $U(N)\times U(N)$ gauge group. It can be realized as a theory based on a brane system consisting of $N$ D3-branes, one NS5-brane, and one $(k,1)$5-brane. All these branes share the directions of 012, which are the coordinates of the 3-dimensional field theory. The $N$ D3-branes are wrapped on the compact direction 9. The NS5-brane and the $(k,1)$5-brane are spread along the 012345 and $012[36]_{\theta_1}[47]_{\theta_2}[58]_{\theta_3}$ directions, respectively, where $[ij]_\theta$ is the direction in the $i$-$j$ plane specified by the angle $\theta$. The angles $\theta_{1,2,3}$ are determined by the BPS conditions.
![Brane configuration for the $U(N)\times U(N)$ Chern-Simons model.[]{data-label="brane.eps"}](brane.eps)
We refer to NS5- and $(k,1)$5-branes as A- and B-branes, respectively. The D3-brane worldvolume is divided into two parts by the intersecting fivebranes (Fig. \[brane.eps\]), and a $U(N)$ vector multiplet exists on each segment. Bifundamental chiral multiplets also arise at the intersections. This brane system is similar to the D4-NS5 system realizing the Klebanov-Witten theory[@Klebanov:1998hh], which is a 4-dimensional ${\cal N}_{(d=4)}=1$ superconformal field theory. In the D4-NS5 system, we have $N$ D4-branes wrapped on ${\mathbb S}^1$, instead of D3-branes, and the A- and B-branes in this case are NS5-branes along different directions.
We generalize the D3-fivebrane system by introducing an arbitrary number of fivebranes. In the case of 4-dimensional ${\cal N}_{(d=4)}=1$ gauge theories, such a generalization is known as an elliptic model, and has been studied in detail[@Uranga:1998vf; @vonUnge:1999hc]. It is known that the moduli spaces of the theories are generalized conifolds. We here carry out a similar analysis in the 3-dimensional case. Let $n_A$ and $n_B$ be the numbers of A- and B-branes, respectively. We denote the total number of fivebranes by $n=n_A+n_B$. Let us label the fivebranes by $I=1,\ldots,n$ according to their order along ${\mathbb S}^1$. We identify $I=n+1$ with $I=1$. On the interval of D3-branes between two fivebranes $I$ and $I+1$, we have a $U(N)$ vector multiplet $V_I$ and an adjoint chiral multiplet $\Phi_I$. (We use the terminology of ${\cal N}_{(d=4)}=(1/2){\cal N}_{(d=3)}=1$ supersymmetry.) The kinetic terms of these multiplets are $$\begin{aligned}
S_V&=&\int d^3x\sum_I\frac{1}{g_I^2}\tr \left[
-\frac{1}{4}(F^I_{\mu\nu})^2-\frac{1}{2}(D_\mu\sigma_I)^2+\frac{1}{2}D_I^2
+\mbox{fermions}\right],
\label{sv}\\
S_\Phi&=&\int d^3xd^4\theta\sum_I \frac{1}{g_I^2}\tr (\Phi_I^*e^{V_I}\Phi_Ie^{-V_I}).\label{sphi}\end{aligned}$$ $\sigma_I$ is the real scalar field in the vector multiplet $V_I$. The adjoint chiral multiplets $\Phi_I$ describe the motion of the D-branes along the fivebranes. When two fivebranes $I$ and $I+1$ are not parallel, the chiral multiplet $\Phi_I$ becomes massive, and the mass term is described by the superpotential $$W=\frac{\mu}{2}\sum_I(q_{I+1}-q_I)\Phi_I^2,
\label{massterm}$$ where $q_I=0$ for A-branes and $q_I=1$ for B-branes. The overall factor $\mu$ is related to the relative angle between A- and B-branes.
We also have bifundamental chiral multiplets $X_I$ and $Y_I$, which arise from open strings stretched between two intervals of D-branes divided by the $I$th fivebrane. (See Fig. \[xyfv.eps\].)
![Brane system and fields.[]{data-label="xyfv.eps"}](xyfv.eps)
These fields belong to the following representations of $U(N)_I\times U(N)_{I-1}$, where $U(N)_I$ is the gauge group associated with the vector multiplet $V_I$: $$X_I:(N,{\overline}N),\quad
Y_I:({\overline}N,N).$$ The kinetic terms of these bifundamental fields are $$\begin{aligned}
S_{XY}
&=&\int d^3x d^4\theta\sum_{I=1}^n\tr\left[
X_I^*e^V_IX_Ie^{-V_{I-1}}
+Y_Ie^{-V_I}Y^*_Ie^{V_{I-1}}
\right]
\nonumber\\
&=&\int d^3x
\sum_{I=1}^n\tr\left[
-D_I(|X_I|^2-|Y_I|^2-|X_{I+1}|^2+|Y_{I+1}|^2)
\right.
\nonumber\\&&
\left.
-(|X_I|^2+|Y_I|^2)(\sigma_I-\sigma_{I-1})^2
+|F^X_I|^2
+|F^Y_I|^2
\right]
+\cdots.\end{aligned}$$ In the component expression we show only the bosonic terms without derivatives. These bifundamental fields couple to the adjoint chiral multiplets through the superpotential $$W=\sum_{I=1}^n\tr\Phi_I(X_IY_I-Y_{I+1}X_{I+1}).
\label{n2term}$$
The difference between the RR-charges of the A- and B-branes generates Chern-Simons terms[@Kitao:1998mf; @Bergman:1999na]. The bosonic part of the ${\cal N}_{d=3}=2$ completion of the Chern-Simons terms is $$S_{\rm CS}
=\sum_{I=1}^n\frac{k_I}{2\pi}\int d^3x\tr\left[
\epsilon^{\mu\nu\rho}
\left(\frac{1}{2}A^I_\mu\partial_\nu A^I_\rho
+\frac{1}{3}A^I_\mu A^I_\nu A^I_\rho\right)
+\sigma_I D_I
\right],
\label{csterm}$$ where the Chern-Simons coupling $k_I$ is given by $$k_I=k(q_{I+1}-q_I).
\label{cscouplings}$$ We assume that $k$ is a positive integer. The Chern-Simons terms in (\[csterm\]) cause some of the vector multiplets to be massive. The masses $\sim k_Ig_I^2$ are proportional to the masses of adjoint chiral multiplets $\Phi_I$. We can promote the supersymmetry of this theory to ${\cal N}_{(d=3)}=3$ by matching the masses of $V_I$ and $\Phi_I$ by setting $\mu=k$.
In $3$-dimensional field theories the coupling constants $g_I$ have mass dimension $1/2$, and taking the low-energy limit is equivalent to taking the strong-coupling limit $g_I\rightarrow\infty$. This makes the masses of $V_I$ and $\Phi_I$ infinity unless $k_I=0$, and we can integrate out the massive adjoint chiral multiplets. After this, the superpotential becomes[^4] $$W=\sum_{q_I=q_{I+1}}\tr\Phi_I(X_IY_I-Y_{I+1}X_{I+1})
+\sum_{q_I\neq q_{I+1}}(q_{I+1}-q_I)\tr(X_IY_IY_{I+1}X_{I+1}).
\label{ellw}$$
Moduli space {#moduli.sec}
============
In this section we investigate the moduli space of the 3-dimensional field theory defined in the previous section. As we mentioned at the end of the previous section we need to take the strong-coupling limit $g_I\rightarrow\infty$ to obtain the conformal theory describing the low-energy limit of M2-branes. Although the dynamics in such a strong coupling region is highly nontrivial, we assume that the vacuum structure is not affected by quantum corrections, and we consider only the classical equations of motion derived from the action given in the previous section. In the strong-coupling limit, the kinetic terms (\[sv\]) and (\[sphi\]) vanish, and the fields $\phi_I$, the scalar components of $\Phi_I$, and $\sigma_I$ become auxiliary fields. The bifundamental chiral multiplets $X_I$ and $Y_I$ are still dynamical, and the moduli space is parameterized by the scalar components of these multiplets.
We are interested in the moduli space for a single M2-brane, and we set $N=1$. Furthermore, we here focus only on the Higgs branch, which describes a mobile M2-brane, and assume $$X_I,
Y_I\neq0.
\label{coulomb}$$
F-term conditions
-----------------
Let us first consider the F-term conditions derived from the superpotential (\[ellw\]). Because the superpotential is the same as the 4-dimensional elliptic model realized by the D4-NS5 brane system, the F-term conditions are also the same. Under the assumption (\[coulomb\]), the F-term conditions give the following solution: $$\Phi_{I\in A}=M_{I\in B}=u,\quad
\Phi_{I\in B}=M_{I\in A}=v,
\label{fcond}$$ where we define the mesonic operators as $M_I=X_IY_I$. $I\in A$ ($I\in B$) means that index $I$ is restricted to the values with $q_I=0$ ($q_I=1$).
Although not directly related to our model, it may be instructive to demonstrate how we can obtain a Calabi-Yau $3$-fold as the moduli space of a 4-dimensional elliptic model in the case of the D4-NS5 system. In this case two complex numbers $u$ and $v$ can be interpreted as the coordinates of the D4-brane along B- and A-branes, respectively. The 4-dimensional theory possesses $U(1)^{n-1}$ gauge symmetry. In addition to the mesonic operators $M_I$, we can construct the gauge-invariant baryonic operators $$x=\prod_{I=1}^nX_I,\quad
y=\prod_{I=1}^nY_I.
\label{baryon}$$ By definition, these gauge-invariant operators are related by $$xy=u^{n_A} v^{n_B}.$$ This algebraic equation defines a Calabi-Yau 3-fold, which is often called a generalized conifold. The toric diagram of this generalized conifold is shown in Fig. \[gc.eps\].
![Toric diagram of a generalized conifold.[]{data-label="gc.eps"}](gc.eps)
D-term conditions {#dterm.sec}
-----------------
In the strong-coupling limit $g_I\rightarrow\infty$, the vector multiplet $V_I$ includes two auxiliary fields $\sigma_I$ and $D_I$. The terms in the action including these auxiliary fields are $$\begin{aligned}
S&=&\sum_{I=1}^n\left[ k_I\sigma_ID_I
-D_I(|X_I|^2-|Y_I|^2-|X_{I+1}|^2+|Y_{I+1}|^2)
\right.
\nonumber\\&&
\left.
-(|X_I|^2+|Y_I|^2)(\sigma_I-\sigma_{I-1})^2
\right].
\label{daction}\end{aligned}$$ In this action, $D_I$ are Lagrange multipliers, and give the constraint $$k_I\sigma_I=|X_I|^2-|Y_I|^2-|X_{I+1}|^2+|Y_{I+1}|^2.
\label{Dconst}$$ If we substitute this into the action (\[daction\]), the first line vanishes and the potential becomes $$V=\sum_{I=1}^n(|X_I|^2+|Y_I|^2)(\sigma_I-\sigma_{I-1})^2.$$ Because of the assumption (\[coulomb\]), vacua are given by $\sigma_I=\sigma_{I-1}$. Namely, all $\sigma_I$ are the same. Let $\sigma$ be the common value of $\sigma_I$. Then the constraint (\[Dconst\]) becomes $$q_I\sigma-(|X_I|^2-|Y_I|^2)
=q_{I+1}\sigma-(|X_{I+1}|^2-|Y_{I+1}|^2).$$ This means that the left- and right-hand sides of this equation do not depend on the index $I$. Thus, we can write $$|X_I|^2-|Y_I|^2=q_I\sigma+c
\label{dterm}$$ with a constant $c$.
Although (\[dterm\]) is not the equation of motion of $D_I$, we can formally interpret it as an ordinary D-term condition associated with a certain symmetry. To rewrite (\[dterm\]) in the form of an ordinary D-term condition, let us define $U(1)$ transformation groups $G_I$ that act only on $X_I$ and $Y_I$ as $$G_I:
X_I\rightarrow e^{i\lambda_I}X_I,\quad
Y_I\rightarrow e^{-i\lambda_I}Y_I,$$ where $\lambda_I$ is a parameter of $G_I$. The groups $G_I$ are different from $U(1)_I$ defined in the previous section. The parameters $\alpha_I$ of $U(1)_I$ and $\lambda_I$ of $G_I$ are related by $$\lambda_I=\alpha_I-\alpha_{I-1}.
\label{lambdaalpha}$$ Although each $G_I$ is not a symmetry of the theory, it is convenient to describe symmetry groups as subgroups of $\prod_IG_I$. For example, the gauge symmetry $G=U(1)^{n-1}$, which does not include the diagonal $U(1)$ decoupling from the theory, is the subgroup of $\prod_IG_I$ that does not rotate the baryonic operators (\[baryon\]).
Let us rewrite (\[dterm\]) in the form of a D-term condition. Equation (\[dterm\]) is equivalent to the condition $$\sum_{I=1}^l\lambda_I(|X_I|^2-|Y_I|^2)=0,
\label{dtermc}$$ for arbitrary $\lambda_I$ satisfying the constraints $$\sum_{I=1}^n\lambda_I=\sum_{I=1}^nq_I\lambda_I=0.
\label{subg}$$ If we regard $\lambda_I$ as the parameters of $G_I$ transformations, the constraints (\[subg\]) imposed on $\lambda_I$ define a subgroup $H=U(1)^{n-2}$ of $\prod_I G_I$. Equation (\[dtermc\]) can be regarded as the D-term condition for $H$.
We emphasize that we do not claim at this point that the gauge symmetry of the theory is $H$ or that relation (\[dterm\]) is obtained as the equations of motion of auxiliary fields in the vector multiplets associated with $H$. We only claim that the vacuum condition (\[dterm\]) is similar to the D-term condition of a gauge theory with the gauge symmetry $H$. In the next subsection, however, we will show that $H$ indeed emerges as the unbroken continuous gauge symmetry.
It is convenient to define the subgroup $H$ in another way. Let us define the baryonic operators $$x_A=\prod_{I\in A}X_I,\quad
y_A=\prod_{I\in A}Y_I,\quad
x_B=\prod_{I\in B}X_I,\quad
y_B=\prod_{I\in B}Y_I.
\label{baryonicops}$$ The group $H$ can be defined as the subgroup of $\prod_IG_I$ that does not rotate these baryonic operators.
Gauge symmetry
--------------
To obtain the moduli space of a gauge theory, we need to remove unphysical degrees of freedom corresponding to gauge symmetries. In the case of Chern-Simons theories, we should carefully take account of symmetry breaking due to the existence of magnetic monopoles. Let us rewrite the abelian Chern-Simons terms in the form $$S_{\rm CS}
=-\frac{k}{2\pi}\sum_{I=1}^nq_I(A^I-A^{I-1})\wedge {\widetilde}F
+(\mbox{quadratic terms of $A^I-A^{I-1}$}),
\label{csdiag}$$ where ${\widetilde}F$ is the field strength of the diagonal $U(1)$ gauge field ${\widetilde}A=(1/n)(A^1+A^2+\cdots+A^n)$. Equation (\[csdiag\]) is obtained by substituting $$A^I={\widetilde}A+(\mbox{linear combination of $A^I-A^{I-1}$})$$ into the Chern-Simons term in (\[csterm\]). The quadratic term of ${\widetilde}A$ vanishes because $\sum_Ik_I=0$. Because the diagonal gauge field ${\widetilde}A$ appears only in the first term of (\[csdiag\]), we can dualize it by adding the term $$\frac{1}{2\pi}\int d\tau\wedge{\widetilde}F,$$ and treating ${\widetilde}F$ as an unconstrained field. The equation of motion of ${\widetilde}F$ gives $$\sum_{I=1}^n k_IA_I=d\tau.$$ Upon the gauge transformation $\delta A_I=d\alpha_I$, the scalar field $\tau$ is transformed as $$\delta \tau=\sum_{I=1}^n k_I\alpha_I.$$ Let us assume that the period of $\tau$ is $2\pi$. This implies that the flux $\oint {\widetilde}F$ is quantized by $$\int{\widetilde}F\in 2\pi{\mathbb Z}.
\label{feqf2}$$ Although we could not show this flux quantization on the field-theory side, we will later show that the moduli space obtained by assuming (\[feqf2\]) coincides with that obtained from the brane configuration by the T-duality and M-theory lift. If we adopt this assumption, the gauge fixing $\tau=0$ partially breaks the gauge symmetry and imposes the following constraint on the parameters $\lambda_I$ and $\alpha_I$: $$\sum_{I=1}^n k_I\alpha_I=k\sum_{I=1}^nq_I\lambda_I\in 2\pi{\mathbb Z}.
\label{ubk}$$ (In the first equality we used (\[cscouplings\]) and (\[lambdaalpha\]).)
Let us first focus on the continuous subgroup. It is generated by parameters satisfying $$\sum_{I=1}^n\lambda_I=\sum_{I=1}^nq_I\lambda_I=0.
\label{sumqlambda}$$ The group defined by (\[sumqlambda\]) is simply group $H$ defined in §\[dterm.sec\]. Because of the emergence of the same group $H$ both in the equations of motion of auxiliary fields and in the unbroken gauge symmetry, we can obtain the moduli space as the coset ${\cal M}/H_{\mathbb C}$ or its orbifold, where ${\cal M}$ is the complex manifold defined by the F-term conditions and $H_{\mathbb C}$ is the complexification of the group $H$. This guarantees that the moduli space is a complex manifold.
In addition to $H$, the group defined by (\[ubk\]) includes the discrete symmetry, which rotates the baryonic operators in (\[baryonicops\]) as $$x_A\rightarrow e^{\frac{2\pi i}{k}}x_A,\quad
y_A\rightarrow e^{-\frac{2\pi i}{k}}y_A,\quad
x_B\rightarrow e^{-\frac{2\pi i}{k}}x_B,\quad
y_B\rightarrow e^{\frac{2\pi i}{k}}y_B.
\label{discrete}$$
Moduli space {#moduli.ssec}
------------
Let us determine the moduli space. We first consider the $k=1$ case. In this case, the discrete gauge symmetry (\[discrete\]) becomes trivial, and we have the gauge-invariant operators $$u,\quad
v,\quad
x_A,\quad
y_A,\quad
x_B,\quad
y_B.
\label{giops}$$ By definition, these operators satisfy the following equations: $$x_Ay_A=u^{n_A},\quad
x_By_B=v^{n_B}.
\label{defeq}$$ These equations define the orbifold ${\mathbb C}^2/{\mathbb Z}_{n_A}\times{\mathbb C}^2/{\mathbb Z}_{n_B}$. Actually, the relation (\[defeq\]) can be solved as $$x_A=z_1^{n_A},\quad
y_A=z_2^{n_A},\quad
u=z_1z_2,\quad
x_B=z_3^{n_B},\quad
y_B=z_4^{n_B},\quad
v=z_3z_4.
\label{zdef}$$ We can identify $z_i$ as the coordinates of ${\mathbb C}^4$, the covering space of the orbifold. None of the variables in (\[giops\]) are changed by the transformations $$(z_1,z_2,z_3,z_4)\rightarrow
(e^{2\pi i/n_A}z_1,
e^{-2\pi i/n_A}z_2,
z_3,z_4)
\label{zna}$$ and $$(z_1,z_2,z_3,z_4)\rightarrow
(z_1,z_2,
e^{2\pi i/n_B}z_3,
e^{-2\pi i/n_B}z_4).
\label{znb}$$ Points in ${\mathbb C}^4$ mapped by these transformations should be identified with each other, and this identification defines the above orbifold.
If $n_A=n_B$, the moduli space agrees with the result in Ref. , in which alternate A- and B-branes are considered. It is interesting that the moduli space does not depend on the order of the two kinds of fivebranes.
Next, let us consider the case when $k>1$. In this case, we should take account of the discrete gauge transformation (\[discrete\]). The transformation of $z_i$ reproducing (\[discrete\]) is $$(z_1,z_2,z_3,z_4)\rightarrow
(e^{2\pi i/kn_A}z_1,e^{-2\pi i/kn_A}z_2,e^{-2\pi i/kn_B}z_3,e^{2\pi i/kn_B}z_4).
\label{znc}$$ The three transformations (\[zna\]), (\[znb\]), and (\[znc\]) generate a discrete subgroup of $U(1)^2$ with $kn_An_B$ elements. Let $\Gamma$ be this discrete group. The moduli space for general $k$ is the abelian orbifold ${\mathbb C}^4/\Gamma$.
M-theory dual {#dual.sec}
=============
In the previous section, we obtained the 4-dimensional orbifold ${\mathbb C}^4/\Gamma$ as the Higgs branch of the moduli space. The purpose of this section is to reproduce the same orbifold by the T-duality transformation and the M-theory lift from the D3-fivebrane system in type IIB string theory.
For simplicity, we first consider a system in which the $(k,1)$5-branes are replaced by D5-branes. After determining the mapping from type IIB string theory to M-theory for NS5- and D5-branes, the dual object for the bound state of these two kinds of branes is easily obtained by superposing the objects for NS5- and D5-branes. Although the tilted angle of the branes should be appropriately chosen according to the charges of branes to preserve supersymmetry, we do not do this because the toric data do not change upon continuous deformations of the manifold and because we can determine the toric data of the dual geometry by using only the topological information. We start from the brane configuration for type IIB string theory in Table \[d5ns5.tbl\].
0 1 2 3 4 5 6 7 8 9
----- --------- --------- --------- --------- --------- --------- --------- --------- --------- ---------
D3 $\circ$ $\circ$ $\circ$ $\circ$
D5 $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ $\circ$
NS5 $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ $\circ$
: Brane configuration in type IIB string theory[]{data-label="d5ns5.tbl"}
Direction $9$ is compactified on ${\mathbb S}^1$. We replaced the $(k,1)$5-brane with the D5-brane and use a coordinate system in which the D5-brane is spread along 012678. In general, the $(k,1)$5-branes are not perpendicular to the NS5-brane, thus we use slanted coordinates.
We first rearrange the coordinates in $4578$ space by using the Hopf fibration. We define $r_a$ ($a=1,2,3$) by $$r_a=u^\dagger \sigma_au,\quad
u=\left(\begin{array}{c}x^4+ix^5\\x^7+ix^8\end{array}\right),$$ and we let $\psi$ be the coordinate of the ${\mathbb S}^1$ fiber. Then the NS5 and D5 worldvolumes are on the positive and negative parts of the $r_3$ axis, respectively, in the $r_a$ space. See Table \[d5ns52.tbl\]. The $\psi$ cycle shrinks at the center of the $r_a$ space, which is shown in the table as “KKM”. “s” in the table represents the shrinking cycle.
0 1 2 3 6 $r_3$ $r_1$ $r_2$ $\psi$ 9
----- --------- --------- --------- --------- --------- ------- ------- ------- --------- ---------
D3 $\circ$ $\circ$ $\circ$ $\circ$
D5 $\circ$ $\circ$ $\circ$ $\circ$ $-$ $\circ$
NS5 $\circ$ $\circ$ $\circ$ $\circ$ $+$ $\circ$
KKM $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ s $\circ$
: The same configuration as Table \[d5ns5.tbl\] with a different coordinate system. “s” represents the shrinking cycle and $+$ and $-$ mean that the branes are spread along the positive or negative part of the axis, respectively.[]{data-label="d5ns52.tbl"}
Let us perform the T-duality transformation along direction $9$, and lift the configuration into M-theory.
0 1 2 3 6 $r_3$ $r_1$ $r_2$ $\psi$ 9 M
----------------------- --------- --------- --------- --------- --------- ------- ------- ------- --------- --------- ---------
(D3$\rightarrow$)M2 $\circ$ $\circ$ $\circ$
(D5$\rightarrow$)KKM $\circ$ $\circ$ $\circ$ $\circ$ $-$ $\circ$ s $\circ$
(NS5$\rightarrow$)KKM $\circ$ $\circ$ $\circ$ $\circ$ $+$ $\circ$ $\circ$ s
KKM $\circ$ $\circ$ $\circ$ $\circ$ $\circ$ s $\circ$ $\circ$
: M-theory dual of the brane configuration.[]{data-label="d5ns53.tbl"}
The D3-branes are mapped to M2-branes as shown in Table \[d5ns53.tbl\]. A single NS5-brane and a single D5-brane become KKM-branes associated with the $(1,0,0)$ and $(0,1,0)$ cycles, respectively, where the first, second, and last components correspond to the $M$, $9$, and $\psi$ coordinates, respectively. If we start with a $(k,1)$5-brane, which is the bound state of $k$ D5-branes and one NS5-brane, we obtain a single KKM-brane with $(0,1,k)$ cycle shrinking.
In addition to these, we have one more KKM-brane, which originates from the special choice of the coordinates. The existence of the other KKM-branes make the shrinking cycle of the last KKM-brane ambiguous, and only the last component of the shrinking cycle has a definite value of $1$. The intersection with other branes changes the shrinking cycle, and the cycle should be determined according to the “charge conservation” of the KKM branes. See Fig. \[c4.eps\](a).
![(a) M-theory dual of the D3-fivebrane system in $36r_3$ space. This can be regarded as a webdiagram of the toric geometry. The corresponding toric diagram is shown in (b).[]{data-label="c4.eps"}](c4.eps)
This system of KKM-branes in $36r_3$ space is simply a webdiagram describing a 4-dimensional toric manifold. We can easily obtain the toric diagram as a dual graph of the webdiagram. (Fig. \[c4.eps\](b)) The toric variety described by this diagram is in fact the orbifold we obtained in §\[moduli.ssec\], as we show in the rest of this section.
The structure of a toric variety is mostly determined by the toric data, which are a set of generators of shrinking cycles. The generators are usually represented as vectors $\vec v_i$ in the lattice associated with the toric fiber. The toric data of ${\mathbb C}^4$ are given by $\vec v_i=\vec e_i$ ($i=1,2,3,4$), where $\vec e_i$ are the unit vectors in the 4-dimensional lattice. $$\vec e_1=(1,0,0,0),\quad
\vec e_2=(0,1,0,0),\quad
\vec e_3=(0,0,1,0),\quad
\vec e_4=(0,0,0,1).$$ The orbifolding of a toric variety is realized by refining the lattice by adding new generators. In the case of the orbifold defined by (\[zna\])–(\[znc\]), we add three generators $$\begin{aligned}
\vec e_5&=&\left(\frac{1}{n_A},-\frac{1}{n_A},0,0\right),\nonumber\\
\vec e_6&=&\left(0,0,\frac{1}{n_B},-\frac{1}{n_B}\right),\nonumber\\
\vec e_7&=&\left(\frac{1}{n_Ak},-\frac{1}{n_Ak},-\frac{1}{n_Bk},\frac{1}{n_Bk}\right).\end{aligned}$$ Of course, the seven vectors $\vec e_1,\ldots,\vec e_7$ are not linearly independent. Let us choose the following linearly independent basis: $$\begin{aligned}
\vec f_1&=&-\vec e_5=\left(-\frac{1}{n_A},\frac{1}{n_A},0,0\right),\nonumber\\
\vec f_2&=&\vec e_7=\left(\frac{1}{n_Ak},-\frac{1}{n_Ak},-\frac{1}{n_Bk},\frac{1}{n_Bk}\right),\nonumber\\
\vec f_3&=&\vec e_3-\vec e_1=(-1,0,1,0),\nonumber\\
\vec f_4&=&\vec e_1=(1,0,0,0).
\label{basisf}\end{aligned}$$ Using this basis, the toric data become $$\vec v_1=[0,0,0,1]_{\vec f},\quad
\vec v_2=[n_A,0,0,1]_{\vec f},\quad
\vec v_3=[0,0,1,1]_{\vec f},\quad
\vec v_4=[n_B,kn_B,1,1]_{\vec f},$$ where $[a_1,\cdots,a_4]_{\vec f}=\sum_ia_i\vec f_i$. We have chosen basis (\[basisf\]) so that the toric data become the standard form in which the last components of the vectors are $1$. We can draw the toric diagram using the first three components of these vectors $\vec v_i$, which coincides with that in Fig. \[c4.eps\](b).
Further generalization {#more.sec}
======================
Up to now we have considered a brane system with two kinds of fivebranes. It is also possible to introduce more than two kinds of fivebranes. To represent the types of branes we used $q_I=0$ and $1$. In this section we allow $q_I$ to be an arbitrary integer. In this case, we do not need to introduce the coefficient $k$ in (\[cscouplings\]) and we set $k=1$. This means that the $I$th fivebrane is a $(q_I,1)$5-brane, and the Chern-Simons couplings are given by $$\label{kInonzero}
k_I=q_{I+1}-q_I.$$ For simplicity we assume that the Chern-Simons couplings do not vanish. This implies that all the adjoint chiral multiplets $\Phi_I$ and the vector multiplets $V_I$ become massive. It is easy to show that even if some of the $k_I$ vanish we obtain the same moduli space as derived below.
By integrating out $\Phi_I$, we obtain the superpotential $$W=-\sum_{I=1}^n\frac{1}{2(q_{I+1}-q_I)}(X_IY_I-Y_{I+1}X_{I+1})^2.$$ (When we obtained the superpotential (\[ellw\]) we used $q_I=0$ and $1$, although we cannot use it here.) From the assumption (\[coulomb\]), the F-term conditions for $X_I$ and $Y_I$ give $$\frac{X_{I+1}Y_{I+1}-X_IY_I}{q_{I+1}-q_I}
=\frac{X_IY_I-X_{I-1}Y_{I-1}}{q_I-q_{I-1}},$$ and this is solved as $$\label{XIYIaqIb}
X_IY_I=a+q_Ib,$$ where $a$ and $b$ are arbitrary complex numbers.
The equations of motion of the auxiliary fields $\sigma_I$ are solved by (\[dtermc\]) with the parameters $\lambda_I$ constrained by (\[subg\]). The constraint (\[subg\]) defines group $H$ and Eq. (\[dtermc\]) has the form of the D-term condition associated with group $H$. This group is identical to the continuous part of the unbroken gauge symmetry, which is given by (\[ubk\]) with $k=1$. The constraints imposed on the parameters are $$\sum_{I=1}^n\lambda_I=0,\quad
\sum_{I=1}^nq_I\lambda_I\in\frac{1}{2\pi}{\mathbb Z}.
\label{hp}$$
The following “baryonic operators” are invariant under gauge symmetries satisfying (\[hp\]), $$x=\prod_{I=1}^nX_I,\quad
x_A=\prod_{I=1}^nX_I^{q_{\max}-q_I},\quad
x_B=\prod_{I=1}^nX_I^{q_I-q_{\min}},$$ $$y=\prod_{I=1}^nY_I,\quad
y_A=\prod_{I=1}^nY_I^{q_{\max}-q_I},\quad
y_B=\prod_{I=1}^nY_I^{q_I-q_{\min}},$$ where $q_{\min}$ and $q_{\max}$ are the minimum and maximum of $q_I$, respectively. Any gauge-invariant monomial of $X_I$ and $Y_I$ can be represented as a monomial of the mesonic operators $M_I=X_IY_I$ and these baryonic operators.
We now have the following $8$ gauge-invariant variables: $$a, b, x, x_A, x_B, y, y_A, y_B.$$ These operators satisfy $$\label{x0y0}
xy=\prod_{I=1}^n(a+q_Ib),\quad
x_Ay_A=\prod_{I=1}^n(a+q_Ib)^{q_{\max}-q_I},\quad
x_By_B=\prod_{I=1}^n(a+q_Ib)^{q_I-q_{\min}},$$ $$\label{x0x1x}
x_Ax_B=x^{q_{\max}-q_{\min}},\quad y_Ay_B=y^{q_{\max}-q_{\min}}.$$ Because the first equation in (\[x0y0\]) is not independent of the other two due to relations (\[x0x1x\]), these relations decrease the number of independent degrees of freedom by four, and the moduli space becomes a complex 4-dimensional space. If the $q_I$ take more than two different values, the equations in (\[x0y0\]) are not binary relations of monomials, and the moduli space is nontoric.
Discussion
==========
In this paper we studied the Higgs branch of Maxwell-Chern-Simons theories described by circular quiver diagrams. We first considered the model realized by the D3-NS5-$(k,1)$5-brane system with an arbitrary number of fivebranes, and showed that the moduli space is the orbifold ${\mathbb C}^4/\Gamma$, where $\Gamma$ is the discrete group generated by (\[zna\])–(\[znc\]). When we determined the orbifold group $\Gamma$, we made an assumption for the flux quantization (\[feqf2\]). Our result was confirmed by comparing it to the M-theory dual of the brane configuration. We also discussed the model realized by a brane system with more than two kinds of fivebranes, and we obtained a 4-dimensional nontoric moduli space.
Note that our result is different from that expected from the orbifold method. In general, a quiver gauge theory obtained by the orbifold method introduced in Ref. includes $n$ copies of fields of the parent theory, where $n$ is the order of the orbifolding group. Such analysis is carried out in Ref. for the model proposed in Ref. , and a theory was obtained in which the number of $U(N)$ factors in the gauge group is proportional to the order of the corresponding orbifolding group. On the other hand, our construction gives $n=n_A+n_B$ copies of fields, whereas the order of the orbifolding group is proportional to the product $n_An_B$. Because the brane construction and orbifold method are both important methods for constructing field theories in string theory, it is very important to understand the reason for this discrepancy.
The moduli spaces we obtained in this paper are completely determined by the number of fivebranes of each type. In the case of the brane system with A- and B-branes, the moduli space depends only on the level $k$ and the numbers of fivebranes $n_A$ and $n_B$. The orders of A- and B-branes along the compact direction do not affect the moduli space. This is also the case in the brane system with more than two types of fivebranes discussed in §\[more.sec\]. This may be interpreted as a duality similar to the Seiberg duality in the 4-dimensional ${\cal N}_{(d=4)}=1$ supersymmetric gauge theories[@Seiberg:1994pq]. In the 4-dimensional case, this duality can be understood as the exchange of the two types of branes[@Elitzur:1997fh]. In the brane system we consider in this paper, the exchange of A- and B-branes generates new D3-branes by the Hanany-Witten effect[@Hanany:1996ie]. It will be an interesting problem to clarify the relation among theories realized by brane systems with different orders of fivebranes.
The models we considered in this paper are expected to flow to conformal fixed points in the low-energy limit, and thus the AdS/CFT correspondence is expected to be useful for studying low-energy dynamics. When we discuss the AdS/CFT correspondence, it is necessary to establish the correspondence between geometries and the UV description of quiver gauge theories. In the case of 4-dimensional ${\cal N}=1$ superconformal theories, brane tiling[@Hanany:2005ve; @Franco:2005rj; @Franco:2005sm] is a convenient tool for finding this correspondence in the toric Calabi-Yau case. Although the generalization of brane tiling to 3-dimensional gauge theories has been proposed[@Lee:2006hw; @Lee:2007kv; @Kim:2007ic], much less is known about the duality in the 4-dimensional case due to the small number of examples. We hope that the examples in this paper will be useful for investigating the general relation between four-manifolds and quiver Chern-Simons gauge theories.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank T. Eguchi for valuable discussions. We would also like to acknowledge the helpful comments of K. Ohta. Y. I. is partially supported by a Grant-in-Aid for Young Scientists (B) (\#19740122) from the Japan Ministry of Education, Culture, Sports, Science and Technology.
[99]{}
J. Bagger and N. Lambert, ; hep-th/0611108. J. Bagger and N. Lambert, ; arXiv:0711.0955. J. Bagger and N. Lambert, ; arXiv:0712.3738.
A. Gustavsson, arXiv:0709.1260.
A. Gustavsson, ; arXiv:0802.3456.
P. M. Ho, R. C. Hou and Y. Matsuo, arXiv:0804.2110.
G. Papadopoulos, ; arXiv:0804.2662. J. P. Gauntlett and J. B. Gutowski, arXiv:0804.3078. M. Van Raamsdonk, ; arXiv:0803.3803.
N. Lambert and D. Tong, arXiv:0804.1114. J. Distler, S. Mukhi, C. Papageorgakis and M. Van Raamsdonk, ; arXiv:0804.1256. J. Gomis, G. Milanesi and J. G. Russo, arXiv:0805.1012. S. Benvenuti, D. Rodriguez-Gomez, E. Tonni and H. Verlinde, arXiv:0805.1087. P. M. Ho, Y. Imamura and Y. Matsuo, arXiv:0805.1202.
Y. Honma, S. Iso, Y. Sumitomo and S. Zhang, arXiv:0805.1895.
M. A. Bandres, A. E. Lipstein and J. H. Schwarz, arXiv:0806.0054. J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk and H. Verlinde, arXiv:0806.0738.
B. Ezhuthachan, S. Mukhi and C. Papageorgakis, arXiv:0806.1639.
S. Mukhi and C. Papageorgakis, ; arXiv:0803.3218. D. Gaiotto and E. Witten, arXiv:0804.2907.
K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, arXiv:0805.3662. S. Lee, ; hep-th/0610204. S. Lee, S. Lee and J. Park, ; hep-th/0702120. S. Kim, S. Lee, S. Lee and J. Park, ; arXiv:0705.3540. O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, arXiv:0806.1218. I. R. Klebanov and E. Witten, ; hep-th/9807080. A. M. Uranga, ; hep-th/9811004. R. von Unge, ; hep-th/9901091.
T. Kitao, K. Ohta and N. Ohta, ; hep-th/9808111. O. Bergman, A. Hanany, A. Karch and B. Kol, ; hep-th/9908075.
M. R. Douglas and G. W. Moore, hep-th/9603167. M. Benna, I. Klebanov, T. Klose and M. Smedback, arXiv:0806.1519. N. Seiberg, ; hep-th/9411149.
S. Elitzur, A. Giveon and D. Kutasov, ; hep-th/9702014.
A. Hanany and E. Witten, ; hep-th/9611230.
A. Hanany and K. D. Kennaway, hep-th/0503149. S. Franco, A. Hanany, K. D. Kennaway, D. Vegh and B. Wecht, ; hep-th/0504110. S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, ; hep-th/0505211.
[^1]: E-mail: imamura@hep-th.phys.s.u-tokyo.ac.jp
[^2]: E-mail: kimura@hep-th.phys.s.u-tokyo.ac.jp
[^3]: The possibility that the existence of magnetic monopoles causes a discrete indentification in the orbifold is also mentioned.
[^4]: We shift the field $\Phi_I$ by $(q_I-1/2)(X_IY_I+Y_{I+1}X_{I+1})$ and set $\mu=1$ to simplify the equations.
|
---
abstract: 'The distance dependence of the probability of observing two photons in a waveguide is investigated. The Glauber correlation functions of the entangled photons in waveguides are considered and the spatial and temporal dependence of the correlation functions is evaluated. We derive upper bounds to the distance dependence of the probability of observing two photons. These inequalities should be possible to observe in experiments.'
author:
- |
A. Khrennikov, B. Nilsson, S. Nordebo\
International Centre for Mathematical Modelling\
in Physics and Cognitive Sciences\
School of Computer Science, Physics and Mathematics\
Linnæus University, SE-3159 Växjö, Sweden\
I.V. Volovich\
Steklov Mathematical Institute\
Russian Academy of Sciences\
Gubkin St. 8, 117966, GSP-1, Moscow, Russia
title: Bounds for the Distance Dependence of Correlation Functions of Entangled Photons in Waveguides
---
Introduction
============
The transmission of light in waveguides, in particular its quantum properties, is a topic of great interest in optics. Investigations of quantum correlations and entanglement among photons have been in focus of in the foundations of quantum theory and its applications to quantum information science and metrology.
With the emergence of quantum communication links over long distances [@LSS; @LHB; @UTS; @HVL], there is a need for a detailed study of the dependence of correlations and entanglement among photons on distance.
Since direct interactions between photons in free space are extremely weak, generation of correlated photons generally requires nonlinear media such as the parametric down conversion. Recently, studies of two-photon scattering from a two-level system inside a one-dimensional waveguide have reported various features of photon correlation [@KHT; @SF; @SF2; @Roy]. In particular a formal scattering theory to study multi-photon transport in a waveguide was employed [@SS]. We also mention a general model describing nonlinear effects in propagation to and from the system in the quantum state, based on one-dimensinal model of field-atom interaction [@HOFF], see even [@HOFF1]. It describes spatiotemporal quantum coherence for the case of spontaneous emission from a single excited atom. In [@KHT] this model was applied to the two-photon input wave packets. This field of research, spatiotemporal behavior of correlations of two photons propagating in nonlinear media, is closely related to studies on nonlinear response of a single atom to an input of two photons, e.g., from a single photon source [@Yam]. This response can be observed in the correlations between the two output photons. Here it is also very important to understand spatiotemporal dependence of correlations.
There are thus sound reasons to study how correlation functions of entangled photons in hollow waveguides behave in space and time. The physical mechanism to model is dispersion that spreads pulses in space and time causing attenuation with distance. The spreading limits also the bit rate for a given waveguide length because of mixing of pulses. Repeaters can be used at some length intervals, which cause higher costs and problems with preserving the quantum state through the repeater. Information on space and time properties of the correlation functions is thus of engineering interest. Such information is provided in the current paper by modelling the effect of modal dispersion.
The study is also a preparation for more elaborate models, including material dispersion in hollow waveguides and fibres, cf. e.g. [@BB], [@BB1]. To describe the situation, a brief background is presented [@Agrawal2002] on optic fibres. For a fibre, it is possible to purify the material to the extent that losses from scattering from the impurities can be neglected in some wavelength bands. Examples of such bands are the 1.3 $\mu$m and 1.55 $\mu$m bands. The dispersion can be reduced to a negligible level for a certain wavelength within such a band, and a low dispersion can then be reached for a narrow wavelength band. Such a reduction of dispersion can be reached [@Agrawal2002] with a dispersion compensator which is a special fibre with tuned length, after the transmission fibre, having the opposite dispersion to the transmission fibre. Another method [@Agrawal2002] is dispersion shift, i.e., a choice of the transversal fibre dimensions. The chromatic dispersion, defined as the combination of modal and material dispersion, can be made to vanish for both methods at the design wavelength.
It is worth mentioning the existence of repeaters for classical optic fibres using optical rather than previously used electrical methods [@Kartalopoulos2002] and fibre switches that preserve the quantum state of the photon [@HallAltepeterKuma2010].
We shall study the asymptotic behaviour of the Glauber correlation functions for the entangled states of two photons in waveguides and show their vanishing for large distances. We estimate the rate of decrease of correlations and present upper bounds for the correlation functions.
To estimate the correlation function we shall use some results on the properties of solutions of the (1+1)-dimensional Klein-Gordon equation analogously used in the Haag-Ruelle scattering theory.
We prove that the probability density $P(z_1,t_1,z_2,t_2)$ observing one photon at point $z_1$ along the waveguide at time $t_1$ and another photon at point $z_2$ at time $t_2$ satisfies the following inequality $$\label{bound}
P(z_1,t_1,z_2,t_2)\leq\frac{C}{(t_0+|t_1|)(t_0+|t_2|)}$$ for some $t_0$ and all $z_1,t_1,z_2,t_2$.
The bound (\[bound\]) and the bounds (\[K-G222\]), (\[K-G223\]) (see below) should be possible to observe in experiments.
Note that the decreasing of the correlations for entangled states in empty space was found in [@Vol], see also discussion in [@Vol1], [@OV]. In this paper we have considered the waveguides and found the universal bound for the correlations, see also [@OLD] for preliminary considerations.
The photon probability density
==============================
Let $E_j(\bm{r},t)$ be the $j$-th component $(j=1,2,3)$ of the electric field operator at the space time point ${\bf r},t$. The operator can be written as the sum of the positive and negative frequency parts: $E_j({\bm r},t)=E_j^{(+)}({\bm r},t)+ E_j^{(-)}({\bf r},t).
$ The probabilities of photo detection are given by Glauber‘s formulas, [@MW]. In particular, the probability that a state $\psi$ of the radiation field will lead to the detection at time $t$ of a photon with the polarization along the direction $j$ by a detector atom placed at point ${\bm r}$ is proportional to the first-order correlation function $P_{\psi}({\bm r},t,j)=\langle\psi|E_j^{(-)}({\bm r},t)
E_j^{(+)}({\bm r},t)|\psi\rangle.
$ The joint probability of observing one photo ionization with polarization $j_{1} $ at point ${\bm r}_{1}$ at time between $t_{1}$ and $t_{1} +dt_{1}$ and another one with polarization $j_{2}$ at point ${\bm r}_{2}$ between $t_{2}$ and $t_{2} +dt_{2}$ with $t_{1}\leq t_{2}$ is proportional to $P_{\psi}({\bm r}_{1},t_{1},j_{1};{\bm r}_{2},t_{2},j_{2})dt_{1}dt_{2}$, where the second-order correlation function is $$\label{i3}P_{\psi}({\bm r}_{1},t_{1},j_{1};{\bm r}_{2},t_{2},j_{2})
=\langle\psi|E_{j_{1}}^{(-)}({\bm r}_{1},t_{1})
E_{j_{2}}^{(-)}({\bm r}_{2}%
,t_{2})
E_{j_{2}}^{(+)}({\bm r}_{2},t_{2})
E_{j_{1}}^{(+)} ({\bm r}_{1}%
,t_{1})|\psi\rangle$$
The waveguide is a hollow conducting cylindrical tube $\Gamma\subset \mathbb{R}^{3}$ along the $z$ axis with boundary surface $S$ and with a cross section $\Omega$ with the bounding curve $\partial\Omega$ in the $xy$-plane. It will be assumed that the walls have infinite conductivity. Appropriate boundary conditions are posed: $E_{t}|_{S}=0,~~H_{n}|_{S}=0,
$ where $E_{t}$ is the component of electric field ${\bm E}$ tangential to the boundary of the waveguide and $H_{n}$ is the component of magnetic field ${\bm H}$ normal to the boundary.
It is well known that in the interior of the waveguide the solutions of the Maxwell equations without sources can be divided into two sets of solutions, the so called $TM$ modes with $H_{z}=0$ and the $TE$ modes with $E_{z}=0$ [@LL; @Col]. A general solution is a linear combination of the $TM$ and $TE$ modes.
The general solution of the Maxwell equations in the waveguide can be written as follows [@Kri]. Let $\varphi_{n\nu}(z,t)$ be any function satisfying the Klein-Gordon equation $$\label{GS1}
(\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}
{\partial z^{2}%
}+m_{n\nu}^{2})\varphi_{n\nu}(z,t)=0.%$$ Here $n=1,2,...$ and $\nu=TM$ or $TE$. We define for $\nu=TM$ $$\label{GS2}
{\bm E}_{n\nu}({\bm r},t)={\bm e}_{n\nu}(x,y)m_{n\nu}^{-1}%
\frac{\partial}{\partial z}\varphi_{n\nu}(z,t)$$$$+{\bm n}_{z}m_{n\nu}v_{n}%
(x,y)\varphi_{n\nu}(z,t),$$$${\bm H}_{n\nu}({\bm r},t)=-{\bm h}_{n\nu}(x,y)m_{n\nu}^{-1}%
\frac{\partial}{\partial t}\varphi_{n\nu}(z,t)$$ where $v_n$ is the solution of the eigenvalue problem
$$\label{s4}(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}} +m_{n}^{2})v_{n}(x,y)=0,~~~~(x,y)\in\Omega,$$
$$v_{n}|_{\partial\Omega}=0$$ with the properties $$\label{s5}\int_{\Omega}v_{n}(x,y)v_{n^{\prime}}(x,y)dxdy=\delta_{nn^{\prime}},$$ $$\label{s6}\sum_{n}v_{n}(x,y)v_{n}(x^{\prime},y^{\prime})=\delta(x-x^{\prime})
\delta(y-y^{\prime}),$$ where $m_{n}^{2}>0$ are the eigenvalues.
It is defined for $\nu=TM$ $$\bm{e}_{n\nu}(x,y)=\nabla_{T}v_{n}(x,y), \label{g5}%$$$$\bm{h}_{n\nu}(x,y)=\bm{n}_{z}\times\nabla_{T}v_{n}(x,y)=\bm{n}_{z}%
\times\bm{e}_{n\nu}(x,y).$$ ${\bm E}_{n\nu}({\bm r},t)$ and ${\bm H}_{n\nu}({\bm r},t)$ are defined in an analogous manner [@OLD] for $\nu=TE$.
The general solution of the Maxwell equations in the waveguide can now be written in the form $$\label{g8}{\bm E}({\bm r},t) =\sum_{n\nu}{\bm E}_{n\nu}%
({\bm r},t),
{\bm H}({\bm r},t) =\sqrt{\frac{\epsilon_0}{\mu_0}}
\sum_{n\nu}{\bm H}_{n\nu}({\bm r},t).$$
We write the solution of the Klein-Gordon Eq. (\[GS1\]) in the form $$\label{s6a}\varphi_{n} (z,t)=\int\frac{d k}{2\sqrt{2\pi\omega_{n}(k)}}%
(a^{+}_{n}(k) e^{i\omega_{n}(k) t-ikz}$$ $$+a_{n}(k)
e^{-i\omega_{n}(k) t+ikz}),$$ where $\omega_{n}(k)=\sqrt{k^{2}+m_{n}^{2}},
$ and quantize it by taking $a_{n}(k), a_{n}^{+}(k)$ as the annihilation and creation operators.
Now the quantum electromagnetic field in the waveguide is reduced to a set of massive (1+1)-dimensional Klein-Gordon fields. Let us consider one of the modes. We define a one particle state $$\label{onep}
|\psi_1\rangle=\int g(k)a_{k}^{\dag}|0\rangle,$$ where $|0\rangle$ is the Fock vacuum. The probability density to detect the photon at the point $z$ along the waveguide at time $t$ is proportional to $$\label{pro1}
P(z,t)=\langle \psi_1|\varphi^{(-)}(z,t)
\varphi^{(+)}(z,t)
|\psi_1\rangle,$$ where $$\label{fie}
\varphi^{(+)} (z,t)=
\frac{1}{(2\pi)^{1/2}}\int_{\mathbb{R}}
\frac{dk}{\sqrt{2\omega_k}}
a_ke^{-i\omega_kt+ikz},\varphi^{(-)}(z,t)=h.c.,$$ and $\omega_k=\sqrt{k^2+m^2}$,$m>0$.
The expression (\[pro1\]) can be written as $$\label{pros12}
P(z,t)=|A(z,t)|^2,$$ where $$\label{pros13}
A(z,t)=\langle 0|\varphi^{(+)}(z,t)
|\psi_1\rangle=
\int
dk
g(k)\frac{e^{-i\omega_{k}t+ikz}}
{2\sqrt{2\pi\omega_{k}}}.$$ The function $A(z,t)$ is a solution of the Klein-Gordon equation. Let us consider the question of how the solution behaves in the limit of large $t$ in a frame of reference moving with a constant velocity $v\leq V<1$. In this frame, corresponding to the time $x(1-v)/v$ after the wave front at $t=x$, the field is given by $$\label{pros14}
A(vt,t)=
\int dk
g(k)\frac{e^{-it(\omega_{k}-kv)}}
{2\sqrt{2\pi\omega_{k}}}.$$ For sufficiently smooth function $g(k)$ one can use the stationary phase method to get $$\label{pros15}
A(vt,t)=
g(k_0)\frac{e^{-it(\omega_{k_0}-k_0v)-i\pi/4}}
{2\sqrt{2\pi\omega_{k_0}}}\sqrt{\frac{2\pi}{t\omega^{''}(k_0)}}
+O(\frac{1}{t}).$$ Here $k_0$ is the solution of the equation $\omega^{'}(k)=v$, i.e. $k_0=mv/ \sqrt{1-v^2}$ and one has $\omega^{''}(k_0)>0$. For the probability $P(z,t)$ we obtain $$\label{pros17}
P(vt,t)=
\frac{1}{t}\frac{|g(k_0)|^2}
{4\omega_{k_0}\omega^{''}(k_0)}
+O(\frac{1}{t^{3/2}}).$$ More elaborated results on asymptotic expansions of the solution to the Klein-Gordon equation are given by Hörmander [@Hor].
The entangled photon correlation function with distance dependence
==================================================================
Now we define a two particle entangled state (biphoton) $$\label{ent}
|\psi\rangle=\int f(k_1,k_2)a_{k_1}^{\dag}a_{k_2}^{\dag}|0\rangle,$$ where $|0\rangle$ is the Fock vacuum and $f(k_1,k_2)$ is the two-photon wave function which is a symmetric function, $f(k_1,k_2)=f(k_2,k_1)$ because we deal with bosons.
The probability to detect one particle at the point $z_1$ along the waveguide at time $t_1$ and another particle at the space point $z_2$ at time $t_2$ is proportional to $$\label{pro}
P(z_1,t_1,z_2,t_2)=\langle \psi|\varphi^{(-)}(z_1,t_1)\varphi^{(-)}(z_2,t_2)
\varphi^{(+)}(z_2,t_2)
\varphi^{(+)}(z_1,t_1)
|\psi\rangle.$$
The expression for $P(z_1,t_1,z_2,t_2)$ (\[pro\]) can be written as $$\label{pros1}
P(z_1,t_1,z_2,t_2)=|A(z_1,t_1,z_2,t_2)|^2,$$ where $$\label{pros2}
A(z_1,t_1,z_2,t_2)=\langle 0|\varphi^{(+)}(z_1,t_1)
\varphi^{(+)}(z_2,t_2)
|\psi\rangle=
\int
dk_1dk_2$$ $$\{\frac{e^{-i\omega_{k_2}t_2+ik_2z_2}}
{2\sqrt{2\pi\omega_{k_2}}}
\frac{e^{-i\omega_{k_1}t_1+ik_1z_1}}{2\sqrt{2\pi\omega_{k_1}}}
f(k_1,k_2)+(k_1\leftrightarrow k_2)\}.$$ Note that the function $A(z_1,t_1,z_2,t_2)$ satisfies the Klein-Gordon equation with respect to $z_1,t_1$ and $z_2,t_2$.
Let us suppose that one of the photons is observed in a frame of reference moving with velocity $v_1$ and another photon is observed in a frame of reference moving with velocity $v_2$. By using the stationary phase method we obtain for large $t_1$ and $t_2$: $$\label{pros21}
A(z_1,t_1,z_2,t_2)=
\sqrt{\frac{2\pi}{t_1\omega^{''}(k_{10})}}
\sqrt{\frac{2\pi}{t_2\omega^{''}(k_{20})}}
f(k_{10},k_{20})\frac{1}{2\sqrt{2\pi\omega_{k_{10}}}}
\frac{1}{2\sqrt{2\pi\omega_{k_{20}}}}e^{-i\pi/2}$$ $$\{ e^{-it_1(\omega_{k_{10}}-ik_{10}v_1)}
e^{-it_2(\omega_{k_{02}}-ik_{20}v_2)}
+(k_1\leftrightarrow k_2)\}.$$ Here $k_{10}=mv_1/ \sqrt{1-v_1^2}$, $k_{20}=mv_2/ \sqrt{1-v_2^2}$, $z_1=v_1t_1$ and $z_2=v_2t_2$.
Therefore $$\label{pros12}
P(z_1,t_1,z_2,t_2)=
\frac{|f(k_{10},k_{20})|^2}
{16t_1t_2\omega^{''}(k_{10})\omega^{''}(k_{20})
\omega_{k_{10}}\omega_{k_{20}}}.$$ $$|\{ e^{-it_1(\omega_{k_{10}}-ik_{10}v_1)}
e^{-it_2(\omega_{k_{02}}-ik_{20}v_2)}
+(k_1\leftrightarrow k_2)\}|^2$$ It is interesting to see the difference between the entangled wave function $f(k_{10},k_{20})$ and the separable one by looking to it with an explicitly indicated dependence on the spacetime coordinates: $$\label{pros123}
f(k_{10},k_{20})=f(m\frac{z_1}{t_1}/\sqrt{1-
\frac{z_1^2}{t_1^2}},m\frac{z_2}{t_2}/\sqrt{1-
\frac{z_2^2}{t_2^2}})$$
Let the wave function of two photons $f(k_1,k_2)$ in a waveguide be a smooth fast decreasing function. Then the probability of observing two photons in the waveguide should satisfy the following bounds. For any $n_1,n_2=0,1,2,3,...$ there exist constants $C_{n_1n_2}$ such that for $$\label{K-G222}
|z_1|\geq |t_1|,\,|z_2|\geq |t_2|,$$ one has $$\label{K-G223}
P(z_1,t_1,z_2,t_2)
\leq\frac{C_{n_1n_2}}{(1+|z_1|)^{n_1}(1+|z_1|)^{n_2}}.$$ Furthermore there exists a constant $C$ such that $$\label{K-G321}
P(z_1,t_1,z_2,t_2)\leq\frac{C}{|t_1||t_2|},$$ for all $z_1,t_1,z_2,t_2$. An asymptotic estimate for $C$, valid for large $t_1$ or $t_2$, is provided by (\[pros12\]).
An explicit expression for the wave function of the biphotons in a special case is given in [@YLS]: $$\label{expl}
f(k_1,k_2)=\frac{i}{k_0^2}f_P(k_1+k_2)\sqrt{6k_1k_2(k_1+k_2)},$$ where $f_P(k_1+k_2)$ is a Gaussian function describing the pumping photons. By using this form of the wave function we obtain the asymptotic formula for the probability in this special case.
To conclude, the main result of this paper is the bounds to the probability density (\[bound\]) and (\[K-G222\]), (\[K-G223\]) which, in principle, should be possible to test in experiments.
[**Acknowledgements**]{}. One of the authors (I. Volovich) would like to thank the International Centre for Mathematical Modelling in Physics and Cognitive Sciences for the support during his visit to Linnæus University.
[9]{}
Lloyd S, Shahriar M S, Shapiro J H and Hemmer P R 2001 Phys. Rev. Lett. 87 167903.
Landry O, van Houwelingen J A W, Beveratos A, Zbinden H and Gisin N 2007 J. Opt. Soc. Am. B 24 398, and references therein.
Ursin R, Tiefenbacher F, Schmitt-Manderbach T, Weier H, Scheidl T, Lindenthal M, Blauensteiner B, Jennewein T, Perdigues J, Trojek P, Ömer B, Fürst M, Meyenburg M, Rarity J, Sodnik Z, Barbieri C, Weinfurter H and Zeilinger A 2007 Nature Phys. 3 481.
Heubel H, Vanner M R, Lederer T, Blauensteiner B, Loreunser T, Poppe A and Zeilinger A 2007 Opt. Express 15 7853.
K. Kojima, H. F. Hofmann, S. Takeuchi, and K. Sasaki, Phys. Rev. A 68, 013803 (2003)
J. T. Shen and S. Fan, Phys. Rev. Lett. 98, 153003 (2007); Phys. Rev. A 76, 062709 (2007).
D. Roy, Phys. Rev. B 81, 155117 (2010).
T. Shi and C. P. Sun, Phys. Rev. B 79, 205111 (2009).
H.F. Hofamann and G. Mahler, Quantum Semiclassic. Opt, 86, 3903 (2001).
H. F. Hofmann, Quantum Noise and Spontaneous Emission in Semiconductor Laser Devices. Institut für Technische Physik, Deutsches Zentrum für Luft und Raumfahrt, Stuttgard (1999).
J. Kim, O. Benson, H. Kim, and Y. Yamamoto, Nature (london) 397, 500 (1999).
I. Bialynicki-Birula, Acta Physica Polonica A 86, 97 (1994).
I. Bialynicki-Birula, Coherence and Quantum Optics VII, Eds. J.H.Eberly, L.Mandel, and E.Wolf., Plenum, New York, 1996, p. 313.
G. P. Agrawal, *Fiber-optic communication systems.* New York: John Wiley & Sons, 2002.
S. Kartalopoulos, *[DWDDM]{}: [N]{}etworks, [D]{}evices, and [T]{}echnology.* Piscataway, NJ: IEEE Press, 2002.
M. Hall, J. Altepeter, and P. Kumar, Ultrafast switching of photonic entanglement. arXiv:1008.4879v1 \[quant-ph\] 28 Aug 2010.
L. Mandel, E. Wolf, *Optical Coherence and Quantum Optics*, Cambridge University Press, Cambridge, 1995.
G. Kristensson,Transient Electromagnetic Wave Propagation in Wave Guides, *Journal of Electromagnetic Waves and Applications*, [**9**]{}, 645-672, 1995.
L. Landau, E. Lifshitz, *Electrodynamics of Continuous Media*, Pergamon Press, Oxford, 1984.
R. Collin, *Field Theory of Guided Waves*, IEEE Press, New York, 1991.
Z.Yang, M.Liscidini, J.E.Sipe, Phys.Rev. A 77, 033808 (2008).
L. Hörmander, Remarks on the Klein-Gordon equation, Journées Équations aux dérivées partielles, p. 1-9, 1987.
|
---
abstract: |
Background
: Predicting the properties of neutron-rich nuclei far from the valley of stability is one of the major challenges of modern nuclear theory. In heavy and superheavy nuclei, a difference of only a few neutrons is sufficient to change the dominant fission mode. A theoretical approach capable of predicting such rapid transitions for neutron-rich systems would be a valuable tool to better understand $r$-process nucleosynthesis or the decay of super-heavy elements.
Purpose
: In this work, we investigate for the first time the transition from asymmetric to symmetric fission through the calculation of primary fission yields with the time-dependent generator coordinate method ([[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}). We choose here the transition in neutron-rich Fermium isotopes, which was the first to be observed experimentally in the late seventies and is often used as a benchmark for theoretical studies.
Methods
: We compute the primary fission fragment mass and charge yields for 254,256 and 258 from the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{} under the Gaussian overlap approximation. The static part of the calculation (generation of a potential energy surface) consists in a series of constrained Hartree-Fock-Bogoliubov calculations based on the [[<span style="font-variant:small-caps;">d1s</span>]{}]{}, [[<span style="font-variant:small-caps;">d1m</span>]{}]{} or [[<span style="font-variant:small-caps;">d1n</span>]{}]{} parameterizations of the Gogny effective interaction in a two-center harmonic oscillator basis. The 2-dimensional dynamics in the collective space spanned by the quadrupole and octupole moments $(\hat{Q}_{20}, \hat{Q}_{30})$ is then computed with the finite element solver [[F<span style="font-variant:small-caps;">elix</span>]{}]{}-2.0.
Results
: The available experimental data and the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{} post-dictions are consistent and agree especially on the position in the Fermium isotopic chain at which the transition occurs. In addition, the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{} predicts two distinct asymmetric modes for the fission of 254.
Conclusions
: Thanks to its intrinsic accounting of shell effects and to its ability to describe the dynamics of the system up to configurations close to scission, the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{} is able to describe qualitatively the fission yield transition in the neutron-rich Fermium isotopes. This makes it a promising tool to study the evolution of the fission yields far from the valley of stability. The main limitation of the method lies in the presence of discontinuities in the 2-dimensional manifold of generator states.
author:
- 'D. Regnier'
- 'N. Dubray'
- 'N. Schunck'
bibliography:
- 'prc\_fm.bib'
title: 'From Asymmetric to Symmetric Fission in the Fermium Isotopes within the Time-Dependent GCM Formalism'
---
Introduction
============
One of the goals of nuclear theory is to provide models that not only reproduce a large set of available experimental data in the neighborhood of the valley of stability, but also have predictive power when computing properties of nuclei far from this region. On-going efforts to better understand the $r$-process of nucleosynthesis or the decay of super-heavy elements give a particular stake to the neutron-rich part of the nuclear chart. In the special case of the fission process, it was shown in Ref. [@goriely_first_2009] that the discrepancies between models used to determine the fission fragment yields of the neutron-rich systems involved in the $r$-process may significantly impact the predictions of the abundances in the region of rare earth peak. This is an incentive to develop a theoretical framework capable of predicting the fission properties for a wide range of neutron-rich nuclei.
At the same time, it is also known that the properties of fission fragments may vary drastically with the number of neutrons and protons of the fissioning system. Historically, a series of experiments conducted in the 70-80’s showed that adding only a few neutrons to 254 could totally change the dominant low-energy fission mode. In these experiments, the post-neutron emission fragments were characterized either by radio-chemistry or directly by measuring their kinetic energy. For 254, the mass yields were obtained from spontaneous fission [@harbour_mass_1973; @gindler_distribution_1977] and clearly showed a mostly asymmetric behavior. When adding a few neutrons, this asymmetric feature is less sharp. The fission yields of 256 both from the neutron-induced channel [@ragaini_symmetric_1974; @flynn_mass_1975] and the spontaneous fission channel [@flynn_distribution_1972; @bemis_mass_1977] all exhibit a mostly asymmetric behavior but the group of Ragani [*et al.*]{} detected in addition the presence of an appreciable symmetric component. For the spontaneous fission of 257, two papers by Balagna [*et al.*]{} [@Balagna_mass_1971] and John [*et al.*]{} [@john_symmetric_1971] reported contradictory results on the dominant fission mode. Finally, symmetric fission clearly dominates in 258 as reported in three papers covering both the neutron-induced and spontaneous fission [@flynn_distribution_1975; @hulet_spontaneous_1980; @hulet_spontaneous_1989]. The group of Hulet [*et al.*]{} even probed the fission of 259 produced by [257]{}(t,p) and found out mostly symmetric yields. Later on, several experiments relying on inverse kinematic beams [@schmidt_relativistic_2000; @martin_studies_2015] highlighted many similar transitions both in the neutron-rich and neutron-deficient sides of the valley of stability. A common feature is that the transition often occurs within a range of just a few nucleons.
Understanding and reproducing these sharp transitions presents a real challenge for nuclear theory and different kinds of approaches have been proposed to tackle this issue. A common starting point is often the computation of the potential energy surface for the fissioning system as a function of a small set of collective degrees of freedom. In 1980, Lustig [*et al.*]{} were the first to study the asymmetric/symmetric transition of mass yields in Fermiums. They adopted a purely static picture and computed the energy of the deformed nucleus within a macroscopic-microscopic model [@lustig_transitions_1980]. Later on, a similar work performed by Cwiok [*et al.*]{} [@cwiok_two_1989] in a five-dimensional deformation space revealed the existence of an elongated and a compact fission mode for 258. More recently, studies of static deformation properties of Fermium isotopes were also performed within a self-consistent mean-field framework based on Gogny, Skyrme and covariant energy density functionals ([[<span style="font-variant:small-caps;">edf</span>]{}]{}) [@warda_self-consistent_2002; @warda_spontaneous_2006; @bonneau_fission_2006; @dubray_structure_2008; @staszczak_microscopic_2009; @zhao_multidimensionally-constrained_2016]. All these papers emphasized the multi-modal character of the fission of Fermium isotopes near $A=256$ and highlighted the presence of three major modes: symmetric compact, symmetric elongated, and asymmetric. Although these static approaches pinpointed the major fission modes that are energetically favored in low-energy fission, they did not provide information about the actual probability to populate each of these modes.
One way to predict fission yields without an explicit treatment of nuclear dynamics is to assume that static nuclear configurations close to scission are populated statistically during the fission process. Such scission-point models have been applied with different choices for the deformed nuclear configurations [@schmidt_review_2018; @pasca_charge_2018; @ichikawa_origin_2009; @lemaitre_new_2015]. These models were able to reproduce the main features of the symmetric/asymmetric transitions of the fission yields for instance in the Thorium and Fermium isotopic chains. However, one of the major limitations of scission point models is the somewhat arbitrary definition of the ensemble of scission configurations which are thermally populated. One should also keep in mind that they ignore any possible “memory effect” of the nucleus as it travels through the potential energy landscape.
Another class of approaches to determine fission yields involve using static nuclear properties as inputs to the explicit modeling of nuclear dynamics. Following this idea, Asano [*et al.*]{} performed Langevin calculations in three-dimensional collective spaces [@asano_dynamical_2004]. This represented the first theoretical attempt to obtain the yields of 256,258,264 through the proper simulation of the time-evolution of the system. However, the calculation failed to reproduce the observed transition from asymmetric to symmetric mass yields between 256 and 258.
A fully quantum mechanical alternative to describe nuclear dynamics is the time-dependent generator coordinate method ([[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}) with the Gaussian overlap approximation ([[<span style="font-variant:small-caps;">goa</span>]{}]{}). Goutte *et al.* used this framework for the first time in 2005 to compute fission yields [@goutte_microscopic_2005]. Since then, it has been successfully applied to several fissioning systems in the actinide region [@younes_fragment_2012; @regnier_fission_2016; @zdeb_fission_2017] including proton-rich Thorium isotopes [@tao_microscopic_2017]. However, the reliability of this method in the neutron-rich sector of the nuclear chart and its ability to predict rapid structural changes in fission yields is yet to be established.
The goal of this paper is to investigate the robustness of the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} approach in reproducing the symmetric/asymmetric yield transition in neutron-rich Fermium isotopes. In particular, we will examine in details the dependence of the results on the various inputs to the calculations: parametrization of the energy density functional, initial conditions, form of the collective inertia tensor, and definition of scission configurations.
In Sec. \[sec:theory\] we briefly recall the formal and numerical methods used to compute fission yields within the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{}. The Sec. \[sec:results\] is devoted to the discussion of the static properties obtained for the Fermium chain and to the comparison between the computed fission yields and associated experimental data. In Sec. \[sec:discussion\] we focus on the reliability of this approach by testing the sensitivity of our results to various input ingredients. We also analyze the impact of discontinuities in the 2-dimensional manifold of generator states.
Methodology {#sec:theory}
===========
A comprehensive presentation of the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} theory and of our implementation of it can be found in Ref. [@regnier_fission_2016]. In this section we only summarize the necessary ingredients of the method and refer the reader to our previous work for further details.
Theoretical Framework
---------------------
In the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{} approach, the evolution of the many-body quantum state $|\Psi(t)\rangle$ describing the fissioning system is determined by a variational approximation of the many-body dynamics. At any time, the many-body wave function takes the form of a continuous and linear superposition of constrained [[<span style="font-variant:small-caps;">hfb</span>]{}]{} states parametrized by a set of collective coordinates ${\mathbf{q}}$, $$\label{eq:gcmApprox}
|\Psi(t)\rangle \equiv \int_{{\mathbf{q}}} f({\mathbf{q}}, t) |\Phi_{{\mathbf{q}}} \rangle \, \text{d}{\mathbf{q}}.$$ Instead of solving the non-local Hill-Wheeler equation resulting from the application of the time-dependent variational principle, we invoke in addition the Gaussian overlap approximation ([[<span style="font-variant:small-caps;">goa</span>]{}]{}) [@brink_generator-coordinate_1968; @reinhard_generator-coordinate_1987; @krappe_theory_2012]. This standard scheme reduces the problem to a local Schrödinger-like equation, $$\label{eq:tdgcmgoa}
i\hbar \frac{\partial g({\mathbf{q}},t)}{\partial t} = \hat{H}_\text{coll}({\mathbf{q}}) \, g({\mathbf{q}},t).$$ The complex function $g({\mathbf{q}},t)$ is the unknown of the equation. It is related to the weight function $f({\mathbf{q}}, t)$ appearing in (\[eq:gcmApprox\]) and contains all the information about the dynamics of the system. The collective Hamiltonian $\hat{H}_\text{coll}({\mathbf{q}})$ is a local linear operator acting on $g({\mathbf{q}},t)$, $$\label{eq:Hcoll}
\hat{H}_\text{coll}({\mathbf{q}}) \equiv
-\frac{\hbar ^2}{2\gamma^{1/2}({\mathbf{q}})} \sum_{ij} \frac{\partial}{\partial q_i} \gamma^{1/2}({\mathbf{q}}) B_{ij}({\mathbf{q}}) \frac{\partial}{\partial q_j} + V({\mathbf{q}}).$$ This operator contains a collective kinetic part characterized by the inertia tensor $\mathsf{B}({\mathbf{q}}) \equiv B_{ij}({\mathbf{q}})$ and a potential term $V({\mathbf{q}})$. In a generalized version of the [[<span style="font-variant:small-caps;">goa</span>]{}]{} [@kamlah_derivation_1973; @onishi_local_1975; @gozdz_extended_1985], it also involves a real and positive metric $\gamma({\mathbf{q}})$. Taking into account this metric leads to a better reproduction of the exact overlaps with Gaussian functions by letting the width of the Gaussian kernels explicitly depend on the position in the collective space. The locality of the collective Hamiltonian implies a continuity equation for the square modulus of the collective wave function $|g({\mathbf{q}},t)|^2$, $$\label{eq:continuity}
\frac{\partial}{\partial t} |g({\mathbf{q}}, t)|^2 \gamma^{1/2}({\mathbf{q}}) = -\nabla \cdot \mathbf{J}({\mathbf{q}}, t),$$ where $\mathbf{J}({\mathbf{q}}, t)$ is the collective current defined from $g({\mathbf{q}},t)$.
To compute the fission yields from the solution of Eq. , we define a frontier line that marks the limit between (i) an inner domain of the collective space where we still have a compound nucleus and (ii) an outer domain containing eventually all the split configurations. Within this picture, each infinitesimal element of the frontier line corresponds to the entrance point of one possible output channel of the fission reaction with a given mass and charge for the two primary fragments. Ideally, the frontier should be chosen in such a way that output channels are completely decoupled from one another. In this situation, the collective dynamics in the inner domain would simulate the evolution up to configurations where the two fragments could not exchange particles any more. The quantum probability to measure a mass split $A_H/A_L$ would then be given by the projection of the final [[<span style="font-variant:small-caps;">gcm</span>]{}]{} state over all output channels leading to this mass split. This is nothing but the integral of $|g({\mathbf{q}}, t)|^2\gamma^{1/2}({\mathbf{q}})$ over a set of outer collective areas, each associated with one output channel. Leveraging the continuity equation Eq. , it can be recast into a sum of time-integrated flux of probability $F(\xi,t)$ to cross an infinitesimal element $\xi$ of the frontier. $$F(\xi,t) = \int_{t=0}^{t} dt \int_{{\mathbf{q}}\in\xi} \mathbf{J}({\mathbf{q}},t)\cdot d\mathbf{S}.
\label{eq:fluxDef}$$ For the fragmentation $A_H/A_L$, the sum runs over all elements $\xi$ in which the [[<span style="font-variant:small-caps;">hfb</span>]{}]{} states have $A_L/A_H$ particles in the light/heavy fragment. In practice, the choice of the frontier is subject to several constraints discussed in Sec. \[sec:front\] and \[subsec:discont\]. In our calculations, the configurations at the frontier are often characterized by a non-negligible interaction energy between the pre-fragments [@younes_nuclear_2011]. This means that the [[<span style="font-variant:small-caps;">hfb</span>]{}]{} states on the frontier do not yet fully belong to one or the other of the output channels. In other words, a realistic evolution of such a state may lead to several mass splits close to our averaged estimate at the frontier. To take this into account, several prescriptions have been proposed in the literature such as convoluting the raw yields with a Gaussian [@younes_fragment_2012] or using a more sophisticated random neck rupture model [@zdeb_fission_2017]. In this work, we retain a simple prescription and adopt a Gaussian convolution with a constant width. Doing so, we introduce the width of the Gaussian used in the convolution as a necessary arbitrary parameter. All final yields are normalized to 200%.
Determination of the GCM+GOA Collective Hamiltonian {#sec:genpes}
---------------------------------------------------
The first step to build the collective Hamiltonian consists in building the manifold of generator states. In practice, it implies performing a series of [[<span style="font-variant:small-caps;">hfb</span>]{}]{} calculations for the compound nucleus with constraints on the expectation value of the two collective coordinates $\hat{Q}_{20}$ and $\hat{Q}_{30}$, which are here defined with the same conventions as in Ref. [@regnier_fission_2016]. We computed each point in the regular grid spanning $[0,450]\times[0,100]$ (in barn units) with the mesh steps $h_{20}=2$ b, and $h_{30}=1$ b$^{3/2}$. Each [[<span style="font-variant:small-caps;">hfb</span>]{}]{} calculation is performed by an iterative solver relying on a two-center [[<span style="font-variant:small-caps;">ho</span>]{}]{} basis to discretize the single particle wave functions. The parameters of this basis are optimized at each deformation point using a new method based on Gaussian processes. This new method, which will be described in details in a future paper, allowed us to speed up the basis parameter optimization procedure by a factor 5, compared to the previous numerical procedure. The [[<span style="font-variant:small-caps;">hfb</span>]{}]{} calculations have been performed with the [[<span style="font-variant:small-caps;">d1s</span>]{}]{}, [[<span style="font-variant:small-caps;">d1n</span>]{}]{} and [[<span style="font-variant:small-caps;">d1m</span>]{}]{} parameterizations of the Gogny effective interaction for each of the three Fermium isotopes.
It is well known that generating a potential energy surface which minimizes the total binding energy (as is the case using self-consistent methods) may lead to some issues related to the imperfect nature of the minimization (local minima) and to the underestimation of some barrier heights (restricted collective space) as discussed in Ref. [@dubray_numerical_2012]. To fully avoid the issue of spurious local minima, a special retro-propagation scheme is used, which ensures that all [[<span style="font-variant:small-caps;">hfb</span>]{}]{} solutions of the potential energy surface are global minima. From the ensemble of [[<span style="font-variant:small-caps;">hfb</span>]{}]{} solutions, the last step is to determine the collective fields involved in Eq. \[eq:Hcoll\]. In the [[<span style="font-variant:small-caps;">gcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} formalism, the inertia tensor is related to the second order derivatives of the reduced Hamiltonian kernel with respect to the collective coordinates. In this work, the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} inertia and metric are calculated at the perturbative cranking approximation; see [@schunck2016] for details. The potential term provided by the [[<span style="font-variant:small-caps;">gcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} approach contains the total [[<span style="font-variant:small-caps;">hfb</span>]{}]{} energy of the constrained state corrected by a vibrational zero-point energy associated with our collective degrees of freedom. The formula used to compute the fields can be found in Eqs. (9)-(15) of Ref. [@regnier_fission_2016].
Solution to the Collective Schrödinger Equation {#sec:dynamics}
-----------------------------------------------
After the calculation of the static properties of the system, we numerically solve the collective Schrödinger-like equation, Eq. , with the version 2.0 of the code [[F<span style="font-variant:small-caps;">elix</span>]{}]{} [@regnier_felix-1.0:_2016; @regnier_felix-2.0:_2018]. We simulate the collective evolution in a symmetric domain for the octupole moment with absorption boundary conditions to avoid spurious reflections. To get the best numerical efficiency, the problem is not discretized on the regular mesh used to compute the static properties but on a refined and adapted finite element mesh. The technical details on the simulation domain, boundary conditions, generation of the mesh and spectral elements basis are reported in App. \[ap:setup\_param\].
The initial state is built as prescribed in [@regnier_fission_2016] as a superposition of collective eigen-modes in an extrapolated first potential well. The weights of this mixture have a Gaussian shape as a function of the eigen-energies of the modes. The width of the Gaussian is fixed to $\sigma_i=0.5$ MeV and its first moment is chosen so that the initial energy lies 1 MeV above the first potential barrier. This choice for the initial collective state distribution simulates the low-energy induced fission while it allows a significant part of the wave packet to escape from the first potential well and evolve toward scission. The sensitivity of our results to the parameter $\sigma_i$ and the initial energy is discussed in Sec. \[sec:ei\].
Starting from this initial condition, the evolution equation is integrated in time using a Krylov approximation scheme for the exponential propagator. We use a dimension-10 Krylov space along with a time step $dt=2\times 10^{-4}$ zs ($10^{-21}$s). The propagation runs up to a time of 20 zs, after which the fission yields are stable with time. According to our previous benchmark on 256, the absolute numerical convergence of the resulting mass yields (normalized to 200%) is expected to be of the order of 0.06%.
Extraction of Fission Mass Distributions {#subsubsec:yields}
----------------------------------------
The frontier used to compute the fission yields is defined by the isoline $Q_{\rm N}=7.5$ of the neck operator [@younes_fragment_2012] $$\hat{Q}_N = \operatorname{exp}\left( - \frac{ (z - z_N)^2}{a_N^2} \right),$$ with $a_N = 1$ fm, $z$ the coordinate along the main axis of the system and $z_N$ is the position of the neck. This line is chosen as one of the lowest-value neck isoline that lies above the fission/fusion valley crossing. This choice is discussed in more details in Sec. \[sec:front\]. In practice, the isoline is discretized as a succession of square cells edges four times smaller than the finite element mesh cell edges. The raw yields extracted from the time-integrated flux through the frontier are convoluted with a normal distribution as already done in Ref. [@regnier_fission_2016]. Such a convolution implements our lack of knowledge on the exact number of particles in the fragments due to several features that we briefly recall below:
- After solving the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} evolution, a proper quantum estimation of the number of particles in each fragment would require first disentangling the two fragments [@younes_nuclear_2011; @schunck2014], and then projecting on states with a good particle number, e.g. as in [@scamps_superfluid_2015]. Since in this work, we only estimate particle numbers based on the integration of the one-body density, we therefore miss some of the quantum fluctuations.
- By construction, the [[<span style="font-variant:small-caps;">hfb</span>]{}]{} theory used to determine the generator states breaks the symmetry associated with the total number of particles in the fissioning system. This implies that at the frontier where the yields are computed, the total wave function of the fissioning system is the superposition of wave functions with different numbers of particles. Once again, a better approach would involve projecting this wave function on good particle number and extracting the characteristics of the fragments from the projected density.
- For the nuclear configurations at the frontier, the nuclear interaction between the fragments can easily be of the order of dozens of MeV; see, e.g., estimates in [@schunck2014]. This implies that several nucleons could be exchanged between the two prefragments. Each configuration at the frontier therefore contributes to several neighboring fragmentations.
- Finally, the experimental fission yields that we have used in this study were measured with a detector resolution of 4-5 mass units (full width at half maximum). This corresponds to a convolution of the raw yields with a normal distribution parameterized with a width $\sigma \simeq 2$. A similar convolution of the theoretical results should normally be applied in order to make consistent comparisons.
Addressing these limitations goes beyond the scope of this article. At the moment, we therefore make the pragmatic choice of taking effectively into account these effects by reducing the resolution of our predictions. To do so we convolute the raw fission yields with a Gaussian of width $\sigma=4$ mass units. The choice of this parameter can be justified based on various physical arguments. Indeed, we have $Q_{\rm N} = 7.5$ for configurations at the frontier, which means that roughly 8 particles are located in a plane within $\pm 1$ fm around the neck position. If the radial total density is constant in this region, and if we assume a random rupture of the neck with a normal probability distribution $P(x_{\text{neck}} + \delta x)$ for the split to happen at $x_{\text{neck}} + \delta x$, then we obtain a spreading with the same width ($\sigma\simeq 4$ mass units). In others words making such a convolution on the fragment mass is equivalent to considering that the neck is randomly cut with a probability following a normal distribution of width 1 fm. While this reasoning provides a qualitative motivation for the choice of the convolution width, it should be clear that the precise quantitative value of the convolution width $\sigma$ is still arbitrary. Hopefully, our previous study [@regnier_fission_2016] shows that changing this value does not impact significantly the main characteristics of the fission modes.
Results {#sec:results}
=======
In this section we present the static and dynamic properties of $^{254,256,258}$Fm obtained within the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} approach.
Main Static Properties
----------------------
For each nucleus, we first computed the generator states with the [[<span style="font-variant:small-caps;">d1s</span>]{}]{} parameterization of the Gogny effective interaction. The fitting process of this parameterization includes information on the fission barrier of 240, which makes it a reference effective interaction for fission studies in general. Unless specified otherwise, the calculations presented in this section are based on Gogny [[<span style="font-variant:small-caps;">d1s</span>]{}]{}.
### Global Topology
The figure \[fig:fmtrans\_pes\] shows the potential energy landscape obtained for the three nuclei under study. Note that the potential includes the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} zero-point energy.
![Potential energy surfaces of the 254, 256 and 258 Fermium isotopes determined from the [[<span style="font-variant:small-caps;">d1s</span>]{}]{} Gogny energy density functional. The potential corresponds to the [[<span style="font-variant:small-caps;">hfb</span>]{}]{} energy corrected from the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} zero-point energy. The color scale is shifted by 10 MeV between consecutive plots. The red continuous line represents the isoline $Q_{\rm N} = 7.5$ of the neck operator.[]{data-label="fig:fmtrans_pes"}](fmtrans_pes.pdf){width="50.00000%"}
The overall topology of theses potential energy surfaces ([[<span style="font-variant:small-caps;">pes</span>]{}]{}) is very similar for the three nuclei. The energy minimum in the first potential well is characterized by $Q_{20}\approx 30$ b and $Q_{30}\approx 0$ b$^{3/2}$. This is typical of the actinide region. Going toward more elongated shapes, there is a first potential barrier whose height depends on the specific nucleus. In our 2-dimensional collective space, two main fission modes are clearly visible. The first one is a rather broad valley (in the $Q_{30}$ direction) leading to asymmetric fragmentations. It reaches neck values $Q_{\rm N}=7.5$ at large elongations $Q_{20}\in [350,400]$ b and corresponds to what is called the asymmetric elongated fission mode. The second fission mode is a tiny valley that follows symmetric configurations and reaches the same neck value at much lower elongation ($Q_{20}\simeq 220$ b). Beyond this line, a rapid change in the energy slope happens around $Q_{20}\simeq 260$ b. The collective potential energy decreases rapidly and the expectation value of the neck operator also vanishes suddenly. It corresponds to the symmetric compact fission mode discussed in Ref. [@bonneau_fission_2006; @staszczak_microscopic_2009]. For 258, a third symmetric elongated mode has also been described in these previous papers but it is not visible here. This is because our 2-dimensional [[<span style="font-variant:small-caps;">pes</span>]{}]{} can only show the lowest energy modes in a given range of $Q_{20},Q_{30}$ whereas the symmetric elongated and compact fission modes span the same range for these collective variables. As shown in Ref. [@staszczak_microscopic_2009], introducing an additional dimension through the $Q_{40}$ collective coordinate would enable us to capture both symmetric modes. Note that calculations in 3-dimensional collective spaces [@bonneau_fission_2006; @staszczak_microscopic_2009] suggest that the symmetric elongated mode lies quite higher in energy than the symmetric compact mode. Therefore, its contribution to the formation of the symmetric peak in fission fragment distributions should not be significant.
$V_{\text{min}}$ $E_{0,\text{GCM}}$ B$_I$ B$_{I, \text{GCM}}$
----- ------------------ -------------------- ------- ---------------------
254 -1886.2 -1883.0 13.3 10.2
256 -1896.6 -1893.6 12.4 9.4
258 -1906.8 -1903.7 11.8 8.7
: Characteristics of the Gogny [[<span style="font-variant:small-caps;">d1s</span>]{}]{} potential energy surfaces for $^{254,256,258}$Fm. The minimum of the potential ($V_{\text{min}}$) in the first well is given in MeV along with the energy of the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} ground-state ($E_{0,\text{GCM}}$). The height of the inner fission barrier (B$_I$) is in MeV relative to $V_{\text{min}}$. The quantity B$_{I, \text{GCM}}$ is the energy that should be brought to the system in its [[<span style="font-variant:small-caps;">gcm</span>]{}]{} ground state in order to fission without tunnel effect.[]{data-label="tab:barriers"}
In details, the heights of the different barriers and ridges significantly differ from one nucleus to another. We show in Table \[tab:barriers\] the first fission barrier heights relative to the minimum energy in the first potential well. A quantity that has more physical relevance than the barrier is the quantity of energy that must be injected in the compound nucleus so that the fission process may happen without tunneling. In our framework this is given by the difference between the potential at the saddle point and the energy of the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} ground-state in the first potential well. We report this quantity as B$_{I,\text{GCM}}$ and show that it is lower than the ’classical’ barrier by a few MeV. This “collective” barrier could be further reduced by a few MeV if axial symmetry were not imposed in our [[<span style="font-variant:small-caps;">hfb</span>]{}]{} calculations.
### Competing Fission Modes
In the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} picture, the presence of valleys in the potential energy landscape favors the diffusion of the collective wave packet towards specific sets of configurations at scission. As discussed in the previous section, two major valleys have been found in the present calculations (see Fig. \[fig:fmtrans\_pes\]). These two valleys are separated by a potential ridge with a shape and height that varies with the nucleus. This ridge is indeed quite pronounced for 258 but progressively disappears as we go toward the lighter isotopes.
![Slice of the [[<span style="font-variant:small-caps;">d1s</span>]{}]{} potential energy surfaces for the three Fermium isotopes at the elongations $Q_{20}=140,180,225$ b. To emphasize the difference of topology between nuclei, all the curves are shifted so that $V(Q_{30}=0) = 0$.[]{data-label="fig:vcut"}](vcut.pdf){width="45.00000%"}
We quantify this behavior in Fig. \[fig:vcut\], which shows slices of the three [[<span style="font-variant:small-caps;">pes</span>]{}]{} at the constant quadrupole moment values $Q_{20} = 140, 180, 225$ b. At $Q_{20}=140$ b, symmetric configurations are largely favored energetically in all three isotopes. Around $Q_{20}=180$ b, the symmetric path is favored in 258 but, in contrast, the asymmetric mode is lower in energy for 254. Since there is no significant potential barrier between the two valleys, the system can diffuse from symmetric configurations to asymmetric configurations at the mouth of the asymmetric valley. In 256, the [[<span style="font-variant:small-caps;">pes</span>]{}]{} is rather flat which provides the opportunity for a collective wave packet to spread over the two valleys populating both modes. At larger elongations $Q_{20}=225$ b, even if the asymmetric mode becomes energetically more favored in 258, a ridge of 4 MeV separates it from the symmetric path and hinders the transition toward the asymmetric elongated mode. Such changes in the topology of the [[<span style="font-variant:small-caps;">pes</span>]{}]{} are likely to be highly correlated with the appearance of gaps in the single particle energy spectra as a function of the collective deformations. Although this analysis based on the static potential energy is not yet quantitative, most of the physics of the transition can already be guessed at that level.
Fission Fragment Distributions
------------------------------
We computed the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} evolution of the 254, 256 and 258 over a period of 20 zs ($10^{-21}$s). During the propagation, the collective wave function $g({\mathbf{q}})$ escapes the first potential well to populate the available fission valleys. After crossing the frontier, it is then absorbed by the artificial imaginary term in the Hamiltonian in the absorption band. After 20 zs, 34%, 34%, and 28% of the total norm crossed the frontier for the $A=254$, 256 and 258 Fermium isotopes respectively. During the last 1 zs, the yields are nearly stable and we have $|| \mathbf{Y}(t)- \mathbf{Y}(t_f)||_{\infty} < 0.4\%$ for the intermediate 256 and $|| \mathbf{Y}(t)- \mathbf{Y}(t_f)||_{\infty} < 0.1\%$ for the others. This means that although some of the wave packet is still leaking from the first potential well, 20 zs is enough time to obtain the qualitative features of the yields.
Figure \[fig:theovsexp\] presents the primary fission mass yields obtained for the fermium isotopic chain compared to a series of experimental data.
![Primary fragment mass yields obtained with the Gogny [[<span style="font-variant:small-caps;">d1s</span>]{}]{} effective interaction and compared with various experimental data sets taken from Ref. [@harbour_mass_1973; @gindler_distribution_1977; @flynn_distribution_1972; @flynn_distribution_1975; @hoffman_12.3-min_1980; @hulet_spontaneous_1989]. All the yields are normalized to 200%. The experimental data points all represent post-neutron evaporation mass yields. The open symbols stand for experimental data associated to spontaneous fission whereas full symbols are related to thermal neutron-induced fission.[]{data-label="fig:theovsexp"}](theovsexp.pdf){width="45.00000%"}
When adding only four neutrons to the compound system, the behavior radically changes from a mostly asymmetric to a mostly symmetric fission. The [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} dynamics applied with the Gogny [[<span style="font-variant:small-caps;">d1s</span>]{}]{} effective interaction successfully captures this rapid transition. The number of neutrons at which this transition is predicted, $N=156$, matches the experimental observations. On the other hand, the mass-by-mass values of the yields sometimes differ from the experimental by up to 2% (in absolute value). In particular, the results obtained for the intermediate nucleus 256 do not reproduce the double-humped shape of the experimental data.
There are several reasons why this comparison between theory and experiment must be kept at the qualitative level. First, for all experimental data the mass of the fragments is measured after the evaporation of prompt neutrons. Taking into account the neutron evaporation would shift our predictions by a few units toward lighter masses as well as bring additional structure and asymmetry between the light and heavy peaks. It could partly be responsible for the light peak of 254 being roughly 7 mass units too high. The shift of the light peak depends non trivially on the fragmentation, and a first account of neutron evaporation would at least require the knowledge of the average neutron multiplicity as a function of the fragment mass. We did not apply such a correction here. A second important effect that also impacts the comparison with experiment is the initial energy of the fissioning system. Some of the experimental data sets are from spontaneous fission whereas others come from induced fission. In the actinide region, where fission is mostly asymmetric, adding more energy to the system is known to enhance the symmetric component of the yields [@duke_fission-fragment_2016; @al-adili_fragment-mass_2016; @gooden_energy_2016]. Such behavior may explain the difference between the two data sets of Flynn for 256 [@flynn_distribution_1972; @flynn_distribution_1975], as well as the high symmetric yields obtained in 254 compared with spontaneous fission experiments. For 258, the situation is the opposite: increasing the energy is expected to flatten the main symmetric peak. This is actually the behavior that we obtain when increasing the energy of the initial state of our dynamic calculation (see Sec. \[sec:ei\]). As the energy increases, the wave packet spreads more easily and populates the modes that are not the most energetically favorable. This is consistent with the fact that the experimental data by Hoffman and Hulet associated with spontaneous fission are much more peaked compared with the data of Flynn and our results for induced fission.
![Total energy as a function of the heavy fragment charge along the isoline $Q_{\rm N}=7.5$.[]{data-label="fig:front"}](front.pdf){width="45.00000%"}
One should emphasize that changes in the principal fission modes can not be detected when looking only at the structure of scission configurations along the frontier where the flux is computed. At first glance, this seems totally inconsistent with the fact that scission-point models such as Ref. [@lemaitre_new_2015; @pasca_charge_2018] could be able to reproduce this transition between symmetric and asymmetric yields for Fermium isotopes. In such models, the statistical population of a given mass and charge split is often given by a Boltzman factor that depends on the free energy at the scission configurations of interest. Fig. \[fig:front\] shows that the collective potential energy as a function of the proton number of the heavy fragment is remarkably similar for all three isotopes (notwithstanding a trivial shift due to the binding energy of the extra neutrons). A thermal occupation of these ’scission’ configurations (along the $Q_N=7.5$ isoline) would be very similar and should result in mostly asymmetric yields for the three Fermium isotopes. The fact that statistical models are somewhat capable to reproduce the experimental transition therefore suggests that the scission configurations they are using are rather different from the ones we observe along the $Q_N=7.5$ isoline. More precisely, we should expect that these configurations correspond to geometrical shapes that are somewhat equivalent to the shapes we observe in our calculations in the area around $Q_{20}=180$ b, where the system “chooses” between the two different modes.
Structure of Asymmetric Modes in 254 {#sec:two_modes}
------------------------------------
Looking more closely at the fission of 254, we found that the large asymmetric peak in the mass yields is actually coming from two well-separated valleys. This is particularly visible in Fig. \[fig:fmtrans\_charge\], where we show the charge yields obtained without any convolution with a Gaussian form factor. One asymmetric mode is centered at $Z=57$ while the other one lies around $Z\simeq 54$. The first one corresponds to the output of the large asymmetric valley corresponding to configurations around $(350, 50)$ in barn units. It corresponds to rather elongated configurations. The other one corresponds to a tiny valley starting at lower elongation and asymmetry, around $(270,25)$ barn units.
![Primary fragment charge yields (normalized to 200%) obtained with the Gogny [[<span style="font-variant:small-caps;">d1s</span>]{}]{} effective interaction. The black doted line represents raw results directly obtained from the flux through the frontier, whereas the red full line accounts for the convolution of the raw results with a Gaussian of width $\sigma=4.0\times Z/A$.[]{data-label="fig:fmtrans_charge"}](fmtrans_charge.pdf){width="45.00000%"}
Looking at the evolution of the raw yields as a function of neutron number, we find that the most asymmetric mode in the Fermium chain (the one centered on $Z=57$ for the heavy fragment) is pretty stable. When the number of neutron increases, it is the less asymmetric mode that vanishes and becomes the symmetric mode. We may speculate that these two asymmetric modes could well be related to the standard-1 and standard-2 modes widely used to fit actinides fission yields [@brosa_nuclear_1990].
Stability of the Results {#sec:discussion}
========================
Parametrizations of the Gogny EDF
---------------------------------
The main input of the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} approach for the determination of fission yields is the energy density functional underpinning all the calculations. Although this input should ultimately be related to the bare interaction between nucleons, practical applications in heavy nuclei rely on empirical parameterizations fitted on various key nuclear observables. In the case of the Gogny effective interaction, the three major parameterizations differ in the methods adopted for the fitting procedure. Although it is the oldest one, the [[<span style="font-variant:small-caps;">d1s</span>]{}]{} parameterization [@decharge_hartree-fock-bogolyubov_1980; @berger_time-dependent_1991] includes constraints on the fission barrier height of 240 estimated at the [[<span style="font-variant:small-caps;">hfb</span>]{}]{} level and can give a rather good description of most nuclear properties. For this reason, we have used it in this work as a reference. The [[<span style="font-variant:small-caps;">d1n</span>]{}]{} parameterization [@chappert_towards_2008] was designed to better reproduce the properties of neutron matter at the [[<span style="font-variant:small-caps;">hfb</span>]{}]{} level and is therefore expected to perform better in the neutron-rich sector of the nuclear chart. Finally, [[<span style="font-variant:small-caps;">d1m</span>]{}]{} [@goriely_first_2009] was especially designed to reproduce the masses and radii of the entire nuclear chart at the 5DCH level, i.e. within a static [[<span style="font-variant:small-caps;">gcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} framework including all quadrupole degrees of freedom. The impact of the choice of parameterization of the Gogny interaction on some fission properties such as barrier heights and half-lives has been investigated in Ref. [@rodriguez-guzman_microscopic_2014]. Although it is clear that significant differences appears between parameterizations, e.g. [[<span style="font-variant:small-caps;">d1s</span>]{}]{} underestimates nuclear binding energies compared with the two others, the topology of the least-action fission paths are qualitatively similar. Therefore, the impact on the fission yields can only be tested in a fully dynamical calculation.
In this section, we compare the fission yields obtained from the three parameterizations. All the codes and numerical parameters are exactly the same for each calculation, which provides for the first time a clean view of the sensitivity of the yields to the Gogny parameterization. The results are plotted in Fig. \[fig:fmtrans\_inter\].
![Comparison of the primary fragment mass yields obtained with the [[<span style="font-variant:small-caps;">d1s</span>]{}]{}, [[<span style="font-variant:small-caps;">d1n</span>]{}]{} and [[<span style="font-variant:small-caps;">d1m</span>]{}]{} parameterizations of the Gogny force. All the yields are normalized to 200%.[]{data-label="fig:fmtrans_inter"}](fmtrans_inter.pdf){width="45.00000%"}
The most important conclusion of this study is that the transition from asymmetric to symmetric fission in Fermium isotopes holds for all three interactions. In fact, results from the different parameterizations in 254 and 258, where one of the modes is strongly favored, are remarkably close. This suggests that for this kind of nuclei, the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} method provides a robust method to predict the qualitative feature of the yields. On the other hand, the yields obtained for 256 differ significantly. The [[<span style="font-variant:small-caps;">d1n</span>]{}]{} effective interaction gives a wide symmetric peak whereas the yields are pretty flat for [[<span style="font-variant:small-caps;">d1s</span>]{}]{} and [[<span style="font-variant:small-caps;">d1m</span>]{}]{}. In this transition nucleus, the sensitivity to the details of the energy functional is much more pronounced. Since the yields result from the competition between several modes, results are much more sensitive to the small changes in the [[<span style="font-variant:small-caps;">pes</span>]{}]{} topology that different parameterizations can induce. In such nuclei, the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} is much less predictive, mostly because of our lack of constraints on the underlying [[<span style="font-variant:small-caps;">edf</span>]{}]{}. On the other hand, if all other limitations of the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} could finally be taken care of, these transition nuclei could provide good test benches to validate energy density functionals.
Initial state {#sec:ei}
-------------
The goal of this section is twofold: first to study the impact of the initial energy of the fissioning system and try to assess how meaningful the comparison with experimental data shown in Fig. \[fig:theovsexp\] is; second to check that changing the Gaussian width $\sigma_i$ used to build the initial state within a reasonable range does not affect our global conclusions.
![Evolution of the primary fragment mass yields as a function of the initial energy for 254 and 258. The energy is given in MeV relative to the energy of the saddle point of the first fission barrier.[]{data-label="fig:fm258_ya_ei"}](fm254_ya_ei.pdf "fig:"){width="44.00000%"} ![Evolution of the primary fragment mass yields as a function of the initial energy for 254 and 258. The energy is given in MeV relative to the energy of the saddle point of the first fission barrier.[]{data-label="fig:fm258_ya_ei"}](fm258_ya_ei.pdf "fig:"){width="45.00000%"}
For thermal neutron-induced fission, the initial energy should be the neutron separation energy of the studied Fermium. This is typically $S_{n}\simeq 6$ MeV according to the ENSDF database [@noauthor_evaluated_nodate]. In addition, the neutron-induced fission of 256 and 258 is known to occur already significantly with a thermal neutron beam as reported in Ref. [@flynn_distribution_1975]. This means that the initial energy of the system is higher than the fission barrier, $\Delta E_i = (E^*_i - B_I ) > 0$. Since the fission barrier energy is positive for these systems, it means that the initial energy relative to the fission barrier should be in the range $$S_n > \Delta E_i > 0$$ This is typically the range of energies that the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} calculation can probe. To assert the sensitivity of the fission yields in this energy range, we performed a series of calculations with various initial energies. The results are reported in Fig. \[fig:fm258\_ya\_ei\]. For 254, the main effect of an increase of the initial energy is a progressive shift of the asymmetric peak toward more asymmetric fragmentations. For the extreme case of $E_i = B_I +10$ MeV, the fission yields become completely different, which is a consequence of the collective wave packet spreading without being so much influenced by the topology of the [[<span style="font-variant:small-caps;">pes</span>]{}]{}. In the case of 258, the increase of the collective energy also implies a spreading of the fission yields. This is consistent with the experimental data showing strongly-peaked yields for spontaneous fission and much more smoothed ones for induced fission. It is important to emphasize that the major modes predicted do not change when varying the initial energy in a range of a few MeV around the fission barrier.
![Primary fragment mass yields computed with different Gaussian width $\sigma_i$ to build the initial wave packet. The width is given in MeV, and all yields are normalized to 200%.[]{data-label="fig:fm258_ya_isig"}](fm254_ya_isig.pdf "fig:"){width="44.00000%"} ![Primary fragment mass yields computed with different Gaussian width $\sigma_i$ to build the initial wave packet. The width is given in MeV, and all yields are normalized to 200%.[]{data-label="fig:fm258_ya_isig"}](fm258_ya_isig.pdf "fig:"){width="45.00000%"}
Although the initial energy of the system may be known in fission experiments, the quantum state of the compound system is not. We assumed here that the deformation of the initial state should be close to the one of the ground state with some fluctuation related to its excitation energy. To be conclusive, our results should however not depend too much on the details of the initial state. The impact of the choice of the initial state on [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} yields has already been explored in the fission of 240 and in Ref. [@regnier_fission_2016; @zdeb_fission_2017]. In these two cases, the characteristics of the main fission mode were not drastically affected by initializing the dynamics with different types of collective states (boosted Gaussian, Gaussian or Fermi mixing of eigen-modes of an extrapolated first potential well). To check the robustness of our conclusion in the case of the Fermium isotopes, we performed calculations with different values of the Gaussian width used to build the initial wave packet $\sigma_i = 0.1, 0.5, 1.0$ and $2.0$ MeV. In Fig. \[fig:fm258\_ya\_isig\] we show that the symmetric/asymmetric transition predicted holds whatever the value of $\sigma_i$. The most notable change occurs for $\sigma_i=0.1$ where the initial state reduces to a single eigen-mode of the extrapolated first well. In this extreme (and somewhat unrealistic) case the yields become indeed more sensitive to the characteristics of the selected eigen-mode.
Theory of Collective Motion
---------------------------
It is well known that building the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} on a basis made of only time-even generator states fails to capture some aspects of the dynamics of the system [@ring_nuclear_2004]. In the special case of translational motion, this leads to underestimating the collective inertia. To mitigate this issue, one possibility is to simulate the collective dynamics of the fissioning nucleus within the requantized adiabatic time dependent Hartree-Fock-Bogoliubov ([[<span style="font-variant:small-caps;">atdhf</span>]{}]{}) theory [@baranger_adiabatic_1978]. Indeed, the Pauli requantization scheme yields an evolution equation formally identical to Eq. , where the inertia becomes the [[<span style="font-variant:small-caps;">atdhf</span>]{}]{} inertia, the metric is the determinant of this inertia tensor, and the collective potential does not contain any zero-point energy contribution.
![Comparison of the primary mass yields obtained with the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} (full red line) and the [[<span style="font-variant:small-caps;">atdhf</span>]{}]{} (dashed black line) collective motion approaches. For completeness, we also give the results obtained with the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} approach without the zero point energy correction on the potential landscape (doted blue line). All the yields are normalized to 200%.[]{data-label="fig:fmtrans_mass"}](fmtrans_mass.pdf){width="45.00000%"}
Figure \[fig:fmtrans\_mass\] shows the comparison of primary mass yields in Fermium isotopes between the [[<span style="font-variant:small-caps;">atdhf</span>]{}]{} and [[<span style="font-variant:small-caps;">gcm</span>]{}]{} prescriptions. Although, the fission yields are significantly impacted by this change in the many-body method, the asymmetric/symmetric transition is still predicted at the correct neutron number for both of them. However, it is clear that the discrepancies resulting from different treatments of the collective dynamics ([[<span style="font-variant:small-caps;">atdhf</span>]{}]{} versus [[<span style="font-variant:small-caps;">gcm</span>]{}]{}) are more pronounced than the ones resulting from different choices of effective interactions, cf. Fig. \[fig:fmtrans\_inter\]. To pinpoint even more precisely the origin of these discrepancies, we computed the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} dynamics without including the zero-point energy correction ([[<span style="font-variant:small-caps;">zpe</span>]{}]{}) to the potential; see Fig. \[fig:fmtrans\_mass\]. Clearly, the energy correction plays a marginal role in determining the fission fragment mass distribution, and the main source of differences between [[<span style="font-variant:small-caps;">atdhf</span>]{}]{} and [[<span style="font-variant:small-caps;">gcm</span>]{}]{} results is the collective mass tensor. This suggests that including the physics of time-odd components into the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} (for instance as proposed in Ref. [@goeke_generator-coordinate-method_1980]) should be a priority if one is to improve the accuracy of these predictions.
Position of the Frontier {#sec:front}
------------------------
The definition of the frontier in our approach is strongly constrained by the discontinuity between the fission and fusion valleys. For the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} to be valid, the dynamics should only take place in a continuous manifold of generator states. As a consequence, the frontier must be before the transition from the fission to the fusion valley. In this paper we choose to compute the frontier as an isoline of the neck particle operator $\hat{Q}_N$. For this isoline to be before the fusion valley, we find that $Q_{\rm N}$ must be at least greater than 7.0. To simulate the dynamics up to configurations that are as close as possible to scission, the best choice is to put the frontier at the lowest possible isovalues of $Q_{\rm N}$ and we therefore choose $Q_{\rm N}=7.5$.
As mentioned in Section \[subsubsec:yields\], this definition of the frontier implies calculating the yields from configurations that still contain a sizeable number of particles in the neck, which results in a non-negligible nuclear interaction energy between the fragments. This is an intrinsic limitation of our 2-dimensional collective description of the process. Going beyond would require adding some missing intermediate states close to scission into the [[<span style="font-variant:small-caps;">gcm</span>]{}]{}. This could be achieved either by systematically adding some collective degrees of freedom, at the price of an exponential increase of the numerical cost, or by finding a better manifold of states connecting continuously the fission and fusion valleys. In both cases, such a study is beyond the scope of this work.
To assess the uncertainty coming from the arbitrary position of the frontier, we computed the yields for different frontiers defined by $Q_{\rm N}=7.0, 7.5, 8.0, 8.5$. Figure \[fig:ya\_qn\] shows the location of these frontiers on the [[<span style="font-variant:small-caps;">pes</span>]{}]{} as well as the yields obtained for the case of 254.
![(a): Isolines $Q_{\rm N}=7.0,7.5,8.0,8.5$ of the Gaussian neck operator used as frontiers to compute the fission yields of 254. (b): Variation of the primary fragment mass yields of 254 with the neck operator isoline used as frontier.[]{data-label="fig:ya_qn"}](fm254_front.pdf "fig:"){width="45.00000%"} ![(a): Isolines $Q_{\rm N}=7.0,7.5,8.0,8.5$ of the Gaussian neck operator used as frontiers to compute the fission yields of 254. (b): Variation of the primary fragment mass yields of 254 with the neck operator isoline used as frontier.[]{data-label="fig:ya_qn"}](fm254_ya_qn.pdf "fig:"){width="45.00000%"}
Although the modification of the frontier does impact the details of the resulting yields, the asymmetric fission picture remains unchanged. This is consistent with the fact that the dominant fission mode is determined at rather low quadrupole deformations ($Q_{20} \simeq 180$ b), much before reaching these frontiers. Similar results were found for the fission of 256 and 258. One might use the results of Fig.\[fig:ya\_qn\] to estimate the yields at the asymptotic limit of vanishing values of the neck.
Discontinuities in the $(Q_{20},Q_{30})$ Manifold {#subsec:discont}
-------------------------------------------------
To be mathematically valid, the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{}+[[<span style="font-variant:small-caps;">goa</span>]{}]{} formalism requires a continuous and twice differentiable manifold of generator states. In practice, the [[<span style="font-variant:small-caps;">pes</span>]{}]{} obtained by series of constrained [[<span style="font-variant:small-caps;">hfb</span>]{}]{} calculations does not necessarily satisfy this property. As stated in Ref. [@dubray_numerical_2012] a [[<span style="font-variant:small-caps;">pes</span>]{}]{} may contain discontinuous [[<span style="font-variant:small-caps;">hfb</span>]{}]{} states. To detect the presence of such discontinuities, we need to define a distance between [[<span style="font-variant:small-caps;">hfb</span>]{}]{} states. A fully quantum-mechanical distance could be provided based on the calculation of the overlaps between any pair of states [@verriere_fission_2017; @verriere_description_2017]. However, in this paper we use a much simpler metric $D$ based on the one-body local density, $$D({\mathbf{q}},{\mathbf{q}}') = \int |\rho_{{\mathbf{q}}}({\mathbf{r}}) - \rho_{{\mathbf{q}}'}({\mathbf{r}})| d^3{\mathbf{r}},$$ where ${\mathbf{q}}$ and ${\mathbf{q}}'$ refer to two [[<span style="font-variant:small-caps;">hfb</span>]{}]{} states of the [[<span style="font-variant:small-caps;">pes</span>]{}]{} and $\rho_{{\mathbf{q}}}({\mathbf{r}})$ and $\rho_{{\mathbf{q}}'}({\mathbf{r}})$ are their respective local, one-body local densities. This distance is only sensitive to the diagonal one body-density and does not involve the anomalous density. As a consequence, this metric may miss some discontinuities, in particular the ones related to pairing correlations.
To check the quality and validity of our 2-dimensional [[<span style="font-variant:small-caps;">pes</span>]{}]{}, we compute for each [[<span style="font-variant:small-caps;">hfb</span>]{}]{} state ${\mathbf{q}}$ the discontinuity indicator $I({\mathbf{q}})$, $$I({\mathbf{q}}) = \text{max} \left\{ D({\mathbf{q}},{\mathbf{q}}') \, |\, \forall\, {\mathbf{q}}' \text{neighbor of }{\mathbf{q}}\right\} .$$ We show in Fig. \[fig:fm256\_dist\] how this indicator allows us to identify discontinuities between neighboring areas of the potential energy surface for 256.
![Discontinuity indicator plotted on top of the 256 [[<span style="font-variant:small-caps;">pes</span>]{}]{}. The red color scale represents the value of the discontinuity indicator $I(k)$. Values below 5.5 are not plotted. The background color map represents the potential energy surface. Finally, the black dashed line is the frontier corresponding to $Q_{\rm N}=7.5$. For the sake of legibility, we removed the points belonging to the non-converged island of high energies in the fusion valley (see Fig. \[fig:fmtrans\_pes\] for comparison).[]{data-label="fig:fm256_dist"}](fm256_dist.pdf){width="50.00000%"}
By plotting the highest values of the discontinuity indicator, we clearly see various lines which correspond to sharp discontinuities between neighboring [[<span style="font-variant:small-caps;">hfb</span>]{}]{} states. The topology and geometry of these lines were found to be similar for the three Fermium isotopes and they could be classified as follows:
#### North-west sector of the [[<span style="font-variant:small-caps;">pes</span>]{}]{}
In this region of exotic shapes, the density of discontinuities is rather high. However, the potential energy associated with such configurations is at least 10 MeV above the energy of the [[<span style="font-variant:small-caps;">gcm</span>]{}]{} ground state and quite far from the principal fission valleys. The collective wave packet does not populate this area during the evolution and the associated discontinuities therefore have no impact on the resulting yields.
#### Top of the first potential barrier
A discontinuity line is present at the top of the first fission barrier at elongations around $Q_{20}=70$ b. This typically indicates that our 2-dimensional description underestimates the height of this first potential barrier. While this could impact significantly the calculation of fission half-lives for example, we expect that it is less important for the mass distributions, since this discontinuity does not affect the competition between the different fission modes. To gain some information on the potential influence of this discontinuity on the fission yields, a 3-dimensional study involving the hexadecapole mass moment operator $\hat{Q}_{40}$ should be done in the future.
#### Ridge between the main symmetric and asymmetric valleys
Between the two main asymmetric and symmetric valleys also lies a discontinuity line at the top of the potential ridge. The values of $Q_{40}$ are lower in the symmetric valley than in the asymmetric valleys. At the frontier between the symmetric and asymmetric valleys, a gap in $Q_{40}$ values can be seen. This is a signature of two separated valleys in the 3-dimensional space $(Q_{20},Q_{30},Q_{40})$ that are now overlapping in our 2-dimensional working space. However, this discontinuity line is roughly parallel to the direction of the main flow of the collective wave packet, which follows the bottom of the valleys. In the case of 254 and 258, the collective wave packet follows mostly one valley and we expect that the spurious flux crossing this line is small compared to the total flux crossing the frontier. In this scenario, the fission yields would not be so much impacted. On the other hand, for 256, this discontinuity could drastically affect the competition between the symmetric and asymmetric modes. This could be partly responsible for the strong symmetric component found in 256 for which the potential energy in this region is pretty flat.
#### Fission/fusion transition
Finally, a discontinuity line starting around $Q_{20}=250$ b for symmetric configurations and going up to large asymmetries corresponds to the transition from the fission valley to split configurations. This ’scission’ discontinuity has already been extensively discussed in the literature (see for instance Ref. [@dubray_structure_2008; @younes_microscopic_2009]). It is one of the main limitations of our approach as it imposes to select a frontier on its left hand side and therefore compute the yields on a set of configurations with a high neck operator value.
To conclude on the subject of discontinuities, we clearly see that they are present in the fission valleys in our 2-dimensional description. With the exception of the ’scission’ one, these discontinuities are mostly signaled by a jump in the value of the $Q_{40}$ multipole moment. Adding this variable into our dynamical description would therefore remove most of these “internal” discontinuities.
Conclusion
==========
We computed the primary fragment mass and charge yields for the low-energy induced fission of $^{254,256,258}$Fm within the [[<span style="font-variant:small-caps;">tdgcm</span>]{}]{} under the Gaussian overlap approximation. The results obtained with the [[<span style="font-variant:small-caps;">d1s</span>]{}]{} parameterization of the Gogny effective interaction successfully reproduce the expected transition from a mostly asymmetric fission for 254 to a mostly symmetric one for 258. This transition is interpreted in the framework of collective dynamics as a competition between different modes that depends on the number of neutrons in the system. Most of the physics of the transition can already be inferred from the static analysis of the [[<span style="font-variant:small-caps;">pes</span>]{}]{} and we show that the bifurcation point responsible for the transition happens at quite low elongations $Q_{20}\simeq 180$ b. In addition, our calculations suggest two asymmetric modes for the fission of 254. The sensitivity of our results to all the inputs of the calculation has been tested and we find that the qualitative picture is robust. Finally, we show that one of the main limitations of this approach is the presence of discontinuities that appear even at low deformation inside the fission valley. In the case of the Fermium isotopes considered in this study, most of these discontinuities are signaled by an abrupt change in the $Q_{40}$ multipole moment value. Extending the calculations to 3-dimensional collective spaces may be sufficient to solve this problem.
Acknowledgements
================
We would like to thank H. Pasca for fruitful discussions about statistical fission models. Support for this work was partly provided through the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Nuclear Physics. It was partly performed under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Computing support for this work came from the Lawrence Livermore National Laboratory (LLNL) Institutional Computing Grand Challenge program.
Spectral element discretization of the collective dynamics {#ap:setup_param}
==========================================================
The first step in the numerical resolution of the collective dynamics consists in building the spectral element basis spanning the collective space of interest, and expanding the collective Hamiltonian on this basis. To do so, we used a slightly modified version of the tool [flx-setup]{} provided with the [[F<span style="font-variant:small-caps;">elix</span>]{}]{} package. This tool proceeds through several steps to transform the information contained in a raw ensemble of constrained [[<span style="font-variant:small-caps;">hfb</span>]{}]{} generator states into relevant inputs for the dynamics. For the sake of reproducibility, we summarize in this section the main steps of this setup and report in Tab. \[tab:setup\_param\] the exhaustive inputs to [flx-setup]{}. The full details on this setup procedure can be found in Ref. [@regnier_felix-2.0:_2018].
Starting from a ensemble of $Q_{30}>0$ configurations, the setup tool first select only states having a neck operator value above a certain threshold. In order to keep only the fission valley and avoid the discontinuity between the fission and fusion valleys we choose the criterion $Q_{\rm N}>7.0$. This choice is discussed in more details in Sec. \[sec:front\]. The deformation domain is then augmented with an absorption band of width 30 in barn units. As presented in our previous work, this band contains an additional Hamiltonian term to absorb progressively the collective wave packet and avoid reflections on the boundaries of the domain. The absorption is parametrized by an absorption rate $r=100$ zs$^{-1}$ and a characteristic width $w=30$ in barn units (cf. [@regnier_felix-1.0:_2016]).
Guided by the numerical convergence benchmarks performed on 256 in Ref. [@regnier_felix-2.0:_2018], we choose to discretize the collective Schrödinger equation on a spectral element basis built with degree-3 polynomials. The spatial domain is partitioned as a mesh of squared cells of size $h_{20}=4.24$ b, $h_{30}=1.41$ b$^{3/2}$. Within a distance 50 (in barn units) to the ground-state, we perform one step of h-refinement for those cells for which the energy at the center is lower than 40 MeV above the ground state. This refinement in the first potential well, where the collective wave function has its most rapid variations, accelerates the numerical convergence of the solution with respect to the dimension of the spectral element basis. Inside the initial domain (defined by $Q_{\rm N}>7.0$), the fields of the collective Hamiltonian are estimated at the nodes of the finite element basis by linear interpolation between constrained [[<span style="font-variant:small-caps;">hfb</span>]{}]{} results. In the absorption band, all the fields are extrapolated continuously based on their distance to the initial domain in the same way as in Ref. [@regnier_fission_2016].
Once the finite element basis and all the necessary fields are determined in the $Q_{30}>0$ region, the whole domain is symmetrized so that the dynamics is performed in a box containing configurations with both positive and negative octupole moments. The collective Hamiltonian is assumed to be symmetric with respect to the $z \rightarrow -z$ transformation. One can show that this assumption implies the symmetry of the fields involved in the collective Hamiltonian and the anti-symmetry of the non-diagonal elements of the inertia tensor, under the action of this transformation. Note that in the [[F<span style="font-variant:small-caps;">elix</span>]{}]{}-2.0 release, the [flx-setup]{} tool assumes for this operation that all the fields are symmetric. We had to modify this behavior here so that the non-diagonal part of the inertia are instead anti-symmetrized during this step. This was the only modification brought to [flx-setup]{}.
Option for [flx-setup]{} Value
-------------------------- ---------------
abs-rate 10
abs-width 30
alpha 5
cell hcube
deg 3
eigen-nstates 100
eigen-tol 1e-13
eigen-vmax 50.
extrapol-width 30.
gs 30.,0.0
gs-extrapol-radius 50.
gs-hrefine-vmax 40.
mesh-step 4.24,1.41
outer-well 100.,0.0
qN-cut 7.0
quad-h gaussLegendre
quad-m gaussLobatto
saddle-vmax 30.
scale 1.,1.
v-slope 4e-2
: Inputs used for the setup of the dynamics with [[F<span style="font-variant:small-caps;">elix</span>]{}]{}-2.0.[]{data-label="tab:setup_param"}
|
---
abstract: 'For continuous maps on a compact manifold $M$, particularly for those that do not preserve the Lebesgue measure $m$, we define the *observable invariant probability measures *as a generalization of the physical measures. We prove that any continuous map has observable measures, and characterize those that are physical in terms of the observability. We prove that there exist physical measures whose basins cover Lebesgue a.e, if and only if the set of all observable measures is finite or infinite numerable. We define for any continuous map, its *generalized attractors *using the set of observable invariant measures where there is no physical measure, and prove that any continuous map defines a decomposition of the space in up to infinitely many generalized attractors whose basins cover Lebesgue a.e. We apply the results to the $C^1$ expanding maps $f$ in the circle, proving that the set of all observable measures (even if $f$ is not $C^{1 + \alpha}$) is a subset of the set of all the equilibrium states of $-\log |f''|$.****'
author:
- 'Eleonora Catsigeras and Heber Enrich [^1]'
date: 'February 18th., 2010.'
title: 'Observable invariant measures.'
---
Introduction
============
It is an old problem to find good" probability measures for a map $f\colon M \mapsto M$, meaning for that, an invariant probability that resume in some sense, the dynamics by iterations of the map. Sometimes, the map is born with a good measure, as in the case of maps preserving a Lebesgue ergodic measure. But this is not true in general, and it is not an easy question to determine, in most examples, a single or a few probability measures representing the dynamics of the map.
There have been proposed several ideas to define a good"invariant probability measure $\mu$:
1. \[phys\] Lebesgue a.e. point in a set is generic with respect $\mu$, that is, $\displaystyle \mu =\lim_{n\to
\infty}\frac1n\sum_{j=0}^{n-1} \delta _ f^j(x)$, where the convergence is in the weak$^*$ topology of the space ${\cal P}$ of probabilities on $M$.
2. \[2srb\] The conditional measures of $\mu$ on unstable manifolds are absolutely continuous respect to the Lebesgue measure along those manifolds.
3. \[pesin\] $\mu$ verifies the Pesin-Ledrappier-Young (PLY) equality: $$h_\mu (f) =\int \sum_i
\lambda^+_i(x) \dim E_i(x) \, d\mu(x)$$ where $h_{\mu}(f)$ is the entropy of $\mu$, and $\dim E_i(x)$ is the multiplicity of the positive Lyapunov exponent $\lambda^+(x)$ in the Oseledec’s decomposition.
4. \[stoch\] The measure is the limit of measures which are invariant under stochastic perturbations.
It is a remarkable property that the four definitions above are equivalent for Axiom A attractors. Moreover, Ledrappier and Young (see [@ledrappieryoung] and [@ledrappieryoung2]), under suitable hypothesis of differentiability, proved that a measure verifies property \[2srb\] if it verifies property \[pesin\] while the converse result is the well known result of Pesin’s entropy formula. Ergodic measures verifying \[2srb\] (or, equivalently, \[pesin\]) with no zero Lyapunov exponents describe chaotic behavior, and are accompanied by rich geometric and dynamical structures (see [@young]). Nevertheless, the other measures listed above are also interesting, because reveal statistical aspects of the behavior of the future iterations of the map. For instance, a measure verifying definitions \[phys\] or \[stoch\] is concentred in the part of the space which is statistically more visited.
We will call physical measure, a probability measure verifying \[phys\], and stochastically stable, a probability measure verifying \[stoch\]. We will call SRB (Sinai-Ruelle-Bowen) measure a probability verifying \[2srb\], and a PLY measure, a probability verifying \[pesin\]. In this work, we propose another concept of good" probabilities, which we call observable measures. The following question was the motivation of this work: Is it possible to describe probabilistically in the space, in some *minimal way, and in a very general regular or irregular setting, *the asymptotic behavior of the time averages of Lebesgue a.e. orbit? We answer this question in Theorem \[toeremaminimal0\].**
Generalized ergodic attractors and observable measures, that we define and theoretically develop along this work, do always exist for any continuous map (Theorems \[teoremaexistencia0\] and \[teoremaDescomposicionGenErgAttr0\]). On the other hand, physical measures and ergodic attractors do not necessarily exist (see examples \[ejemplodossillas\] and \[ejemploExpanding\]).
It is largely known the difficulties to characterize, or just find, non hyperbolic or non $C^{1+ \alpha}$ maps that do have physical measures. This is a hard problem even in some systems whose iterated topological behavior is known ([@carvalho], [@enrich], [@hu], [@huyoung]). The difficulties appear when trying to apply to a non hyperbolic setting, or to a non $C^{1+ \alpha}$ map, the known techniques for constructing the physical measures of hyperbolic $C^{1 + \alpha}$ maps ([@pesin], [@sinai], [@anosov]). The $C^{1 + \alpha}$ hypothesis allows the existence of SRB measures ([@bowen3], [@ruelle], [@sinai]) and are relevant and widely studied occupying an important focus of interest in the ergodic differentiable theory of dynamical systems ([@anosov], [@pesinsinai], [@pughshub2], [@viana], [@bonattiviana]). But if having a weak or non uniform hyperbolic setting, the obstruction usually resides in the irregularity of the invariant manifolds, which technically translate the non trivial relations between topologic, measurable and differentiable properties of the system ([@pughshub2]). The difficulties arise even for maps posed in a very regular setting as Lewowicz diffeomorphisms in the two-torus (see [@lewowicz]). For them, the differentiable regularity of the given transformation (they are analytic maps) and the topological known behavior of the iterated system (they are conjugated to Anosov maps), were not enough, up to the moment, to prove the existence of physical measures and ergodic attractors, except in some meager set of examples [@nosotros]. Before stating the results we need to formalize some definitions that we will use all along this work: Let $f\colon M \mapsto M$ be a continuous map with $M$ a compact, finite-dimensional manifold. Let $m$ be a probability Lebesgue measure, and not necessarily $f$-invariant. We denote $\cal P$ the set of all Borel probability measures in $M$, provided with the weak$^*$ topology, and a metric structure inducing this topology.
For any point $x \in M $ we denote $p\omega (x)$ to the set of the Borel probabilities in $M $ that are the partial limits of the (not necessarily convergent) sequence $$\label{1} \left\{ \frac{1}{n} \; \sum_{j=0}^{n-1} \delta
_{f^j(x)} \right\} _{ n \in \mathbb{N}}$$ where $\delta_y$ is the Dirac delta probability measure supported in $y
\in M$.
The set $p\omega (x) \subset \cal P$ is the collection of the spatial probability measures describing the asymptotic time average (given by (\[1\])) of the system states, provided the initial state is $x$. If the sequence (\[1\]) converges then we denote $p\omega (x)
= \{\mu _x\}$. To include also those cases for which the sequence (\[1\]) is not convergent (for a set of orbits with positive Lebesgue measure) we consider, for a given measure $\mu$, the set of points $x \in M$ such that the minimum distance between $\mu$ and the set of partial limits of the sequence (\[1\]) is small. We define:
*\[definicionobservable\] \[definicion1\] [**(Observable probability measures.)**]{} A probability measure $\mu \in \cal P$ is *observable *if for all $\varepsilon
>0 $ the set ${A}_{\varepsilon }= \{x \in M:
\mbox{ dist}^* ( p\omega(x), \mu )< \varepsilon \}$ has positive Lebesgue measure. The set ${A}_{\varepsilon} = A_{\varepsilon}(\mu) \subset M$ is called *the $\varepsilon$- basin of partial attraction *of the probability $\mu$.*****
We note that the definition above is independent of the choice of the distance in $\cal P$, provided that the metric structure induces its weak$^*$ topology. Observable measures are $f$-invariant, and usually at most a few part of the space of invariant measures for $f$ are observable measures (see the examples in Section \[ejemplos2\]). We remark that for observable measures, the condition $m(A_\varepsilon) >0$ must be verified not only *for some *but *for all *$\varepsilon >0$.****
*\[definicionsrb\] [**(Physical probability measures.)**]{} A probability measure $\mu \in \cal P$ is *physical *(even if it is not ergodic), if the set $B = \{x \in M:
p\omega(x)= \{\mu\} \}$ has positive Lebesgue measure. The set $B= {B} (\mu) \subset M$ is called the *basin of attraction *of $\mu$.*****
Therefore, all physical measures are observable, but not all observable measures are physical (Examples \[ejemplodossillas\] and \[ejemploExpanding\]).
We state the following starting results:
\[teoremaexistencia0\] *[ **(Existence of observable measures and physical measures.)** ]{} ***
[**a)**]{} For any continuous map $f$, the set $\cal O$ of all observable probability measures for $f$ is non-empty and weak$^*$- compact.
[**b)**]{} $\cal O$ is finite or countably infinite if and only if there exist *(resp. finitely or countable infinitely many) *physical measures of $f$ such that the union of their basins of attraction cover Lebesgue a.e.**
The first statement of this theorem is proved in paragraph \[teoremaexistencia\] and the second one in paragraph \[srb\].
The $p\omega(x)$ limit set of convergent subsequences of (\[1\]) may have many different partial limit measures. Nevertheless, we prove that $p\omega (x)$ is formed only with observable measures, for Lebesgue almost all $x \in M$, as stated in the following Theorem \[toeremaminimal0\].
*\[definicionbasin\] [**(Basin of attraction.)**]{}*
The *basin of attraction $B({\cal K})$ *of a compact subset ${\cal K}$ of the space ${\cal P}$ of all the Borel probability measures in $M$, is the (maybe empty) subset of $M$ defined as: $$B({\cal K}) = \{x \in M: p\omega(x) \subset {\cal K} \}$$**
If the purpose is to study the asymptotic to the future time average behaviors of Lebesgue almost all points in $M$, then the *consideration of the set $\cal O$ of all the observable measures for $f$, is the necessary and sufficient condition. *In fact we have the following:**
\[toeremaminimal0\] *[**(Attracting minimality property of the set of observable measures.)**]{} *The set $\cal O$ of all observable measures for $f$ is the minimal compact subset of the space $\cal P$ whose basin of attraction has total Lebesgue measure.**
We prove this theorem in paragraph \[pruebaGenErgAttract\].
The theory about the observable measures describe the statistical asymptotic behavior of time averages, instead of the theory of physical measures when these last probabilities do not exist. In particular it is suitable to study the statistics of the future iterations of maps, disregarding their regularity, that do not preserve a Lebesgue equivalent measure, or that do preserve it but are not ergodic. In other words, the results about the observable measures, independently if they are or not useful to find physical measures in some concrete examples, substitute the physical measures, and do that in a wide setting of dynamical systems (all the continuous maps), without loosing their statistical meaning (see Proposition \[propositionAtraeEnMedia\].)
In Section \[seccion2\] we define *attractor $A$, *to the support in $M$ of a physical, non necessarily ergodic measure $\nu$, similarly as done by Pugh and Shub in [@pughshub]. Analogously, we call *generalized attractor, *to the union $A$ of the supports of an adequately reduced weak$^*$-compact family ${\cal O}_1$ of observable measures. We will not require strict topological attraction to $A$, but weak topological: in the average the orbit lays in an arbitrarily small neighborhood of $A$ as much time as wanted. (Proposition \[propositionAtraeEnMedia\].) In Theorem \[teoremaDescomposicionGenErgAttr0\] we prove that there always exist up to countable many generalized attractors whose basins cover Lebesgue a.e. The following open question refers to the existence and finiteness of physical measures and to the convergence of the sequence (\[1\]) of time averages of the system for a set of initial states with total Lebesgue measure. It is possed in [@palis]:****
*\[Palis1\] [**Palis Conjecture** ]{} *Most dynamical systems have up to finitely many physical measures (or ergodic attractors) such that their basins of attraction cover Lebesgue almost all points. ****
This conjecture admits the following equivalent statement, that seems weaker. (In fact, the definition \[definicionobservable\] of observability is certainly weaker than the definition \[definicionsrb\] of physical measures.)
*\[Palis2\] [**Equivalent formulation of Palis Conjecture:**]{}*
*For most dynamical systems the set of observable measures is finite. ***
Note: To prove the equivalence of statements \[Palis1\] and \[Palis2\] it is enough to join the results of Theorems \[teoremaexistencia0\].b and \[toeremaminimal0\].
For systems preserving the Lebesgue measure the main question is their ergodicity. It is immediate the following result:
\[teoremaLebesgue0\] *[**(Observability and ergodicity.)**]{}*
If $f$ preserves the Lebesgue measure $m$ then the following assertions are equivalent:
*(i) *$f$ is ergodic respect to $m$.**
*(ii) *There exists a unique observable measure $\mu$ for $f$.**
*(iii) *There exists a unique physical measure $\nu $ for $f$ attracting Lebesgue a.e.**
Besides, if the assertions above are verified, then $m = \mu =
\nu$
Given a Lebesgue measure preserving map $f$, the question if $f$ has a physical measure is mostly open, for differentiable maps that do not have some kind of uniform total or partial hyperbolicity [@viana]. The existence of a physical measure attracting Lebesgue a.e. is equivalent to the ergodicity of the map, and is also an open question if most of conservative maps are ergodic ([@pughshub2], [@bonattivianawilk]). The key difficult point resides in those maps that do not have any kind of uniform total or partial $C^1$ hyperbolicity in the space, as in the inspiring Lewowicz examples in the two torus [@lewowicz]. Due to Remark \[teoremaLebesgue0\] those open questions are equivalent to the *uniqueness of the observable measure. ***
Attractors {#seccion2}
==========
Due to the conjecture in \[Palis1\] and Theorem \[teoremaexistencia0\], we are interested in partitioning the set $\cal O$ of observable measures, or to reduce it as much as possible, into different compact subsets whose basins of attractions have positive Lebesgue measure. Due to Theorem \[toeremaminimal0\], no proper compact part of $\cal O$ has a total Lebesgue basin. We define:
\[definicionreduccion0\] *[**(Generalized Attractors - Reductions of the space ${\cal O}$.)**]{} A *generalized attractor $(A, \cal A) \subset M \times {\cal O}$, *(or a *reduction *${\cal A}$ of the space ${\cal O } $ of all observable measures for $f$), is a compact subset $(A, \cal A)$ such that the basin of attraction $B({\cal A})= \{x \in M: p \omega
(x) \subset {\cal A}\}$ has positive Lebesgue measure in $M$, and $A $ is the (minimal) compact support in $M$ of all the probability measures in ${\cal A}$. We call $(A, \{\mu\})$ an *attractor *if it is a generalized attractor with a single invariant probability $\mu$, i.e. ${\mu}$ is a physical measure.*******
*Sometimes we refer to a generalized attractor only to $A$ or only to ${\cal A}$. The *irreducible *generalized attractors (if they exist), attract the time averages distributions and are minimal in some sense, but are not formed necessarily with ergodic measures (Example \[ejemplodossillas\]). In spite a system could not exhibit a physical measure, still the reductions of the space of observable measures divide the manifold in the basins of *generalized attractors. *Each reduction ${\cal A}$ has a basin $C = C({\cal A})$ with positive Lebesgue measure and is minimal respect to $C$. We state this result as follows:*****
*[**(Minimality of generalized attractors.)**]{} *\[teoremaminimal0r\] Any generalized attractor ${\cal A}$ is the minimal compact set of observable measures attracting its basin $C({\cal A})$. More precisely $$m (C({\cal A}) \setminus C({\cal K}) ) >0 \ \ \ \mbox{
for all compact subset } \ \ \ {\cal K} \subset {\cal A}.$$**
We prove Theorem \[teoremaminimal0r\] in paragraph \[pruebaGenErgAttract\].
*[**Irreducibility** ]{} \[definicionirreducible0\]*
A generalized attractor ${\cal A}\subset {\cal P} $ is *irreducible *if it does not contain proper compact subsets that are also generalized attractors.**
It is *trivial *or *trivially irreducible *if its diameter in ${\cal P }$ is zero , or in other words, if ${\cal A}$ has a unique observable measure $\mu$.****
In other words: physical measures are trivially irreducible and conversely.
The following result is much weaker than, but related with, the Palis’ conjecture stated in paragraph \[Palis1\]:
\[teoremaDescomposicionGenErgAttr0\] [**(Decomposition Theorem)**]{}
For any continuous map $f\colon M \mapsto M$ there exist a collection of (up to countable infinitely many) generalized attractors whose basins of attraction are pairwise Lebesgue-almost disjoint and cover Lebesgue-almost all $M$.
The space of all continuous maps divide in two disjoint classes:
$\bullet$ The generalized attractors of the decomposition are all irreducible and then the decomposition is unique.
$\bullet $ For all $\varepsilon >0$ there exists a decomposition for which the reducible generalized attractors have all diameter (in the weak$^*$ space of probabilities), smaller than $\varepsilon$ **
We prove this theorem in paragraph \[teoremadescomposicionGenArgAttract\]. Note that, in the second class of systems, as the reducible generalized attractors have in the space of probabities a small diameter, for a rough observer, each of those attractors acts as a physical measure.
In Theorem \[teoremaPhysicalMeasuresAndChains\] and Corollary \[corolarioCo-cadenaSRB\] we characterize those maps whose generalized attractors are the support of physical measures, as asked in the statement of Palis conjecture.
*[**Milnor attractor.**]{} [@milnor] \[definicionMilnor\] \[Milnor2\] A Milnor attractor is a compact set $A \subset M$ such that its topological basin of attraction $$B(A) = \{ x \in M: \omega (x) \subset A \}$$ has positive Lebesgue measure, and for any compact proper subset $K \subset A $ the set $$B(A) \setminus B(K) = \{x \in M: \omega (x) \subset A ,\; \omega (x)
\not \subset K \}$$ also has positive Lebesgue measure.*
Note: Here $\omega (x)$ is the $\omega$-limit set in $M$ of the orbit of $x$, i.e. the set of limit points in $M$ of the orbit with initial state $x$.
The generalized attractors $(A, {\cal A})$ where ${\cal A} \subset {\cal P}$, were inspired in the definition \[definicionMilnor\]. They play in ${\cal P}$ a similar topological role that Milnor attractors play in $M$. In particular, the physical measures in $\cal P $ play the role that sinks do in $M$, those first considered as punctual attractors of time averages probabilities, and these last considered as punctual attractors of the points in $M$, along the future orbits.
Proofs of Theorems \[teoremaexistencia0\] (a) and \[toeremaminimal0\].
======================================================================
*[**(Weak$^*$ topology in the space of probability measures.)** ]{} \[definicion0debilestrella\]*
The weak$^*$ topologic structure in the space $\cal P$ is defined as: $$\mu_n \rightarrow \mu \; \;
\mbox{in } {\cal P} \;\; \; \mbox{ iff } \;\; \; \lim \int \phi \, d
\mu _n = \int \phi \, d \mu \;\; \mbox{ for all } \phi \in C^0(M, \mathbb{R})$$ where $C^0(M, \mathbb{R})$ denotes the space of the continuous real functions in $M$.
A classic basic theorem on Topology states that the space $\cal P$ is compact and metrizable when endowed with the weak$^*$ topology [@mane]. Let us denote as ${\cal P}_f \subset {\cal P}$ the set of the Borel probability measures in $M$ that are $f$-invariant, that is $\mu ({f^{-1}(B)}) = \mu (B)$ for all Borel set $B \subset M$. Note that the Lebesgue measure $m$ does not necessarily belong to ${\cal
P}_f$. Fix any metric in $\cal P$ giving its weak$^*$ topology structure. We denote as ${\cal B}_\varepsilon (\mu)$ to the open ball in $\cal P$ centered in $\mu \in \cal P$ and with radius $\varepsilon
>0$.
\[definicion0\] *[**The $p \omega$- limit sets.**]{}*
At the beginning of this paper we defined, for each initial state $x \in M$, the set $p\omega (x)$ in the space $\cal P $ as the partial limits, in the weak$^*$ topology in $\cal P$, of the sequence (\[1\]) of time averages. In other words: $$p\omega (x) = \left\{\mu \in {\cal P}: \lim \frac{1}{n_i} \sum_{j=0}^{n_i-1}
\phi({f^j(x)}) = \int \phi \, d \mu \; \forall \; \phi \in C^0(M,
\mathbb{R}) \; \mbox { for some } n_i \rightarrow + \infty\right\}$$ For further uses we state here the following property for the $p\omega$-limit sets:
*\[teoremaconvexo\] [**Convex-like property.**]{}*
For any point $x \in M$
i\) *If $\mu, \nu \in p\omega(x)$ then for each real number $ 0
\leq \lambda \leq 1$ there exists a measure $\mu_{\lambda} \in
p\omega (x)$ such that $${\mbox{$\,$dist$\,$}}(\mu_{\lambda}, \mu) = \lambda
{\mbox{$\,$dist$\,$}}(\mu, \nu)$$*
ii\) *The set $p\omega(x)$ *either has a single element or non-countable infinitely many.**
[*Proof:*]{} See \[prueba\] in the appendix.
*\[teoremaexistencia\] [**Proof of Theorem \[teoremaexistencia0\] (a):** ]{} (The proof of the assertion (b) of Theorem \[teoremaexistencia0\] is delayed until paragraph \[srb\].)*
Let us prove that, given any continuous map $f$, the set $ {\cal O
}_f $ of the probability measures that are $\varepsilon$-observable for all $\varepsilon
>0$ is a non-empty compact subset of the (weak$^*$ topologic) space ${\cal
P}_f$ of the $f$-invariant measures.
The key question is that ${\cal O}_f$ is not empty, which we prove at the end.
Let us first prove that ${\cal O}_f \subset {\cal P}
_f$. In fact, given $\mu \in {\cal O}_f$ then, for any $\varepsilon
= 1/n
>0$ there exists some $\mu _n \in {\cal B}_{\varepsilon} (\mu)$ which is the limit of a convergent subsequence of (\[1\]) for some $x \in M$. As the limits of all convergent subsequences of (\[1\]) are $f$-invariant, we have that $\mu _n \in {\cal P}_f \subset {\cal P}$ for all natural number $n$, and $\mu _n \rightarrow \mu$ with the weak$^*$ topologic structure of $\cal P$. The space ${\cal P}_f$ is a compact subspace of $\cal P$ with the weak$^*$ topologic structure, so $\mu \in {\cal P}_f$ as wanted.
Second, let us prove that ${\cal O} = {\cal O }_f$ is compact. The complement ${\cal O}^c$ of $\cal O$ in $\cal P$ is the set of all probability measures $\mu$(not necessarily $f$-invariant) such that for some $\varepsilon = \varepsilon (\mu) >0$ the set $\{x \in M:
p\omega(x) \cap {\cal B}_{\varepsilon}(\mu) \neq \emptyset\}$ has zero Lebesgue measure. Therefore ${\cal O}^c$ is open in $\cal P$, and ${\cal O}$ is a closed subspace of $\cal P$. As $\cal P$ is compact we deduce that $\cal O$ is compact as wanted.
Third, let us prove that $\cal O$ is not empty. Suppose by contradiction that it is empty. Then ${\cal O}^c = {\cal P}$, and for every $\mu \in {\cal P}$ there exists some $\varepsilon =
\varepsilon (\mu) >0$ such that the set $A= \{x \in M: p\omega(x)
\subset ({\cal B}_{\varepsilon}(\mu))^c \}$ has total Lebesgue probability.
As $\cal P$ is compact, let us consider a finite covering of $\cal
P$ with such open balls ${\cal B}_{\varepsilon}(\mu)$, say $B_1,
B_2, \ldots B_k$, and their respective sets $A_1, A_2, \ldots A_k$ defined as above. As $m(A_i) = 1$ for all $i= 1, 2, \ldots, k$ we have that the intersection $B= \cap_{i=1}^k A_i$ is not empty. By construction, for all $x \in B$ the $p\omega$-limit of $x$ is contained in the complement of $B_i$ for all $i = 1, 2 \ldots, k$, and so it would not be contained in $\cal P$, that is the contradiction ending the proof. $ \Box $
Note that the proof of Theorem \[teoremaexistencia0\] (a) does not use any property of the Lebesgue measure on $M$ different from those that any Borel probability $m$ on $M$ also has. The same result works (but maybe defining a different subset of observable measures) if any other, non necessarily invariant probability measure $m$ in ${\cal P}$ is used, as the reference probability distribution for the choice of the initial state $x$, instead of the Lebesgue measure. If so, the concept of *physical *measure also changes accordingly. Nevertheless, along this work, we are calling $m$ to the Lebesgue measure, i.e. the volume form measure, given by the Riemannian metric on the manifold $M$, adequately rescaled to be a probability: $m(M) = 1$.**
*\[definicionObservSize\] [**Observability size.**]{} If $\mu \in {\cal P}$ is a (non necessarily invariant) probability measure (in particular if it is an observable measure, see Definition \[definicion1\]), we call *observability size of $\mu$ *to the non negative real function $o = o_{\mu}\colon
\overline{\mathbb{R}^+} \mapsto \overline{\mathbb{R}^+}$ defined as $$o_{\mu } (\varepsilon)= m (A(\varepsilon ,\mu ))$$ where $m$ is the Lebesgue measure in $M$ and $A(\varepsilon ,\mu )$ is the set $$A(\varepsilon ,\mu ) = \{x \in M: p\omega (x) \cap {\cal B}_{\varepsilon }(\mu )
\neq \emptyset \} \; \; \; \; \; \ \mbox{ being } \; \; \; \ {\cal
B}_{\varepsilon }(\mu) = \{\nu \in {\cal P}: \mbox{ dist}^*(\nu,
\mu) < \varepsilon\}$$***
For some fixed $\varepsilon >0$, we say that $\mu \in {\cal P}$ is $\varepsilon-$ observable if $o_{\mu }(\varepsilon) >0$.
*For any probability measure $\mu$, its observability size function $o (\varepsilon )$ *is positive and decreasing with $\varepsilon
>0$. *Then $o(\varepsilon )$ has always a non-negative limit value when $\varepsilon \rightarrow 0^+$. We reformulate Definition \[definicionobservable\] of observability of measures, in the following equivalent terms:***
\[definicion2\] *[**Remark:** ]{} [**(Observability revisited.)**]{} $\mu \in {\cal P} $ is *observable *for $f$, if and only if it is $\varepsilon$-observable (see \[definicion1\]) for all $\varepsilon
>0$. In particular, $\mu$ is physical if and only if $\lim _{\varepsilon \rightarrow 0} o_{\mu}(\varepsilon) >0$.***
The characterization of those continuous maps having physical-measures as those whose sets of observable measures, or some reductions of them, are finite or countable infinite (Theorem \[srb\]), derives the attention to try to define and find sufficient conditions to reduce as much as possible the set of observable measures.
Besides, the *reductions *of the space of observable measures will work as *Generalized Ergodic Attractors, *even in the case that this reduction can not be done as much as to obtain physical measures. We first prove that the reducibility of the set $\cal O$ of observable measures for $f$ must be defined carefully, because in the following sense, this set $\cal O$ is minimal. ******
\[teoremaminimal\] *[**(Reformulation of Theorem \[toeremaminimal0\])**]{}*
*Let $f\colon M \mapsto M$ be any given continuous map in the compact manifold $M$.*
The set ${\cal O }_f $ of all its observable measures belongs to the family $$\aleph= \{ {\cal K} \subset {\cal P}: {\cal K} \mbox{ is compact
and } p\omega (x) \subset {\cal K} \mbox{ for Lebesgue almost
every point } x \in M \}$$
Moreover $${\cal O}_f = \bigcap _{{\cal K} \in \aleph} {\cal
K}$$ and thus ${\cal O}_f$ is the unique minimal set in $\aleph$.
[*Proof:*]{} For simplicity let us denote ${\cal O}={\cal O}_f$. Given any subset $\cal K \subset \cal P$ (this $\cal K$ is neither necessarily in $\aleph$ nor necessarily compact), let us consider: $$\label{2}
A({\cal K})= \{x \in M: p\omega(x)\cap
{\cal K} \neq \emptyset \}, \;\;\;\;\;\;\; C({\cal K})= \{x \in M:
p\omega(x)\subset {\cal K} \}$$
It is enough to prove that $m(C({\cal O})) =1$ and that ${\cal K} \supset
\cal O$ for all ${\cal K} \in \aleph$. Let us first prove the second assertion.
To prove that ${\cal K} \supset \cal O$ it is enough to show that $\mu \not \in {\cal O}$ if $\mu \not \in {\cal K} \in \aleph$.
If $\mu \not \in {\cal K}$ take $\varepsilon = {\mbox{$\,$dist$\,$}}(\mu, {\cal
K})>0$. For all $x \in C({\cal K})$ the set $p\omega (x)\subset
{\cal K} $ is disjoint with the ball $B_{\varepsilon }(\mu)$. But almost all Lebesgue point $x \in C({\cal K})$, because ${\cal K} \in
\aleph$. Therefore $p\omega (x) \cap B_{\varepsilon }(\mu) =
\emptyset$ Lebesgue a.e. This last assertion and Definition $\ref{definicionobservable}$ and paragraph \[definicion2\] imply that $\mu \not \in {\cal O}$, as wanted.
Now let us prove that $m(C({\cal O})) =1$, which is the key matter of this theorem. We know $\cal O$ is compact and not empty. So, for any $\mu \not \in {\cal O}$ it is defined the distance ${\mbox{$\,$dist$\,$}}(\mu,
{\cal O})>0$. Observe that the complement ${\cal O}^c$ of $\cal O$ in $\cal P$ can be written as the increasing union of compacts sets ${\cal K}_n$ (not in the family $\aleph $) as follows:$$\label{3} {\cal O}^c = \bigcup
_{n=1}^{\infty} {\cal K}_n, \;\;\;\;\;\;\;\;{\cal K}_n = \{\mu \in
{\cal P}: {\mbox{$\,$dist$\,$}}(\mu, {\cal O}) \geq 1/n \} \; \supset \; {\cal
K}_{n+1}$$ Let us take the sequence $A_n= A({\cal K}_n)$ of sets in $M$ defined in (\[2\]) at the beginning of this proof, and denote $A_{\infty}= A({\cal O}^c)$. We deduce from (\[2\]) and (\[3\]) that: $$A_{\infty } = \bigcup _{n=1}^{\infty} A_n, \;\;\;\; m(A_n)\rightarrow m(A_{\infty}), \;\;\;\;
A_{\infty} = A({\cal O}^c)$$ To finish the proof is thus enough to show that $m(A_n)=0$ for all $n \in \mathbb{N}$.
In fact, $A_n= A({\cal K}_n)$ and ${\cal K}_n$ is compact and contained in ${\cal O}^c$. By Definition \[definicionobservable\] and paragraph \[definicion2\] there exists a finite covering of ${\cal K}_n$ with open balls ${\cal B}_1, {\cal B}_2, \ldots, {\cal
B}_k$ such that $$\label{4}
m(A({\cal B}_i))= 0 \;\;\;
\mbox{for all } i = 1, 2, \ldots , k$$ By (\[2\]) the finite collection of sets $A({\cal B}_i); \; i= 1,2, \ldots, k$ cover $A_n$ and therefore (\[4\]) implies $m(A_n) = 0$ ending the proof. $\Box$
As shown in the examples of Section \[ejemplos2\], there exist maps whose spaces $\cal O$ of observable measures are irreducible and maps for which they are reducible. Also there exist maps that do not have irreducible subsets in $\cal O$. In section \[chains\] we define chains and co-chains of reductions. Those systems having physical measures can be characterized also according to the existence of adequate decreasing sequences (chains) of generalized attractors.
For further uses we define:
*\[definicionAttracSize\] [**(Diameter and Attracting Size.)**]{} Let $\cal O$ be the set of the observable measures for $f$. Let ${\cal O}_1 $ be a reduction or generalized attractor of $\cal O$.*
*The diameter of ${\cal O}_1$ *is $\max\{dist (\mu, \nu): \mu, \nu \in
{\cal O}_1\}$. *The attracting size of ${\cal O}_1 $ *is $m(B({\cal
O}_1))$, where $B({\cal O}_1)$ is the basin of attraction of ${\cal O}_1$ (see definition \[definicionbasin\]).****
By definition of generalized attractor its attracting size is positive. If the basin of attraction $B$ of some compact subset of ${\cal
O}$ has positive Lebesgue measure, then there exists a compact set $K \subset B $ with positive Lebesgue measure. Thus we obtain the following characterization of all the reductions of $\cal O$, as a consequence of Egoroff Theorem:
*\[teoremaCU\] [**(Generalized attractors and uniform convergence.)**]{} ***
The subspace ${\cal
O}_1$ is a reduction of the space $\cal O$ of the observable measures for $f$, *(i.e. ${\cal O}_1$ is a generalized attractor for $f$), *if and only if there exists a positive Lebesgue measure set $K \subset M$ such that $$\mbox{\em dist} \left(\frac{1}{n}\sum_{j=0}^{n-1}\delta _{f^j(x)}, \;
{\cal O}_1 \right) \rightarrow 0 \;\; \mbox {uniformly in } x \in
K$$ *Note: The set $K \subset M$ is not necessarily $f$-invariant.***
[*Proof:* ]{} Let us call $B$ to the basin of attraction of ${\cal
O}_1$ (see definition \[definicionbasin\]). We have $m(B)>0$ and therefore, the sequence in \[teoremaCU\] converges to 0 $m$-a.e. $ x \in B$. The direct result is now a straightforward consequence of Egoroff Theorem and its converse is obvious. $\Box$
The following is other characterization of the reductions of the space of observable measures for $f \in C^0(M)$, in terms of the invariant subsets in $M$ that have positive Lebesgue measure:
*\[proposicionrestriccion\] [**(Restricting the map to reduce the set of observable measures.)**]{}*
*The subspace ${\cal O}_1$ is a reduction of the space ${\cal
O }_f$ of the observable measures for $f$ (i.e. ${\cal O}_1$ is a generalized attractor for $f$) if and only if ${\cal
O}_1={\cal O }_{f_1}$, where ${\cal O}_{f_1}$ is the set of all observable measures of the map $f_1 = f|_C$, obtained when $f$ is restricted to some invariant set $C \subset M$ that has positive Lebesgue measure. Besides $C$ can be chosen as the basin attraction ${B({\cal
O}_1)}$ of ${\cal O}_1$.*
[*Proof:* ]{} This Theorem is a corollary of Theorem \[teoremaminimal\]. In fact, to prove the converse statement apply \[teoremaminimal\] to $f|_C$ instead of $f$, taking $C={B({\cal
O}_1)} $ where ${\cal O}_1$ is the given reduction of ${\cal O}_f$. To prove the direct result apply also \[teoremaminimal\] to $f|_C$, but now taking $C$ as the given invariant subset in $M$ with positive Lebesgue measure. $\Box$
*\[pruebaGenErgAttract\] [**- Proof of Theorem \[toeremaminimal0\]**]{}: By Proposition \[proposicionrestriccion\] and Theorem \[teoremaminimal\] applied to $f |_ C$ where $C = B({\cal O}_1)$, we have that Lebesgue almost all $x \in C$ verifies $p\omega(x) \subset {\cal O}_1$, and any $\mu \in {\cal O}_1$ is observable for $f|_C$. Take any $\mu \in
{\cal O}_1 \setminus {\cal K}$. There exists an open ball $B_{\varepsilon }(\mu )$ that does not intersect ${\cal K}$. As $\mu$ is observable for $f|_C$, (see Definition \[definicionobservable\]), $p\omega(x)$ is not contained in ${\cal
K}$ for a set of $x \in C$ with positive Lebesgue measure. Therefore $m (C \setminus B({\cal K})) >0$ as wanted. $\Box$*
Examples. {#ejemplos2}
=========
The examples in this section are well known or very simple, but give a scenario of the possible dynamics, in terms of the statistics given by the time mean sequence (\[1\]). In fact, they are paradigmatic of some different classes of continuous dynamical systems $f \in C^0(M)$. After Theorem \[teoremaDescomposicionGenErgAttr0\]), some of these examples may appear joint with the others, in each of the basins of the (up to countable infinitely many) generalized attractors, in which the complete set ${\cal O}_f$ of observable measures decomposes. In a general case, the complete topological dynamics may be much more complicated, since the basins of the infinitely many generalized attractors of $f$, may be topologically riddled in $M$, as explained in the discussion at the end of this section.
*\[ejemploPozo\] For a map with a single periodic point $x_0$, being a topological sink whose topological basin is $M$ almost all point, the set ${\cal O}$ has a unique measure that is the $\delta$-Dirac measure supported on $x_0$.*
*\[ejemploAnosov\] For any transitive Anosov $C^{1 + \alpha}$ diffeomorphism the set $\cal O$ is irreducible containing uniquely the SRB measure $\mu$. But there are also infinitely many other ergodic and non ergodic invariant probabilities, that are not observable (for instance the equally distributed Dirac delta measures combination supported on a periodic orbit). In particular, if $f $ preserves a probability $\mu$ equivalent to the Lebesgue measure, then $f$ has $\mu$ as the unique observable, and thus physical, probability. This result is generalized also for some subclass of $C^{3}$ diffeomorphisms on the two-torus, conjugated to transitive Anosov, with a non hyperbolic fixed point in which the derivative of $f$ has double eigenvalue 1 and a single eigendirection ([@nosotros]). But is is mostly open for other conjugated to transitive Anosov, even if they are analytic.*
*\[ejemploHu\] In [@huyoung] it is studied the class of $C^2$ diffeomorphisms $f$ in the two-torus obtained from a transitive Anosov, and in the same conjugation class of the Anosov, when the unstable eigenvalue of a fixed point $x_0$ is weakened to be 1, maintaining its stable eigenvalue strictly smaller than 1 and maintaining also the uniform hyperbolicity in each iterate outside a neighborhood (non invariant) of the fixed point. It is proved that $f$ has a single physical measure that is the Dirac delta supported on $x_0$ and that its basin has total Lebesgue measure. Therefore this is the single observable measure for $f$, although there are infinitely many other ergodic invariant measures. As the physical measure is supported in a fixed point $x_0$, statistically $x_0$ acts as a sink, attracting the sequences of time averages of Lebesgue almost all orbit. Nevertheless, as $f$ is conjugated to Anosov, it is topologically chaotic (i.e. expansive, or sensible to initial conditions).*
*\[ejemploCao\] The diffeomorphism $f\colon [0,1]^2 \mapsto [0,1]^2; \;\; f(x,y) =
(x/2, y)$ has the set ${\cal O}$ of observable measures as the set of Dirac delta measures $\delta _{(0,y)}$ for all $ y \in [0,1]$. In this case $\cal O$ coincides with the set of all ergodic invariant measures for $f$, it is infinitely reducible (i.e. $\cal
O$ is reducible and any reduction of ${\cal O}$ is also reducible). Not all $f$-invariant measure $\mu$ for $f$ is observable: for instance, the one-dimension Lebesgue measure on the interval $[0]\times [0,1]$ is invariant and is not observable. This example shows that the set ${\cal O}$ is not necessarily convex.*
\[ejemploInfinitosPozos\] *The maps exhibiting infinitely many simultaneous hyperbolic sinks, constructed from Newhouse’s theorem ([@newhouse]) has a space ${\cal O}$ of observable measures that is reducible. But it has infinitely many reductions (the Dirac delta supported on the hyperbolic sinks), each of them being irreducible. Also the maps exhibiting infinitely Hénon-like attractors, constructed by Colli in [@colli], has a space of observable measures that is reducible, having infinitely many reductions (the physical measures supported on the Hénon-like attractors), each one that is irreducible.*
*\[ejemploFeigenbaum\] Consider the quadratic family $\{f_t\}_t$ in the interval $I$, and in this family, the map $f $ where the first cascade or period doubling bifurcating maps converge. It has a single attractor $A$ which is Feigenbaum-Coullet-Tresser. This attractor $A$ is formed by a single orbit, whose closure $K$ is a Cantor set having as extremes of its gaps, the future orbit of the critical point. For all $x \in A$, $f^n(x)$ moves quasi-periodically in a single orbit (with quasi-periods $2^n$ for all $n \geq 1$) and attracts topologically all the points of $I$, except those of countable many periodic hyperbolic repellors (with periods $2^n$, for all $n \geq 0$). The map $f$ is infinitely doubling renormalizable and has a single observable probability $\mu$ (and thus physical measure) supported on $K$. This physical measure is constructed as follows: For any fixed $n \geq 1$ call $\{K_{i,n}: \, { 0 \leq i \leq 2^n-1} \}$ to the family of $2^n$ atoms of generation $n$, i.e. the sets $ K_{i,n} : \; 0 \leq i (mod \ 2^n) \leq 2^n-1 $ are the pairwise disjoint compact invervals such that $f(K_{i,n}) \subset K_{i+1,n}$, and $K = \bigcap _{n \geq 1}
\bigcup_{i= 0}^{2^n-1} K_{i,n}$. Then define the probability $\mu$ such that $\mu(K_{i,n}) = 1/2^n$ for all $n \geq 1$ and for all $0 \leq i \leq 2^n-1$.*
*\[grad\] (Example 1 in [@araujo2]). Define the function $\phi \colon [- \, \frac1\pi,\frac1\pi] \mapsto
\mathbb{R}$, $\phi(s)=s^4 \sin(\frac1s)$ and identify the extremes of the interval to obtain $\varphi \colon S^1 \mapsto \mathbb{R}$. Consider the time one of the gradient flow given by $\dot{x}= \nabla \varphi(x)$. This map has infinitely many sources and sinks, which accumulate at 0. The physical measures are the infinitely many sinks. The Dirac delta on the accumulation point of the sinks and sources, is not a physical measure, but it is an observable measure. It is besides the unique stochastically stationary measure. We conclude that physical measures, even if their basins attract Lebesgue a.e., do not necessarily include the stochastically stationary probabilities but, at least in this example, observable measures include them.*
*\[ejemplodossillas\] The following example, due to Bowen (see also example 2 in [@araujo2]), shows that the space of observable measures may be formed by measures that are partial limits of the sequences of time averages of the system states and that this sequence may be not convergent for Lebesgue almost all points. Consider a diffeomorphism $f$ in a ball of $\mathbb{R}^2$ with two hyperbolic saddle points $A$ and $B$ such that the unstable global manifold $W^u(A)$ of $A$ is a embedded arc that coincides (except for $A$ and $B$) with the stable global manifold $W^s(B)$ of $B$, and the unstable global manifold $W^u(B)$ of $B$ is also an embedded arc that coincides with the stable manifold $W^s(A)$ of $A$. Let us take $f$ such that there exists a source $C$ in the open ball $U$ with boundary $W^u(A)\cup
W^u(B) $, and all orbits in that ball $U$ have $\alpha$-limit set $C$ and $\omega $-limit set $W^u(A) \cup W^u(B)$. If the eigenvalues of the derivative of $f$ at $A$ and $B$ are well chosen, then one can get that the time average sequences of the orbits in $U
\setminus \{C\}$ are not convergent, have at least one subsequence convergent to the Dirac delta $\delta_A$ on $A$ and have other subsequence convergent to the Dirac delta $\delta _B$ on $B$.*
Due to Theorem \[teoremaconvexo\], for each $x \in U
\setminus\{C\}$ there are non countably many probability measures which are the limit measures of the time average sequence of the future orbit starting on $x$. All these measures are invariant under $f$ and therefore, due to Poincaré Recurrence Theorem (see [@mane]), all of them are supported on $\{A\} \cup \{B\}$. Due to this last observation and due to Theorem \[teoremaconvexo\] all the convex combinations of $\delta _A$ and $\delta _B$ are limit measures of the sequence of time averages of any orbit starting at $U \setminus \{C\}$ and conversely.
Therefore the set ${\cal O}$ of observable measures for $f$ coincides with the set of convex combinations of $\delta _A$ and $\delta _B$. The set ${\cal O}$ is irreducible and formed by non-countable many probability measures. It is not the set of all invariant measures; in fact the measure $\delta _C$ is not observable.
This example also shows that the observable measures are not necessarily ergodic.
A different exact adjustment in the eigenvalues of the two saddles $A$ and $B$ allows a different example, for which all the convergent subsequences of (\[1\]) converge to the same exact previously chosen convex combination $\mu$ of $\delta_A$ and $\delta_B$. So, there is a physical measure that attracts the time averages of the orbits of $U$ and that is not ergodic. This proves that physical measures are not necessarily ergodic.
In [@araujo2] it is shown that this physical measure, which is a convex combination $\mu$ of $\delta_A$ and $\delta_B$, is stochastically stable, even being non ergodic.
\[ejemploExpanding\] *The $C^{1 + \varepsilon}$ expanding maps $f\colon S^1 \mapsto S^1$ in the circle (i.e. $f'(x) >1 \; \forall \; x \in S^1$), have been extensively studied, they present a unique SRB and physical measure attracting Lebesgue a.e. In [@cq] it is shown that also in the $C^1$ topology, generically $f$ has a unique physical measure. Nevertheless this physical measure is not SRB (it is not absolute continuous respect to Lebesgue) in those generic $C^1$ (not $C^{1+
\varepsilon})$ examples. They show that, for $C^1$ uniformly hyperbolic maps, (that are besides topologically mixing), if there is a unique observable (and thus physical) probability, this measure is not necessarily SRB.*
On the other hand, there exists $C^1$ examples of expanding maps for which a non ergodic invariant measure $\mu$ is equivalent to Lebesgue ([@Q3]). Thus, there is not a unique observable measure. This shows that, out of the $C^{1 + \varepsilon}$ case, when uniform hyperbolicity and topological mixing hold, the set of observable measures is not necessarily reduced to a single probability, even if the sequence (\[1\]) converges Lebesgue a.e.
The case of $f \in C^0$ stands in contrast with the former situation. In [@misiurewicz] (see Theorem 3.4) is proved in particular that there exist maps $f\colon S^1 \to S^1$ topologically conjugate with $g_d\colon S^1 \to S^1$, $g_d(x)=dx$ such that for Lebesgue almost every point $x$ in $S^1$ and every $f$-invariant measure $\mu $, some subsequence of the sequence (\[1\]) converges to $\mu$.
When formulating the theorems about chains and co-chains of generalized attractors in Section \[seccionphysical\], we have in mind the three paradigmatic different statistical dynamical behaviors that $C^0$ systems may exhibit (in relation to the limit probabilities of the sequence (\[1\])):
First, in Examples \[ejemploCao\] Lebesgue a.e. orbit defines a convergent subsequence (\[1\]) of probabilities, but no physical measure exist (at least for a subset of initial states with positive Lebesgue measure).
Second, in Examples \[ejemplodossillas\] Lebesgue a.e. orbit defines a non convergent subsequence (\[1\]).
Third, in Examples \[ejemploExpanding\] (the $C^2$ or the $C^1$ generic expanding maps), \[ejemploFeigenbaum\] (the Feigenbaum attractor), \[ejemploInfinitosPozos\] (infinitely many coexisting sinks or Hénon like attractors), \[ejemploAnosov\] and \[ejemploHu\] ($C^2$ conjugated to Anosov in the two-torus), Lebesgue almost all orbit defines a convergent subsequence (\[1\]) that converges to a physical measure, and there are at most countable many such physical measures.
To state the Theorems of Section \[seccionphysical\], we are thinking in those different statistical dynamical behaviors, but not as dynamical systems that are topological isolated one from the others. Precisely, we are considering that the basins of the generalized attractors of all those examples may be topologically immersed in a larger dimension compact manifold $\widehat M$, in such a way that become mutually riddled (i.e. they are dense subsets of $\widehat M$).
Equilibrium states and observable measures for $C^1$ expanding maps.
====================================================================
We will develop the theorems in this section for order preserving expanding maps in the circle $S^1$, but the proofs also work for order reversing expanding maps in $S^1$. Some of the results also work for expanding maps, similarly defined, in appropiate manifolds of dimension larger than one.
*A $C^1$ (order preserving) map $f: S^1 \mapsto S^1$ is expanding if $f'(x) >1$ for all $x \in S^1$. (If $f$ is order reversing then it is expanding if $-f'(x) >1$.)*
In particular, for all integer $d >1$ we denote $g_d : S^1 \mapsto S^1$ to the linear expanding map $$g_d(x) = dx \; \forall \, x \in S^1$$ If $f$ is expanding, then there exist a unique integer $d >1$, called the degree of $f$, such that $f$ is topologically conjugated to $g_d$. Thus, all the topological properties of $g_d$ are translated to $f$.
We denote ${\mathcal E}^1 \subset C^1 (S^1)$ to the family of $C^1$ expanding maps in the circle $S^1$. For all $r \geq 1$, we denote ${\mathcal E}^r \subset {\mathcal{E}^1}$ to the $C^r$ expanding maps in $S^1$. Denote $\mbox{Homeo}(S^1)$ to the set of homeomorphisms on the circle $S^1$ with the $C^0$ topology. Finally, for all $f \in
{\mathcal{E}^1}$, denote $\mbox{Conj}(f) \subset \mbox{Homeo}(S^1)
$ to the set of all the conjugacies between $f$ and $g_d$.
For a seek of completeness we state here a Lemma from [@cq]. For each $x \in S^1$ denote $U_x$ to the $C^1 $ open and dense set of all expanding maps $f$ in $S^1$ such that $f(x) \neq x.$
*[****]{} \[conj\] For each $x \in S^1$ there is a continuous map $\Pi_x\colon U_x \to \mbox{Homeo}(S^1)$ such that $\Pi_x(f) \in \mbox{Conj} (f)$ for each $f$. In particular, given $f \in \mathcal{E}^1$ of degree $d$, there is a neighborhood $U$ of $f$ on which there is a continuous choice of conjugacies to the map $g_d$.*
[*Proof:* ]{} See Lemma 1 of [@cq] and Theorem 2.4.6. of [@kh].
[**Pressure and equilibrium states.**]{} *\[definicionPresure\]*
Let $f $ a $C^1$- expanding map in the circle. We denote $\psi = -\log f' \in C^0 (S^1, \mathbb{R})$ and ${\cal P}_f$ to the set of $f-$invariant Borel probabilities in $S^1$. For $\nu \in {\cal P}_f$ we denote $h_{\nu}(f)$ to the entropy of $\nu$.
Let $\varphi \in C^0(S^1, \mathbb{R})$. The pressure of $\varphi $, is $$P_f(\varphi)=\sup_{\nu\in \mathcal{P}_f} \left\{h_\nu(f) + \int \varphi \, d\nu\right\}.$$
A measure $\mu \in \mathcal{P}_f$ is *an equilibrium state for $\varphi$ *if $$h_\mu(f)
+ \int \varphi \, d\mu = P_f(\varphi).$$ In particular $\mu \in {\cal P}_f$ is *an equilibrium state for $\psi = - \log f'$ *if $$h_\mu(f)
= \int \log f'(x) \, d\mu(x) + P_f(-\log f').$$ We denote as $ES_f \subset {\cal P}_f$ to the set of $f$-invariant probabilities that are equilibrium state for $\psi = - \log f'$.****
For a seek of completeness we recall well known results in the following Theorem \[teoremaRuelle\] and Corollary \[corolarioRuelle\]:
\[teoremaRuelle\] [**Ruelle inequality and Pesin-Ledrappier-Young Equality for $C^{1 + \alpha}$ expanding maps.**]{} If $f \in {\mathcal E}^{1 + \alpha}$ *(i.e. $f$ is a $C^1$ expanding map in the circle such that $f'$ is $\alpha >0$-Hölder continuous), *then there exists a unique $f$-invariant and ergodic measure $\mu$ that is the equilibrium state for $\psi = - \log f'$. Besides $\mu$ is the unique $f$-invariant measure that is absolute continuous respect to the Lebesgue measure $m$, and the pressure $P_f(\log f') = 0$. In other words, for all $\nu \in {\cal P}_f$ the inequality of Ruelle holds: $$h _{\nu}(f) \leq \int \log f'(x) \; d \nu.$$ Besides, this last is an equality if and only if $\nu \ll m$, and this holds if and only if $\nu = \mu$.**
[*Proof:* ]{} See Theorem 6.3.8 of [@ke].
\[corolarioRuelle\] [**Ruelle inequality for $C^1$ expanding maps.**]{} If $f \in {\mathcal E}^{1 }$ *(i.e. $f$ is a $C^1$ expanding map in the circle), *then the pressure $P_f(\log f') = 0$. In other words, for all $\nu \in {\cal P}_f$ the inequality of Ruelle holds: $$h _{\nu}(f) \leq \int \log f'(x) \; d \nu .$$**
[*Proof:* ]{} We reproduce the proof of Lemma 2 in [@cq]. After the PLY Equality of Theorem \[teoremaRuelle\] it follows that $$P_f (-\log f') = h_{\mu} (f) - \int \log f' \; d \mu = 0 \ \mbox{ if } \
f \in \mathcal{E}^2$$ for the unique $f$-invariant measure $\mu $ which is absolute continuous respect to the Lebesgue measure $m$.
If $f \in \mathcal{E}^1$ has degree $d$, after the lemma \[conj\], there exists a neighborhood $U \subset \mathcal{E}^1$ of $f$, and for all $g \in U$, conjugacies $\gamma_g \in \mbox{Conj}(g)$, between $g \in U$ and the linear expanding map $g_d$ of degree $d$, such that the application $g \to \gamma_g$ is continuous on $U$. Denote $\psi_{g} (x) = - \log g' (x)$ for all $g \in U$. Take $\{f_i\}_{i\in \mathbb{N} }
\subset \mathcal{E}^2$ such that $f_i \to f \in \mathcal{E}^1$ (the convergence is in the $C^1$ topology). Then $\psi_{f_i}\circ \gamma_{f_i}
\to \psi_f\circ \gamma_f \in C^0(S^1, \mathbb{R})$. Note that the pressure $p_g(\varphi)$, for any fixed $g \in {\cal E}^1$, depends continuously on $\varphi \in C^0(S^1, \mathbb{R})$. Then $$P_{g_d}
(\psi_{f_i} \circ \gamma_{f_i}) = P_{g_d} (\psi _f \circ \gamma_f).$$
Also note that, if $\gamma \in \mbox{Conj}(g)$ for some $g \in
{\mathbb{E}^1}$ and if $\varphi \in C^0(S_1, \mathbb{R})$, then $$P_{g_d} (\varphi \circ \gamma)
= P_g(\varphi).$$ Indeed, $\gamma$ induces a bijection $\gamma^*$ between the $g_d$ invariant measures $\nu \in {\cal P}_{g_d}$, and the $g$ invariant measures $\gamma^* \nu \in {\cal P}_g$. Then $\int \varphi \circ \gamma \; d \nu =
\int \varphi \; d \gamma^* \nu$ for all $\nu \in {\cal P}_{g_d}$. Besides, since $\gamma^*$ is a measure-theoretic isomorphism, the entropies coincide $h_{\nu}(g_d)= h_{\gamma^* \nu} (g)$.
We conclude that $$0=P_{f_i} (\psi_{f_i}) = P_{g_d}(\psi_{f_i}\circ
\gamma_{f_i})=P_{g_d} (\psi_{f }\circ \gamma_{f })
= P_f (\psi_{f }). \ \; \; \Box$$
**
If $f $ is a $C^1$ expanding map in the circle that is not $C^{1 + \alpha}$, then (except the inequality of Ruelle), the thesis of Theorem \[teoremaRuelle\] does not necessarily hold. In fact, the uniqueness of the equilibrium state of $-\log f'$, or its absolute continuity respect to Lebesgue, may fail, as in the following cases:
On one hand, the PLY equality may hold for a unique equilibrium state that is a physical measure but singular respect to Lebesgue ([@cq]).
On the other hand, the PLY equality may hold for some invariant probability that is absolute continuous respect to Lebesgue, but not ergodic: see [@Q3] and Theorem of Ledrappier (Theorem 2 in [@walters]). Therefore its ergodic components are also equilibrium states of $-\log f'$.
The following theorem states that any observable measure is an equilibrium state. It is a stronger version of Theorem 6.1.8 of the book of Keller [@ke][[^2]]{}, but its proof is in essence the same.
\[teoremon\] [****]{}
Let $f$ be a $C^1$ expanding map on the circle $S^1$. Then, any observable measure of $f$ and any convex combination of observable measures of $f$, is an equilibrium state for $\psi= - \log f'$, and its pressure is equal to 0, i.e. any observable measure $\mu$ of $f
$ satisfies the PLY equality for the entropy: $$h_\mu(f)= \int \log f' \, d\mu.$$
We prove Theorem \[teoremon\] in the subsection \[pruebadirectoTeoremon\]. Let us now state its Corollaries. The first Corollary is a well known result. Nevertheless a new point of view for its proof rises from Theorems \[teoremon\], \[teoremaexistencia0\] and \[toeremaminimal0\].
\[corolarioAgregado\] A $C^1$ expanding map on the circle always has a non empty set $ES_f$ of probability measures that verify the PLY equality of the entropy. Besides, $f$ has a physical measure $\mu$ attracting Lebesgue a.e. if and only if the observable measure is unique, and this happens if there is a unique probability $\mu \in ES_f$.
[*Proof:* ]{} After Theorem \[teoremaexistencia0\]: ${\cal O}_f \neq \emptyset$. After Theorem \[teoremon\], the closed convex hull of ${\cal O}_f$ is contained in $ ES_f$, so $ES_f \neq \emptyset$. Finally, from Theorem \[toeremaminimal0\], $f$ has a physical measure $\mu$ attracting Lebesgue a.e. if and only if ${\cal O}_f = \{\mu\}$ and this trivially holds if $ES_f = \{\mu\} \ \ \Box$
We say that a probability measure is atomic if it is supported on a finite set.
\[corolarioteoremon\] There is no atomic ergodic observable measure of a $C^1$ expanding map.
[*Proof of the corollary \[corolarioteoremon\]:*]{} By contradiction, if $\mu $ is an atomic ergodic observable measure of an expanding map $f$, then $h_\mu(f)=0$. As $ \psi = -\log f' <0$, and $\mu $ is an equilibrium state for $\psi$, then the pressure $P_f(\psi)$ is negative, contradicting Corollary \[corolarioRuelle\]. $\ \Box$
In the following definitions and lemmas, we reproduce those of the book of Keller ([@ke]) about the equilibrium states in its section 4.4, applied in particular to expansive $C^1$ maps $f$ in the circle $S^1$. Recall that if $f \in {\cal E}^1$, the set of equilibrium states of $\psi= - \log f'$ is $ES_f \subset {\cal P}_f$, i.e. $\mu \in ES_f$ if and only if the PLY equality of the entropy holds: $h_{\mu}(f) = \int \log f' \; d \mu$.
[**Notation:**]{} *\[notacionKsubr\] For all $\nu \in {\cal P}_f$ we denote $$V_f (\nu) = h_{\nu}(f) -\int
\log f'(x) \; d \nu(x)$$ Due to Ruelle inequality (Corollary \[corolarioRuelle\]): $$V_f (\nu ) \leq 0 \; \; \; \forall \nu \in {\cal P}_f
.$$ For all $r \geq 0$ we denote $${\cal K}_r = \{\nu \in {\cal P}_f: \;
V_f (\nu) \geq - r\}$$ In particular ${\cal K}_0 = ES_f$. For all $r \geq 0$ the set ${\cal K}_r$ is non empty, compact (in the weak$^*$ topology) and convex (join the proof of Theorem 4.2.3 of the book in [@ke], with Theorem 4.2.4 and Remark 6.1.10 of the same book).*
For all integer $n \geq 1$ and all $x \in S^1$ denote $\sigma_{n,x}$ to the (non necessarily $f$ invariant) probability of the sequence (\[1\]), called the *empirical distribution of the future orbit of $x$ up to time $n$, *i.e.: $$\sigma_{n,x} = \frac{1}{n} \sum_{j= 0}^{n-1} \delta_{f^j(x)}$$ where $\delta_x$ is the Dirac-delta probability supported on the point $x$.**
As stated from the beginning of this paper, for $x \in S_1$ fixed, we denote $$pw(x) \subset {\cal P}_f$$ to the set of all probabilities that are the weak$^*$-partial limits (limits of the convergent subsequences) of the sequence $\{\sigma_{n,x}\}_{n \geq 1}$ of the empirical distributions of the future orbit of $x$.
\[lemma1teoremon\] Let $f$ be a $C^1$ expanding map of the circle $S^1$. Let ${\cal K}_r $ the compact set of $f$ invariant probabilities defined in \[notacionKsubr\]. For all $r \geq 0$ and for all open neighborhood ${\cal V} $ of ${\cal K}_r$ in the space ${\cal P}$ of all the (not necessarily $f$-invariant) probabilities, the following inequality holds: $$\limsup_{n \rightarrow + \infty} \frac{1}{n} \log m \{x \in S_1: \sigma_{n,x} \not \in {\cal V}\}
\; \leq \; - r,$$ where $m$ is the Lebesgue probability in the circle $S^1$.
[*Proof:*]{} This Lemma is the Proposition 6.1.11 of the book of Keller [@ke]. All the hypothesis of that proposition[^3] hold in the case that $f$ is a $C^1$ expanding map of the circle $S^1$: see Remark 6.1.10 of [@ke].
\[pruebadirectoTeoremon\] [**Proof of Theorem \[teoremon\]:**]{} *As $ES_f$ is convex, it is enough to prove that ${\cal O}_f\subset ES_f $. Consider, for any $r >0$ the compact set $K_r \subset {\cal P}_f$ defined in \[notacionKsubr\]. By definition, the intersection of the (decreasing with $r$) family $\{K_r\}_{r >0}$ is the non empty compact set $${\cal K}_0 = \bigcap_{ n=1}^\infty {\cal K}_{\frac1n}$$*
It is enough to prove that its basin of attraction $${B({\cal K}_0)}
= \{x \in S_1: pw(x) \subset {\cal K}_0\}$$ has total Lebesgue measure. In fact, if we prove that $m(B({\cal K}_0)) = 1$, then applying Theorem \[toeremaminimal0\], ${\cal O}_f \subset {\cal
K}_0 = ES_f$ as wanted.
Now, we reproduce the proof of the part a. of Theorem 6.1.8 of [@ke]. Let $r >0$. We fix $0<\varepsilon < r $. After Lemma \[lemma1teoremon\], for any weak$^*$ neighborhood ${\cal N}$ of ${\cal K}_r \subset{\cal P}$, there exists $n_0$ such that for any $n>n_0$: $m \{x:\sigma_{n,x} \in \mathcal{P} \setminus {\cal N}\}\leq
e^{-n( r-\varepsilon)}.$ This implies that $\sum_{n=1}^\infty m (x:\sigma_{n,x}
\in \mathcal{P} \setminus {\cal N})< +\infty.$ After the Borel-Cantelli Lemma it follows that $$m\left(\bigcap_{n_0=1}^\infty \bigcup_{n=n_0}^\infty \{x:\sigma_{n,x} \in \mathcal{P}
\setminus {\cal N}\}\right)=0.$$ In other words, for each open neighborhood ${\cal N}$ of ${\cal K}_r$, for $m$-a.e. $x \in S^1$ there exists $n_0 \geq 1 $ such that $\varepsilon_{n,x}\in {\cal N}$ for all $n\geq n_0$. Hence, $pw(x) \subset {\cal K}
_r$ for $m$-a.e. $x \in S^1$. It follows that $pw(x) \subset \bigcap_{1 \leq n\in \mathbb{N}} {\cal K}_{1/n}
= {\cal K}_0 =
\{ \mu \in \mathcal{P}_f: V_f(\mu) = 0\} $ for $m$-a.e.$x \in S^1$ as wanted. $\Box$
Cardinality and decomposition of ${\cal O}$. {#seccionphysical}
============================================
The first aim of this section is to state some results that characterize the maps having physical measures whose basins attract Lebesgue a.e., in terms of the cardinality of the set of its observable measures. In particular we prove the part (b) of Theorem \[teoremaexistencia0\] and Theorem \[teoremaDescomposicionGenErgAttr0\], that were delayed up to this section.
The second aim of this section, for a seek of completeness, is to cover in the theory, all the possible cases in $C^0(M)$, including those maps that are singular respect to Lebesgue. To do that we analyze the chains and co-chains of reductions of the space of observable measures, even if no physical measures exists, or if one or more than one physical measure exists, but the union of their basins of attraction do not cover Lebesgue a.e.
*\[teoremaCardinalidad\] [**Cardinality of ${\cal O}$ and physical measures.**]{}*
Let $f\colon M \mapsto M$ be any continuous map in the compact manifold $M$.
*If the set $\cal O$ of the observable measures for $f$, or some proper reduction ${\cal O}_1$ of $\cal O$, is finite or countable infinite then there exists physical measures $\mu$ for $f$, precisely of $f|_{B({\cal O}_1)}$, where $B({\cal O}_1)$ denotes the basin of attraction in $M$ of ${\cal O}_1$.*
Conversely, if there exists a physical measure $\mu$ for $f$ then, either the space $\cal O$ has a single probability measure $\mu$, or it is reducible and there exists a proper reduction ${\cal O}_2$ of $\cal O$ with a single element.
[*Proof:*]{} The converse assertion is immediate. In fact, if $\mu
$ is physical then the basin of attraction of $\mu$ has positive Lebesgue measure, and thus $\{\mu\}$ is a trivial reduction of ${\cal O}$. It is either a proper reduction or not. If not then ${\cal O}=\{\mu\}$.
Let us prove the direct assertion. Denote extensively as $\{\mu
_n: n \in N\}$ the finite or countable infinite reduction ${\cal
O}_1$ (proper or not) that is given in the hypothesis. (If it has only a finite cardinality, then repeat one or more of its elements in the extensive notation, but include all of them at least once).
By Proposition \[proposicionrestriccion\] the space ${\cal O}_1$ is the set of all observable measures for the restriction $f|_C$ of $f$ to some forward invariant set with positive Lebesgue measure, say $m(C)
>0$. Thus $C \subset B({\cal O}_1)$ and it is not restrictive to assume that $C = B({\cal O}_1)$.
Rename if necessary $f|_C$ as $f$, ${\cal O}_1$ as $\cal O$, and rename as $m$ the Lebesgue measure in $C $, (i.e: the restriction to $C$ of the Lebesgue measure in $M$, which is then renormalized to be a probability measure in $C$). Resuming: $$\label{[7]}
\mbox{For every } x \in C: p\omega (x ) \subset {\cal O}=\{\mu_n: n
\in \mathbb{N}
\}$$ Let us define $C_n \in C$ to be candidates of the basins of attraction for the measures $\mu _n$, and relate their respective Lebesgue measures $m(C_n)$ as follows: $$\label{[8]}C_n = \{x \in C: \mu_n \in p\omega(x)\}; \;\;\;\;\; C=
\bigcup_{n=1}^{\infty} C_n; \;\;\;\;\; \sum_{n=1}^{\infty}m (C_n)
\geq m(C) = 1$$ So $m(C_n)
>0$ for some $n \in \mathbb{N}$.
To end the proof we shall show that for all $x \in C_n: \{\mu_n\}= p\omega (x)$. Due to (\[\[7\]\]) and (\[\[8\]\]), it is enough to prove that $C_n \cap C_k = \emptyset$ if $\mu_n
\neq \mu_k$.
By contradiction, suppose that for some $\mu_n \neq \mu_k$ there exists a point $x \in C_n \cap C_k \subset C$. Then, from (\[\[8\]\]) we have $\mu_n , \mu_k \in p\omega (x)$. Now we apply Theorem \[teoremaconvexo\] and (\[\[7\]\]) to conclude that the space $\cal O$ is non-countably infinite. $\Box$
*\[srb\] \[corolarioCo-cadenaphysical\] [**Proof of Theorem \[teoremaexistencia0\] (b)**]{} It is straightforward consequence of Theorems \[toeremaminimal0\] and \[teoremaCardinalidad\]. $\Box$*
Now, let us analyze in an abstract theory, the complementary case: the set of all observable measures is a non countable infinite compact subset of the space of invariant probabilities.
[**(Chains of reductions.)**]{} \[chains\] A chain of reductions *of the space ${\cal O}$ of the observable measures for $f$ is a (finite or countable infinite) sequence $\{{\cal O}_n \} _{ n \in I \subset \mathbb{N}}$ of reductions or *generalized attractors *(see Definition \[definicionreduccion0\]) such that ${\cal O}_{i} \varsubsetneq{\cal
O}_j$ if $i > j$ in the set $I$ of natural indexes.***
We call *length of the chain *to its finite or countable infinite cardinality $\# I$.**
Recall that, by definition of reductions (i.e. generalized attractors) each ${{\cal O}_n}$ is a compact part of the set of the observable measures.
For any chain $\{{\cal O}_n \}_{n \in I \subset \mathbb{N}}$ of reductions of the space of observable measures for $f$, let $$d_n = \mbox{diam}
({\cal O}_n); \;\;\;\;\;\;\;\;\;\; s_n = \mbox{attrSize}({\cal O}_n)$$ where $\mbox{diam}$ and attrSize denote respectively the diameter and the attracting size, defined in \[definicionAttracSize\].
Observe that $d_n$ and $s_n$ are non negative decreasing sequences. We denote $\underline{d} = \lim d_n$ and $\underline s = \lim s_n$.
*[**(Physical measures and chains.)**]{} \[teoremaPhysicalMeasuresAndChains\] *There exists a physical measure for $f$ if and only if there exists a chain ${\cal O }_n$ of reductions of the space $\cal O$ of the observable measures of $f$, such that the sequence of its diameters converges to zero and the sequence of its attracting sizes converges to some $\alpha
>0$.**
[*Proof:* ]{} The converse statement is immediate defining the length 1-chain $\{\{\mu\}\}$, where $\mu$ is the given physical measure. The direct result is also immediate if the length of the given chain is finite. Let us see now the case when the chain ${\cal O}_n; \; n \in \mathbb{N}$ is infinite. As the sequence of its diameters converges to zero, then $\cap _{n \in \mathbb{N}}
{\cal O}_n = \{\mu \}$ for some $\mu$. It is enough to show that the attracting size of $\mu$ is positive. Note that from the construction of such $\mu$ we have that $C(\{\mu\})= \cap _{n \in
\mathbb{N}} C_n $ where $C_n$ denotes the basins of attractions $C({\cal O}_n)$. These basins are a countably infinite decreasing family of sets in $M$ with positive Lebesgue measures $s_n$. Therefore, $attrSize (\{\mu\}) = m (C(\{\mu\}))= \lim s_n = \alpha
>0$, as wanted. $\Box$
[**(Independence of generalized attractors and chains.)**]{} *We say that two generalized attractors or reductions of the space of observable measures *are independent if the basin of attraction of their intersection has zero Lebesgue measure. ****
We note from Definition \[definicionbasin\] that the basin of attractions of two reductions ${\cal O}_1$ and ${\cal O}_2$ intersect exactly in the basin of attraction of ${\cal O}_1 \cap
{\cal O}_2$. Therefore:
*Two ergodic attractors are independent if and only if the intersection of their basins has zero Lebesgue measure.***
We say that *two chains of reductions are independent *if each one of the chains has a reduction that is independent with *some *reduction of the other chain.****
[**(Co-chains of reductions.)**]{} \[teoremaCadena\] A co-chain of reductions *of the space ${\cal O}$ of the observable measures for $f$ is a (finite or countable infinite) family $\{{\cal O}_n; \; n \in I \subset \mathbb{N} \}$ of reductions or *generalized attractors *(see Definition \[definicionreduccion0\]) *that are pairwise independent. ******
We call *length of the co-chain *to its finite or countable infinite cardinality $\# I$.**
[**Remark:**]{} If the space $\cal O$ of all the observable measures for $f$ is irreducible, then $\{{\cal O}\}$ is the unique chain of reductions and also the unique co-chain.
Now we state a slightly generalized version of a known result in the theory of Discrete Mathematics, the Theorem of Dilworth ([@liu]), applied to the chains and co-chains of generalized attractors:
[**(Reformulation of Dilworth Theorem.)**]{} \[teoremaDilworth\] For any continuous map $f$ the supreme $k$ of the lengths of the co-chains of reductions in the space of observable measures for $f$, is equal to the supreme $h$ of the number of pairwise independent chains.
Moreover: for any co-chain of length ${ l}$ there is a family of $\ { l}$ pairwise independent chains, and conversely.
[*Proof:* ]{} Any co-chain $\{{\cal O}_j, \; j\in J\}$ with length $\; l = \# J$ can be seen as a collection $ \{\{{\cal O}_j\}, \; j
\in J \} $ of $l$ pairwise independent chains $P_j = \{{\cal
O}_j\}$, each chain $P_j$ with length one. So $k \leq h$.
Conversely, given any collection $\{ P_j, \; j \in J\}$ of pairwise independent chains, take a reduction ${\cal O}_1 \in P_1$ independent to some $\widehat{\cal O}_2 \in P_2$, and take $\breve
{\cal O}_2 \in P_2$ independent with $\widehat{\cal O}_3 \in P_3$. As both reductions $\widehat{\cal O}_2 $ and $ \breve{\cal O}_2$ belong to the chain $P_2$, one of them must be contained in the other; thus their intersection, say ${\cal O}_2$, is also a reduction of the chain $P_2$. Besides ${\cal O}_2$ is independent with ${\cal O}_1 \in P_1$ and with $\widehat {\cal O} _3 \in P_3$. Analogously construct by induction a (finite or infinite) sequence $\{{\cal O}_j: j \in J\}$ of pairwise independent reductions, such that ${\cal O}_j \in P_j$. This sequence of reductions is by definition a co-chain. Therefore, $h \leq k$. $\Box$
When no physical measure exist, or when some of them exist but their basins do not cover Lebesgue almost every point of the phase space $M$, we will still state a equivalent condition for the space being partitioned in (up to countably many) irreducible generalized attractors, whose basins cover Lebesgue all orbit.
\[teoremaCo-cadena\] *[**(Co-Chains and irreducible attractors.)**]{}*
1. *A map $f\colon M \mapsto M$ has *(up to countable infinitely many) *irreducible attractors whose basins of attraction cover Lebesgue almost all point in $M$ if and only if there exist a co-chain of reductions of the space $\cal O$ of the observable measures for $f$ such that the *(finite or countable infinite) *sum of its attracting sizes is 1.*****
In this case:
2. The irreducible attractors are all physical measures if and only if the diameter of the reductions are all zero.
3. The *(finite or countable infinite) *number of such irreducible generalized attractors is equal to the supreme $k$ of the lengths of the co-chains of reductions for $f$ and to the supreme $h$ of the number of independent chains for $f$.**
[*Proof:* ]{} From Definition $\ref{definicionAttracSize}$ we obtain that $f$ has generalized attractors whose basins cover Lebesgue almost all points, if and only if the following statement holds:
1. \[countab\] There exist (up to countable many) ${\cal O}_n \in {\cal O}$ such that $s_n = attrSize ({\cal O}_n) >0$ and $\sum s_n = 1$.
Note that two different trivial reductions of $\cal O$ are always mutually independent. Therefore, \[countab\] is equivalent to the following:
2. The family ${\cal O}_n$ is a co-chain of trivial reductions such that $\sum s_n = 1$.
So the first assertion of Theorem \[teoremaCo-cadena\] is proved.
The second assertion is trivial from the characterization of physical measures as those reductions of the space of observable measures that have zero diameter.
To prove the third assertion observe that each reduction of any co-chain must contain at least one of the generalized attractors ${\cal O}_n$ because $\sum s_n = 1$, and that two different reductions of the same co-chain can not contain any common reduction, because they must be independent. Then $k$ is smaller than or equal to the number of independent attractors. On the other hand, the set of all the independent reductions form itself a co-chain, so $k$ is greater than or equal to the number of independent attractors. Finally apply Theorem \[teoremaDilworth\] to show that the number of irreducible independent attractors is also equal to $h$. $\Box$
\[corolarioCo-cadenaSRB\] A map $f\colon M \mapsto M$ has *(up to countable infinitely many) *physical measures whose basins of attraction cover Lebesgue almost all point in $M$, if and only if there exist a *(finite or infinite) *family $$\{{\cal O}_i ^j, \; i \in I \subset \mathbb{N}, \; j \in J \subset
\mathbb{N} \}$$ of generalized attractors ${\cal O}_i^j$ for $f$ such that *for all $i \in I$ and $j, k \in J, \; j \neq k $: $${\cal O}_{i+1}^j \subset {\cal O}_i^j, \; \;\;
\lim _ {i } d_i^j \, = \, 0 \; \; \; \lim _{i
} s_i^{j,k} = 0 \;\;\;\; \mbox{and}
\;\;\;\lim _{i } \sum_{j \in J} s_i^j
\, \geq \, 1$$ where $d_i^j$ and $s_i^j$ denote respectively the diameter and attracting size of ${\cal O}_i ^j$ and $s_i^{j,k}
$ is the Lebesgue measure of the basin of attraction of ${\cal
O}_i^j \cap {\cal O}_i^k$.*****
[*Proof:* ]{} If there exist such physical measures $\mu _j$ for $j \in
J \subset N$, simply define the family ${\cal O}_i ^j = \{ \mu
_j\}$ for $i= 1$ and $j \in J$. This family of ergodic attractors verify all stated conditions.
To prove the converse statement let us first apply Theorem \[teoremaCadena\] to the chains $\{ {\cal O}_i, i
\in \widehat I \subset I\}$, for each fixed $j \in J$ such that $\lim _{i } s_i^j
>0$, (while $\lim _i d_ i^j = 0$). Then each of such chains has a intersection $\{\mu _j \}$ where $\mu _j$ is a physical measure.
For $j \neq k$ the basins of attraction of $\mu _j$ and $\mu_k$ are Lebesgue almost disjoint, because its Lebesgue measure is $\lim s_i^{j,k} = 0 $. Thus $\mu
_j \neq \mu _k$. Finally consider the co-chain $\{ \{\mu_j\}, j
\in J\}$ and apply Theorem \[teoremaCo-cadena\]. $\Box$
*\[teoremadescomposicionGenArgAttract\] [**Proof of Theorem \[teoremaDescomposicionGenErgAttr0\]: Decomposition in independent generalized attractors.**]{} We will prove Theorem \[teoremaDescomposicionGenErgAttr0\] in the following version \[teoremaDescomposicionGenAttr1\], that gives an upper bound to the number of generalized independent ergodic attractors in which the space $\cal P$ can be decomposed, and besides states a sufficient condition for the decomposition be unique. From the following Theorem \[teoremaDescomposicionGenAttr1\], it follows also the last assertion of Theorem \[teoremaDescomposicionGenErgAttr0\], using that the space ${\cal
O}$ of all observable measures is compact, and thus, for all $\varepsilon >0$ it can be covered by a finite number of weak$^*$ balls of size $\varepsilon >0$.*
*[**(Reformulation of Theorem \[teoremaDescomposicionGenErgAttr0\].)**]{} *\[teoremaDescomposicionGenAttr1\]**
Any continuous map $f\colon M \mapsto M$ has a collection $S$ formed by *(up to countable many) *pairwise independent generalized attractors *(that are not necessarily irreducible) *whose basins of attraction cover Lebesgue almost all points in $M$.****
The supreme $a$ of the number of such generalized attractors verifies ${a} \leq {k} $ *(where ${k}$, may be infinite, is the supreme of the lengths of the co-chains of reductions in the space of the observable measures for $f$). ***
If there exists such a collection $S$ whose generalized attractors are besides all irreducible, then such $S$ is unique and besides $a
= k = l$, where $l$ is the cardinality of $S$.
[**Proof:** ]{} To prove the first and second statements note that, by definition of the independence of the reductions, any co-chain $ P$ of reductions of the space ${\cal O}$ of the observable measures, verifies $\sum s_j \leq 1$, where $s_j$ denotes the attracting size of the reduction ${\cal O}_j \in P$. Now take the family ${\cal F}$ of all the co-chains $S $ such that $\sum s_j = 1$. (There exists always at least one such co-chain: in fact the length-1 co-chain $\{{\cal O}\}$ verifies $\sum s_j = s({\cal O }) = 1$, due to the results of Theorem \[teoremaminimal\]). By construction each $S \in {\cal F}$ verifies the wanted conditions. As the family ${\cal F}$ is a subfamily of all the co-chains of reductions, we obtain $a \leq k$.
Let us prove now the last assertions of the theorem. If there exists in $\cal F$ a co-chain $S= \{\widehat {\cal O}_h: \, h \in H \subset \mathbb{N}\}$ with cardinality $l = \#H$ and whose reductions $\widehat {\cal O}_h$ are all irreducible, to prove that $a = k = l$ it is enough to show than $l \geq k$.
Take any co-chain $P = \{{\cal O}_ j , \; j \in J \subset \mathbb{N}\} $ ($P$ is not necessarily in ${\cal F}$).
It is enough to exhibit an injective application from each $j \in
J$ to some $h \in H$.
In fact, let us fix any $j \in J $ and consider the basin of attraction $C({\cal O}_j)$. By definition of reduction, this basin has positive Lebesgue measure $m(C({\cal O}_j))$. But $S \in {\cal F}$, so $\sum _{h \in H} m(C(\widehat
{\cal O}_h)) = 1$. Then we deduce that $$0< m (C({\cal O}_j))= \sum
_{h \in H} m \left( C({\cal O}_j) \cap C(\widehat {\cal O}_h) \right) = \sum
_{h \in H} m \left( C({\cal O}_j \cap \widehat {\cal O}_h) \right)$$ Therefore some of the intersections in the sum above at right has positive Lebesgue measure. We obtain that for all $j \in J $ there exist some $h = h(j) \in
H$ such that ${\cal O}_j \cap \widehat {\cal O}_h $ is a reduction. But as $\widehat {\cal O}_h $ is irreducible then $ {\cal O}_j \supset \widehat {\cal
O}_h$.
To end the proof it remains to show that for $j \neq i \in J$ the sets $\widehat {\cal
O}_{h (i)}$ and $\widehat {\cal
O}_{h (j)}$ in $S$ are different reductions. By contradiction, if they were the same reduction in $S$, they both would be contained in two different reductions ${\cal O}_i$ and ${\cal O}_j$ in the chain $P$ and therefore these two last reductions would not be independent and $P$ would not be a co-chain.
Let us prove now the unicity of $S$, if there exists one, such that $S= \{\widehat {\cal O}_h: \, h \in H \subset \mathbb{N}\} \in {\cal F}$ and ${\cal O}_h$ are all irreducible. If there were two such collections $S_1$ and $S_2$, then repeating the construction above in this proof with $S = S_1$ and $P = S_2$ we conclude any ${\cal O}_j \in S_2$ contains one and only one $\widehat {\cal O}_{h(j)} \in S_1$. But as all $ {\cal O}_j \in S_2$ are irreducible we must have $\widehat {\cal O}_{h(j)}= {\cal O}_{j} \in
S_1$. So $S_2 \subset S_1$. The symmetric relation is obtained taking $S= S_2$ and $P = S_1$. $\Box$
Appendix.
=========
*[**Topological attraction in mean of the generalized attractors.**]{}*
Let $f \in C^0(M)$. Recall Definition \[definicionreduccion0\] of Generalized Attractor. Let $(A, {\cal A}) \subset M \times {\cal O}$ be a generalized Attractor, and let $B \subset A$ be its basin of attraction, which by definition has positive Lebesgue measure $m(B)
>0$. Recall that $A \subset M$ is the minimum compact subset in $M$ that contains the support of all the observable measures $\mu \in
{\cal A}$. In the next statement we will call $A \subset M$ as the attractor.
*\[propositionAtraeEnMedia\]*
For all $\varepsilon >0$ there exists $N \geq 1$ and a subset $C
\subset B$ of the basin of attraction $B$ of the generalized attractor $A$ such that $m(B\setminus C)< \varepsilon \, m(B)$, and for all $x \in C$ and for all $n \geq N$, more than $(1-\varepsilon)100$% of the iterates of the finite piece $\{f^j(x)\}_{0 \leq j \leq n-1}$ of the future orbit of $x$, lay in the $\varepsilon$-neighborhood of the attractor $A$.
[*Proof:*]{} The attractor $A$ is compact. Call $V \supset A$ to the $\varepsilon$-neighborhood of $A$ Construct a continuous function $\phi\colon M \mapsto [0,1]$ such that $\phi(x) = 1$ for all $x \in
A$ and $\phi(x) = 0$ for all $x \not \in V$. By definition of the generalized attractor, for all $x \in B$ the convergent subsequences of (\[1\]) converge to a probability supported in $A$. So $
\frac{1}{n} \sum_{j=0}^{n-1} \phi(f^j(x))$ has all its convergent subsequences converging to 1. Then: $$B \subset \bigcup_{N \geq 1} \bigcap_{n \geq N} \{x \in M:
\frac{1}{n} \sum_{j=0}^{n-1} \phi(f^j(x)) > 1 - \varepsilon\}$$ $$m(B) \leq \lim_{N \rightarrow + \infty} m
(\bigcap_{n \geq N} \{x \in M: \frac{1}{n} \sum_{j=0}^{n-1} \phi(f^j(x)) > 1 - \varepsilon\} \ )$$ Therefore, there exists $N \geq 1$ such that $m(B \setminus C) > 1
-\varepsilon$ where: $$C = \bigcap_{n \geq N} \{x \in B: \frac{1}{n} \sum_{j=0}^{n-1} \phi(f^j(x)) > 1 - \varepsilon\}$$ Call $\chi_V$ to the characteristic function of the open set $V$. By construction $0 \leq \chi_V \leq \phi$. Then, for all $n \geq N$, and for all $x \in C$: $$\frac{\# \{0 \leq j \leq n-1: f^j(x) \in V\}}{n} = \frac{1}{n} \sum_{j=0}^{n-1} \chi(f^j(x))
\geq \frac{1}{n} \sum_{j=0}^{n-1} \phi(f^j(x)) > 1 - \varepsilon\; \; \Box$$
*\[prueba\] [**Proof of Theorem \[teoremaconvexo\].**]{}*
We shall prove the following:
1. \[prime\] If $\mu, \nu \in p\omega(x)$ then for each real number $ 0 \leq
\lambda \leq 1$ there exists a measure $\mu_{\lambda} \in p\omega
(x)$ such that $${\mbox{$\,$dist$\,$}}(\mu_{\lambda}, \mu) = \lambda {\mbox{$\,$dist$\,$}}(\mu, \nu)$$
2. \[segu\] The set $p\omega(x)$ either has a single element or non-countable infinitely many.
[*Proof:*]{} First let us deduce \[segu\] from \[prime\]: Suppose that $p\omega (x)$ has at least two different values $\mu$ and $\nu$. It is enough to note that the application $\lambda \in
[0,1] \mapsto \mu_\lambda \in p\omega(x)$ that verifies thesis \[prime\], is injective. Therefore $p\omega(x)$ has non-countable infinitely many elements.
To prove \[prime\] consider the sequence (\[1\]) $ \ \mu_n=
\left\{ \frac1n \; \sum_{j=0}^{n-1} \delta _{f^j(x)}\right\} _{ n
\in \mathbb{N}} $ of time averages. Either it is convergent, or has at least two convergent subsequences, say $\mu_{m_j} \rightarrow \mu
$ and $\mu _{n_j} \rightarrow \nu $, with $\mu \neq \nu$. It is enough to exhibit in the case $\mu \neq \nu$ a convergent subsequence of (\[1\]) whose limit $\mu_{\lambda } $ verifies the thesis \[prime\].
[**Assertion A:** ]{} *For any given $\varepsilon >0 $ and $K >0$ there exists a natural number $h = h(\varepsilon, K)>K$ such that *$$|{\mbox{$\,$dist$\,$}}(\mu_h, \mu) -\lambda {\mbox{$\,$dist$\,$}}(\nu, \mu)| \leq \varepsilon$$**
Let us first prove that Assertion A implies thesis \[prime\]: Take in assertion $A$: $h_0 = 1$ and by induction, for $j \geq 1$ take $h_j$ given $\varepsilon _j = 1/j$ and $K_j = h_{j-1}$. Then we obtain a sequence $\mu _{h_j}$, subsequence of (\[1\]), that verifies ${\mbox{$\,$dist$\,$}}(\mu_{h_j}, \mu)
\rightarrow \lambda ({\mbox{$\,$dist$\,$}}(\nu, \mu))$. Any convergent subsequence of $\mu _{h_j}$ (that do exist $\cal P$ is compact in the weak$^*$ topology) verify \[prime\].
Now, let us prove Assertion A:
As $\mu_{m_j} \rightarrow \mu$ and $\mu_{n_j} \rightarrow \nu $ let us choose first $m_j$ and then $n_j$ such that $$m_j >K; \;\; \; \frac{1}{m_j} < \varepsilon /4; \;\; \; {\mbox{$\,$dist$\,$}}(\mu, \mu_{m_j}) < \varepsilon /4; \; \;\;
n_j > m_j; \; \;\; {\mbox{$\,$dist$\,$}}(\nu, \mu_{n_j}) < \varepsilon /4$$ To exhibit the computations let us explicit some metric structure giving the weak$^*$ topology of $\cal P$. We will use for instance the following distance: $${\mbox{$\,$dist$\,$}}(\rho, \delta) =
\sum_{i=1}^{\infty}\frac {1}{2^i} \; \left| \int g_i \, d \rho
- \int g_i \, d \delta \; \right|$$ for any $\rho, \delta \in {\cal P}$, where $\{g_i\}_{i \in \mathbb{N}}$ is a countable set of functions $g_i \in
C(M)$ such that $|g_i|\leq 1$, dense in the unitary ball of $C(M)$.
Note from the sequence (\[1\]) that $|\int g \, d \mu _n - \int g \, d \mu
_{n+1}| \leq (1/n) ||g||$ for all $g \in C(M)$ and all $n \geq
1$. Then in particular for $n = m_j + k$, we obtain $$\label{inequ}
{\mbox{$\,$dist$\,$}}(\mu_{m_j + k}, \mu _{m_j + k + 1}) \leq \frac{1}{m_j} < \varepsilon /4 \;
\;\; \mbox { for all }
k \geq 0$$ Now let us choose a natural number $0 \leq k \leq n_j - m_j$ such that $$\left|{\mbox{$\,$dist$\,$}}(\mu_{m_j}, \mu _{m_j + k})- \lambda {\mbox{$\,$dist$\,$}}(\mu_{m_j}, \mu
_{n_j}) \right|< \varepsilon /4 \;\;\mbox{ for the given } \lambda
\in [0,1]$$ Such $k$ does exist because inequality (\[inequ\]) is verified for all $k \geq 0$ and besides:
$\bullet$ If $ k= 0 \mbox{ then } {\mbox{$\,$dist$\,$}}(\mu_{m_j}, \mu _{m_j + k})=
0$ and
$\bullet$ if $ k= n_j -m_j \mbox{ then } {\mbox{$\,$dist$\,$}}(\mu_{m_j}, \mu _{m_j + k})=
{\mbox{$\,$dist$\,$}}(\mu_{m_j}, \mu _{n_j })$
Now renaming $h = m_j + k$, joining all the inequalities above, and applying the triangular property, we deduce: $$\left| {\mbox{$\,$dist$\,$}}(\mu_{h}, \mu) - \lambda {\mbox{$\,$dist$\,$}}(\nu, \mu)\right| \leq
\left| {\mbox{$\,$dist$\,$}}(\mu_{h}, \mu_{m_j}) - \lambda {\mbox{$\,$dist$\,$}}(\mu_{m_j},
\mu_{n_j})\right| +$$ $$+ \left| {\mbox{$\,$dist$\,$}}(\mu_{h}, \mu) -
{\mbox{$\,$dist$\,$}}(\mu_{h}, \mu_{m_j})\right| +$$ $$+ \lambda \left| {\mbox{$\,$dist$\,$}}(\mu_{m_j}, \mu_{n_j}) - {\mbox{$\,$dist$\,$}}(\mu_{m_j}, \nu)\right| + \lambda
\left| {\mbox{$\,$dist$\,$}}(\mu_{m_j}, \nu) - {\mbox{$\,$dist$\,$}}(\mu, \nu)\right| <
\varepsilon \;\;\;\;\; \Box$$
[**Acknowledgements.**]{}
We thank the referee for his remarks and suggestions to the first version of this paper.
[XXXX 22]{}
Anosov, D. V.: [*Geodesic flow on closed Riemanninan manifolds of negative curvature.* ]{} [Proc.Steklov.Inst.Math [ **90**]{} 1967.]{}
Araújo, V.: [*Infinitely many stochastically stable attractors.* ]{} [Nonlinearity [ **14.**]{} 2001.[ pp. 583-596]{}]{}
Bonatti, C.; Matheus, C.; Viana, M.; Wilkinson, A.: [*Abundance of stable ergodicity.* ]{} [Comment. Math. Helv. [ **79.** ]{}2003.[ pp. 753–757]{}]{}
Bonatti, C.; Viana, M.: [*SRB measures for partially hyperbolic systems whose central direction is mostly contractive.* ]{} [Israel J. Math.[ **115.** ]{}2000.[ pp. 157-194]{}]{} Bowen, R., Ruelle, D.: [*The ergodic theory of Axiom A flows.*]{} [Invent.Math.[ **29.**]{} 1975.[ pp. 181-202]{}]{}
Campbell, J.; Quas, A.: [*A generic $C^1$ expanding map has a singular S-R-B measure.*]{} [Commun. Math. Phys. [ **349**]{} 2001[ pp. 221-335]{}]{}
Carvalho, M.: [*Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms.* ]{} [Ergod. Th. and Dyn. Sys.[ **13.**]{} 1993.[ pp. 21-44]{}]{}
Catsigeras, E.; Enrich, H.: [*SRB measures of certain almost hyperbolic diffeomorphisms with a tengency.* ]{} [Disc. and Cont. Dyn. Sys. [ **7.**]{} 2001.[ pp. 177-202]{}]{}
Colli, E.: [*Infinitely many coexisting strange attractors.*]{} [Ann. de l’IHP.An. non linéaire [ **15.**]{} 1998.[ pp. 539-579]{}]{}
Enrich, H.: [*A heteroclinic biffurcation of Anosov diffeomorphisms.* ]{} [Ergod. Th. and Dyn. Sys. [ **18.**]{} 1998.[ pp. 567-608]{}]{}
Hu, H.: [*Conditions for the existence of SRB measures for Almost Anosov diffeomorphisms.* ]{} [Trans. Amer.Math.Soc. [ **352** ]{} 2000.[ pp. 2331-2367]{}]{}
Hu, H.; Young, L. S.: [*Nonexistence of SRB measures for some diffeomorphisms that are almost Anosov* ]{} [Ergod. Th. and Dyn. Sys. [ **15**]{} 1995.[ pp. 67-76]{}]{}
Katok, A.; Hasselblatt, B.: [*Introduction to the modern theory of dynamical systems.*]{} [Ency. of Math. and its Appl. Vol 54 [ Cambridge University Press]{} Cambridge 1995]{}
Keller, G.: [*Equilibrium states in ergodic theory.*]{} [London Mathematics Society Student Texts.[ Cambridge Universtity Press ]{} 1998[ ]{}]{} Ledrappier, F.; Young, L. S.: [*The metric entropy of diffeomorphisms. Part I: Characterization of measures satisfying Pesin’s entropy formula.* ]{} [Annals of Mathematics [ **122**]{} 1985.[ pp. 509-574]{}]{}
Ledrappier, F.; Young, L. S.: [*The metric entropy of diffeomorphisms. Part II: Relations between entropy, exponents and dimension.* ]{} [Annals of Mathematics [ **122**]{} 1985.[ pp. 540-539]{}]{}
Lewowicz, J.: [*Lyapunov functions and topological stability.* ]{} [Journ. of Diff. Eq. [ **38**]{} 1980.[ pp. 192-209]{}]{}
Liu, C. L.: [*Elements of Discrete Mathematics.*]{} [Mc.Graw-Hill[ ****]{} 1985 [ ]{}]{} Mañé, R.: [*Ergodic Theory and Differentiable Dynamics.*]{} [Springer-Verlag.[ ]{} 1989.[ ]{}]{}
Milnor, J.: [*On the concept of attractor.* ]{} [Commun. of Math. Phys. [ **99**]{} 1985[ pp. 177-195]{}]{}
Misiurewicz, M.: [*Ergodic natural measures.*]{} [in Contemporary Mathematics [bf 385]{} Algebraic and topological dynamics, edts: Kolyada, S.; Manin, Y.; Ward, T. .[ ]{} 2004.[ pp. 1-6]{}]{}
Newhouse, S.: [*Diffeomorphisms with infinitely many sinks.*]{} [Topology. [ **13.**]{} 1974.[ pp. 9-18]{}]{}
Palis, J.: [*A global view of Dynimcs and a conjecture on the denseness of finitude of attractors.* ]{} [Astérisque [ **261** ]{} 1999.[ pp. 339-351]{}]{}
Pesin, Ya. B.: [*Characteristic Lyapunov exponents and smooth ergodic theory.* ]{} [Russian Math. Surveys[ **32**]{} 1977.[ pp. 55-112]{}]{}
Pesin, Ya. B.; Sinai, Ya. G.: [*Gibbs measures for partially hyperbolic attractors.* ]{} [Ergod. Th. and Dyn. Sys. [ **2**]{} 1982.[ pp. 417-438]{}]{}
Pugh, C.; Shub, M.: [*Ergodic Attractors.* ]{} [Trans. Amer.Math.Soc.[ **312**]{} 1989 [ pp. 1-54]{}]{}
Pugh, C.; Shub, M.: [*Stable ergodicity.* ]{} [Bull. Amer.Math.Soc. [ **41**]{} 2004.[ pp. 1-41]{}]{} Quas, A.: [*Non ergodicity for $C^1$ expanding maps and g-measures.*]{} [Ergodic Systemas Dynamical Systems [ **16**]{} 1996[ pp. 531-543]{}]{}
Ruelle, D.: [*A measure associated with axiom A attractors.* ]{} [Amer. Journ. of Math. [**98**]{} 1976. [ pp. 619-654]{}]{}
Sinai, Ya. G.: [*Gibbs measure in ergodic theory.*]{} [Russ. Math. Surveys [ **27**]{} 1972.[ pp. 21-69]{}]{}
Viana, M.: [*Dynamics: a probabilistic and geometric perspective.* ]{} [Proceedings of the International Congress of Mathematics at Berlin. Doc. Math. Extra [ **Vol. I**]{} 1998 [ pp. 395-416]{}]{}
Walters, P.: [*Ruelle’s operator theorem and g-measures.*]{} [Trans. of the A.M.S.[**214**]{} 1975 [pp.375-387]{}]{} Young, L. S.: [*What are SRB measures and which dynamical systems have them?* ]{} [J. Stat. Phys. [ **108-5**]{} 2002 [ pp. 733-754]{}]{}
[^1]: Both authors: Instituto de Matemática y Estadística Rafael Laguardia (IMERL), Fac. Ingeniería, Universidad de la República, Uruguay. E-mails: eleonora@fing.edu.uy and enrich@fing.edu.uy Address: Herrera y Reissig 565. Montevideo. Uruguay.
[^2]: Our definition of observability is weaker than the definition in [@ke]
[^3]: The needed hypothesis are restated according to Corrections to Equilibrium states in ergodic theory" in http://www.mi.uni-erlangen.de/ keller/publications/equibook-corrections.pdf published by the author of the book.
|
---
abstract: |
We continue our program of single-site observations of pulsating subdwarf B (sdB) stars and present the results of extensive time series photometry of HS 0039+4302 and HS 0444+0458. Both were observed at MDM Observatory during the fall of 2005. We extend the number of known frequencies for HS 0039+4302 from 4 to 14 and discover one additional frequency for HS 0444+0458, bringing the total to three. We perform standard tests to search for multiplet structure, measure amplitude variations, and examine the frequency density to constrain the mode degree $\ell$.
Including the two stars in this paper, 23 pulsating sdB stars have received follow-up observations designed to decipher their pulsation spectra. It is worth an examination of what has been detected. We compare and contrast the frequency content in terms of richness and range and the amplitudes with regards to variability and diversity. We use this information to examine observational correlations with the proposed $\kappa$ pulsation mechanism as well as alternative theories.
author:
- |
M. D. Reed,$^1$[^1] D. M. Terndrup,$^2$ A.-Y. Zhou,$^1$ C. T. Unterborn,$^2$ D. An,$^2$ and J. R. Eggen$^1$\
$^1$Department of Physics, Astronomy and Materials Science, Missouri State University, 901 S. National, Springfield, MO 65897 USA\
$^2$Department of Astronomy, The Ohio State University, 140 W. 18th Ave., Columbus, OH 43210 USA
date: Accepted Received
title: 'Resolving the pulsations of subdwarf B stars: HS 0039+4302, HS 0444+0458, and an examination of the group properties of resolved pulsators'
---
Stars: oscillations – stars: variables – stars: individual (HS 0039+4302, HS 0444+0857) – Stars: subdwarfs
Introduction
============
Subdwarf B (sdB) stars are horizontal-branch stars with masses $\approx 0.5 M_\odot$, thin ($< 10^{-2} M_\odot$) hydrogen shells, and temperatures from $22\,000$ to $40\,000$ K [@heber; @saf94]. Pulsating sdB stars come in two varieties: short period (90 to 600 seconds), and long period (45 minutes to 2 hours). This work concentrates on the short-period pulsators, which are named EC 14026-2647 stars after the prototype [@kill97]; they are also known as V361 Hya stars or sdBV stars. They typically have pulsation amplitudes near 1%, and detailed studies reveal a few to dozens of frequencies. The longer period pulsators are known as PG 1716 pulsators after that prototype and typically have amplitudes less then 0.1% [@grn03]. They are also cooler then the EC 14026-type pulsators, though there is some overlap, and they are most likely $g-$mode pulsators [@font].
Asteroseismology of pulsating sdB stars can potentially probe the interior structure and provide estimates of total mass, shell mass, luminosity, helium fusion cross sections, and coefficients for radiative levitation and gravitational diffusion. To apply the tools of asteroseismology, however, it is necessary to resolve the pulsation frequencies. This usually requires extensive photometric campaigns, preferably at several sites spaced in longitude to reduce day/night aliasing. Generally, discovery surveys have simply identified variables and detected only the highest-amplitude pulsations, while multisite campaigns have observed few sdBV stars.
We have been engaged in a long-term program to resolve poorly-studied sdB pulsators, principally from single-site data. This method has proven useful for several sdBV stars [@me2; @reed06a; @reed07; @zhou]. Here, we report the results of our observations of HS 0039+4302 (hereafter HS 0039) and HS 0444+0458 (hereafter HS 0444). HS 0039 ($B = 15.6$) and HS 0444 ($B = 16.5$) were discovered to be members of the EC 14026 class by @ost01a [hereafter [[Ø]{}01]{}]. Their observations of HS 0039 consisted of three runs of 1, 2, and 3 hours, and they obtained three $\approx 1.5$ hour runs for HS 0444. Even with these short runs, they were able to detect four frequencies in HS 0039 and two in HS 0444. [Ø]{}01 also obtained spectra of both stars, from which they determined $T_{\rm eff} = 33~100$ K and $34~500$ K, and $\log g = 6.0$ and $6.1$ (with $g$ in cgs units of $cm\cdot s^{-2}$) for HS 0039 and HS 0444, respectively, and that neither star is a spectroscopic binary. In §2 we describe our new observations of these stars, in §3 we analyze the pulsation frequencies, and in §4 we discuss our findings and apply asteroseismic tests.
With the addition of these two stars, 23 sdBV stars have received follow-up observations, including 18 stars for which our program has directly contributed data. In §5, we will discuss the observational properties of all 23 stars, concentrating on pulsation stability, amplitudes, and a comparison with known driving mechanisms.
Observations
============
Data were obtained at MDM Observatory’s 1.3 m telescope using an Apogee Alta U47+ CCD camera. MDM Observatory is located on the southwest ridge of Kitt Peak, Arizona and is operated by a consortium of five universities, including the Ohio State University. Images were transferred via USB2.0 for high-speed readout; our binned ($2\times 2$) images had an average dead-time of one second. The observations used a red cut-off filter (BG38), so the effective bandpass covers the $B$ and $V$ filters and is essentially that of a blue-sensitive photomultiplier tube. Such a setup allows us to maximize light throughput while maintaining compatibility with observations obtained with photomultipliers. Tables \[tab01\] and \[tab02\] provide the details of our observations including date, start time, run length, and integration time. The observations total nearly 100 hours of data for HS 0039 and more than 60 hours for HS 0444.
Standard image reduction procedures, including bias subtraction, dark current and flat field correction, were followed using IRAF[^2] packages. Intensities were extracted using aperture photometry. Extinction and cloud corrections were obtained from the normalized intensities of several field stars. Because sdB stars are substantially hotter than typical field stars, differential light curves are not flat due to atmospheric reddening. A low-order polynomial was fit to remove nightly trends from the data. Finally, the lightcurves were normalized by their average flux and centered around zero so the reported differential intensities are $\Delta I = \left( I /\langle I\rangle\right) -1 $. Amplitudes are given as milli-modulation amplitudes (mma), with 10 mma corresponding to 1.0% or 9.2 millimagnitudes. Sample lightcurves are shown in Fig. \[fig01\].
----------- ---------- --------- -------- -------------
Run UT Start UT Date Length Integration
(h:m:s) 2005 (hr) (s)
mdm111505 01:46:00 15 Nov. 7.9 15
mdm111605 01:17:00 16 Nov. 8.4 12
mdm111705 01:18:00 17 Nov. 8.4 12
mdm111805 01:16:00 18 Nov. 8.5 12
mdm111905 01:15:00 19 Nov. 8.5 12
mdm112005 01:23:00 20 Nov. 8.3 12
mdm112105 01:23:00 21 Nov. 8.2 10
mdm112205 01:11:00 22 Nov. 8.6 10
mdm112505 02:35:00 25 Nov. 6.5 12
mdm112605 01:06:00 26 Nov. 4.0 12
mdm112705 01:11:00 27 Nov. 0.6 12
mdm112805 01:11:00 28 Nov. 6.2 10
mdm112905 01:06:00 29 Nov. 6.4 10
mdm113005 01:90:00 30 Nov. 3.4 10
mdm120905 01:24:00 09 Dec. 4.7 8
mdm121405 01:14:00 14 Dec. 1.1 8
----------- ---------- --------- -------- -------------
: Observations of HS 0039 \[tab01\]
--------------- ---------- --------- -------- -------------
Run UT Start UT Date Length Integration
(h:m:s) 2005 (hr) (s)
hs04mdm111505 09:54:00 15 Nov. 3.0 15
hs04mdm111605 09:52:00 16 Nov. 3.1 12
hs04mdm111705 09:53:00 17 Nov. 3.0 12
hs04mdm111805 09:52:00 18 Nov. 2.9 12
hs04mdm111905 09:53:00 19 Nov. 2.8 12
hs04mdm112005 09:57:00 20 Nov. 2.7 12
hs04mdm112105 09:48:00 21 Nov. 2.8 12
hs04mdm112205 10:52:00 22 Nov. 1.6 12
hs04mdm112605 05:26:00 26 Nov. 6.8 12
hs04mdm112705 08:59:00 27 Nov. 3.1 12
hs04mdm112805 07:23:10 28 Nov. 4.8 10
hs04mdm112905 07:32:00 29 Nov. 4.5 10
hs04mdm113005 04:26:00 30 Nov. 4.7 10
hs04mdm120905 06:26:40 09 Dec. 2.6 15
hs04mdm121005 01:30:30 10 Dec. 9.5 10
hs04mdm121405 05:45:10 14 Dec. 5.3 12
--------------- ---------- --------- -------- -------------
: Observations of HS 0444 \[tab02\]
Pulsation Analysis
==================
[**HS 0039:** ]{} A quick analysis during observations alerted us that the amplitudes of HS 0039 were not stable. This is easily seen in the final nightly reductions, twelve of which are shown in Fig. \[fig02\]. In this figure, we show the pulsation spectra (Fourier transforms; FTs) from adjacent nights, except for the short night on 27 November. Only the frequency near 4270 $\mu$Hz appears stable in amplitude, while those near 5175, 5482, and 7348 $\mu$Hz show substantial variation. We therefore examined the amplitudes and phases for indications of closely spaced multiplets, which would produce roughly sinusoidal amplitude variations and phase changes near the median amplitude [see @dmp]. Figure \[fig03\] shows our non-linear least-squares fits for the four dominant frequencies.
The frequency near 4270 $\mu$Hz is the most stable, both in amplitude and phase. The remaining three frequencies show significant amplitude variation, but only a little variation in phase. The phase for the peak near 5175 $\mu$Hz shows an unusual bimodality at the beginning of the campaign, with a steady, intermediate value at the end. We therefore separated the data into two subsets composed of data from the first seven runs (15 – 21 Nov.) and the last six runs (25 – 30 Nov.). Figure \[fig04\] shows the region near 5175 $\mu$Hz for all the November data and the two subsets. The FT of the first seven runs allowed us to interpret the phase information: the frequency at 5175 $\mu$Hz is composed of a close doublet separated by $\sim 17 \mu$Hz. The separation between frequencies is just shorter than 1 day ($11.6$ $\mu$Hz) and so nightly runs will not resolve these into two separate frequencies. Evidently the timing was just right near the beginning of the campaign that the phase switched between the two frequencies of the doublet on alternate nights. This was not the case later in the run, although the very last phase might show that the pattern was reestablishing itself.
The frequency near 5482 $\mu$Hz shows a similar variation in amplitude to that near 5175 $\mu$Hz, but not the bimodal phase. We grouped the data into the same subsets as in Fig. \[fig04\], but did not see any clear sign of a close doublet. If the amplitude variations were intrinsic to that peak, the discovery of frequencies by prewhitening would be more complicated. We produced an accurate window function as in @reed06a; the window function is the FT of a noise-free sinusoidal single frequency sampled at the same times as the data. The window function matched the multi-peaked structure of the temporal spectra around 5482 $\mu$Hz, leading us to conclude that the amplitude and phase variations are intrinsic to that frequency and it is not a closely-spaced multiplet. We also examined the region near 7348 $\mu$Hz in the complete data and in the subsets. There is no indication of closely spaced multiplets, nor in the the phases (Fig. \[fig03\]). We conclude that this frequency also has intrinsic amplitude variations, but is a temporally resolved frequency.
We do not see any such indications in the remaining frequencies, though most have amplitudes that are too small to be detected in individual runs. Armed with five resolved frequencies, we continued simultaneously least-squares fitting and removing peaks (prewhitening) in the combined data set. This process is shown in Fig. \[fig05\], where the top panel is the original FT and the next three panels show residuals after prewhitening by 5, 10, and 14 frequencies (from top to bottom). The solid (blue in the on-line version) line in the figure indicates the $4\sigma$ noise limit, below which we do not fit any peaks. After fitting 14 frequencies, we concluded that any remaining power in the residuals was caused by amplitude and/or phase variations and so does not represent remaining unfit frequencies. The solution to our fit, listing frequencies, periods, and average amplitudes is provided in Table \[tab03\].
------- ------------------ ------------------- -------------
ID Frequency Period Amplitude
($\mu$Hz) (s) (mma)
$f1$ 4135.559 (0.042) 241.8052 (0.0024) 0.85 (0.09)
$f2$ 4271.481 (0.008) 234.1108 (0.0004) 4.35 (0.09)
$f3$ 4952.858 (0.078) 201.9035 (0.0031) 0.46 (0.09)
$f4$ 5130.474 (0.036) 194.9137 (0.0013) 1.00 (0.09)
$f5$ 5175.466 (0.009) 193.2193 (0.0003) 3.99 (0.09)
$f6$ 5192.660 (0.010) 192.5794 (0.0003) 3.73 (0.09)
$f7$ 5482.319 (0.016) 182.4045 (0.0005) 2.16 (0.09)
$f8$ 5550.170 (0.047) 180.1746 (0.0015) 0.77 (0.09)
$f9$ 5705.677 (0.044) 175.2640 (0.0013) 0.82 (0.09)
$f10$ 5751.316 (0.081) 173.8732 (0.0024) 0.45 (0.09)
$f11$ 5756.651 (0.068) 173.7120 (0.0020) 0.53 (0.09)
$f12$ 7348.444 (0.031) 136.0832 (0.0006) 1.15 (0.09)
$f13$ 7449.256 (0.064) 134.2415 (0.0011) 0.65 (0.09)
$f14$ 7459.765 (0.063) 134.0524 (0.0011) 0.67 (0.09)
------- ------------------ ------------------- -------------
: Frequencies, periods, and average amplitudes for HS 0039. Formal least-squares errors are in parentheses.\[tab03\][]{data-label="tab03"}
[**HS 0444:**]{} Early in our campaign, we concentrated on HS 0039, leaving HS 0444 as a secondary target. Even from the relatively short runs (Table \[tab02\]), we could tell that HS 0444 was a simple pulsator with just a few frequencies. In December, when HS 0444 was better placed in the sky, we obtained longer individual runs to ensure that we did not miss any low amplitude peaks. The temporal spectrum of HS 0444 is shown in Fig. \[fig06\]. The top panel shows the original FT, the bottom panel the residuals after prewhitening by three frequencies, and the inset is the window function. The frequencies, periods and amplitudes from our least-squares solution are provided in Table \[tab04\]. In Fig. \[fig07\], we show the amplitudes and phases of the two frequencies which were detectable each night. The amplitudes and phases are relatively stable and do not show any indications of additional unresolved frequencies.
------ ------------------ --------------------- ------------
ID Frequency Period Amplitude
($\mu$Hz) (s) (mma)
$f1$ 5902.511 (0.012) 169.41940 (0.00035) 2.5 (0.2)
$f2$ 6553.484 (0.037) 152.59056 (0.00086) 0.8 (0.2)
$f3$ 7311.728 (0.003) 136.76657 (0.00005) 11.1 (0.2)
------ ------------------ --------------------- ------------
: Periods, frequencies, and amplitudes for HS 0444. Formal least-squares errors are in parentheses. []{data-label="tab04"}
Discussion
==========
Comparison with the discovery data
----------------------------------
The goal of our observational program is to resolve the pulsation frequencies for asteroseismic analysis. For the sake of comparison with the discovery data, we calculate the temporal resolution as $1/\Delta t$ with $\Delta t$ being the extent of the observations in time [see @kill99]. For the discovery data, we can determine the temporal resolution from information provided in Ø01, and estimate the detection limit as twice the top of the noise level in their FTs outside of the pulsations and window functions.
For HS 0039, our observations have a temporal resolution of 0.8 $\mu$Hz (excluding the December runs), which is $\approx
6.5\times$ better than the discovery data. The detection limit is 0.4 mma, which is about $5\times$ better than in Ø01. It is difficult to determine whether the frequencies in HS 0039 have changed since the discovery observations. Ø01’s reported frequencies (5.14, 5.48, and 4.27 mHz) roughly correspond to our $f4$ or $f5$, $f7$, and $f2$, respectively, but Ø01’s last frequency of 5.21 mHz, detected in only a single observing run, is not found in our data. As this frequency is wedged between higher-amplitude frequencies, and Ø01’s window function is complex, it is difficult to judge the significance of its detection, though it does look reasonable in their figures. However, the main difficulty is the abundance of frequencies, many of which have low amplitudes. The discovery data only detected four frequencies, whereas we detect 14. We estimate Ø01’s noise limit as $\approx 2$ mma; if our limit were as high, we also would have detected four frequencies. It is therefore not possible to ascertain the long-term stability of the pulsations from just these two sets of data.
Additional data on HS 0039 were obtained using the ULTRACAM multicolor instrument on the 4.2 m William Herschel Telescope in 2002 by Jeffery et al. (2004; hereafter J04). Two long ($\approx 8$ hr) runs with very high signal-to-noise were obtained on consecutive nights. We recover J04’s eight frequency detections, six of which are independent. Two are aliases of $f7$ and one is an alias of $f13$. J04 used multicolour photometry to estimate that $f8$ is an $\ell =4$ mode. Though geometric cancellation should reduce amplitudes for higher degree modes, the $(\ell,m) = (4,1)$ mode can be reduced by as little as 30%, depending on orientation [@me1]. Since $f8$ is 18% the amplitude of $f2$ (the highest amplitude frequency), there is no problem in ascribing it as an $\ell = 4$ mode. Additionally, as this region of the FT is relatively uncrowded, the data obtained by J04 should have been sufficient to make this determination. Although they do not claim this identification with certainty, it seems reasonable and would certainly be worth additional multicolour photometry or time-series spectroscopy for confirmation.
For HS 0444, our observations have a temporal resolution of 0.4 $\mu$Hz, which is $\approx 14\times$ better than the discovery data. The detection limit is 0.6 mma, which is about $3\times$ better than in Ø01. Additionally, we have recovered the two frequencies detected in the discovery data, to within the errors, and uncover a single new frequency, at an amplitude below their detection limit.
Constraints on the pulsation modes
----------------------------------
In addition to improving the known pulsation spectra of these stars, we wish to place observational constraints on the pulsation modes. The modes are mathematically described by spherical harmonics with three quantum numbers, $n$ (or $k$), $\ell$, and $m$. Rotation can break the $m$ degeneracy by separating each degree $\ell$ into a multiplet of $2\ell +1$ components, so multiplet structure is a very useful tool for observationally constraining pulsation degree [see @wing91; @me2; @simon].
For slow rotators, like most sdB stars are thought to be [@heber99; @heber00], rotationally-split multiplets should be nearly equally spaced in frequency. Such structure is, however, seldom observed in sdBV stars and in neither case are multiplets detected in our observations. Even though HS 0039 has 14 frequencies, the frequency spacings are not regular. Instead, the spacings are distributed from 5 to 3325 $\mu$Hz, with no obvious groupings. For HS 0444, there are only three detected frequencies, but the spacings are not similar, so there is no multiplet structure in this star.
Another tool that can be used is the density of frequencies within a given range. In resolved sdBV stars, we sometimes observe many more pulsation modes than $\ell = 0$, 1, and 2 can provide, independent of the number of inferred $m \neq 0$ frequencies. Higher $\ell$ modes may be needed, but if so they must have a larger amplitude than is measured because of the large degree of geometric cancellation [@charp05a; @me1]. A general guideline would be one $n$ order per $\ell$ degree per 1000 $\mu$Hz (@char02 find an average spacing near 1440 $\mu$Hz), so the temporal spectrum can accommodate three frequencies per 1000 $\mu$Hz without the necessity of invoking high-$\ell$ values if no multiplet structure is observed. Filling all possible $m$ values, the limit becomes nine frequencies per 1000 $\mu$Hz..
Obviously there is no need to invoke high-$\ell$ modes for HS 0044. Between 4900 and 5800 $\mu$Hz, HS 0039 has 9 of its 14 frequencies. Therefore, HS 0039 has too large a frequency density to exclude $\ell\geq 3$ modes, particularly with the absence of any obvious multiplets. This supports the identification of an $\ell = 4$ mode by J04.
Group Properties
================
Data sources
------------
Since the discovery of the EC 14026 class of pulsating sdB stars in 1997 [@kill97], there have been three areas of emphasis for observations: 1) to discover more pulsators; 2) to resolve the pulsations using long time-base campaigns, sometimes at multiple sites; and 3) to obtain high signal-to-noise observations over short time intervals. In recent years, multicolour photometry and time-series spectroscopy have been obtained as additional tools for mode identification.
For this paper, we will concentrate on the second point above, and examine pulsators for which a considerable effort has been expended to resolve the pulsation frequencies. We do this because it has been our area of emphasis, we feel that it is an important component in applying asteroseismology to sdB stars, and most importantly, we have data for all the stars except those from Kilkenny et al. (2002, 2006a, 2006b). Though this will not be a complete sample of sdBV follow-up observations, we can perform uniform tests upon them (except as noted above) for intercomparison. Table \[tab05\] provides a list of studies that have thoroughly investigated the pulsations of sdBV stars. Some stars, such as Feige 48, PG 0014, PG 1219, and PG 1605, have received extensive observations over the course of many years, while most have only been observed during a single campaign. Column 1 of that table lists the full name of each pulsator. Columns 2 and 3 display the dates of observations and the number of hours observed. The final two columns display the observing sites and references for each star. The references in Table \[tab05\] are the basis for our analysis below, but are not necessarily a complete record of the detailed observations of each star.
Target Inclusive Dates Hours Observed Sites References
--------------------------- ------------------------- ---------------- ----------------------- ------------
Balloon090100001 17 Aug. - 19 Sep. 2004 125 1 a
$^{\dagger}$ 8 Aug. - 30 Sep. 2005 NA 2, 12 U
EC 05217-3914$^{\dagger}$ 6 - 15 Nov. 1999 59 3, 4 b
EC 14026-2647 July 2003 NA 4 c
EC 20338-1925 23 Jul. - 26 Sep., 1998 45.9 4 d
June 2004 (2 nights) 12 4 c
Feige 48$^{\dagger}$ 1998 – 2006 $>500$ 2, 5, 6, 7, 8, 10, 11 e
HS 0039+4302$^{\dagger}$ 15 Nov. - 14 Dec. 2005 91 6 f
HS 0444+0458$^{\dagger}$ 15 Nov. - 14 Dec. 2005 63 6 f
HS 1824+5745$^{\dagger}$ 25 May - 11 Jul. 2005 127 6, 9 g
HS 2149+0847 July 2003 (4 nights) NA 4 d
June 2004 (10 nights) NA 4 d
HS 2151+0857$^{\dagger}$ 18 Jun. - 11 Jul. 2005 42 6, 9 g
HS 2201+2610$^{\dagger}$ 17 Sep. - 4 Oct. 2000 95.0 5, 12 h
KPD 1930+2752$^{\dagger}$ 11 - 16 Jul. 2002 38 7, 11 i
$^{\dagger}$ 15 Aug. - 9 Sep. 2003 246.5 10 U
KPD 2109+4401$^{\dagger}$ 12 Sep. - 14 Oct. 2004 182.6 2, 6, 11 j
PB 8783 8 - 22 Oct. 1996 183 12 k
PG 0014+182$^{\dagger}$ 8 - 20 Oct. 2004 142 6, 10 l
PG 0048+091$^{\dagger}$ 26 Sep. - 11 Oct., 2005 167 9, 11 m
PG 0154+182$^{\dagger}$ 6 - 14 Oct. 2004 28.4 6 g
PG 1047+003 17 Feb. - 2 Mar. 1998 98 12 n
PG 1219+534$^{\dagger}$ 2003 – 2006 $>200$ 2, 6, 7 o
PG 1325+101$^{\dagger}$ 3 Mar. - 3 Apr. 2003 264 2, 12 p
PG 1336-018$^{\dagger}$ 3 - 20 April, 1999 172 10 q
$^{\dagger}$ 14 Apr. - 1 May 2001 288 10 U
PG 1605+072$^{\dagger}$ 1997 – 2002 $>400$ 4, 5, 10, 11 r
PG 1618+563$^{\dagger}$ 17 Mar. - 1 May 2005 200.5 2, 7, 9, 11 m
: List of follow-up observations of pulsating sdB stars. A dagger$^{\dagger}$ indicates observations that we were involved in and NA indicates information that was not available. Observing sites: 1) Suhora 0.6 m; 2) Baker 0.4 m; 3) CTIO 1.5 m; 4) SAAO 1.9 m; 5) Fick 0.6 m; 6) MDM 1.3 m; 7) McDonald 2.1 m; 8) McDonald 0.9 m; 9) MDM 2.4 m; 10) Whole Earth Telescope Campaign, 11) Multisite campaign by Reed et al., 12) Other campaign. Column 5 references are: a) Baran et al. 2005; b) Reed et al. 2006b; c) Kilkenny et al. 2006a; d) Kilkenny et al. 2006b; e) Reed et al. 2004; f) this work; g) Reed et al. 2006a; h) Silvotti et al. 2002a; i) Reed et al. 2006c; j) Zhou et al. 2006; k) O’Donoghue et al. 1998a; l) Vucković et al. 2006; m) Reed et al. 2007; n) Kilkenny et al. 2002; o) Harms, Reed, O’Toole 2006; p) Silvotti et al. 2006; q) Kilkenny et al. 2003; r) Kilkenny et al. 1999; U) unpublished.\[tab05\]
The pulsation content
---------------------
In Table \[tab06\], we assemble data on the various sdBV stars. Column 1 lists an abbreviated name, which we will use hereafter. Columns 2 – 4 give the total number of detected frequencies, and the numbers with high and low amplitudes (these and other quantities in the table are discussed more thoroughly below). In columns 5 and 6, we show the temporal resolution $1 / {\Delta t}$ and the noise limit in mma as inferred from the analysis in each paper (§4.1). Column 7 displays our judgment about whether the frequencies were completely resolved. Of the 23 stars in Table \[tab07\], 18 are likely resolved. The other five either have too little data (EC 20338 and PB 8783), or have (some) pulsation amplitudes that are variable on time-scales too short for frequency resolution (BA09, KPD 1930, and PG 0048). The effective temperatures and gravities are listed in Columns 8 and 9; quantities in parentheses are estimates (described in §5.3). The next two columns display the total power in the resolved frequencies and the largest detected amplitude. The last column provides references for the spectroscopically determined values of $\log g$ and $T_{\rm eff}$.
Figure \[fig08\] displays $\log g$ against $T_{\rm eff}$ for the 20 stars in Table \[tab06\] with previously determined values. These are shown as filled circles. In addition, PG 1716-type pulsators are shown as filled (blue) triangles, and non-pulsators are shown as open circles. The solid (black) line is the zero-age helium main sequence with masses marked, the dashed line is the zero-age extended horizontal branch, and evolutionary tracks [from @me2] are shown as solid (blue) lines. The coolest track has a hydrogen envelope thick enough for shell fusion, while the hotter two do not.
Figures \[fig09\] and \[fig10\] show schematic representations of the temporal spectra of all 23 stars, ordered by $\log g$, with the three stars without spectroscopically constrained values at the end. The dotted arrows (blue in the electronic version) indicate frequencies which are only observed occasionally. PG 0014 has dashed arrows (green in the electronic version), indicating frequencies that were only observed using ULTRACAM. To make low-amplitude frequencies visible in the plots, the vertical axes may begin below zero and are scaled so that the highest amplitude peak touches the top line, except for PG 1605, BA09, PG 1325, and EC 20338. All of the latter have one high-amplitude peak that would make the others too small on the plot. Those frequencies are indicated with arrows which pass beyond the top of the plot, though still not to scale. Note that all panels are plotted at different scales. BA09 shows a mixture of short- (EC 14026-type), and long-period (PG 1716-type) oscillations; only the former are shown.
---------- ------- ------ ------ -------------- ------- ----------- --------------- -------------------- ----------- --------------- ------
Star Total High Low $1/\Delta T$ Limit Resolved? $T_{\rm eff}$ $\log g$ Power A$_{\rm max}$ Refs
(\#) (\#) (\#) ($\mu$Hz) (mma) (K) ($cm\cdot s^{-2}$) (mma$^2$) (mma)
BA09 19 3 16 0.4 0.5 No 29446 5.33 4103.92 57.7 1
EC 05217 8 6 2 0.9 1.3 No 32000 5.73 29.36 3.9 2
EC 14026 3 1 2 NA NA Yes 34700 6.10 164 12 3
EC 20338 5 3 2 1.2 0.8 No (35500) (5.8) 774.43 26.6 U
Feige 48 8 3 5 0.8 0.1 Yes 29500 5.50 69.73 6.4 4
HS 0039 14 6 8 0.8 0.4 Yes 32400 5.70 60.15 4.4 5
HS 0444 3 2 1 0.4 0.6 Yes 33800 5.60 130.1 4.4 5
HS 1824 1 1 0 0.25 0.48 Yes 33100 6.03 11.56 3.4 5
HS 2149 6 6 6 NA NA Yes 35600 5.90 28.67 7.0 6
HS 2151 5 5 0 0.5 0.53 Yes 34500 6.13 30.02 3.8 5
HS 2201 5 2 3 0.01 0.5 Yes 29300 5.40 119.94 10.8 6
KPD 1930 39 31 8 0.5 0.8 No 33300 5.61 74.54 3.6 7
KPD 2109 8 6 2 0.4 0.29 Yes 31800 5.79 97.0 6.4 8
PB 8783 10 6 4 0.8 0.1 No 35700 5.54 16.13 2.1 3
PG 0014 13 8 5 0.9 0.48 Yes 34130 5.77 20.37 3.9 9
PG 0048 30 29 1 0.28 0.8 Yes (34000) (5.75) 42.16 2.3 U
PG 0154 6 4 2 1.4 0.76 Yes (35000) (5.8) 129.34 9.5 U
PG 1047 18 6 12 0.8 0.2 Yes 33150 5.80 87.0 6.7 10
PG 1219 6 4 2 0.8 0.6 Yes 33600 5.81 108.18 6.6 11
PG 1325 14 6 8 0.5 0.8 Yes 34800 5.81 744.27 27.1 12
PG 1336 27 13 14 1.0 0.25 Yes 33000 5.70 83.75 4.7 13
PG 1605 55 5 50 0.8 0.5 Yes 32300 5.25 1802.1 27.4 14
PG 1618 6 6 0 0.3 0.59 Yes 33900 5.80 18.53 2.2 15
---------- ------- ------ ------ -------------- ------- ----------- --------------- -------------------- ----------- --------------- ------
: Pulsation properties of EC 14026-type pulsators for which follow-up data has been obtained. References for the spectroscopic measurements are as follows: 1) Oreiro et al. 2004; 2) Koen et al. 1999a; 3) O’Donoghue et al. 1997; 4) Koen et al. 1998b; 5) Østensen et al. 2001a; 6) Østensen et al. 2000b; 7) Bill' eres et al. 2000; 8) Koen 1998; 9) Brassard et al. 2001; 10) O’Donoghue et al. 1998b; 11) Koen et al. 1999b; 12) Silvotti et al. 2002b; 13) Kilkenney et al. 1998; 14) Koen et al. 1998a; 15) Silvotti et al. 2000; U) Unpublished. Parameters in parentheses are inferred using color and pulsation frequencies. NA indicates information that is not available. BA09 also has longer period (PG 1716-type) pulsations and combination frequencies which are not included in this table.\[tab06\]
These two figures show the enormous variety of amplitudes and frequencies detected in sdBV stars. There are only four high-amplitude (here $A > 20$ mma) pulsators known and they have a great range (for sdB stars) of gravities and temperatures. There are pulsators with 20+ frequencies that have similar temperatures and gravities to stars with 5 frequencies (e.g., BA09 & HS 2201). If one looks at only the range of gravities from $\log g = 5.6$ to 5.7 (about $1\sigma$ in error), there are two stars with more than 20 frequencies yet one star with only three frequencies (but higher amplitudes!).
Observational tests and trends
------------------------------
In an attempt to bring order to the class as a whole and to find trends in the observational properties, we have organized the data in several ways which have benefited studies of other variable stars. In this subsection we will show the results along with some motivation, but leave in-depth interpretations to the next subsection.
[**Frequency groupings:**]{} Our first arrangement was to put the frequencies shown in Figs. \[fig09\] and \[fig10\] onto a common frequency scale and make a correction for gravity. Because $p-$mode periods are inversely proportional to the square root of the density, we expect that stars with lower $\log
g$ have longer periods (shorter frequencies). This is largely observed in the left panel of Fig. \[fig11\]. In the right panel, we have rescaled pulsation frequencies by $1/g^{0.75}$ (assuming constant mass) which is an adjustment for both size and density using just $\log g$. For the three sdBV stars without measured gravities, we estimated the gravity from the position of the shortest frequency compared to pulsators with known gravities. These assumed gravities are provided in parentheses in Fig. \[fig11\] and Table \[tab06\]. Such a study of pulsating white dwarfs revealed groups of frequencies which could then be related to individual modes [@clem94]. However, as evidenced by the summation of the right panel, no such groupings occur. The horizontal line just above the summation frequencies (right panel of Fig. \[fig11\]) shows the effect of an error of $\log g = 0.05$ in the rescaling. It is therefore possible that any groups are being smeared out by measurement errors in $\log g$. We attempted to correct for this by fixing the lowest modified frequency to a given value, but no corrections or fixed reference values of $\log g$ show reasonably-separated grouping that could be of use.
[**Relative pulsation amplitudes:**]{} The line lengths in Figs. \[fig09\] through \[fig11\] indicate another feature that is observed about half the time; one or two amplitudes are significantly higher than the rest. To parameterize this phenomenon, we have denoted frequencies whose amplitudes are within a factor of five of the highest amplitude as “high” amplitudes; the remainder are “low” amplitudes. The numbers that fall into each category, along with the highest amplitude observed ($A_{\rm max}$) for each star, are provided in Table \[tab06\]. The factor of five was chosen so that all amplitudes greater than 10 mma for PG 1605 and BA09 would fall into the high category. Additionally, we use the published (typically average) amplitudes, even though some vary by large amounts (discussed below).
The left panel of Fig. \[fig12\] shows the ratio of high (H) to total (T) number of frequencies against the total number of frequencies; the values of which are also provided in Table \[tab06\]. One might suspect that if a star has one high amplitude, all amplitudes are relatively higher and thus easier to detect. This appears not to be the case, since the points make a scatter diagram. In the right panel of Fig. \[fig12\], we show the H/T ratio against $\log g$. It might be expected that gravity plays a factor, in that it is easier to pulsate radially at lower gravities. The values of H/T are indeed positively correlated with gravity, but the correlation coefficient is only 0.59.
A more quantitative approach is shown in Fig. \[fig13\], where we analyse the distribution of amplitudes and power in each star. For each star, the frequencies were sorted by increasing amplitude and the cumulative distribution of amplitudes (or power) was computed at intervals of 10% of the total. This process is shown in Fig. \[fig13a\] for the star HS 0039. The 14 frequencies from Table \[tab03\] are shown as squares with their amplitudes given by the right-hand Y-axis. The circles indicate the cumulative fractional amplitude for each frequency, connected by the solid line, with their scale given by the left-hand Y-axis. The points for HS 0039 in the left panel of Fig. \[fig13\] correspond to the locations in Fig. \[fig13a\] where the dotted lines intersect the solid line. The top solid line in Fig. \[fig13\] connects the points for the 90$^{\rm th}$ percentile, the next line for the 80$^{\rm th}$ percentile, and so on. The left panel of Fig. \[fig13\] displays these points, or amplitude distribution, for the 20 stars with more than five frequencies. Their designations are provided along the bottom in order of increasing gravity. Low values indicate that the top one or two amplitudes contain a large fraction of the total amplitude (Fig. \[fig13a\] would show a sharply peaked line). The right panel displays the distribution of power which we define as the sum of the squares of the amplitudes; the power and the maximum amplitude $A_{\rm max}$ are provided in Table \[tab06\]. On average, the lowest-amplitude half of the frequencies contribute about 25% of the total amplitude and 9.6% of the total power.
If the amplitudes or power were evenly distributed (equal-height amplitudes) then $f_i/f_T$ would equal $A_i/A_T$ (or $\sqrt{P_i/P_T}$). The corresponding distribution in Fig. \[fig13a\] would be a straight diagonal line. There is a trend in Fig. \[fig13\] in that the contours are closest at low gravities and most diffuse at high gravities. However, these trends are dominated by four stars (two at each end) and are not representative of the majority of the class. In the middle of each panel, there is a large dispersion in values, with neighbouring stars transmitting most of their power through one frequency, or distributing it more equally.
In each panel, the dashed line indicates the fractional amplitude or power emitted by the frequency with the highest amplitude. For PG 1325, the one highest amplitude has 99% of the total, while for PG 0048 it is only $\sim 12$%. The dashed line in Fig. \[fig13a\] indicates that for HS 0039, the highest amplitude frequency has 20% of the combined amplitudes. While it is difficult to assign significance to this plot as the number of pulsators is still relatively low, it is suggestive that in lower gravity stars the pulsation energy is channeled into relatively few (or one) frequencies even though many frequencies are available.
[**Multiplet structure:**]{} As the prototype for multiplet structure in pulsating white dwarfs, PG 1159 showed that observational determination of the modes can lead to tight constraints on the models [@wing91]. With detailed studies of 23 pulsating sdBV stars, it would be hoped that a similar star of this class would have been detected. However, this has not been the case: multiplet structure has been conspicuously absent, even from rich pulsators. The only confirmed case of multiplets caused by rotational splitting is Feige 48 [@me2; @simon], while another likely candidate is BA09 [@andy]. There are also marginal cases for rotationally induced multiplet structure in HS 2201, PB 8783, PG 0014, PG 1047, and PG 1605 none of which have been confirmed by additional observations. KPD 1930 and PG 1336 are both known to be in short period binaries and frequency splittings are commensurate with the binary period. An initial interpretation did not attribute these to rotational splitting, but rather to tidal effects induced by the companion [@reed06c; @reed06d].
A possible reason for the lack of observed multiplet structure was proposed by @kaw05. Their picture invokes sharp differential rotation in rapidly spinning cores to modify the frequency spacings. Exceptions would be for those stars in close binaries where rotations are tidally locked.
Another multiplet pattern that has emerged lately is the “Kawaler-scheme” (Kawaler et al. 2006) a purely mathematical formalism based on an asymptotic-like relationship for the frequencies: $f(i,j)=f_o+i\times\delta +j\times\Delta$ where $i$ has integer values, $j$ is limited to values of $-1,\,0,$ and $1$, $\delta$ is usually a small spacing and $\Delta$ is usually a large spacing. Their Table 3 indicates the significance of their predicted frequencies to those observed in several stars. However, it is also known that the Kawaler-scheme does not fit several pulsators (including HS 0039) and so we merely make note of the scheme but await a full report in a forthcoming paper.
[**Frequency density:**]{} As discussed in §4.2, another tool at our disposal is the mode density. Although the mode density does not help to assign modes to individual frequencies, we can set limits on the number of degrees ($\ell$) per order ($n$) required to create the observed frequency density from currently available models of @char01 [hereafter CFB01] and of @me2.
Figure \[fig14\] shows the mode density, plotted against $\log g$, with the dotted line indicating 3 frequencies per 1000 $\mu$Hz (all $\ell\leq 2,m=0$ modes) and the dashed line indicating 9 frequencies per 1000 $\mu$Hz (all possible $2\ell +1$ $m$ modes). Starred points indicate those pulsators for which we have inferred the $\log g$ values. Less than 20% of sdBV stars have mode densities below 3 per 1000 $\mu$Hz, while more than 25% have mode densities too high to be reconciled with $\ell \leq 2$ modes even if all possible $m$ values are used. It therefore seems reasonable to conclude that $\ell \geq 3$ modes must be excited.
[**Amplitude variability:**]{} A criterion outlined in Christensen-Dalsgaard et al (2001; hereafter JCD01) and previously applied to several sdBV stars (Pereira & Lopes 2005; Zhou et al. 2006; Reed et al. 2006a, 2007) is to compare the average amplitude $\langle
A\rangle$ to the standard deviation of the amplitudes, $\sigma
(A)$. For stochastically excited pulsations, this ratio should have a value near 0.5. For all resolvable frequencies in our target stars, we have calculated both parameters and their ratios which are given in columns 3 to 5 of Table \[tab07\] and plotted in Fig. \[fig17\]. Columns 6 and 7 of Table \[tab07\] list the maximum and minimum observed amplitudes and column 8 gives the maximum time-scale over which the observations are sensitive to amplitude variations. In the Figure, triangles (squares) indicate frequencies known to have stable (not-stable) phases, circles indicate frequencies with ambiguous or no phase information and stars indicate the PG 0048 frequencies, which are known to have stochastic-like properties (Reed et al. 2007). Stochastic oscillations do not have stable pulsation phases and so frequencies with stable phases should be driven rather than stochastically excited. Amplitudes for HS 2201 came from Silvotti et al. (2002a) and no errors were published, so no errorbars appear in the Figure.
Just like their average amplitudes and frequency density, the only conclusion we can draw from Fig. \[fig17\] and Table \[tab07\] is sdBV stars show a large variety of amplitude variations. Of course it is known that many classes of pulsators show large amplitude differences and that correlations between excitation rates and amplitudes are weak at best, but these results indicate that there is no clear separation in amplitude variability for phase-stable and unstable frequencies. As such, it follows that the JCD01 criterion is likely not applicable for these stars. One last note is that our calculations do not include frequencies that have been observed only one time (as in PG 1219 and Feige 48) or stars for which we do not have the data. Both EC 14026 and EC 20338 are reported to have sufficient amplitude variability that frequencies completely disappear between observing seasons (Kilkenny et al. 2006b).
Star Frequency $\langle A\rangle$ $\sigma (A)$ $\sigma (A)/\langle A\rangle$ $A_{max}$ $A_{min}$ Timescale
---------- ----------- -------------------- -------------- ------------------------------- ----------- ----------- -----------
BA09 2807 45.76 1.36 $0.03\pm 0.004$ 52.89 44.30 53 days
2823 10.30 2.15 $0.21\pm 0.03$ 20.68 8.20 53 days
2824 14.19 0.68 $0.05\pm 0.02$ 15.2 11.50 53 days
2827 3.60 0.24 $0.07\pm 0.03$ 4.86 2.4 53 days
3776 1.58 0.57 $0.36\pm 0.06$ 4.35 1.4 53 days
3791 1.26 0.50 $0.39\pm 0.07$ 2.39 0.14 53 days
EC 05217 4595 3.04 0.79 $0.26\pm 0.10$ 4.59 2.74 9 days
4629 4.32 0.73 $0.17\pm 0.06$ 5.16 3.05 9 days
Feige 48 2851 3.83 2.85 $0.74\pm 0.20$ 9.46 1.31 8 years
2877 5.80 2.14 $0.37\pm 0.08$ 10.4 1.89 8 years
2906 4.10 1.44 $0.35\pm 0.08$ 5.28 0.58 8 years
HS 0039 4271 4.45 0.17 $0.04\pm 0.03$ 5.03 3.82 30 days
5482 2.35 1.30 $0.56\pm 0.17$ 4.44 0.12 30 days
7348 1.09 0.77 $0.70\pm 0.24$ 3.17 0.37 30 days
HS 0444 5903 2.59 0.44 $0.17\pm 0.01$ 3.16 1.76 30 days
7312 11.04 0.61 $0.06\pm 0.02$ 12.39 10.11 30 days
HS 1824 7190 3.01 0.92 $0.30\pm 0.06$ 5.26 0.87 47 days
HS 2151 6616 3.89 0.47 $0.12\pm 0.05$ 5.14 3.36 23 days
6859 1.73 0.42 $0.24\pm 0.10$ 2.45 1.31 23 days
7424 1.19 0.22 $0.18\pm 0.12$ 1.66 0.95 23 days
HS 2201 2738 0.39$\dagger$ 0.08 0.20 0.48 0.34 1 year
2824 4.85$\dagger$ 0.54 0.13 5.65 4.23 1 year
2861 10.31$\dagger$ 0.54 0.05 10.88 9.77 1 year
2881 1.16$\dagger$ 0.16 0.13 1.34 1.00 1 year
2922 0.6$\dagger$ 0.04 0.07 0.64 0.56 1 year
KPD 2109 5045 2.64 1.05 $0.40\pm 0.11$ 4.80 0.59 32 days
5093 6.45 0.75 $0.12\pm 0.10$ 8.09 5.21 32 days
5212 1.65 0.54 $0.32\pm 0.09$ 3.14 0.31 32 days
5481 6.21 0.45 $0.07\pm 0.10$ 7.41 4.96 32 days
PB 8783 7870 1.68 0.27 $0.16\pm 0.09$ 2.4 1.4 14 days
8092 1.19 0.24 $0.20\pm 0.11$ 1.6 0.5 14 days
PG 0048 5245 1.74 0.58 $0.33\pm 0.09$ 2.44 0.93 15 days
7237 1.45 0.40 $0.25\pm 0.10$ 2.13 0.27 15 days
PG 0154 6090 9.57 0.21 $0.02\pm 0.03$ 10.28 8.97 8 days
6785 3.75 1.16 $0.31\pm 0.11$ 5.31 1.50 8 days
7032 3.57 0.83 $0.23\pm 0.09$ 5.05 1.33 8 days
7688 2.61 0.28 $0.11\pm 0.20$ 3.28 0.84 8 days
8362 1.12 0.22 $0.19\pm 0.16$ 1.86 0.13 8 days
9015 1.03 0.21 $0.21\pm 0.16$ 2.53 0.51 8 days
PG 1219 6722 3.35 0.49 $0.15\pm 0.03$ 4.72 2.15 4 years
6961 7.00 0.46 $0.07\pm 0.01$ 8.06 5.73 4 years
7490 5.19 1.74 $0.33\pm 0.07$ 8.48 2.94 4 years
7808 6.35 1.65 $0.26\pm 0.05$ 9.85 4.44 4 years
PG 1325 7253 23.21 2.12 $0.09\pm 0.03$ 26.04 18.58 14 days
Star Frequency $\langle A\rangle$ $\sigma (A)$ $\sigma (A)/\langle A\rangle$ $A_{max}$ $A_{min}$ Timescale
---------- ----------- -------------------- -------------- ------------------------------- ----------- ----------- -----------
PG 1605 1891 9.6683 3.38 $0.35\pm 0.12$ 16.30 7.80 7 years
1986 12.06 4.62 $0.38\pm 13$ 14.77 2.0 7 years
2076 24.51 23.74 $0.97\pm 0.44$ 56.92 8.4 7 years
2102 29.24 12.32 $0.42\pm 0.15$ 48.9 13.4 7 years
2392 3.52 1.54 $0.44\pm 0.16$ 6.65 2.2 7 years
2743 14.61 8.48 $0.58\pm 0.22$ 29.0 5.02 7 years
2763 6.80 2.88 $0.42\pm 0.15$ 10.89 2.5 7 years
2845 4.87 1.71 $0.35\pm 0.12$ 7.1 2.0 7 years
PG 1618B 7191 2.13 0.44 $0.21\pm 0.12$ 3.6 1.14 45 days
7755 1.75 0.50 $0.28\pm 0.14$ 3.50 0.76 45 days
[**Comparison with theoretical instability contours:**]{} We can make a direct comparison between the instability zone of the second-generation pulsation models of @char01 and observed pulsation properties. Figures \[fig15\] and \[fig16\] are used for these discussions. In all the plots, filled circles are the sdBV stars of this study (stars indicate EC 20338, PG 0048, and PG 0154[^3]), open circles are sdB stars (and inferred to be non-pulsators) from the Hamburg-Schmidt survey [@edel03], Moehler et al (1990), and Saffer et al. (1994), and filled (blue) triangles are PG 1716-type pulsators (Green et al. 2003). The contours are reproduced from CFB01 with the outside contour representing one unstable $\ell=0$ frequency with each interior contour representing an additional unstable $\ell=0$ frequency up to $N=7$.
In the left panel of Fig. \[fig15\], we determine how discriminating the instability zone is by examining the ratio of pulsators to non-pulsators within each contour. However, we need to have an exception for the red (cool) edge of the instability zone. There is an indication that sdB stars may *switch* from EC 14026- to PG 1716-type pulsators in this region. As such, this region (separated with a dotted line) should be excluded; particularly since it is stated that “all cool sdB stars of low gravity may be PG 1716 pulsators” (Fontaine et al. 2006). Working from the inside ($N= 7$) contour outward, the fraction of pulsating to non-pulsating sdB stars is: $N\geq 7$, 50%; $N\geq 6$, 25%; $N\geq 5$, 25%; $N\geq4$, 22%; $N\geq 3$, 21%; $N\geq 2$, 20%; $N\geq 1$, 20% and the fraction of *all* sdB stars within the contours is $N\geq 7$, 12%; $N\geq 6$, 52%; $N\geq 5$, 80%, $N\geq4$, 90%; $N\geq 3$, 92%; $N\geq 2$, 95%; $N\geq 1$, 95%; $N<1$, 100%. There does appear to be a relationship between the interior instability ($N=7$) contour and fraction of pulsators. Yet outside of the first contour, the ratio only changes by 5% and all of the EC 14026-type pulsators are within the first three contours. Yet so are 80% of *all* sdB stars. So while the contours match where the pulsators are, they also match where *most* sdB stars with $T_{\rm
eff}\geq$30 000 K are.
The remaining panels of Figs. \[fig15\] and \[fig16\] show the same points, but the size of the dots correlate with the following observed properties; the number of pulsation frequencies (right panel of Fig. \[fig15\]), the maximum amplitude ($A_{max}$), and total pulsation power. $A_{max}$ and total pulsation power are related as power is dominated by a few high-amplitude frequencies. Stars such as PG 0048 with many, but low amplitude frequencies cannot match the power of a single $A\geq 20$ mma frequency. We would expect a correlation of the instability contours with the *number* of frequencies detected, as each interior contour increases the number of theoretically unstable frequencies, but such is not the case. From §5.2, we know that it is not a detection issue as rich pulsators occur with both low and high amplitudes. In fact there seems to be no correlation whatsoever with the largest points (largest number of frequencies, highest amplitude and most pulsation power) occupying multiple regions of the $\log
g - T_{\rm eff}$ diagrams and similar results for the smallest points. As such the group properties do not add observational support to the driving theory.
Conclusions
===========
From extensive follow-up data acquired at MDM observatory, we are confident that we have resolved the pulsation spectra of two additional pulsating sdB stars. For HS 0039, we detect 10 additional frequencies bringing the total to 14 and for HS 0444, we confirm the two frequencies of the discovery data and detect an additional low amplitude frequency. We have also noted that while the amplitudes and phases of HS 0444 appear very steady over the duration of our observations, those in HS 0039 did not, but rather have varied considerably. This is illustrative of the variety observed in sdBV stars where some stars can have very simple and/or stable pulsation spectra while others can be quite rich, with tens of frequencies that may change amplitudes on short time scales.
Since the discovery of sdBV stars in 1996, more than 23 of the 34 known EC 14026-type pulsators have received follow-up observations. We have examined these 23 stars for which extended timebase (and often multisite) observations have been acquired. We have searched for trends in frequency groupings, the number of high (H) amplitude to total (T) frequencies, and frequency density as a function of gravity and note that the only trend seems to be a weak relationship between the H/T ratio and gravity: The lowest gravity pulsators have the smallest H/T ratio (a few very high amplitude frequencies) while the highest gravity pulsators have H/T$\approx 1$ (relatively even amplitudes distributed amongst the observed frequencies).
We examined amplitude stability, which has been used to infer stochastically excited oscillations in other variable classes and previously applied to sdB pulsators. And while many pulsation frequencies fit the JCD01 value, the distribution between those and amplitude-stable and presumably driven is relatively smooth with no clear separations. However, phase-stable frequencies, some with quite large variability, preclude them from being stochastically excited and so the simplest conclusion is that the JCD01 criteria is not appropriate for sdBV stars.
We compared pulsators to theoretical instability contours to search for relationships that correlate with current driving theory. While there is a weak concentration of pulsators precisely where expected, these same contours include more than 80% of *all* sdB stars in our sample with $T_{\rm eff}\geq
30\,000$ K (an inferred cut-off for PG 1716-type pulsators) and there is no correlation between the contours and the number of pulsation frequencies, nor their maximum amplitude or total pulsation power. It would be nice if we could note that any correlation is canceled by stars with lower gravities requiring less energy to drive higher amplitudes which results in more frequencies being detected, but this is also not the case as the amplitudes of neighboring stars *anywhere* in the $\log
g - T_{\rm eff}$ plane can have amplitudes that differ by more than an order of magnitude.
In the end, what we have uncovered in our examination of sdBV stars is the large variety they encompass. Subdwarf B pulsators seem to exist at *all* temperatures and gravities (though we only examined the EC 14026-type pulsators at $T_{\rm eff}\geq
30\,000$ K) that sdB stars do and at roughly the same concentrations. Regardless of temperature or gravity (and thus evolutionary age and/or total mass and/or envelope mass and/or metallicity) pulsators can have relatively high ($\geq 20$ mma) or low ($\leq 2$ mma), stable or unstable amplitudes and the frequency density can be quite high or very low. We hope that our investigation of these 23 well-studied sdBV stars will be useful and bring new insight to their theoretical perspectives. Additionally, we also hope that this study will be useful for observers pursuing multicolour photometry and time-series spectroscopy. We anticipate that such works will be required for an overall understanding of sdB pulsations, yet these more detailed observations require that the pulsation characteristics of the star be known beforehand. So even though 23 of 34 known sdBV stars have received follow-up observations, there is still a long way to go both observationally and theoretically.
ACKNOWLEDGMENTS: We would like to thank the MDM and McDonald observatory TACs for generous time allocations, without which this work would not have been possible. We would also like to thank Dave Mills for his help with the Linux camera drivers, Darragh O’Donoghue, Chris Koen, Dave Kilkenny and Andrzej Baran for kindly sharing their data. We also thank the many observers who have provided support for our campaigns, particularly those at Lulin and Suhora observatories who have helped on many campaigns. Support for DMT came in part from funds provided by the Ohio State University Department of Astronomy. This material is based in part upon work supported by the National Science Foundation under Grant Number AST007480. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
refsmnras.tex
[^1]: E-mail:MikeReed@missouristate.edu
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^3]: We have inferred $\log g$ from the shortest pulsation frequency and $T_{\rm eff}$ from $B-V$ and $J-K_s$ colours for EC 20338, PG 0048, and PG 0154. These are very crude estimates and should be considered as such.
|
---
abstract: 'The factorial moments analyses are performed to study the scaling properties of the dynamical fluctuations of contacts and nodes in temporal networks based on empirical data sets. The intermittent behaviors are observed in the fluctuations for all orders of the moments. It indicates that the interaction has self-similarity structure in time interval and the fluctuations are not purely random but dynamical and correlated. The scaling exponents for contacts in [***Prostitution***]{} data and nodes in [***Conference***]{} data are very close to that for 2D Ising model undergoing a second-order phase transition.'
author:
- Liping Chi
- Chunbin Yang
bibliography:
- 'bibliography.bib'
date: 'August 8, 2015'
title: The scaling properties of dynamical fluctuations in temporal networks
---
Introduction
============
Interactions in complex systems are not static but change over time, which can be modelled in terms of temporal networks [@Holme2012]. Temporal network consists of a set of contacts $(n_i, n_j, t)$, emphasizing on the time [*when*]{} node $i$ and $j$ have a connection. The addition of time dimension provides a new sight into the framework of complex network theory. In temporal networks, both structural properties and spreading dynamics crucially depend on the time-ordering of links.
The research of temporal networks has attracted great attention and it mainly focuses on two major aspects from the point view of time dimension. One is corresponding to the strategy of time aggregation especially when the topological characteristics are more relevant than the temporal properties. The topological structure of temporal network is achieved through aggregating contacts over a certain time interval and the temporal network is then represented as a series of snapshots of static graphs. Consequently, many existing concepts and tools of static graphs can be adopted to analyze temporal networks, since it is usually easier to analyze static networks. For example, the degree of a node $k_i(t)$ is described as the number of links that it has to other nodes within the time window $[t, t+\Delta t]$. The error and attack strategies in static networks have been applied to evaluate the temporal vulnerability [@Trajanovski], and so on. In order to understand the structure of temporal networks, it plays a crucial role to choose an optimal time interval $\Delta t$ to construct static graphs from temporal networks. Krings [*et al.*]{} studied the influences of time intervals when aggregating the mobile phone network over time [@Krings]. Holme analyzed three ways of constructing static snapshots from temporal networks [@Holme2013], but no candidate weighing out as a best choice. It is now still an open question on how to choose the time interval to represent temporal networks.
The other aspect is related to using dynamical processes to probe into the influence of time series on temporal network. We should take into account the time-ordering of each contact and the inter-event time between two consecutive contacts. The inter-event time distribution in temporal network follows a power-law, which is also called burstiness [@Barabasi2005]. Although it is recognized that time-ordering and bursty characters have strong influences on the dynamical processes of temporal networks, numerous studies have appeared to arrive at contradictory results. Lambiotte [*et al.*]{} had stressed that time-ordering and burstiness of contacts were critical in spreading process, which leaded to slow down spreading [@Lambiotte]. In the work of Rocha [*et al*]{}, they concluded that temporal correlations accelerated outbreaks [@Rocha2011] in SI and SIR model. Miritello [*et al.*]{} demonstrated that bursts hindered propagation at large scales, but group conversations favored local rapid cascades [@Miritello].
Despite the promoting results in temporal networks, this field is still in its early stages about how temporal effect and topological structure interplay and hence affect the dynamical process. In this paper, based on empirical data sets, we will investigate the scaling properties of the dynamical fluctuations of contacts and nodes in temporal networks by using the factorial moments. We are aiming at extracting the fundamental properties from the large amount of data and revealing the influences of time effects on temporal networks from a new perspective.
The rest of the paper is organized as follows. Section II briefly introduces the method of factorial moments. In Section III we give a brief description of the data sets and present the corresponding results, especially the scaling properties of fluctuations for contacts and nodes in the empirical data sets. Conclusions are offered in the final section.
Method of factorial moments
===========================
Temporal network consists of a sequence of contacts $(n_i,n_j,t)$, representing that node $i$ and node $j$ has a contact at time $t$. The number of contacts characterizes the frequency that individuals are connected with each other and the number of nodes describe the activeness that individuals are involved. In this paper, factorial moments will be used to study the dynamical fluctuations of contacts and nodes in temporal networks and the scaling properties of those fluctuations in the system.
Factorial moments are originally introduced in nuclear physics to study the multiplicity fluctuation of hadrons produced during the high energy collisions[@Bialas1986]. The fluctuations and correlations in multiplicity distributions provide a general and sensitive method to characterize the dynamical interactions. Here we will focus on the multiplicity of contacts and nodes in temporal networks. Consider the time series of contacts (or nodes) $y(t)$, where $t$ is the time that contacts happen and $t$ ranges from 0 to $T$. We divide the whole time range $T$ into $M$ equal bins (the remainders are discarded). So the time interval in each bin is $\Delta t = T/M$. Within each bin window $m$ ($m=1,2,...,M$), denote the number of contacts (or nodes) as $n_m$. Of course, $n_m$ fluctuates for different bin windows. To measure the fluctuations and correlations, the $q-th$ order factorial moment is introduced as, $$\begin{aligned}
f_q & = \frac{1}{M}\sum\limits_{m=1}^{M}n_m(n_m-1)...(n_m-q+1) \notag \\
& = \langle n_m(n_m-1)...(n_m-q+1) \rangle.
\label{factmoment}\end{aligned}$$
In factorial moments, $f_1=\langle n \rangle$ is the mean number of contacts (or nodes) under a certain bin size, averaged over all the bins $m$. Note that $n_m$ must be greater than $q$ $(n_m > q)$ in order to contribute to $f_q$ , and $q$ is usually an integer. As $M$ increases, $\Delta t$ is decreased and the average multiplicity $\langle n \rangle$ in a bin decreases. This may lead to $ n_m < q$ which is not allowed. Thus high $q$ corresponds to higher $n_m$ in the bin under consideration, i.e., large fluctuations from $\langle n \rangle$ [@Hwa1998].
Normalized factorial moments are more generally used, $$F_q=\frac{f_q}{f_1^q}.$$
It can be proved that $F_q$ can filter out the statistical fluctuations. The method of factorial moments has been applied to analyze different complex systems, such as multiplicity of produced hadrons [@Chunbin1998], human electroencephalogram and gait series in biology [@Hwa2002; @Yang2002], financial price series [@Schoeffel], critical fluctuations in Bak-Sneppen model [@Xiao], spectra analysis of complex networks [@Yang2005], to name a few. Specially it indicates that the fluctuations in the system have self-similarity when $F_q$ has a power-law dependence on the bin size $M$. $$F_q \propto M^{\alpha_q}, \qquad \alpha_q>0.$$
This phenomenon is referred to as the intermittency. Intermittency basically means random deviations from smooth or regular behavior. Intermittent behavior is expected in a variety of statistical systems at the phase transition point of the second-order type. Hence the existence of intermittency suggests that the fluctuations are not purely Poisson distribution, but the indication of dynamical processes in the fluctuations.
Results and discussions
=======================
In this paper the factorial moments analyses are performed to uncover the scaling properties of the fluctuations in temporal networks based on the following two empirical data sets.
- [***Prostitution:***]{} The data set consists of sexual contacts between sex buyers and sellers from a Brazilian web forum [@Rocha2010]. The time resolution is 1 day and the whole time range is $T = 2232 $ days.
- [***Conference:***]{} The data set was collected at a 3-day conference from face-to-face interactions between conference participants. A contact is recorded every 20-second intervals if two individuals are within range of 1.5m [@Isella]. The whole time range is $T= 212 340$ seconds.
We now divide the whole time range $T$ into $M$ bins and count the number of contacts and nodes in each bin window. Calculate $f_q$ and $F_q$ according to Eq. (1) and (2), respectively. It is noticed that $f_q$ is averaged over all bins (known as the horizontal average).
Figure 1 presents the log-log plot of $F_q$ as a function of $M$ for contacts (open circles) and nodes (filled circles) in [***Prostitution***]{} data. With $M$ ranging from about 3 to 60 bins, it means that the time interval $\Delta t$ extends approximately from 30 to 750 days. We find that $\ln F_q$ increases linearly with $\ln M$ for both contacts and nodes by $q$ varying from 2 to 6. The slopes of nodes are a little larger than that of contacts. The same phenomena have also been observed in [*Conference*]{} data in Fig. 2.
The increase of bin size $M$ means that the fluctuations of arbitrary sizes can appear in the system, and consequently leading to the growth of factorial moment $F_q$ with $M$. The scaling relationship between $F_q$ and $M$, $F_q \sim M^{\alpha_q}$, indicates the existence of intermittency. As stated in Ref. [@Schoeffel], for uncorrelated Poissonian or Gaussian distributions, $F_q=1$ for all orders $q$; whereas for correlated contacts or nodes distributions, $F_q$ should increase with the growth of bin size $M$. Hence the intermittent behavior implies that the fluctuations of contacts and nodes in both [*Prostitution*]{} and [*Conference*]{} data have self-similar structures and the fluctuations are not random Poisson distribution but have dynamical and correlated behaviors inside.
![Log-log plot of factorial moments $F_q$ as a function of bin size $M$ for [***Prostitution***]{} data with $q$ varying from 2 to 6. (a) The fluctuations of contacts (open circles). (b) The fluctuations of nodes (filled circles). The symbols are the same below.](fig1_prost.pdf){width="45.00000%"}
![Log-log plot of $F_q$ as a function of $M$ for [***Conference***]{} data with the range of $q$ from 2 to 6. (a) for contacts; (b) for nodes.](fig1_confer.pdf){width="45.00000%"}
Further investigations have been performed on $F_q$ and $F_2$. The scaling between $F_q$ and $F_2$ is more general than intermittency, which could be true even under the condition that intermittency does not exist.
We plot $F_q$ as a function of $F_2$ on the log-log scale for [*Prostitution*]{} and [*Conference*]{} data sets in Fig. 3 and 4, respectively. The scaling relationship between $F_q$ and $F_2$ can be clearly observed in both figures. $$F_q \propto F_2^{\beta_q},$$
where $\beta_q = \alpha_q/\alpha_2$ for the case of intermittency. We are interested more in the dependence of $\beta_q$ on $q$. The plot of $\beta_q$ as a function of $(q-1)$ is presented on a log-log scale in Fig. 5 for [*Prostitution*]{} data and in Fig. 6 for [*Conference*]{} data. There is a remarkably linear relationship between $\beta_q$ and $(q-1)$ for all $q$. Now one has $$\beta_q \propto (q-1)^{\gamma}.$$
The linear fits are also plotted in the figures. In [*Prostitution*]{} data, $\gamma$ are 1.341 for contacts and 1.104 for nodes. In [*Conference*]{} data $\gamma$ are 0.992 and 1.345 for contacts and nodes, respectively. It should be recognized that the power-law relationship in Eq. (5) implies that the exponents $\beta_q$ are independent of bin size $M$. It suggests a common feature of scaling invariance in temporal networks.
![Log-log plot of $F_q$ as a function of $F_2$ for [***Prostitution***]{} data. (a) for contacts; and (b) for nodes.](fig2_prost.pdf){width="45.00000%"}
![Log-log plot of $F_q$ as a function of $F_2$ for [*Conference*]{} data. (a) for contacts; and (b) for nodes.](fig2_confer.pdf){width="45.00000%"}
It is known that $\gamma$ is approximately 1.3 for 2D Ising model undergoing a second-order phase transition [@HwaPRL]. The exponents $\gamma$ of contacts in [*Prostitution*]{} data $(\gamma = 1.341)$ and of nodes in [*Conference*]{} data $(\gamma = 1.345)$ are very close to this value.
![Scaling properties between $\beta_q$ and $(q-1)$ for [*Prostitution*]{} data. The scaling exponents are about: (a) 1.341 for contacts; and (b) 1.104 for nodes.](fig3_prost.pdf){width="45.00000%"}
![Scaling properties between $\beta_q$ and $(q-1)$ for [*Conference*]{} data. The scaling exponents are about: (a) 0.992 for contacts; and (b) 1.345 for nodes.](fig3_confer.pdf){width="45.00000%"}
Conclusions
===========
The factorial moments analyses are performed to study the scaling properties for fluctuations of contacts and nodes in temporal networks based on empirical data sets. The phenomena of intermittency $F_q \sim M^{\alpha_q}$ have been observed for all orders $q$ in the fluctuations of contacts and nodes for both [*Prostitution*]{} and [*Conference*]{} data sets. The result indicates that the system has self-similar structure and the fluctuations are not purely random, but have dynamical and correlated behaviors embedded in the system. A more general scaling relationship between $F_q$ and $F_2$ has also been presented, $F_q \sim F_2^{\beta_q}$. We further find that $\beta_q$ scales with $q$ as $\beta_q \sim (q-1)^{\gamma}$. The exponents $\gamma$ for nodes in the [*Prostitution*]{} data and for contacts in the [*Conference*]{} data are very close to that for 2D Ising model. The other exponents $\gamma$ are not.
Still, there are some issues to be addressed. First, what is the driving mechanism(s) behind these scaling properties of fluctuations in temporal networks? Second, why are some scaling exponents close to that of Ginzburg-Landau second-order phase transition? Are they belong to the same universal class? All these topics cannot be covered in this paper and will be discussed later.
The scaling invariances of fluctuations shed light on the temporal correlations of contact series and provide a new sight into understanding the influence of time dimension in temporal networks.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Professor Petter Holme for providing the data. This work was supported in part by the programme of Introducing Talents of Discipline to Universities under No. B08033, and by the self-determined research funds of CCNU from the colleges’ basic research and operation of MOE under No. CCNU2015A05046.
|
---
abstract: 'A pulsed atom laser derived from a Bose-Einstein condensate is used to probe a second target condensate. The target condensate scatters the incident atom laser pulse. From the spatial distribution of scattered atoms, one can infer important properties of the target condensate and its interaction with the probe pulse. As an example, we measure the $s$-wave scattering length that, in low energy collisions, describes the interaction between the $\left|F=1,m_F=-1\right\rangle$ and $\left|F=2,m_F=0\right\rangle$ hyperfine ground states in $^{87}\mbox{Rb}$.'
address: |
Australian Research Council Centre of Excellence for Quantum-Atom Optics,\
Physics Department, The Australian National University,\
Canberra, 0200, Australia
author:
- 'D. Döring, N. P. Robins, C. Figl, and J. D. Close'
title: 'Probing a Bose-Einstein condensate with an atom laser'
---
[99]{}
M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, “Output Coupler for Bose-Einstein Condensed Atoms,” , 582-585 (1997).
E. W. Hagley, L. Deng, M. Kozuma, J. Wen, K. Helmerson, S. L. Rolston, W. D. Phillips, “A Well-Collimated Quasi-Continuous Atom Laser,” Science [**283**]{}, 1706-1709 (1999).
Immanuel Bloch, Theodor W. Hänsch, and Tilman Esslinger, “Atom Laser with a cw Output Coupler,” , 3008-3011 (1999).
Y. Le Coq, J. H. Thywissen, S. A. Rangwala, F. Gerbier, S. Richard, G. Delannoy, P. Bouyer, and A. Aspect, “Atom Laser Divergence,” , 170403 (2001).
N. P. Robins, C. Figl, S. A. Haine, A. K. Morrison, M. Jeppesen, J. J. Hope, and J. D. Close, “Achieving Peak Brightness in an Atom Laser,” , 140403 (2006).
J.-F. Riou, W. Guerin, Y. Le Coq, M. Fauquembergue, V. Josse, P. Bouyer, and A. Aspect, “Beam Quality of a Nonideal Atom Laser,” , 070404 (2006).
Anton Öttl, Stephan Ritter, Michael Köhl, and Tilman Esslinger, “Correlations and Counting Statistics of an Atom Laser,” , 090404 (2005).
, G. Scoles, ed., (Oxford University Press, New York, 1992), Vol. 2.
G. Brusdeylins, R. Bruce Doak, and J. Peter Toennies, “Measurement of the Dispersion Relation for Rayleigh Surface Phonons of LiF(001) by Inelastic Scattering of He Atoms,” , 437-439 (1981).
, E. Hulpke, ed., Springer Series in Surface Sciences (Springer-Verlag, Heidelberg, 1992), Vol. 27.
D. Farias and K. H. Rieder, “Atomic beam diffraction from solid surfaces,” Rep. Prog. Phys. [**61**]{}, 1575-1664 (1998).
L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, “ Four-wave mixing with matter waves,” , 218-220 (1999).
A. Perrin, H. Chang, V. Krachmalnicoff, M. Schellekens, D. Boiron, A. Aspect, and C. I. Westbrook, “Observation of Atom Pairs in Spontaneous Four-Wave Mixing of Two Colliding Bose-Einstein Condensates,” , 150405 (2007).
J. M. Vogels, J. K. Chin, and W. Ketterle, “Coherent Collisions between Bose-Einstein Condensates,” , 030403 (2003).
Olaf Mandel, Markus Greiner, Artur Widera, Tim Rom, Theodor W. Hänsch, and Immanuel Bloch, “Controlled collisions for multiparticle entanglement of optically trapped atoms,” , 937-940 (2003).
C. Samuelis, E. Tiesinga, T. Laue, M. Elbs, H. Knöckel, and E. Tiemann, “Cold atomic collisions studied by molecular spectroscopy,” , 012710 (2000).
John Weiner, Vanderlei S. Bagnato, Sergio Zilio, and Paul S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. [**71**]{}, 1-85 (1999).
E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, and B. J. Verhaar, “Interisotope Determination of Ultracold Rubidium Interactions from Three High-Precision Experiments,” , 093201 (2002).
Ch. Buggle, J. Léonard, W. von Klitzing, and J. T. M. Walraven, “Interferometric Determination of the $s$ and $d$-Wave Scattering Amplitudes in $^{87}\mbox{Rb}$,” , 173202 (2004).
Nicholas R. Thomas, Niels Kjærgaard, Paul S. Julienne, and Andrew C. Wilson, “Imaging of $s$ and $d$ Partial-Wave Interference in Quantum Scattering of Identical Bosonic Atoms,” , 173201 (2004).
Angela S. Mellish, Niels Kjærgaard, Paul S. Julienne, and Andrew C. Wilson, “Quantum scattering of distinguishable bosons using an ultracold-atom collider,” , 020701(R) (2007).
Due to the rf ouput-coupling process we cannot avoid a weak density cloud of atoms in the $\left|2,1\right\rangle$ state. In the numerical analysis, the position of these atoms during the scattering process is approximated to be identical to the position of the source condensate.
Th. Busch, M. Köhl, T. Esslinger, and K. Mølmer, “Transverse mode of an atom laser,” , 043615 (2002).
M. J. Bijlsma and H. T. C. Stoof, “Coherently scattering atoms from an excited Bose-Einstein condensate,” , 013605 (2000).
Uffe V. Poulsen and Klaus Mølmer, “Scattering of atoms on a Bose-Einstein condensate,” , 013610 (2003).
Introduction
============
Since the realization of Bose-Einstein condensation in ultracold atomic gases, the atom laser, a highly coherent, freely propagating beam of low energy atoms has been developed by several groups [@M.1997; @E.1999; @Immanuel1999; @Y.2001]. To form an atom laser, a beam of atoms is coherently output-coupled from a trapped Bose-Einstein condensate to a state that does not interact with the trapping potential. The atoms fall away from the trap producing a coherent de Broglie matter wave that is the atom laser beam. Atom lasers are the direct analogue of optical lasers. Both devices rely on Bose-enhanced scattering for their operation, and both produce coherent beams derived from a macroscopically populated trapped state. The flux, the spatial mode, the coherence properties and the quantum statistics have all been studied both experimentally and theoretically [@N.2006; @J.-F.2006; @Anton2005].
Despite the promise shown by the atom laser as a bright source of coherent atoms, an atom laser has not yet been used as a measurement device. In contrast, fast atomic and molecular beams derived from supersonic nozzle expansions have found widespread use in physics and physical chemistry to determine important properties such as molecular potential surfaces. In such experiments, two beams collide and scatter inside a vacuum chamber. Analyses of the angular distribution of scattering events provides information on the potentials describing the interaction between the collision partners [@G.1992]. This information is a basic input into many calculations in quantum chemistry. Similarly, fast atomic beams are widely used in surface science to measure properties such as the surface geometry of adsorbates, underlying surface crystal structures and the density of states of surface phonons. In these experiments, a fast beam strikes the surface and is scattered. The angular distribution and energy of the scattered atoms is analysed and the desired surface property is extracted [@G.1981; @E.1992; @D.1998].
It is an intriguing idea to explore the atom laser for analogous applications at very low collision energies. In this Letter, we present results from such an experiment. We measure the $s$-wave scattering length describing the interaction between the $\left|F=2, m_F=0\right\rangle$ and $\left|F=1, m_F=-1\right\rangle$ hyperfine ground states of $^{87}\mbox{Rb}$ by scattering an atom laser beam off a target Bose-Einstein condensate.
The $s$-wave scattering length is an important quantity that describes atomic interactions in low energy collisions in ultracold bosonic systems. Interactions in these systems have been used to study four-wave mixing of matter waves [@L.1999; @A.2007] and to demonstrate matter wave amplification [@J.M.2003a]. Low energy collisions can be exploited to produce multi-particle entanglement [@Olaf2003] and have potential applications in quantum computation and in producing squeezed states for precision measurements. Precise measurements of $s$-wave scattering lengths of alkali atoms have been conducted using, e.g., Raman [@C.2000] and photoassociative spectroscopy [@J.1999]. For $^{87}\mbox{Rb}$, such measurements have been accomplished in a highly accurate way. The present uncertainty in the $s$-wave scattering lengths is of the order of $0.1\,\%$ [@E.2002]. As rightly pointed out by Buggle *et al.* [@Ch.2004], these methods are based on refined knowledge of the molecular potentials. The experiment we present here does not rely on this information and offers a more direct and less complex way of measuring the interactions between two different hyperfine states in $^{87}\mbox{Rb}$. The precision of our measurement is $3\,\%$ and can most likely be improved by a more detailed theoretical analysis. In a conceptually identical setup, the technique could be adapted to study inter-species scattering and measure currently unknown interaction parameters.
Unlike recent experiments studying the scattering properties of two colliding condensates in the same [@Ch.2004; @Nicholas2004] or in different [@Angela2007] internal states, we investigate scattering in an energy regime where it is only necessary to consider $s$-wave collisions. The probing atom laser is in the $\left|F=2, m_F=0\right\rangle$ state, which is to first order magnetically insensitive and therefore reduces the impact of unstable magnetic fields on the measurement.
Experimental methods
====================
To study the low energy atomic interactions in $^{87}\mbox{Rb}$, we derive an atom laser pulse in the $\left|F=2, m_F=0\right\rangle$ hyperfine ground state from a lasing or source condensate and collide it with a second target condensate in the $\left|F=1, m_F=-1\right\rangle$ state. A schematic diagram of the experiment is shown in Fig. \[fig:scheme\]. The center-of-mass energy of the colliding atoms lies below $1\,\mu\mbox{K}$. Analyses of the distribution of scattered atoms allow us to determine the $s$-wave scattering length describing the interaction between atoms in these states in the low energy regime.
![\[fig:scheme\] (Color online) Schematic setup of the experiment. The two condensates are situated at different vertical positions in the same magnetic trap. (a) The atom laser is output-coupled from the source condensate and (b) scattered off the target condensate. (c) The scattered probe pulse falls under gravity and (d) is imaged by near-resonant light incident along the long axis of the condensates. The confining magnetic fields are switched off before the image is taken, letting the condensates expand for $5\,\mbox{ms}$.](figu1_scheme.eps)
Producing two separated Bose-Einstein condensates
-------------------------------------------------
Our experimental apparatus for producing two spatially separated atom clouds is based around an ultrahigh vacuum chamber operating at a pressure of $10^{-11}\,\mbox{Torr}$. In this chamber, we operate a three-dimensional magneto-optical trap (MOT) which is loaded with $10^{10}$ $^{87}\mbox{Rb}$ atoms. The loading process occurs via a cold atomic beam derived from a two-dimensional MOT. After loading, the MOT is compressed, and the confining magnetic fields are switched off. The atoms are polarization-gradient cooled in the remaining optical molasses to a temperature of $40\,\mu\mbox{K}$. We then apply a short intense laser pulse of circular polarization and resonant with the $^{87}\mbox{Rb}$ $\left|F=2\right\rangle \rightarrow \left|F'=2\right\rangle$ $\mbox{D}_2$-transition to optically pump the atoms into the $\left|F=1,m_F=-1\right\rangle$ state. By precise control of the length of this pulse (with an accuracy of $0.5\,\mu\mbox{s}$), we retain a fraction of the atoms (up to $5\times 10^9$) in the $\left|F=2,m_F=2\right\rangle$ state. Typical lengths of the optical pumping pulse are between $20\,\mu\mbox{s}$ and $40\,\mu\mbox{s}$. After preparing the internal state, a magnetic quadrupole field with a gradient of $200\,\mbox{G}/\mbox{cm}$ is switched on, confining both atoms in the $\left|1,-1\right\rangle$ and in the $\left|2,2\right\rangle$ state. By means of a mechanical translation stage, the magnetically trapped atoms are transported over a distance of $20\,\mbox{cm}$ and transferred into a harmonic magnetic trap. The temperature is further reduced by radio-frequency (rf) induced evaporative cooling. Because of the different magnetic moments of the trapped states, the cooling only works efficiently for atoms in the $\left|1,-1\right\rangle$ state. However, atoms in the more tightly confined $\left|2,2\right\rangle$ state are sympathetically cooled by elastic collisions with the $\left|1,-1\right\rangle$ atoms. Gravity shifts the potential minimum to a position vertically below the magnetic field minimum. The more tightly confined atoms in the $\left|2,2\right\rangle$ source condensate are situated above the $\left|1,-1\right\rangle$ target cloud (see Fig. \[fig:scheme\]). The separation between the centers of the two condensates is $7.3\,\mu\mbox{m}$. The Thomas-Fermi radius of each of the clouds depends only weakly on the atom number and lies between $4\,\mu\mbox{m}$ and $6\,\mu\mbox{m}$ (in the plane perpendicular to the long condensate axis). By turning off the repumping light during the imaging process, we verify that, before the output-coupling process, there is no cloud of atoms in the $\left|2,1\right\rangle$ state which would spatially overlap with the $\left|1,-1\right\rangle$ condensate.
Output-coupling, scattering and imaging the probe pulse
-------------------------------------------------------
Two magnetic coils are placed near the vacuum chamber, allowing us to output-couple atoms to the untrapped ($m_F=0$)-state from either condensate via rf transitions between Zeeman levels. One of the coils is used to deplete the target ($\left|1,-1\right\rangle$) condensate before the scattering experiment is carried out. We can precisely control the size of this condensate by adjusting the power of the output-coupling pulse. Due to the finite frequency width of the pulse, we cannot avoid the side effect of a partial depletion of the source ($\left|2,2\right\rangle$) condensate. We compensate for this by simultaneously adjusting the optical pumping time, so that the number of atoms in the source condensate is the same for each data point. After adjusting the number of atoms in the target condensate, the desired atom number is output-coupled from the source cloud for $80\,\mu\mbox{s}$ using the second magnetic coil [@comm]. The short output-coupling process allows for a high signal-to-noise ratio in the image of the probe pulse. The atoms constituting the pulse are in the $\left|2,0\right\rangle$ state. The atom laser pulse scatters as it propagates through the target condensate and is detected via absorption imaging. The circularly polarized imaging laser is resonant with the $\left|F=2\right\rangle \rightarrow \left|F'=3\right\rangle$ transition. In order to image the atoms in the $\left|F=1\right\rangle$ state, we apply a short ($1\,\mbox{ms}$) repumping pulse immediately before the image is taken. To observe an effect of the lower condensate on the propagating atom laser, it is crucial to choose the right axis for the imaging beam. The geometric factor determining the influence of the target condensate on the atom laser pulse is the gradient of the atomic density in the condensate. The highest gradient and the strongest scattering is to be expected in the plane perpendicular to the long axis of the condensate, which is therefore used as the imaging direction (see Fig. \[fig:scheme\](d)).
Results and discussion
======================
The interactions between the target condensate and the probe pulse are analyzed using the spatial density distribution in the pulse for different sizes of the target condensate. We present experimental results and make a comparison to numerical simulations to get a measurement of the $s$-wave scattering length between the two states involved in the scattering process.
![\[fig:results\] (Color online) Non-averaged absorption pictures of the scattered probe pulse and the two condensates for atom numbers of (a) $0.05\times 10^6$, (b) $0.16\times 10^6$, (c) $0.45\times 10^6$, (d) $0.87\times 10^6$, (e) $1.25\times 10^6$, and (f) $1.38\times 10^6$ atoms in the target condensate. The three images in each section are taken $0\,\mbox{ms}$, $9\,\mbox{ms}$ and $22\,\mbox{ms}$ after output-coupling the atom laser pulse. The ballistic expansion time for the two condensates is $5\,\mbox{ms}$. The graphs below each section show the density profile of the probe pulse, integrated over the width of the pulse along the propagation ($y$-) direction.](figu2_results.eps){width="8.35cm"}
Experimental data
-----------------
Increasing the size of the target condensate leads to a clear increase in the width and a change in the form of the atom laser pulse (see Fig. \[fig:results\]). Whereas we observe the characteristic ‘horseshoe’-shaped pattern for the case with almost no target condensate present, this pattern changes towards a flattened profile with peaks in the atomic density on the sides of the pulse when the size of the target condensate is increased. The effect can be described by the altered momentum distribution of the probe atom laser pulse. We image the probe pulse in what can be seen as the analogue of the far field limit in classical optics. The density distribution is dominated by the momentum distribution as opposed to the change in position (which is of the order of a few $\mu\mbox{m}$) when the atom laser pulse traverses the target condensate. The effect of the altered momentum distribution on the width of the probe pulse depends proportionally on its fall time. It is crucial to optimize the fall time (to give the required spatial resolution in the image) while still maintaining a reasonable signal-to-noise ratio in the absorption image. We choose expansion times up to $22\,\mbox{ms}$. The measure used for the quantitative analysis of the scattering process is the width (FWHM) in $x$-direction of the density of the atom laser pulse, integrated along the $y$-axis (see Fig. \[fig:scheme\]). For each set of experimental parameters, the pulse width and the atom number is averaged over five images. The measured pulse widths are shown in Fig. \[fig:graph\].
The collision energy of the scattering atoms is determined by the velocity that an atom has gained before it reaches the target condensate. For the fall distances considered here (the laser pulse falls less than $10\,\mu\mbox{m}$ before it reaches the target condensate), the collision energy lies below $1\,\mu\mbox{K}$. This energy is low enough to assume pure $s$-wave scattering and neglect contributions from higher order partial waves (see , e.g., [@Ch.2004]).
![\[fig:graph\] (Color online) Width of the scattered atom laser pulse (FWHM) as a function of the atom number in the target condensate. The red dots indicate the results of the measurements, the blue circles show the best fit of the numerical simulations to the experimental data. The fall time of the atom pulse before measuring the pulse width is $22\,\mbox{ms}$.](fig3_graph.eps)
Numerical simulations
---------------------
The experiment allows us to extract the $s$-wave scattering length between the two states involved in the scattering process, $\left|F=1,m_F=-1\right\rangle$ and $\left|F=2,m_F=0\right\rangle$. We use a classical two-dimensional model and numerically simulate the scattering process. Quantum mechanical path interferences are neglected and can be assumed to add a fringe pattern to the classical pulse shape without significantly affecting the width of the pulse [@Busch2002]. The fact that we do not observe this fringe pattern in the experiment is due to the integration effect when imaging the atom laser pulse.
In our numerical simulations, we assume the initial density profile of the atom laser pulse to be a two-dimensional Thomas-Fermi distribution, modified by a Gaussian output-coupling efficiency profile. The potential experienced by the pulse is given by Eq. (1): $$U(x,y)=U_t(x,y)+U_s(x,y)+mgy \, ,$$ where $U_t$ and $U_s$ are the potentials generated by the target and the source condensate. We obtain $$U_t(x,y)= g_{12}\frac{\mu_t}{g_{11}}\left(1-\frac{(y-y_t)^2+x^2}{r_t^2}\right)\theta(r_t^2-((y-y_t)^2+x^2))\,,$$ and an expression equivalent to Eq. (2) for the source condensate. Here, $\mu_t$ is the chemical potential, $r_t$ the Thomas-Fermi radius and $y_t$ the $y$-coordinate of the center of the target condensate. $g_{11}$ describes the coupling constant for interactions between identical atoms in the $\left|1,-1\right\rangle$ state. The inter-state coupling constant $g_{12}$ refers to scattering between atoms in the $\left|1,-1\right\rangle$ and $\left|2,0\right\rangle$ states and is related to the scattering length $a_{12}$ via $g_{12}=4\pi\hbar^2a_{12}/m$. Due to the step-function $\theta(x)$ the potential is zero outside the Thomas-Fermi radius of each condensate.
The atom numbers of the source and the target condensate are determined from long expansion time images of the two atom clouds. The $80\,\mu\mbox{s}$ output-coupling pulse has a Fourier-limited frequency width of $16\,\mbox{kHz}$ (FWHM), and the atom laser pulse is output-coupled from a large fraction of the source condensate. There is no significant influence of the curvature of the output-coupling region, and we neglect this effect in the analysis. The number of atoms in the probe pulse is small compared to the total number of atoms in the system (the ratio is $\sim 0.05$), and we approximate the mean-field potential of both the source and the target cloud to be constant during the scattering process. The scattering length $a_{12}$ is varied in $15$ steps between $90a_0$ and $105a_0$, where $a_0$ is the Bohr radius. As the vertical position of the output-coupling region is difficult to determine accurately from the experiment, it enters the simulations as a free parameter. The best fit to the experimental data (see Fig. \[fig:graph\]) is obtained for the center of the output-coupling region situated $2\,\mu\mbox{m}$ above the condensate center, confirming our expectation of output-coupling mainly from the central condensate region and giving a scattering length of $a_{12}=94(3)a_0$. In comparison, the values for the singlet and triplet scattering lengths in $^{87}\mbox{Rb}$ are $a_S=90.4a_0$ and $a_T=98.98a_0$ [@E.2002]. The stated uncertainty of $3a_0$ includes the statistical confidence region of the fit as well as the estimated systematic uncertainty due to neglecting the third dimension (along the long condensate axis) in the simulation. However, as the gradient in the atomic density along this axis is two orders of magnitude smaller than in the radial direction, the scattering in this dimension is far less distinct.
Conclusion
==========
The work presented in this paper is the first experiment using an atom laser to probe the properties of a second independent Bose-Einstein condensate. Such a technique can take full advantage of the coherent nature of ultracold atomic samples. It is an intriguing challenge to use the coherence of a wavelength tunable (slow to fast) atom laser for the investigation of molecular potentials or for applications in surface science.
The analysis gives a measurement with an uncertainty of $3\,\%$ of the scattering length $a_{12}$ between $^{87}\mbox{Rb}$ atoms in the $\left|1,-1\right\rangle$ and $\left|2,0\right\rangle$ ground states. Theoretical analysis of the scattering of atoms on Bose-Einstein condensates has been conducted by different groups (see , e.g., [@Bijlsma2000a; @Poulsen2003]), and we believe that a more detailed theoretical analysis of measurements like the one presented could lead to a more precise value of the scattering length $a_{12}$. The method could be extended to using a longer probe pulse or even a quasi-continuous atom laser beam. This would increase the spatial selectivity and potentially widen the applications of the conducted experiment. The use of short pulses in the work described here offers the advantage of a high signal-to-noise ratio as compared to a quasi-continuous atom laser.
Using the state-of-the-art knowledge of the singlet and triplet scattering lengths and molecular potentials, one could accurately calculate $a_{12}$. However, the method presented above does not rely on this knowledge and offers a more direct way of measuring the scattering length. In an experimental setup designed for multi-species trapping, it could be used for the study of inter-species interactions and for the measurement of currently unknown interaction parameters.
|
---
abstract: |
We study the extremal solution for the problem $(-\Delta)^s u=\lambda f(u)$ in $\Omega$, $u\equiv0$ in $\R^n\setminus\Omega$, where $\lambda>0$ is a parameter and $s\in(0,1)$. We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions $n<4s$. We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever $n<10s$. In the limit $s\uparrow1$, $n<10$ is optimal. In addition, we show that the extremal solution is $H^s(\R^n)$ in any dimension whenever the domain is convex.
To obtain some of these results we need $L^q$ estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with $L^p$ data. We prove optimal $L^q$ and $C^\beta$ estimates, depending on the value of $p$. These estimates follow from classical embedding results for the Riesz potential in $\R^n$.
Finally, to prove the $H^s$ regularity of the extremal solution we need an $L^\infty$ estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.
address:
- 'Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona, Spain'
- 'Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Diagonal 647, 08028 Barcelona, Spain'
author:
- 'Xavier Ros-Oton'
- Joaquim Serra
title: The extremal solution for the fractional Laplacian
---
[^1]
Introduction and results
========================
Let $\Omega \subset\mathbb R^n$ be a bounded smooth domain and $s\in(0,1)$, and consider the problem $$\label{pb}
\left\{ \begin{array}{rcll} (-\Delta)^s u &=&\lambda f(u)&\textrm{in }\Omega \\
u&=&0&\textrm{in }\mathbb R^n\backslash\Omega,\end{array}\right.$$ where $\lambda$ is a positive parameter and $f:[0,\infty)\longrightarrow\mathbb R$ satisfies $$\label{condicions}f\textrm{ is } C^1\ \textrm{and nondecreasing},\ f(0)>0,\ \textrm{and}\ \lim_{t\rightarrow+\infty}\frac{f(t)}{t}=+\infty.$$ Here, $(-\Delta)^s$ is the fractional Laplacian, defined for $s\in(0,1)$ by $$\label{laps}(-\Delta)^s u (x)= c_{n,s}{\rm PV}\int_{\R^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,$$ where $c_{n,s}$ is a constant.
It is well known —see [@BV] or the excellent monograph [@D] and references therein— that in the classical case $s=1$ there exists a finite extremal parameter $\lambda^*$ such that if $0<\lambda<\lambda^*$ then problem (\[pb\]) admits a minimal classical solution $u_\lambda$, while for $\lambda>\lambda^*$ it has no solution, even in the weak sense. Moreover, the family of functions $\{u_\lambda:0<\lambda<\lambda^*\}$ is increasing in $\lambda$, and its pointwise limit $u^*=\lim_{\lambda\uparrow \lambda^*}u_\lambda$ is a weak solution of problem (\[pb\]) with $\lambda=\lambda^*$. It is called the extremal solution of (\[pb\]).
When $f(u)=e^u$, we have that $u^*\in L^\infty(\Omega)$ if $n\leq9$ [@CR], while $u^*(x)=\log\frac{1}{|x|^2}$ if $n\geq10$ and $\Omega=B_1$ [@JL]. An analogous result holds for other nonlinearities such as powers $f(u)=(1+u)^p$ and also for functions $f$ satisfying a limit condition at infinity; see [@S]. In the nineties H. Brezis and J.L. Vázquez [@BV] raised the question of determining the regularity of $u^*$, depending on the dimension $n$, for general nonlinearities $f$ satisfying (\[condicions\]). The first result in this direction was proved by G. Nedev [@N], who obtained that the extremal solution is bounded in dimensions $n\leq3$ whenever $f$ is convex. Some years later, X. Cabré and A. Capella [@CC] studied the radial case. They showed that when $\Omega=B_1$ the extremal solution is bounded for all nonlinearities $f$ whenever $n\leq9$. For general nonlinearities, the best known result at the moment is due to X. Cabré [@C4], and states that in dimensions $n\leq4$ then the extremal solution is bounded for any convex domain $\Omega$. Recently, S. Villegas [@V] have proved, using the results in [@C4], the boundedness of the extremal solution in dimension $n=4$ for all domains, not necessarily convex. The problem is still open in dimensions $5\leq n\leq9$.
The aim of this paper is to study the extremal solution for the fractional Laplacian, that is, to study problem for $s\in(0,1)$.
The closest result to ours was obtained by Capella-Dávila-Dupaigne-Sire [@CDDS]. They studied the extremal solution in $\Omega=B_1$ for the [spectral]{} fractional Laplacian $A^s$. The operator $A^s$, defined via the Dirichlet eigenvalues of the Laplacian in $\Omega$, is related to (but different from) the fractional Laplacian . We will state their result later on in this introduction.
Let us start defining weak solutions to problem .
\[def\] We say that $u\in L^1(\Omega)$ is a *weak solution* of if $$\label{def1}
f(u)\delta^s\in L^1(\Omega),$$ where $\delta(x)={\rm dist}(x,\partial\Omega)$, and $$\label{def2}
\int_\Omega u(-\Delta)^s\zeta dx=\int_\Omega\lambda f(u)\zeta dx$$ for all $\zeta$ such that $\zeta$ and $(-\Delta)^s\zeta$ are bounded in $\Omega$ and $\zeta\equiv0$ on $\partial\Omega$.
Any bounded weak solution is a *classical solution*, in the sense that it is regular in the interior of $\Omega$, continuous up to the boundary, and holds pointwise; see Remark \[solucions\].
Note that for $s=1$ the above notion of weak solution is exactly the one used in [@BCMR; @BV].
In the classical case (that is, when $s=1$), the analysis of singular extremal solutions involves an intermediate class of solutions, those belonging to $H^1(\Omega)$; see [@BV; @MP]. These solutions are called [@BV] energy solutions. As proved by Nedev [@N2], when the domain $\Omega$ is convex the extremal solution belongs to $H^1(\Omega)$, and hence it is an energy solution; see [@CS] for the statement and proofs of the results in [@N2].
Similarly, here we say that a weak solution $u$ is an *energy solution* of when $u\in H^s(\R^n)$. This is equivalent to saying that $u$ is a critical point of the energy functional $$\label{energy}
\mathcal E(u)=\frac12\|u\|_{\accentset{\circ}{H}^s}^2-\int_\Omega \lambda F(u)dx,\qquad F'=f,$$ where $$\label{Hs-norm}
\|u\|_{\accentset{\circ}{H}^s}^2=\int_{\R^n}\left|(-\Delta)^{s/2} u\right|^2dx=\frac{c_{n,s}}{2}\int_{\R^n}\int_{\R^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}dxdy=(u,u)_{\accentset{\circ}{H}^s}$$ and $$\label{prod}
(u,v)_{\accentset{\circ}{H}^s}=\int_{\R^n}\hspace{-1mm}(-\Delta)^{s/2} u(-\Delta)^{s/2} v\,dx=\frac{c_{n,s}}{2}\hspace{-1mm}\int_{\R^n}\hspace{-1mm}\int_{\R^n} \hspace{-2mm} \frac{\bigl(u(x)-u(y)\bigr)\bigl(v(x)-v(y)\bigr)}{|x-y|^{n+2s}}dxdy.$$
Our first result, stated next, concerns the existence of a minimal branch of solutions, $\{u_\lambda,\ 0<\lambda<\lambda^*\}$, with the same properties as in the case $s=1$. These solutions are proved to be positive, bounded, increasing in $\lambda$, and semistable. Recall that a weak solution $u$ of is said to be *semistable* if $$\label{semistable}
\int_\Omega \lambda f'(u)\eta^2dx\leq \|\eta\|_{\accentset{\circ}{H}^s}^2$$ for all $\eta\in H^s(\R^n)$ with $\eta\equiv0$ in $\R^n\setminus\Omega$. When $u$ is an energy solution this is equivalent to saying that the second variation of energy $\mathcal E$ at $u$ is nonnegative.
\[existence\] Let $\Omega\subset \mathbb R^n$ be a bounded smooth domain, $s\in(0,1)$, and $f$ be a function satisfying . Then, there exists a parameter $\lambda^*\in(0,\infty)$ such that:
- If $0<\lambda<\lambda^*$, problem admits a minimal classical solution $u_\lambda$.
- The family of functions $\{u_\lambda:0<\lambda<\lambda^*\}$ is increasing in $\lambda$, and its pointwise limit $u^*=\lim_{\lambda\uparrow \lambda^*}u_\lambda$ is a weak solution of with $\lambda=\lambda^*$.
- For $\lambda>\lambda^*$, problem admits no classical solution.
- These solutions $u_\lambda$, as well as $u^*$, are semistable.
The weak solution $u^*$ is called the extremal solution of problem .
As explained above, the main question about the extremal solution $u^*$ is to decide whether it is bounded or not. Once the extremal solution is bounded then it is a classical solution, in the sense that it satisfies equation pointwise. For example, if $f\in C^\infty$ then $u^*$ bounded yields $u^*\in C^\infty(\Omega)\cap C^s(\overline\Omega)$.
Our main result, stated next, concerns the regularity of the extremal solution for problem . To our knowledge this is the first result concerning extremal solutions for . In particular, the following are new results even for the unit ball $\Omega=B_1$ and for the exponential nonlinearity $f(u)=e^u$.
\[S\] Let $\Omega$ be a bounded smooth domain in $\mathbb R^n$, $s\in(0,1)$, $f$ be a function satisfying , and $u^*$ be the extremal solution of .
- Assume that $f$ is convex. Then, $u^*$ is bounded whenever $n<4s$.
- Assume that $f$ is $C^2$ and that the following limit exists: $$\label{tau}
\tau:=\lim_{t\rightarrow+\infty}\frac{f(t)f''(t)}{f'(t)^2}.$$ Then, $u^*$ is bounded whenever $n<10s$.
- Assume that $\Omega$ is convex. Then, $u^*$ belongs to $H^s(\R^n)$ for all $n\geq1$ and all $s\in(0,1)$.
Note that the exponential and power nonlinearities $e^u$ and $(1+u)^p$, with $p>1$, satisfy the hypothesis in part (ii) whenever $n<10s$. In the limit $s\uparrow1$, $n<10$ is optimal, since the extremal solution may be singular for $s=1$ and $n=10$ (as explained before in this introduction).
Note that the results in parts (i) and (ii) of Theorem \[S\] do not provide any estimate when $s$ is small (more precisely, when $s\leq 1/4$ and $s\leq1/10$, respectively). The boundedness of the extremal solution for small $s$ seems to require different methods from the ones that we present here. Our computations in Section \[sec-exp\] suggest that the extremal solution for the fractional Laplacian should be bounded in dimensions $n\leq 7$ for all $s\in(0,1)$, at least for the exponential nonlinearity $f(u)=e^u$. As commented above, Capella-Dávila-Dupaigne-Sire [@CDDS] studied the extremal solution for the *spectral* fractional Laplacian $A^s$ in $\Omega=B_1$. They obtained an $L^\infty$ bound for the extremal solution in a ball in dimensions $n<2\left(2+s+\sqrt{2s+2}\right)$, and hence they proved the boundedness of the extremal solution in dimensions $n\leq 6$ for all $s\in(0,1)$.
To prove part (i) of Theorem \[S\] we borrow the ideas of [@N], where Nedev proved the boundedness of the extremal solution for $s=1$ and $n\leq3$. To prove part (ii) we follow the approach of M. Sanchón in [@S]. When we try to repeat the same arguments for the fractional Laplacian, we find that some identities that in the case $s=1$ come from local integration by parts are no longer available for $s<1$. Instead, we succeed to replace them by appropriate inequalities. These inequalities are sharp as $s\uparrow1$, but not for small $s$. Finally, part (iii) is proved by an argument of Nedev [@N2], which for $s<1$ requires the Pohozaev identity for the fractional Laplacian, recently established by the authors in [@RS]. This argument requires also some boundary estimates, which we prove using the moving planes method; see Proposition \[bdyestimates\] at the end of this introduction.
An important tool in the proofs of the results of Nedev [@N] and Sanchón [@S] is the classical $L^p$ to $W^{2,p}$ estimate for the Laplace equation. Namely, if $u$ is the solution of $-\Delta u =g$ in $\Omega$, $u=0$ in $\partial\Omega$, with $g\in L^p(\Omega)$, $1<p<\infty$, then $$\|u\|_{W^{2,p}(\Omega)}\leq C\|g\|_{L^p(\Omega)}.$$ This estimate and the Sobolev embeddings lead to $L^q(\Omega)$ or $C^\alpha(\overline\Omega)$ estimates for the solution $u$, depending on whether $1<p<\frac n2$ or $p>\frac n2$, respectively.
Here, to prove Theorem \[S\] we need similar estimates but for the fractional Laplacian, in the sense that from $(-\Delta)^s u\in L^p(\Omega)$ we want to deduce $u\in L^q(\Omega)$ or $u\in C^\alpha(\overline\Omega)$. However, $L^p$ to $W^{2s,p}$ estimates for the fractional Laplace equation, in which $-\Delta$ is replaced by the fractional Laplacian $(-\Delta)^s$, are not available for all $p$, even when $\Omega=\R^n$; see Remarks \[rem1\] and \[rem11\].
Although the $L^p$ to $W^{2s,p}$ estimate does not hold for all $p$ in this fractional framework, what will be indeed true is the following result. This is a crucial ingredient in the proof of Theorem \[S\].
\[th2\] Let $\Omega\subset \R^n$ be a bounded $C^{1,1}$ domain, $s\in(0,1)$, $n>2s$, $g\in C(\overline\Omega)$, and $u$ be the solution of $$\label{Omega}
\left\{ \begin{array}{rcll} (-\Delta)^s u &=&g&\textrm{in }\Omega \\
u&=&0&\textrm{in }\mathbb R^n\backslash\Omega.\end{array}\right.$$
- For each $1\leq r<\frac{n}{n-2s}$ there exists a constant $C$, depending only on $n$, $s$, $r$, and $|\Omega|$, such that $$\|u\|_{L^r(\Omega)}\leq C\|g\|_{L^1(\Omega)},\quad r<\frac{n}{n-2s}.$$
- Let $1<p<\frac{n}{2s}$. Then there exists a constant $C$, depending only on $n$, $s$, and $p$, such that $$\|u\|_{L^q(\Omega)}\leq C\|g\|_{L^p(\Omega)},\quad \textrm{where}\quad q=\frac{np}{n-2ps}.$$
- Let $\frac{n}{2s}<p<\infty$. Then, there exists a constant $C$, depending only on $n$, $s$, $p$, and $\Omega$, such that $$\|u\|_{C^{\beta}(\R^n)}\leq C\|g\|_{L^p(\Omega)},\quad \textrm{where}\quad \beta=\min\left\{s,2s-\frac{n}{p}\right\}.$$
We will use parts (i), (ii), and (iii) of Proposition \[th2\] in the proof of Theorem \[S\]. However, we will only use part (iii) to obtain an $L^\infty$ estimate for $u$, we will not need the $C^\beta$ bound. Still, for completeness we prove the $C^\beta$ estimate, with the optimal exponent $\beta$ (depending on $p$).
\[n=1\] Proposition \[th2\] does not provide any estimate for $n\leq 2s$. Since $s\in(0,1)$, then $n\leq2s$ yields $n=1$ and $s\geq 1/2$. In this case, any bounded domain is of the form $\Omega=(a,b)$, and the Green function $G(x,y)$ for problem is explicit; see [@BGR]. Then, by using this expression it is not difficult to show that $G(\cdot,y)$ is $L^\infty(\Omega)$ in case $s>1/2$ and $L^p(\Omega)$ for all $p<\infty$ in case $s=1/2$. Hence, in case $n<2s$ it follows that $\|u\|_{L^\infty(\Omega)}\leq C\|g\|_{L^1(\Omega)}$, while in case $n=2s$ it follows that $\|u\|_{L^q(\Omega)}\leq C\|g\|_{L^1(\Omega)}$ for all $q<\infty$ and $\|u\|_{L^\infty(\Omega)}\leq C\|g\|_{L^p(\Omega)}$ for $p>1$.
Proposition \[th2\] follows from Theorem \[th1\] and Proposition \[prop:u-is-Cbeta\] below. The first one contains some classical results concerning embeddings for the Riesz potential, and reads as follows.
\[th1\] Let $s\in(0,1)$, $n>2s$, and $g$ and $u$ be such that $$\label{R^n}
u=(-\Delta)^{-s}g\ \ \textrm{in}\ \R^n,$$ in the sense that $u$ is the Riesz potential of order $2s$ of $g$. Assume that $u$ and $g$ belong to $L^p(\R^n)$, with $1\leq p<\infty$.
- If $p=1$, then there exists a constant $C$, depending only on $n$ and $s$, such that $$\|u\|_{L^q_{{\rm weak}}(\R^n)}\leq C\|g\|_{L^1(\R^n)},\quad \textrm{where}\quad q=\frac{n}{n-2s}.$$
- If $1<p<\frac{n}{2s}$, then there exists a constant $C$, depending only on $n$, $s$, and $p$, such that $$\|u\|_{L^q(\R^n)}\leq C\|g\|_{L^p(\R^n)},\quad \textrm{where}\quad q=\frac{np}{n-2ps}.$$
- If $\frac{n}{2s}<p<\infty$, then there exists a constant $C$, depending only on $n$, $s$, and $p$, such that $$[u]_{C^{\alpha}(\R^n)}\leq C\|g\|_{L^p(\R^n)},\quad \textrm{where}\quad \alpha=2s-\frac{n}{p},$$ where $[\,\cdot\,]_{C^{\alpha}(\R^n)}$ denotes the $C^\alpha$ seminorm.
Parts (i) and (ii) of Theorem \[th1\] are proved in the book of Stein [@Stein Chapter V]. Part (iii) is also a classical result, but it seems to be more difficult to find an exact reference for it. Although it is not explicitly stated in [@Stein], it follows for example from the inclusions $$I_{2s}(L^p)=I_{2s-n/p}(I_{n/p}(L^p))\subset I_{2s-n/p}(\textrm{BMO})\subset C^{2s-\frac{n}{p}},$$ which are commented in [@Stein p.164]. In the more general framework of spaces with non-doubling $n$-dimensional measures, a short proof of this result can also be found in [@GG].
Having Theorem \[th1\] available, to prove Proposition \[th2\] we will argue as follows. Assume $1< p<\frac{n}{2s}$ and consider the solution $v$ of the problem $$(-\Delta)^s v=|g|\ \ {\rm in}\ \R^n,$$ where $g$ is extended by zero outside $\Omega$. On the one hand, the maximum principle yields $-v\leq u\leq v$ in $\R^n$, and by Theorem \[th1\] we have that $v\in L^q(\R^n)$. From this, parts (i) and (ii) of the proposition follow. On the other hand, if $p>\frac{n}{2s}$ we write $u=\tilde v+w$, where $\tilde v$ solves $(-\Delta)^s \tilde v=g$ in $\R^n$ and $w$ is the solution of $$\left\{ \begin{array}{rcll} (-\Delta)^s w &=&0&\textrm{in }\Omega \\
w&=&\tilde v&\textrm{in }\mathbb R^n\backslash\Omega.\end{array}\right.$$ As before, by Theorem \[th1\] we will have that $\tilde v\in C^\alpha(\R^n)$, where $\alpha=2s-\frac{n}{p}$. Then, the $C^\beta$ regularity of $u$ will follow from the following new result.
\[prop:u-is-Cbeta\] Let $\Omega$ be a bounded $C^{1,1}$ domain, $s\in(0,1)$, $h\in C^\alpha(\R^n\setminus\Omega)$ for some $\alpha>0$, and $u$ be the solution of $$\label{g}
\left\{ \begin{array}{rcll} (-\Delta)^s u &=&0&\textrm{in }\Omega \\
u&=&h&\textrm{in }\mathbb R^n\backslash\Omega.\end{array}\right.$$ Then, $u\in C^\beta(\R^n)$, with $\beta=\min\{s,\alpha\}$, and $$\|u\|_{C^{\beta}(\R^n)}\le C \|h\|_{C^\alpha(\R^n\setminus\Omega)},$$ where $C$ is a constant depending only on $\Omega$, $\alpha$, and $s$.
To prove Proposition \[prop:u-is-Cbeta\] we use similar ideas as in [@RS-Dir]. Namely, since $u$ is harmonic then it is smooth inside $\Omega$. Hence, we only have to prove $C^\beta$ estimates near the boundary. To do it, we use an appropriate barrier to show that $$|u(x)-u(x_0)|\leq C\|h\|_{C^{\alpha}}\delta(x)^\beta\quad {\rm in}\ \Omega,$$ where $x_0$ is the nearest point to $x$ on $\partial\Omega$, $\delta(x)={\rm dist}(x,\partial\Omega)$, and $\beta=\min\{s,\alpha\}$. Combining this with the interior estimates, we obtain $C^\beta$ estimates up to the boundary of $\Omega$.
Finally, as explained before, to show that when the domain is convex the extremal solution belongs to the energy class $H^s(\R^n)$ —which is part (iii) of Theorem \[S\]— we need the following boundary estimates.
\[bdyestimates\] Let $\Omega\subset\R^n$ be a bounded convex domain, $s\in(0,1)$, $f$ be a locally Lipschitz function, and $u$ be a bounded positive solution of $$\label{Omega}
\left\{ \begin{array}{rcll} (-\Delta)^s u &=&f(u)&\textrm{in }\Omega \\
u&=&0&\textrm{in }\mathbb R^n\backslash\Omega.\end{array}\right.$$ Then, there exists constants $\delta>0$ and $C$, depending only on $\Omega$, such that $$\|u\|_{L^{\infty}(\Omega_\delta)}\leq C\|u\|_{L^1(\Omega)},$$ where $\Omega_\delta=\{x\in\Omega\,:\, {\rm dist}(x,\partial\Omega)<\delta\}$.
This estimate follows, as in the classical result of de Figueiredo-Lions-Nussbaum [@FLN], from the moving planes method. There are different versions of the moving planes method for the fractional Laplacian (using the Caffarelli-Silvestre extension, the Riesz potential, the Hopf lemma, etc.). A particularly clean version uses the maximum principle in small domains for the fractional Laplacian, recently proved by Jarohs and Weth in [@JW]. Here, we follow their approach and we show that this maximum principle holds also for integro-differential operators with decreasing kernels.
The paper is organized as follows. In Section \[sec-exist\] we prove Proposition \[existence\]. In Section \[sec-exp\] we study the regularity of the extremal solution in the case $f(u)=e^u$. In Section \[sec-reg\] we prove Theorem \[S\] (i)-(ii). In Section \[sec-bdry\] we show the maximum principle in small domains and use the moving planes method to establish Proposition \[bdyestimates\]. In Section \[sec-Hs\] we prove Theorem \[S\] (iii). Finally, in Section \[sec3\] we prove Proposition \[th2\].
Existence of the extremal solution {#sec-exist}
==================================
In this section we prove Proposition \[existence\]. For it, we follow the argument from Proposition 5.1 in [@CC]; see also [@D].
*Step 1.* We first prove that there is no weak solution for large $\lambda$.
Let $\lambda_1>0$ be the first eigenvalue of $(-\Delta)^s$ in $\Omega$ and $\varphi_1>0$ the corresponding eigenfunction, that is, $$\left\{\begin{array}{rcll}
(-\Delta)^s \varphi_1 &=&\lambda_1\varphi_1&\ \textrm{in}\ \Omega\\
\varphi_1&>&0&\ \textrm{in}\ \Omega\\
\varphi_1&=&0&\ \textrm{in}\ \R^n\setminus\Omega.
\end{array}\right.$$ The existence, simplicity, and boundedness of the first eigenfunction is proved in [@SV2 Proposition 5] and [@SV3 Proposition 4]. Assume that $u$ is a weak solution of . Then, using $\varphi_1$ as a test function for problem (see Definition \[def\]), we obtain $$\label{contradiction}
\int_\Omega \lambda_1u\,\varphi_1 dx=\int_\Omega u(-\Delta)^s\varphi_1 dx=\int_\Omega\lambda f(u)\varphi_1 dx.$$ But since $f$ is superlinear at infinity and positive in $[0,\infty)$, it follows that $\lambda f(u)>\lambda_1 u$ if $\lambda$ is large enough, a contradiction with .
*Step 2.* Next we prove the existence of a classical solution to for small $\lambda$. Since $f(0)>0$, $\underline{u}\equiv0$ is a strict subsolution of for every $\lambda>0$. The solution $\overline u$ of $$\label{f=1}
\left\{ \begin{array}{rcll} (-\Delta)^s \overline u &=&1&\textrm{in }\Omega \\
\overline u&=&0&\textrm{on }\mathbb R^n\backslash\Omega\end{array}\right.$$ is a bounded supersolution of for small $\lambda$, more precisely whenever $\lambda f(\max \overline u)<1$. For such values of $\lambda$, a classical solution $u_\lambda$ is obtained by monotone iteration starting from zero; see for example [@D].
*Step 3.* We next prove that there exists a finite parameter $\lambda^*$ such that for $\lambda<\lambda^*$ there is a classical solution while for $\lambda>\lambda^*$ there does not exist classical solution.
Define $\lambda^*$ as the supremum of all $\lambda>0$ for which admits a classical solution. By Steps 1 and 2, it follows that $0<\lambda^*<\infty$. Now, for each $\lambda<\lambda^*$ there exists $\mu\in (\lambda,\lambda^*)$ such that admits a classical solution $u_\mu$. Since $f>0$, $u_\mu$ is a bounded supersolution of , and hence the monotone iteration procedure shows that admits a classical solution $u_\lambda$ with $u_\lambda\leq u_\mu$. Note that the iteration procedure, and hence the solution that it produces, are independent of the supersolution $u_\mu$. In addition, by the same reason $u_\lambda$ is smaller than any bounded supersolution of . It follows that $u_\lambda$ is minimal (i.e., the smallest solution) and that $u_\lambda<u_\mu$.
*Step 4.* We show now that these minimal solutions $u_\lambda$, $0<\lambda<\lambda^*$, are semistable.
Note that the energy functional for problem in the set $\{u\in H^s(\R^n)\,:\, u\equiv0\ \textrm{in}\ \R^n\setminus\Omega,\ 0\leq u\leq u_\lambda\}$ admits an absolute minimizer $u_{\textrm{min}}$. Then, using that $u_\lambda$ is the minimal solution and that $f$ is positive and increasing, it is not difficult to see that $u_{\textrm{min}}$ must coincide with $u_\lambda$. Considering the second variation of energy (with respect to nonpositive perturbations) we see that $u_{\textrm{min}}$ is a semistable solution of . But since $u_{\textrm{min}}$ agrees with $u_\lambda$, then $u_\lambda$ is semistable. Thus $u_\lambda$ is semistable.
*Step 5.* We now prove that the pointwise limit $u^*=\lim_{\lambda\uparrow\lambda^*} u_\lambda$ is a weak solution of for $\lambda=\lambda^*$ and that this solution $u^*$ is semistable.
As above, let $\lambda_1>0$ the first eigenvalue of $(-\Delta)^s$, and $\varphi_1>0$ be the corresponding eigenfunction. Since $f$ is superlinear at infinity, there exists a constant $C>0$ such that $$\label{ineq-f}
\frac{2\lambda_1}{\lambda^*}t\leq f(t)+C\quad \textrm{for all}\quad t\geq0.$$ Using $\varphi_1$ as a test function in for $u_\lambda$, we find $$\int_\Omega \lambda f(u_\lambda)\varphi_1dx=\int_\Omega \lambda_1 u_\lambda \varphi_1dx\leq \frac{\lambda^*}{2}\int_\Omega\left( f(u_\lambda)+C\right)\varphi_1dx.$$ In the last inequality we have used . Taking $\lambda\geq \frac34 \lambda^*$, we see that $f(u_\lambda)\varphi_1$ is uniformly bounded in $L^1(\Omega)$. In addition, it follows from the results in [@RS-Dir] that $$c_1\delta^s\leq \varphi_1\leq C_2\delta^s\ \ {\rm in}\ \Omega$$ for some positive constants $c_1$ and $C_2$, where $\delta(x)={\rm dist}(x,\partial\Omega)$. Hence, we have that $$\lambda\int_\Omega f(u_\lambda)\delta^sdx\leq C$$ for some constant $C$ that does not depend on $\lambda$. Use now $\overline u$, the solution of , as a test function. We obtain that $$\int_\Omega u_\lambda dx=\lambda\int_\Omega f(u_\lambda)\overline u dx\leq C_3\lambda\int_\Omega f(u_\lambda)\delta^sdx\leq C$$ for some constant $C$ depending only on $f$ and $\Omega$. Here we have used that $\overline u\leq C_3\delta^s$ in $\Omega$ for some constant $C_3>0$, which also follows from [@RS-Dir].
Thus, both sequences, $u_\lambda$ and $\lambda f(u_\lambda)\delta^s$ are increasing in $\lambda$ and uniformly bounded in $L^1(\Omega)$ for $\lambda<\lambda^*$. By monotone convergence, we conclude that $u^*\in L^1(\Omega)$ is a weak solution of for $\lambda=\lambda^*$.
Finally, for $\lambda<\lambda^*$ we have $\int_\Omega \lambda f'(u_\lambda)|\eta|^2dx\leq \|\eta\|_{\accentset{\circ}{H}^s}^2$, where $\|\eta\|_{\accentset{\circ}{H}^s}^2$ is defined by , for all $\eta\in H^s(\R^n)$ with $\eta\equiv0$ in $\R^n\setminus\Omega$. Since $f'\geq0$, Fatou’s lemma leads to $$\int_\Omega \lambda^* f'(u^*)|\eta|^2dx\leq \|\eta\|_{\accentset{\circ}{H}^s}^2,$$ and hence $u^*$ is semistable.
\[solucions\] As said in the introduction, the study of extremal solutions involves three classes of solutions: classical, energy, and weak solutions; see Definition \[def\]. It follows from their definitions that any classical solution is an energy solution, and that any energy solution is a weak solution.
Moreover, any weak solution $u$ which is bounded is a classical solution. This can be seen as follows. First, by considering $u\ast\eta^\epsilon$ and $f(u)\ast\eta^\epsilon$, where $\eta^\epsilon$ is a standard mollifier, it is not difficult to see that $u$ is regular in the interior of $\Omega$. Moreover, by scaling, we find that $|(-\Delta)^{s/2}u|\leq C\delta^{-s}$, where $\delta(x)=\textrm{dist}(x,\partial\Omega)$. Then, if $\zeta\in C^\infty_c(\Omega)$, we can integrate by parts in to obtain $$\label{eq-rem}
(u,\zeta)_{\accentset{\circ}{H}^s}=\int_{\R^n}\int_{\R^n}\frac{\bigl(u(x)-u(y)\bigr)\bigl(\zeta(x)-\zeta(y)\bigr)}{|x-y|^{n+2s}}dx\,dy=\int_\Omega \lambda f(u)\zeta dx$$ for all $\zeta\in C^\infty_c(\Omega)$. Hence, since $f(u)\in L^\infty$, by density holds for all $\zeta\in H^s(\R^n)$ such that $\zeta\equiv0$ in $\R^n\setminus\Omega$, and therefore $u$ is an energy solution. Finally, bounded energy solutions are classical solutions; see Remark 2.11 in [@RS-Dir] and [@SV].
An example case: the exponential nonlinearity {#sec-exp}
=============================================
In this section we study the regularity of the extremal solution for the nonlinearity $f(u)=e^u$. Although the results of this section follow from Theorem \[S\] (ii), we exhibit this case separately because the proofs are much simpler. Furthermore, this exponential case has the advantage that we have an explicit unbounded solution to the equation in the whole $\R^n$, and we can compute the values of $n$ and $s$ for which this singular solution is semistable.
The main result of this section is the following.
\[exp\] Let $\Omega$ be a smooth and bounded domain in $\mathbb R^n$, and let $u^*$ the extremal solution of . Assume that $f(u)=e^u$ and $n<10s$. Then, $u^*$ is bounded.
Let $\alpha$ be a positive number to be chosen later. Setting $\eta=e^{\alpha u_\lambda}-1$ in the stability condition (\[semistable\]) (note that $\eta\equiv0$ in $\R^n\setminus\Omega$), we obtain that $$\label{exp1}
\int_\Omega \lambda e^{u_\lambda}(e^{\alpha u_\lambda}-1)^2dx\leq \left\|e^{\alpha u_\lambda}-1\right\|_{\accentset{\circ}{H}^s}^2.$$
Next we use that $$\label{ineqexp}
\left(e^{b}-e^{a}\right)^2\leq \frac{1}{2}\left(e^{2 b}-e^{2 a}\right)(b-a)$$ for all real numbers $a$ and $b$. This inequality can be deduced easily from the Cauchy-Schwarz inequality, as follows $$\left(e^{b}-e^{a}\right)^2=\left(\int_a^b e^{t}dt\right)^2\leq (b-a)\int_a^b e^{2 t}dt=\frac{1}{2}\left(e^{2 b}-e^{2 a}\right)(b-a).$$ Using , , and integrating by parts, we deduce $$\begin{aligned}
\left\|e^{\alpha u_\lambda}-1\right\|_{\accentset{\circ}{H}^s}^2&=&\frac{c_{n,s}}{2}\int_{\R^n}\int_{\R^n}\frac{\left(e^{\alpha u_\lambda(x)} - e^{\alpha u_\lambda(y)}\right)^2}{|x-y|^{n+2s}}dxdy\\
&\leq& \frac{c_{n,s}}{2}\int_{\R^n}\int_{\R^n}\frac{\frac{1}{2}\left(e^{2\alpha u_\lambda(x)} - e^{2\alpha u_\lambda(y)}\right)\left(\alpha u_\lambda(x)-\alpha u_\lambda(y)\right)}{|x-y|^{n+2s}}dxdy\\
&=&\frac{\alpha}{2}\int_{\Omega} e^{2\alpha u_\lambda}(-\Delta)^su_\lambda dx.\end{aligned}$$ Thus, using that $(-\Delta)^s u_\lambda=\lambda e^{u_\lambda}$, we find $$\label{exp2}
\left\|e^{\alpha u_\lambda}-1\right\|_{\accentset{\circ}{H}^s}^2\leq \frac{\alpha}{2}\int_{\Omega} e^{2\alpha u_\lambda}(-\Delta)^su_\lambda dx=\frac{\alpha}{2}\int_\Omega \lambda e^{(2\alpha+1)u_\lambda}dx.$$
Therefore, combining and , and rearranging terms, we get $$\left(1-\frac\alpha2\right)\int_\Omega e^{(2\alpha+1)u_\lambda}-2\int_\Omega e^{(\alpha+1)u_\lambda}+\int_\Omega e^{\alpha u_\lambda}\leq0.$$ From this, it follows from Hölder’s inequality that for each $\alpha<2$ $$\label{L^5}
\|e^{u_\lambda}\|_{L^{2\alpha+1}}\leq C$$ for some constant $C$ which depends only on $\alpha$ and $|\Omega|$.
Finally, given $n<10s$ we can choose $\alpha<2$ such that $\frac{n}{2s}<2\alpha+1<5$. Then, taking $p=2\alpha+1$ in Proposition \[th2\] (iii) (see also Remark \[n=1\]) and using we obtain $$\|u_\lambda\|_{L^\infty(\Omega)}\leq C_1\|(-\Delta)^s u_\lambda\|_{L^p(\Omega)}= C_1\lambda\|e^{u_\lambda}\|_{L^p(\Omega)}\leq C$$ for some constant $C$ that depends only on $n$, $s$, and $\Omega$. Letting $\lambda\uparrow \lambda^*$ we find that the extremal solution $u^*$ is bounded, as desired.
The following result concerns the stability of the explicit singular solution $\log\frac{1}{|x|^{2s}}$ to equation $(-\Delta)^s u=\lambda e^u$ in the whole $\R^n$.
\[stability-exp\] Let $s\in(0,1)$, and let $$u_0(x)=\log\frac{1}{|x|^{2s}}.$$ Then, $u_0$ is a solution of $(-\Delta)^su=\lambda_0e^{u}$ in all of $\R^n$ for some $\lambda_0>0$. Moreover, $u_0$ is semistable if and only if $$\label{semistable-condition-Gammas}
\frac{\Gamma\left(\frac{n}{2}\right)\Gamma(1+s)}{\Gamma\left(\frac{n-2s}{2}\right)}\leq \frac{\Gamma^2\left(\frac{n+2s}{4}\right)}{\Gamma^2\left(\frac{n-2s}{4}\right)}.$$ As a consequence:
- If $n\leq7$, then $u$ is unstable for all $s\in(0,1)$.
- If $n=8$, then $u$ is semistable if and only if $s\lesssim0'28206...$.
- If $n=9$, then $u$ is semistable if and only if $s\lesssim0'63237...$.
- If $n\geq10$, then $u$ is semistable for all $s\in(0,1)$.
Proposition \[stability-exp\] suggests that the extremal solution for the fractional Laplacian should be bounded whenever $$\label{ineq-gammas}
\frac{\Gamma\left(\frac{n}{2}\right)\Gamma(1+s)}{\Gamma\left(\frac{n-2s}{2}\right)}> \frac{\Gamma^2\left(\frac{n+2s}{4}\right)}{\Gamma^2\left(\frac{n-2s}{4}\right)},$$ at least for the exponential nonlinearity $f(u)=e^u$. In particular, $u^*$ should be bounded for all $s\in(0,1)$ whenever $n\leq7$. This is an open problem.
When $s=1$ and when $s=2$, inequality coincides with the expected optimal dimensions for which the extremal solution is bounded for the Laplacian $\Delta$ and for the bilaplacian $\Delta^2$, respectively. In the unit ball $\Omega=B_1$, it is well known that the extremal solution for $s=1$ is bounded whenever $n\leq9$ and may be singular if $n\geq10$ [@CC], while the extremal solution for $s=2$ is bounded whenever $n\leq12$ and may be singular if $n\geq13$ [@DDGM]. Taking $s=1$ and $s=2$ in , one can see that the inequality is equivalent to $n<10$ and $n\lesssim12.5653...$, respectively.
We next give the
First, using the Fourier transform, it is not difficult to compute $$(-\Delta)^su_0=(-\Delta)^s\log\frac{1}{|x|^{2s}}=\frac{\lambda_0}{|x|^{2s}},$$ where $$\lambda_0=2^{2s}\frac{\Gamma\left(\frac{n}{2}\right)\Gamma(1+s)}{\Gamma\left(\frac{n-2s}{2}\right)}.$$ Thus, $u_0$ is a solution of $(-\Delta)^su_0=\lambda_0 e^{u_0}$.
Now, since $f(u)=e^u$, by we have that $u_0$ is semistable in $\Omega=\R^n$ if and only if $$\lambda_0\int_{\R^n} \frac{\eta^2}{|x|^{2s}}dx\leq \int_{\R^n}\left|(-\Delta)^{s/2}\eta\right|^2dx$$ for all $\eta\in H^s(\R^n)$.
The inequality $$\int_\Omega \frac{\eta^2}{|x|^{2s}}dx\leq H_{n,s}^{-1}\int_{\R^n}\left|(-\Delta)^{s/2}\eta\right|^2dx$$ is known as the fractional Hardy inequality, and the best constant $$H_{n,s}=2^{2s}\frac{\Gamma^2\left(\frac{n+2s}{4}\right)}{\Gamma^2\left(\frac{n-2s}{4}\right)}$$ was obtained by Herbst [@H] in 1977; see also [@FLS]. Therefore, it follows that $u_0$ is semistable if and only if $$\lambda_0\leq H_{n,s},$$ which is the same as .
Boundedness of the extremal solution in low dimensions {#sec-reg}
======================================================
In this section we prove Theorem \[S\] (i)-(ii).
We start with a lemma, which is the generalization of inequality . It will be used in the proof of both parts (i) and (ii) of Theorem \[S\].
\[po\] Let $f$ be a $C^1([0,\infty))$ function, $\widetilde f(t)=f(t)-f(0)$, $\gamma>0$, and $$\label{functiong}
g(t)=\int_0^t \widetilde f(s)^{2\gamma-2}f'(s)^2ds.$$ Then, $$\left(\widetilde f(a)^\gamma-\widetilde f(b)^\gamma\right)^2\leq \gamma^2\bigl(g(a)-g(b)\bigr)(a-b)$$ for all nonnegative numbers $a$ and $b$.
We can assume $a\leq b$. Then, since $\frac{d}{dt}\left\{\widetilde f(t)^\gamma\right\}=\gamma \widetilde f(t)^{\gamma-1}f'(t)$, the inequality can be written as $$\left(\int_a^b\gamma \widetilde f(t)^{\gamma-1}f'(t)dt\right)^2\leq \gamma^2(b-a)\int_a^b \widetilde f(t)^{2\gamma-2}f'(t)^2dt,$$ which follows from the Cauchy-Schwarz inequality.
The proof of part (ii) of Theorem \[S\] will be split in two cases. Namely, $\tau\geq1$ and $\tau<1$, where $\tau$ is given by . For the case $\tau\geq1$, Lemma \[lemamanel\] below will be an important tool. Instead, for the case $\tau<1$ we will use Lemma \[lemapotencia\]. Both lemmas are proved by Sanchón in [@S], where the extremal solution for the $p$-Laplacian operator is studied.
\[lemamanel\] Let $f$ be a function satisfying , and assume that the limit in exists. Assume in addition that $$\tau=\lim_{t\rightarrow\infty}\frac{f(t)f''(t)}{f'(t)^2}\geq1.$$ Then, any $\gamma\in(1,1+\sqrt{\tau})$ satisfies $$\label{condgamma}
\limsup_{t\rightarrow+\infty}\frac{\gamma^2g(t)}{f(t)^{2\gamma-1}f'(t)}<1,$$ where $g$ is given by .
\[lemapotencia\] Let $f$ be a function satisfying , and assume that the limit in exists. Assume in addition that $$\tau=\lim_{t\rightarrow\infty}\frac{f(t)f''(t)}{f'(t)^2}<1.$$ Then, for every $\epsilon\in(0,1-\tau)$ there exists a positive constant $C$ such that $$f(t)\leq C(1+t)^{\frac{1}{1-(\tau+\epsilon)}},\qquad \mbox{for all}\ \ t>0.$$ The constant $C$ depends only on $\tau$ and $\epsilon$.
The first step in the proof of Theorem \[S\] (ii) in case $\tau\geq1$ is the following result.
\[propgamma\] Let $f$ be a function satisfying . Assume that $\gamma\geq1$ satisfies , where $g$ is given by . Let $u_\lambda$ be the solution of given by Proposition \[existence\] (i), where $\lambda<\lambda^*$. Then, $$\|f(u_\lambda)^{2\gamma}f'(u_\lambda)\|_{L^1(\Omega)}\leq C$$ for some constant $C$ which does not depend on $\lambda$.
Recall that the seminorm $\|\,\cdot\,\|_{\accentset{\circ}{H}^s}$ is defined by . Using Lemma \[po\], , and integrating by parts, $$\begin{split}\label{fr}
\left\|\widetilde f(u_\lambda)^\gamma\right\|_{\accentset{\circ}{H}^s}^2&=\frac{c_{n,s}}{2}\int_{\R^n}\int_{\R^n}\frac{\left(\widetilde f(u_\lambda(x))^\gamma - \widetilde f(u_\lambda(y))^\gamma\right)^2}{|x-y|^{n+2s}}dxdy\\
&\leq \gamma^2 \frac{c_{n,s}}{2}\int_{\R^n}\int_{\R^n}\frac{\bigl(g(u_\lambda(x)) - g( u_\lambda(y))\bigr)\left(u_\lambda(x)-u_\lambda(y)\right)}{|x-y|^{n+2s}}dxdy\\
&= \gamma^2 \int_{\R^n}(-\Delta)^{s/2}g(u_\lambda)(-\Delta)^{s/2}u_\lambda\,dx\\
&=\gamma^2\int_{\Omega} g(u_\lambda)(-\Delta)^su_\lambda\,dx\\
&=\gamma^2\int_{\Omega} f(u_\lambda)g(u_\lambda)dx.\end{split}$$
Moreover, the stability condition applied with $\eta=\widetilde f(u_\lambda)^\gamma$ yields $$\int_\Omega f'(u_\lambda)\widetilde f(u_\lambda)^{2\gamma} \leq \left\|\widetilde f(u_\lambda)^\gamma\right\|_{\accentset{\circ}{H}^s}^2.$$ This, combined with , gives $$\label{4nedev4}
\int_\Omega f'(u_\lambda)\widetilde f(u_\lambda)^{2\gamma}\leq \gamma^2\int_\Omega f(u_\lambda)g(u_\lambda).$$
Finally, by and since $\widetilde f(t)/f(t)\rightarrow1$ as $t\rightarrow+\infty$, it follows from that $$\label{h1}
\int_\Omega f(u_\lambda)^{2\gamma}f'(u_\lambda)\leq C$$ for some constant $C$ that does not depend on $\lambda$, and thus the proposition is proved.
We next give the proof of Theorem \[S\] (ii).
Assume first that $\tau\geq1$, where $$\tau=\lim_{t\rightarrow\infty}\frac{f(t)f''(t)}{f'(t)^2}.$$ By Lemma \[propgamma\] and Lemma \[lemamanel\], we have that $$\label{visL^1}
\int_{\Omega}f(u_\lambda)^{2\gamma}f'(u_\lambda)dx\leq C$$ for each $\gamma\in(1,1+\sqrt{\tau})$.
Now, for any such $\gamma$, we have that ${\widetilde f}^{2\gamma}$ is increasing and convex (since $2\gamma\geq1$), and thus $$\widetilde f(a)^{2\gamma}-\widetilde f(b)^{2\gamma}\leq 2\gamma f'(a)\widetilde f(a)^{2\gamma-1}(a-b).$$ Therefore, we have that $$\begin{aligned}
(-\Delta)^s\widetilde f(u_\lambda)^{2\gamma}(x) &=& c_{n,s}\int_{\R^n}\frac{\widetilde f(u_\lambda(x))^{2\gamma}-\widetilde f(u_\lambda(y))^{2\gamma}}{|x-y|^{n+2s}}dy\\
&\leq& 2\gamma f'(u_\lambda(x))\widetilde f(u_\lambda(x))^{2\gamma-1}c_{n,s}\int_{\R^n}\frac{u_\lambda(x)-u_\lambda(y)}{|x-y|^{n+2s}}dy\\
&=&2\gamma f'(u_\lambda(x))\widetilde f(u_\lambda(x))^{2\gamma-1}(-\Delta)^s u_\lambda(x)\\
&\leq&2\gamma\lambda f'(u_\lambda(x))f(u_\lambda(x))^{2\gamma},\end{aligned}$$ and thus, $$\label{defv}
(-\Delta)^s\widetilde f(u_\lambda)^{2\gamma}\leq 2\gamma \lambda f'(u_\lambda)f(u_\lambda)^{2\gamma}:=v(x).$$
Let now $w$ be the solution of the problem $$\label{hv2}
\left\{ \begin{array}{rcll} (-\Delta)^s w &=&v&\textrm{in }\Omega \\
w&=&0&\textrm{in }\mathbb R^n\backslash\Omega,
\end{array}\right.$$ where $v$ is given by . Then, by and Proposition \[th2\] (i) (see also Remark \[n=1\]), $$\|w\|_{L^p(\Omega)}\leq \|v\|_{L^1(\Omega)}\leq C\quad \textrm{for each}\ p<\frac{n}{n-2s}.$$ Since $\widetilde f(u_\lambda)^{2\gamma}$ is a subsolution of —by —, it follows that $$0\leq \widetilde f(u_\lambda)^{2\gamma}\leq w.$$ Therefore, $\|f(u_\lambda)\|_{L^p}\leq C$ for all $p<2\gamma\,\frac{n}{n-2s}$, where $C$ is a constant that does not depend on $\lambda$. This can be done for any $\gamma\in(1,1+\sqrt{\tau})$, and thus we find $$\label{f(u)tau-}
\|f(u_\lambda)\|_{L^p}\leq C\ \ \textrm{for each}\ \ p<\frac{2n(1+\sqrt{\tau})}{n-2s}.$$
Hence, using Proposition \[th2\] (iii) and letting $\lambda\uparrow\lambda^*$ it follows that $$u^*\in L^\infty(\Omega)\quad \mbox{whenever}\quad n<6s+4s\sqrt{\tau}.$$ Hence, the extremal solution is bounded whenever $n<10s$.
Assume now $\tau<1$. In this case, Lemma \[lemapotencia\] ensures that for each $\epsilon\in(0,1-\tau)$ there exist a constant $C$ such that $$\label{m}
f(t)\leq C(1+t)^{m},\qquad m=\frac{1}{1-(\tau+\epsilon)}.$$ Then, by we have that $\|f(u_\lambda)\|_{L^p}\leq C$ for each $p<p_0:=\frac{2n(1+\sqrt{\tau})}{n-2s}$.
Next we show that if $n<10s$ by a bootstrap argument we obtain $u^*\in L^\infty(\Omega)$. Indeed, by Proposition \[th2\] (ii) and we have $$f(u^*)\in L^p\quad \Longleftrightarrow\quad (-\Delta)^s u^*\in L^p\quad \Longrightarrow\quad u^*\in L^q
\quad \Longrightarrow\quad f(u^*)\in L^{q/m},$$ where $q=\frac{np}{n-2s p}$. Now, we define recursively $$p_{k+1}:=\frac{np_k}{m(n-2s p_k)},\qquad p_0=\frac{2n(1+\sqrt{\tau})}{n-2s}.$$ Now, since $$p_{k+1}-p_k=\frac{p_k}{n-2s p_k}\left(2s p_k-\frac{m-1}{m}n\right),$$ then the bootstrap argument yields $u^*\in L^\infty(\Omega)$ in a finite number of steps provided that $(m-1)n/m<2s p_0$. This condition is equivalent to $n<2s+4s\frac{1+\sqrt{\tau}}{\tau+\epsilon}$, which is satisfied for $\epsilon$ small enough whenever $n\leq10s$, since $\frac{1+\sqrt{\tau}}{\tau}>2$ for $\tau<1$. Thus, the result is proved.
Before proving Theorem \[S\] (i), we need the following lemma, proved by Nedev in [@N].
\[lemah\] Let $f$ be a convex function satisfying , and let $$\label{defg}
g(t)=\int_0^t f'(\tau)^2d\tau.$$ Then, $$\lim_{t\rightarrow +\infty}\frac{f'(t)\widetilde f(t)^2-\widetilde f(t)g(t)}{f(t)f'(t)}=+\infty,$$ where $\widetilde f(t)=f(t)-f(0)$.
As said above, this lemma is proved in [@N]. More precisely, see equation (6) in the proof of Theorem 1 in [@N] and recall that $\widetilde f/f\rightarrow1$ at infinity.
We can now give the
Let $g$ be given by . Using Lemma \[po\] with $\gamma=1$ and integrating by parts, we find $$\label{ineqnedevf(u)}
\begin{split}
\left\|f(u_\lambda)\right\|_{\accentset{\circ}{H}^s}^2 &= \frac{c_{n,s}}{2}\int_{\R^n}\int_{\R^n}\frac{\left(f(u_\lambda(x))-f(u_\lambda(y))\right)^2}{|x-y|^{n+2s}}dxdy\\
&\leq \frac{c_{n,s}}{2}\int_{\R^n}\int_{\R^n}\frac{\left(g(u_\lambda(x))-g(u_\lambda(y))\right)\left(u_\lambda(x)-u_\lambda(y)\right)}{|x-y|^{n+2s}}dxdy\\
&= \int_{\R^n}(-\Delta)^{s/2}g(u_\lambda)(-\Delta)^{s/2}u_\lambda dx\\
&= \int_{\R^n}g(u_\lambda)(-\Delta)^s u_\lambda dx\\
&=\int_\Omega f(u_\lambda)g(u_\lambda).
\end{split}$$
The stability condition (\[semistable\]) applied with $\eta=\widetilde f(u_\lambda)$ yields $$\int_\Omega f'(u_\lambda)\widetilde f(u_\lambda)^2 \leq \|\widetilde f(u_\lambda)\|_{\accentset{\circ}{H}^s}^2,$$ which combined with gives $$\label{4nedev}
\int_\Omega f'(u_\lambda)\widetilde f(u_\lambda)^2\leq \int_\Omega f(u_\lambda)g(u_\lambda).$$ This inequality can be written as $$\int_\Omega\left\{f'(u_\lambda)\widetilde f(u_\lambda)^2-\widetilde f(u_\lambda)g(u_\lambda)\right\}\leq f(0)\int_\Omega g(u_\lambda).$$ In addition, since $f$ is convex we have $$g(t)=\int_0^t f'(s)^2ds\leq f'(t)\int_0^tf'(s)ds\leq f'(t)f(t),$$ and thus, $$\int_\Omega\left\{f'(u_\lambda)\widetilde f(u_\lambda)^2-\widetilde f(u_\lambda)g(u_\lambda)\right\}\leq f(0)\int_\Omega f'(u_\lambda)f(u_\lambda).$$ Hence, by Lemma \[lemah\] we obtain $$\label{h1}
\int_\Omega f(u_\lambda)f'(u_\lambda)\leq C.$$
Now, on the one hand we have that $$f(a)-f(b)\leq f'(a)(a-b),$$ since $f$ is increasing and convex. This yields, as in , $$(-\Delta)^s\widetilde f(u_\lambda)\leq f'(u_\lambda)(-\Delta)^su_\lambda=f'(u_\lambda)f(u_\lambda):=v(x).$$
On the other hand, let $w$ the solution of the problem $$\label{hv}
\left\{ \begin{array}{rcll} (-\Delta)^s w &=&v&\textrm{in }\Omega \\
w&=&0&\textrm{on }\partial\Omega.
\end{array}\right.$$ By and Proposition \[th2\] (i) (see also Remark \[n=1\]), $$\|w\|_{L^p(\Omega)}\leq \|v\|_{L^1(\Omega)}\leq C\textrm{ for each }p<\frac{n}{n-2s}.$$ Since $\widetilde f(u_\lambda)$ is a subsolution of , then $0\leq \widetilde f(u_\lambda)\leq w$. Therefore, $$\|f(u^*)\|_{L^p(\Omega)}\leq C\ \ \textrm{for each}\ \ p<\frac{n}{n-2s},$$ and using Proposition \[th2\] (iii), we find $$u^*\in L^\infty(\Omega)\ \ \textrm{whenever}\ \ n<4s,$$ as desired.
Boundary estimates: the moving planes method {#sec-bdry}
============================================
In this section we prove Proposition \[bdyestimates\]. This will be done with the celebrated moving planes method [@GNN], as in the classical boundary estimates for the Laplacian of de Figueiredo-Lions-Nussbaum [@FLN].
The moving planes method has been applied to problems involving the fractional Laplacian by different authors; see for example [@CLO; @BMW; @FW]. However, some of these results use the specific properties of the fractional Laplacian —such as the extension problem of Caffarelli-Silvestre [@CS-ext], or the Riesz potential expression for $(-\Delta)^{-s}$—, and it is not clear how to apply the method to more general integro-differential operators. Here, we follow a different approach that allows more general nonlocal operators.
The main tool in the proof is the following maximum principle in small domains.
Recently, Jarohs and Weth [@JW] obtained a parabolic version of the maximum principle in small domains for the fractional Laplacian; see Proposition 2.4 in [@JW]. The proof of their result is essentially the same that we present in this section. Still, we think that it may be of interest to write here the proof for integro-differential operators with decreasing kernels.
\[mpsd\] Let $\Omega\subset\R^n$ be a domain satisfying $\Omega\subset \R^n_+=\{x_1>0\}$. Let $K$ be a nonnegative function in $\R^n$, radially symmetric and decreasing, and satisfying $$K(z)\geq c|z|^{-n-\nu}\quad\textrm{for all}\quad z\in B_1$$ for some positive constants $c$ and $\nu$, and let $$L_K u(x)=\int_{\R^n}\bigl(u(y)-u(x)\bigr)K(x-y)dy.$$
Let $V\in L^\infty(\Omega)$ be any bounded function, and $w\in H^s(\R^n)$ be a bounded function satisfying $$\label{pbmpsd}
\left\{ \begin{array}{rcll} L_K w &=&V(x)w&\textrm{in }\Omega \\
w&\geq&0&\textrm{in }\R^n_+\setminus\Omega\\
w(x)&\geq&-w(x^*)&\textrm{in }\R^n_+,\end{array}\right.$$ where $x^*$ is the symmetric to $x$ with respect to the hyperplane $\{x_1=0\}$. Then, there exists a positive constant $C_0$ such that if $$\label{cond-mpsd}
\left(1+\|V^-\|_{L^\infty(\Omega)}\right)|\Omega|^{\frac{\nu}{n}}\leq C_0,$$ then $w\geq0$ in $\Omega$.
When $L_K$ is the fractional Laplacian $(-\Delta)^s$, then the condition can be replaced by $\|V^-\|_{L^\infty}|\Omega|^{\frac{2s}{n}}\leq C_0$.
The identity $L_K w =V(x)w$ in $\Omega$ written in weak form is $$\label{mpsd1}
(\varphi,w)_K:=\int\int_{\R^{2n}\setminus(\R^n\setminus\Omega)^2}{(\varphi(x)-\varphi(y))(w(x)-w(y))}K(x-y)dx\,dy=\int_\Omega Vw\varphi$$ for all $\varphi$ such that $\varphi\equiv0$ in $\R^n\setminus\Omega$ and $\int_{\R^n}\bigl(\varphi(x)-\varphi(y)\bigr)^2K(x-y)dx\,dy<\infty$. Note that the left hand side of can be written as $$\begin{split}
(\varphi,w)_K=&\int_\Omega\int_\Omega {(\varphi(x)-\varphi(y))(w(x)-w(y))}K(x-y)dx\,dy\\
&+2\int_{\Omega}\int_{\R^n_+\setminus\Omega} {\varphi(x)(w(x)-w(y))}K(x-y)dx\,dy\\
&+2\int_{\Omega}\int_{\R^n_+} {\varphi(x)(w(x)-w(y^*))}K(x-y^*)dx\,dy,\end{split}$$ where $y^*$ denotes the symmetric of $y$ with respect to the hyperplane $\{x_1=0\}$.
Choose $\varphi=-w^-\chi_\Omega$, where $w^-$ is the negative part of $w$, i.e., $w=w^+-w^-$. Then, we claim that $$\label{claim}
\int\int_{\R^{2n}\setminus(\R^n\setminus\Omega)^2} {(w^-(x)\chi_\Omega(x)-w^-(y)\chi_\Omega(y))^2}K(x-y)dx\,dy\leq (-w^-\chi_\Omega,w)_K.$$ Indeed, first, we have $$\begin{split}
(-w^-\chi_\Omega,w)_K&= \int_\Omega\int_\Omega \{(w^-(x)\hspace{-1mm}-\hspace{-1mm}w^-(y))^2\hspace{-1mm}+\hspace{-1mm}w^-(x)w^+(y)\hspace{-1mm}+\hspace{-1mm}w^+(x)w^-(y)\}K(x\hspace{-1mm}-\hspace{-1mm}y)dxdy+\\
&+2\int_{\Omega}\int_{\R^n_+\setminus\Omega} \{w^-(x)(w^-(x)-w^-(y))+w^-(x)w^+(y)\}K(x-y)dx\,dy\\
&+2\int_{\Omega}\int_{\R^n_+} \{w^-(x)(w^-(x)-w^-(y^*))+w^-(x)w^+(y^*)\}K(x-y^*)dx\,dy,
\end{split}$$ where we have used that $w^+(x)w^-(x)=0$ for all $x\in \R^n$.
Thus, rearranging terms and using that $w^-\equiv0$ in $\R^n_+\setminus\Omega$, $$\begin{split}
(-w^-\chi_\Omega,w)_K=& \int\int_{\R^{2n}\setminus(\R^n\setminus\Omega)^2} {(w^-(x)\chi_\Omega(x)-w^-(y)\chi_\Omega(y))^2}K(x-y)dx\,dy\\
&+\int_\Omega\int_\Omega {2w^-(x)w^+(y)}K(x-y)dx\,dy+\\
&+2\int_{\Omega}\int_{\R^n_+\setminus\Omega} \{w^-(x)w^+(y)-w^-(x)w^-(y)\}K(x-y)dx\,dy\\
&+2\int_{\Omega}\int_{\R^n_+} \{w^-(x)w^+(y^*)-w^-(x)w^-(y^*)\}K(x-y^*)dx\,dy\\
\geq&\int\int_{\R^{2n}\setminus(\R^n\setminus\Omega)^2} {(w^-(x)\chi_\Omega(x)-w^-(y)\chi_\Omega(y))^2}K(x-y)dx\,dy+\\
&+2\int_\Omega\int_{\R^n_+} {w^-(x)w^+(y)}K(x-y)dx\,dy+\\
&+2\int_{\Omega}\int_{\R^n_+} {-w^-(x)w^-(y^*)}K(x-y^*)dx\,dy.
\end{split}$$ We next use that, since $K$ is radially symmetric and decreasing, $K(x-y^*)\leq K(x-y)$ for all $x$ and $y$ in $\R^n_+$. We deduce $$\begin{split}(-w^-\chi_\Omega,w)_K\geq&
\int\int_{\R^{2n}\setminus(\R^n\setminus\Omega)^2} {(w^-(x)\chi_\Omega(x)-w^-(y)\chi_\Omega(y))^2}K(x-y)dx\,dy+\\
&+2\int_{\Omega}\int_{\R^n_+} {w^-(x)w^+(y)-w^-(x)w^-(y^*)}K(x-y)dx\,dy,
\end{split}$$ and since $w^-(y^*)\leq w^+(y)$ for all $y$ in $\R^n_+$ by assumption, we obtain .
Now, on the one hand note that from we find $$\int_{\Omega}\int_{\Omega} {(w^-(x)-w^-(y))^2}K(x-y)dx\,dy\leq (-w^-\chi_\Omega,w)_K.$$ Moreover, since $K(z)\geq c|z|^{-n-\nu}\chi_{B_1}(z)$, then $$\begin{split}
\|w^-\|^2_{\accentset{\circ}{H}^{\nu/2}(\Omega)}&:=\frac{c_{n,s}}{2}\int_{\Omega}\int_{\Omega} \frac{(w^-(x)-w^-(y))^2}{|x-y|^{-n-\nu}}dx\,dy\\
&\leq C\|w^-\|_{L^2(\Omega)}+C\int_{\Omega}\int_{\Omega} {\bigl(w^-(x)-w^-(y)\bigr)^2}K(x-y)dx\,dy,
\end{split}$$ and therefore $$\label{mpsd3}
\|w^-\|^2_{\accentset{\circ}{H}^{\nu/2}(\Omega)}\leq C_1\|w^-\|_{L^2(\Omega)}+C_1(-w^-\chi_\Omega,w)_K.$$
On the other hand, it is clear that $$\label{mpsd2}
\int_\Omega Vww^-=\int_\Omega V(w^-)^2\leq \|V^-\|_{L^\infty(\Omega)}\|w^-\|_{L^2(\Omega)}.$$
Thus, it follows from , , and that $$\|w^-\|_{\accentset{\circ}{H}^{\nu/2}(\Omega)}^2\leq C_1\left(1+\|V^-\|_{L^\infty}\right)\|w^-\|_{L^2(\Omega)}.$$ Finally, by the Hölder and the fractional Sobolev inequalities, we have $$\|w^-\|_{L^2(\Omega)}^2\leq |\Omega|^{\frac{\nu}{n}}\|w^-\|_{L^q(\Omega)}^2\leq C_2|\Omega|^{\frac{\nu}{n}}\|w^-\|_{\accentset{\circ}{H}^{\nu/2}(\Omega)}^2,$$ where $q=\frac{2n}{n-\nu}$. Thus, taking $C_0$ such that $C_0<(C_1C_2)^{-1}$ the lemma follows.
Now, once we have the nonlocal version of the maximum principle in small domains, the moving planes method can be applied exactly as in the classical case.
Replacing the classical maximum principle in small domains by Lemma \[mpsd\], we can apply the moving planes method to deduce $\|u\|_{L^\infty(\Omega_\delta)}\leq C\|u\|_{L^1(\Omega)}$ for some constants $C$ and $\delta>0$ that depend only on $\Omega$, as in de Figueiredo-Lions-Nussbaum [@FLN]; see also [@B].
Let us recall this argument. Assume first that all curvatures of $\partial\Omega$ are positive. Let $\nu(y)$ be the unit outward normal to $\Omega$ at $y$. Then, there exist positive constants $s_0$ and $\alpha$ depending only on the convex domain $\Omega$ such that, for every $y\in\partial\Omega$ and every $e\in \R^n$ with $|e| = 1$ and $e\cdot \nu(y)\geq\alpha$, $u(y-se)$ is nondecreasing in $s\in[0, s_0]$. This fact follows from the moving planes method applied to planes close to those tangent to $\Omega$ at $\partial\Omega$. By the convexity of $\Omega$, the reflected caps will be contained in $\Omega$. The previous monotonicity fact leads to the existence of a set $I_x$, for each $x\in\Omega_\delta$, and a constant $\gamma>0$ that depend only on $\Omega$, such that $$|I_x|\geq \gamma,\qquad u(x)\leq u(y)\quad \textrm{for all}\quad y\in I_x.$$ The set $I_x$ is a truncated open cone with vertex at $x$.
As mentioned in page 45 of de Figuereido-Lions-Nussbaum [@FLN], the same can also be proved for general convex domains with a little more of care.
When $\Omega=B_1$, Proposition \[bdyestimates\] follows from the results in [@BMW], where Birkner, López-Mimbela, and Wakolbinger used the moving planes method to show that any nonnegative bounded solution of $$\label{semilinear}
\left\{\begin{array}{rcll}
(-\Delta)^s u &=&f(u)&\quad \textrm{in}\ B_1\\
u&=&0&\quad \textrm{in}\ \R^n\setminus B_1
\end{array}\right.$$ is radially symmetric and decreasing.
When $u$ is a bounded semistable solution of , there is an alternative way to show that $u$ is radially symmetric. This alternative proof applies to all solutions (not necessarily positive), but does not give monotonicity. Indeed, one can easily show that, for any $i\neq j$, the function $w=x_iu_{x_j}-x_j u_{x_i}$ is a solution of the linearized problem $$\left\{\begin{array}{rcll}
(-\Delta)^s w &=&f'(u)w&\quad \textrm{in}\ B_1\\
w&=&0&\quad \textrm{in}\ \R^n\setminus B_1.
\end{array}\right.$$ Then, since $\lambda_1\left((-\Delta)^s-f'(u);B_1\right)\geq0$ by assumption, it follows that either $w\equiv0$ or $\lambda_1=0$ and $w$ is a multiple of the first eigenfunction, which is positive —see the proof of Proposition 9 in [@SV2 Appendix A]. But since $w$ is a tangential derivative then it can not have constant sign along a circumference $\{|x|=r\}$, $r\in(0,1)$, and thus it has to be $w\equiv0$. Therefore, all the tangential derivatives $\partial_t u=x_iu_{x_j}-x_j u_{x_i}$ equal zero, and thus $u$ is radially symmetric.
$H^s$ regularity of the extremal solution in convex domains {#sec-Hs}
===========================================================
In this section we prove Theorem \[S\] (iii). A key tool in this proof is the Pohozaev identity for the fractional Laplacian, recently obtained by the authors in [@RS]. This identity allows us to compare the interior $H^s$ norm of the extremal solution $u^*$ with a boundary term involving $u^*/\delta^s$, where $\delta$ is the distance to $\partial\Omega$. Then, this boundary term can be bounded by using the results of the previous section by the $L^1$ norm of $u^*$, which is finite.
We first prove the boundedness of $u^*/\delta^s$ near the boundary.
\[bdyestimates-quotient\] Let $\Omega$ be a convex domain, $u$ be a bounded solution of , and $\delta(x)=\textrm{dist}(x,\partial\Omega)$. Assume that $$\|u\|_{L^1(\Omega)}\leq c_1$$ for some $c_1>0$. Then, there exists constants $\delta>0$, $c_2$, and $C$ such that $$\|u/\delta^s\|_{L^{\infty}(\Omega_\delta)}\leq C\left(c_2+\|f\|_{L^\infty([0,c_2])}\right),$$ where $\Omega_\delta=\{x\in\Omega\,:\, {\rm dist}(x,\partial\Omega)<\delta\}$. Moreover, the constants $\delta$, $c_2$, and $C$ depend only on $\Omega$ and $c_1$.
The result can be deduced from the boundary regularity results in [@RS-Dir] and Proposition \[bdyestimates\], as follows.
Let $\delta>0$ be given by Proposition \[bdyestimates\], and let $\eta$ be a smooth cutoff function satisfying $\eta\equiv0$ in $\Omega\setminus\Omega_{2\delta/3}$ and $\eta\equiv1$ in $\Omega_{\delta/3}$. Then, $u\eta\in L^\infty(\Omega)$ and $u\eta\equiv0$ in $\R^n\setminus\Omega$. Moreover, we claim that $$\label{eqt}
(-\Delta)^s(u\eta)=f(u)\chi_{\Omega_{\delta/4}}+g\qquad \textrm{in}\ \Omega$$ for some function $g\in L^\infty(\Omega)$, with the estimate $$\label{eqt2}
\|g\|_{L^\infty(\Omega)}\leq C\left(\|u\|_{C^{1+s}(\Omega_{4\delta/5}\setminus\Omega_{\delta/5})}+\|u\|_{L^1(\Omega)}\right).$$
To prove that holds pointwise we argue separately in $\Omega_{\delta/4}$, in $\Omega_{3\delta/4}\setminus\Omega_{\delta/4}$, and in $\Omega\setminus\Omega_{3\delta/4}$, as follows:
- In $\Omega_{\delta/4}$, $g=(-\Delta)^s (u\eta)-(-\Delta)^s u$. Since $u\eta-u$ vanishes in $\Omega_{\delta/3}$ and also outside $\Omega$, $g$ is bounded and satisfies .
- In $\Omega_{3\delta/4}\setminus\Omega_{\delta/4}$, $g=(-\Delta)^s(u\eta)$. Then, using $$\|(-\Delta)^s(u\eta)\|_{L^\infty(\Omega_{3\delta/4}\setminus\Omega_{\delta/4})}\leq C\left(\|u\eta\|_{C^{1+s}(\Omega_{4\delta/5}\setminus\Omega_{\delta/5})}+\|u\eta\|_{L^1(\R^n)}\right)$$ and that $\eta$ is smooth, we find that $g$ is bounded and satisfies .
- In $\Omega\setminus\Omega_{3\delta/4}$, $g=(-\Delta)^s(u\eta)$. Since $u\eta$ vanishes in $\Omega\setminus\Omega_{2\delta/3}$, $g$ is bounded and satisfies .
Now, since $u$ is a solution of , by classical interior estimates we have $$\label{eqt3}
\|u\|_{C^{1+s}(\Omega_{4\delta/5}\setminus\Omega_{\delta/5})}\leq C\left(\|u\|_{L^\infty(\Omega_\delta)}+\|u\|_{L^1(\Omega)}\right);$$ see for instance [@RS-Dir]. Hence, by and Theorem 1.2 in [@RS-Dir], $u\eta/\delta^s\in C^{\alpha}(\overline\Omega)$ for some $\alpha>0$ and $$\|u\eta/\delta^s\|_{C^{\alpha}(\overline\Omega)}\leq C\|f(u)\chi_{\Omega_{\delta/4}}+g\|_{L^\infty(\Omega)}.$$ Thus, $$\begin{split}
\|u/\delta^s\|_{L^\infty(\Omega_{\delta/3})}&
\leq \|u\eta/\delta^s\|_{C^{\alpha}(\overline\Omega)}\leq C\left(\|g\|_{L^\infty(\Omega)}+\|f(u)\|_{L^{\infty}(\Omega_{\delta/4})}\right)\\
&\leq C\left(\|u\|_{L^1(\Omega)}+\|u\|_{L^\infty(\Omega_\delta)}+\|f(u)\|_{L^\infty(\Omega_{\delta/4})}\right).
\end{split}$$ In the last inequality we have used and . Then, the result follows from Proposition \[bdyestimates\].
We can now give the
Recall that $u_\lambda$ minimizes the energy $\mathcal E$ in the set $\{u\in H^s(\R^n)\,:\,0\leq u\leq u_\lambda\}$ (see Step 4 in the proof of Proposition \[existence\] in Section \[sec-exist\]). Hence, $$\label{p1}
\|u_\lambda\|_{\accentset{\circ}{H}^s}^2-\int_\Omega \lambda F(u_\lambda)=\mathcal E(u_\lambda)\leq \mathcal E(0)=0.$$ Now, the Pohozaev identity for the fractional Laplacian can be written as $$\label{p2}s\|u_\lambda\|_{\accentset{\circ}{H}^s}^2-n\mathcal E(u_\lambda)=\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\left(\frac{u_\lambda}{\delta^s}\right)^2(x\cdot \nu)d\sigma,$$ see [@RS page 2]. Therefore, it follows from and that $$\|u_\lambda\|_{\accentset{\circ}{H}^s}^2\leq \frac{\Gamma(1+s)^2}{2s}\int_{\partial\Omega}\left(\frac{u_\lambda}{\delta^s}\right)^2(x\cdot \nu)d\sigma.$$
Now, by Proposition \[bdyestimates-quotient\], we have that $$\int_{\partial\Omega}\left(\frac{u_\lambda}{\delta^s}\right)^2(x\cdot \nu)d\sigma\leq C$$ for some constant $C$ that depends only on $\Omega$ and $\|u_\lambda\|_{L^1(\Omega)}$. Thus, $\|u_\lambda\|_{\accentset{\circ}{H}^s}\leq C$, and since $u^*\in L^1(\Omega)$, letting $\lambda\uparrow \lambda^*$ we find $$\|u^*\|_{\accentset{\circ}{H}^s}<\infty,$$ as desired.
$L^p$ and $C^\beta$ estimates for the linear Dirichlet problem {#sec3}
==============================================================
The aim of this section is to prove Propositions \[th2\] and \[prop:u-is-Cbeta\]. We prove first Proposition \[th2\].
\(i) It is clear that we can assume $\|g\|_{L^1(\Omega)}=1$.
Consider the solution $v$ of $$(-\Delta)^s v=|g|\ \ {\rm in}\ \R^n$$ given by the Riesz potential $v=(-\Delta)^{-s}|g|$. Here, $g$ is extended by 0 outside $\Omega$.
Since $v\geq0$ in $\R^n\setminus\Omega$, by the maximum principle we have that $|u|\leq v$ in $\Omega$. Then, it follows from Theorem \[th1\] that $$\|u\|_{L^q_{\textrm{weak}}(\Omega)}\leq C,\quad\textrm{where}\quad q=\frac{n}{n-2s},$$ and hence we find that $$\|u\|_{L^r(\Omega)}\leq C\quad\textrm{for all}\quad r<\frac{n}{n-2s}$$ for some constant that depends only on $n$, $s$, and $|\Omega|$.
\(ii) The proof is analogous to the one of part (i). In this case, the constant does not depend on the domain $\Omega$.
\(iii) As before, we assume $\|g\|_{L^p(\Omega)}=1$. Write $u=\tilde v+w$, where $\tilde v$ and $w$ are given by $$\label{v}
\tilde v=(-\Delta)^{-s} g\ \ {\rm in}\ \R^n,$$ and $$\label{w}
\left\{ \begin{array}{rcll} (-\Delta)^s w &=&0&\textrm{in }\Omega \\
w&=&\tilde v&\textrm{in }\mathbb R^n\backslash\Omega.\end{array}\right.$$ Then, from and Theorem \[th1\] we deduce that $$\label{frt}
[\tilde v]_{C^\alpha(\R^n)}\leq C,\quad\textrm{where}\quad \alpha=2s-\frac{n}{p}.$$ Moreover, since the domain $\Omega$ is bounded, then $g$ has compact support and hence $\tilde v$ decays at infinity. Thus, we find $$\label{ineqv}
\|\tilde v\|_{C^\alpha(\R^n)}\leq C$$ for some constant $C$ that depends only on $n$, $s$, $p$, and $\Omega$.
Now, we apply Proposition \[prop:u-is-Cbeta\] to equation . We find $$\label{ineqw}
\|w\|_{C^\beta(\R^n)}\leq C\|\tilde v\|_{C^\alpha(\R^n)},$$ where $\beta=\min\{\alpha,s\}$. Thus, combining , and the result follows.
Note that we have only used Proposition \[prop:u-is-Cbeta\] to obtain the $C^\beta$ estimate in part (iii). If one only needs an $L^\infty$ estimate instead of the $C^\beta$ one, Proposition \[prop:u-is-Cbeta\] is not needed, since the $L^\infty$ bound follows from the maximum principle.
As said in the introduction, the $L^p$ to $W^{2s,p}$ estimates for the fractional Laplace equation, in which $-\Delta$ is replaced by the fractional Laplacian $(-\Delta)^s$, are not true for all $p$, even when $\Omega=\R^n$. This is illustrated in the following two remarks.
Recall the definition of the fractional Sobolev space $W^{\sigma,p}(\Omega)$ which, for $\sigma\in(0,1)$, consists of all functions $u\in L^p(\Omega)$ such that $$\|u\|_{W^{\sigma,p}(\Omega)}=\|u\|_{L^p(\Omega)}+\left(\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+p\sigma}}\,dx\,dy\right)^{\frac1p}$$ is finite; see for example [@DPV] for more information on these spaces.
\[rem1\] Let $s\in(0,1)$. Assume that $u$ and $g$ belong to $L^p(\R^n)$, with $1<p<\infty$, and that $$(-\Delta)^su=g\ \ \textrm{in}\ \R^n.$$
- If $p\geq2$, then $u\in W^{2s,p}(\R^n)$.
- If $p<2$ and $2s\neq1$ then $u$ may not belong to $W^{2s,p}(\R^n)$. Instead, $u\in B^{2s}_{p,2}(\R^n)$, where $B^\sigma_{p,q}$ is the Besov space of order $\sigma$ and parameters $p$ and $q$.
For more details see the books of Stein [@Stein] and Triebel [@T2].
By the preceding remark we see that the $L^p$ to $W^{2s,p}$ estimate does not hold in $\R^n$ whenever $p<2$ and $s\neq \frac12$. The following remark shows that in bounded domains $\Omega$ this estimate do not hold even for $p\geq2$.
\[rem11\] Let us consider the solution of $(-\Delta)^su=g$ in $\Omega$, $u\equiv0$ in $\R^n\setminus\Omega$. When $\Omega=B_1$ and $g\equiv1$, the solution to this problem is $$u_0(x)=\left(1-|x|^2\right)^s\chi_{B_1}(x);$$ see [@G]. For $p$ large enough one can see that $u_0$ does not belong to $W^{2s,p}(B_1)$, while $g\equiv1$ belongs to $L^p(B_1)$ for all $p$. For example, when $s=\frac12$ by computing $|\nabla u_0|$ we see that $u_0$ does not belong to $W^{1,p}(B_1)$ for $p\geq2$.
We next prove Proposition \[prop:u-is-Cbeta\]. For it, we will proceed similarly to the $C^s$ estimates obtained in [@RS-Dir Section 2] for the Dirichlet problem for the fractional Laplacian with $L^\infty$ data.
The first step is the following:
\[bound-u\] Let $\Omega$ be a bounded domain satisfying the exterior ball condition, $s\in(0,1)$, $h$ be a $C^\alpha(\R^n\setminus\Omega)$ function for some $\alpha>0$, and $u$ be the solution of . Then $$|u(x)-u(x_0)|\leq C\|h\|_{C^\alpha(\R^n\setminus\Omega)}\delta(x)^\beta\ \ {\rm in}\ \Omega,$$ where $x_0$ is the nearest point to $x$ on $\partial\Omega$, $\beta=\min\{s,\alpha\}$, and $\delta(x)={\rm dist}(x,\partial\Omega)$. The constant $C$ depends only on $n$, $s$, and $\alpha$.
Lemma \[bound-u\] will be proved using the following supersolution. Next lemma (and its proof) is very similar to Lemma 2.6 in [@RS-Dir].
\[supersolution\] Let $s\in(0,1)$. Then, there exist constants $\epsilon$, $c_1$, and $C_2$, and a continuous radial function $\varphi$ satisfying $$\label{supers}
\begin{cases}
(-\Delta)^s \varphi \ge 0 &\mbox{in }B_2\setminus B_1\\
\varphi \equiv 0 \quad &\mbox{in }B_1 \\
c_1(|x|-1)^s\le\varphi \le C_2(|x|-1)^s &\mbox{in }\R^n\setminus B_1.
\end{cases}$$ The constants $c_1$ and $C_2$ depend only on $n$, $s$, and $\beta$.
We follow the proof of Lemma 2.6 in [@RS-Dir]. Consider the function $$u_0(x)=(1-|x|^2)^s_+.$$ It is a classical result (see [@G]) that this function satisfies $$(-\Delta)^su_0= \kappa_{n,s}\ \ {\rm in}\ B_1$$ for some positive constant $\kappa_{n,s}$.
Thus, the fractional Kelvin transform of $u_0$, that we denote by $u_0^*$, satisfies $$(-\Delta)^s u_0^*(x)=|x|^{-2s-n}(-\Delta)^su_0\left(\frac{x}{|x|^2}\right)\geq c_0\ \ {\rm in}\ B_2\setminus B_1.$$ Recall that the Kelvin transform $u_0^*$ of $u_0$ is defined by $$u_0^*(x)=|x|^{2s-n}u_0\left(\frac{x}{|x|^2}\right).$$ Then, it is clear that $$a_1(|x|-1)^s\leq u_0^*(x)\leq A_2(|x|-1)^s\ \ {\rm in}\ B_{2}\setminus B_1,$$ while $u_0^*$ is bounded at infinity.
Let us consider now a smooth function $\eta$ satisfying $\eta\equiv0$ in $B_3$ and $$A_1(|x|-1)^s\leq \eta\leq A_2(|x|-1)^s\ \ {\rm in}\ \R^n\setminus B_4.$$ Observe that $(-\Delta)^s\eta$ is bounded in $B_2$, since $\eta(x)(1+|x|)^{-n-2s}\in L^1$. Then, the function $$\varphi=Cu_0^*+\eta,$$ for some big constant $C>0$, satisfies $$\begin{cases}
(-\Delta)^s \varphi \ge 1 &\mbox{in }B_{2}\setminus B_1\\
\varphi \equiv 0 \quad &\mbox{in }B_1 \\
c_1(|x|-1)^s\le\varphi \le C_2(|x|-1)^s &\mbox{in }\R^n\setminus B_1.
\end{cases}$$ Indeed, it is clear that $\varphi\equiv0$ in $B_1$. Moreover, taking $C$ big enough it is clear that we have that $(-\Delta)^s \varphi \ge 1$. In addition, the condition $c_1(|x|-1)^s\le\varphi \le C_2(|x|-1)^s$ is satisfied by construction. Thus, $\varphi$ satisfies , and the proof is finished.
Once we have constructed the supersolution, we can give the
First, we can assume that $\|h\|_{C^\alpha(\R^n\setminus\Omega)}=1$. Then, by the maximum principle we have that $\|u\|_{L^\infty(\R^n)}=\|h\|_{L^\infty(\R^n)}\leq1$. We can also assume that $\alpha\leq s$, since $$\|h\|_{C^s(\R^n)}\leq C\|h\|_{C^\alpha(\R^n\setminus\Omega)}\quad\textrm{whenever}\ s<\alpha.$$
Let $x_0\in \partial\Omega$ and $R>0$ be small enough. Let $B_R$ be a ball of radius $R$, exterior to $\Omega$, and touching $\partial\Omega$ at $x_0$. Let us see that $|u(x)-u(x_0)|$ is bounded by $CR^\beta$ in $\Omega\cap B_{2R}$.
By Lemma \[supersolution\], we find that there exist constants $c_1$ and $C_2$, and a radial continuous function $\varphi$ satisfying $$\label{supers}
\begin{cases}
(-\Delta)^s \varphi \ge 0 &\mbox{in }B_2\setminus B_1\\
\varphi \equiv 0 \quad &\mbox{in }B_1 \\
c_1(|x|-1)^s\le\varphi \le C_2(|x|-1)^s &\mbox{in }\R^n\setminus B_1.
\end{cases}$$
![\[figura2\] ](dibuix_fractional_extremal.pdf)
Let $x_1$ be the center of the ball $B_R$. Since $\|h\|_{C^\alpha(\R^n\setminus\Omega)}=1$, it is clear that the function $$\varphi_R(x)=h(x_0)+3R^\alpha+C_3R^s\varphi\left(\frac{x-x_1}{R}\right),$$ with $C_3$ big enough, satisfies $$\label{supersR}
\begin{cases}
(-\Delta)^s \varphi_R \ge 0 &\mbox{in }B_{2R}\setminus B_R\\
\varphi_R \equiv h(x_0)+3R^\alpha \quad &\mbox{in }B_R \\
h(x_0)+|x-x_0|^\alpha\leq \varphi_R&\mbox{in }\R^n\setminus B_{2R}\\
\varphi_R \le h(x_0)+C_0R^\alpha &\mbox{in }B_{2R}\setminus B_R.
\end{cases}$$ Here we have used that $\alpha\leq s$.
Then, since $$(-\Delta)^s u\equiv0\leq (-\Delta)^s \varphi_R\quad \textrm{in}\ \Omega\cap B_{2R},$$ $$h\leq h(x_0)+3R^\alpha\equiv\varphi_R\quad \textrm{in}\ B_{2R}\setminus\Omega,$$ and $$h(x)\leq h(x_0)+|x-x_0|^\alpha\leq\varphi_R\quad \textrm{in}\ \R^n\setminus B_{2R},$$ it follows from the comparison principle that $$u\leq \varphi_R\ \ {\rm in}\ \Omega\cap B_{2R}.$$ Therefore, since $\varphi_R \le h(x_0)+C_0R^\alpha$ in $B_{2R}\setminus B_R$, $$\label{eqhatu}
u(x)- h(x_0)\leq C_0R^\alpha\ \ {\rm in}\ \Omega\cap B_{2R}.$$
Moreover, since this can be done for each $x_0$ on $\partial\Omega$, $h(x_0)=u(x_0)$, and we have $\|u\|_{L^\infty(\Omega)}\leq 1$, we find that $$\label{equ}
u(x)-u(x_0)\leq C\delta^\beta\ \ {\rm in}\ \Omega,$$ where $x_0$ is the projection on $\partial\Omega$ of $x$.
Repeating the same argument with $u$ and $h$ replaced by $-u$ and $-h$, we obtain the same bound for $h(x_0)-u(x)$, and thus the lemma follows.
The following result will be used to obtain $C^\beta$ estimates for $u$ inside $\Omega$. For a proof of this lemma see for example Corollary 2.4 in [@RS-Dir].
\[int-est-brick2\] Let $s\in(0,1)$, and let $w$ be a solution of $(-\Delta)^s w = 0$ in $B_2$. Then, for every $\gamma\in(0,2s)$ $$\|w\|_{C^{\gamma}(\overline{B_{1/2}})} \le C\biggl(\|(1+|x|)^{-n-2s}w(x)\|_{L^1(\R^n)} + \|w\|_{L^\infty(B_2)}\biggr),$$ where the constant $C$ depends only on $n$, $s$, and $\gamma$.
Now, we use Lemmas \[bound-u\] and \[int-est-brick2\] to obtain interior $C^\beta$ estimates for the solution of .
\[Cbetabounds\] Let $\Omega$ be a bounded domain satisfying the exterior ball condition, $h\in C^\alpha(\R^n\setminus\Omega)$ for some $\alpha>0$, and $u$ be the solution of . Then, for all $x\in \Omega$ we have the following estimate in $B_R(x)=B_{\delta(x)/2}(x)$ $$\label{seminorm-estimate-u}
\|u\|_{C^\beta(\overline{B_{R}(x)})}\le C\|h\|_{C^\alpha(\R^n\setminus\Omega)},$$ where $\beta=\min\{\alpha,s\}$ and $C$ is a constant depending only on $\Omega$, $s$, and $\alpha$.
Note that $B_R(x)\subset B_{2R}(x)\subset \Omega$. Let $\tilde u(y)= u(x+Ry)-u(x)$. We have that $$\label{Rs1}
(-\Delta)^s \tilde u(y) = 0\quad \mbox{in } B_1\,.$$ Moreover, using Lemma \[bound-u\] we obtain $$\label{Rs2}
\|\tilde u\|_{L^\infty(B_1)}\le C\|h\|_{C^\alpha(\R^n\setminus\Omega)} R^\beta.$$ Furthermore, observing that $|\tilde u(y)|\le C\|h\|_{C^\alpha(\R^n\setminus\Omega)} R^\beta(1+|y|^\beta)$ in all of $\R^n$, we find $$\label{Rs3}
\|(1+|y|)^{-n-2s}\tilde u(y)\|_{L^1(\R^n)}\le C \|h\|_{C^\alpha(\R^n\setminus\Omega)} R^\beta,$$ with $C$ depending only on $\Omega$, $s$, and $\alpha$.
Now, using Lemma \[int-est-brick2\] with $\gamma=\beta$, and taking into account , , and , we deduce $$\|\tilde u\|_{C^{\beta}\left(\overline{B_{1/4}}\right)} \le C \|h\|_{C^\alpha(\R^n\setminus\Omega)}R^\beta,$$ where $C=C(\Omega,s,\beta)$.
Finally, we observe that $$[u]_{C^\beta\left(\overline{B_{R/4}(x)}\right)}=R^{-\beta}[\tilde u]_{C^\beta\left(\overline{B_{1/4}}\right)}.$$ Hence, by an standard covering argument, we find the estimate for the $C^\beta$ norm of $u$ in $\overline {B_{R}(x)}$.
Now, Proposition \[prop:u-is-Cbeta\] follows immediately from Lemma \[Cbetabounds\], as in Proposition 1.1 in [@RS-Dir].
This proof is completely analogous to the proof of Proposition 1.1 in [@RS-Dir]. One only have to replace the $s$ in that proof by $\beta$, and use the estimate from the present Lemma \[Cbetabounds\] instead of the one from [@RS-Dir Lemma 2.9].
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Xavier Cabré for his guidance and useful discussions on the topic of this paper.
We also thank Louis Dupaigne and Manel Sanchón for interesting discussions on the topic of this paper.
[00]{}
M. Birkner, J. A. López-Mimbela, A. Wakolbinger, *Comparison results and steady states for the Fujita equation with fractional Laplacian*, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 83-97.
R. M. Blumenthal, R. K. Getoor, D. B. Ray, *On the distribution of first hits for the symmetric stable processes*, Trans. Amer. Math. Soc. 99 (1961), 540-554.
H. Brezis, *Symmetry in nonlinear PDE’s*, Differential equations: La Pietra 1996 (Florence), 1-12, Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Providence, RI, 1999.
H. Brezis, J. L. Vázquez, *Blow-up solutions of some nonlinear elliptic problems*, Rev. Mat. Univ. Complut. Madrid 10 (1997), 443-469.
H. Brezis, T. Cazenave, Y. Martel, A. Ramiandrisoa, *Blow up for $u_t-\Delta u=g(u)$ revisited*, Advences in PDE 1 (1996), 73-90.
X. Cabré, *Regularity of minimizers of semilinear elliptic problems up to dimension four*, Comm. Pure Appl. Math. 63 (2010), 1362-1380.
X. Cabré, A. Capella, *Regularity of radial minimizers and extremal solutions of semi-linear elliptic equations*, J. Funct. Anal. 238 (2006), 709-733.
X. Cabré, M. Sanchón, *Geometric-type Hardy-Sobolev inequalities and applications to regularity of minimizers*, J. Funct. Anal. 264 (2013), 303-325.
L. Caffarelli, L. Silvestre, *An extension problem related to the fractional Laplacian*, Comm. Partial Differential Equations 32 (2007), 1245-1260.
A. Capella, J. Davila, L. Dupaigne, Y. Sire, *Regularity of radial extremal solutions for some non local semilinear equations*, Comm. Partial Differential Equations 36 (2011), 1353-1384.
W. Chen, C. Li, B. Ou, *Classification of solutions to an integral equation*, Comm. Pure Appl. Math. 59 (2006), 330-343.
M. G. Crandall, P. H. Rabinowitz, *Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems*, Arch. Rational Mech. Anal. 58 (1975), 207-218.
J. Dávila, L. Dupaigne, I. Guerra, M. Montenegro, *Stable solutions for the bilaplacian with exponential nonlinearity*, SIAM J. Math. Anal. 39 (2007), 565-592.
D. de Figueiredo, P.-L. Lions, R. D. Nussbaum, *A priori estimates and existence of positive solutions of semilinear elliptic equations*, J. Math. Pures Appl. 61 (1982), 41-63.
E. Di Nezza, G. Palatucci, E. Valdinoci, *Hitchhiker’s guide to the fractional Sobolev spaces*, Bull. Sci. math. 136 (2012), 521-573.
L. Dupaigne, *Stable Solutions to Elliptic Partial Differential Equations*, Chapman $\&$ Hall, 2011.
M. M. Fall, T. Weth, *Nonexistence results for a class of fractional elliptic boundary value problems*, J. Funct. Anal. 263 (2012), 2205-2227.
R. Frank, E. H. Lieb, R. Seiringer, *Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators*, J. Amer. Math. Soc. 21 (2008), 925-950.
J. García-Cuerva, A.E. Gatto, *Boundedness properties of fractional integral operators associated to non-doubling measures*, Studia Mathematica 162 (2004), 245-261.
R. K. Getoor, *First passage times for symmetric stable processes in space*, Trans. Amer. Math. Soc. 101 (1961), 75-90.
B. Gidas, W. M. Ni, L. Nirenberg, *Symmetry and related properties via the maximum principle*, Comm. Math. Phys. 68 (1979), 209-243.
S. Jarohs, T. Weth, *Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations*, arXiv:1301.1811v2.
D. D. Joseph, T. S. Lundgren, *Quasilinear Dirichlet problems driven by positive sources*, Arch. Rational Mech. Anal. 49 (1973), 241-269.
I. W. Herbst, *Spectral theory of the operator $(p^2+m^2)^{1/2} - Ze^2/r$*. Comm. Math. Phys. 53 (1977), 285-294.
F. Mignot, J. P. Puel, *Solution radiale singulière de $-\Delta u=e^u$*, C. R. Acad. Sci. Paris 307 (1988), 379-382.
G. Nedev, *Regularity of the extremal solution of semilinear elliptic equations*, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 997-1002.
G. Nedev, *Extremal solutions of semilinear elliptic equations*, unpublished preprint, 2001.
X. Ros-Oton, J. Serra, *The Dirichlet problem for the fractional Laplacian: regularity up to the boundary*, J. Math. Pures Appl., to appear.
X. Ros-Oton, J. Serra, *The Pohozaev identity for the fractional laplacian*, arXiv:1207.5986.
M. Sanchón, *Boundedness of the extremal solution of some $p$-Laplacian problems*, Nonlinear analysis 67 (2007), 281-294.
R. Servadei, E. Valdinoci, *Variational methods for non-local operators of elliptic type*, Discrete Contin. Dyn. Syst. 33 (2013), 2105-2137.
R. Servadei, E. Valdinoci, *A Brezis-Nirenberg result for non-local critical equations in low dimension*, to appear in Commun. Pure Appl. Anal.
R. Servadei, E. Valdinoci, *Weak and viscosity solutions of the fractional Laplace equation*, to appear in Publ. Mat.
E. Stein, *Singular Integrals And Differentiability Properties Of Functions*, Princeton Mathematical Series No. 30, 1970.
H. Triebel, *Theory Of Function Spaces*, Monographs in Mathematics 78, Birkhauser, 1983.
S. Villegas, *Boundedness of the extremal solutions in dimension 4*, Adv. Math. 235 (2013), 126-133.
[^1]: The authors were supported by grants MINECO MTM2011-27739-C04-01 and GENCAT 2009SGR-345.
|
---
abstract: 'In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the “roughness” and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds to the Nash-Moser principle combined with a new concept of “higher order averaging operators along highly fractal stochastic curves”. This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.'
address:
- 'Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053 Blindern, 0316 Oslo, Norway.'
- 'Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053 Blindern, 0316 Oslo, Norway.'
- 'Department of Mathematics, University of Oslo, Moltke Moes vei 35, P.O. Box 1053 Blindern, 0316 Oslo, Norway.'
author:
- Oussama Amine
- David Baños
- Frank Proske
title: '$C^{\infty }$-Regularization By Noise of Singular ODE’s'
---
Introduction
============
Consider the ordinary differential equation (ODE)$$\frac{d}{dt}X_{t}^{x}=b(t,X_{t}^{x}),\quad X_{0}=x,\quad 0\leq t\leq T
\label{ODE}$$for a vector field $b:[0,T]\times \mathbb{R}^{d}\longrightarrow \mathbb{R}^{d}$.
It is well-known that the ODE admits the existence of a unique solution $X_{t}$, $0\leq t\leq T$, if $b$ is a Lipschitz function of linear growth, uniformly in time. Further, if in addition $b\in C^{k}([0,T]\times
\mathbb{R}^{d};\mathbb{R}^{d})$, $k\geq 1$, then the flow associated with the ODE inherits the regularity from the vector field, that is $$\begin{aligned}
(x\longmapsto X_{t}^{x})\in C^{k}(\mathbb{R}^{d};\mathbb{R}^{d}).\end{aligned}$$ However, well-posedness of the ODE in the sense of existence, uniqueness and the regularity of solutions or flow may fail, if the driving vector field $b$ lacks regularity, that is if $b$ e.g. is not Lipschitzian or discontinuous.
In this article we aim at studying the restoration of well-posedness of the ODE in the above sense by perturbing the equation via a specific noise process $\mathbb{B}_{t}$, $0\leq t\leq T$, that is we are interested to analyze strong solutions to the following stochastic differential equation (SDE)$$\label{SDE}
X_{t}^{x}=x+\int_{0}^{t}b(t,X_{s}^{x})ds+\mathbb{B}_{t},\quad 0\leq t\leq T,$$where the driving process $\mathbb{B}_{t},$ $0\leq t\leq T$ is a stationary Gaussian process with non-Hölder continuous paths given by $$\label{B}
\mathbb{B}_{t}=\sum_{n\geq 1}\lambda _{n}B_{t}^{H_{n},n}.$$Here $B_{\cdot }^{H_{n},n}$, $n\geq 1$ are independent fractional Brownian motions in $\mathbb{R}^{d}$ with Hurst parameters $H_{n}\in (0,\frac{1}{2}),n\geq 1$ such that $$H_{n}\searrow 0$$for $n\longrightarrow \infty $. Further, $\sum_{n\geq 1}\left\vert \lambda
_{n}\right\vert <\infty $ for $\lambda _{n}\in \mathbb{R},n\geq 1$.
In fact, on the other hand, the SDE (\[SDE\]) can be also naturally recast for $Y_{t}^{x}:=X_{t}^{x}-\mathbb{B}_{t}$ in terms of the ODE$$Y_{t}^{x}=x+\int_{0}^{t}b^{\ast }(t,Y_{s}^{x})ds, \label{RODE}$$where $b^{\ast }(t,y):=b(t,y+\mathbb{B}_{t})$ is a “randomization” of the input vector field $b$.
We recall (for $d=1$) that a fractional Brownian motion $B_{\cdot }^{H}$ with Hurst parameter $H\in (0,1)$ is a centered Gaussian process on some probability space with a covariance structure $R_{H}(t,s)$ given by $$R_{H}(t,s)=E[B_{t}^{H}B_{s}^{H}]=\frac{1}{2}(s^{2H}+t^{2H}+\left\vert
t-s\right\vert ^{2H}),\quad t,s\geq 0.$$We mention that $B_{\cdot }^{H}$ has a version with Hölder continuous paths with exponent strictly smaller than $H$. The fractional Brownian motion coincides with the Brownian motion for $H=\frac{1}{2}$, but is neither a semimartingale nor a Markov process, if $H\neq \frac{1}{2}$. We also recall here that a fractional Brownian motion $B_{\cdot }^{H}$ has a representation in terms of a stochastic integral as$$B_{t}^{H}=\int_{0}^{t}K_{H}(t,u)dW_{u}, \label{Rep}$$where $W_{\cdot }$ is a Wiener process and where $K_{H}(t,\cdot )$ is an integrable kernel. See e.g. [@Nua10] and the references therein for more information about fractional Brownian motion.
Using Malliavin calculus combined with integration-by-parts techniques based on Fourier analysis, we want to show in this paper the existence of a unique global strong solution $X_{\cdot }^{x}$ to with a stochastic flow which is *smooth*, that is $$\label{Smooth}
(x\longmapsto X_{t}^{x})\in C^{\infty }(\mathbb{R}^{d};\mathbb{R}^{d})\quad \mbox{a.e. for all}\quad t,$$when the driving vector field $b$ is *singular*, that is more precisely, when $$b\in \mathcal{L}_{2,p}^{q}:=L^{q}([0,T];L^{p}(\mathbb{R}^{d};\mathbb{R}^{d}))\cap L^{1}({\mathbb R}^{d};L^{\infty }([0,T];\mathbb{R}^{d}))$$for $p,q\in (2,\infty ]$.
We think that the latter result is rather surprising since it seems to contradict the paradigm in the theory of (stochastic) dynamical systems that solutions to ODE’s or SDE’s inherit their regularity from the driving vector fields.
Further, we expect that the regularizing effect of the noise in will also pay off dividends in PDE theory and in the study of dynamical systems with respect to singular SDE’s:
For example, if $X_{\cdot }^{x}$ is a solution to the ODE on $[0,\infty )$, then $X:[0,\infty )\times \mathbb{R}^{d}\longrightarrow
\mathbb{R}^{d}$ may have the interpretation of a flow of a fluid with respect to the velocity field $u=b$ of an incompressible inviscid fluid, which is described by a solution to an incompressible Euler equation$$\begin{aligned}
\label{Euler}
u_{t}+(Du)u+\triangledown P=0,\text{ }\triangledown \cdot u=0,\end{aligned}$$ where $P:[0,\infty )\times \mathbb{R}^{d}\longrightarrow \mathbb{R}^{d}$ is the pressure field.
Since solutions to may be singular, a deeper analysis of the regularity of such solutions also necessitates the study of ODE’s with irregular vector fields. See e.g. Di Perna, Lions [@DPL89] or Ambrosio [@ambrosio.04] in connection with the construction of (generalized) flows associated with singular ODE’s.
In the context of stochastic regularization of the ODE in the sense of , however, the obtained results in this article naturally give rise to the question, whether the constructed smooth stochastic flow in may be used for the study of regular solutions of a stochastic version of the Euler equation .
Regarding applications to the theory of stochastic dynamical systems one may study the behaviour of orbits with respect to solutions to SDE’s with singular vector fields at sections on a $2$-dimensional sphere (Theorem of Poincaré-Bendixson). Another application may pertain to stability results in the sense of a modified version of the Theorem of Kupka-Smale [@Smale.63]. We mention that well-posedness in the sense of existence and uniqueness of strong solutions to via regularization of noise was first found by Zvonkin [@Zvon74] in the early 1970ties in the one-dimensional case for a driving process given by the Brownian motion, when the vector field $b$ is merely bounded and measurable. Subsequently the latter result, which can be considered a milestone in SDE theory, was extended to the multidimensional case by Veretennikov [@Ver79].
Other more recent results on this topic in the case of Brownian motion were e.g. obtained by Krylov, Röckner [@KR05], where the authors established existence and uniqueness of strong solutions under some integrability conditions on $b$. See also the works of Gyöngy, Krylov [@GyK96] and Gyöngy, Martinez [@GyM01]. As for a generalization of the result of Zvonkin [@Zvon74] to the case of stochastic evolution equations on a Hilbert space, we also mention the striking paper of Da Prato, Flandoli, Priola, Röckner [@DPFPR13], who constructed strong solutions for bounded and measurable drift coefficients by employing solutions of infinite-dimensional Kolmogorov equations in connection with a technique known as the “Itô-Tanaka-Zvonkin trick”.
The common general approach used by the above mentioned authors for the construction of strong solutions is based on the so-called Yamada-Watanabe principle [@YW71]: The authors prove the existence of a weak solution (by means of e.g. Skorokhod’s or Girsanov’s theorem) and combine it with the property of pathwise uniqueness of solutions, which is shown by using solutions to (parabolic) PDE’s, to eventually obtain strong uniqueness. As for this approach in the case of certain classes of Lévy processes the reader may consult Priola [@Priola12] or Zhang [@Zhang13] and the references therein.
Let us comment on here that the methods of the above authors, which are essentially limited to equations with Markovian noise, cannot be directly used in connection with our SDE . The reason for this is that the initial noise in is not a Markov process. Furthermore, it is even not a semimartingale due to the properties of a fractional Brownian motion.
In addition, we point out that our approach is diametrically opposed to the Yamada-Watanabe principle: We first construct a strong solution to by using Mallliavin calculus. Then we verify uniqueness in law of solutions, which enables us to establish strong uniqueness, that is we use the following principle:
$$\begin{aligned}
\fbox{Strong existence}+\text{\fbox{Uniqueness in law}}\Rightarrow \text{
\fbox{Strong uniqueness}.}\end{aligned}$$
Finally, let us also mention some results in the literature on the existence and uniqueness of strong solutions of singular SDE’s driven by a non-Markovian noise in the case of fractional Brownian motion:
The first results in this direction were obtained by Nualart, Ouknine [nualart.ouknine.02,nualart.ouknine.03]{} for one-dimensional SDE’s with additive noise. For example, using the comparison theorem, the authors in [@nualart.ouknine.02] are able to derive unique strong solutions to such equations for locally unbounded drift coefficients and Hurst parameters $H<\frac{1}{2}$.
More recently, Catellier, Gubinelli [@CG] developed a construction method for strong solutions of multi-dimensional singular SDE’s with additive fractional noise and $H\in (0,1)$ for vector fields $b$ in the Besov-Hölder space $B_{\infty ,\infty }^{\alpha +1},\alpha \in \mathbb{R}
$. Here the solutions obtained are even *path-by-path* in the sense of Davie [@Da07] and the construction technique of the authors rely on the Leray-Schauder-Tychonoff fixed point theorem and a comparison principle based on an average translation operator.
Another recent result which is based on Malliavin techniques very similar to our paper can be found in Baños, Nilssen, Proske [@BNP.17]. Here the authors proved the existence of unique strong solutions for coefficients $$\begin{aligned}
b\in L_{\infty ,\infty }^{1,\infty }:=L^{1}({\mathbb R}^d;L^{\infty }([0,T];\mathbb{R}^{d}))\cap L^{\infty }({\mathbb R}^d;L^{\infty }([0,T];\mathbb{R}^{d}))\end{aligned}$$ for sufficiently small $H\in (0,\frac{1}{2})$.
The approach in [@BNP.17] is different from the above mentioned ones and the results for vector fields $b\in L_{\infty ,\infty }^{1,\infty }$ are not in the scope of the techniques in [@CG]. See also [@BOPP.17] in the case fractional noise driven SDE’s with distributional drift.
Let us now turn to results in the literature on the well-posedness of singular SDE’s under the aspect of the regularity of stochastic flows:
If we assume that the vector field $b$ in the ODE is not smooth, but merely require that $b\in W^{1,p}$ and $\triangledown \cdot b\in
L^{\infty },$ then it was shown in [@DPL89] the existence of a unique generalized flow $X$ associated with the ODE . See also [ambrosio.04]{} for a generalization of the latter result to the case of vector fields of bounded variation.
On the other hand, if $b$ in ODE is less regular than required [@DPL89; @ambrosio.04], then a flow may even not exist in a generalized sense.
However, the situation changes, if we regularize the ODE by an (additive) noise:
For example, if the driving noise in the SDE is chosen to be a Brownian noise, or more precisely if we consider the SDE $$\begin{aligned}
dX_{t}=u(t,X_{t})dt+dB_{t},\quad s,t\geq 0,\quad X_{s}=x\in \mathbb{R}^{d}\end{aligned}$$ with the associated stochastic flow $\varphi _{s,t}:\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$, the authors in [@MNP14] could prove for merely bounded and measurable vector fields $b$ a regularizing effect of the Brownian motion on the ODE that is they could show that $\varphi
_{s,t}$ is a stochastic flow of Sobolev diffeomorphisms with $$\begin{aligned}
\varphi _{s,t},\varphi _{s,t}^{-1}\in L^{2}(\Omega ;W^{1,p}(\mathbb{R}^{d};w))\text{ }\end{aligned}$$ for all $s,t$ and $p\in (1,\infty)$, where $W^{1,p}(\mathbb{R}^{d};w)$ is a weighted Sobolev space with weight function $w:\mathbb{R}^{d}\rightarrow
\lbrack 0,\infty )$. Further, as an application of the latter result, which rests on techniques similar to those used in this paper, the authors also study solutions of a singular stochastic transport equation with multiplicative noise of Stratonovich type.
Another work in this direction with applications to Navier-Stokes equations, which invokes similar techniques as introduced in [@MNP14], deals with globally integrable $u\in L^{r,q}$ for $r/d+2/q<1$ ($r$ stands here for the spatial variable and $q$ for the temporal variable). In this context, we also mention the paper [@FedFlan.13], where the authors present an alternative method to the above mentioned ones based on solutions to backward Kolmogorov equations. See also [@FedFlan10]. We also refer to [@Priola12] and [@Zhang13] in the case of $\alpha$-stable processes.
On the other hand if we consider a noise in the SDE , which is rougher than Brownian motion with respect to the path properties and given by fractional Brownian motion for small Hurst parameters, one can even observe a stronger regularization by noise effect on the ODE : For example, using Malliavin techniques very similar to those in our paper, the authors in [@BNP.17] are able to show for vector fields $b\in
L_{\infty ,\infty }^{1,\infty }$ the existence of higher order Fréchet differentiable stochastic flows $$\begin{aligned}
(x\mapsto X_{t}^{x})\in C^{k}(\mathbb{R}^{d})\quad \text{a.e. for all} \quad
t,\end{aligned}$$provided $H=H(k)$ is sufficient small.
Another work in connection with fractional Brownian motion is that of Catellier, Gubinelli [@CG], where the authors under certain conditions obtain Lipschitz continuity of the associated stochastic flow for drift coefficients $b$ in the Besov-Hölder space $B_{\infty,\infty }^{\alpha
+1},\alpha \in \mathbb{R}$.
We again stress that our approach for the construction of strong solutions of singular SDE’s in connection with smooth stochastic flows is not based on the Yamada-Watanabe principle or techniques from Markov or semimartingale theory as commonly used in the literature. In fact, our construction method has its roots in a series of papers [@MMNPZ10], [MBP06]{}, [@MBP10], [@BNP.17]. See also [@HaaPros.14] in the case of SDE’s driven by Lévy processes, [@FNP.13], [@MNP14] regarding the study of singular stochastic partial differential equations or [BOPP.17]{}, [@BHP.17] in the case of functional SDE’s.
The method we aim at employing in this paper for the construction of strong solutions rests on a compactness criterion for square integrable functionals of a cylindrical Brownian motion from Malliavin calculus, which is a generalization of that in [@DPMN92], applied to solutions $X_{\cdot }^{x,n}$ $$\begin{aligned}
dX_{t}^{x,n}=b_{n}(t,X_{t}^{x,n})dt+d\mathbb{B}_{t},\quad
X_{0}^{x,n}=x,\quad n\geq 1,\end{aligned}$$ where $b_{n},n\geq 0$ are smooth vector fields converging to $b\in \mathcal{L}_{2,p}^{q}$. Then using variational techniques based on Fourier analysis, we prove that $X_{t}^{x}$ as a solution to is the strong $L^{2}-$limit of $X_{t}^{x,n}$ for all $t$.
To be more specific (in the case of time-homogeneous vector fields), we “linearize” the problem of finding strong solutions by applying Malliavin derivatives $D^{i}$ in the direction of Wiener processes $W^{i}$ with respect to the corresponding representations of $B_{\cdot }^{H_{i},i}$ in (\[Rep\]) in connection with (\[B\]) and get the linear equation$$D_{t}^{i}X_{u}^{x,n}=\int_{t}^{u}b_{n}^{\shortmid
}(X_{s}^{x,n})D_{t}^{i}X_{s}^{x,n}ds+K_{H}(u,t)I_{d},0\leq t<u,n\geq 1,
\label{LinearD}$$where $b_{n}^{\shortmid }$ denotes the spatial derivative of $b_{n}$, $K_{H}$ the kernel in (\[Rep\]) and $I_{d}\in \mathbb{R}^{d\times d}$ the unit matrix. Picard iteration then yields$$D_{t}^{i}X_{u}^{x,n}=K_{H}(u,t)I_{d}+\sum_{m\geq
1}\int_{t<s_{1}<...<s_{m}<u}b_{n}^{\shortmid
}(X_{s_{m}}^{x,n})...b_{n}^{\shortmid
}(X_{s_{1}}^{x,n})K_{H}(s_{1},t)I_{d}ds_{1}...ds_{m}. \label{MD}$$In a next step, in order to “get rid of” the derivatives of $b_{n}$ in ([MD]{}), we use Girsanov’s change of measure in connection with the following “local time variational calculus” argument:$$\int_{0<s_{1}<...<s_{n}<t}\kappa (s)D^{\alpha }f(\mathbb{B}_{s})ds=\int_{\mathbb{R}^{dn}}D^{\alpha }f(z)L_{\kappa }^{n}(t,z)dz=(-1)^{\left\vert
\alpha \right\vert }\int_{\mathbb{R}^{dn}}f(z)D^{\alpha }L_{\kappa
}^{n}(t,z)dz, \label{localtime}$$for $\mathbb{B}_{s}:=(\mathbb{B}_{s_{1}},...,\mathbb{B}_{s_{n}})$ and smooth functions $f:\mathbb{R}^{dn}\longrightarrow \mathbb{R}$ with compact support, where $D^{\alpha }$ stands for a partial derivative of order $\left\vert \alpha \right\vert $ for a multi-index $\alpha $). Here, $L_{\kappa }^{n}(t,z)$ is a spatially differentiable local time of $\mathbb{B}_{\cdot }$ on a simplex scaled by non-negative integrable function $\kappa
(s)=$ $\kappa _{1}(s)...\kappa _{n}(s)$.
Using the latter enables us to derive upper bounds based on Malliavin derivatives $D^{i}$ of the solutions in terms of continuous functions of $\left\Vert b_{n}\right\Vert _{\mathcal{L}_{2,p}^{q}}$, which we can use in conncetion with a compactness criterion for square integrable functionals of a cylindrical Brownian motion to obtain the strong solution as a $L^{2}-$limit of approximating solutions.
Based on similar previous arguments we also verify that the flow associated with for $b\in \mathcal{L}_{2,p}^{q}$ is smooth by using an estimate of the form $$\sup_{t}\sup_{x\in U}E\left[ \left\Vert \frac{\partial ^{k}}{\partial x^{k}}X_{t}^{x,n}\right\Vert ^{\alpha }\right] \leq C_{p,q,d,H,k,\alpha ,T}\left(
\left\Vert b_{n}\right\Vert _{\mathcal{L}_{2,p}^{q}}\right) ,n\geq 1$$for arbitrary $k\geq 1,$ where $C_{p,q,d,H,k,\alpha ,T}:[0,\infty
)\rightarrow \lbrack 0,\infty )$ is a continuous function, depending on $p,q,d,H=\{H_{n}\}_{n\geq 1},k,\alpha ,T$ for $\alpha \geq 1$ and $U\subset
\mathbb{R}^{d}$ a fixed bounded domain. See Theorem \[VI\_derivative\].
We also mention that the method used in this article significantly differs from that in [@BNP.17] and related works, since the underlying noise of $\mathbb{B}_{\cdot }$ in is of infinite-dimensional nature, that is a cylindrical Brownian motion. The latter however, requires in this paper the application of an infinite-dimensional version of the compactness criterion in [@DPMN92] tailored to the driving noise $\mathbb{B}_{\cdot
} $.
It is crucial to note here that the above technique explained in the case of perturbed ODE’s of the form (\[SDE\]) reveals or strongly hints at a general principle, which could be used to study important classes of PDE’s in connection with conservation laws or fluid dynamics. In fact, we believe that the following underlying principles may play a major role in the analysis of solutions to PDE’s:
**1.** *Nash-Moser principle*: The idea of this principle, which goes back to J. Nash [@Nash] and and J. Moser [@Moser], can be (roughly) explained as follows:
Assume a function $\Phi $ of class $C^{k}$. Then the Nash-Moser technique pertains to the study of solutions $u$ to the equation$$\Phi (u)=\Phi (u_{0})+f, \label{NMeq}$$where $u_{0}\in C^{\infty }$ is given and where $f$ is a “small” perturbation.
In the setting of our paper, the latter equation corresponds to the SDE ([SDE]{}) with a (non-deterministic) perturbation given by $f=\mathbb{B}_{\cdot
}$ (or $\varepsilon \mathbb{B}_{\cdot }$ for small $\varepsilon >0$). Then, using this principle, the problem of studying solutions to (\[NMeq\]) is “linearized” by analyzing solutions to the linear equation$$\Phi ^{\shortmid }(u)v=g, \label{NMlinear}$$where $\Phi ^{\shortmid }$ stands for the Fréchet derivative of $\Phi $. The study of the latter problem, however, usually comes along with a “loss of derivatives”, which can be measured by “tame” estimates based on a (decreasing) family of Banach spaces $E_{s},0\leq s<\infty $ with norms $\left\vert \cdot \right\vert _{s}$ such that $\cap _{s\geq 0}E_{s}=C^{\infty
}$. Typically, $E_{s}=C^{s}$ (Hölder spaces) or $E_{s}=H^{s}$ (Sobolev spaces).
In our situation, equation (\[NMlinear\]) has its analogon in ([LinearD]{}) with respect to the (stochastic Sobolev) derivative $D^{i}$ (or the Fréchet derivative $D$ in connection with flows).
Roughly speaking, in the case of Hölder spaces, assume that $$\Phi ^{\shortmid }(u)\psi (u)=Id$$for a linear mapping $\psi (u)$, which satisfies the “tame” estimate:$$\left\vert \psi (u)g\right\vert _{\alpha }\leq C(\left\vert g\right\vert
_{\alpha +\lambda }+\left\vert g\right\vert _{\lambda }(1+\left\vert
u\right\vert _{\alpha +r}))$$for numbers $\lambda ,r\geq 0$ and $\alpha \geq 0$. In addition, require a similar estimate with respect to $\Phi ^{\shortmid \shortmid }(u)$. Then, there exists in a certain neighbourhood $W$ of the origin such that for $f\in W$ equation (\[NMeq\]) has a solution $u(f)\in C^{\alpha }$. Solution here means that there exists a sequence $u_{j},j\geq 1$ in $C^{\infty }$ such that for all $\varepsilon >0$, $u_{j}\longrightarrow u$ in $C^{\alpha
-\varepsilon }$ and $\Phi (u_{j})\longrightarrow \Phi (u_{0})+f$ in $C^{\alpha +\lambda -\varepsilon }$ for $j\longrightarrow \infty $. The proof of the latter result rests on a Newton approximation scheme and results from Littlewood-Paley theory. See also [@AlG] and the references therein.
**2.** *Signature of higher order averaging operators along a highly fractal stochastic curve*: In fact another, but to the best of our knowledge new principle, which comes into play in connection with our technique for the study of perturbed ODE’s, is the “extraction” of information from “signatures” of *higher order averaging operators* along a highly irregular or fractal stochastic curve $\gamma _{t}=\mathbb{B}_{t}$ of the form $$\begin{aligned}
&&(T_{t}^{0,\gamma ,l_{1},...,l_{k}}(b)(x),T_{t}^{1,\gamma
,l_{1},...,l_{k}}(b)(x),T_{t}^{2,\gamma ,l_{1},...,l_{k}}(b)(x),...) \notag \\
&=&(I_{d},\int_{\mathbb{R}^{d}}b(x^{(1)}+z_{1})\Gamma _{\kappa
}^{1,l_{1},...,l_{k}}(z_{1})dz_{1}, \notag \\
&&\int_{\mathbb{R}^{2d}}b^{\otimes 2}(x^{(2)}+z_{2})\Gamma _{\kappa
}^{2,l_{1},...,l_{k}}(z_{2})dz_{2},\int_{\mathbb{R}^{3d}}b^{\otimes
3}(x^{(3)}+z_{3})\Gamma _{\kappa }^{3,l_{1},...,l_{k}}(z_{3})dz_{3},...)
\notag \\
&\in &\mathbb{R}^{d\times d}\times \mathbb{R}^{d}\times \mathbb{R}^{d\times d}\times...
\label{S}\end{aligned}$$where $b:\mathbb{R}^{d}\longrightarrow \mathbb{R}^{d}$ is a “rough”, that is a merely (locally integrable) Borel measurable vector field and $$\Gamma _{\kappa }^{n,l_{1},...,l_{k}}(z_{n})=(D^{\alpha
^{j_{1},...,j_{n-1},j,l_{1},...,l_{k}}}L_{\kappa }^{n}(t,z_{n}))_{1\leq
j_{1},...,j_{n-1},j\leq d}$$for multi-indices $\alpha ^{j_{1},...,j_{n-1},j,l_{1},...,l_{k}}\in \mathbb{N}_{0}^{nd}$ of order $\left\vert \alpha
^{j_{1},...,j_{n-1},j,l_{1},...,l_{k}}\right\vert =n+k-1$ for all (fixed) $l_{1},...,l_{k}\in \{1,...,d\}$, $k\geq 0$ and $x^{(n)}:=(x,...,x)\in
\mathbb{R}^{nd}$. Here $L_{\kappa }^{n}$ is the local time from ([localtime]{}) and the multiplication of $b^{\otimes n}(z_{n})$ and $\Gamma
_{\kappa }^{n,l_{1},...,l_{k}}(z_{n})$ in the above signature is defined via tensor contraction as$$(b^{\otimes n}(z_{n})\Gamma _{\kappa
}^{n,l_{1},...,l_{k}}(z_{n}))_{ij}=\sum_{j_{1},...,j_{n-1}=1}^{d}(b^{\otimes
n}(z_{n}))_{ij_{1},...,j_{n-1}}(\Gamma _{\kappa
}^{n,l_{1},...,l_{k}}(z_{n}))_{j_{1},...,j_{n-1}j}, n\geq 2\text{.}$$If $k=0$, we simply set$$T_{t}^{n,\gamma ,l_{1},...,l_{k}}(b)(x)=T_{t}^{n,\gamma }(b)(x)=\int_{\mathbb{R}^{d}}b(z)L_{\kappa }^{1}(t,z)dz$$for all $n\geq 1$.
The motivation for the concept (\[S\]) for rough vector fields $b$ comes from the integration by parts formula (\[localtime\]) applied to each summand of (\[MD\]) (under a change of measure), which can be written in terms of $T_{u}^{n,\gamma ,l_{1},...,l_{k}}(b)(x)$ for $k=1$.
Higher order derivatives $(D^{i})^{k}$ (or alternatively Fréchet derivatives $D^{k}$ of order $k$) in connection with (\[MD\]) give rise to the definition of operators $T_{u}^{n,\gamma ,l_{1},...,l_{k}}(b)(x)$ for general $k\geq 1$ (see Section $5$).
For example, if $n=1$, $k=2$, $\kappa \equiv 1$, then we have for (smooth) $b
$ that $$\begin{aligned}
\int_{0}^{t}b^{\shortmid \shortmid }(x+\gamma _{s})ds
&=&\int_{0}^{t}b^{\shortmid \shortmid }(x+\mathbb{B}_{s})ds \notag \\
&=&(\int_{\mathbb{R}^{d}}b(x^{(1)}+z_{1})(D^{2}L_{\kappa
}^{1}(t,z_{1}))_{l_{1},l_{2}}dz_{1})_{1\leq l_{1},l_{2}\leq d} \notag \\
&=&(\int_{\mathbb{R}^{d}}b(x^{(1)}+z_{1})\Gamma _{\kappa
}^{1,l_{1},l_{2}}(z_{1})dz_{1})_{_{1\leq l_{1},l_{2}\leq d}} \notag \\
&=&(T_{t}^{1,\gamma ,l_{1},l_{2}}(b)(x))_{1\leq l_{1},l_{2}\leq d}\in
\mathbb{R}^{d}\otimes \mathbb{R}^{d}\text{.} \label{Example}\end{aligned}$$In the case, when $n=1$, $k=0$, $\kappa \equiv 1$ and $\gamma _{t}=B_{t}^{H}$ a fractional Brownian motion for $H<\frac{1}{2},$ the first order averaging operator $T_{t}^{1,\gamma }$ along the curve $\gamma _{t}$ in (\[S\]) coincides with that in Catellier, Gubinelli [@CG] given by$$T_{t}^{\gamma }(b)(x)=\int_{0}^{t}b(x+B_{s}^{H})ds,$$which was used by the authors- as mentioned before- to study the regularization effect of $\gamma _{t}$ on ODE’s perturbed by such curves. For example, if $b\in B_{\infty ,\infty }^{\alpha +1}$ (Besov-Hölder space) with $\alpha >2-\frac{1}{2H}$, then the corresponding SDE (\[SDE\]) driven by $B_{\cdot }^{H}$ admits a unique Lipschitz flow. The reason- and this is important to mention here- why the latter authors “only” obtain Lipschitz flows and not higher regularity is that they do not take into account in their analysis information coming from higher order averaging operators $T_{t}^{n,\gamma ,l_{1},...,l_{k}}$ for $n>1$, $k\geq 1$. Here in this article, we rely in fact on the information based on such higher order averaging operators to be able to study $C^{\infty }-$regularization effects with respect to flows.
Let us also mention here that T. Tao, J. Wright [@TW] actually introduced averaging operators of the type $T_{t}^{\gamma }$ along (smooth) *deterministic* curves $\gamma _{t}$ for improving bounds of such operators on $L^{p}$ along such curves. See also the recent work of [@Gressman] and the references therein.
On the other hand, in view of the possibility of a geometric study of the regularity of solutions to ODE’s or PDE’s, it would be (motivated by ([Example]{}) natural to replace the signatures in (\[S\]) by the following family of signatures for rough vector fields $b$:$$\begin{aligned}
S_{t}^{n}(b)(x) &:&=(1,T_{t}^{n,\gamma }(b)(x),(T_{t}^{n,\gamma
,l_{1}}(b)(x))_{1\leq l_{1}\leq d},(T_{t}^{n,\gamma
,l_{1},l_{2}}(b)(x))_{1\leq l_{1},l_{2}\leq d},...) \\
&\in &T(\mathbb{R}^{d}):=\prod_{k\geq 0}(\otimes _{i=1}^{k}\mathbb{R}^{d}),n\geq 1,\end{aligned}$$where we use the convention $\otimes _{i=1}^{0}\mathbb{R}^{d}=\mathbb{R}$. The space $T(\mathbb{R}^{d})$ becomes an associative algebra under tensor multiplication. Then the regularity of solutions to ODE’s or PDE’s can be analyzed by means of such signatures in connection with Lie groups $\mathfrak{G}\subset T_{1}(\mathbb{R}^{d}):=\{(g_{0},g_{1},...)\in T(\mathbb{R}^{d}):g_{0}=1\}$.
In this context, it would be conceivable to be able to derive a Chen-Strichartz type of formula by means of $S_{t}^{n}(b)$ in connection with a sub-Riemannian geometry for the study of flows. See [@Baudoin] and the references therein.
**3.** *Removal of a “thin” set of “worst case” input data via noisy perturbation*: As explained before well-posedness of the ODE (\[ODE\]) can be restored by “randomization” or perturbation of the input vector field $b$ in (\[RODE\]). The latter suggests that this procedure leads to a removal of a “thin” set of “worst case” input data, which do not allow for regularization or the restoration of well-posedness. It would be interesting here to develop methods for the measurement of the size of such “thin” sets
The organization of our article is as follows: In Section \[frameset\] we discuss the mathematical framework of this paper. Further, in Section [monstersection]{} we derive important estimates via variational techniques based on Fourier analysis, which are needed later on for the proofs of the main results of this paper. Section \[strongsol\] is devoted to the construction of unique strong solutions to the SDE . Finally, in Section \[flowsection\] we show $C^{\infty }-$regularization by noise $\mathbb{B}_{\cdot }$ of the singular ODE .
Notation
--------
Throughout the article, we will usually denote by $C$ a generic constant. If $\pi$ is a collection of parameters then $C_{\pi}$ will denote a collection of constants depending only on the collection $\pi$. Given differential structures $M$ and $N$, we denote by $C_c^{\infty}(M;N)$ the space of infinitely many times continuously differentiable function from $M$ to $N$ with compact support. For a complex number $z\in \mathbb{C}$, $\overline{z}$ denotes the conjugate of $z$ and $\boldsymbol{i}$ the imaginary unit. Let $E$ be a vector space, we denote by $|x|$, $x\in E$ the Euclidean norm. For a matrix $A$, we denote $|A|$ its determinant and $\|A\|_\infty$ its maximum norm.
Framework and Setting {#frameset}
=====================
In this section we recollect some specifics on Fourier analysis, shuffle products, fractional calculus and fractional Brownian motion which will be extensively used throughout the article. The reader might consult [Mall97]{}, [@Mall78] or [@DOP08] for a general theory on Malliavin calculus for Brownian motion and [@Nua10 Chapter 5] for fractional Brownian motion. For more detailed theory on harmonic analysis and Fourier transform the reader is referred to [@grafakos.08].
Fourier Transform
-----------------
In the course of the paper we will make use of the Fourier transform. There are several definitions in the literature. In the present article we have taken the following: let $f\in L^1({\mathbb R}^d)$ then we define its *Fourier tranform*, denoted it by $\widehat{f}$, by $$\begin{aligned}
\label{Fourier}
\widehat{f}(\xi) = \int_{{\mathbb R}^d} f(x) e^{-2\pi \boldsymbol{i} \langle
x,\xi\rangle_{{\mathbb R}^d}} dx, \quad \xi \in {\mathbb R}^d.\end{aligned}$$
The above definition can be actually extended to functions in $L^2({\mathbb R}^d)$ and it makes the operator $L^2({\mathbb R}^d) \ni f \mapsto \widehat{f}\in L^2({\mathbb R}^d)$ a linear isometry which, by polarization, implies $$\langle \widehat{f},\widehat{g}\rangle_{L^2({\mathbb R}^d)} = \langle f,g\rangle_{L^2({\mathbb R}^d)},\quad f,g\in L^2({\mathbb R}^d),$$ where $$\langle f,g\rangle_{L^2({\mathbb R}^d)} = \int_{{\mathbb R}^d} f(z)\overline{g(z)} dz,\quad
f,g\in L^2({\mathbb R}^d).$$
Shuffles {#VI_shuffles}
--------
Let $k\in \mathbb{N}$. For given $m_1,\dots, m_k\in \mathbb{N}$, denote $$m_{1:j} := \sum_{i=1}^j m_i,$$ e.g. $m_{1:k} = m_1+\cdots +m_k$ and set $m_0:=0$. Denote by $S_{m}
=\{\sigma: \{1,\dots, m\}\rightarrow \{1,\dots,m\} \}$ the set of permutations of length $m \in \mathbb{N}$. Define the set of *shuffle permutations* of length $m_{1:k} = m_1+\cdots m_k$ as $$S(m_1,\dots, m_k) := \{\sigma\in S_{m_{1:k}}: \, \sigma(m_{1:i} +1)<\cdots
<\sigma(m_{1:i+1}), \, i=0,\dots,k-1\},$$ and the $m$-dimensional simplex in $[0,T]^m$ as $$\Delta_{t_0,t}^m:=\{(s_1,\dots,s_m)\in [0,T]^m : \, t_0<s_1<\cdots <
s_m<t\}, \quad t_0,t\in [0,T], \quad t_0<t.$$ Let $f_i:[0,T] \rightarrow [0,\infty)$, $i=1,\dots,m_{1:k}$ be integrable functions. Then, we have $$\begin{aligned}
\label{VI_shuffle}
\begin{split}
\prod_{i=0}^{k-1} \int_{\Delta_{t_0,t}^{m_i}} f_{m_{1:i}+1}(s_{m_{1:i}+1})
&\cdots f_{m_{1:i+1}}(s_{m_{1:i+1}}) ds_{m_{1:i}+1}\cdots ds_{m_{1:i+1}} \\
&= \sum_{\sigma^{-1}\in S(m_1,\dots, m_k)} \int_{\Delta_{t_0,t}^{m_{1:k}}}
\prod_{i=1}^{m_{1:k}} f_{\sigma(i)}(w_i) dw_1\cdots dw_{m_{1:k}}.
\end{split}$$ The above is a trivial generalisation of the case $k=2$ where $$\begin{aligned}
\label{shuffleIntegral}
\begin{split}
\int_{\substack{ t_0<s_1\cdots <s_{m_1}<t \\ t_0<s_{m_1+1}<\cdots
<s_{m_1+m_2}<t}} &\prod_{i=1}^{m_1+m_2} f_i(s_i) \, ds_1 \cdots ds_{m_1+m_2}
\\
&\hspace{-1cm}= \sum_{\sigma^{-1}\in S(m_1,m_2)} \int_{t_0<w_1<\cdots
<w_{m_1+m_2}<t} \prod_{i=1}^{m_1+m_2} f_{\sigma(i)} (w_i) dw_1\cdots
dw_{m_1+m_2}
\end{split},\end{aligned}$$ which can be for instance found in [@LCL.04].
We will also need the following formula. Given indices $j_0,j_1,\dots,
j_{k-1}\in \mathbb{N}$ such that $1\leq j_i\leq m_{i+1}$, $i=1,\dots,k-1$ and we set $j_0:=m_1+1$. Introduce the subset $S_{j_1,\dots,j_{k-1}}(m_1,\dots,m_k)$ of $S(m_1,\dots, m_k)$ defined as $$\begin{aligned}
S_{j_1,\dots, j_{k-1}}(m_1,\dots,m_k):=& \, \Big\{\sigma \in
S(m_1,\dots,m_k): \, \sigma(m_{1:i}+1)<\cdots <\sigma(m_{1:i} + j_i -1), \\
&\sigma(l)=l, \, m_{1:i} + j_i \leq l \leq m_{1:i+1} , \, i=0,\dots,k-1 \Big\}.\end{aligned}$$ We have $$\begin{aligned}
\label{VI_shuffle2}
\begin{split}
&\int_{\Delta_{t_0,t}^{m_k} \times
\Delta_{t_0,s_{m_{1:k-1}+j_{k-1}}}^{m_{k-1}} \times \cdots \times
\Delta_{t_0, s_{m_1+j_1}}^{m_1}} \prod_{i=1}^{m_{1:k}} f_i (s_i) \,
ds_1\cdots ds_{m_{1:k}} \\
& \hspace{1cm} = \int_{\substack{ t_0< s_1<\cdots <s_{m_1}< s_{m_1+j_1} \\ t_0<s_{m_1+m_2+1}<\cdots < s_{m_1+m_2}< s_{m_1+m_2+j_2} \\ \vdots \\ t_0<s_{m_1+\cdots m_{k-1}+1}<\cdots <s_{m_1+\cdots +m_k}< t}}
\prod_{i=1}^{m_{1:k}} f_i (s_i) \, ds_1\cdots ds_{m_{1:k}} \\
& \hspace{1cm} = \sum_{\sigma^{-1}\in S_{j_1,\dots, j_{k-1}}(m_1,\dots,
m_k)} \int_{t_0<w_1<\cdots <w_{m_{1:k}}<t} \prod_{i=1}^{m_{1:k}}
f_{\sigma(i)}(w_i) \, dw_1\cdots dw_{m_{1:k}}.
\end{split}.\end{aligned}$$
$$\# S(m_1,\dots,m_k) = \frac{(m_1+\cdots+m_k)!}{m_1! \cdots m_k!},$$
where $\#$ denotes the number of elements in the given set. Then by using Stirling’s approximation, one can show that $$\# S(m_1,\dots,m_k) \leq C^{m_1+\cdots+m_k}$$ for a large enough constant $C>0$. Moreover, $$\# S_{j_1,\dots,j_{k-1}}(m_1,\dots,m_k) \leq \# S(m_1,\dots,m_k).$$
Fractional Calculus {#VI_fraccal}
-------------------
We pass in review here some basic definitions and properties on fractional calculus. The reader may consult [samko.et.al.93]{} and [@lizorkin.01] for more information about this subject.
Suppose $a,b\in {\mathbb R}$ with $a<b$. Further, let $f\in L^{p}([a,b])$ with $p\geq
1$ and $\alpha >0$. Introduce the *left-* and *right-sided Riemann-Liouville fractional integrals* by $$I_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int_{a}^{x}(x-y)^{\alpha
-1}f(y)dy$$and $$I_{b^{-}}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int_{x}^{b}(y-x)^{\alpha
-1}f(y)dy$$for almost all $x\in \lbrack a,b]$, where $\Gamma $ stands for the Gamma function.
Furthermore, for an integer $p\geq 1$, denote by $I_{a^{+}}^{\alpha }(L^{p})$ (resp. $I_{b^{-}}^{\alpha }(L^{p})$) the image of $L^{p}([a,b])$ of the operator $I_{a^{+}}^{\alpha }$ (resp. $I_{b^{-}}^{\alpha }$). If $f\in
I_{a^{+}}^{\alpha }(L^{p})$ (resp. $f\in I_{b^{-}}^{\alpha }(L^{p})$) and $0<\alpha <1$ then we define the *left-* and *right-sided Riemann-Liouville fractional derivatives* by $$D_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\frac{{\mbox{d}}}{{\mbox{d}}x}\int_{a}^{x}\frac{f(y)}{(x-y)^{\alpha }}dy$$and $$D_{b^{-}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\frac{{\mbox{d}}}{{\mbox{d}}x}\int_{x}^{b}\frac{f(y)}{(y-x)^{\alpha }}dy.$$
The above left- and right-sided derivatives of $f$ can be represented as follows: $$D_{a^{+}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\left( \frac{f(x)}{(x-a)^{\alpha }}+\alpha \int_{a}^{x}\frac{f(x)-f(y)}{(x-y)^{\alpha +1}}dy\right) ,$$ $$D_{b^{-}}^{\alpha }f(x)=\frac{1}{\Gamma (1-\alpha )}\left( \frac{f(x)}{(b-x)^{\alpha }}+\alpha \int_{x}^{b}\frac{f(x)-f(y)}{(y-x)^{\alpha +1}}dy\right) .$$
By construction one also finds the relations $$I_{a^{+}}^{\alpha }(D_{a^{+}}^{\alpha }f)=f$$for all $f\in I_{a^{+}}^{\alpha }(L^{p})$ and $$D_{a^{+}}^{\alpha }(I_{a^{+}}^{\alpha }f)=f$$for all $f\in L^{p}([a,b])$ and similarly for $I_{b^{-}}^{\alpha }$ and $D_{b^{-}}^{\alpha }$.
Fractional Brownian motion
--------------------------
Consider $d$-dimensional *fractional Brownian motion* $B_{t}^{H}=(B_{t}^{H,(1)},...,B_{t}^{H,(d)}),$ $0\leq t\leq T$ with Hurst parameter $H\in (0,1/2)$. So $B_{\cdot }^{H}$ is a centered Gaussian process with covariance structure $$(R_{H}(t,s))_{i,j}:=E[B_{t}^{H,(i)}B_{s}^{H,(j)}]=\delta _{i,j}\frac{1}{2}\left( t^{2H}+s^{2H}-|t-s|^{2H}\right) ,\quad i,j=1,\dots ,d,$$where $\delta _{i,j}=1$ if $i=j$ and $\delta _{i,j}=0$ otherwise.
One finds that $E[|B_{t}^{H}-B_{s}^{H}|^{2}]=d|t-s|^{2H}$. The latter implies that $B_{\cdot }^{H}$ has stationary increments and Hölder continuous trajectories of index $H-\varepsilon $ for all $\varepsilon \in
(0,H)$. In addition, one also checks that the increments of $B_{\cdot }^{H}$, $H\in (0,1/2)$ are not independent. This fact however, complicates the study of e.g. SDE’s driven by the such processes compared to the Wiener setting. Another difficulty one is faced with in connection with such processes is that they are not semimartingales, see e.g. [@Nua10 Proposition 5.1.1].
In what follows let us briefly discuss the construction of fractional Brownian motion via an isometry. In fact, this construction can be done componentwise. Therefore, for convenience we confine ourselves to the one-dimensional case. We refer to [@Nua10] for further details.
Let us denote by $\mathcal{E}$ the set of step functions on $[0,T]$ and by $\mathcal{H}$ the Hilbert space, which is obtained by the closure of $\mathcal{E}$ with respect to the inner product $$\langle 1_{[0,t]},1_{[0,s]}\rangle _{\mathcal{H}}=R_{H}(t,s).$$The mapping $1_{[0,t]}\mapsto B_{t}^{H}$ has an extension to an isometry between $\mathcal{H}$ and the Gaussian subspace of $L^{2}(\Omega )$ associated with $B^{H}$. We denote the isometry by $\varphi \mapsto
B^{H}(\varphi )$.
The following result, which can be found in (see [Nua10]{} ), provides an integral representation of $R_{H}(t,s)$, when $H<1/2$:
Let $H<1/2$. The kernel $$K_H(t,s)= c_H \left[\left( \frac{t}{s}\right)^{H- \frac{1}{2}} (t-s)^{H-
\frac{1}{2}} + \left( \frac{1}{2}-H\right) s^{\frac{1}{2}-H} \int_s^t u^{H-\frac{3}{2}} (u-s)^{H-\frac{1}{2}} du\right],$$ where $c_H = \sqrt{\frac{2H}{(1-2H) \beta(1-2H , H+1/2)}}$ being $\beta$ the Beta function, satisfies $$\begin{aligned}
\label{VI_RH}
R_H(t,s) = \int_0^{t\wedge s} K_H(t,u)K_H(s,u)du.\end{aligned}$$
The kernel $K_{H}$ also has a representation in terms of a fractional derivative as follows $$K_{H}(t,s)=c_{H}\Gamma \left( H+\frac{1}{2}\right) s^{\frac{1}{2}-H}\left(
D_{t^{-}}^{\frac{1}{2}-H}u^{H-\frac{1}{2}}\right) (s).$$
Let us now introduce a linear operator $K_{H}^{\ast }:\mathcal{E}\rightarrow
L^{2}([0,T])$ by $$(K_{H}^{\ast }\varphi )(s)=K_{H}(T,s)\varphi (s)+\int_{s}^{T}(\varphi
(t)-\varphi (s))\frac{\partial K_{H}}{\partial t}(t,s)dt$$for every $\varphi \in \mathcal{E}$. We see that $(K_{H}^{\ast
}1_{[0,t]})(s)=K_{H}(t,s)1_{[0,t]}(s)$. From this and we obtain that $K_{H}^{\ast }$ is an isometry between $\mathcal{E}$ and $L^{2}([0,T])$ which has an extension to the Hilbert space $\mathcal{H}$.
For a $\varphi \in \mathcal{H}$ one proves the following representations for $K_{H}^{\ast }$: $$(K_{H}^{\ast }\varphi )(s)=c_{H}\Gamma \left( H+\frac{1}{2}\right) s^{\frac{1}{2}-H}\left( D_{T^{-}}^{\frac{1}{2}-H}u^{H-\frac{1}{2}}\varphi (u)\right)
(s),$$ $$\begin{aligned}
(K_{H}^{\ast }\varphi )(s)=& \,c_{H}\Gamma \left( H+\frac{1}{2}\right)
\left( D_{T^{-}}^{\frac{1}{2}-H}\varphi (s)\right) (s) \\
& +c_{H}\left( \frac{1}{2}-H\right) \int_{s}^{T}\varphi (t)(t-s)^{H-\frac{3}{2}}\left( 1-\left( \frac{t}{s}\right) ^{H-\frac{1}{2}}\right) dt.\end{aligned}$$
On the other hand one also gets the relation $\mathcal{H}=I_{T^{-}}^{\frac{1}{2}-H}(L^{2})$ (see [@decreu.ustunel.98] and [alos.mazet.nualart.01]{}).
Using the fact that $K_{H}^{\ast }$ is an isometry from $\mathcal{H}$ into $L^{2}([0,T])$, the $d$-dimensional process $W=\{W_{t},t\in \lbrack 0,T]\}$ given by $$W_{t}:=B^{H}((K_{H}^{\ast })^{-1}(1_{[0,t]}))$$is a Wiener process and the process $B^{H}$ can be represented as $$B_{t}^{H}=\int_{0}^{t}K_{H}(t,s)dW_{s}\text{.}
\label{VI_BHW}$$See [@alos.mazet.nualart.01].
In the sequel, we denote by $W_{\cdot }$ a standard Wiener process on a given probability space endowed with the natural filtration generated by $W$ augmented by all $P$-null sets. Further, $B_{\cdot }:=B_{\cdot }^{H}$ stands for the fractional Brownian motion with Hurst parameter $H\in (0,1/2)$ given by the representation .
In the following, we need a version of Girsanov’s theorem for fractional Brownian motion which goes back to [@decreu.ustunel.98 Theorem 4.9]. Here we state the version given in [@nualart.ouknine.02 Theorem 3.1]. In preparation of this, we introduce an isomorphism $K_{H}$ from $L^{2}([0,T])$ onto $I_{0+}^{H+\frac{1}{2}}(L^{2})$ associated with the kernel $K_{H}(t,s)$ in terms of the fractional integrals as follows, see [@decreu.ustunel.98 Theorem 2.1] $$(K_{H}\varphi )(s)=I_{0^{+}}^{2H}s^{\frac{1}{2}-H}I_{0^{+}}^{\frac{1}{2}-H}s^{H-\frac{1}{2}}\varphi ,\quad \varphi \in L^{2}([0,T]).$$
Using the latter and the properties of the Riemann-Liouville fractional integrals and derivatives, one finds that the inverse of $K_{H}$ is given by $$\label{opK_H-1}
(K_{H}^{-1}\varphi )(s)=s^{\frac{1}{2}-H}D_{0^{+}}^{\frac{1}{2}-H}s^{H-\frac{1}{2}}D_{0^{+}}^{2H}\varphi (s),\quad \varphi \in I_{0+}^{H+\frac{1}{2}}(L^{2}).$$
Hence, if $\varphi $ is absolutely continuous, see [@nualart.ouknine.02], one can prove that $$\label{VI_inverseKH}
(K_{H}^{-1}\varphi )(s)=s^{H-\frac{1}{2}}I_{0^{+}}^{\frac{1}{2}-H}s^{\frac{1}{2}-H}\varphi ^{\prime }(s),\quad a.e.$$
\[VI\_girsanov\] Let $u=\{u_t, t\in [0,T]\}$ be an $\mathcal{F}$-adapted process with integrable trajectories and set $\widetilde{B}_t^H = B_t^H +
\int_0^t u_s ds, \quad t\in [0,T].$ Assume that
- $\int_0^{\cdot} u_s ds \in I_{0+}^{H+\frac{1}{2}} (L^2 ([0,T]))$, $P$-a.s.
- $E[\xi_T]=1$ where $$\xi_T := \exp\left\{-\int_0^T K_H^{-1}\left( \int_0^{\cdot} u_r
dr\right)(s)dW_s - \frac{1}{2} \int_0^T K_H^{-1} \left( \int_0^{\cdot} u_r
dr \right)^2(s)ds \right\}.$$
Then the shifted process $\widetilde{B}^H$ is an $\mathcal{F}$-fractional Brownian motion with Hurst parameter $H$ under the new probability $\widetilde{P}$ defined by $\frac{d\widetilde{P}}{dP}=\xi_T$.
For the multidimensional case, define $$(K_H \varphi)(s):= ( (K_H \varphi^{(1)} )(s), \dots, (K_H
\varphi^{(d)})(s))^{\ast}, \quad \varphi \in L^2([0,T];{\mathbb R}^d),$$ where $\ast$ denotes transposition. Similarly for $K_H^{-1}$ and $K_H^{\ast}$.
Finally, we mention a crucial property of the fractional Brownian motion which was proven by [@pitt.78] for general Gaussian vector fields.
Let $m\in \mathbb{N}$ and $0=:t_0<t_1<\cdots <t_m<T$. Then for every $\xi_1,\dots, \xi_m\in {\mathbb R}^d$ there exists a positive finite constant $C>0$ (depending on $m$) such that $$\begin{aligned}
\label{VI_SLND}
Var\left[ \sum_{j=1}^m \langle\xi_j, B_{t_j}^H-B_{t_{j-1}}^H\rangle_{{\mathbb R}^d}\right] \geq C \sum_{j=1}^m |\xi_j|^2 E\left[|B_{t_j}^H-B_{t_{j-1}}^H|^2\right].\end{aligned}$$
The above property is known as the *local non-determinism* property of the fractional Brownian motion. A stronger version of the local non-determinism, which we want to make use of in this paper and which is referred to as *two sided strong local non-determinism* in the literature, is also satisfied by the fractional Brownian motion: There exists a constant $K>0$, depending only on $H$ and $T$, such that for any $t\in \lbrack 0,T]$, $0<r<t$, $$Var\left[ B_{t}^{H}|\ \{B_{s}^{H}:|t-s|\geq r\}\right] \geq Kr^{2H}.
\label{2sided}$$The reader may e.g. consult [@pitt.78] or [@xiao.11] for more information on this property.
A New Regularizing Process {#monstersection}
==========================
Throughout this article we operate on a probability space $(\Omega ,\mathfrak{A},P)$ equipped with a filtration $\mathcal{F}:=\{\mathcal{F}_{t}\}_{t\in \lbrack 0,T]}$ where $T>0$ is fixed, generated by a process $\mathbb{B}_{\cdot }=\mathbb{B}_{\cdot }^{H}=\{\mathbb{B}_{t}^{H},t\in
\lbrack 0,T]\}$ to be defined later and here $\mathfrak{A}:=\mathcal{F}_{T}$.
Let $H=\{H_{n}\}_{n\geq 1}\subset (0,1/2)$ be a sequence of numbers such that $H_{n}\searrow 0$ for $n\longrightarrow \infty $. Also, consider $\lambda =\{\lambda _{n}\}_{n\geq 1}\subset {\mathbb R}$ a sequence of real numbers such that there exists a bijection $$\{n:\lambda _{n}\neq 0\}\rightarrow \mathbb{N} \label{lambdacond}$$and $$\sum_{n=1}^{\infty }|\lambda _{n}|\in (0,\infty ). \label{lambdacond2}$$
Let $\{W_{\cdot }^{n}\}_{n\geq 1}$ be a sequence of independent $d$-dimensional standard Brownian motions taking values in ${\mathbb R}^{d}$ and define for every $n\geq 1$, $$\label{compfBm}
B_{t}^{H_{n},n}=\int_{0}^{t}K_{H_{n}}(t,s)dW_{s}^{n}=\left(
\int_{0}^{t}K_{H_{n}}(t,s)dW_{s}^{n,1},\dots
,\int_{0}^{t}K_{H_{n}}(t,s)dW_{s}^{n,d}\right) ^{\ast }.$$
By construction, $B_{\cdot }^{H_{n},n}$, $n\geq 1$ are pairwise independent $d$-dimensional fractional Brownian motions with Hurst parameters $H_{n}$. Observe that $W_{\cdot }^{n}$ and $B_{\cdot }^{H_{n},n}$ generate the same filtrations, see [@Nua10 Chapter 5, p. 280]. We will be interested in the following stochastic process $$\mathbb{B}_{t}^{H}=\sum_{n=1}^{\infty }\lambda _{n}B_{t}^{H_{n},n},\quad
t\in \lbrack 0,T]. \label{monster}$$
Finally, we need another technical condition on the sequence $\lambda
=\{\lambda _{n}\}_{n\geq 1}$, which is used to ensure continuity of the sample paths of $\mathbb{B}_{\cdot }^{H}$: $$\sum_{n=1}^{\infty }|\lambda _{n}|E\left[ \sup_{0\leq s\leq
1}|B_{s}^{H_{n},n}|\right] <\infty , \label{contcond}$$where $\sup_{0\leq s\leq 1}|B_{s}^{H_{n},n}|\in L^{1}(\Omega )$ indeed, see e.g. [@berman.89].
The following theorem gives a precise definition of the process $\mathbb{B}_{\cdot }^{H}$ and some of its relevant properties.
\[monsterprocess\] Let $H=\{H_{n}\}_{n\geq 1}\subset (0,1/2)$ be a sequence of real numbers such that $H_{n}\searrow 0$ for $n\longrightarrow
\infty $ and $\lambda =\{\lambda _{n}\}_{n\geq 1}\subset {\mathbb R}$ satisfying , and . Let $\{B_{\cdot }^{H_{n},n}\}_{n=1}^{\infty }$ be a sequence of $d$-dimensional independent fractional Brownian motions with Hurst parameters $H_{n}$, $n\geq 1$, defined as in . Define the process $$\mathbb{B}_{t}^{H}:=\sum_{n=1}^{\infty }\lambda _{n}B_{t}^{H_{n},n},\quad
t\in \lbrack 0,T],$$where the convergence is $P$-a.s. and $\mathbb{B}_{t}^{H}$ is a well defined object in $L^{2}(\Omega )$ for every $t\in \lbrack 0,T]$.
Moreover, $\mathbb{B}_t^H$ is normally distributed with zero mean and covariance given by $$E[\mathbb{B}_t^H (\mathbb{B}_s^H)^\ast] = \sum_{n=1}^{\infty} \lambda_n^2
R_{H_n}(t,s)I_d,$$ where $\ast$ denotes transposition, $I_d$ is the $d$-dimensional identity matrix and $R_{H^n}(t,s):= \frac{1}{2}\left(s^{2H_n}+t^{2H_n}-|t-s|^{2H_n}\right)$ denotes the covariance function of the components of the fractional Brownian motions $B_t^{H_n,n}$.
The process $\mathbb{B}_{\cdot }^{H}$ has stationary increments. It does not admit any version with Hölder continuous paths of any order. $\mathbb{B}_{\cdot }^{H}$ has no finite $p$-variation for any order $p>0$, hence $\mathbb{B}_{\cdot }^{H}$ is not a semimartingale. It is not a Markov process and hence it does not possess independent increments.
Finally, under condition , $\mathbb{B}_{\cdot }^{H}$ has $P$-a.s. continuous sample paths.
One can verify, employing Kolmogorov’s three series theorem, that the series converges $P$-a.s. and we easily see that $$E[|\mathbb{B}_t^H|^2] = d\sum_{n=1}^{\infty}\lambda_n^2 t^{2H_n}\leq d(1+t)
\sum_{n=1}^{\infty}\lambda_n^2<\infty,$$ where we used that $x^{\alpha}\leq 1+x$ for all $x\geq 0$ and any $\alpha\in[0,1]$.
The Gaussianity of $\mathbb{B}_{t}^{H}$ follows simply by observing that for every $\theta \in {\mathbb R}^{d}$, $$E\left[ \exp \left\{ \boldsymbol{i}\langle \theta ,\mathbb{B}_{t}^{H}\rangle
_{{\mathbb R}^{d}}\right\} \right] =e^{-\frac{1}{2}\sum_{n=1}^{\infty
}\sum_{j=1}^{d}\lambda _{n}^{2}t^{2H_{n}}\theta ^{2}},$$where we used the independence of $B_{t}^{H_{n},n}$ for every $n\geq 1$. The covariance formula follows easily again by independence of $B_{t}^{H_{n},n}$.
The stationarity follows by the fact that $B^{H_n,n}$ are independent and stationary for all $n\geq 1$.
The process $\mathbb{B}_{\cdot }^{H}$ could *a priori* be very irregular. Since $\mathbb{B}_{\cdot }^{H}$ is a stochastically continuous separable process with stationary increments, we know by [@MR.06 Theorem 5.3.10] that either $\mathbb{B}^{H}$ has $P$-a.s. continuous sample paths on all open subsets of $[0,T]$ or $\mathbb{B}^{H}$ is $P$-a.s. unbounded on all open subsets on $[0,T]$. Under condition and using the self-similarity of the fractional Brownian motions we see that $$\begin{aligned}
E\left[ \sup_{s\in \lbrack 0,T]}|\mathbb{B}_{s}^{H}|\right] & \leq
\sum_{n=1}^{\infty }|\lambda _{n}|T^{H_{n}}E\left[ \sup_{s\in \lbrack
0,1]}|B_{s}^{H_{n},n}|\right] \\
& \hspace{2cm}\leq (1+T)\sum_{n=1}^{\infty }|\lambda _{n}|E\left[ \sup_{s\in
\lbrack 0,1]}|B_{s}^{H_{n},n}|\right] <\infty\end{aligned}$$and hence by Belyaev’s dichotomy for separable stochastically continuous processes with stationary increments (see e.g. [@MR.06 Theorem 5.3.10]) there exists a version of $\mathbb{B}_{\cdot }^{H}$ with continuous sample paths.
Trivially, $\mathbb{B}_{\cdot }^{H}$ is never Hölder continuous since for arbitrary small $\alpha >0$ there is always $n_{0}\geq 1$ such that $H_{n}<\alpha $ for all $n\geq n_{0}$ and since the sequence $\lambda $ satisfies cancellations are not possible. Further, one also argues that $\mathbb{B}_{\cdot }^{H}$ is neither Markov nor has finite variation of any order $p>0$ which then implies that $\mathbb{B}_{\cdot
}^{H} $ is not a semimartingale.
We will refer to as a *regularizing* cylindrical fractional Brownian motion with associated Hurst sequence $H$ or simply a regularizing fBm.
Next, we state a version of Girsanov’s theorem which actually shows that equation admits a weak solution. Its proof is mainly based on the classical Girsanov theorem for a standard Brownian motion in Theorem [VI\_girsanov]{}.
\[girsanov\] Let $u:[0,T]\times \Omega \rightarrow {\mathbb R}^d$ be a (jointly measurable) $\mathcal{F}$-adapted process with integrable trajectories such that $t\mapsto \int_0^t u_s ds$ belongs to the domain of the operator $K_{H_{n_0}}^{-1}$ from for some $n_0\geq 1$.
Define the ${\mathbb R}^d$-valued process $$\widetilde{\mathbb{B}}_t^H := \mathbb{B}_t^H + \int_0^t u_s ds.$$
Define the probability $\widetilde{P}_{n_0}$ in terms of the Radon-Nikodym derivative $$\frac{d\widetilde{P}_{n_0}}{dP_{n_0}}:=\xi_T,$$ where $$\xi_T^{n_0} := \exp \left\{-\int_0^T K_{H_{n_0}}^{-1}\left( \frac{1}{\lambda_{n_0}} \int_0^{\cdot} u_s ds \right) (s) dW_s^{n_0} -\frac{1}{2}\int_0^T \left|K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}
\int_0^{\cdot}u_s ds \right) (s)\right|^2 ds\right\}.$$
If $E[\xi _{T}^{n_{0}}]=1$, then $\widetilde{\mathbb{B}_{\cdot }}^{H}$ is a regularizing ${\mathbb R}^{d}$-valued cylindrical fractional Brownian motion with respect to $\mathcal{F}$ under the new measure $\widetilde{P}_{n_{0}}$ with Hurst sequence $H$.
Indeed, write $$\begin{aligned}
\widetilde{\mathbb{B}}_t^H &= \int_0^t u_s ds +
\lambda_{n_0}B_t^{H_{n_0},n_0}+\sum_{n\neq n_0}^{\infty} \lambda_n
B_t^{H_n,n} \\
&= \lambda_{n_0}\left(\frac{1}{\lambda_{n_0}}\int_0^t u_s ds +
B_t^{H_{n_0},n_0}\right) + \sum_{n\neq n_0}^{\infty} \lambda_n B_t^{H_n,n} \\
&= \lambda_{n_0}\left(\frac{1}{\lambda_{n_0}}\int_0^t u_s ds + \int_0^t
K_{H_{n_0}}(t,s) dW_s^{n_0}\right) + \sum_{n\neq n_0}^{\infty} \lambda_n
B_t^{H_n,n} \\
&= \lambda_{n_0}\left(\int_0^t K_{H_{n_0}}(t,s) d\widetilde{W}_s^{n_0}\right) + \sum_{n\neq n_0}^{\infty} \lambda_n B_t^{H_n,n},\end{aligned}$$ where $$\widetilde{W}_t^{n_0} := W_t^{n_0} + \int_0^t K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}} \int_0^{\cdot} u_r dr\right)(s)ds.$$
Then it follows from Theorem \[VI\_girsanov\] or [nualart.ouknine.03]{} that $$\widetilde{B}_t^{H_{n_0},n_0}:= \int_0^t K_{H_{n_0}}(t,s) d\widetilde{W}_t^{n_0}$$ is a fractional Brownian motion with Hurst parameter $H_{n_0}$ under the measure $$\frac{d\widetilde{P}_{n_0}}{dP_{n_0}} = \exp \left\{-\int_0^T
K_{H_{n_0}}^{-1}\left( \frac{1}{\lambda_{n_0}} \int_0^{\cdot} u_s ds \right)
(s) dW_s^{n_0} -\frac{1}{2}\int_0^T \left|K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}} \int_0^{\cdot} u_s ds \right) (s)\right|^2 ds\right\}.$$
Hence, $$\widetilde{\mathbb{B}}_t^H = \sum_{n=1}^{\infty} \lambda_n\widetilde{B}_t^{H_n,n},$$ where $$\widetilde{B}_t^{H_n,n} =
\begin{cases}
B_t^{H_n,n} \quad \mbox{if}\quad n\neq n_0, \\
\widetilde{B}_t^{H_{n_0},n_0} \quad \mbox{if}\quad n= n_0\end{cases},$$ defines a regularizing ${\mathbb R}^d$-valued cylindrical fractional Brownian motion under $\widetilde{P}_{n_0}$.
In the above Girsanov theorem we just modify the law of the drift plus one selected fractional Brownian motion with Hurst parameter $H_{n_0}$. In our proof later, we show that actually $t\mapsto \int_0^t b(s,\mathbb{B}_s^H)ds$ belongs to the domain of the operators $K_{H_n}^{-1}$ for any $n\geq 1$ but only large $n\geq 1$ satisfy Novikov’s condition for arbitrary selected values of $p,q\in (2,\infty]$.
Consider now the following stochastic differential equation with the driving noise $\mathbb{B}_{\cdot }^{H}$, introduced earlier: $$\label{eqsmooth}
X_{t}=x+\int_{0}^{t}b(s,X_{s})ds+\mathbb{B}_{t}^{H},\quad t\in \lbrack 0,T],$$where $x\in {\mathbb R}^{d}$ and $b$ is regular.
The following result summarises the classical existence and uniqueness theorem and some of the properties of the solution. Existence and uniqueness can be conducted using the classical arguments of $L^{2}([0,T]\times \Omega
) $-completeness in connection with a Picard iteration argument.
Let $b:[0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}^{d}$ be continuously differentiable in ${\mathbb R}^{d}$ with bounded derivative uniformly in $t\in \lbrack 0,T]$ and such that there exists a finite constant $C>0$ independent of $t$ such that $|b(t,x)|\leq C(1+|x|)$ for every $(t,x)\in \lbrack 0,T]\times {\mathbb R}^{d}$. Then equation admits a unique global strong solution which is $P$-a.s. continuously differentiable in $x$ and Malliavin differentiable in each direction $W^{i}$, $i\geq 1$ of $\mathbb{B}_{\cdot }^{H}$. Moreover, the space derivative and Malliavin derivatives of $X$ satisfy the following linear equations $$\frac{\partial }{\partial x}X_{t}=I_{d}+\int_{0}^{t}b^{\prime }(s,X_{s})\frac{\partial }{\partial x}X_{s}ds,\quad t\in \lbrack 0,T]$$and $$D_{t_{0}}^{i}X_{t}=\lambda
_{i}K_{H_{i}}(t,t_{0})I_{d}+\int_{t_{0}}^{t}b^{\prime
}(s,X_{s})D_{t_{0}}^{i}X_{s}ds,\quad i\geq 1,\quad t_{0},t\in \lbrack
0,T],\quad t_{0}<t,$$where $b^{\prime }$ denotes the space Jacobian matrix of $b$, $I_{d}$ the $d$-dimensional identity matrix and $D_{t_{0}}^{i}$ the Malliavin derivative along $W^{i}$, $i\geq 1$. Here, the last identity is meant in the $L^{p}$-sense $[0,T]$.
Construction of the Solution {#strongsol}
============================
We aim at constructing a Malliavin differentiable unique global $\mathcal{F}$-strong solution to the following equation $$\label{maineq}
dX_{t}=b(t,X_{t})dt+d\mathbb{B}_{t}^{H},\quad X_{0}=x\in {\mathbb R}^{d},\quad t\in
\lbrack 0,T],$$where the differential is interpreted formally in such a way that if admits a solution $X_{\cdot }$, then $$X_{t}=x+\int_{0}^{t}b(s,X_{s})ds+\mathbb{B}_{t}^{H},t\in \lbrack 0,T],$$whenever it makes sense. Denote by $L_{p}^{q}:=L^{q}([0,T];L^{p}({\mathbb R}^{d};{\mathbb R}^{d}))$, $p,q\in \lbrack 1,\infty ]$ the Banach space of integrable functions such that $$\Vert f\Vert _{L_{p}^{q}}:=\left( \int_{0}^{T}\left( \int_{{\mathbb R}^{d}}|f(t,z)|^{p}dz\right) ^{q/p}dt\right) ^{1/q}<\infty ,$$where we take the essential supremum’s norm in the cases $p=\infty $ and $q=\infty $.
In this paper, we want to reach the class of discontinuous coefficients $b:[0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}^{d}$ in the Banach space $$\mathcal{L}_{2,p}^{q}:=L^{q}([0,T];L^{p}({\mathbb R}^{d};{\mathbb R}^{d}))\cap L^{1}({\mathbb R}^{d};L^{\infty }([0,T];\mathbb{R}^{d})),\quad p,q\in (2,\infty ],$$of functions $f:[0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}^{d}$ with the norm $$\Vert f\Vert _{\mathcal{L}_{2,p}^{q}}=\Vert f\Vert _{L_{p}^{q}}+\Vert f\Vert
_{L_{\infty }^{1}}$$for chosen $p,q\in (2,\infty ]$, where $$L_{\infty }^{1}:=L^{1}({\mathbb R}^{d};L^{\infty }([0,T];\mathbb{R}^{d})).$$
Hence, our computations also show the result for uniformly bounded coefficients that are square-integrable.
We will show existence and uniqueness of strong solutions of equation driven by a $d$-dimensional regularizing fractional Brownian motion with Hurst sequence $H$ with coefficients $b$ belonging to the class $\mathcal{L}_{2,p}^{q}$. Moreover, we will prove that such solution is Malliavin differentiable and infinitely many times differentiable with respect to the initial value $x$, where $d\geq 1$, $p,q\in (2,\infty ]$ are arbitrary.
We would like to remark that with the method employed in the present article, the existence of weak solutions and the uniqueness in law, holds for drift coefficients in the space $L_{p}^{q}$. In fact, as we will see later on, we need the additional space $L_{\infty }^{1}$ to obtain unique strong solutions.
This solution is neither a semimartingale, nor a Markov process, and it has very irregular paths. We show in this paper that the process $\mathbb{B}_{\cdot }^{H}$ is a right noise to use in order to produce infinitely classically differentiable flows of for highly irregular coefficients.
To construct a solution the main key is to approximate $b$ by a sequence of smooth functions $b_n$ a.e. and denoting by $X^n = \{X_t^n, t\in [0,T]\}$ the approximating solutions, we aim at using an *ad hoc* compactness argument to conclude that the set $\{X_t^n\}_{n\geq 1}\subset L^2(\Omega)$ for fixed $t\in [0,T]$ is relatively compact.
As for the regularity of the mapping $x\mapsto X_{t}^{x}$, we are interested in proving that it is infinitely many times differentiable. It is known that the SDE $dX_{t}=b(t,X_{t})dt+dB_{t}^{H}$, $X_{0}=x\in {\mathbb R}^{d}$ admits a unique strong solution for irregular vector fields $b\in L_{\infty ,\infty
}^{1,\infty }$ and that the mapping $x\mapsto X_{t}^{x}$ belongs, $P$-a.s., to $C^{k}$ if $H=H(k,d)<1/2$ is small enough. Hence, by adding the noise $\mathbb{B}_{\cdot }^{H}$, we should expect the solution of to have a smooth flow.
Hereunder, we establish the following main result, which will be stated later on in this Section in a more precise form* *(see Theorem [VI\_mainthm]{}):
*Let* $b\in \mathcal{L}_{2,p}^{q}$*,* $p,q\in (2,\infty ]$*and assume that* $\lambda =\{\lambda _{i}\}_{i\geq 1}$* in (*[monster]{}*) satisfies certain growth conditions to be specified later on. Then there exists a unique (global) strong solution* $X=\{X_{t},t\in
\lbrack 0,T]\}$* of equation* *. Moreover, for every* $t\in \lbrack 0,T]$*,* $X_{t}$* is Malliavin differentiable in each direction of the Brownian motions* $W^{n}$*,* $n\geq 1$* in* .
The proof of Theorem \[VI\_mainthm\] consists of the following steps:
1. First, we give the construction of a weak solution $X_{\cdot }$ to by means of Girsanov’s theorem for the process $\mathbb{B}_{\cdot }^{H}$, that is we introduce a probability space $(\Omega ,\mathfrak{A},P)$, on which a regularizing fractional Brownian motion $\mathbb{B}_{\cdot }^{H}$ and a process $X_{\cdot }$ are defined, satisfying the SDE . However, a priori $X_{\cdot }$ is not adapted to the natural filtration $\mathcal{F}=\{\mathcal{F}_{t}\}_{t\in \lbrack 0,T]}$ with respect to $\mathbb{B}_{\cdot }^{H}$.
2. In the next step, consider an approximation of the drift coefficient $b $ by a sequence of compactly supported and infinitely continuously differentiable functions (which always exists by standard approximation results) $b_{n}:[0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}^{d}$, $n\geq 0$ such that $b_{n}(t,x)\rightarrow b(t,x)$ for a.e. $(t,x)\in \lbrack 0,T]\times {\mathbb R}^{d}$ and such that $\sup_{n\geq 0}\Vert b_{n}\Vert _{\mathcal{L}_{2,p}^{q}}\leq M$ for some finite constant $M>0$. Then by the previous Section we know that for each smooth coefficient $b_{n}$, $n\geq 0$, there exists unique strong solution $X^{n}=\{X_{t}^{n},t\in \lbrack 0,T]\}$ to the SDE $$\label{VI_Xn}
dX_{t}^{n}=b_{n}(t,X_{t}^{n})du+d\mathbb{B}_{t}^{H},\,\,0\leq t\leq
T,\,\,\,X_{0}^{n}=x\in \mathbb{R}^{d}\,.$$Then we prove that for each $t\in \lbrack 0,T]$ the sequence $X_{t}^{n}$ converges weakly to the conditional expectation $E[X_{t}|\mathcal{F}_{t}]$ in the space $L^{2}(\Omega )$ of square integrable random variables.
3. By the previous Section we have that for each $t\in \lbrack 0,T]$ the strong solution $X_{t}^{n}$, $n\geq 0$, is Malliavin differentiable, and that the Malliavin derivatives $D_{s}^{i}X_{t}^{n}$, $i\geq 1$, $0\leq s\leq
t$, with respect to $W^{i}$ in satisfy $$D_{s}^{i}X_{t}^{n}=\lambda _{i}K_{H_{i}}(t,s)I_{d}+\int_{s}^{t}b_{n}^{\prime
}(u,X_{u}^{n})D_{s}^{i}X_{u}^{n}du,$$for every $i\geq 1$ where $b_{n}^{\prime }$ is the Jacobian of $b_{n}$ and $I_{d}$ the identity matrix in ${\mathbb R}^{d\times d}$. Then, we apply an infinite-dimensional compactness criterion for square integrable functionals of a cylindrical Wiener process based on Malliavin calculus to show that for every $t\in \lbrack 0,T]$ the set of random variables $\{X_{t}^{n}\}_{n\geq
0}$ is relatively compact in $L^{2}(\Omega )$. The latter, however, enables us to prove that $X_{t}^{n}$ converges strongly in $L^{2}(\Omega )$ to $E[X_{t}|\mathcal{F}_{t}]$. Further we find that $E[X_{t}|\mathcal{F}_{t}]$ is Malliavin differentiable as a consequence of the compactness criterion.
4. We verify that $E[X_{t}|\mathcal{F}_{t}]=X_{t}$. So it follows that $X_{t}$ is $\mathcal{F}_{t}$-measurable and thus a strong solution on our specific probability space.
5. Uniqueness in law is enough to guarantee pathwise uniqueness.
In view of the above scheme, we go ahead with step (1) by first providing some preparatory lemmas in order to verify Novikov’s condition for $\mathbb{B}_{\cdot }^{H}$. Consequently, a weak solution can be constructed via a change of measure.
\[interlemma\] Let $\mathbb{B}_{\cdot }^{H}$ be a $d$-dimensional regularizing fBm and $p,q\in \lbrack 1,\infty ]$. Then for every Borel measurable function $h:[0,T]\times {\mathbb R}^{d}\rightarrow \lbrack 0,\infty )$ we have $$\label{estimateh}
E\left[ \int_{0}^{T}h(t,\mathbb{B}_{t}^{H})dt\right] \leq C\Vert h\Vert
_{L_{p}^{q}},$$where $C>0$ is a constant depending on $p$, $q$, $d$ and $H$. Also, $$\label{estimatehexp}
E\left[ \exp \left\{ \int_{0}^{T}h(t,\mathbb{B}_{t}^{H})dt\right\} \right]
\leq A(\Vert h\Vert _{L_{p}^{q}}),$$where $A$ is an analytic function depending on $p$, $q$, $d$ and $H$.
Let $\mathbb{B}_{\cdot }^{H}$ be a $d$-dimensional regularizing fBm, then $$\mathbb{B}_{t}^{H}-E\left[ \mathbb{B}_{t}^{H}|\mathcal{F}_{t_{0}}\right]
=\sum_{n=1}^{\infty }\lambda _{n}\int_{t_{0}}^{t}K_{H_{n}}(t,s)dW_{s}^{n}.$$So because of the independence of the increments of the Brownian motion, we find that$$Var\left[ \mathbb{B}_{t}^{H}|\mathcal{F}_{t_{0}}\right] =Var[\mathbb{B}_{t}^{H}-E\left[ \mathbb{B}_{t}^{H}|\mathcal{F}_{t_{0}}\right] ].$$On the other the strong local non-determinism of the fractional Brownian motion yields$$Var[\mathbb{B}_{t}^{H}-E\left[ \mathbb{B}_{t}^{H}|\mathcal{F}_{t_{0}}\right]
]=Var\left[ \mathbb{B}_{t}^{H}|\mathcal{F}_{t_{0}}\right] \geq
\sum_{n=1}^{\infty }\lambda _{n}^{2}C_{n}(t-t_{0})^{2H_{n}},$$where $C_{n}$ are the constants depending on $H_{n}$.
Hence, by a conditioning argument it is easy to see that for every Borel measurable function $h$ we have $$\begin{aligned}
E& \left[ \int_{t_{0}}^{T}h(t_{1},\mathbb{B}_{t_{1}}^{H})dt_{1}\bigg|\mathcal{F}_{t_{0}}\right] \\
& \leq \int_{t_{0}}^{T}\int_{{\mathbb R}^{d}}h(t_{1},Y+z)(2\pi )^{-d/2}\sigma
_{t_{0},t_{1}}^{-d}\exp \left( -\frac{|z|^{2}}{2\sigma _{t_{0},t_{1}}^{2}}\right) dzdt_{1}\bigg|_{Y=\sum_{n=1}^{\infty }\lambda
_{n}\int_{0}^{t_{0}}K_{H_{n}}(t,s)dW_{s}^{n}},\end{aligned}$$where $$\sigma _{t_{0},t_{1}}^{2}:=\sum_{n=1}^{\infty }\lambda
_{n}^{2}C_{n}|t_{1}-t_{0}|^{2H_{n}}.$$Applying Hölder’s inequality, first w.r.t. $z$ and then w.r.t. $t_{1}$ we arrive at $$\begin{aligned}
E& \left[ \int_{t_{0}}^{T}h(t_{1},\mathbb{B}_{t_{1}}^{H})dt_{1}\bigg|\mathcal{F}_{t_{0}}\right] \leq \\
& \leq C\left( \int_{t_{0}}^{T}\left( \int_{{\mathbb R}^{d}}h(t_{1},x_{1})^{p}dx_{1}\right) ^{q/p}dt_{1}\right) ^{1/q}\left( \int_{t_{0}}^{T}\left( \sigma
_{t_{0},t_{1}}^{2}\right) ^{-dq^{\prime }(p^{\prime }-1)/2p^{\prime
}}dt_{1}\right) ^{1/q^{\prime }},\end{aligned}$$for some finite constant $C>0$. The time integral is finite for arbitrary values of $d,q^{\prime }$ and $p^{\prime }$. To see this, use the bound $\sum_{n}a_{n}\geq a_{n_{0}}$ for $a_{n}\geq 0$ and for all $n_{0}\geq 1$. Hence, $$\begin{aligned}
\int_{t_{0}}^{T}& \left( \sum_{n=1}^{\infty }\lambda
_{n}^{2}C_{n}(t_{1}-t_{0})^{2H_{n}}\right) ^{-dq^{\prime }(p^{\prime
}-1)/2p^{\prime }}dt_{1} \\
& \leq \left( \lambda _{n_{0}}^{2}C_{n_{0}}\right) ^{-dq^{\prime }(p^{\prime
}-1)/2p^{\prime }}\int_{t_{0}}^{T}(t_{1}-t_{0})^{-H_{n_{0}}dq^{\prime
}(p^{\prime }-1)/p^{\prime }}dt_{1},\end{aligned}$$then for fixed $d,q^{\prime }$ and $p^{\prime }$ choose $n_{0}$ so that $H_{n_{0}}dq^{\prime }(p^{\prime }-1)/p^{\prime }<1$. Actually, the above estimate already implies that all exponential moments are finite by [@Por90 Lemma 1.1]. Here, though we need to derive the explicit dependence on the norm of $h$.
Altogether, $$\begin{aligned}
\label{conditionalest}
E\left[\int_{t_0}^T h(t_1,\mathbb{B}_{t_1}^H) dt_1\bigg| \mathcal{F}_{t_0} \right] \leq C \left(\int_{t_0}^T \left(\int_{{\mathbb R}^d} h(t_1,x_1)^p
dx_1\right)^{q/p} dt_1\right)^{1/q},\end{aligned}$$ and setting $t_0=0$ this proves .
In order to prove , Taylor’s expansion yields $$E\left[ \exp \left\{ \int_{0}^{T}h(t,\mathbb{B}_{t}^{H})dt\right\} \right]
=1+\sum_{m=1}^{\infty }E\left[ \int_{0}^{T}\int_{t_{1}}^{T}\cdots
\int_{t_{m-1}}^{T}\prod_{j=1}^{m}h(t_{j},\mathbb{B}_{t_{j}}^{H})dt_{m}\cdots
dt_{1}\right] .$$Using iteratively we have $$E\left[ \exp \left\{ \int_{0}^{T}h(t,\mathbb{B}_{t}^{H})dt\right\} \right]
\leq \frac{C^{m}}{(m!)^{1/q}}\left( \int_{0}^{T}\left( \int_{{\mathbb R}^{d}}h(t,x)^{p}dx\right) ^{q/p}dt\right) ^{m/q}=\frac{C^{m}\Vert h\Vert
_{L_{p}^{q}}^{m}}{(m!)^{1/q}},$$and the result follows with $A(x):=\sum_{m=1}^{\infty }\frac{C^{m}}{(m!)^{1/q}}x^{m}$.
\[domainKH\] Let $\mathbb{B}_{\cdot }^{H}$ be a $d$-dimensional regularizing fBm and assume $b\in L_{p}^{q}$, $p,q\in \lbrack 2,\infty ]$. Then for every $n\geq 1$, $$t\mapsto \int_{0}^{t}b(s,\mathbb{B}_{s}^{H})ds\in I_{0+}^{H_{n}+\frac{1}{2}}(L^{2}([0,T])),\quad P-a.s.,$$i.e. the process $t\mapsto \int_{0}^{t}b(s,\mathbb{B}_{s}^{H})ds$ belongs to the domain of the operator $K_{H_{n}}^{-1}$ for every $n\geq 1$, $P$-a.s.
Using the property that $D_{0^{+}}^{H+\frac{1}{2}}I_{0^{+}}^{H+\frac{1}{2}}(f)=f$ for $f\in L^{2}([0,T])$ we need to show that for every $n\geq 1$, $$D_{0^{+}}^{H_{n}+\frac{1}{2}}\int_{0}^{\cdot }|b(s,\mathbb{B}_{s}^{H})|ds\in
L^{2}([0,T]),\quad P-a.s.$$Indeed, $$\begin{aligned}
\left\vert D_{0^{+}}^{H_{n}+\frac{1}{2}}\left( \int_{0}^{\cdot }|b(s,\mathbb{B}_{s}^{H})|ds\right) (t)\right\vert =& \frac{1}{\Gamma \left( \frac{1}{2}-H_{n}\right) }\Bigg(\frac{1}{t^{H_{n}+\frac{1}{2}}}\int_{0}^{t}|b(u,\mathbb{B}_{u}^{H})|du \\
& +\,\left( H+\frac{1}{2}\right) \int_{0}^{t}(t-s)^{-H_{n}-\frac{3}{2}}\int_{s}^{t}|b(u,\mathbb{B}_{u}^{H})|duds\Bigg) \\
& \hspace{-4cm}\leq \frac{1}{\Gamma \left( \frac{1}{2}-H_{n}\right) }\Bigg(\frac{1}{t^{H_{n}+\frac{1}{2}}}+\,\left( H+\frac{1}{2}\right)
\int_{0}^{t}(t-s)^{-H_{n}-\frac{3}{2}}ds\Bigg)\int_{0}^{t}|b(u,\mathbb{B}_{u}^{H})|du.\end{aligned}$$Hence, for some finite constant $C_{H,T}>0$ we have $$\left\vert D_{0^{+}}^{H+\frac{1}{2}}\left( \int_{0}^{\cdot }|b(s,\mathbb{\tilde{B}}_{s}^{H})|ds\right) (t)\right\vert ^{2}\leq
C_{H,T}\int_{0}^{T}|b(u,\mathbb{B}_{u}^{H})|^{2}du$$and taking expectation the result follows by Lemma \[interlemma\] applied to $|b|^{2}$.
We are now in a position to show that Novikov’s condition is met if $n$ is large enough.
\[novikov\] Let $\mathbb{B}_t^H$ be a $d$-dimensional regularizing fractional Brownian motion with Hurst sequence $H$. Assume $b\in L_p^q$, $p,q\in (2,\infty]$. Then for every $\mu \in {\mathbb R}$, there exists $n_0$ with $H_{n}< \frac{1}{2}-\frac{1}{p}$ for every $n\geq n_0$ and such that for every $n\geq n_0$ we have $$E\left[ \exp\left\{\mu \int_0^T \left|K_{H_n}^{-1}\left( \frac{1}{\lambda_n}\int_0^{\cdot} b(r,\mathbb{B}_r^H) dr\right) (s)\right|^2 ds\right\} \right]
\leq C_{\lambda_n,H_n,d,\mu,T}(\|b\|_{L_{p}^{q}})$$ for some real analytic function $C_{\lambda_n,H_n,d,\mu,T}$ depending only on $\lambda_n$, $H_n$, $d$, $T$ and $\mu$.
In particular, there is also some real analytic function $\widetilde{C}_{\lambda_n,H_n,d,\mu,T}$ depending only on $\lambda_n$, $H_n$, $d$, $T$ and $\mu$ such that $$E\left[ \mathcal{E}\left(\int_0^T K_{H_n}^{-1}\left(\frac{1}{\lambda_n}\int_0^{\cdot} b(r,\mathbb{B}_r^H) dr\right)^{\ast} (s) dW_s^n\right)^\mu \right] \leq \widetilde{C}_{H,d,\mu,T}(\|b\|_{L_{p}^{q}}),$$ for every $\mu \in {\mathbb R}$.
By Lemma \[domainKH\] both random variables appearing in the statement are well defined. Then, fix $n\geq n_0$ and denote $\theta_s^n :=
K_{H_n}^{-1}\left(\frac{1}{\lambda_n}\int_0^{\cdot} |b(r,\mathbb{B}_r^H)|
dr\right) (s)$. Then using relation we have $$\begin{aligned}
\label{thetan}
|\theta_s^n| =& \left|\frac{1}{\lambda_n}s^{H_n-\frac{1}{2}} I_{0^+}^{\frac{1}{2}-H_n} s^{\frac{1}{2}-H_n} |b(s,\mathbb{B}_s^H)|\right| \notag \\
=&\frac{1/|\lambda_n|}{\Gamma \left(\frac{1}{2}-H_n\right)} s^{H_n- \frac{1}{2}} \int_0^s (s-r)^{-\frac{1}{2}-H_n} r^{\frac{1}{2}-H_n} |b(r,\mathbb{B}_r^H)|dr.\end{aligned}$$ Observe that since $H_n< \frac{1}{2}-\frac{1}{p}$, $p\in (2,\infty]$ we may take $\varepsilon\in [0,1)$ such that $H_n<\frac{1}{1+\varepsilon}-\frac{1}{2}$ and apply Hölder’s inequality with exponents $1+\varepsilon$ and $\frac{1+\varepsilon}{\varepsilon}$, where the case $\varepsilon=0$ corresponds to the case where $b$ is bounded. Then we get
$$\begin{aligned}
\label{thetabound}
|\theta_s^n| \leq C_{\varepsilon,\lambda_n,H_n} s^{\frac{1}{1+\varepsilon}-H_n-\frac{1}{2}} \left(\int_0^s |b(r,\mathbb{B}_r^H)|^{\frac{1+\varepsilon}{\varepsilon}}dr\right)^{\frac{\varepsilon}{1+\varepsilon}},\end{aligned}$$
where $$C_{\varepsilon,\lambda_n, H_n}:=\frac{\Gamma\left(1-(1+\varepsilon)(H_n+1/2)\right)^{\frac{1}{1+\varepsilon}}\Gamma\left(1+(1+\varepsilon)(1/2-H_n)\right)^{\frac{1}{1+\varepsilon}} }{\lambda_n \Gamma \left(\frac{1}{2}-H_n\right) \Gamma \left(2(1-(1+\varepsilon)H_n)\right)^{\frac{1}{1+\varepsilon}}}.$$
Squaring both sides and using the fact that $|b|\geq 0$ we have the following estimate $$\begin{aligned}
|\theta_s^n|^2 \leq C_{\varepsilon,\lambda_n,H_n}^2 s^{\frac{2}{1+\varepsilon}-2H_n-1} \left(\int_0^T |b(r,\mathbb{B}_r^H)|^{\frac{1+\varepsilon}{\varepsilon}}dr\right)^{\frac{2\varepsilon}{1+\varepsilon}}, \quad P-a.s.\end{aligned}$$ Since $0<\frac{2\varepsilon}{1+\varepsilon}<1$ and $|x|^{\alpha}\leq \max
\{\alpha,1-\alpha\}(1+|x|)$ for any $x\in {\mathbb R}$ and $\alpha\in (0,1)$ we have $$\begin{aligned}
\label{VI_fracL2}
\int_0^T |\theta_s^n|^2 ds \leq C_{\varepsilon,\lambda_n,H_n,T} \left(1+
\int_0^T |b(r,\mathbb{B}_r^H)|^{\frac{1+\varepsilon}{\varepsilon}}dr\right),
\quad P-a.s.\end{aligned}$$ for some constant $C_{\varepsilon,\lambda_n, H_n,T}>0$. Then estimate from Lemma \[interlemma\] with $h =
C_{\varepsilon,\lambda_n,H_n,T} \ \mu \ b^{\frac{1+\varepsilon}{\varepsilon}} $ with $\varepsilon\in [0,1)$ arbitrarily close to one yields the result for $p,q\in (2,\infty]$.
Let $(\Omega ,\mathfrak{A},\widetilde{P})$ be some given probability space which carries a regularizing fractional Brownian motion $\widetilde{\mathbb{B}_{\cdot }}^{H}$ with Hurst sequence $H=\{H_{n}\}_{n\geq 1}$ and set $X_{t}:=x+\widetilde{\mathbb{B}}_{t}^{H}$, , $x\in {\mathbb R}^{d}$. Set $\theta _{t}^{n_{0}}:=\left( K_{H_{n_{0}}}^{-1}\left( \frac{1}{\lambda
_{n_{0}}}\int_{0}^{\cdot }b(r,X_{r})dr\right) \right) (t)$ for some fixed $n_{0}\geq 1$ such that Proposition \[novikov\] can be applied and consider the new measure defined by $$\frac{dP_{n_{0}}}{d\widetilde{P}_{n_{0}}}=Z_{T}^{n_{0}},$$where $$Z_{t}^{n_{0}}:=\prod_{n=1}^{\infty }\mathcal{E}\left( \theta _{\cdot
}^{n_{0}}\right) _{t}:=\exp \left\{ \int_{0}^{t}\left( \theta
_{s}^{n_{0}}\right) ^{\ast }dW_{s}^{n_{0}}-\frac{1}{2}\int_{0}^{t}|\theta
_{s}^{n_{0}}|^{2}ds\right\} ,\quad t\in \lbrack 0,T].$$
In view of Proposition \[novikov\] the above random variable defines a new probability measure and by Girsanov’s theorem, see Theorem \[girsanov\], the process $$\label{VI_weak}
\mathbb{B}_{t}^{H}:=X_{t}-x-\int_{0}^{t}b(s,X_{s})ds,\quad t\in \lbrack 0,T]$$is a regularizing fractional Brownian motion on $(\Omega ,\mathfrak{A},P_{n_{0}})$ with Hurst sequence $H$. Hence, because of , the couple $(X,\mathbb{B}_{\cdot }^{H})$ is a weak solution of on $(\Omega ,\mathfrak{A},P_{n_{0}})$. Since $n_{0}\geq 1$ is fixed we will omit the notation $P_{n_{0}}$ and simply write $P$.
Henceforth, we confine ourselves to the filtered probability space $(\Omega ,\mathfrak{A},P)$, $\mathcal{F}=\{\mathcal{F}_{t}\}_{t\in \lbrack 0,T]}$ which carries the weak solution $(X,\mathbb{B}_{\cdot }^{H})$ of .
\[VI\_stochbasisrmk\] In order to establish existence of a strong solution, the main difficulty here is that $X_{\cdot }$ is $\mathcal{F}$-adapted. In fact, in this case $X_{t}=F_{t}(\mathbb{B}_{\cdot }^{H})$ for some family of progressively measurable functional $F_{t}$, $t\in \lbrack
0,T]$ on $C([0,T];{\mathbb R}^{d})$ and for any other stochastic basis $(\hat{\Omega},\hat{\mathfrak{A}},\hat{P},\hat{\mathbb{B}})$ one gets that $X_{t}:=F_{t}(\hat{\mathbb{B}}_{\cdot })$, $t\in \lbrack 0,T]$, is a solution to SDE , which is adapted with respect to the natural filtration of $\hat{\mathbb{B}}_{\cdot }$. But this exactly gives the existence of a strong solution to SDE .
We take a weak solution $X_{\cdot }$ of and consider $E[X_{t}|\mathcal{F}_{t}]$. The next result corresponds to step (2) of our program.
\[VI\_weakconv\] Let $b_n:[0,T]\times {\mathbb R}^d\rightarrow {\mathbb R}^d$, $n\geq 1 $, be a sequence of compactly supported smooth functions converging a.e. to $b$ such that $\sup_{n\geq 1} \|b_n\|_{L_p^q}<\infty$. Let $t\in [0,T]$ and $X_t^n$ denote the solution of when we replace $b$ by $b_n$. Then for every $t\in [0,T]$ and continuous function $\varphi:{\mathbb R}^d
\rightarrow {\mathbb R}$ of at most linear growth we have that $$\varphi(X_t^{n}) \xrightarrow{n \to \infty} E\left[ \varphi(X_t) |\mathcal{F}_t \right],$$ weakly in $L^2(\Omega)$.
Let us assume, without loss of generality, that $x=0$. In the course of the proof we always assume that for fixed $p,q\in (2,\infty ]$ then $n_{0}\geq 1$ is such that $H_{n_{0}}<\frac{1}{2}-\frac{1}{p}$ and hence Proposition [novikov]{} can be applied.
First we show that $$\begin{aligned}
\label{VI_doleansDadeConvergence}
\mathcal{E}\left(\frac{1}{\lambda_{n_0}} \int_0^t
K_{H_{n_0}}^{-1}\left(\int_0^{\cdot} b_n(r,\mathbb{B}^H_r)dr\right)^{\ast}(s) dW_s^{n_0} \right) \rightarrow \mathcal{E}\left( \int_0^t
K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot} b(r,\mathbb{B}^H_r)dr\right)^{\ast}(s) dW_s^{n_0}\right)\end{aligned}$$ in $L^p(\Omega)$ for all $p \geq 1$. To see this, note that $$K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot} b_n(r,\mathbb{B}^H_r)dr\right)(s) \rightarrow K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot} b(r,\mathbb{B}^H_r)dr\right)(s)$$ in probability for all $s$. Indeed, from we have a constant $C_{\varepsilon,\lambda_{n_0},H_{n_0}}>0$ such that $$\begin{aligned}
E\Bigg[\Big|& K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot}
b_n(r,\mathbb{B}^H_r)dr\right)(s) - K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot} b(r,\mathbb{B}^H_r)dr\right)(s)\Big| \Bigg] \\
&\leq C_{\varepsilon,,\lambda_{n_0},H_{n_0}} s^{\frac{1}{1+\varepsilon}-H_{n_0}-\frac{1}{2}} \left(\int_0^s |b_n(r,\mathbb{B}_r^H) -b(r,\mathbb{B}_r^H) |^{\frac{1+\varepsilon}{\varepsilon}} dr\right)^{\frac{\varepsilon}{1+\varepsilon}} \rightarrow 0\end{aligned}$$ as $n \rightarrow \infty$ by Lemma \[interlemma\].
Moreover, $\left\{ K_{H_{n_0}}^{-1}(\frac{1}{\lambda_{n_0}}\int_0^{\cdot}
b_n(r,\mathbb{B}_r^H)dr) \right\}_{n \geq 0}$ is bounded in $L^2([0,t]
\times \Omega; \mathbb{R}^d)$. This is directly seen from (\[VI\_fracL2\]) in Proposition \[novikov\].
Consequently $$\int_0^t K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot} b_n(r,\mathbb{B}_r^H)dr \right)^{\ast}(s) dW_s^{n_0} \rightarrow \int_0^t
K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot} b(r,\mathbb{B}_r^H)dr\right)^{\ast}(s) dW_s^{n_0}$$ and $$\int_0^t \left|K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot}
b_n(r,\mathbb{B}_r^H)dr\right)(s)\right|^2 ds \rightarrow \int_0^t
\left|K_{H_{n_0}}^{-1}\left(\frac{1}{\lambda_{n_0}}\int_0^{\cdot} b(r,\mathbb{B}_r^H)dr\right)(s)\right|^2 ds$$ in $L^2(\Omega)$ since the latter is bounded $L^p(\Omega)$ for any $p \geq 1$, see Proposition \[novikov\].
By applying the estimate $|e^{x}-e^{y}|\leq e^{x+y}|x-y|$, Hölder’s inequality and the bounds in Proposition \[novikov\] in connection with Lemma \[interlemma\] we see that (\[VI\_doleansDadeConvergence\]) holds.
Similarly, one finds that $$\exp \left\{ \left\langle \alpha ,\int_{s}^{t}b_{n}(r,\mathbb{B}_{r}^{H})dr\right\rangle \right\} \rightarrow \exp \left\{ \left\langle
\alpha ,\int_{s}^{t}b(r,\mathbb{B}_{r}^{H})dr\right\rangle \right\}$$in $L^{p}(\Omega )$ for all $p\geq 1$, $0\leq s\leq t\leq T$, $\alpha \in {\mathbb R}^{d}$.
In order to complete the proof, we note that the set $$\Sigma _{t}:=\left\{ \exp \{\sum_{j=1}^{k}\langle \alpha _{j},\mathbb{B}_{t_{j}}^{H}-\mathbb{B}_{t_{j-1}}^{H}\rangle \}:\{\alpha
_{j}\}_{j=1}^{k}\subset \mathbb{R}^{d},0=t_{0}<\dots <t_{k}=t,k\geq 1\right\}$$is a total subspace of $L^{2}(\Omega ,\mathcal{F}_{t},P)$ and therefore it is sufficient to prove the convergence $$\lim_{n\rightarrow \infty }E\left[ \left( \varphi (X_{t}^{n})-E[\varphi
(X_{t})|\mathcal{F}_{t}]\right) \xi \right] =0$$for all $\xi \in \Sigma _{t}$. In doing so, we notice that $\varphi $ is of linear growth and hence $\varphi (\mathbb{B}_{t}^{H})$ has all moments. Thus, we obtain the following convergence $$E\left[ \varphi (X_{t}^{n})\exp \left\{ \sum_{j=1}^{k}\langle \alpha _{j},\mathbb{B}_{t_{j}}^{H}-\mathbb{B}_{t_{j-1}}^{H}\rangle \right\} \right]$$$$=E\left[ \varphi (X_{t}^{n})\exp \left\{ \sum_{j=1}^{k}\langle \alpha
_{j},X_{t_{j}}^{n}-X_{t_{j-1}}^{n}-\int_{t_{j-1}}^{t_{j}}b_{n}(s,X_{s}^{n})ds\rangle \right\} \right]$$$$=E[\varphi (\mathbb{B}_{t}^{H})\exp \{\sum_{j=1}^{k}\langle \alpha _{j},\mathbb{B}_{t_{j}}^{H}-\mathbb{B}_{t_{j-1}}^{H}-\int_{t_{j-1}}^{t_{j}}b_{n}(s,\mathbb{B}_{s}^{H})ds\rangle \}\mathcal{E}\left( \int_{0}^{t}K_{H_{n_{0}}}^{-1}\left( \frac{1}{\lambda _{n_{0}}}\int_{0}^{\cdot }b_{n}(r,\mathbb{B}_{r}^{H})dr\right) ^{\ast
}(s)dW_{s}^{n_{0}}\right) ]$$$$\rightarrow E[\varphi (\mathbb{B}_{t}^{H})\exp \{\sum_{j=1}^{k}\langle
\alpha _{j},\mathbb{B}_{t_{j}}^{H}-\mathbb{B}_{t_{j-1}}^{H}-\int_{t_{j-1}}^{t_{j}}b(s,\mathbb{B}_{s}^{H})ds\rangle \}\mathcal{E}\left(
\int_{0}^{t}K_{H_{n_{0}}}^{-1}\left( \frac{1}{\lambda _{n_{0}}}\int_{0}^{\cdot }b(r,\mathbb{B}_{r}^{H})dr\right) ^{\ast
}(s)dW_{s}^{n_{0}}\right) ]$$$$=E[\varphi (X_{t})\exp \{\sum_{j=1}^{k}\langle \alpha _{j},\mathbb{B}_{t_{j}}^{H}-\mathbb{B}_{t_{j-1}}^{H}\rangle \}]$$$$=E[E[\varphi (X_{t})|\mathcal{F}_{t}]\exp \{\sum_{j=1}^{k}\langle \alpha
_{j},\mathbb{B}_{t_{j}}^{H}-\mathbb{B}_{t_{j-1}}^{H}\rangle \}].$$
We now turn to step (3) of our program. For its completion we need to derive some crucial estimates.
In preparation of those estimates, we introduce some notation and definitions:
Let $m$ be an integer and let the function $f:[0,T]^{m}\times ({\mathbb R}^{d})^{m}\rightarrow {\mathbb R}$ be of the form $$f(s,z)=\prod_{j=1}^{m}f_{j}(s_{j},z_{j}),\quad s=(s_{1},\dots ,s_{m})\in
\lbrack 0,T]^{m},\quad z=(z_{1},\dots ,z_{m})\in ({\mathbb R}^{d})^{m}, \label{f}$$where $f_{j}:[0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}$, $j=1,\dots ,m$ are smooth functions with compact support. Further, let $\varkappa
:[0,T]^{m}\rightarrow {\mathbb R}$ a function of the form $$\varkappa (s)=\prod_{j=1}^{m}\varkappa _{j}(s_{j}),\quad s\in \lbrack
0,T]^{m}, \label{kappa}$$where $\varkappa _{j}:[0,T]\rightarrow {\mathbb R}$, $j=1,\dots ,m$ are integrable functions.
Let $\alpha _{j}$ be a multi-index and denote by $D^{\alpha _{j}}$ its corresponding differential operator. For $\alpha =(\alpha _{1},\dots ,\alpha
_{m})$ viewed as an element of $\mathbb{N}_{0}^{d\times m}$ we define $|\alpha |=\sum_{j=1}^{m}\sum_{l=1}^{d}\alpha _{j}^{(l)}$ and write $$D^{\alpha }f(s,z)=\prod_{j=1}^{m}D^{\alpha _{j}}f_{j}(s_{j},z_{j}).$$
The objective of this section is to establish an integration by parts formula of the form $$\int_{\Delta _{\theta ,t}^{m}}D^{\alpha }f(s,\mathbb{B}_{s})ds=\int_{({\mathbb R}^{d})^{m}}\Lambda _{\alpha }^{f}(\theta ,t,z)dz, \label{ibp}$$where $\mathbb{B}:=\mathbb{B}_{\cdot }^{H}$, for a random field $\Lambda
_{\alpha }^{f}$. In fact, we can choose $\Lambda _{\alpha }^{f}$ by $$\Lambda _{\alpha }^{f}(\theta ,t,z)=(2\pi )^{-dm}\int_{({\mathbb R}^{d})^{m}}\int_{\Delta _{\theta
,t}^{m}}\prod_{j=1}^{m}f_{j}(s_{j},z_{j})(-iu_{j})^{\alpha _{j}}\exp
\{-i\langle u_{j},\mathbb{B}_{s_{j}}-z_{j}\rangle \}dsdu. \label{LambdaDef}$$
Let us strat by *defining* $\Lambda _{\alpha }^{f}(\theta ,t,z)$ as above and show that it is a well-defined element of $L^{2}(\Omega )$.
We also need the following notation: Given $(s,z)=(s_{1},\dots
,s_{m},z_{1}\dots ,z_{m})\in \lbrack 0,T]^{m}\times ({\mathbb R}^{d})^{m}$ and a shuffle $\sigma \in S(m,m)$ we define $$f_{\sigma }(s,z):=\prod_{j=1}^{2m}f_{[\sigma (j)]}(s_{j},z_{[\sigma (j)]})$$and $$\varkappa _{\sigma }(s):=\prod_{j=1}^{2m}\varkappa _{\lbrack \sigma
(j)]}(s_{j}),$$where $[j]$ is equal to $j$ if $1\leq j\leq m$ and $j-m$ if $m+1\leq j\leq
2m $.
For a multiindex $\alpha $, define
$$\begin{aligned}
&&\Psi _{\alpha }^{f}(\theta ,t,z,H) \\
&:&=\prod_{l=1}^{d}\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}\sum_{\sigma \in S(m,m)}\int_{\Delta _{0,t}^{2m}}\left\vert f_{\sigma
}(s,z)\right\vert \prod_{j=1}^{2m}\frac{1}{\left\vert
s_{j}-s_{j-1}\right\vert ^{H(d+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma
(j)]}^{(l)})}}ds_{1}...ds_{2m}\end{aligned}$$
respectively, $$\begin{aligned}
&&\Psi _{\alpha }^{\varkappa }(\theta ,t,H) \\
&:&=\prod_{l=1}^{d}\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}\sum_{\sigma \in S(m,m)}\int_{\Delta _{0,t}^{2m}}\left\vert \varkappa
_{\sigma }(s)\right\vert \prod_{j=1}^{2m}\frac{1}{\left\vert
s_{j}-s_{j-1}\right\vert ^{H(d+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma
(j)]}^{(l)})}}ds_{1}...ds_{2m}.\end{aligned}$$
\[mainthmlocaltime\] Suppose that $\Psi _{\alpha }^{f}(\theta
,t,z,H_{r}),\Psi _{\alpha }^{\varkappa }(\theta ,t,H_{r})<\infty $ for some $r\geq r_{0}$. Then, $\Lambda _{\alpha }^{f}(\theta ,t,z)$ as in is a random variable in $L^{2}(\Omega )$. Further, there exists a universal constant $C_{r}=C(T,H_{r},d)>0$ such that$$E[\left\vert \Lambda _{\alpha }^{f}(\theta ,t,z)\right\vert ^{2}]\leq \frac{1}{\lambda _{r}^{2md}}C_{r}^{m+\left\vert \alpha \right\vert }\Psi _{\alpha
}^{f}(\theta ,t,z,H_{r}). \label{supestL}$$Moreover, we have$$\left\vert E[\int_{(\mathbb{R}^{d})^{m}}\Lambda _{\alpha }^{f}(\theta
,t,z)dz]\right\vert \leq \frac{1}{\lambda _{r}^{md}}C_{r}^{m/2+\left\vert
\alpha \right\vert /2}\prod_{j=1}^{m}\left\Vert f_{j}\right\Vert _{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))}(\Psi _{\alpha }^{\varkappa }(\theta
,t,H_{r}))^{1/2}. \label{intestL}$$
For notational simplicity we consider $\theta =0$ and set $\mathbb{B}_{\cdot
}=\mathbb{B}_{\cdot }^{H}$, $\Lambda _{\alpha }^{f}(t,z)=\Lambda _{\alpha
}^{f}(0,t,z).$
For an integrable function $g:(\mathbb{R}^{d})^{m}\longrightarrow \mathbb{C}$ we get that$$\begin{aligned}
&&\left\vert \int_{(\mathbb{R}^{d})^{m}}g(u_{1},...,u_{m})du_{1}...du_{m}\right\vert ^{2} \\
&=&\int_{(\mathbb{R}^{d})^{m}}g(u_{1},...,u_{m})du_{1}...du_{m}\int_{(\mathbb{R}^{d})^{m}}\overline{g(u_{m+1},...,u_{2m})}du_{m+1}...du_{2m} \\
&=&\int_{(\mathbb{R}^{d})^{m}}g(u_{1},...,u_{m})du_{1}...du_{m}(-1)^{dm}\int_{(\mathbb{R}^{d})^{m}}\overline{g(-u_{m+1},...,-u_{2m})}du_{m+1}...du_{2m},\end{aligned}$$where we employed the change of variables $(u_{m+1},...,u_{2m})\longmapsto
(-u_{m+1},...,-u_{2m})$ in the last equality.
This yields$$\begin{aligned}
&&\left\vert \Lambda _{\alpha }^{f}(t,z)\right\vert ^{2} \\
&=&(2\pi )^{-2dm}(-1)^{dm}\int_{(\mathbb{R}^{d})^{2m}}\int_{\Delta
_{0,t}^{m}}\prod_{j=1}^{m}f_{j}(s_{j},z_{j})(-iu_{j})^{\alpha
_{j}}e^{-i\left\langle u_{j},\mathbb{B}_{s_{j}}-z_{j}\right\rangle
}ds_{1}...ds_{m} \\
&&\times \int_{\Delta
_{0,t}^{m}}\prod_{j=m+1}^{2m}f_{[j]}(s_{j},z_{[j]})(-iu_{j})^{\alpha
_{\lbrack j]}}e^{-i\left\langle u_{j},\mathbb{B}_{s_{j}}-z_{[j]}\right\rangle }ds_{m+1}...ds_{2m}du_{1}...du_{2m} \\
&=&(2\pi )^{-2dm}(-1)^{dm}\sum_{\sigma \in S(m,m)}\int_{(\mathbb{R}^{d})^{2m}}\left( \prod_{j=1}^{m}e^{-i\left\langle
z_{j},u_{j}+u_{j+m}\right\rangle }\right) \\
&&\times \int_{\Delta _{0,t}^{2m}}f_{\sigma }(s,z)\prod_{j=1}^{2m}u_{\sigma
(j)}^{\alpha _{\lbrack \sigma (j)]}}\exp \left\{
-\sum_{j=1}^{2m}\left\langle u_{\sigma (j)},\mathbb{B}_{s_{j}}\right\rangle
\right\} ds_{1}...ds_{2m}du_{1}...du_{2m},\end{aligned}$$where we applied shuffling in connection with Section \[VI\_shuffles\] in the last step.
By taking the expectation on both sides in connection with the assumption that the fractional Brownian motions $B_{\cdot }^{i,H_{i}},i\geq 1$ are independent we find that$$\begin{aligned}
\label{Lambda}
\begin{split}
&E[\left| \Lambda _{\alpha }^{f}(t,z)\right|^{2}]
\\
&=(2\pi )^{-2dm}(-1)^{dm}\sum_{\sigma \in S(m,m)}\int_{(\mathbb{R}^{d})^{2m}}\left( \prod_{j=1}^{m}e^{-i\left\langle
z_{j},u_{j}+u_{j+m}\right\rangle }\right) \\
&\times \int_{\Delta _{0,t}^{2m}}f_{\sigma }(s,z)\prod_{j=1}^{2m}u_{\sigma (j)}^{\alpha _{\lbrack \sigma (j)]}}\exp \left\{ -\frac{1}{2}
Var[\sum_{j=1}^{2m}\left\langle u_{\sigma (j)},\mathbb{B}_{s_{j}}\right
\rangle ]\right\} ds_{1}...ds_{2m}du_{1}...du_{2m} \\
&=(2\pi )^{-2dm}(-1)^{dm}\sum_{\sigma \in S(m,m)}\int_{(\mathbb{R}
^{d})^{2m}}\left( \prod_{j=1}^{m}e^{-i\left\langle
z_{j},u_{j}+u_{j+m}\right\rangle }\right) \\
&\times \int_{\Delta _{0,t}^{2m}}f_{\sigma }(s,z)\prod_{j=1}^{2m}u_{\sigma (j)}^{\alpha _{\lbrack \sigma (j)]}}\exp \left\{ -\frac{1}{2}\sum_{n\geq 1}\lambda _{n}^{2}\sum_{l=1}^{d}Var[\sum_{j=1}^{2m}u_{\sigma
(j)}^{(l)}B_{s_{j}}^{(l),n,H_{n}}]\right\}
ds_{1}\dots ds_{2m}du_{1}^{(1)}\dots du_{2m}^{(1)} \\
&\dots du_{1}^{(d)}\dots du_{2m}^{(d)} \\
&=(2\pi )^{-2dm}(-1)^{dm}\sum_{\sigma \in S(m,m)}\int_{(\mathbb{R}^{d})^{2m}}\left( \prod_{j=1}^{m}e^{-i\left\langle
z_{j},u_{j}+u_{j+m}\right\rangle }\right)
\\
&\times \int_{\Delta _{0,t}^{2m}}f_{\sigma }(s,z)\prod_{j=1}^{2m}u_{\sigma(j)}^{\alpha _{\lbrack \sigma (j)]}}\prod_{n\geq
1}\prod_{l=1}^{d}\exp \left\{ -\frac{1}{2}\lambda _{n}^{2}((u_{\sigma
(j)}^{(l)})_{1\leq j\leq 2m})^{\ast }Q_{n}((u_{\sigma (j)}^{(l)})_{1\leq
j\leq 2m})\right\} ds_{1}\dots ds_{2m} \\
&du_{\sigma (1)}^{(1)}\dots du_{\sigma (2m)}^{(1)}\dots du_{\sigma
(1)}^{(d)}\dots du_{\sigma (2m)}^{(d)},
\end{split}\end{aligned}$$ where $\ast $ stands for transposition and where $$Q_{n}=Q_{n}(s):=(E[B_{s_{i}}^{(1)}B_{s_{j}}^{(1)}])_{1\leq i,j\leq 2m}.$$Further, we get that$$\begin{aligned}
\label{Lambda2}
\begin{split}
&\int_{\Delta _{0,t}^{2m}}\left| f_{\sigma }(s,z)\right| \int_{(\mathbb{R}^{d})^{2m}}\prod_{j=1}^{2m}\prod_{l=1}^{d}\left| u_{\sigma
(j)}^{(l)}\right|^{\alpha _{\lbrack \sigma
(j)]}^{(l)}}\prod_{n\geq 1}\prod_{l=1}^{d}\exp \left\{ -\frac{1}{2}\lambda _{n}^{2}((u_{\sigma (j)}^{(l)})_{1\leq j\leq 2m})^{\ast
}Q_{n}((u_{\sigma (j)}^{(l)})_{1\leq j\leq 2m})\right\} \\
&du_{\sigma (1)}^{(1)}\dots du_{\sigma (2m)}^{(1)}\dots du_{\sigma
(1)}^{(d)}\dots du_{\sigma (2m)}^{(d)}ds_{1}\dots ds_{2m} \\
&\leq \int_{\Delta _{0,t}^{2m}}\left| f_{\sigma }(s,z)\right|
\int_{(\mathbb{R}^{d})^{2m}}\prod_{j=1}^{2m}\prod_{l=1}^{d}\left|
u_{j}^{(l)}\right|^{\alpha _{\lbrack \sigma (j)]}^{(l)}} \\
&\times \prod_{l=1}^{d}\exp \left\{ -\frac{1}{2}\lambda_{r}^{2}\left\langle Q_{r}u^{(l)},u^{(l)}\right\rangle \right\} \\
&du_{1}^{(1)}\dots du_{2m}^{(1)}\dots du_{1}^{(d)}\dots du_{2m}^{(d)}ds_{1}\dots ds_{2m} \\
&=\int_{\Delta _{0,t}^{2m}}\left| f_{\sigma }(s,z)\right|
\prod_{l=1}^{d}\int_{\mathbb{R}^{2m}}(\prod_{j=1}^{2m}\left|
u_{j}^{(l)}\right|^{\alpha _{\lbrack \sigma (j)]}^{(l)}})\exp \left\{-
\frac{1}{2}\lambda _{r}^{2}\left\langle Q_{r}u^{(l)},u^{(l)}\right\rangle
\right\} du_{1}^{(l)}\dots du_{2m}^{(l)}ds_{1}\dots ds_{2m},
\end{split}\end{aligned}$$ where $$u^{(l)}:=(u_{j}^{(l)})_{1\leq j\leq 2m}.$$We obtain that$$\begin{aligned}
&&\int_{\mathbb{R}^{2m}}(\prod_{j=1}^{2m}\left\vert u_{j}^{(l)}\right\vert
^{\alpha _{\lbrack \sigma (j)]}^{(l)}})\exp \left\{ -\frac{1}{2}\lambda
_{r}^{2}\left\langle Q_{r}u^{(l)},u^{(l)}\right\rangle \right\}
du_{1}^{(l)}...du_{2m}^{(l)} \\
&=&\frac{1}{\lambda _{r}^{2m}}\frac{1}{(\det Q_{r})^{1/2}}\int_{\mathbb{R}^{2m}}(\prod_{j=1}^{2m}\left\vert \left\langle
Q_{r}^{-1/2}u^{(l)},e_{j}\right\rangle \right\vert ^{\alpha _{\lbrack \sigma
(j)]}^{(l)}})\exp \left\{ -\frac{1}{2}\left\langle
u^{(l)},u^{(l)}\right\rangle \right\} du_{1}^{(l)}...du_{2m}^{(l)},\end{aligned}$$where $e_{i},i=1,...,2m$ is the standard ONB of $\mathbb{R}^{2m}$.
We also have that$$\begin{aligned}
&&\int_{\mathbb{R}^{2m}}(\prod_{j=1}^{2m}\left\vert \left\langle
Q_{r}^{-1/2}u^{(l)},e_{j}\right\rangle \right\vert ^{\alpha _{\lbrack \sigma
(j)]}^{(l)}})\exp \left\{ -\frac{1}{2}\left\langle
u^{(l)},u^{(l)}\right\rangle \right\} du_{1}^{(l)}...du_{2m}^{(l)} \\
&=&(2\pi )^{m}E[\prod_{j=1}^{2m}\left\vert \left\langle
Q_{r}^{-1/2}Z,e_{j}\right\rangle \right\vert ^{\alpha _{\lbrack \sigma
(j)]}^{(l)}}],\end{aligned}$$where$$Z\sim \mathcal{N}(\mathcal{O},I_{2m\times 2m}).$$On the other hand, it follows from Lemma \[LiWei\], which is a type of Brascamp-Lieb inequality, that$$\begin{aligned}
&&E[\prod_{j=1}^{2m}\left\vert \left\langle Q_{r}^{-1/2}Z,e_{j}\right\rangle
\right\vert ^{\alpha _{\lbrack \sigma (j)]}^{(l)}}] \\
&\leq &\sqrt{perm(\sum )}=\sqrt{\sum_{\pi \in S_{2\left\vert \alpha
^{(l)}\right\vert }}\prod_{i=1}^{2\left\vert \alpha ^{(l)}\right\vert
}a_{i\pi (i)}},\end{aligned}$$where $perm(\sum )$ is the permanent of the covariance matrix $\sum
=(a_{ij}) $ of the Gaussian random vector$$\underset{\alpha _{\lbrack \sigma (1)]}^{(l)}\text{ times}}{\underbrace{(\left\langle Q^{-1/2}Z,e_{1}\right\rangle ,...,\left\langle
Q^{-1/2}Z,e_{1}\right\rangle }},\underset{\alpha _{\lbrack \sigma (2)]}^{(l)}\text{ times}}{\underbrace{\left\langle Q^{-1/2}Z,e_{2}\right\rangle
,...,\left\langle Q^{-1/2}Z,e_{2}\right\rangle }},...,\underset{\alpha
_{\lbrack \sigma (2m)]}^{(l)}\text{ times}}{\underbrace{\left\langle
Q^{-1/2}Z,e_{2m}\right\rangle ,...,\left\langle
Q^{-1/2}Z,e_{2m}\right\rangle }}),$$$\left\vert \alpha ^{(l)}\right\vert :=\sum_{j=1}^{m}\alpha _{j}^{(l)}$ and where $S_{n}$ denotes the permutation group of size $n$.
Furthermore, using an upper bound for the permanent of positive semidefinite matrices (see [@AG]) or direct computations, we find that$$perm(\sum )=\sum_{\pi \in S_{2\left\vert \alpha ^{(l)}\right\vert
}}\prod_{i=1}^{2\left\vert \alpha ^{(l)}\right\vert }a_{i\pi (i)}\leq
(2\left\vert \alpha ^{(l)}\right\vert )!\prod_{i=1}^{2\left\vert \alpha
^{(l)}\right\vert }a_{ii}. \label{PSD}$$
Let now $i\in \lbrack \sum_{k=1}^{j-1}\alpha _{\lbrack \sigma
(k)]}^{(l)}+1,\sum_{k=1}^{j}\alpha _{\lbrack \sigma (k)]}^{(l)}]$ for some arbitrary fixed $j\in \{1,...,2m\}$. Then$$a_{ii}=E[\left\langle Q_{r}^{-1/2}Z,e_{j}\right\rangle \left\langle
Q_{r}^{-1/2}Z,e_{j}\right\rangle ].$$
Further, substitution yields$$\begin{aligned}
&&E[\left\langle Q_{r}^{-1/2}Z,e_{j}\right\rangle \left\langle
Q_{r}^{-1/2}Z,e_{j}\right\rangle ] \\
&=&(\det Q_{r})^{1/2}\frac{1}{(2\pi )^{m}}\int_{\mathbb{R}^{2m}}\left\langle
u,e_{j}\right\rangle ^{2}\exp (-\frac{1}{2}\left\langle
Q_{r}u,u\right\rangle )du_{1}...du_{2m} \\
&=&(\det Q_{r})^{1/2}\frac{1}{(2\pi )^{m}}\int_{\mathbb{R}^{2m}}u_{j}^{2}\exp (-\frac{1}{2}\left\langle Q_{r}u,u\right\rangle
)du_{1}...du_{2m}\end{aligned}$$
In the next step, we want to apply Lemma \[CD\]. Then we obtain that$$\begin{aligned}
&&\int_{\mathbb{R}^{2m}}u_{j}^{2}\exp (-\frac{1}{2}\left\langle
Q_{r}u,u\right\rangle )du_{1}...du_{m} \\
&=&\frac{(2\pi )^{(2m-1)/2}}{(\det Q_{r})^{1/2}}\int_{\mathbb{R}}v^{2}\exp (-\frac{1}{2}v^{2})dv\frac{1}{\sigma _{j}^{2}} \\
&=&\frac{(2\pi )^{m}}{(\det Q_{r})^{1/2}}\frac{1}{\sigma _{j}^{2}},\end{aligned}$$where $\sigma _{j}^{2}:=Var[B_{s_{j}}^{H_{r}}\left\vert
B_{s_{1}}^{H_{r}},...,B_{s_{2m}}^{H_{r}}\text{ without }B_{s_{j}}^{H_{r}}\right] .$
We now aim at using strong local non-determinism of the form (see ([2sided]{})): For all $t\in \lbrack 0,T],$ $0<r<t:$$$Var[B_{t}^{H_{r}}\left\vert B_{s}^{H_{r}},\left\vert t-s\right\vert \geq r
\right] \geq Kr^{2H_{r}}$$for a constant $K$ depending on $H_{r}$ and $T$.
The latter entails that $$(\det Q_{r}(s))^{1/2}\geq K^{(2m-1)/2}\left\vert s_{1}\right\vert
^{H_{r}}\left\vert s_{2}-s_{1}\right\vert ^{H_{r}}...\left\vert
s_{2m}-s_{2m-1}\right\vert ^{H_{r}}$$as well as$$\sigma _{j}^{2}\geq K\min \{\left\vert s_{j}-s_{j-1}\right\vert
^{2H_{r}},\left\vert s_{j+1}-s_{j}\right\vert ^{2H_{r}}\}.$$Hence$$\begin{aligned}
\prod_{j=1}^{2m}\sigma _{j}^{-2\alpha _{\lbrack \sigma (j)]}^{(l)}} &\leq
&K^{-2m}\prod_{j=1}^{2m}\frac{1}{\min \{\left\vert s_{j}-s_{j-1}\right\vert
^{2H_{r}\alpha _{\lbrack \sigma (j)]}^{(l)}},\left\vert
s_{j+1}-s_{j}\right\vert ^{2H_{r}\alpha _{\lbrack \sigma (j)]}^{(l)}}\}} \\
&\leq &C^{m+\left\vert \alpha ^{(l)}\right\vert }\prod_{j=1}^{2m}\frac{1}{\left\vert s_{j}-s_{j-1}\right\vert ^{4H_{r}\alpha _{\lbrack \sigma
(j)]}^{(l)}}}\end{aligned}$$for a constant $C$ only depending on $H_{r}$ and $T$.
So we conclude from (\[PSD\]) that$$\begin{aligned}
perm(\sum ) &\leq &(2\left\vert \alpha ^{(l)}\right\vert
)!\prod_{i=1}^{2\left\vert \alpha ^{(l)}\right\vert }a_{ii} \\
&\leq &(2\left\vert \alpha ^{(l)}\right\vert )!\prod_{j=1}^{2m}((\det
Q_{r})^{1/2}\frac{1}{(2\pi )^{m}}\frac{(2\pi )^{m}}{(\det Q_{r})^{1/2}}\frac{1}{\sigma _{j}^{2}})^{\alpha _{\lbrack \sigma (j)]}^{(l)}} \\
&\leq &(2\left\vert \alpha ^{(l)}\right\vert )!C^{m+\left\vert \alpha
^{(l)}\right\vert }\prod_{j=1}^{2m}\frac{1}{\left\vert
s_{j}-s_{j-1}\right\vert ^{4H_{r}\alpha _{\lbrack \sigma (j)]}^{(l)}}}.\end{aligned}$$Thus$$\begin{aligned}
&&E[\prod_{j=1}^{2m}\left\vert \left\langle Q_{r}^{-1/2}Z,e_{j}\right\rangle
\right\vert ^{\alpha _{\lbrack \sigma (j)]}^{(l)}}]\leq \sqrt{perm(\sum )} \\
&\leq &\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}C^{m+\left\vert \alpha
^{(l)}\right\vert }\prod_{j=1}^{2m}\frac{1}{\left\vert
s_{j}-s_{j-1}\right\vert ^{2H_{r}\alpha _{\lbrack \sigma (j)]}^{(l)}}}.\end{aligned}$$Therefore we see from (\[Lambda\]) and (\[Lambda2\]) that$$\begin{aligned}
&&E[\left\vert \Lambda _{\alpha }^{f}(\theta ,t,z)\right\vert ^{2}] \\
&\leq &C^{m}\sum_{\sigma \in S(m,m)}\int_{\Delta _{0,t}^{2m}}\left\vert
f_{\sigma }(s,z)\right\vert \prod_{l=1}^{d}\int_{\mathbb{R}^{2m}}(\prod_{j=1}^{2m}\left\vert u_{j}^{(l)}\right\vert ^{\alpha _{\lbrack
\sigma (j)]}^{(l)}})\exp \left\{ -\frac{1}{2}\left\langle
Q_{r}u^{(l)},u^{(l)}\right\rangle \right\}
du_{1}^{(l)}...du_{2m}^{(l)}ds_{1}...ds_{2m} \\
&\leq &M^{m}\sum_{\sigma \in S(m,m)}\int_{\Delta _{0,t}^{2m}}\left\vert
f_{\sigma }(s,z)\right\vert \frac{1}{\lambda _{r}^{2md}}\frac{1}{(\det
Q(s))^{d/2}}\prod_{l=1}^{d}\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}C^{m+\left\vert \alpha ^{(l)}\right\vert }\prod_{j=1}^{2m}\frac{1}{\left\vert s_{j}-s_{j-1}\right\vert ^{2H_{r}\alpha _{\lbrack \sigma
(j)]}^{(l)}}}ds_{1}...ds_{2m} \\
&=&\frac{1}{\lambda _{r}^{2md}}M^{m}C^{md+\left\vert \alpha \right\vert
}\prod_{l=1}^{d}\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}\sum_{\sigma
\in S(m,m)}\int_{\Delta _{0,t}^{2m}}\left\vert f_{\sigma }(s,z)\right\vert
\prod_{j=1}^{2m}\frac{1}{\left\vert s_{j}-s_{j-1}\right\vert
^{H_{r}(d+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma (j)]}^{(l)})}}ds_{1}...ds_{2m}\end{aligned}$$for a constant $M$ depending on $d$.
In the final step, we want to prove estimate (\[intestL\]). Using the inequality (\[supestL\]), we get that$$\begin{aligned}
&&\left\vert E\left[ \int_{(\mathbb{R}^{d})^{m}}\Lambda _{\alpha
}^{\varkappa f}(\theta ,t,z)dz\right] \right\vert \\
&\leq &\int_{(\mathbb{R}^{d})^{m}}(E[\left\vert \Lambda _{\alpha
}^{\varkappa f}(\theta ,t,z)\right\vert ^{2})^{1/2}dz\leq \frac{1}{\lambda
_{r}^{md}}C^{m/2+\left\vert \alpha \right\vert /2}\int_{(\mathbb{R}^{d})^{m}}(\Psi _{\alpha }^{\varkappa f}(\theta ,t,z,H_{r}))^{1/2}dz.\end{aligned}$$By taking the supremum over $[0,T]$ with respect to each function $f_{j}$, i.e.$$\left\vert f_{[\sigma (j)]}(s_{j},z_{[\sigma (j)]})\right\vert \leq
\sup_{s_{j}\in \lbrack 0,T]}\left\vert f_{[\sigma (j)]}(s_{j},z_{[\sigma
(j)]})\right\vert ,j=1,...,2m$$we find that$$\begin{aligned}
&&\left\vert E\left[ \int_{(\mathbb{R}^{d})^{m}}\Lambda _{\alpha
}^{\varkappa f}(\theta ,t,z)dz\right] \right\vert \\
&\leq &\frac{1}{\lambda _{r}^{md}}C^{m/2+\left\vert \alpha \right\vert
/2}\max_{\sigma \in S(m,m)}\int_{(\mathbb{R}^{d})^{m}}\left(
\prod_{l=1}^{2m}\left\Vert f_{[\sigma (l)]}(\cdot ,z_{[\sigma
(l)]})\right\Vert _{L^{\infty }([0,T])}\right) ^{1/2}dz \\
&&\times (\prod_{l=1}^{d}\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}\sum_{\sigma \in S(m,m)}\int_{\Delta _{0,t}^{2m}}\left\vert \varkappa
_{\sigma }(s)\right\vert \prod_{j=1}^{2m}\frac{1}{\left\vert
s_{j}-s_{j-1}\right\vert ^{H(d+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma
(j)]}^{(l)})}}ds_{1}...ds_{2m})^{1/2} \\
&=&\frac{1}{\lambda _{r}^{md}}C^{m/2+\left\vert \alpha \right\vert
/2}\max_{\sigma \in S(m,m)}\int_{(\mathbb{R}^{d})^{m}}\left(
\prod_{l=1}^{2m}\left\Vert f_{[\sigma (l)]}(\cdot ,z_{[\sigma
(l)]})\right\Vert _{L^{\infty }([0,T])}\right) ^{1/2}dz\cdot (\Psi _{\alpha
}^{\varkappa }(\theta ,t,H_{r}))^{1/2} \\
&=&\frac{1}{\lambda _{r}^{md}}C^{m/2+\left\vert \alpha \right\vert /2}\int_{(\mathbb{R}^{d})^{m}}\prod_{j=1}^{m}\left\Vert f_{j}(\cdot ,z_{j})\right\Vert
_{L^{\infty }([0,T])}dz\cdot (\Psi _{\alpha }^{\varkappa }(\theta
,t,H_{r}))^{1/2} \\
&=&\frac{1}{\lambda _{r}^{md}}C^{m/2+\left\vert \alpha \right\vert
/2}\prod_{j=1}^{m}\left\Vert f_{j}(\cdot ,z_{j})\right\Vert _{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))}\cdot (\Psi _{\alpha }^{\varkappa }(\theta
,t,H_{r}))^{1/2}.\end{aligned}$$
Using Theorem \[mainthmlocaltime\] we obtain the following crucial estimate (compare [@BNP.17], [@BOPP.17], [@ABP] and [@ACHP]):
\[mainestimate1\] Let the functions $f$ and $\varkappa $ be as in ([f]{}), respectively as in (\[kappa\]). Further, let $\theta ,\theta \prime
,t\in \lbrack 0,T],\theta \prime <\theta <t$ and$$\varkappa _{j}(s)=(K_{H_{r_{0}}}(s,\theta )-K_{H_{r_{0}}}(s,\theta \prime
))^{\varepsilon _{j}},\theta <s<t$$for every $j=1,...,m$ with $(\varepsilon _{1},...,\varepsilon _{m})\in
\{0,1\}^{m}$ for $\theta ,\theta \prime \in \lbrack 0,T]$ with $\theta
\prime <\theta .$ Let $\alpha \in (\mathbb{N}_{0}^{d})^{m}$ be a multi-index. If for some $r\geq r_{0}$ $$H_{r}<\frac{\frac{1}{2}-\gamma _{r_{0}}}{(d-1+2\sum_{l=1}^{d}\alpha
_{j}^{(l)})}$$holds for all $j$, where $\gamma _{r_{0}}\in (0,H_{r_{0}})$ is sufficiently small, then there exists a universal constant $C_{r_{0}}$ (depending on $H_{r_{0}}$, $T$ and $d$, but independent of $m$, $\{f_{i}\}_{i=1,...,m}$ and $\alpha $) such that for any $\theta ,t\in \lbrack 0,T]$ with $\theta <t$ we have$$\begin{aligned}
&&\left\vert E\int_{\Delta _{\theta ,t}^{m}}\left( \prod_{j=1}^{m}D^{\alpha
_{j}}f_{j}(s_{j},\mathbb{B}_{s_{j}})\varkappa _{j}(s_{j})\right)
ds\right\vert \\
&\leq &\frac{1}{\lambda _{r}^{md}}C_{r_{0}}^{m+\left\vert \alpha \right\vert
}\prod_{j=1}^{m}\left\Vert f_{j}(\cdot ,z_{j})\right\Vert _{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))}\left( \frac{\theta -\theta \prime }{\theta \theta
\prime }\right) ^{\gamma _{r_{0}}\sum_{j=1}^{m}\varepsilon _{j}}\theta
^{(H_{r_{0}}-\frac{1}{2}-\gamma _{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}} \\
&&\times \frac{(\prod_{l=1}^{d}(2\left\vert \alpha ^{(l)}\right\vert
)!)^{1/4}(t-\theta )^{-H_{r}(md+2\left\vert \alpha \right\vert )+(H_{r_{0}}-\frac{1}{2}-\gamma _{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}+m}}{\Gamma
(-H_{r}(2md+4\left\vert \alpha \right\vert )+2(H_{r_{0}}-\frac{1}{2}-\gamma
_{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}+2m)^{1/2}}.\end{aligned}$$
From the definition of $\Lambda _{\alpha }^{\varkappa f}$ (\[LambdaDef\]) we see that the integral in our proposition can be expressed as$$\int_{\Delta _{\theta ,t}^{m}}\left( \prod_{j=1}^{m}D^{\alpha
_{j}}f_{j}(s_{j},B_{s_{j}}^{H})\varkappa _{j}(s_{j})\right) ds=\int_{\mathbb{R}^{dm}}\Lambda _{\alpha }^{\varkappa f}(\theta ,t,z)dz.$$By taking expectation and using Theorem \[mainthmlocaltime\] we get that$$\left\vert E\int_{\Delta _{\theta ,t}^{m}}\left( \prod_{j=1}^{m}D^{\alpha
_{j}}f_{j}(s_{j},B_{s_{j}}^{H})\varkappa _{j}(s_{j})\right) ds\right\vert
\leq \frac{1}{\lambda _{r}^{md}}C_{r}^{m/2+\left\vert \alpha \right\vert
/2}\prod_{j=1}^{m}\left\Vert f_{j}(\cdot ,z_{j})\right\Vert _{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))}\cdot (\Psi _{\alpha }^{\varkappa }(\theta
,t,H_{r}))^{1/2},$$where in this case $$\begin{aligned}
&&\Psi _{k}^{\varkappa }(\theta ,t,H_{r}) \\
&:&=\prod_{l=1}^{d}\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}\sum_{\sigma \in S(m,m)}\int_{\Delta
_{0,t}^{2m}}\prod_{j=1}^{2m}(K_{H_{r}}(s_{j},\theta )-K_{H_{r}}(s_{j},\theta
\prime ))^{\varepsilon _{\lbrack \sigma (j)]}} \\
&&\frac{1}{\left\vert s_{j}-s_{j-1}\right\vert
^{H_{r}(d+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma (j)]}^{(l)})}}ds_{1}...ds_{2m}.\end{aligned}$$We wish to use Lemma \[VI\_iterativeInt\]. For this purpose, we need that $-H_{r}(d+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma (j)]}^{(l)})+(H_{r_{0}}-\frac{1}{2}-\gamma _{r_{0}})\varepsilon _{\lbrack \sigma (j)]}>-1$ for all $j=1,...,2m.$ The worst case is, when $\varepsilon _{\lbrack \sigma (j)]}=1$ for all $j$. So $H_{r}<\frac{\frac{1}{2}-\gamma _{r}}{(d-1+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma (j)]}^{(l)})}$ for all $j$, since $H_{r_{0}}\geq
H_{r}$. Therfore, we get that$$\begin{aligned}
\Psi _{\alpha }^{\varkappa }(\theta ,t,H_{r}) &\leq
&C_{r_{0}}^{2m}\sum_{\sigma \in S(m,m)}\left( \frac{\theta -\theta \prime }{\theta \theta \prime }\right) ^{\gamma _{r_{0}}\sum_{j=1}^{2m}\varepsilon
_{\lbrack \sigma (j)]}}\theta ^{(H_{r_{0}}-\frac{1}{2}-\gamma
_{r_{0}})\sum_{j=1}^{2m}\varepsilon _{\lbrack \sigma (j)]}} \\
&&\times \prod_{l=1}^{d}\sqrt{(2\left\vert \alpha ^{(l)}\right\vert )!}\Pi
_{\gamma }(2m)(t-\theta )^{-H_{r}(2md+4\left\vert \alpha \right\vert
)+(H_{r}-\frac{1}{2}-\gamma _{r})\sum_{j=1}^{2m}\varepsilon _{\lbrack \sigma
(j)]}+2m},\end{aligned}$$where $\Pi _{\gamma }(m)$ is defined as in Lemma \[VI\_iterativeInt\] and where $C_{r_{0}}$ is a constant, which only depends on $H_{r_{0}}$ and $T$. The factor $\Pi _{\gamma }(m)$ has the following upper bound: $$\Pi _{\gamma }(2m)\leq \frac{\prod_{j=1}^{2m}\Gamma
(1-H_{r}(d+2\sum_{l=1}^{d}\alpha _{\lbrack \sigma (j)]}^{(l)}))}{\Gamma
(-H_{r}(2md+4\left\vert \alpha \right\vert )+(H_{r_{0}}-\frac{1}{2}-\gamma
_{r_{0}})\sum_{j=1}^{2m}\varepsilon _{\lbrack \sigma (j)]}+2m)}.$$Note that $\sum_{j=1}^{2m}\varepsilon _{\lbrack \sigma
(j)]}=2\sum_{j=1}^{m}\varepsilon _{j}.$ Hence, it follows that$$\begin{aligned}
&&(\Psi _{k}^{\varkappa }(\theta ,t,H_{r}))^{1/2} \\
&\leq &C_{r_{0}}^{m}\left( \frac{\theta -\theta \prime }{\theta \theta
\prime }\right) ^{\gamma _{r_{0}}\sum_{j=1}^{m}\varepsilon _{j}}\theta
^{(H_{r}-\frac{1}{2}-\gamma _{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}} \\
&&\times \frac{(\prod_{l=1}^{d}(2\left\vert \alpha ^{(l)}\right\vert
)!)^{1/4}(t-\theta )^{-H_{r}(md+2\left\vert \alpha \right\vert )-(H_{r_{0}}-\frac{1}{2}-\gamma _{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}+m}}{\Gamma
(-H_{r}(2md+4\left\vert \alpha \right\vert )+2(H_{r_{0}}-\frac{1}{2}-\gamma
_{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}+2m)^{1/2}},\end{aligned}$$where we used $\prod_{j=1}^{2m}\Gamma (1-H_{r}(d+2\sum_{l=1}^{d}\alpha
_{\lbrack \sigma (j)]}^{(l)})\leq K^{m}$ for a constant $K=K(\gamma
_{r_{0}})>0$ and $\sqrt{a_{1}+...+a_{m}}\leq \sqrt{a_{1}}+...\sqrt{a_{m}}$ for arbitrary non-negative numbers $a_{1},...,a_{m}$.
\[mainestimate2\] Let the functions $f$ and $\varkappa $ be as in ([f]{}), respectively as in (\[kappa\]). Let $\theta ,t\in \lbrack 0,T]$ with $\theta <t$ and$$\varkappa _{j}(s)=(K_{H_{r_{0}}}(s,\theta ))^{\varepsilon _{j}},\theta <s<t$$for every $j=1,...,m$ with $(\varepsilon _{1},...,\varepsilon _{m})\in
\{0,1\}^{m}$. Let $\alpha \in (\mathbb{N}_{0}^{d})^{m}$ be a multi-index. If for some $r\geq r_{0}$ $$H_{r}<\frac{\frac{1}{2}-\gamma _{r_{0}}}{(d-1+2\sum_{l=1}^{d}\alpha
_{j}^{(l)})}$$holds for all $j$, where $\gamma _{r_{0}}\in (0,H_{r_{0}})$ is sufficiently small, then there exists a universal constant $C_{r_{0}}$ (depending on $H_{r_{0}}$, $T$ and $d$, but independent of $m$, $\{f_{i}\}_{i=1,...,m}$ and $\alpha $) such that for any $\theta ,t\in \lbrack 0,T]$ with $\theta <t$ we have$$\begin{aligned}
&&\left\vert E\int_{\Delta _{\theta ,t}^{m}}\left( \prod_{j=1}^{m}D^{\alpha
_{j}}f_{j}(s_{j},\mathbb{B}_{s_{j}})\varkappa _{j}(s_{j})\right)
ds\right\vert \\
&\leq &\frac{1}{\lambda _{r}^{md}}C_{r_{0}}^{m+\left\vert \alpha \right\vert
}\prod_{j=1}^{m}\left\Vert f_{j}(\cdot ,z_{j})\right\Vert _{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))}\theta ^{(H_{r_{0}}-\frac{1}{2})\sum_{j=1}^{m}\varepsilon _{j}} \\
&&\times \frac{(\prod_{l=1}^{d}(2\left\vert \alpha ^{(l)}\right\vert
)!)^{1/4}(t-\theta )^{-H_{r}(md+2\left\vert \alpha \right\vert )+(H_{r_{0}}-\frac{1}{2}-\gamma _{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}+m}}{\Gamma
(-H_{r}(2md+4\left\vert \alpha \right\vert )+2(H_{r_{0}}-\frac{1}{2}-\gamma
_{r_{0}})\sum_{j=1}^{m}\varepsilon _{j}+2m)^{1/2}}.\end{aligned}$$
The proof is similar to the previous proposition.
\[Remark 3.4\] We mention that$$\prod_{l=1}^{d}(2\left\vert \alpha ^{(l)}\right\vert )!\leq (2\left\vert
\alpha \right\vert )!C^{\left\vert \alpha \right\vert }$$for a constant $C$ depending on $d$. Later on in the paper, when we deal with the existence of strong solutions, we will consider the case$$\alpha _{j}^{(l)}\in \{0,1\}\text{ for all }j,l$$with$$\left\vert \alpha \right\vert =m.$$
The next proposition is a verification of the sufficient condition needed to guarantee relative compactness of the approximating sequence $\{X_{t}^{n}\}_{n\geq 1}$.
\[Holderintegral\] Let $b_{n}:[0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}^{d}$, $n\geq 1$, be a sequence of compactly supported smooth functions converging a.e. to $b$ such that $\sup_{n\geq 1}\Vert b_{n}\Vert _{\mathcal{L}_{2,p}^{q}}<\infty $, $p,q\in (2,\infty ]$. Let $X_{\cdot }^{n}$ denote the solution of when we replace $b$ by $b_{n}$. Further, let $C_{i}$ for $r_{0}=i$ be the (same) constant (depending only on $H_{i}$, $T$ and $d$) in the estimates of Proposition \[mainestimate1\] and [mainestimate2]{}. Then there exist sequences $\{\alpha _{i}\}_{i=1}^{\infty }$, $\beta =\{\beta _{i}\}_{i=1}^{\infty }$ (depending only on $\{H_{i}\}_{i=1}^{\infty }$) with $0<\alpha _{i}<\beta _{i}<\frac{1}{2}$, $\delta =\{\delta _{i}\}_{i=1}^{\infty }$ as in Theorem \[compinf\] and $\lambda =\{\lambda _{i}\}_{i=1}^{\infty }$ in (\[monster\]), which satisfies (\[lambdacond\]), (\[lambdacond2\]), (\[contcond\]) and which is of the form $\lambda _{i}={\varphi}_{i}\cdot \varphi (C_{i})$ being independent of the size of $\sup_{n\geq 1}\Vert b_{n}\Vert _{\mathcal{L}_{2,p}^{q}}$ for a sequence $\{{\varphi}_{i}\}_{i=1}^{\infty }$ and a bounded function $\varphi$ , such that $$\sum_{i=1}^{\infty }\frac{|{\varphi}_{i}|^{2}}{1-2^{-2(\beta _{i}-\alpha
_{i})}\delta _{i}^{2}}<\infty , \label{Finite}$$$$\sup_{n\geq 1}E[\Vert X_{t}^{n}\Vert ^{2}]<\infty ,$$$$\sup_{n\geq 1}\sum_{i=1}^{\infty }\frac{1}{\delta _{i}^{2}}\int_{0}^{t}E[\Vert D_{t_{0}}^{i}X_{t}^{n}\Vert ^{2}]dt_{0}\leq
C_{1}(\sup_{n\geq 1}\Vert b_{n}\Vert _{\mathcal{L}_{2,p}^{q}})<\infty ,$$and $$\begin{aligned}
&&\sup_{n\geq 1}\sum_{i=1}^{\infty }\frac{1}{(1-2^{-2(\beta _{i}-\alpha
_{i})})\delta _{i}^{2}}\int_{0}^{t}\int_{0}^{t}\frac{E[\Vert
D_{t_{0}}^{i}X_{t}^{n}-D_{t_{0}^{\prime }}^{i}X_{t}^{n}\Vert ^{2}]}{|t_{0}-t_{0}^{\prime }|^{1+2\beta _{i}}}dt_{0}dt_{0}^{\prime } \\
&\leq &C_{2}(\sup_{n\geq 1}\Vert b_{n}\Vert _{\mathcal{L}_{2,p}^{q}})<\infty \end{aligned}$$for all $t\in \lbrack 0,T]$, where $C_{j}:[0,\infty )\longrightarrow \lbrack
0,\infty ),$ $j=1,2$ are continuous functions depending on $\{H_{i}\}_{i=1}^{\infty }$, $p$, $q$, $d$, $T$ and where $D^{i}$ denotes the Malliavin derivative in the direction of the standard Brownian motion $W^{i}$, $i\geq 1$. Here, $\Vert \cdot \Vert $ denotes any matrix norm.
\[Phi\]The proof Proposition \[Holderintegral\] shows that one may for example choose $\lambda _{i}={\varphi}_{i}\cdot \varphi (C_{i})$ in ([monster]{}) for $\varphi (x)=\exp (-x^{100})$ and $\{{\varphi}_{i}\}_{i=1}^{\infty }$ satisfying (\[Finite\]).
The most challenging estimate is the last one, the two others can be proven easily. Take $t_{0},t_{0}^{\prime }>0$ such that $0<t_{0}^{\prime }<t_{0}<t$. Using the chain rule for the Malliavin derivative, see [@Nua10 Proposition 1.2.3], we have $$D_{t_{0}}^{i}X_{t}^{n}=\lambda
_{i}K_{H_{i}}(t,t_{0})I_{d}+\int_{t_{0}}^{t}b_{n}^{\prime
}(t_{1},X_{t_{1}}^{n})D_{t_{0}}X_{t_{1}}^{n}dt_{1}$$$P$-a.s. for all $0\leq t_{0}\leq t$ where $b_{n}^{\prime }(t,z)=\left(
\frac{\partial }{\partial z_{j}}b_{n}^{(i)}(t,z)\right) _{i,j=1,\dots ,d}$ denotes the Jacobian matrix of $b_{n}$ at a point $(t,z)$ and $I_{d}$ the identity matrix in ${\mathbb R}^{d\times d}$. Thus we have
$$\begin{aligned}
D_{t_{0}}^{i}X_{t}^{n}-& D_{t_{0}^{\prime }}^{i}X_{t}^{n}=\lambda
_{i}(K_{H_{i}}(t,t_{0})I_{d}-K_{H_{i}}(t,t_{0}^{\prime })I_{d}) \\
& +\int_{t_{0}}^{t}b_{n}^{\prime
}(t_{1},X_{t_{1}}^{n})D_{t_{0}}^{i}X_{t_{1}}^{n}dt_{1}-\int_{t_{0}^{\prime
}}^{t}b_{n}^{\prime }(t_{1},X_{t_{1}}^{n})D_{t_{0}^{\prime
}}^{i}X_{t_{1}}^{n}dt_{1} \\
=& \lambda _{i}(K_{H_{i}}(t,t_{0})I_{d}-K_{H_{i}}(t,t_{0}^{\prime })I_{d}) \\
& -\int_{t_{0}^{\prime }}^{t_{0}}b_{n}^{\prime
}(t_{1},X_{t_{1}}^{n})D_{t_{0}^{\prime
}}^{i}X_{t_{1}}^{n}dt_{1}+\int_{t_{0}}^{t}b_{n}^{\prime
}(t_{1},X_{t_{1}}^{n})(D_{t_{0}}^{i}X_{t_{1}}^{n}-D_{t_{0}^{\prime
}}^{i}X_{t_{1}}^{n})dt_{1} \\
=& \lambda _{i}\mathcal{K}_{t_{0},t_{0}^{\prime
}}^{H_{i}}(t)I_{d}-(D_{t_{0}^{\prime }}^{i}X_{t_{0}}^{n}-\lambda
_{i}K_{H_{i}}(t_{0},t_{0}^{\prime })I_{d}) \\
& +\int_{t_{0}}^{t}b_{n}^{\prime
}(t_{1},X_{t_{1}}^{n})(D_{t_{0}}^{i}X_{t_{1}}^{n}-D_{t_{0}^{\prime
}}^{i}X_{t_{1}}^{n})dt_{1},\end{aligned}$$
where as in Proposition \[mainestimate1\] we define $$\mathcal{K}_{t_{0},t_{0}^{\prime
}}^{H_{i}}(t)=K_{H_{i}}(t,t_{0})-K_{H_{i}}(t,t_{0}^{\prime }).$$
Iterating the above equation we arrive at
$$\begin{aligned}
D_{t_{0}}^{i}X_{t}^{n}-& D_{t_{0}^{\prime }}^{i}X_{t}^{n}=\lambda _{i}\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(t)I_{d} \\
& +\lambda _{i}\sum_{m=1}^{\infty }\int_{\Delta
_{t_{0},t}^{m}}\prod_{j=1}^{m}b_{n}^{\prime }(t_{j},X_{t_{j}}^{n})\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(t_{m})I_{d}dt_{m}\cdots dt_{1} \\
& -\left( I_{d}+\sum_{m=1}^{\infty }\int_{\Delta
_{t_{0},t}^{m}}\prod_{j=1}^{m}b_{n}^{\prime
}(t_{j},X_{t_{j}}^{n})dt_{m}\cdots dt_{1}\right) \left( D_{t_{0}^{\prime
}}^{i}X_{t_{0}}^{n}-\lambda _{i}K_{H_{i}}(t_{0},t_{0}^{\prime })I_{d}\right)
.\end{aligned}$$
On the other hand, observe that one may again write $$D_{t_{0}^{\prime }}^{i}X_{t_{0}}^{n}-\lambda
_{i}K_{H_{i}}(t_{0},t_{0}^{\prime })I_{d}=\lambda _{i}\sum_{m=1}^{\infty
}\int_{\Delta _{t_{0}^{\prime },t_{0}}^{m}}\prod_{j=1}^{m}b_{n}^{\prime
}(t_{j},X_{t_{j}}^{n})(K_{H_{i}}(t_{m},t_{0}^{\prime })I_{d})\,dt_{m}\cdots
dt_{1}.$$In summary, $$D_{t_{0}}^{i}X_{t}^{n}-D_{t_{0}^{\prime }}^{i}X_{t}^{n}=\lambda
_{i}I_{1}(t_{0}^{\prime },t_{0})+\lambda _{i}I_{2}^{n}(t_{0}^{\prime
},t_{0})+\lambda _{i}I_{3}^{n}(t_{0}^{\prime },t_{0}),$$where $$\begin{aligned}
I_{1}(t_{0}^{\prime },t_{0}):=& \mathcal{K}_{t_{0},t_{0}^{\prime
}}^{H_{i}}(t)I_{d}=K_{H_{i}}(t,t_{0})I_{d}-K_{H_{i}}(t,t_{0}^{\prime })I_{d}
\\
I_{2}^{n}(t_{0}^{\prime },t_{0}):=& \sum_{m=1}^{\infty }\int_{\Delta
_{t_{0},t}^{m}}\prod_{j=1}^{m}b_{n}^{\prime }(t_{j},X_{t_{j}}^{n})\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(t_{m})I_{d}\ dt_{m}\cdots dt_{1} \\
I_{3}^{n}(t_{0}^{\prime },t_{0}):=& -\left( I_{d}+\sum_{m=1}^{\infty
}\int_{\Delta _{t_{0},t}^{m}}\prod_{j=1}^{m}b_{n}^{\prime
}(t_{j},X_{t_{j}}^{n})dt_{m}\cdots dt_{1}\right) \\
& \times \left( \sum_{m=1}^{\infty }\int_{\Delta _{t_{0}^{\prime
},t_{0}}^{m}}\prod_{j=1}^{m}b_{n}^{\prime
}(t_{j},X_{t_{j}}^{n})(K_{H_{i}}(t_{m},t_{0}^{\prime })I_{d})dt_{m}\cdots
dt_{1}.\right) .\end{aligned}$$
Hence, $$E[\Vert D_{t_{0}}^{i}X_{t}^{n}-D_{t_{0}^{\prime }}^{i}X_{t}^{n}\Vert
^{2}]\leq C\lambda _{i}^{2}\left( E[\Vert I_{1}(t_{0}^{\prime },t_{0})\Vert
^{2}]+E[\Vert I_{2}^{n}(t_{0}^{\prime },t_{0})\Vert ^{2}]+E[\Vert
I_{3}^{n}(t_{0}^{\prime },t_{0})\Vert ^{2}]\right) .$$
It follows from Lemma \[VI\_doubleint\] and condition that $$\begin{aligned}
\sum_{i=1}^{\infty }\frac{\lambda _{i}^{2}}{1-2^{-2(\beta _{i}-\alpha
_{i})}\delta _{i}^{2}}& \int_{0}^{t}\int_{0}^{t}\frac{\Vert
I_{1}(t_{0}^{\prime },t_{0})\Vert _{L^{2}(\Omega )}^{2}}{|t_{0}-t_{0}^{\prime }|^{1+2\beta _{i}}}dt_{0}dt_{0}^{\prime } \\
& \leq \sum_{i=1}^{\infty }\frac{\lambda _{i}^{2}}{1-2^{-2(\beta _{i}-\alpha
_{i})}\delta _{i}^{2}}t^{4H_{i}-6\gamma _{i}-2\beta _{i}-1}<\infty\end{aligned}$$for a suitable choice of sequence $\{\beta _{i}\}_{i\geq 1}\subset (0,1/2)$.
Let us continue with the term $I_{2}^{n}(t_{0}^{\prime },t_{0})$. Then Theorem \[girsanov\], Cauchy-Schwarz inequality and Lemma \[novikov\] imply $$\begin{aligned}
E[& \Vert I_{2}^{n}(t_{0}^{\prime },t_{0})\Vert ^{2}] \\
& \leq C(\Vert b_{n}\Vert _{L_{p}^{q}})E\left[ \left\Vert \sum_{m=1}^{\infty
}\int_{\Delta _{t_{0},t}^{m}}\prod_{j=1}^{m}b_{n}^{\prime }(t_{j},x+\mathbb{B}_{t_{j}}^{H})\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(t_{m})I_{d}\
dt_{m}\cdots dt_{1}\right\Vert ^{4}\right] ^{1/2},\end{aligned}$$where $C:[0,\infty )\rightarrow \lbrack 0,\infty )$ is the function from Lemma \[novikov\]. Taking the supremum over $n$ we have $$\sup_{n\geq 0}C(\Vert b_{n}\Vert _{L_{p}^{q}})=:C_{1}<\infty .$$
Let $\Vert \cdot \Vert $ from now on denote the matrix norm in ${\mathbb R}^{d\times
d}$ such that $\Vert A\Vert =\sum_{i,j=1}^{d}|a_{ij}|$ for a matrix $A=\{a_{ij}\}_{i,j=1,\dots ,d}$, then we have
$$\begin{aligned}
& E[\Vert I_{2}^{n}(t_{0}^{\prime },t_{0})\Vert ^{2}]\leq C_{1}\Bigg(\sum_{m=1}^{\infty }\sum_{j,k=1}^{d}\sum_{l_{1},\dots ,l_{m-1}=1}^{d}\Bigg\|\int_{\Delta _{t_{0},t}^{m}}\frac{\partial }{\partial x_{l_{1}}}b_{n}^{(j)}(t_{1},x+\mathbb{B}_{t_{1}}^{H}) \notag \\
& \times \frac{\partial }{\partial x_{l_{2}}}b_{n}^{(l_{1})}(t_{2},x+\mathbb{B}_{t_{2}}^{H})\cdots \frac{\partial }{\partial x_{k}}b_{n}^{(l_{m-1})}(t_{m},x+\mathbb{B}_{t_{m}}^{H})\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(t_{m})dt_{m}\cdots dt_{1}\Bigg\|_{L^{4}(\Omega ,{\mathbb R})}\Bigg)^{2}. \label{I2}\end{aligned}$$
Now, the aim is to shuffle the four integrals above. Denote $$\label{VI_I}
J_{2}^{n}(t_{0}^{\prime },t_{0}):=\int_{\Delta _{t_{0},t}^{m}}\frac{\partial
}{\partial x_{l_{1}}}b_{n}^{(j)}(t_{1},x+\mathbb{B}_{t_{1}}^{H})\cdots \frac{\partial }{\partial x_{k}}b_{n}^{(l_{m-1})}(t_{m},x+\mathbb{B}_{t_{m}}^{H})\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(t_{m})dt.$$
Then, shuffling $J_{2}^{n}(t_{0}^{\prime },t_{0})$ as shown in , one can write $(J_{2}^{n}(t_{0}^{\prime
},t_{0}))^{2}$ as a sum of at most $2^{2m}$ summands of length $2m$ of the form $$\int_{\Delta _{t_{0},t}^{2m}}g_{1}^{n}(t_{1},x+\mathbb{B}_{t_{1}}^{H})\cdots
g_{2m}^{n}(t_{2m},x+\mathbb{B}_{t_{2m}}^{H})dt_{2m}\cdots dt_{1},$$where for each $l=1,\dots ,2m$, $$g_{l}^{n}(\cdot ,x+\mathbb{B}_{\cdot }^{H})\in \left\{ \frac{\partial }{\partial x_{k}}b_{n}^{(j)}(\cdot ,x+\mathbb{B}_{\cdot }^{H}),\frac{\partial
}{\partial x_{k}}b_{n}^{(j)}(\cdot ,x+\mathbb{B}_{\cdot }^{H})\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(\cdot ),\,j,k=1,\dots ,d\right\} .$$
Repeating this argument once again, we find that $J_{2}^{n}(t_{0}^{\prime
},t_{0})^{4}$ can be expressed as a sum of, at most, $2^{8m}$ summands of length $4m$ of the form $$\label{VI_III}
\int_{\Delta _{t_{0},t}^{4m}}g_{1}^{n}(t_{1},x+\mathbb{B}_{t_{1}}^{H})\cdots
g_{4m}^{n}(t_{4m},x+\mathbb{B}_{t_{4m}}^{H})dt_{4m}\cdots dt_{1},$$where for each $l=1,\dots ,4m$, $$g_{l}^{n}(\cdot ,x+\mathbb{B}_{\cdot }^{H})\in \left\{ \frac{\partial }{\partial x_{k}}b_{n}^{(j)}(\cdot ,x+\mathbb{B}_{\cdot }^{x}H\frac{\partial }{\partial x_{k}}b_{n}^{(j)}(\cdot ,x+\mathbb{B}_{\cdot }^{H})\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(\cdot ),\,j,k=1,\dots ,d\right\} .$$
It is important to note that the function $\mathcal{K}_{t_{0},t_{0}^{\prime
}}^{H_{i}}(\cdot )$ appears only once in term and hence only four times in term . So there are indices $j_{1},\dots
,j_{4}\in \{1,\dots ,4m\}$ such that we can write as $$\int_{\Delta _{t_{0},t}^{4m}}\left( \prod_{j=1}^{4m}b_{j}^{n}(t_{j},x+\mathbb{B}_{t_{j}}^{H})\right) \prod_{l=1}^{4}\mathcal{K}_{t_{0},t_{0}^{\prime }}^{H_{i}}(t_{j_{l}})dt_{4m}\cdots dt_{1},$$where $$b_{l}^{n}(\cdot ,x+\mathbb{B}_{\cdot }^{H})\in \left\{ \frac{\partial }{\partial x_{k}}b_{n}^{(j)}(\cdot ,x+\mathbb{B}_{\cdot }^{H}),\,j,k=1,\dots
,d\right\} ,\quad l=1,\dots ,4m.$$
The latter enables us to use the estimate from Proposition [mainestimate1]{} for $\sum_{r=1}^{4m}\varepsilon _{r}=4,$ $\left\vert \alpha
\right\vert =4m$, $\sum_{l=1}^{d}\alpha _{j}^{(l)}=1$ for all $l,$ $H_{r}<\frac{1}{2(d+2)}$ for some $r\geq i$ combined with Remark \[Remark 3.4\]. Thus we obtain that
$$\begin{aligned}
\left( E(J_{2}^{n}(t_{0}^{\prime },t_{0}))^{4}\right) ^{1/4} &\leq & \\
&&\frac{1}{\lambda _{r}^{md}}C_{i}^{2m}\left\Vert b_{n}\right\Vert _{L^{1}(\mathbb{R}^{d};L^{\infty }([0,T]))}^{m}\left\vert \frac{t_{0}-t_{0}^{\prime }}{t_{0}t_{0}^{\prime }}\right\vert ^{\gamma _{i}}t_{0}^{(H_{i}-\frac{1}{2}-\gamma _{i})} \\
&&\times \frac{C(d)^{m}((8m)!)^{1/16}\left\vert t-t_{0}\right\vert
^{-H_{r}(md+2m)+(H_{i}-\frac{1}{2}-\gamma _{i})+m}}{\Gamma (-H_{r}(2\cdot
4md+4\cdot 4m)+2(H_{i}-\frac{1}{2}-\gamma _{i})+8m)^{1/8}}\end{aligned}$$
for a constant $C(d)$ depending only on $d.$
Then the series in (\[I2\]) is summable over $j,k$, $l_{1},\dots ,l_{m-1}$ and $m$. Hence, we just need to verify that the double integral is finite for suitable $\gamma _{i}$’s and $\beta _{i}$’s. Indeed, $$\int_{0}^{t}\int_{0}^{t}\frac{\left\vert t_{0}-t_{0}^{\prime }\right\vert
^{2\gamma _{i}-1-2\beta _{i}}}{\left\vert t_{0}t_{0}^{\prime }\right\vert
^{2\gamma _{i}}}t_{0}^{2\left( H_{i}-\frac{1}{2}-\gamma _{i}\right)
}|t-t_{0}|^{-2\left( H_{i}-\frac{1}{2}-\gamma _{i}\right)
}dt_{0}dt_{0}^{\prime }<\infty ,$$whenever $2\left( H_{i}-\frac{1}{2}-\gamma _{i}\right) >-1$, $2\gamma
_{i}-1-2\beta _{i}>-1$ and $2\left( H_{i}-\frac{1}{2}-\gamma _{i}\right)
-2\gamma _{i}>-1$ which is fulfilled if for instance $\gamma _{i}<H_{i}/4$ and $0<\beta _{i}<\gamma _{i}$.
Now we may choose for example a function $\varphi $ with $\varphi (x)=\exp
(-x^{100})$. In this case, we find that$$C_{i}^{2m}\lambda _{i}={\varphi}_{i}C_{i}^{2m}\varphi (C_{i})\leq {\varphi}_{i}\left( \frac{1}{50}\right) ^{\frac{m}{50}}m^{\frac{m}{50}}$$So, finally, if $H_{r}$ for a fixed $r\geq i$ is sufficiently small, the sums over $i\geq 1$ also converge since we have ${\varphi}_{i}$ satisfying \[Finite\].
For the term $I_{3}^{n}$ we may use Theorem \[girsanov\], Cauchy-Schwarz inequality twice and observe that the first factor of $I_{3}^{n}$ is bounded uniformly in $t_{0},t\in \lbrack 0,T]$ by a simple application of Proposition \[mainestimate2\] with $\varepsilon _{j}=0$ for all $j$. Then, the remaining estimate is fairly similar to the case of $I_{2}^{n}$ by using Proposition \[mainestimate2\] again. As for the estimate for the Malliavin derivative the reader may agree that the arguments are analogous.
The following is a consequence of combining Lemma \[VI\_weakconv\] and Proposition \[Holderintegral\].
\[VI\_L2conv\] For every $t\in \lbrack 0,T]$ and continuous function $\varphi :{\mathbb R}^{d}\rightarrow {\mathbb R}$ with at most linear growth we have $$\varphi (X_{t}^{n})\xrightarrow{n\to \infty}\varphi (E[X_{t}|\mathcal{F}_{t}])$$strongly in $L^{2}(\Omega )$. In addition, $E[X_{t}|\mathcal{F}_{t}]$ is Malliavin differentiable along any direction $W^{i}$, $i\geq 1$ of $\mathbb{B}_{\cdot }^{H}$. Moreover, the solution $X$ is $\mathcal{F}$-adapted, thus being a strong solution.
This is a direct consequence of the relative compactness from Theorem [compinf]{} combined with Proposition \[Holderintegral\] and by Lemma [VI\_weakconv]{}, we can identify the limit as $E[X_{t}|\mathcal{F}_{t}].$ Then the convergence holds for any bounded continuous functions as well. The Malliavin differentiability of $E[X_{t}|\mathcal{F}_{t}]$ is verified by taking $\varphi =I_{d}$ and the second estimate in Proposition [Holderintegral]{} in connection with [@Nua10 Proposition 1.2.3].
Finally, we can complete step (4) of our scheme.
The constructed solution $X_{\cdot }$ of is strong.
We have to show that $X_{t}$ is $\mathcal{F}_{t}$-measurable for every $t\in
\lbrack 0,T]$ and by Remark \[VI\_stochbasisrmk\] we see that there exists a strong solution in the usual sense, which is Malliavin differentiable. In proving this, let $\varphi $ be a globally Lipschitz continuous function. Then it follows from Corollary \[VI\_L2conv\] that there exists a subsequence $n_{k}$, $k\geq 0$, that $$\varphi (X_{t}^{n_{k}})\rightarrow \varphi (E[X_{t}|\mathcal{F}_{t}]),\ \
P-a.s.$$as $k\rightarrow \infty $.
Further, by Lemma \[VI\_weakconv\] we also know that $$\varphi (X_{t}^{n})\rightarrow E\left[ \varphi (X_{t})|\mathcal{F}_{t}\right]$$weakly in $L^{2}(\Omega )$. By the uniqueness of the limit we immediately obtain that $$\varphi \left( E[X_{t}|\mathcal{F}_{t}]\right) =E\left[ \varphi (X_{t})|\mathcal{F}_{t}\right] ,\ \ P-a.s.$$which implies that $X_{t}$ is $\mathcal{F}_{t}$-measurable for every $t\in
\lbrack 0,T]$.
Finally, we turn to step (5) and complete this Section by showing pathwise uniqueness. Following the same argument as in [@RY2004 Chapter IX, Exercise (1.20)] we see that strong existence and uniqueness in law implies pathwise uniqueness. The argument does not rely on the process being a semimartingale. Hence, uniqueness in law is enough. The following Lemma actually implies the desired uniqueness by estimate in connection with [@LS.77 Theorem 7.7].
Let $X$ be a strong solution of where $b\in L_{p}^{q}$, $p,q\in (2,\infty ]$. Then the estimates and hold for $X$ in place of $\mathbb{B}_{\cdot }^{H}$. As a consequence, uniqueness in law holds for equation and since $X$ strong, pathwise uniqueness follows.
Assume first that $b$ is bounded. Fix any $n\geq 1$ and set $$\eta _{s}^{n}=K_{H_{n}}^{-1}\left( \frac{1}{\lambda _{n}}\int_{0}^{\cdot
}b(r,X_{r})dr\right) (s).$$Since $b$ is bounded it is easy to see from by changing $\mathbb{B}_{\cdot }^{H}$ with $X$ and bounding $b$ that for every $\kappa
\in {\mathbb R}$, $$\label{exp1}
E_{\widetilde{P}}\left[ \exp \left\{ -2\kappa \int_{0}^{T}(\eta
_{s}^{n})^{\ast }dW_{s}^{n}-2\kappa ^{2}\int_{0}^{T}|\eta
_{s}^{n}|^{2}ds\right\} \right] =1,$$where $$\frac{d\widetilde{P}}{dP}=\exp \left\{ -\int_{0}^{T}(\eta _{s}^{n})^{\ast
}dW_{s}^{n}-\frac{1}{2}\int_{0}^{T}|\eta _{s}^{n}|^{2}ds\right\} .$$Hence, $X_{t}-x$ is a regularizing fractional Brownian motion with Hurst sequence $H$ under $\widetilde{P}$. Define $$\xi _{T}^{\kappa }:=\exp \left\{ -\kappa \int_{0}^{T}(\eta _{s}^{n})^{\ast
}dW_{s}^{n}-\frac{\kappa }{2}\int_{0}^{T}|\eta _{s}^{n}|^{2}ds\right\} .$$Then, $$\begin{aligned}
E_{\widetilde{P}}\left[ \xi _{T}^{\kappa }\right] & =E_{\widetilde{P}}\left[
\exp \left\{ -\kappa \int_{0}^{T}(\eta _{s}^{n})^{\ast }dW_{s}^{n}-\frac{\kappa }{2}\int_{0}^{T}|\eta _{s}^{n}|^{2}ds\right\} \right] \\
& =E_{\widetilde{P}}\left[ \exp \left\{ -\kappa \int_{0}^{T}(\eta
_{s}^{n})^{\ast }dW_{s}^{n}-\kappa ^{2}\int_{0}^{T}|\eta
_{s}^{n}|^{2}ds\right\} \exp \left\{ \left( \kappa ^{2}+\frac{\kappa }{2}\right) \int_{0}^{T}|\eta _{s}^{n}|^{2}ds\right\} \right] \\
& \leq \left( E_{\widetilde{P}}\left[ \exp \left\{ 2\left\vert \kappa ^{2}+\frac{\kappa }{2}\right\vert \int_{0}^{T}|\eta _{s}^{n}|^{2}ds\right\} \right] \right) ^{1/2}\end{aligned}$$in view of .
On the other hand, using with $X$ in place of $\mathbb{B}_{\cdot }^{H}$ we have $$\int_{0}^{T}|\eta _{s}|^{2}ds\leq C_{\varepsilon ,\lambda
_{n},H_{n},T}\left( 1+\int_{0}^{T}|b(r,X_{r})|^{\frac{1+\varepsilon }{\varepsilon }}dr\right) ,\quad P-a.s.$$for any $\varepsilon \in (0,1)$. Hence, applying Lemma \[interlemma\] we get $$E_{\widetilde{P}}\left[ \xi _{T}^{\kappa }\right] \leq e^{\left\vert \kappa
^{2}+\frac{\kappa }{2}\right\vert C_{\varepsilon ,\lambda
_{n},H_{n},T}}\left( A\left( C_{\varepsilon ,\lambda _{n},H_{n},T}\left\vert
\kappa ^{2}+\frac{\kappa }{2}\right\vert \Vert |b|^{\frac{1+\varepsilon }{\varepsilon }}\Vert _{L_{p}^{q}}\right) \right) ^{1/2},$$where $A$ is the analytic function from Lemma \[interlemma\].
Furthermore, observe that for every $\kappa\in {\mathbb R}$ we have $$\begin{aligned}
\label{sumexp}
E_P[\xi_T^{\kappa}] = E_{\widetilde{P}}[\xi_T^{\kappa-1}].\end{aligned}$$ In fact, holds for any $b\in L_p^q$ by considering $b_n:=b\mathbf{1}_{\{|b|\leq n\}}$, $n\geq 1$ and then letting $n\to \infty$.
Finally, let $\delta \in (0,1)$ and apply Hölder’s inequality in order to get $$E_{P}\left[ \int_{0}^{T}h(t,X_{t})dt\right] \leq T^{\delta }\left( E_{\widetilde{P}}[(\xi _{T}^{1})^{\frac{1+\delta }{\delta }}]\right) ^{\frac{\delta }{1+\delta }}\left( E_{\widetilde{P}}\left[
\int_{0}^{T}h(t,X_{t})^{1+\delta }dt\right] \right) ^{\frac{1}{1+\delta }},$$and $$E_{P}\left[ \exp \left\{ \int_{0}^{T}h(t,X_{t})dt\right\} \right] \leq
T^{\delta }\left( E_{\widetilde{P}}[(\xi _{T}^{1})^{\frac{1+\delta }{\delta }}]\right) ^{\frac{\delta }{1+\delta }}\left( E_{\widetilde{P}}\left[ \exp
\left\{ (1+\delta )\int_{0}^{T}h(t,X_{t})dt\right\} \right] \right) ^{\frac{1}{1+\delta }},$$for every Borel measurable function. Since we know that $X_{t}-x$ is a regularizing fractional Brownian motion with Hurst sequence $H$ under $\widetilde{P}$, the result follows by Lemma \[interlemma\] by choosing $\delta $ close enough to 0.
Using the all the previous intermediate results, we are now able to state the main result of this Section:
\[VI\_mainthm\] Retain the conditions for $\lambda =\{\lambda
_{i}\}_{i\geq 1}$ with respect to $\mathbb{B}_{\cdot }^{H}$ in Theorem [Holderintegral]{}. Let* *$b\in \mathcal{L}_{2,p}^{q}$*,* $p,q\in
(2,\infty ]$. Then there exists a unique (global) strong solution* *$X_{t},0\leq t\leq T$ of equation . Moreover, for every* *$t\in \lbrack 0,T]$*,* $X_{t}$* *is Malliavin differentiable in each direction of the Brownian motions* *$W^{n}$*,* $n\geq 1$* *in* *.
Infinitely Differentiable Flows for Irregular Vector Fields {#flowsection}
============================================================
From now on, we denote by $X_{t}^{s,x}$ the solution to the following SDE driven by a regularizing fractional Brownian motion $\mathbb{B}_{\cdot }^{H}$ with Hurst sequence $H$: $$dX_{t}^{s,x}=b(t,X_{t}^{s,x})dt+d\mathbb{B}_{t}^{H},\quad s,t\in \lbrack
0,T],\quad s\leq t,\quad X_{s}^{s,x}=x\in {\mathbb R}^{d}.$$
We will then assume the hypotheses from Theorem \[VI\_mainthm\] on $b$ and $H$.
The next estimate essentially tells us that the stochastic mapping $x\mapsto
X_{t}^{s,x}$ is $P$-a.s. infinitely many times continuously differentiable. In particular, it shows that the strong solution constructed in the former section, in addition to being Malliavin differentiable, is also smooth in $x$ and, although we will not prove it explicitly here, it is also smooth in the Malliavin sense, and since Hörmander’s condition is met then implies that the densities of the marginals are also smooth.
\[VI\_derivative\] Let $b\in C_{c}^{\infty }((0,T)\times {\mathbb R}^{d})$. Fix integers $p\geq 2$ and $k\geq 1$. Choose a $r$ such that $H_{r}<\frac{1}{(d-1+2k)}$. Then there exists a continuous function $C_{k,d,H_{r},p,\overline{p},\overline{q},T}:[0,\infty )^{2}\rightarrow \lbrack 0,\infty )$, depending on $k,d,H_{r},p,\overline{p},\overline{q}$ and $T$. $$\sup_{s,t\in \lbrack 0,T]}\sup_{x\in {\mathbb R}^{d}}\text{\emph{E}}\left[ \left\Vert
\frac{\partial ^{k}}{\partial x^{k}}X_{t}^{s,x}\right\Vert ^{p}\right] \leq
C_{k,d,H_{r},p,\overline{p},\overline{q},T}(\Vert b\Vert _{L_{\overline{p}}^{\overline{q}}},\Vert b\Vert _{L_{\infty }^{1}}).$$
For notational simplicity, let $s=0$, $\mathbb{B}_{\cdot }=\mathbb{B}_{\cdot
}^{H}$ and let $X_{t}^{x},$ $0\leq t\leq T$ be the solution with respect to the vector field $b\in C_{c}^{\infty }((0,T)\times \mathbb{R}^{d})$. We know that the stochastic flow associated with the smooth vector field $b$ is smooth, too (compare to e.g. [@Kunita]). Hence, we get that$$\frac{\partial }{\partial x}X_{t}^{x}=I_{d}+\int_{s}^{t}Db(u,X_{u}^{x})\cdot
\frac{\partial }{\partial x}X_{u}^{x}du,$$where $Db(u,\cdot ):\mathbb{R}^{d}\longrightarrow L(\mathbb{R}^{d},\mathbb{R}^{d})$ is the derivative of $b$ with respect to the space variable.
By using Picard iteration, we see that$$\frac{\partial }{\partial x}X_{t}^{x}=I_{d}+\sum_{m\geq 1}\int_{\Delta
_{0,t}^{m}}Db(u,X_{u_{1}}^{x})...Db(u,X_{u_{m}}^{x})du_{m}...du_{1},
\label{FirstOrder}$$where$$\Delta _{s,t}^{m}=\{(u_{m},...u_{1})\in \lbrack 0,T]^{m}:\theta
<u_{m}<...<u_{1}<t\}.$$
By applying dominated convergence, we can differentiate both sides with respect to $x$ and find that$$\frac{\partial ^{2}}{\partial x^{2}}X_{t}^{x}=\sum_{m\geq 1}\int_{\Delta
_{0,t}^{m}}\frac{\partial }{\partial x}[Db(u,X_{u_{1}}^{x})...Db(u,X_{u_{m}}^{x})]du_{m}...du_{1}.$$Further, the Leibniz and chain rule yield$$\begin{aligned}
&&\frac{\partial }{\partial x}[Db(u_{1},X_{u_{1}}^{x})...Db(u_{m},X_{u_{m}}^{x})] \\
&=&\sum_{r=1}^{m}Db(u_{1},X_{u_{1}}^{x})...D^{2}b(u_{r},X_{u_{r}}^{x})\frac{\partial }{\partial x}X_{u_{r}}^{x}...Db(u_{m},X_{u_{m}}^{x}),\end{aligned}$$where $D^{2}b(u,\cdot )=D(Db(u,\cdot )):\mathbb{R}^{d}\longrightarrow L(\mathbb{R}^{d},L(\mathbb{R}^{d},\mathbb{R}^{d}))$.
Therefore (\[FirstOrder\]) entails$$\begin{aligned}
\frac{\partial ^{2}}{\partial x^{2}}X_{t}^{x} &=&\sum_{m_{1}\geq
1}\int_{\Delta
_{0,t}^{m_{1}}}\sum_{r=1}^{m_{1}}Db(u_{1},X_{u_{1}}^{x})...D^{2}b(u_{r},X_{u_{r}}^{x})
\notag \\
&&\times \left( I_{d}+\sum_{m_{2}\geq 1}\int_{\Delta
_{0,u_{r}}^{m_{2}}}Db(v_{1},X_{v_{1}}^{x})...Db(v_{m_{2}},X_{v_{m_{2}}}^{x})dv_{m_{2}}...dv_{1}\right)
\notag \\
&&\times
Db(u_{r+1},X_{u_{r+1}}^{x})...Db(u_{m_{1}},X_{u_{m_{1}}}^{x})du_{m_{1}}...du_{1}
\notag \\
&=&\sum_{m_{1}\geq 1}\sum_{r=1}^{m_{1}}\int_{\Delta
_{0,t}^{m_{1}}}Db(u_{1},X_{u_{1}}^{x})...D^{2}b(u_{r},X_{u_{r}}^{x})...Db(u_{m_{1}},X_{u_{m_{1}}}^{x})du_{m_{1}}...du_{1}
\notag \\
&&+\sum_{m_{1}\geq 1}\sum_{r=1}^{m_{1}}\sum_{m_{2}\geq 1}\int_{\Delta
_{0,t}^{m_{1}}}\int_{\Delta
_{0,u_{r}}^{m_{2}}}Db(u_{1},X_{u_{1}}^{x})...D^{2}b(u_{r},X_{u_{r}}^{x})
\notag \\
&&\times
Db(v_{1},X_{v_{1}}^{x})...Db(v_{m_{2}}X_{v_{m_{2}}}^{x})Db(u_{r+1},X_{u_{r+1}}^{x})...Db(u_{m_{1}},X_{u_{m_{1}}}^{x})
\notag \\
&&dv_{m_{2}}...dv_{1}du_{m_{1}}...du_{1} \notag \\
&=&:I_{1}+I_{2}. \label{SecondOrder}\end{aligned}$$
In the next step, we wish to employ Lemma \[OrderDerivatives\] (in connection with shuffling in Section \[VI\_shuffles\]) to the term $I_{2}$ in (\[SecondOrder\]) and get that$$I_{2}=\sum_{m_{1}\geq 1}\sum_{r=1}^{m_{1}}\sum_{m_{2}\geq 1}\int_{\Delta
_{0,t}^{m_{1}+m_{2}}}\mathcal{H}_{m_{1}+m_{2}}^{X}(u)du_{m_{1}+m_{2}}...du_{1} \label{l2}$$for $u=(u_{1},...,u_{m_{1}+m_{2}}),$ where the integrand $\mathcal{H}_{m_{1}+m_{2}}^{X}(u)\in \mathbb{R}^{d}\otimes \mathbb{R}^{d}\otimes \mathbb{R}^{d}$ has entries given by sums of at most $C(d)^{m_{1}+m_{2}}$ terms, which are products of length $m_{1}+m_{2}$ of functions being elements of the set$$\left\{ \frac{\partial ^{\gamma ^{(1)}+...+\gamma ^{(d)}}}{\partial ^{\gamma
^{(1)}}x_{1}...\partial ^{\gamma ^{(d)}}x_{d}}b^{(r)}(u,X_{u}^{x}),\text{ }r=1,...,d,\text{ }\gamma ^{(1)}+...+\gamma ^{(d)}\leq 2,\text{ }\gamma
^{(l)}\in \mathbb{N}_{0},\text{ }l=1,...,d\right\} .$$Here it is important to mention that second order derivatives of functions in those products of functions on $\Delta _{0,t}^{m_{1}+m_{2}}$ in (\[l2\]) only occur once. Hence the total order of derivatives $\left\vert \alpha
\right\vert $ of those products of functions in connection with Lemma [OrderDerivatives]{} in the Appendix is$$\left\vert \alpha \right\vert =m_{1}+m_{2}+1.$$Let us now choose $p,c,r\in \lbrack 1,\infty )$ such that $cp=2^{q}$ for some integer $q$ and $\frac{1}{r}+\frac{1}{c}=1.$ Then we can employ Hölder’s inequality and Girsanov’s theorem (see Theorem \[VI\_girsanov\]) combined with Lemma \[novikov\] and obtain that$$\begin{aligned}
&&E[\left\Vert I_{2}\right\Vert ^{p}] \notag \\
&\leq &C(\left\Vert b\right\Vert _{L_{\overline{p}}^{\overline{q}}})\left(
\sum_{m_{1}\geq 1}\sum_{r=1}^{m_{1}}\sum_{m_{2}\geq 1}\sum_{i\in
I}\left\Vert \int_{\Delta _{0,t}^{m_{1}+m_{2}}}\mathcal{H}_{i}^{\mathbb{B}}(u)du_{m_{1}+m_{2}}...du_{1}\right\Vert _{L^{2^{q}}(\Omega ;\mathbb{R})}\right) ^{p}, \label{Lp}\end{aligned}$$where $C:[0,\infty )\longrightarrow \lbrack 0,\infty )$ is a continuous function depending on $p,\overline{p}$ and $\overline{q}$. Here $\#I\leq
K^{m_{1}+m_{2}}$ for a constant $K=K(d)$ and the integrands $\mathcal{H}_{i}^{\mathbb{B}}(u)$ are of the form $$\mathcal{H}_{i}^{B^{H}}(u)=\prod_{l=1}^{m_{1}+m_{2}}h_{l}(u_{l}),h_{l}\in
\Lambda ,l=1,...,m_{1}+m_{2}$$where $$\Lambda :=\left\{
\begin{array}{c}
\frac{\partial ^{\gamma ^{(1)}+...+\gamma ^{(d)}}}{\partial ^{\gamma
^{(1)}}x_{1}...\partial ^{\gamma ^{(d)}}x_{d}}b^{(r)}(u,x+\mathbb{B}_{u}),\text{ }r=1,...,d, \\
\gamma ^{(1)}+...+\gamma ^{(d)}\leq 2,\text{ }\gamma ^{(l)}\in \mathbb{N}_{0},\text{ }l=1,...,d\end{array}\right\} .$$As above we observe that functions with second order derivatives only occur once in those products.
Let $$J=\left( \int_{\Delta _{0,t}^{m_{1}+m_{2}}}\mathcal{H}_{i}^{\mathbb{B}}(u)du_{m_{1}+m_{2}}...du_{1}\right) ^{2^{q}}.$$By using shuffling (see Section \[VI\_shuffles\]) once more, successively, we find that $J$ has a reprsentation as a sum of, at most of length $K(q)^{m_{1}+m_{2}}$ with summands of the form$$\int_{\Delta
_{0,t}^{2^{q}(m_{1}+m_{2})}}\prod_{l=1}^{2^{q}(m_{1}+m_{2})}f_{l}(u_{l})du_{2^{q}(m_{1}+m_{2})}...du_{1},
\label{f}$$where $f_{l}\in \Lambda $ for all $l$.
Note that the number of factors $f_{l}$ in the above product, which have a second order derivative, is exactly $2^{q}$. Hence the total order of the derivatives in (\[f\]) in connection with Lemma \[OrderDerivatives\] (where one in that Lemma formally replaces $X_{u}^{x}$ by $x+\mathbb{B}_{u}$ in the corresponding terms) is $$\left\vert \alpha \right\vert =2^{q}(m_{1}+m_{2}+1). \label{alpha2}$$
We now aim at using Theorem \[mainestimate2\] for $m=2^{q}(m_{1}+m_{2})$ and $\varepsilon _{j}=0$ and find that$$\begin{aligned}
&&\left\vert E\left[ \int_{\Delta
_{0,t}^{2^{q}(m_{1}+m_{2})}}\prod_{l=1}^{2^{q}(m_{1}+m_{2})}f_{l}(u_{l})du_{2^{q}(m_{1}+m_{2})}...du_{1}\right] \right\vert \\
&\leq &C^{m_{1}+m_{2}}(\left\Vert b\right\Vert _{L_{\infty
}^{1}})^{2^{q}(m_{1}+m_{2})} \\
&&\times \frac{((2(2^{q}(m_{1}+m_{2}+1))!)^{1/4}}{\Gamma
(-H_{r}(2d2^{q}(m_{1}+m_{2})+42^{q}(m_{1}+m_{2}+1))+22^{q}(m_{1}+m_{2}))^{1/2}}\end{aligned}$$for a constant $C$ depending on $H_{r},T,d$ and $q$.
Therefore the latter combined with (\[Lp\]) implies that$$\begin{aligned}
&&E[\left\Vert I_{2}\right\Vert ^{p}] \\
&\leq &C(\left\Vert b\right\Vert _{L_{\overline{p}}^{\overline{q}}})\left(
\sum_{m_{1}\geq 1}\sum_{m_{2}\geq 1}K^{m_{1}+m_{2}}(\left\Vert b\right\Vert
_{L_{\infty }^{1}})^{2^{q}(m_{1}+m_{2})}\right. \\
&&\left. \times \frac{((2(2^{q}(m_{1}+m_{2}+1))!)^{1/4}}{\Gamma
(-H_{r}(2d2^{q}(m_{1}+m_{2})+42^{q}(m_{1}+m_{2}+1))+22^{q}(m_{1}+m_{2}))^{1/2}})^{1/2^{q}}\right) ^{p}\end{aligned}$$for a constant $K$ depending on $H_{r},$ $T,$ $d,$ $p$ and $q$.
Since $\frac{1}{2(d+3)}\leq \frac{1}{2(d+2\frac{m_{1}+m_{2}+1}{m_{1}+m_{2}})}
$ for $m_{1},$ $m_{2}\geq 1$, one concludes that the above sum converges, whenever $H_{r}<\frac{1}{2(d+3)}$.
Further, one gets an estimate for $E[\left\Vert I_{1}\right\Vert ^{p}]$ by using similar reasonings as above. In summary, we obtain the proof for $k=2$.
We now give an explanation how we can generalize the previous line of reasoning to the case $k\geq 2$: In this case, we we have that$$\frac{\partial ^{k}}{\partial x^{k}}X_{t}^{x}=I_{1}+...+I_{2^{k-1}},
\label{Ik}$$where each $I_{i},$ $i=1,...,2^{k-1}$ is a sum of iterated integrals over simplices of the form $\Delta _{0,u}^{m_{j}},$ $0<u<t,$ $j=1,...,k$ with integrands having at most one product factor $D^{k}b$, while the other factors are of the form $D^{j}b,j\leq k-1$.
In the following we need the following notation: For multi-indices $m.=(m_{1},...,m_{k})$ and $r:=(r_{1},...,r_{k-1})$, set$$m_{j}^{-}:=\sum_{i=1}^{j}m_{i}\text{ }$$and$$\sum_{\substack{ m\geq 1 \\ r_{l}\leq m_{l}^{-} \\ l=1,...,k-1}}:=\sum_{m_{1}\geq 1}\sum_{r_{1}=1}^{m_{1}}\sum_{m_{2}\geq
1}\sum_{r_{2}=1}^{m_{2}^{-}}...\sum_{r_{k-1}=1}^{m_{k-1}^{-}}\sum_{m_{k}\geq
1}.$$In what follows, without loss of generality we confine ourselves to deriving an estimate with respect to the summand $I_{2^{k-1}}$ in (\[Ik\]). Just as in the case $k=2,$ we obtain by employing Lemma \[OrderDerivatives\] (in connection with shuffling in Section \[VI\_shuffles\]) that $$I_{2^{k-1}}=\sum_{\substack{ m\geq 1 \\ r_{l}\leq m_{l}^{-} \\ l=1,...,k-1}}\int_{\Delta _{0,t}^{m_{1}+...+m_{k}}}\mathcal{H}_{m_{1}+...+m_{k}}^{X}(u)du_{m_{1}+m_{2}}...du_{1}$$for $u=(u_{m_{1}+...+m_{k}},...,u_{1}),$ where the integrand $\mathcal{H}_{m_{1}+...+m_{k}}^{X}(u)\in \otimes _{j=1}^{k+1}\mathbb{R}^{d}$ has entries, which are given by sums of at most $C(d)^{m_{1}+...+m_{k}}$ terms. Those terms are given by products of length $m_{1}+...m_{k}$ of functions, which are elements of the set$$\left\{
\begin{array}{c}
\frac{\partial ^{\gamma ^{(1)}+...+\gamma ^{(d)}}}{\partial ^{\gamma
^{(1)}}x_{1}...\partial ^{\gamma ^{(d)}}x_{d}}b^{(r)}(u,X_{u}^{x}),r=1,...,d,
\\
\gamma ^{(1)}+...+\gamma ^{(d)}\leq k,\gamma ^{(l)}\in \mathbb{N}_{0},l=1,...,d\end{array}\right\} .$$Exactly as in the case $k=2$ we can invoke Lemma \[OrderDerivatives\] in the Appendix and get that the total order of derivatives $\left\vert \alpha
\right\vert $ of those products of functions is $$\left\vert \alpha \right\vert =m_{1}+...+m_{k}+k-1.$$Then we can adopt the line of reasoning as before and choose $p,c,r\in
\lbrack 1,\infty )$ such that $cp=2^{q}$ for some integer $q$ and $\frac{1}{r}+\frac{1}{c}=1$ and find by applying Hölder’s inequality and Girsanov’s theorem (see Theorem \[VI\_girsanov\]) combined with Lemma \[novikov\] that$$\begin{aligned}
&&E[\left\Vert I_{2^{k-1}}\right\Vert ^{p}] \notag \\
&\leq &C(\left\Vert b\right\Vert _{L_{\overline{p}}^{\overline{q}}})\left(
\sum_{\substack{ m\geq 1 \\ r_{l}\leq m_{l}^{-} \\ l=1,...,k-1}}\sum_{i\in
I}\left\Vert \int_{\Delta _{0,t}^{m_{1}+m_{2}}}\mathcal{H}_{i}^{\mathbb{B}}(u)du_{m_{1}+...+m_{k}}...du_{1}\right\Vert _{L^{2^{q}}(\Omega ;\mathbb{R})}\right) ^{p}, \label{Lp2}\end{aligned}$$where $C:[0,\infty )\longrightarrow \lbrack 0,\infty )$ is a continuous function depending on $p,\overline{p}$ and $\overline{q}$. Here $\#I\leq
K^{m_{1}+...+m_{k}}$ for a constant $K=K(d)$ and the integrands $\mathcal{H}_{i}^{\mathbb{B}}(u)$ take the form $$\mathcal{H}_{i}^{\mathbb{B}}(u)=\prod_{l=1}^{m_{1}+...+m_{k}}h_{l}(u_{l}),\text{ }h_{l}\in \Lambda ,\text{ }l=1,...,m_{1}+...+m_{k},$$where $$\Lambda :=\left\{
\begin{array}{c}
\frac{\partial ^{\gamma ^{(1)}+...+\gamma ^{(d)}}}{\partial ^{\gamma
^{(1)}}x_{1}...\partial ^{\gamma ^{(d)}}x_{d}}b^{(r)}(u,x+\mathbb{B}_{u}),\text{ }r=1,...,d, \\
\gamma ^{(1)}+...+\gamma ^{(d)}\leq k,\text{ }\gamma ^{(l)}\in \mathbb{N}_{0},\text{ }l=1,...,d\end{array}\right\} .$$Define $$J=\left( \int_{\Delta _{0,t}^{m_{1}+...+m_{k}}}\mathcal{H}_{i}^{\mathbb{B}}(u)du_{m_{1}+...+m_{k}}...du_{1}\right) ^{2^{q}}.$$Once more, repeated shuffling (see Section \[VI\_shuffles\]) shows that $J$ can be represented as a sum of, at most of length $K(q)^{m_{1}+....m_{k}}$ with summands of the form$$\int_{\Delta
_{0,t}^{2^{q}(m_{1}+...+m_{k})}}\prod_{l=1}^{2^{q}(m_{1}+...+m_{k})}f_{l}(u_{l})du_{2^{q}(m_{1}+....+m_{k})}...du_{1},
\label{f2}$$where $f_{l}\in \Lambda $ for all $l$.
By applying Lemma \[OrderDerivatives\] again (where one in that Lemma formally replaces $X_{u}^{x}$ by $x+B_{u}^{H}$ in the corresponding expressions) we obtain that the total order of the derivatives in the products of functions in (\[f2\]) is given by$$\left\vert \alpha \right\vert =2^{q}(m_{1}+...+m_{k}+k-1).$$
Then Proposition \[mainestimate2\] for $m=2^{q}(m_{1}+...+m_{k})$ and $\varepsilon _{j}=0$ yields that$$\begin{aligned}
&&\left\vert E\left[ \int_{\Delta
_{0,t}^{2^{q}(m_{1}+...+m_{k})}}\prod_{l=1}^{2^{q}(m_{1}+...+m_{k})}f_{l}(u_{l})du_{2^{q}(m_{1}+...+m_{k})}...du_{1}\right] \right\vert \\
&\leq &C^{m_{1}+...+m_{k}}(\left\Vert b\right\Vert _{L_{\infty
}^{1}})^{2^{q}(m_{1}+...+m_{k})} \\
&&\times \frac{((2(2^{q}(m_{1}+...+m_{k}+k-1))!)^{1/4}}{\Gamma
(-H_{r}(2d2^{q}(m_{1}+...+m_{k})+42^{q}(m_{1}+...+m_{k}+k-1))+22^{q}(m_{1}+...+m_{k}))^{1/2}}\end{aligned}$$for a constant $C$ depending on $H_{r},$ $T,$ $d$ and $q$.
Thus we can conclude from (\[Lp2\]) that$$\begin{aligned}
&&E[\left\Vert I_{2^{k-1}}\right\Vert ^{p}] \\
&\leq &C(\left\Vert b\right\Vert _{L_{\overline{p}}^{\overline{q}}})\left(
\sum_{m_{1}\geq 1}...\sum_{m_{k}\geq 1}K^{m_{1}+...+m_{k}}(\left\Vert
b\right\Vert _{L_{\infty }^{1}})^{2^{q}(m_{1}+...+m_{k})}\right. \\
&&\left. \times \frac{((2(2^{q}(m_{1}+...+m_{k}+k-1))!)^{1/4}}{\Gamma
(-H_{r}(2d2^{q}(m_{1}+...+m_{k})+42^{q}(m_{1}+...+m_{k}+k-1))+22^{q}(m_{1}+...+m_{k}))^{1/2}})^{1/2^{q}}\right) ^{p} \\
&\leq &C(\left\Vert b\right\Vert _{L_{\overline{p}}^{\overline{q}}}\left(
\sum_{m\geq 1}\sum_{\substack{ l_{1},...,l_{k}\geq 0: \\ l_{1}+...+l_{k}=m}}K^{m}(\left\Vert b\right\Vert _{L_{\infty }^{1}})^{2^{q}m}\right. \\
&&\left. \times \frac{((2(2^{q}(m+k-1))!)^{1/4}}{\Gamma
(-H_{r}(2d2^{q}m+42^{q}(m+k-1))+22^{q}m)^{1/2}})^{1/2^{q}}\right) ^{p}\end{aligned}$$for a constant $K$ depending on $H_{r},$ $T,$ $d,$ $p$ and $q$.
Since $H_{r}<$ $\frac{1}{2(d-1+2k)}$ by assumption, we see that the above sum converges. Hence the proof follows.
The following is the main result of this Section and shows that the regularizing fractional Brownian motion $\mathbb{B}_{\cdot }^{H}$ “produces” an infinitely continuously differentiable stochastic flow $x\mapsto X_{t}^{x}
$, when $b$ merely belongs to $\mathcal{L}_{2,p}^{q}$ for any $p,q\in
(2,\infty ]$.
Assume that the conditions for $\lambda =\{\lambda _{i}\}_{i=1}^{\infty }$ with respect to $\mathbb{B}_{\cdot }^{H}$ in Theorem \[VI\_mainthm\] hold. Suppose that $b\in \mathcal{L}_{2,p}^{q}$, $p,q\in (2,\infty ]$. Let $U\subset {\mathbb R}^{d}$ be an open and bounded set and $X_{t},$ $0\leq t\leq T$ the solution of . Then for all $t\in \lbrack 0,T]$ we have that $$X_{t}^{\cdot }\in \bigcap_{k\geq 1}\bigcap_{\alpha >2}L^{2}(\Omega
,W^{k,\alpha }(U)).$$
First, we approximate the irregular drift vector field $b$ by a sequence of functions $b_{n}:[0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}^{d}$, $n\geq 0$ in $C_{c}^{\infty }((0,T)\times {\mathbb R}^{d},{\mathbb R}^{d})$ in the sense of . Let $X^{n,x}=\{X_{t}^{n,x},t\in \lbrack 0,T]\}$ be the solution to with initial value $x\in {\mathbb R}^{d}$ associated with $b_{n}$.
We find that for any test function $\varphi \in C_{c}^{\infty }(U,{\mathbb R}^{d})$ and fixed $t\in \lbrack 0,T]$ the set of random variables $$\langle X_{t}^{n,\cdot },\varphi \rangle :=\int_{U}\langle
X_{t}^{n,x},\varphi (x)\rangle _{{\mathbb R}^{d}}dx,\quad n\geq 0$$is relatively compact in $L^{2}(\Omega )$. In proving this, we want to apply the compactness criterion Theorem \[compinf\] in terms of the Malliavin derivative in the Appendix. Using the sequence $\{\delta
_{i}\}_{i=1}^{\infty }$ in Proposition \[Holderintegral\], we get that
$$\begin{aligned}
\sum_{i=1}^{\infty }\frac{1}{\delta _{i}^{2}}E[\int_{0}^{T}|D_{s}^{i,(j)}\langle X_{t}^{n,\cdot },\varphi \rangle |^{2}ds]=& \sum_{l=1}^{d}\left(
\int_{U}E[D_{s}^{i,(j)}X_{t}^{n,x,(l)}]\varphi _{l}(x)dx\right) ^{2} \\
\leq & d\Vert \varphi \Vert _{L^{2}({\mathbb R}^{d},{\mathbb R}^{d})}^{2}\lambda \{\mbox{supp }(\varphi )\}\sup_{x\in U}\sum_{i=1}^{\infty }\frac{1}{\delta _{i}^{2}}E\left[
\int_{0}^{T}\Vert D_{s}^{i}X_{t}^{n,x}\Vert ^{2}ds\right] ,\end{aligned}$$
where $D^{i,(j)}$ denotes the Malliavin derivative in the direction of $W^{i,(j)}$ where $W^{i}$ is the $d$-dimensional standard Brownian motion defining $B^{H_{i},i}$ and $W^{i,(j)}$ its $j$-th component, $\lambda $ the Lebesgue measure on ${\mathbb R}^{d}$, $\mbox{supp }(\varphi )$ the support of $\varphi $ and $\Vert \cdot \Vert $ a matrix norm. So it follows from the estimates in Proposition \[Holderintegral\] that $$\sup_{n\geq 0}\sum_{i=1}^{\infty }\frac{1}{\delta _{i}^{2}}\Vert D_{\cdot
}^{i}\langle X_{t}^{n,\cdot },\varphi \rangle \Vert _{L^{2}(\Omega \times
\lbrack 0,T])}^{2}\leq C\Vert \varphi \Vert _{L^{2}({\mathbb R}^{d},{\mathbb R}^{d})}^{2}\lambda \{\mbox{supp }(\varphi )\}.$$Similarly, we get that $$\sup_{n\geq 0}\sum_{i=1}^{\infty }\frac{1}{(1-2^{-2(\beta _{i}-\alpha
_{i})})\delta _{i}^{2}}\int_{0}^{T}\int_{0}^{T}\frac{E[\Vert D_{s^{\prime
}}^{i}\langle X_{t}^{n,\cdot },\varphi \rangle -D_{s}^{i}\langle
X_{t}^{n,\cdot },\varphi \rangle \Vert ^{2}]}{|s^{\prime }-s|^{1+2\beta _{i}}}<\infty$$for some sequences $\{\alpha _{i}\}_{i=1}^{\infty }$, $\{\beta
_{i}\}_{i=1}^{\infty }$ as in Proposition \[Holderintegral\]. Hence $\langle X_{t}^{n,\cdot },\varphi \rangle $, $n\geq 0$ is relatively compact in $L^{2}(\Omega )$. Denote by $Y_{t}(\varphi )$ its limit after taking (if necessary) a subsequence.
By adopting the same reasoning as in Lemma \[VI\_weakconv\] one proves that $$\langle X_{t}^{n,\cdot },\varphi \rangle \xrightarrow{n \to \infty}\langle
X_{t}^{\cdot },\varphi \rangle$$weakly in $L^{2}(\Omega )$. Then by uniqueness of the limit we see that $$\langle X_{t}^{n,\cdot },\varphi \rangle \underset{n\longrightarrow \infty }{\longrightarrow }Y_{t}(\varphi )=\langle X_{t}^{\cdot },\varphi \rangle$$in $L^{2}(\Omega )$ for all $t$ (without using a subsequence).
We observe that $X_{t}^{n,\cdot },n\geq 0$ is bounded in the Sobolev norm $L^{2}(\Omega ,W^{k,\alpha }(U))$ for each $n\geq 0$ and $k\geq 1$. Indeed, from Proposition \[VI\_derivative\] it follows that $$\begin{aligned}
\sup_{n\geq 0}\Vert X_{t}^{n,\cdot }\Vert _{L^{2}(\Omega ,W^{k,\alpha
}(U))}^{2}=& \sup_{n\geq 0}\sum_{i=0}^{k}E\left[ \Vert \frac{\partial ^{i}}{\partial x^{i}}X_{t}^{n,\cdot }\Vert _{L^{\alpha }(U)}^{2}\right] \\
\leq & \sum_{i=0}^{k}\int_{U}\sup_{n\geq 0}E\left[ \Vert \frac{\partial ^{i}}{\partial x^{i}}X_{t}^{n,x}\Vert ^{\alpha }\right] ^{\frac{2}{\alpha }}dx \\
<& \infty .\end{aligned}$$
The space $L^{2}(\Omega ,W^{k,\alpha }(U))$, $\alpha \in (1,\infty )$ is reflexive. So the set $\{X_{t}^{n,x}\}_{n\geq 0}$ is (relatively) weakly compact in $L^{2}(\Omega ,W^{k,\alpha }(U))$ for every $k\geq 1$. Hence, there exists a subsequence $n(j)$, $j\geq 0$ such that $$X_{t}^{n(j),\cdot }\xrightarrow[j\to \infty]{w}Y\in L^{2}(\Omega
,W^{k,\alpha }(U)).$$
We als know that $X_{t}^{n,x}\rightarrow X_{t}^{x}$ strongly in $L^{2}(\Omega )$ for all $t$.
So for all $A\in \mathcal{F}$ and $\varphi \in C_{0}^{\infty }({\mathbb R}^{d},{\mathbb R}^{d}) $ we have for all multi-indices $\gamma $ with $\left\vert \gamma
\right\vert \leq k$ that $$\begin{aligned}
E[1_{A}\langle X_{t}^{\cdot },D^{\gamma }\varphi \rangle ]
&=&\lim_{j\rightarrow \infty }E[1_{A}\langle X_{t}^{n(j),\cdot },D^{\gamma
}\varphi \rangle ] \\
&=&\lim_{j\rightarrow \infty }(-1)^{\left\vert \gamma \right\vert
}E[1_{A}\langle D^{\gamma }X_{t}^{n(j),\cdot },\varphi \rangle
]=(-1)^{\left\vert \gamma \right\vert }E[1_{A}\langle D^{\gamma }Y,\varphi
\rangle ]\end{aligned}$$Using the latter, we can conclude that $$X_{t}^{\cdot }\in L^{2}(\Omega ,W^{k,\alpha }(U)),\ \ P-a.s.$$Since $k\geq 1$ is arbitrary, the proof follows.
A Compactness Criterion for Subsets of $L^{2}(\Omega )$
=======================================================
The following result which is originally due to [@DPMN92] in the finite dimensional case and which can be e.g. found in [@B10], provides a compactness criterion of square integrable functionals of cylindrical Wiener processes on a Hilbert space:
\[General\] Let $B_{t},0\leq t\leq T$ be a cylindrical Wiener process on a separable Hilbert space $H$ with respect to a complete probability space $(\Omega ,\mathcal{F},\mu )$, where $\mathcal{F}$ is generated by $B_{t}$, $0\leq t\leq T$. Further, let $\mathcal{L}_{HS}(H,\mathbb{R})$ be the space of Hilbert-Schmidt operators from $H$ to $\mathbb{R}$ and let $D: \mathbb{D}^{1,2}\longrightarrow L^{2}(\Omega ;L^{2}([0,T])\otimes \mathcal{L}_{HS}(H,\mathbb{R}))$ be the Malliavin derivative in the direction of $B_{t}$, $0\leq t\leq T$, where $\mathbb{D}^{1,2}$ is the space of Malliavin differentiable random variables in $L^{2}(\Omega )$.
Suppose that $C$ is a self-adjoint compact operator on $L^{2}([0,T])\otimes
\mathcal{L}_{HS}(H,\mathbb{R})$ with dense image. Then for any $c>0$ the set $$\begin{aligned}
\mathcal{G}=\left\{ G\in \mathbb{D}^{1,2}:\left\Vert G\right\Vert
_{L^{2}(\Omega )}+\left\Vert C^{-1}DG\right\Vert _{L^{2}(\Omega
;L^{2}([0,T])\otimes \mathcal{L}_{HS}(H,\mathbb{R}))}\leq c\right\}\end{aligned}$$ is relatively compact in $L^{2}(\Omega )$.
In this paper we aim at using a special case of the the previous theorem, which is more suitable for explicit estimations. To this end we need the following auxiliary result from [@DPMN92].
\[Lemma\] Denote by $v_{s}$ ,$s\geq 0$ with $v_{0}=1$ the Haar basis of $L^{2}([0,1])$. Define for any $0<\alpha <\frac{1}{2}$ the operator $A_{\alpha }$ on $L^{2}([0,1])$ by $$\begin{aligned}
A_{\alpha }v_{s}=2^{k\alpha }v_{s},\quad \text{if} \quad s=2^{k}+j, \quad
k\geq 0, \quad 0\leq j\leq 2^{k}\end{aligned}$$ and $$\begin{aligned}
A_{\alpha }1=1.\end{aligned}$$ Then for $\alpha <\beta <\frac{1}{2}$ we have that$$\begin{aligned}
\left\Vert A_{\alpha }f\right\Vert _{L^{2}([0,1])}^{2} \leq 2
\left(\left\Vert f\right\Vert _{L^{2}([0,1])}^{2}+\frac{1}{1-2^{-2(\beta
-\alpha )}}\int_{0}^{1}\int_{0}^{1}\frac{\left\vert f(t)-f(u)\right\vert ^{2}}{\left\vert t-u\right\vert ^{1+2\beta }}dtdu\right).\end{aligned}$$
\[compinf\] Let $D^{i}$ be the Malliavin derivative in the direction of the $i$-th component of $B_{t}$, $0\leq t\leq 1$, $i\geq 1$. In addition, let $0<\alpha_{i}<\beta _{i}<\frac{1}{2}$ and $\delta_{i}>0$ for all $i\geq
1 $. Define the sequence $\lambda _{s,i}=2^{-k\alpha _{i}}\delta _{i}$, if $s=2^{k}+j$, $k\geq 0,0\leq j\leq 2^{k},$ $i\geq 1$. Assume that $\lambda_{s,i}\longrightarrow 0$ for $s,i\longrightarrow \infty$. Let $c>0$ and $\mathcal{G}$ the collection of all $G\in \mathbb{D}^{1,2}$ such that $$\begin{aligned}
\left\Vert G\right\Vert _{L^{2}(\Omega )}\leq c,\end{aligned}$$ $$\begin{aligned}
\sum_{i\geq 1}\delta _{i}^{-2}\left\Vert D^{i}G\right\Vert _{L^{2}(\Omega
;L^{2}([0,1]))}^{2}\leq c\end{aligned}$$ and $$\begin{aligned}
\sum_{i\geq 1}\frac{1}{(1-2^{-2(\beta _{i}-\alpha _{i})})\delta _{i}^{2}}
\int_{0}^{1}\int_{0}^{1}\frac{\left\Vert D_{t}^{i}G-D_{u}^{i}G\right\Vert
_{L^{2}(\Omega )}^{2}}{\left\vert t-u\right\vert ^{1+2\beta _{i}}}dtdu\leq c.\end{aligned}$$ Then $\mathcal{G}$ is relatively compact in $L^{2}(\Omega)$.
As before denote by $v_{s}$, $s\geq 0$ with $v_{0}=1$ the Haar basis of $L^{2}([0,1])$ and by $e_{i}^{\ast}=\langle e_i,\cdot\rangle_H$, $i\geq 1$ an orthonormal basis of $\mathcal{L}_{HS}(H,{\mathbb R})$ ($\cong H^{\ast }$) where $e_i$, $i\geq 1$ is an orthonormal basis of $H$. Define a self-adjoint compact operator $C$ on $L^{2}([0,1])\otimes \mathcal{L}_{HS}(H,\mathbb{R})$ with dense image by $$\begin{aligned}
C(v_{s}\otimes e_{i}^{\ast })=\lambda _{s,i}v_{s}\otimes e_{i}^{\ast },
\quad s\geq 0, \quad i\geq 1.\end{aligned}$$ Then it follows for $G\in \mathbb{D}^{1,2}$ from Lemma \[Lemma\] that $$\begin{aligned}
&\hspace{-2cm}\left\Vert C^{-1}DG\right\Vert _{L^{2}(\Omega
;L^{2}([0,1])\otimes \mathcal{L}_{HS}(H,\mathbb{R}))}^{2} \\
=&\,\sum_{i\geq 1}\sum_{s\geq 0}\lambda _{s,i}^{-2}E[\left\langle
DG,v_{s}\otimes e_{i}^{\ast }\right\rangle _{L^{2}([0,1])\otimes \mathcal{L}_{HS}(H,\mathbb{R}))}^{2}] \\
=&\, \sum_{i\geq 1}\delta_{i}^{-2}\left\Vert A_{\alpha
_{i}}D^{i}G\right\Vert_{L^{2}(\Omega ;L^{2}([0,1]))}^{2} \\
\leq &\, 2\sum_{i\geq 1}\delta_{i}^{-2}\left\Vert
D^{i}G\right\Vert_{L^{2}(\Omega ;L^{2}([0,1]))}^{2} \\
&+2\sum_{i\geq 1}\frac{1}{(1-2^{-2(\beta _{i}-\alpha _{i})})\delta_{i}^{2}}\int_{0}^{1}\int_{0}^{1}\frac{\left\Vert
D_{t}^{i}G-D_{u}^{i}G\right\Vert_{L^{2}(\Omega )}^{2}}{\left\vert
t-u\right\vert ^{1+2\beta _{i}}}dtdu \\
\leq & \, M\end{aligned}$$ for a constant $M<\infty $. So using Theorem \[General\] we obtain the result.
Technical Estimates
===================
The following technical estimate is used in the course of the paper.
\[VI\_doubleint\] Let $H \in (0,1/2)$ and $t\in [0,T]$ be fixed. Then, there exists a $\beta \in (0,1/2)$ such that $$\begin{aligned}
\label{VI_intI}
\int_0^t \int_0^t \frac{|K_H(t,t_0^{\prime}) - K_H(t,t_0)|^2}{|t_0^{\prime}-t_0|^{1+2\beta}}dt_0 dt_0 ^{\prime}< \infty.\end{aligned}$$
Let $t_0,t_0^{\prime}\in [0,t]$, $t_0^{\prime}<t_0$ be fixed. Write $$K_H (t,t_0) - K_H(t,t_0^{\prime}) = c_H\left[f_t(t_0) - f_t(t_0^{\prime}) +
\left(\frac{1}{2}-H\right) \left(g_t(t_0) - g_t(t_0^{\prime})\right)\right],$$ where $f_t (t_0):= \left(\frac{t}{t_0} \right)^{H-\frac{1}{2}} (t-t_0)^{H-\frac{1}{2}}$ and $g_t(t_0) := \int_{t_0}^t \frac{f_u (t_0)}{u}du$, $t_0\in
[0,t]$.
We will proceed to estimating $K_H (t,t_0) - K_H(t,t_0^{\prime})$. First, observe the following fact, $$\frac{y^{-\alpha} -x^{-\alpha}}{(x-y)^{\gamma}} \leq C y^{-\alpha-\gamma}$$ for every $0<y<x<\infty$ and $\alpha :=(\frac{1}{2}-H) \in (0,1/2)$ and $\gamma < \frac{1}{2}-\alpha$. This implies $$\begin{aligned}
f_t(t_0) - f_t(t_0^{\prime}) &= \left(\frac{t}{t_0} (t-t_0)\right)^{H-\frac{1}{2}}-\left(\frac{t}{t_0^{\prime}} (t-t_0^{\prime})\right)^{H-\frac{1}{2}} \\
&\leq C \left(\frac{t}{t_0}(t-t_0 )\right)^{H-\frac{1}{2} -\gamma
}t^{2\gamma }\frac{(t_0-t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }} \\
&\leq C\frac{(t_0 -t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }}(t-t_0 )^{H-\frac{1}{2}-\gamma } \\
&\leq C\frac{(t_0 -t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }}t_0^{H-\frac{1}{2}-\gamma }(t-t_0)^{H-\frac{1}{2}-\gamma }.\end{aligned}$$
Further, $$\begin{aligned}
g_{t}(t_0 )-g_{t}(t_0^{\prime}) &= \int_{t_0 }^{t}\frac{f_{u}(t_0
)-f_{u}(t_0^{\prime})}{u}du-\int_{t_0^{\prime}}^{t_0 }\frac{f_{u}(t_0^{\prime})}{u}du \\
&\leq \int_{t_0 }^{t}\frac{f_{u}(t_0 )-f_{u}(t_0^{\prime})}{u}du \\
&\leq C\frac{(t_0 -t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }}\int_{t_0 }^{t}\frac{(u-t_0 )^{H-\frac{1}{2}-\gamma }}{u}du \\
&\leq C\frac{(t_0 -t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }}t_0^{H-\frac{1}{2}-\gamma }\int_{1}^{\infty }\frac{(u-1)^{H-\frac{1}{2}-\gamma }}{u}du \\
&\leq C\frac{(t_0-t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }}t_0^{H-\frac{1}{2}-\gamma } \\
&\leq C\frac{(t_0 -t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }}t_0^{H-\frac{1}{2}-\gamma }(t-t_0 )^{H-\frac{1}{2}-\gamma }.\end{aligned}$$
As a result, we have for every $\gamma\in (0,H)$, $0<t_0^{\prime}<t_0<t<T$, $$\begin{aligned}
\label{estimateK}
K_{H}(t,t_0)-K_{H}(t,t_0^{\prime})\leq C_{H,T}\frac{(t_0
-t_0^{\prime})^{\gamma }}{(t_0 t_0^{\prime})^{\gamma }}t_0^{H-\frac{1}{2}-\gamma }(t-t_0 )^{H-\frac{1}{2}-\gamma },\end{aligned}$$ for some constant $C_{H,T}>0$ depending only on $H$ and $T$.
Thus $$\begin{aligned}
\int_{0}^{t}\int_{0}^{t_0 }&\frac{(K_{H}(t,t_0)-K_{H}(t,t_0^{\prime}))^{2}}{|t_0 -t_0^{\prime}|^{1+2\beta }}dt_0^{\prime}dt_0 \\
&\leq C\int_{0}^{t}\int_{0}^{t_0}\frac{|t_0 -t_0^{\prime}|^{-1-2\beta
+2\gamma }}{(t_0 t_0^{\prime})^{2\gamma }}t_0^{2H-1-2\gamma }(t-t_0
)^{2H-1-2\gamma }dt_0^{\prime}dt_0 \\
& =C\int_{0}^{t}t_0^{2H-1-4\gamma }(t-t_0 )^{2H-1-2\gamma }\int_{0}^{t_0
}|t_0 -t_0^{\prime}|^{-1-2\beta +2\gamma }(t_0^{\prime})^{-2\gamma
}dt_0^{\prime}dt_0 \\
&= C\int_{0}^{t}t_0^{2H-1-4\gamma }(t-t_0 )^{2H-1-2\gamma }\frac{\Gamma
(-2\beta +2\gamma )\Gamma (-2\gamma +1)}{\Gamma (-2\beta +1)}t_0 ^{-2\beta
}dt_0 \\
&\leq C\int_{0}^{t}t_0^{2H-1-4\gamma -2\beta }(t-t_0 )^{2H-1-2\gamma }dt_0 \\
&=C\frac{\Gamma (2H-2\gamma )\Gamma (2H-4\gamma -2\beta )}{\Gamma
(4H-6\gamma -2\beta )}t^{4H-6\gamma -2\beta -1}<\infty,\end{aligned}$$for appropriately chosen small $\gamma $ and $\beta$.
On the other hand, we have that $$\begin{aligned}
\int_{0}^{t}\int_{t_0}^{t}&\frac{(K_{H}(t,t_0)-K_{H}(t,t_0^{\prime}))^{2}}{|t_0 -t_0^{\prime}|^{1+2\beta }}dt_0^{\prime}dt_0 \\
&\leq C\int_{0}^{t}t_0^{2H-1-4\gamma }(t-t_0)^{2H-1-2\gamma }\int_{t_0}^{t}\frac{|t_0 -t_0^{\prime}|^{-1-2\beta +2\gamma }}{(t_0^{\prime})^{2\gamma }}dt_0^{\prime}dt_0 \\
&\leq C\int_{0}^{t}t_0^{2H-1-6\gamma }(t-t_0)^{2H-1-2\gamma } \int_{t_0}^t
|t_0 -t_0^{\prime}|^{-1-2\beta +2\gamma } dt_0^{\prime}dt_0 \\
&=C\int_{0}^{t}t_0^{2H-1-6\gamma }(t-t_0 )^{2H-1 -2\beta }dt_0 \\
&\leq Ct^{4H-6\gamma -2\beta -1}.\end{aligned}$$Hence $$\begin{aligned}
\int_{0}^{t}\int_{0}^{t}\frac{(K_{H}(t,t_0 )-K_{H}(t,t_0^{\prime}))^{2}}{|t_0 -t_0^{\prime}|^{1+2\beta }}dt_0^{\prime}dt_0 <\infty .\end{aligned}$$
\[VI\_iterativeInt\] Let $H \in (0,1/2)$, $\theta,t\in [0,T]$, $\theta<t$ and $(\varepsilon_1,\dots, \varepsilon_{m})\in \{0,1\}^{m}$ be fixed. Assume $w_j+\left(H-\frac{1}{2}-\gamma\right) \varepsilon_j>-1$ for all $j=1,\dots,m
$. Then exists a finite constant $C=C(H,T)>0$ such that $$\begin{aligned}
\int_{\Delta_{\theta,t}^{m}} &\prod_{j=1}^{m} (K_H(s_j,\theta) -
K_H(s_j,\theta^{\prime}))^{\varepsilon_j} |s_j-s_{j-1}|^{w_j} ds \\
\leq& C^m \left(\frac{\theta-\theta^{\prime}}{\theta \theta^{\prime}}\right)^{\gamma \sum_{j=1}^m \varepsilon_j} \theta^{\left( H-\frac{1}{2}-\gamma\right)\sum_{j=1}^m \varepsilon_j} \, \Pi_{\gamma}(m) \,
(t-\theta)^{\sum_{j=1}^m w_j + \left( H-\frac{1}{2}-\gamma\right)
\sum_{j=1}^m \varepsilon_j +m}\end{aligned}$$ for $\gamma \in (0,H)$, where $$\begin{aligned}
\label{VI_Pi}
\Pi_{\gamma}(m):=\prod_{j=1}^{m-1} \frac{\Gamma \left(\sum_{l=1}^{j} w_l +
\left(H-\frac{1}{2}-\gamma \right)\sum_{l=1}^{j} \varepsilon_l +
j\right)\Gamma \left( w_{j+1}+1\right)}{\Gamma \left( \sum_{l=1}^{j+1} w_l +
\left(H-\frac{1}{2}-\gamma \right)\sum_{l=1}^{j} \varepsilon_l + j+1\right)}.\end{aligned}$$ Observe that if $\varepsilon_j=0$ for all $j=1,\dots, m$ we obtain the classical formula.
\[VI\_remPi\] Observe that $$\begin{aligned}
\Pi_{\gamma}(m)&\leq \frac{\prod_{j=1}^m\Gamma (w_j +1)}{\Gamma
\left(\sum_{j=1}^{m} w_j + \left(H-\frac{1}{2}-\gamma
\right)\sum_{j=1}^{m-1} \varepsilon_j + m \right)} \\
&\leq \frac{\prod_{j=1}^m\Gamma (w_j +1)}{\Gamma \left(\sum_{j=1}^{m} w_j +
\left(H-\frac{1}{2}-\gamma \right)\sum_{j=1}^{m} \varepsilon_j + m \right)},\end{aligned}$$ since the function $\Gamma$ is increasing on $(1,\infty)$.
First, we recall the following well-known formula: for given exponents $a,b>-1$ and some fixed $s_{j+1}>s_j$ we have $$\int_{\theta}^{s_{j+1}} (s_{j+1}-s_j)^{a} (s_j-\theta)^b ds_j =\frac{\Gamma
\left( a+1\right)\Gamma \left( b+1\right)}{\Gamma \left( a+b+2\right)}
(s_{j+1}-\theta)^{a+b+1}.$$ We recall from Lemma \[VI\_intI\] that for every $\gamma\in (0,H)$, $0<\theta^{\prime}<\theta<s_j<T$, $$\begin{aligned}
K_{H}(s_j,\theta )-K_{H}(s_j,\theta^{\prime})\leq C_{H,T}\frac{(\theta
-\theta^{\prime})^{\gamma }}{(\theta \theta^{\prime})^{\gamma }}\theta^{H-\frac{1}{2}-\gamma }(s_j-\theta )^{H-\frac{1}{2}-\gamma },\end{aligned}$$ for some constant $C_{H,T}>0$ depending only on $H$ and $T$. In view of the above arguments we have $$\begin{aligned}
\int_{\theta}^{s_2}
&|K_H(s_1,\theta)-K_H(s_1,\theta^{\prime})|^{\varepsilon_1}
|s_2-s_1|^{w_2}|s_1-\theta|^{w_1}ds_1 \\
&\leq C_{H,T}^{\varepsilon_1} \frac{(\theta
-\theta^{\prime})^{\gamma\varepsilon_1 }}{(\theta \theta^{\prime})^{\gamma
\varepsilon_1}}\theta^{\left(H-\frac{1}{2}-\gamma\right)\varepsilon_1
}\int_{\theta}^{s_2}|s_2-s_1|^{w_2}|s_1-\theta|^{w_1+\left(H-\frac{1}{2}-\gamma\right)\varepsilon_1}ds_1 \\
&= C_{H,T}^{\varepsilon_1} \frac{(\theta
-\theta^{\prime})^{\gamma\varepsilon_1 }}{(\theta \theta^{\prime})^{\gamma
\varepsilon_1}}\theta^{\left(H-\frac{1}{2}-\gamma\right)\varepsilon_1 }
\frac{\Gamma\left(\hat{w}_1\right)\Gamma\left(\hat{w}_2\right)}{\Gamma\left(\hat{w}_1+\hat{w}_2\right)}(s_2-\theta)^{w_1+w_2+\left(H-\frac{1}{2}-\gamma\right)\varepsilon_1+1},\end{aligned}$$ where $$\hat{w}_1 := w_1+\left(H-\frac{1}{2}-\gamma\right)\varepsilon_1+1, \quad
\hat{w}_2:=w_2+1.$$ Integrating iteratively we obtain the desired formula.
Finally, we give a similar estimate which is used in Lemma \[VI\_relcomp\].
\[VI\_iterativeInt2\] Let $H \in (0,1/2)$, $\theta,t\in [0,T]$, $\theta<t$ and $(\varepsilon_1,\dots, \varepsilon_{m})\in \{0,1\}^{m}$ be fixed. Assume $w_j+\left(H-\frac{1}{2}\right) \varepsilon_j>-1$ for all $j=1,\dots,m$. Then exists a finite constant $C>0$ such that $$\begin{aligned}
\int_{\Delta_{\theta,t}^{m}} &\prod_{j=1}^{m}
(K_H(s_j,\theta))^{\varepsilon_j} |s_j-s_{j-1}|^{w_j} ds \\
&\leq C^m \theta^{\left( H-\frac{1}{2}\right)\sum_{j=1}^m \varepsilon_j} \,
\Pi_0(m) \, (t-\theta)^{\sum_{j=1}^m w_j + \left( H-\frac{1}{2}\right)
\sum_{j=1}^m \varepsilon_j +m}\end{aligned}$$ for $\gamma \in (0,H)$, where $\Pi_0$ is given as in . Observe that if $\varepsilon_j=0$ for all $j=1,\dots, m$ we obtain the classical formula.
\[VI\_remPi2\] Observe that $$\Pi_0(m)\leq \frac{\prod_{j=1}^m\Gamma (w_j +1)}{\Gamma \left(\sum_{j=1}^{m}
w_j + \left(H-\frac{1}{2} \right)\sum_{j=1}^m \varepsilon_j + m\right)},$$ due to the fact that $\Gamma$ is increasing on $(1,\infty)$.
By similar arguments as in the proof of Lemma \[VI\_intI\] it is easy to derive the following estimate $$|K_H(s_j,\theta)| \leq C_{H,T} |s_j-\theta|^{H-\frac{1}{2}}\theta^{H-\frac{1}{2}}$$ for every $0<\theta<s_j<T$ and some constant $C_{H,T}>0$. This implies $$\begin{aligned}
&\int_{\theta}^{s_2} (K_H(s_1,\theta))^{\varepsilon_1}
|s_2-s_1|^{w_2}|s_1-\theta|^{w_1}ds_1 \\
&\leq C_{H,T}^{\varepsilon_1} \, \theta^{\left(H-\frac{1}{2}\right)\varepsilon_1} \int_{\theta}^{s_2} |s_2-s_1|^{w_2}
|s_1-\theta|^{w_1+\left(H-\frac{1}{2}\right)\varepsilon_1}ds_1 \\
&= C_{H,T}^{\varepsilon_1} \, \theta^{\left(H-\frac{1}{2}\right)\varepsilon_1} \frac{\Gamma\left(w_1+w_2+\left(H-\frac{1}{2}\right)\varepsilon_1+1\right)\Gamma\left(w_2+1\right)}{\Gamma\left(w_1+w_2+\left(H-\frac{1}{2}\right)\varepsilon_1+2\right)}(s_2-\theta)^{w_1+w_2+\left(H-\frac{1}{2}\right)\varepsilon_1+1}\end{aligned}$$ Integrating iteratively one obtains the desired estimate.
The next auxiliary result can be found in [@LiWei].
\[LiWei\] Assume that $X_{1},...,X_{n}$ are real centered jointly Gaussian random variables, and $\Sigma =(E[X_{j}X_{k}])_{1\leq j,k\leq n}$ is the covariance matrix, then$$E[\left\vert X_{1}\right\vert ...\left\vert X_{n}\right\vert ]\leq \sqrt{perm(\Sigma )},$$where $perm(A)$ is the permanent of a matrix $A=(a_{ij})_{1\leq i,j\leq n}$ defined by$$perm(A)=\sum_{\pi \in S_{n}}\prod_{j=1}^{n}a _{j,\pi (j)}$$for the symmetric group $S_{n}$.
The next result corresponds to Lemma 3.19 in [@CD]:
\[CD\] Let $Z_{1},...,Z_{n}$ be mean zero Gaussian variables which are linearly independent. Then for any measurable function $g:\mathbb{R}\longrightarrow \mathbb{R}_{+}$ we have that$$\int_{\mathbb{R}^{n}}g(v_{1})\exp (-\frac{1}{2}Var[\sum_{j=1}^{n}v_{j}Z_{j}])dv_{1}...dv_{n}=\frac{(2\pi )^{(n-1)/2}}{(\det
Cov(Z_{1},...,Z_{n}))^{1/2}}\int_{\mathbb{R}}g(\frac{v}{\sigma _{1}})\exp (-\frac{1}{2}v^{2})dv,$$where $\sigma _{1}^{2}:=Var[Z_{1}\left\vert Z_{2},...,Z_{n}\right] $.
\[OrderDerivatives\] Let $n,$ $p$ and $k$ be non-negative integers, $k\leq n$. Assume we have functions $f_{j}:[0,T]\rightarrow \mathbb{R}$, $j=1,\dots ,n$ and $g_{i}:[0,T]\rightarrow \mathbb{R}$, $i=1,\dots ,p$ such that $$f_{j}\in \left\{ \frac{\partial ^{\alpha _{j}^{(1)}+...+\alpha _{j}^{(d)}}}{\partial ^{\alpha _{j}^{(1)}}x_{1}...\partial ^{\alpha _{j}^{(d)}}x_{d}}b^{(r)}(u,X_{u}^{x}),\text{ }r=1,...,d\right\} ,\text{ }j=1,...,n$$and $$g_{i}\in \left\{ \frac{\partial ^{\beta _{i}^{(1)}+...+\beta _{i}^{(d)}}}{\partial ^{\beta _{i}^{(1)}}x_{1}...\partial ^{\beta _{i}^{(d)}}x_{d}}b^{(r)}(u,X_{u}^{x}),\text{ }r=1,...,d\right\} ,\text{ }i=1,...,p$$for $\alpha :=(\alpha _{j}^{(l)})\in \mathbb{N}_{0}^{d\times n}$ and $\beta
:=(\beta _{i}^{(l)})\in \mathbb{N}_{0}^{d\times p},$ where $X_{\cdot }^{x}$ is the strong solution to $$X_{t}^{x}=x+\int_{0}^{t}b(u,X_{u}^{x})du+B_{t}^{H},\text{ }0\leq t\leq T$$for $b=(b^{(1)},...,b^{(d)})$ with $b^{(r)}\in C_{c}([0,T]\times \mathbb{R}^{d})\mathcal{\ }$for all $r=1,...,d$. So (as we shall say in the sequel) the product $g_{1}(r_{1})\cdot \dots \cdot g_{p}(r_{p})$ has a total order of derivatives $\left\vert \beta \right\vert
=\sum_{l=1}^{d}\sum_{i=1}^{p}\beta _{i}^{(l)}$. We know from Section [VI\_shuffles]{} that $$\begin{aligned}
& \int_{\Delta _{\theta ,t}^{n}}f_{1}(s_{1})\dots f_{k}(s_{k})\int_{\Delta
_{\theta ,s_{k}}^{p}}g_{1}(r_{1})\dots g_{p}(r_{p})dr_{p}\dots
dr_{1}f_{k+1}(s_{k+1})\dots f_{n}(s_{n})ds_{n}\dots ds_{1} \notag \\
& =\sum_{\sigma \in A_{n,p}}\int_{\Delta _{\theta ,t}^{n+p}}h_{1}^{\sigma
}(w_{1})\dots h_{n+p}^{\sigma }(w_{n+p})dw_{n+p}\dots dw_{1}, \label{h}\end{aligned}$$where $h_{l}^{\sigma }\in \{f_{j},g_{i}:1\leq j\leq n,$ $1\leq i\leq p\}$, $A_{n,p}$ is a subset of permutations of $\{1,\dots ,n+p\}$ such that $\#A_{n,p}\leq C^{n+p}$ for an appropriate constant $C\geq 1$, and $s_{0}=\theta $. Then the products$$h_{1}^{\sigma }(w_{1})\cdot \dots \cdot h_{n+p}^{\sigma }(w_{n+p})$$have a total order of derivatives given by $\left\vert \alpha \right\vert
+\left\vert \beta \right\vert .$
The result is proved by induction on $n$. For $n=1$ and $k=0$ the result is trivial. For $k=1$ we have $$\begin{aligned}
\int_{\theta }^{t}f_{1}(s_{1})\int_{\Delta _{\theta
,s_{1}}^{p}}g_{1}(r_{1})\dots g_{p}(r_{p}) &&dr_{p}\dots dr_{1}ds_{1} \\
&=&\int_{\Delta _{\theta ,t}^{p+1}}f_{1}(w_{1})g_{1}(w_{2})\dots
g_{p}(w_{p+1})dw_{p+1}\dots dw_{1},\end{aligned}$$where we have put $w_{1}=s_{1},$ $w_{2}=r_{1},\dots ,w_{p+1}=r_{p}$. Hence the total order of derivatives involved in the product of the last integral is given by $\sum_{l=1}^{d}\alpha
_{1}^{(l)}+\sum_{l=1}^{d}\sum_{i=1}^{p}\beta _{i}^{(l)}=\left\vert \alpha
\right\vert +\left\vert \beta \right\vert .$
Assume the result holds for $n$ and let us show that this implies that the result is true for $n+1$. Either $k=0,1$ or $2\leq k\leq n+1$. For $k=0$ the result is trivial. For $k=1$ we have $$\begin{aligned}
\int_{\Delta _{\theta ,t}^{n+1}}& f_{1}(s_{1})\int_{\Delta _{\theta
,s_{1}}^{p}}g_{1}(r_{1})\dots g_{p}(r_{p})dr_{p}\dots
dr_{1}f_{2}(s_{2})\dots f_{n+1}(s_{n+1})ds_{n+1}\dots ds_{1} \\
& =\int_{\theta }^{t}f_{1}(s_{1})\left( \int_{\Delta _{\theta
,s_{1}}^{n}}\int_{\Delta _{\theta ,s_{1}}^{p}}g_{1}(r_{1})\dots
g_{p}(r_{p})dr_{p}\dots dr_{1}f_{2}(s_{2})\dots
f_{n+1}(s_{n+1})ds_{n+1}\dots ds_{2}\right) ds_{1}.\end{aligned}$$Using Section \[VI\_shuffles\] we obtain by employing the shuffle permutations that the latter inner double integral on diagonals can be written as a sum of integrals on diagonals of length $p+n$ with products having a total order of derivatives given by $\sum_{l=1}\sum_{j=2}^{n+1}\alpha _{j}^{(l)}+\sum_{l=1}^{d}\sum_{i=1}^{p}\beta _{i}^{(l)}$. Hence we obtain a sum of products, whose total order of derivatives is $\sum_{l=1}^{d}\sum_{j=2}^{n+1}\alpha
_{j}^{(l)}+\sum_{l=1}^{d}\sum_{i=1}^{p}\beta _{i}^{(l)}+\sum_{l=1}^{d}\alpha
_{1}^{(l)}=\left\vert \alpha \right\vert +\left\vert \beta \right\vert .$
For $k\geq 2$ we have (in connection with Section \[VI\_shuffles\]) from the induction hypothesis that $$\begin{aligned}
\int_{\Delta _{\theta ,t}^{n+1}}f_{1}(s_{1})\dots f_{k}(s_{k})\int_{\Delta
_{\theta ,s_{k}}^{p}}g_{1}(r_{1})\dots g_{p}(r_{p})& dr_{p}\dots
dr_{1}f_{k+1}(s_{k+1})\dots f_{n+1}(s_{n+1})ds_{n+1}\dots ds_{1} \\
=\int_{\theta }^{t}f_{1}(s_{1})\int_{\Delta _{\theta
,s_{1}}^{n}}f_{2}(s_{2})\dots f_{k}(s_{k})& \int_{\Delta _{\theta
,s_{k}}^{p}}g_{1}(r_{1})\dots g_{p}(r_{p})dr_{p}\dots dr_{1} \\
& \times f_{k+1}(s_{k+1})\dots f_{n+1}(s_{n+1})ds_{n+1}\dots ds_{2}ds_{1} \\
=\sum_{\sigma \in A_{n,p}}\int_{\theta }^{t}f_{1}(s_{1})\int_{\Delta
_{\theta ,s_{1}}^{n+p}}& h_{1}^{\sigma }(w_{1})\dots h_{n+p}^{\sigma
}(w_{n+p})dw_{n+p}\dots dw_{1}ds_{1},\end{aligned}$$where each of the products $h_{1}^{\sigma }(w_{1})\cdot \dots \cdot
h_{n+p}^{\sigma }(w_{n+p})$ have a total order of derivatives given by $\sum_{l=1}\sum_{j=2}^{n+1}\alpha
_{j}^{(l)}+\sum_{l=1}^{d}\sum_{i=1}^{p}\beta _{i}^{(l)}.$ Thus we get a sum with respect to a set of permutations $A_{n+1,p}$ with products having a total order of derivatives which is$$\sum_{l=1}^{d}\sum_{j=2}^{n+1}\alpha
_{j}^{(l)}+\sum_{l=1}^{d}\sum_{i=1}^{p}\beta _{i}^{(l)}+\sum_{l=1}^{d}\alpha
_{1}^{(l)}=\left\vert \alpha \right\vert +\left\vert \beta \right\vert .$$
[999]{} Alinhac, S., Gérard, P.: Pseudo-Differential Operators and the Nash-Moser Theorem. Graduate Studie in Mathematics, Vol 82, AMS (2007).
E. Alòs, O. Mazet and D. Nualart, *Stochastic calculus with respect to Gaussian processes*. Annals of Probability **29**, (2001), 766–801.
L. Ambrosio, *Transport equation and Cauchy problem for BV vector fields.* Inventiones mathematicae 158, (2004), 227–260.
Amine, O., Baños, D., Proske, F.: Regularity properties of the stochastic flow of a skew fractional Brownian motion. arXiv:1805.04889v1 (2018).
Amine, O., Coffie, E., Harang, F., Proske, F.: A Bismut-Elworthy-Li formula for singular SDE’s driven by fractional Brownian motion and applications to rough volatility modeling. arXiv:1805.11435 (2018).
Anari, N., Gurvits, L., Gharan, S., Saberi, A.: Simply exponential approximation of the permanent of positive semdefinite matrices. 58th Annual IEEE Symposium on Foundations of Computer Science (2017).
D. R. Baños, H. H. Haferkorn, F. Proske, *Strong Uniqueness of Singular Stochastic Delay Equations*, arXiv:1707.02271 \[math.PR\]
D. R. Baños, T. Nilssen, F. Proske, *Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE’s with singular drift*, arXiv:1511.02717 \[math.PR\]
D. R. Baños, S. Ortiz-Latorre, A. Pilipenko, F. Proske, *Strong solutions of SDE’s with generalized drift and multidimensional fractional Brownian initial noise*, arXiv:1705.01616 \[math.PR\] Baudoin, F.: An Introduction to the Geometry of Stochastic Flows. Imperial College Press (2004).
S. M. Berman and N. Kono, *The Maximum of a Process with Nonconstant Variance: A Sharp Bound for the Distribution Tail*, The Annals of Probability, Vol 17, No 2 (1989), pp. 632–650.
V. L. Bogachev, *Differentiable Measures and the Malliavin Calculus.* Mathematical Surveys and Monographs, AMS (2010).
Catellier, R., Gubinelli, M.: *Averaging along irregular curves and regularisation of ODE’s.* Stoch. Proc. and their Appl., Vol. 126, Issue 8, p. 2323-2366 (August 2016).
Cuzick, J., DuPreez, P.: Joint continuity of Gaussian local times. Annals of Prob., Vol. 10, No. 3, 810-817 (1982).
A. M. Davie, *Uniqueness of solutions of stochastic differential equations.* Int. Math. Res. 24, Article ID rnm 124, (2007), 26 P.\]
G. Da Prato, F. Flandoli, E. Priola, M. Röckner, *Strong uniqueness for stochastic evolution equations in Hilbert spaces with bounded and measurable drift.* Ann. of Prob., Vol. 41 (5), (2013), 3306–3344
G. Da Prato, P. Malliavin, D. Nualart, *Compact families of Wiener functionals.* C. R. Acad. Sci. Paris, t. 315, Série I, (1992), 1287–1291.
L. Decreusefond, A. S. Üstünel, *Stochastic analysis of the fractional Brownian motion*. Potential Analysis **10**, (1998), 177–214.
G. Di Nunno, B. Øksendal, F. Proske, *Malliavin Calculus for Lévy Processes with Applications to Finance.* Springer (2008).
R. J. DiPerna, P.L. Lions, *Ordinary differential equations, transport theory and Sobolev spaces.* Inventiones mathematicae 98, (1989), 511–547.
E. Fedrizzi and F. Flandoli, *Hölder Flow and Differentiability for SDEs with Nonregular Drift*, (2010), preprint.
Fedrizzi, E., Flandoli, F. *Noise prevents singularities i linear transport equations*. J. of Funct. Analysis, Vol. 264, No. 6, (2013), 1329–1354.
Flandoli, F., Nilssen, T., Proske, F. *Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces.* Preprint series, University of Oslo, ISSN 0806-2439 (2013).
Gressman, P.T.: Generalized curvature for certain Radon-like operators of intermediate dimension. arXiv:1609.02972 (2016).
L. Grafakos, *Classical Fourier analysis*, second ed., Springer, New York, 2008.
I. Gyöngy, N. V. Krylov, *Existence of strong solutions for Itô’s stochastic equations via approximations.* Probab. Theory Relat. Fields, **105** (1996), 143–158.
I. Gyöngy, T. Martinez, *On stochastic differential equations with locally unbounded drift.* Czechoslovak Mathematical Journal, **51** (4), (2001), 763–783.
Haadem, S., Proske, F. *On the construction and Malliavin differentiability of solutions of Lévy noise driven SDE’s with singular coefficients.* J. of Funct. Analysis, 266 (8), (2014), 5321–5359.
N.V. Krylov, M. Röckner, * Strong solutions of stochastic equations with singular time dependent drift.* Prob. Theory Rel. Fields **131** (2), (2005), 154–196.
Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press (1990).
Li, W., Wei, A.: A Gaussian inequality for expected absolute products. Journal of Theoretical Prob., Vol. 25, Issue 1, pp 92-99 (2012).
Liptser, R.S., Shiryaev, A.N. *Statistics of random processes.* ”Nauka”, Moscow, 1974 in Russian; English translation: Springer-Verlag, Berlin and NewYork, 1977.
P.I. Lizorkin, *Fractional integration and differentiation*, Encyclopedia of Mathematics, Springer, (2001)
T. J. Lyons, M. Caruana, T. Lévy, *Differential Equations Driven by Rough Paths.* École d’Été de Probabilités de Saint-Flour XXXIV–2004.
P. Malliavin, *Stochastic calculus of variations and hypoelliptic operators.* In: Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto 1976, Wiley, (1978), 195–263.
P. Malliavin, *Stochastic Analysis.* Springer (1997).
M. B. Marcus, J. Rosen, *Markov Processes, Gaussian Processes, and Local Times* Cambridge University Press, New York, (2006).
O. Menoukeu-Pamen, T. Meyer-Brandis, T. Nilssen, F. Proske, T. Zhang, *A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s.* Math. Ann. **357** (2), (2013), 761–799.
T. Meyer-Brandis, F. Proske, *On the existence and explicit representability of strong solutions of Lévy noise driven SDE’s.* Communications in Mathematical Sciences, **4** (1) (2006).
T. Meyer-Brandis, F. Proske, *Construction of strong solutions of SDE’s via Malliavin calculus.* Journal of Funct. Anal. **258**, (2010), 3922–3953.
S.E.A. Mohammed, T. Nilssen, F. Proske, *Sobolev Differentiable Stochastic Flows for SDE’s with Singular Coefficients: Applications to the Transport Equation.* Ann. Probab. **43** (3), (2015), 1535–1576. Moser, J.: A rapidly convergent iteration method and nonlinear differential equations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 20, p. 499-535 (1966).
Nash, J.: The imbedding problem for Riemannian manifolds.Ann. Math. 63, p. 20-63 (1956).
D. Nualart and Y. Ouknine, *Regularization of differential equations by fractional noise*. Stochastic Processes and their Applications, **102** (1), (2002), 103–116.
D. Nualart and Y. Ouknine, *Stochastic Differential Equations with Additive Fractional Noise and Locally Unbounded Drift*, Stochastic Inequalities and Applications, Volume 56 of the series Progress in Probability, (2003), 353–365.
D. Nualart, *The Malliavin Calculus and Related Topics.* 2nd Ed. Springer (2010).
L. D. Pitt, *Local times for Gaussian vector fields*, Indiana Univ. Math. J. **27**, (1978), 309–330.
N. I. Portenko, *Generalized Diffusion Processes.* Transl. Math. Monographs, 83, Amer. Math. Soc., Providence, R. I. (1990).
E. Priola, *Pathwise uniqueness for singular SDE’s driven by stable processes.* Osaka J. Math., **49** (2), (2012), 421–447.
D. Revuz, M. Yor, *Continuous Martingales and Brownian Motion.* Third edition, Springer, 1999.
S. G. Samko, A. A. Kilbas and O. L. Marichev, *Fractional Integrals and Derivatives*. Theory and Applications, Gordon and Breach, (1993).
S. Smale, *Stable manifolds for differential equations and diffeomorphisms*. Ann. Scuela. Sup. Pisa, **17**, no. 1–2, (1963), 97–116. Tao, T., Wright, J.: $L^{p}$ improving bounds for averages along curves. Journal of the American Mathematical Society, 16(3):605-638 (2003).
A.Y. Veretennikov, *On the strong solutions of stochastic differential equations.* Theory Probab. Appl. **24**, (1979), 354–366.
Y. X. Xiao, *Fractal and smoothness properties of space–time Gaussian models.* Frontiers of Mathematics in China 6, (2011), 1217–1248.
T. Yamada, S. Watanabe, *On the uniqueness of solutions of stochastic differential equations.* J. Math. Kyoto Univ. II, (1971), 155–167.
X. Zhang, *Stochastic differential equations with Sobolev drifts and driven by $\alpha$-stable processes.* Ann. Inst. H. Poincaré Probab. Statist., **49**, (2013), 915–931.
A.K. Zvonkin, *A transformation of the state space of a diffusion process that removes the drift.* Math.USSR (Sbornik) **22**, (1974), 129–149.
|
---
author:
- '[Balázs Szegedy]{}'
title: 'Structure of finite nilspaces and inverse theorems for the Gowers norms in bounded exponent groups.'
---
Introduction
============
In a recent paper [@Sz4] the author proved a general regularity lemma and inverse theorem for Gowers’s uniformity norms $U_{k+1}$. The results in [@Sz4] connect the theory of certain algebraic structures, called nilspaces [@HKr2],[@NP], with the theory of Gowers norms. Note that nilspaces are parallelepiped structures (introduced in a fundamental paper by Host and Kra [@HKr2]) in which cubes of every dimension are defined.
The main result in [@Sz4] says (very roughly speaking) that every function $f:A\rightarrow\mathbb{C}$ with $|f|\leq 1$ on a compact abelian group can be decomposed as $f=f_s+f_e+f_r$ where $f_e$ is a small error term, $f_s$ is a structured function related to an algebraic morphism $\phi:A\rightarrow N$ where $N$ is a bounded complexity $k$-step nilspace, and $\|f_r\|_{U_{k+1}}$ is very small in terms of the complexity of $N$.
It turns out that if $A$ is chosen from the special family of groups in which the order of every element divides a fixed number $q$ (called $q$ exponent groups) then the nilspace $N$ in the previous decomposition is finite and has exponent $q$ in the sense that it is built up from $k$ abelian groups of exponent $q$ as an iterated bundle. This motivates us to analyze the structure of finite nilspaces.
Our main method is to produce every finite nilspace $N$ as a factor of a nilspace $M$ which is the direct product of [*cyclic nilspaces*]{}. (A cyclic nilspace is a cyclic abelian group endowed with a $k$-degree structure for some natural number $k$.) We show that every morphism $\phi:A\rightarrow N$ from an abelian group $A$ to $N$ can be lifted as a morphism $\psi:B\rightarrow M$ where $A$ is a factor group of $B$.
Using these results combined with the results in [@Sz4] we get inverse theorems for the Gowers norms in bounded exponent groups. The next definition is from [@TZ]
A phase polynomial $\phi:A\rightarrow\{x|x\in\mathbb{C},|x|=1\}$ of degree $k$ on an abelian group $A$ is a function which trivializes after $k+1$ consecutive application of operators $\Delta_t$ defined by $\Delta_t f(x)=f(x)\overline{f(x+t)}$ and $t\in A$.
\[boundexinv\] Let $q$ be a fixed natural number. For every $\epsilon>0$ there is $\delta>0$ such that if $f:A\rightarrow\mathbb{C}$ with $|f|\leq 1$ is a function on the abelian group $A$ of exponent $q$ with $\|f\|_{U_{k+1}}\geq\epsilon$ then there is an extension $B$ of $A$ with the same rank as $A$ and phase polynomial function $\phi:B\rightarrow\mathbb{C}$ of degree $k$ such that $(f,\phi)\geq\delta$.
Note that in the above theorem $f$ is interpreted also as a function on $B$ by composing the projection $B\rightarrow A$ with $f$. This makes it possible to take the scalar product $(f,\phi)$. We can also formulate this inverse theorem in a trivially equivalent but slightly more conventional form if we introduce the notion of a projected phase polynomial.
A function $f$ is called a [**projected phase polynomial**]{} (of degree $k$) on a finite abelian group $A$ if there is an extension $$0\rightarrow C\rightarrow B\stackrel{\tau}{\rightarrow} A\rightarrow 0$$ with $\rk(A)=\rk(B)$ and a phase polynomial $\phi:B\rightarrow\mathbb{C}$ (of degree $k$) such that $$f(a)=|C|^{-1}\sum_{b\in B,\tau(b)=a}\phi(b).$$
Using this definition, theorem \[boundexinv\] says the following.
[*If $\|f\|_{U_{k+1}}\geq\epsilon$ then $f$ correlates with a projected phase polynomial of degree $k$.*]{}
It is clear that projected phase polynomials are “purely structured” functions. This means that correlation with them implies non negligible $U_{k+1}$ norm. To see this, assume that $g$ correlates with a projected phase polynomial. Then on the extended group $B$ the function $g'=\tau\circ g$ correlates with a phase polynomial of degree $k$ and so $\|g'\|_{U_{k+1}}$ is non negligible. On the other hang $\|g'\|_{U_{k+1}}=\|g\|_{U_{k+1}}$.
It will be important that theorem \[boundexinv\] can be strengthened in two ways.
\[boundext\] In theorem \[boundexinv\] we can also assume that
1. $\phi^m=1$ for some $m=q^i$ where $i$ is bounded in terms of $\epsilon$,
2. $\phi=\phi_1\phi_2$ where $\phi_1$ is a degree $k-1$ phase polynomial and $\phi_2$ takes only $q$-th roots of unities.
We will show that theorem \[boundext\] implies the next theorem by Tao and Ziegler [@TZ].
\[TZth\] Let $p$ be a fixed prime and $k\leq p-1$. For every $\epsilon>0$ there is $\delta>0$ such that if $f:A\rightarrow\mathbb{C}$ with $|f|\leq 1$ is a function on the abelian group $A$ of exponent $p$ with $\|f\|_{U_{k+1}}\geq\epsilon$ then there is a phase polynomial function $\phi:A\rightarrow\mathbb{C}$ of degree $k$ and with $\phi^p=1$ such $(f,\phi)\geq\delta$.
Note that in [@TZ] the authors also prove an inverse theorem for $k>p-1$ in which the function correlates with a phase polynomial of degree bounded in terms of $k$.
Structure of finite nilspaces
=============================
Roughly speaking, a nilspace is a structure in which cubes of every dimension are defined and they behave very similarly as cubes in abelian groups. An abstract cube of dimension $n$ is the set $\{0,1\}^n$. A cube of dimension $n$ in an abelian group $A$ is a function $f:\{0,1\}^n\rightarrow A$ which extends to an affine homomorphism (a homomorphism plus a translation) $f':\mathbb{Z}^n\rightarrow A$. Similarly, a morphism $\psi:\{0,1\}^n\rightarrow\{0,1\}^m$ between abstract cubes is a map which extends to an affine morphism from $\mathbb{Z}^n\rightarrow\mathbb{Z}^m$.
A nilspace is a set $N$ and a collection $C^n(N)\subseteq N^{\{0,1\}^n}$ of functions (or cubes) of the form $f:\{0,1\}^n\rightarrow N$ such that the following axioms hold.
1. [**(Composition)**]{} If $\psi:\{0,1\}^n\rightarrow\{0,1\}^m$ is a cube morphism and $f:\{0,1\}^m\rightarrow N$ is in $C^m(N)$ then the composition $\psi\circ f$ is in $C^n(N)$.
2. [**(Ergodictiry)**]{} $C^1(N)=N^{\{0,1\}}$.
3. [**(Gluing)**]{} If a map $f:\{0,1\}^n\setminus\{1^n\}\rightarrow N$ is in $C^{n-1}(N)$ restricted to each $n-1$ dimensional face containing $0^n$ then $f$ extends to the full cube as a map in $C^n(N)$.
If $N$ is a nilspace and in the third axiom the extension is unique for $n=k+1$ then we say that $N$ is a $k$-step nilspace. If a space $N$ satisfies the first axiom (but the last two are not required) then we say that $N$ is a [**cubespace**]{}. A function $f:N_1\rightarrow N_2$ between two cubespaces is called a [**morphism**]{} if $\phi\circ f$ is in $C^n(N_2)$ for every $n$ and function $\phi\in C^n(N_1)$. The set of morphisms between $N_1$ and $N_2$ is denoted by ${\mathrm{Hom}}(N_1,N_2)$. With this notation $C^n(N)={\mathrm{Hom}}(\{0,1\}^n,N)$. If $N$ is a nilspace then every morphism $f:\{0,1\}^n\rightarrow\{0,1\}^m$ induces a map $\hat{f}:C^m(N)\rightarrow C^n(N)$ by simply composing $f$ with maps in $C^m(N)$.
Every abelian group $A$ has a natural nilspace structure in which cubes are maps $f:\{0,1\}^n\rightarrow A$ which extend to affine homomorphisms $f':\mathbb{Z}^n\rightarrow A$. We refer to this as the linear structure on $A$. For every natural number $k$ we can define a nilspace structure $\mathcal{D}_k(A)$ on $A$ that we call the $k$-degree structure. A function $f:\{0,1\}^n\rightarrow A$ is in $C^n(\mathcal{D}_k(A))$ if for every cube morphism $\phi:\{0,1\}^{k+1}\rightarrow\{0,1\}^n$ we have that $$\sum_{v\in\{0,1\}^{k+1}}f(\phi(v))(-1)^{h(v)}=0$$ where $h(v)=\sum_{i=1}^{k+1}v_i$. We will use the next elementary lemma in this paper.
\[derivmorph\] Let $\phi:\mathcal{D}_i(A)\rightarrow\mathcal{D}_j(B)$ be a morphism. Then if $i>j$ then $\phi$ is constant. If $1\leq i\leq j$ then $\phi$ is also a morphism from $\mathcal{D}_1(A)$ to $\mathcal{D}_{j-i+1}(B)$.
This follows directly by applying the operators $\partial_x$ (defined in [@NP]) to the cub structures on $\mathcal{D}_i(A)$ and $\mathcal{D}_j(B)$.
It was shown in [@NP] that higher degree abelian groups are building blocks of every $k$-step nilspace. To state the precise statement we will need the following formalism. Let $A$ be an abelian group and $X$ be an arbitrary set. An $A$ bundle over $X$ is a set $B$ together with a free action of $A$ such that the orbits of $A$ are parametrized by the elements of $X$. This means that there is a projection map $\pi:B\rightarrow X$ such that every fibre is an $A$-orbit. The action of $a\in A$ on $x\in B$ is denoted by $x+a$. Note that if $x,y\in B$ are in the same $A$ orbit then it make sense to talk about the difference $x-y$ which is the unique element $a\in A$ with $y+a=x$. In other words the $A$ orbits can be regarded as affine copies of $A$.
A $k$-fold abelian bundle $X_k$ is a structure which is obtained from a one element set $X_0$ in $k$-steps in a way that in the $i$-th step we produce $X_i$ as an $A_i$ bundle over $X_{i-1}$. The groups $A_i$ are the structure groups of the $k$-fold bundle. We call the spaces $X_i$ the $i$-th factors.
\[bundef\] Let $X_k$ be a $k$-fold abelian bundle with factors $\{X_i\}_{i=1}^k$ and structure groups $\{A_i\}_{i=1}^k$. Let $\pi_i$ denote the projection of $X_k$ to $X_i$. Assume that $X_k$ admits a cubespace structure with cube sets $\{C^n(X_k)\}_{n=1}^\infty$. We say that $X_k$ is a [**$k$-degree bundle**]{} if it satisfies the following conditions
1. $X_{k-1}$ is a $k-1$ degree bundle.
2. Every function $f\in C^n(X_{k-1})$ can be lifted to $f'\in C^n(X_k)$ with $f'\circ\pi_{k-1}=f$.
3. If $f\in C^n(X_k)$ then the fibre of $\pi_{k-1}:C^n(X_k)\rightarrow C^n(X_{k-1})$ containing $f$ is $$\{f+g|g\in C^n(\mathcal{D}_k(A_k))\}.$$
The next theorem form [@NP] says that $k$-degree bundles are the same as $k$-step nilspaces.
Every $k$-degree bundle is a $k$-step nilspace and every $k$-step nilspace arises as a $k$-degree bundle.
We say that a $k$-step nilspace is of exponent $q$ if all the structure groups $A_i$ are of exponent $q$.
Let $x,y$ be two elements in the nilspace $N$. We say that $x\sim_i y$ if the map $c:\{0,1\}^{i+1}\rightarrow N$ with $c(0^{i+1})=x$ and $c(v)=y$ if $v\neq 0^{i+1}$ is an element in $C^{i+1}(N)$. It was proved in [@NP] that $\sim_i$ is an equivalence relation and the classes form a factor of $N$ denoted by $\mathcal{F}_i(N)$. The factor $\mathcal{F}_i$ coincides with the factor $X_i$ in definition \[bundef\].
We introduce special morphisms between $k$-step nilspaces which have very strong surjectivity properties and behave consistently with respect to the equivalence classes $\sim_i$.
Let $N$ and $M$ be $k$-step nilspaces. A morphism $\phi:N\rightarrow M$ is called a factor map if for every $1\leq i\leq k$ the image of every $\sim_i$ class in $N$ is a $\sim_i$ class in $M$.
Notice that every morphism maps a $\sim_i$ class into a $\sim_i$ class. In the above definition we require the surjectivity of these local maps. These type of maps were also investigated in [@NP]. If $N$ and $M$ are compact nilspaces then factor maps are measure preserving which is useful. In this paper we will use this notion to obtain finite nilspaces from free nilspaces as factors.
Extensions
----------
\[kdegext\] Let $N$ be an arbitrary nilspace. A degree $k$-extension of $N$ is an abelian bundle $M$ over $N$ which is a cube space with the following properties.
1. For every $n\in\mathbb{N}$ and $c\in C^n(N)$ there is $c'\in C^n(N)$ such that $\pi(c')=c$,
2. If $c_1\in C^n(M)$ and $c_2:\{0,1\}^n\rightarrow M$ with $\pi(c_1)=\pi(c_2)$ then $c_2\in C^n(M)$ if and only if $c_1-c_2\in C^n(\mathcal{D}_k(A))$.
The map $\pi$ is the projection from $M$ to $N$. The extension $M$ is called a split extension if there is a cube preserving morphism $m:N\rightarrow M$ such that $m\circ\pi$ is the identity map of $N$.
Let $N$ be a $k$-step nilspace. If $M$ is a degree $i$ extension of $N$ by an abelian group $A$ then $\mathcal{F}_i(M)$ is a degree $i$ extension of $\mathcal{F}_i(N)$ by $A$. Furthermore the projection $\tau:\mathcal{F}_i(M)\rightarrow\mathcal{F}_i(N)$ is the composition of the projection $\pi:M\rightarrow N$ by the projection $\pi_i:N\rightarrow\mathcal{F}_i(N)$.
First of all observe that $\mathcal{F}_i(M)$ is some $i$-degree extension of $\mathcal{F}_i(N)$. All we need to show is that the structure group of this extension is $A$. In other words $A$ does not collapse when we look at the situation modulo $\sim_i$. To see this we show that if $x,y$ are in the same fibre of $\pi$ and $x\neq y$ then $x$ is different from $y$ mod $\sim_i$. Let $F$ be the fibre containing $x$ and $y$. Since the structure of $M$ restricted to $F$ is $\mathcal{D}_i(A)$ and $\sim_i$ separates every element in $\mathcal{D}_i(A)$ the proof is complete.
\[subdirect\] Let $N$ be a $k$-step nilspace. If $M$ is a degree $i$ extension of $N$ by an abelian group $A$ (with projection $\pi$) then $M$ is a sub direct product of $N$ with a degree $i$ extension $K$ of $\mathcal{F}_i(N)$ by $A$. The sub direct product is the set of pairs $(a,b)$ such that $\pi_i(a)=\tau(b)$ such that $\pi_i:N\rightarrow\mathcal{F}_i(N)$ and $\tau:K\rightarrow\mathcal{F}_i(N)$ are the projection maps.
Let $K=\mathcal{F}_i(M)$ and $\tau:M\rightarrow\mathcal{F}_i(M)$ be the projection. Let $\phi=\pi\times\tau$ be the morphism from $M$ to $N\times K$. According to the previous lemma the image of $\phi$ gives an isomorphism between $M$ and the subdirect product of $N$ and $K$ defined in the lemma.
Translation groups
------------------
For an arbitrary subset $F$ in $\{0,1\}^n$ and map $\alpha:N\rightarrow N$ we define the map $\alpha^F$ from $C^n(N)$ to $N^{\{0,1\}^n}$ such that $\alpha^F(c)(v)=\alpha(c(v))$ if $v\in F$ and $\alpha^F(c)(v)=c(v)$ if $v\notin F$.
\[transdef\] Let $N$ be a nilspace. A map $\alpha:N\rightarrow N$ is called a translation of hight $i$ if for every natural number $n\geq i$, $n-i$ dimensional face $F\subseteq\{0,1\}^n$ and $c\in C^n(N)$ the map $\alpha^F(c)$ is in $C^n(N)$. We denote the set of hight $i$ translations by ${\mathrm{Trans}}_i(N)$. We will use the short hand notation ${\mathrm{Trans}}(N)$ for ${\mathrm{Trans}}_1(N)$.
It is not hard to see that if $N$ is a $k$-step nilspace then ${\mathrm{Trans}}(N)$ is a $k$-nilpotent group and $\{{\mathrm{Trans}}_i(N)\}_{i=1}^k$ is a central series in ${\mathrm{Trans}}(N)$. In this chapter we are interested in the following question.
[*Let $N$ be an $k$-degree extension of a $k-1$ step nilspace $M$ and let $\alpha\in{\mathrm{Trans}}_i(M)$. Under what circumstances can we lift $\alpha$ to an element $\alpha'\in{\mathrm{Trans}}_i(N)$ such that $\pi(\alpha'(n))=\alpha(\pi(n))$ for every $n\in N$ ($\pi:N\rightarrow M$ is the projection.)?*]{}
We will need the definition of the arrow spaces. Let $f_1,f_2:\{0,1\}^n\rightarrow N$ be two maps. We denote by $(f_1,f_2)_i$ the map $g:\{0,1\}^{n+i}\rightarrow N$ such that $g(v,w)=f_1(v)$ if $w\in\{0,1\}^i\setminus\{1^i\}$ and $g(v,w)=f_2(v)$ if $w=1^i$. If $f:\{0,1\}^n\rightarrow N\times N$ is a single map with components $f_1,f_2$ then we denote by $(f)_i$ the map $(f_1,f_2)_i$. A map $f:\{0,1\}^n\rightarrow N\times N$ is a cube in the $i$-th arrow space if $(f)_i$ is a cube in $N$.
Let $\mathcal{T}=\mathcal{T}(\alpha,N,i)$ be the set of pairs $(x,y)\in N^2$ where $\alpha(\pi_{k-1}(x))=\pi_{k-1}(y)$. We interpret $\mathcal{T}$ as a subset of the $i$-th arrow space over $N$. It is easy to see that if $k\geq i+1$ then $\mathcal{T}$ is an ergodic nilspace with the inherited cubic structure. We define $\mathcal{T}^*$ as $\mathcal{F}_{k-1}(\mathcal{T})$. We have [@NP] that $\mathcal{T}^*$ is a degree $k-i$ extension of $\mathcal{F}_{k-1}(N)$ by $A_k$. The next theorem was proved in [@NP]
\[transext\] Let $N$ be a $k$-step nilspace and $\alpha\in{\mathrm{Trans}}_i(\mathcal{F}_{k-1}(N))$. If $\mathcal{T}^*=\mathcal{T}^*(\alpha,N,i)$ is a split extension then $\alpha$ lifts to an element $\beta\in{\mathrm{Trans}}_i(N)$.
Free nilspaces and their extensions
-----------------------------------
Let $a_1,a_2,\dots,a_k$ be a sequence of natural numbers. We denote by $F(a_1,a_2,\dots,a_k)$ the nilspace $$\prod_{i=1}^k\mathcal{D}_i(\mathbb{Z}^{a_i}).$$ We say that $F(a_1,a_2,\dots,a_k)$ is the free nilspace of rank $(a_1,a_2,\dots,a_k)$.
\[cubcub\] Let $N$ be a nilspace. If $c\in C^n(\mathcal{D}_i(\mathbb{Z}))$, $f\in C^n(N)$ and $\alpha\in{\mathrm{Trans}}_i(N)$. Then $f^c$ defined by $v\rightarrow \alpha^{c(v)}(f(v))$ (where $v\in\{0,1\}^n$) is in $C^n(N)$.
If $c$ has the special structure that it takes a value $a\in\mathbb{Z}$ on a face of co-dimension $i$ and takes $0$ on the rest of $\{0,1\}^n$ then the statement follows directly from the definition of ${\mathrm{Trans}}_i(N)$. Every other cube in $C^n(\mathcal{D}_i(\mathbb{Z}))$ can be generated by such simple cubes, so we get the general case by iterating the special case.
The main result of this chapter is the following.
\[split\] Let $M$ be a degree $d$ extension of the free nilspace $F=F(a_1,a_2,\dots,a_k)$ by an abelian group $A$. Then $M$ is a split extension.
We prove the statement by induction on $d$. If $d=1$ then using lemma \[subdirect\] we get that the space $M$ is the direct product of $F(0,a_2,a_3,\dots,a_k)$ with an abelian group extension of $\mathbb{Z}^{a_1}$ by $A$. Since such an extension splits the case $d=1$ is done.
Assume that the statement holds for $d-1$ and $d\geq 2$. Using lemma \[subdirect\] we get that $M$ is the direct product of $\prod_{i=d+1}^k\mathcal{D}_i(\mathbb{Z}^{a_i})$ with a degree $d$ extension $M_d$ of $F_d=F(a_1,a_2,\dots,a_d)$ by $A$. We also have that the $d$-th structure group $B$ of $M_d$ is a $d$-degree extension of $\mathbb{Z}^{a_d}$ by $A$. It follows that $B$ is the $d$-degree structure on an abelian group extension of $\mathbb{Z}^{a_d}$ by $A$. Such an extension splits so it remains to show that the degree $d$ extension $M_d$ of $F_{d-1}=F(a_1,a_2,\dots,a_{d-1})$ by $B$ splits. In other words we reduced the problem to the case when $k=d-1$. By abusing the notation let us assume that $k=d-1$, $B=A$ and $F=F_{d-1}$.
We use that $\mathcal{D}_i(\mathbb{Z}^{a_i})$ is embedded into $F$ by setting all the other coordinates to $0$. Let $S_i$ be a free generating system in $\mathbb{Z}^{a_i}$ embedded into $F$ this way (for every $1\leq i\leq d-1$). Every element in $g\in S_i$ acts on $F$ by $x\mapsto x+g$ using the abelian group addition in $\prod_{i=1}^{d-1}\mathbb{Z}^{a_i}$. Let us denote this action by $\alpha(g)$. It is clear that $\alpha(g)\in{\mathrm{Trans}}_i(F)$. We claim that $\alpha(g)$ can be lifted to $M$. Let $\mathcal{T}=\mathcal{T}(\alpha(g),M,i)$. We have by proposition \[transext\] that $\mathcal{T}$ is an $d-i$ degree extension of $F_{d-1}$ by $A$. Using our induction this extension splits and so there is a lift $\alpha'(g)$ of $\alpha(g)$ to ${\mathrm{Trans}}_i(M)$.
The last step of the proof is to create a complement of $A$ in $M$ using the group elements $\alpha'(g)$ where $g\in S_i$. Assume that $S_i=\{g_{i,1},g_{i,2},\dots,g_{i,a_i}\}$. We represent every element $x$ in $F$ in a unique way as $$x=\sum_{i=1}^{d-1}\sum_{j=1}^{a_i}\lambda_{i,j}g_{i,j}.$$ Let $m\in M$ be a fixed element in the fibre of $0\in F$ in $M$. We define the map $h:F\rightarrow M$ such that $h(x)$ is the image of $m$ under the transformation $$\label{noncommpr}
\prod_{i=1}^{d-1}\prod_{j=1}^{a_i}\alpha'(g_{i,j})^{\lambda_{i,j}}.$$
Note that the order in the above product is important since we multiply non commuting transformations. It remains to show that $h$ is cube preserving and $h(F)$ is a diagonal embedding of $F$ into $M$. Lemma \[cubcub\] shows that if a product of the form (\[noncommpr\]) gets extended by one more term then it remains cube preserving. By induction on the length of (\[noncommpr\]) we get that $h$ is a morphism. It is clear from its definition that $h$ creates a diagonal embedding.
Finite nilspaces as factors of free nilspaces
---------------------------------------------
In this chapter we establish finite nilspaces as factors of free nilspaces. Note that in this paper free nilspaces are defined to have finite rank.
\[freefact\] For every finite $k$-step nilspace $N$ there is a factor map $h:F\rightarrow N$ from a $k$-step free nilspace $F$ with the following property. If $\phi:\mathbb{Z}^n\rightarrow N$ is a morphism then there is a morphism $\phi':\mathbb{Z}^n\rightarrow F$ such that $\phi'\circ h=\phi$.
We proceed by induction on $k$. If $k=1$ the $N$ is an (affine) abelian group and then the result is classical. Assume that $k\geq 2$ and assume that the statement holds for $k-1$. Let $N$ be a fixed $k$-step nilspace. We can regard $N$ as a $k$-degree extension of a $k-1$ step nilspace $M$ by an abelian group $A$. Let $\pi:N\rightarrow M$ be the projection. We use the induction hypothesis for $M$ and construct a free nilspace $F_{k-1}=F(a_1,a_2,\dots,a_{k-1})$ and factor map $h':F_{k-1}\rightarrow M$ satisfying the requirement of the lemma. Let $Q$ be the subdirect product of $F_{k-1}$ and $N$ in the following way. The set $Q$ consists of the pairs $(a,b)$ such that $a\in F_{k-1}$,$b\in N$ and $h'(a)=\pi(b)$. It is clear that $Q$ as a subset of the nilspace $F_{k-1}\times N$ is a nilspace which is a $k$ degree extension of $F_{k-1}$ by $A$. It follows form theorem \[split\] that $Q$ is a split extension and so $Q\simeq F_{k-1}\times\mathcal{D}_k(A)$. Let $\beta:\mathbb{Z}^r\rightarrow A$ be a surjective homomorphism between abelian groups for some natural number $r$. Then $\beta$ is also a morphism from $\mathcal{D}_k(\mathbb{Z}^r)$ to $\mathcal{D}_k(A)$. Let $F=F_{k-1}\times\mathcal{D}_k(\mathbb{Z}^r)$ and $\gamma:F\rightarrow Q$ be the identity map on $F_{k-1}$ times $\beta$. Let $h:F\rightarrow N$ be the composition of $\gamma$ with the projection from $Q$ to the second coordinate. It is clear that $h$ is a factor map. We claim that $h$ has the desired lifting property.
Let $\phi:\mathbb{Z}^n\rightarrow N$ be a morphism. First we lift $\phi$ to $Q$. According to induction we can lift $\phi\circ\pi$ to a morphism $\phi_2:\mathbb{Z}^n\rightarrow F_{k-1}$. Let $\phi_3=\phi_2\times\phi$. It is clear that $\phi_3$ maps $\mathbb{Z}^n$ to $Q$ and it lifts $\phi$. Now we have to further lift $\phi_3$ from $Q$ to $F$. Since $Q=F_{k-1}\times\mathcal{D}_k(A)$ we can write $\phi_3$ as $\phi_4\times\phi_5$ where $\phi_4:\mathbb{Z}^n\rightarrow F_{k-1}$ and $\phi_5:\mathbb{Z}^n\rightarrow\mathcal{D}_k(A)$. It remains to show that $\phi_5$ can be lifted to $\mathcal{D}_k(A)$. The map $\phi_5$ is a degree $k$ polynomial map from $\mathbb{Z}^n$ to $A$. An easy lemma in [@Sz4] show exactly that such a map can be lifted.
Periodicity of morphisms
------------------------
\[specper\] Let $A$ be a finite abelian group of order $n$. Then for every $k\in\mathbb{N}$ there is a natural number $\alpha$ depending on $n$ and $k$ such that any morphism $\phi:\mathcal{D}_i(\mathbb{Z})\rightarrow\mathcal{D}_k(A)$ is $n^\alpha$ periodic.
We use lemma \[derivmorph\]. If $i>k$ then $\phi$ is constant. If $i\leq k$ then $\phi$ is a morphism from $\mathcal{D}_1(\mathbb{Z})$ to $\mathcal{D}_{k-i+1}(A)$. It was proved in [@Sz4] (but it is also very easy to see) that such a function is of the form $m\rightarrow \sum_{j=0}^{k-i+1}x_j{{m}\choose{j}}$ where $x_i\in A$ for every $i$. Such functions are $n^{k-i+1}$ periodic.
\[per\] Let $N$ be a finite $k$-step nilspace of size $|N|=n$. then there is a natural number $\alpha\in\mathbb{N}$ such that any morphism $\phi:\mathcal{D}_i(\mathbb{Z})\rightarrow N$ is $n^\alpha$ periodic.
We prove the statement by induction on $k$. If $k=1$ then the statement (by lemma \[specper\]) is trivial since in this case $\phi$ is an affine morphism of $\mathcal{D}_i(\mathbb{Z})$ into an abelian group. Assume that $k\geq 2$ and the statement if true for $k-1$. Assume that the structure groups of $N$ are $A_1,A_2,\dots,A_k$. Let $n_2=|\mathcal{F}_{k-1}(N)|=\prod_{i=1}^{k-1}|A_i|$. We have that $n_2$ divides $n$. We also have by induction that any morphism $\phi:\mathbb{Z}\rightarrow N$ composed with the projection map $\pi:N\rightarrow\mathcal{F}_{k-1}(N)$ is $n_2^{\alpha_2}$ periodic for some natural number $\alpha_2$. This means that for any natural number $m$ the sequence $j\mapsto\phi(m+n_2^{\alpha_2}j)$ is a morphism of $\mathcal{D}_i(\mathbb{Z})$ into a single fibre of the projection $\pi$.
The cube structure of a fibre of $\pi$ is $\mathcal{D}_k(A)$. Lemma \[specper\] finishes the proof.
\[moding\] Let $\phi:F\rightarrow N$ be a morphism from a free group $F=F(a_1,a_2,\dots,a_k)$ to a finite nilspace $F$ with $|F|=n$. Then there is a natural number $\alpha$ such that $\phi$ factors through the map $\psi:F\rightarrow F/n^\alpha$ where $F/n^\alpha$ is the space $\prod_{i=1}^k\mathcal{D}_i((\mathbb{Z}/(n^\alpha))^{a_i})$ and $\psi$ is the map which takes every coordinate mod $n^\alpha$.
It follows from lemma \[per\] that if $g$ is an element in $F$ which has only one nonzero coordinate then the map $j\rightarrow\phi(z+g(jn^\alpha))$ on $\mathbb{Z}$ is constant for every $z\in F$ using addition in the abelian group $\prod_{i=1}^k\mathbb{Z}^{a_i}$. This periodicity using all the generators of each component $\mathbb{Z}^{a_i}$ implies the statement.
Lifting morphisms
-----------------
A hight $i$ extension of an abelian group $A$ of rank $r$ and exponent $e$ is an abelian group $B$ which is an extension of $A$ with rank $r$ and exponent dividing $e^i$.
We denote by $F_n(a_1,a_2,\dots,a_k)=\prod_{i=1}^k\mathcal{D}_i(\mathbb{Z}_n^{a_k})$ and we call it the modulo $n$ free nilspace.
\[invprep\] Let $N$ be a finite nilspace of exponent $e$, $A$ be an abelian group of exponent $e$ and $\phi:A\rightarrow N$ be a morphism. Then there is a morphism $\psi:B\rightarrow F$ from some hight $i$ extension $B$ of $A$ to a modulo $e^{\alpha}$ free nilspace $F=F_{e^\alpha}(a_1,a_2,\dots,a_k)$ such that there is a factor map $\beta:F\rightarrow N$ with $\psi\circ\beta=\pi\circ\phi$ where $\pi:B\rightarrow A$ is the projection map. The value $\alpha$ and the number $i$ depends only on the structure of $N$.
Assume that $A$ is of rank $r$. This means that we can write $A$ as a factor group of $\mathbb{Z}^r$. Theorem \[freefact\] and theorem \[moding\] imply that the statement is true if $B$ is replaced by $\mathbb{Z}^d$. It remains to show that a morphisms $\psi':\mathbb{Z}^d\rightarrow F_{e^\alpha}(a_1,a_2,\dots,a_k)$ factors through a bounded hight extension of $A$. This follows form lemma \[per\] since $\psi'$ has to be periodic in each coordinate with a bounded power of $e$.
We get a further strengthening of the previous lemma from using the fact that $\beta$ is a factor map. Since the $\sim_{k-1}$ classes of $F$ are mapped surejectively to the $\sim_{k-1}$ classes in $N$ (and the $\sim_{k-1}$ classes are copies of $\mathcal{D}_k(A_k)$) the map $\beta$ factors through the map $$\label{redukalt}
F_{e^\alpha}(a_1,a_2,\dots,a_k)\rightarrow F_{e^\alpha}(a_1,a_2,\dots,a_{k-1})\times\mathcal{D}_{k}(A_k)=F'$$ where $A_k$ is the $k$-th structure group of $N$. We formulate it as a separate lemma.
\[invprep2\] In Lemma \[invprep\] the nilspace $F$ can be replaced by $F'$ in (\[redukalt\]).
Applications to the Gowers norms
================================
Let $\mathfrak{A}_e$ denote the family of finite abelian groups of exponent $e$. A $\mathfrak{A}_e$ nilspace is a nilspace whose structure groups are all in $\mathfrak{A}_e$. In particular such nilspaces are all finite. A $\mathfrak{A}_e$ nilspace polynomial on $A\in\mathfrak{A}_e$ is a composition of a morphism $\phi:A\rightarrow N$ with $g:N\rightarrow\mathbb{C}$ where $N$ is a $\mathfrak{A}_e$ nilspace and $|g|\leq 1$.
The next regularity lemma was proved in [@Sz4].
\[reglem\] Let $k$ be a fixed number and $F:\mathbb{R}^+\times\mathbb{N}\rightarrow\mathbb{R^+}$ be an arbitrary function. Then for every $\epsilon>0$ there is a number $n=n(\epsilon,F)$ such that for every measurable function $f:A\rightarrow\mathbb{C}$ on $A\in\mathfrak{A}_e$ with $|f|\leq 1$ there is a decomposition $f=f_s+f_e+f_r$ and number $m\leq n$ such that the following conditions hold.
1. $f_s$ is a degree $k$, complexity $m$ and $F(\epsilon,m)$-balanced $\mathfrak{A}_e$-nilspace-polynomial,
2. $\|f_e\|_1\leq\epsilon$,
3. $\|f_r\|_{U_{k+1}}\leq F(\epsilon,m)$ , $|f_r|\leq 1$ and $|(f_r,f_s+f_e)|\leq F(\epsilon,m)$.
The notion of $b$-balanced is not crucial in this paper but it expresses a strong measure preserving property. According to this theorem, if we want to get an inverse theorem in $\mathfrak{A}_e$ all we need to do is to write $\mathfrak{A}_e$ nilspace polynomials as linear combinations in a suitable basis $\mathcal{B}$. If we can have such a linear combination with boundedly many elements then we get correlation with a function from $\mathcal{B}$.
Let $\phi:A\rightarrow N$ be a morphism from $A\in\mathfrak{A}_e$ to a $\mathfrak{A}_e$-nilspace $N$. Assume that $N$ is a $k$-step nilspace and has size at most $m$. We use lemma \[invprep2\] for $\phi$. We obtain a morphism $\psi:B\rightarrow F'$ where $F'$ has the form (\[redukalt\]). The group $B\in\mathfrak{A}_{e^i}$ is a bounded hight extension of $A$. Let $\phi'$ denote the composition of the projection $B\rightarrow A$ with $\phi$. We have that $\phi'$ factors through $\psi$. This means that nilspace polynomials using $\phi'$ can be also obtained as nilspace polynomials using $\psi$. Let $\mathcal{B}$ be the set of functions on $B$ which are obtained by composing a linear character $\chi$ of $\prod_{i=1}^{k-1}\mathbb{Z}_{e^\alpha}^{a_i}\times A_k$ with $\psi$. Let $\chi=\prod_{i=1}^k\chi_i$ where $\chi_i$ is a character of $\mathbb{Z}_{e^\alpha}^{a_i}$ is $1\leq i\leq k-1$ and $\chi_k$ is a character of $A_k$. We have that $\chi(\psi(x))=\prod_{i=1}^k\chi_i(\psi_i(x))$ where $\psi_i$ is a morphism from $B$ to $\mathcal{D}_i(\mathbb{Z}_{e^\alpha}^{a_i})$. In other words $\psi_i$ is a polynomial map of degree $i$ and $x\mapsto\chi_i(\psi_i(x))$ is a phase polynomial of degree $i$. Furthermore since $A_k$ has exponent $e$ we have that $\chi_k(\psi_k)$ takes only $e$-th roots of unities.
This completes the proof of both theorem \[boundexinv\] and theorem \[boundext\]
On the Tao-Ziegler theorem
--------------------------
Let $p$ be a fixed prime number. We apply theorem \[boundext\] inductively on $k$. Assume that the inverse theorem is established for degree $k-1$ and $k\leq p-1$. Let $f$ be a function on $A=\mathbb{Z}_p^n$ such that $|f|\leq 1$ and $\|f\|_{U_{k+1}}\geq\epsilon$. We have by theorem \[boundext\] that there is a group extension $B\rightarrow A$ (where $B$ has rank $n$) and a phase polynomial $\phi$ on $B$ of degree $k$ such that $(\tau\circ f,\phi)>\delta$. Furthermore we have that $\phi=\phi_1\phi_2$ where $\phi_1$ is of degree $k-1$ and $\phi_2$ takes only $p$-th roots of unities.
We claim that $\phi_2=\tau\circ\phi_3$ where $\phi_3$ is a phase polynomial on $A$. Since $B$ has rank $n$ we can write $B$ as a factor group of $\mathbb{Z}^n$ with homomorphism $\tau':\mathbb{Z}^n\rightarrow B$. Let us lift $\phi_2$ to a phase polynomial $\phi_4$ on $\mathbb{Z}^n$ by composing it with $\tau'$. Since $\phi_4^p=1$ we can obtain $\phi_4$ from a polynomial map $l:\mathbb{Z}^n\rightarrow\mathbb{Z}_p$ of degree $k$. Such polynomial maps (see [@Sz4]) are linear combinations of functions $(x_1,x_2\dots,x_n)\rightarrow \prod_{i=1}^n{{x_i}\choose{r_i}}$ where $\sum r_i=k$. This show that the value of any such map depends only of the residue classes of $x_i$ modulo $p$. The claim is proved.
Let $\gamma$ be the projection of $\phi_1\phi_2$ to $A$. Since $\phi_2$ factors through $\tau$ we have that $\gamma=\gamma_2\phi_3$ where $\gamma_2$ is the projection of $\phi_1$. To finish the proof we need to show that $\gamma_2$ can be well approximated by a bounded linear combination of phase polynomials on $A$ of degree $k-1$. Intuitively this follows from our induction hypothesis and the fact that $\gamma_2$ is a “purely structured” degree $k-1$ function. In the rest of the proof we show a lemma which makes this intuition precise.
Let $k$ be a value such that theorem \[TZth\] holds. Then for every $\epsilon_1,\epsilon_2>0$ there is a number $m$ such that if $\|f\|_{U_{k+1}}\geq\epsilon_1$ holds for a projected phase polynomial $f$ on $A$ then $$\|f-\sum_{i=1}^m\lambda_i\beta_i\|_2\leq\epsilon_2$$ holds for some linear combination $\sum\lambda_i\beta_i$ of degree $k$ phase polynomials $\beta_i$ with $|\lambda_i|\leq 1$.
We proceed by contradiction. Let $\epsilon_1,\epsilon_2$ be such that there is a sequence of groups $A_i=\mathbb{Z}_p^{n_i}$ with extensions $B_i$ and phase polynomials $\phi_i$ on $B_i$ such that the projection $f_i$ of $\phi_i$ has $U_{k+1}$ norm at least $\epsilon_1$, but there is no required linear combination with $i$ elements.
Let $\bA$ (resp. $\bB$) be the ultra product of the sequence $\{A_i\}_{i=1}^\infty$ (resp. $\{B_i\}_{i=1}^\infty$). Let $\phi$ be the ultra limit of the sequence $\{\phi_i\}_{i=1}^\infty$ and $f$ be the ultra limit of $\{f_i\}_{i=1}^\infty$. The theory developed in [@Sz1] says that there is a maximal $\sigma$-algebra $\mathcal{F}_k(\bB)$ on $\bB$ such that $U_{k+1}$ is a norm on $L^\infty(\mathcal{F}_k(\bB))$.
The next step is to show that the projection of $\phi$ to $\bA$ (which is equal to $f$) is measurable in $\mathcal{F}_k(\bA)$. We can think about the projection to $\bA$ as the projection to a $\sigma$ algebra generated by the factor map $\bB\rightarrow\bA$. Such $\sigma$-algebras are called coset $\sigma$-algebras in [@Sz1] since measurable sets are unions of cosets of the kernel of the morphism from $\bB$ to $\bA$. It was proved in $\cite{Sz1}$ that the projection of a function measurable in a shift invariant $\sigma$-algebra to a coset $\sigma$ algebra is measurable in the original $\sigma$ algebra. We obtain that a function in $\mathcal{F}_k(\bB)$ projected to $\bA$ is measurable in $\mathcal{F}_k(\bA)$. In particular $f$ is measurable in $\mathcal{F}_k(\bA)$.
Next we observe that $L^2(\mathcal{F}_k(\bA))$ is generated by ultra limits of phase polynomials. Assume that it is not true. Then there is a nonzero function $g$ (with $|g|\leq 1$) measurable in $\mathcal{F}_k(\bA)$ which is orthogonal to the space generated by the ultra limits of phase polynomials. Since $U_{k+1}$ is a norm on $L^\infty(\mathcal{F}_k(\bA))$ we have that $\|g\|_{U_k}=g>0$. Then we choose a sequence of functions $g_i$ on $a_i$ whose ultra limit is $g$. This sequence would contradict the assumption that theorem \[TZth\] holds for $k$.
We obtain that $f=\sum_{i=1}^\infty w_i\lambda_i$ where $w_i$ are ultra limits of phase polynomials $\{w_i^j\}_{j=1}^\infty$ of degree $k$. By repeating terms $w_i$ many times we can assume the each lambda has absolute value at most $1$. Then there is $m$ such that $\|f-\sum_{i=1}^m\lambda_iw_i\|_2\leq\epsilon_2/2$. This gives a contradiction since $\|f_i-\sum_{r=1}^m\lambda_rw_r^i\|_2<\epsilon_2$ holds for infinitely many indices $i$.
[99]{}
O. Camarena, B. Szegedy, [*Nilspaces,nilmanifolds an their morphisms*]{}, arXiv:1009.3825
W.T. Gowers, [*A new proof of Szemerédi’s theorem*]{}, Geom. Funct. Anal. 11 (2001), no 3, 465-588
W.T. Gowers, [*Fourier analysis and Szemerédi’s theorem*]{}, Proceedings of the International Congress of Mathematics, Vol. I (Berlin 1998).
W.T. Gowers, J. Wolf, [*Linear forms and higher degree uniformity for functions on $\mathbb{F}_p^n$*]{}, arXiv:1002.2208
B. Host, B. Kra, [*Parallelepipeds, nilpotent groups and Gowers norms*]{}, Bulletin de la Soci�t� Math�matique de France 136, fascicule 3 (2008), 405-437
B. Szegedy, [*Higher order Fourier analysis as an algebraic theory I.*]{}, arXiv:0903.0897
B. Szegedy, [*Higher order Fourier analysis as an algebraic theory II.*]{}, arXiv:0911.1157
B. Szegedy, [*Higher order Fourier analysis as an algebraic theory III.*]{}, arXiv:1001.4282
B. Szegedy, [*Gowers norms, regularization and limits of functions on abelian groups*]{}, preprint
T. Tao, T. Ziegler, [*The inverse conjecture for the Gowers norm over finite fields via the correspondence principle*]{} arXiv:0810.5527
|
---
abstract: 'We present Joey NMT, a minimalist neural machine translation toolkit based on PyTorch that is specifically designed for novices. [Joey NMT ]{}provides many popular NMT features in a small and simple code base, so that novices can easily and quickly learn to use it and adapt it to their needs. Despite its focus on simplicity, [Joey NMT ]{}supports classic architectures (RNNs, transformers), fast beam search, weight tying, and more, and achieves performance comparable to more complex toolkits on standard benchmarks. We evaluate the accessibility of our toolkit in a user study where novices with general knowledge about Pytorch and NMT and experts work through a self-contained [Joey NMT ]{}tutorial, showing that novices perform almost as well as experts in a subsequent code quiz. [Joey NMT ]{}is available at <https://github.com/joeynmt/joeynmt>.'
author:
- |
Julia Kreutzer\
Computational Linguistics\
Heidelberg University\
[kreutzer@cl.uni-heidelberg.de]{}\
Jasmijn Bastings\
ILLC\
University of Amsterdam\
[bastings@uva.nl]{}\
Stefan Riezler\
Computational Linguistics & IWR\
Heidelberg University\
[riezler@cl.uni-heidelberg.de]{}
bibliography:
- 'references.bib'
title: 'Joey NMT: A Minimalist NMT Toolkit for Novices'
---
Introduction
============
Since the first successes of neural machine translation (NMT), various research groups and industry labs have developed open source toolkits specialized for NMT, based on new open source deep learning platforms. While toolkits like OpenNMT [@OpenNMT], XNMT [@XNMT] and Neural Monkey [@NeuralMonkey:2017] aim at readability and extensibility of their codebase, their target group are researchers with a solid background in machine translation and deep learning, and with experience in navigating, understanding and handling large code bases. However, none of the existing NMT tools has been designed primarily for readability or accessibility for novices, nor has anyone studied quality and accessibility of such code empirically. On the other hand, it is an important challenge for novices to understand how NMT is implemented, what features each toolkit implements exactly, and which toolkit to choose in order to code their own project as fast and simple as possible.
We present an NMT toolkit especially designed for novices, providing clean, well documented, and minimalistic code, that is yet of comparable quality to more complex codebases on standard benchmarks. Our approach is to identify the core features of NMT that have not changed over the last years, and to invest in documentation, simplicity and quality of the code. These core features include standard network architectures (RNN, transformer, different attention mechanisms, input feeding, configurable encoder/decoder bridge), standard learning techniques (dropout, learning rate scheduling, weight tying, early stopping criteria), and visualization/monitoring tools.
We evaluate our codebase in several ways: Firstly, we show that Joey NMT’s comment-to-code ratio is almost twice as high as other toolkits which are roughly 9-10 times larger. Secondly, we present an evaluation on standard benchmarks (WMT17, IWSLT) where we show that the core architectures implemented in Joey NMT achieve comparable performance to more complex state-of-the-art toolkits. Lastly, we conduct a user study where we test the code understanding of novices, i.e. students with basic knowledge about NMT and PyTorch, against expert coders. While novices, after having worked through a self-contained [Joey NMT ]{}tutorial, needed more time to answer each question in an in-depth code quiz, they achieved only marginally lower scores than the experts. To our knowledge, this is the first user study on the accessibility of NMT toolkits.
[Joey NMT ]{} {#sec:\joeynmt}
=============
NMT Architectures {#sec:models}
-----------------
This section formalizes the [Joey NMT ]{}implementation of autoregressive recurrent and fully-attentional models.
In the following, a source sentence of length $l_x$ is represented by a sequence of one-hot encoded vectors $\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_{l_x}$ for each word. Analogously, a target sequence of length $l_y$ is represented by a sequence of one-hot encoded vectors $\mathbf{y}_1, \mathbf{y}_2, \dots, \mathbf{y}_{l_y}$.
### RNN
[Joey NMT ]{}implements the RNN encoder-decoder variant from @LuongETAL:15.
#### Encoder.
The encoder RNN transforms the input sequence $\mathbf{x}_1, \dots, \mathbf{x}_{l_x}$ into a sequence of vectors $\mathbf{h}_1, \dots, \mathbf{h}_{l_x}$ with the help of the embeddings matrix $E_{src}$ and a recurrent computation of states $$\begin{aligned}
\mathbf{h}_i &= \text{RNN}(E_{src}\, \mathbf{x}_i, \mathbf{h}_{i-1}); &\mathbf{h}_0 = \mathbf{0}.\end{aligned}$$ The RNN consists of either GRU or a LSTM units. For a bidirectional RNN, hidden states from both directions are are concatenated to form $\mathbf{h}_i$. The initial encoder hidden state $\mathbf{h}_0$ is a vector of zeros. Multiple layers can be stacked by using each resulting output sequence $\mathbf{h}_1, \dots, \mathbf{h}_{l_x}$ as the input to the next RNN layer.
#### Decoder.
The decoder uses input feeding [@LuongETAL:15] where an attentional vector $\mathbf{\tilde{s}}$ is concatenated with the representation of the previous word as input to the RNN. Decoder states are computed as follows: $$\begin{aligned}
\mathbf{s}_t &= \text{RNN}([E_{trg}\, \mathbf{y}_{t-1}; \mathbf{\tilde{s}}_{t-1}], \mathbf{s}_{t-1})\\
\mathbf{s}_0 &= \begin{cases} \tanh(W_{bridge}\, \mathbf{h}_{l_x} + \mathbf{b}_{bridge}) & \text{if bridge} \\
\mathbf{h}_{l_x} & \text{if last} \\
\mathbf{0} & \text{otherwise}
\end{cases} \\
\mathbf{\tilde{s}}_{t} &= \tanh(W_{att} [\mathbf{s}_{t}; \mathbf{c}_{t}] + \mathbf{b}_{att})\end{aligned}$$ The initial decoder state is configurable to be either a non-linear transformation of the last encoder state (“bridge”), or identical to the last encoder state (“last”), or a vector of zeros.
#### Attention.
The context vector $\mathbf{c}_t$ is computed with an attention mechanism scoring the previous decoder state $\mathbf{s}_{t-1}$ and each encoder state $\mathbf{h}_i$: $$\begin{aligned}
\mathbf{c}_t &= \sum_{i} a_{ti} \cdot \mathbf{h}_i\\
a_{ti} &= \frac{\exp(\text{score}(\mathbf{s}_{t-1}, \mathbf{h}_i))}{\sum_{k} \exp(\text{score}(\mathbf{s}_{t-1}, \mathbf{h}_k))}\end{aligned}$$ where the scoring function is a multi-layer perceptron [@BahdanauETAL:15] or a bilinear transformation [@LuongETAL:15].
#### Output.
The output layer produces a vector $\mathbf{o}_t = W_{out}\, \mathbf{\tilde{s}}_t$, which contains a score for each token in the target vocabulary. Through a softmax transformation, these scores can be interpreted as a probability distribution over the target vocabulary $\mathcal{V}$ that defines an index over target tokens $v_j$. $$\begin{aligned}
p(y_t = v_j \mid x, y_{<t}) = \frac{\exp(\mathbf{o}_t[j])}{\sum_{k=1}^{|\mathcal{V}|}\exp(\mathbf{o}_t[k])}\end{aligned}$$
### Transformer {#sec:transformer}
[Joey NMT ]{}implements the Transformer from @vaswani2017attention, with code based on *The Annotated Transformer* blog [@rush2018annotated].
#### Encoder.
Given an input sequence $\mathbf{x}_1, \dots, \mathbf{x}_{l_x}$, we look up the word embedding for each input word using $E_{src} \mathbf{x}_i$, add a position encoding to it, and stack the resulting sequence of word embeddings to form matrix $X \in \mathbb{R}^{l_x \times d}$, where $l_x$ is the sentence length and $d$ the dimensionality of the embeddings.
We define the following learnable parameters:[^1] $$A \in \mathbb{R}^{d \times d_a} \quad B \in \mathbb{R}^{d \times d_a} \quad C \in \mathbb{R}^{d \times d_o}$$ where $d_a$ is the dimensionality of the attention (inner product) space and $d_o$ the output dimensionality. Transforming the input matrix with these matrices into new word representations $H$ $$H = \underbrace{\text{softmax}\big( X\!A \, B^\top \!\! X^\top \big)}_{\text{self-attention}} \, X\!C$$ which have been updated by attending to all other source words. [Joey NMT ]{}implements multi-headed attention, where this transformation is computed $k$ times, one time for each head with different parameters $A, B, C$.
After computing all $k$ $H$s in parallel, we concatenate them and apply layer normalization and a final feed-forward layer: $$\begin{aligned}
H &= [ H^{(1)}; \dots ; H^{(k)} ] \\
H' &= \text{layer-norm}(H) + X\\
H^{\text{(enc)}} &= \text{feed-forward}(H') + H'\end{aligned}$$ We set $d_o = d / k$, so that $H \in \mathbb{R}^{l_x \times d}$. Multiple of these layers can be stacked by setting $X=H^{\text{(enc)}}$ and repeating the computation.
#### Decoder.
The Transformer decoder operates in a similar way as the encoder, but takes the stacked target embeddings $Y\!\!\in\!\!\mathbb{R}^{l_y \times d}$ as input: $$H = \underbrace{\text{softmax}\big( Y\!A \, B^{\top}\!Y^{\top}\big)}_{\text{masked self-attention}} Y\!C$$ For each target position attention to future input words is inhibited by setting those attention scores to $-inf$ before the $\text{softmax}$. After obtaining $H' = H + Y$, and before the feed-forward layer, we compute multi-headed attention again, but now between intermediate decoder representations $H'$ and final encoder representations $H^{\text{(enc)}}$: $$\begin{aligned}
Z &= \underbrace{\text{softmax}\big( H'A \, B^\top {H^{\text{(enc)}}}^\top \big)}_{\text{src-trg attention}} \, H^{\text{(enc)}}C \\
H^{\text{(dec)}} & = \text{feed-forward}(\text{layer-norm}(H' + Z))\end{aligned}$$ We predict target words with $H^{\text{(dec)}}W_{out}$.
Features {#sec:features}
--------
In the spirit of minimalism, we follow the 80/20 principle [@pareto1896cours] and aim to achieve 80% of the translation quality with 20% of a common toolkit’s code size. For this purpose we identified the most common features (the bare necessities) in recent works and implementations.[^2] It includes standard architectures (see §\[sec:models\]), label smoothing, dropout in multiple places, various attention mechanisms, input feeding, configurable encoder/decoder bridge, learning rate scheduling, weight tying, early stopping criteria, beam search decoding, an interactive translation mode, visualization/monitoring of learning progress and attention, checkpoint averaging, and more.
Documentation {#sec:documentation}
-------------
The code itself is documented with doc-strings and in-line comments (especially for tensor shapes), and modules are tested with unit tests. The documentation website[^3] contains installation instructions, a walk-through tutorial for training, tuning and testing an NMT model on a toy task[^4], an overview of code modules, and a detailed API documentation. In addition, we provide thorough answers to frequently asked questions regarding usage, configuration, debugging, implementation details and code extensions, and recommend resources, such as data collections, PyTorch tutorials and NMT background material.
Code Complexity {#sec:codecomplexity}
---------------
In order to facilitate fast code comprehension and navigation [@wiedenbeck1999comparison], [Joey NMT ]{}objects have at most one level of inheritance. Table \[tab:cloc\] compares [Joey NMT ]{}with OpenNMT-py and XNMT (selected for their extensibility and thoroughness of documentation) in terms of code statistics, i.e. lines of Python code, lines of comments and number of files.[^5] OpenNMT-py and XNMT have roughly 9-10x more lines of code, spread across 4-5x more files than [Joey NMT ]{}. These toolkits cover more than the essential features for NMT (see §\[sec:features\]), in particular for other generation or classification tasks like image captioning and language modeling. However, Joey NMT’s comment-to-code ratio is almost twice as high, which we hope will give code readers better guidance in understanding and extending the code.
Benchmarks {#sec:benchmarks}
----------
Our goal is to achieve a performance that is comparable to other NMT toolkits, so that novices can start off with reliable benchmarks that are trusted by the community. This will allow them to build on [Joey NMT ]{}for their research, should they want to do so. We expect novices to have limited resources available for training, i.e., not more than one GPU for a week, and therefore we focus on benchmarks that are within this scope. Pre-trained models, data preparation scripts and configuration files for the following benchmarks will be made available on <https://github.com/joeynmt/joeynmt>.
#### WMT17.
We use the settings of @hieber2018sockeye, using the exact same data, pre-processing, and evaluation using WMT17-compatible SacreBLEU scores [@sacrebleu].[^6] We consider the setting where toolkits are used out-of-the-box to train a Groundhog-like model (1-layer LSTMs, MLP attention), the ‘best found’ setting where @hieber2018sockeye train each model using the best settings that they could find, and the Transformer base setting.[^7] Table \[tab:wmt17all\] shows that [Joey NMT ]{}performs very well compared against other shallow, deep and Transformer models, despite its simple code base.[^8]
#### IWSLT14.
This is a popular benchmark because of its relatively small size and therefore fast training time. We use the data, pre-processing, and word-based vocabulary of @wiseman-rush-2016-sequence and evaluate with SacreBLEU.[^9] Table \[tab:iwslt-de\] shows that [Joey NMT ]{}performs well here, with both its recurrent and its Transformer model. We also included BPE results for future reference.
User Study
==========
The target group for [Joey NMT ]{}are novices who will use NMT in a seminar project, a thesis, or an internship. Common tasks are to re-implement a paper, extend standard models by a small novel element, or to apply them to a new task. In order to evaluate how well novices understand Joey NMT, we conducted a user study comparing the code comprehension of novices and experts.
Study Design
------------
#### Participants.
The novice group is formed of eight undergraduate students with a Computational Linguistics major that have all passed introductory courses to Python and Machine Learning, three of them also a course about Neural Networks. None of them had practical experience with training or implementing NMT models nor PyTorch, but two reported theoretic understanding of NMT. They attended a 20h crash course introducing NMT and Pytorch basics.[^10] Note that we did not teach [Joey NMT ]{}explicitly in class, but the students independently completed the [Joey NMT ]{}tutorial.
As a control group (the “experts”), six graduate students with NMT as topic of their thesis or research project participated in the study. In contrast to the novices, this group of participants has a solid background in Deep Learning and NMT, had practical experience with NMT. All of them had previously worked with NMT in PyTorch.
#### Conditions.
The participation in the study was voluntary and not graded. Participants were not allowed to work in groups and had a maximum time of 3h to complete the quiz. They had previously locally installed Joey NMT[^11] and could browse the code with the tools of their choice (IDE or text editor). They were instructed to explore the [Joey NMT ]{}code with the help of the quiz, informed about the purpose of the study, and agreed to the use of their data in this study. Both groups of participants had to learn about [Joey NMT ]{}in a self-guided manner, using the same tutorial, code, and documentation. The quiz was executed on the university’s internal e-learning platform. Participants could jump between questions, review their answers before finally submitting all answers and could take breaks (without stopping the timer). Answers to the questions were published after all students had completed the test.
#### Question design.
The questions are not designed to test the participant’s prior knowledge on the topic, but to guide their exploration of the code. The questions are either free text, multiple choice or binary choice. There are three blocks of questions:[^12]
1. **Usage of [Joey NMT ]{}**: nine questions on how to interpret logs, check whether models were saved, interpret attention matrices, pre-/post-process, and to validate whether the model is doing what it is built for.
2. **Configuring [Joey NMT ]{}**: four questions that make the users configure [Joey NMT ]{}in such a way that it works for custom situations, e.g. with custom data, with a constant learning rate, or creating model of desired size.
3. **[Joey NMT ]{}Code**: eighteen questions targeting the detailed understanding of the [Joey NMT ]{}code: the ability to navigate between python modules, identify dependencies, and interpret what individual code lines are doing, hypothesize how specific lines in the code would have to get changed to change the behavior (e.g. working with a different optimizer). The questions in this block were designed in a way that in order to find the correct answers, every python module contained in [Joey NMT ]{}had to be visited at least once.
Every question is awarded one point if answered correctly. Some questions require manual grading, most of them have one correct answer. We record overall completion time and time per question.[^13]
Analysis
--------
#### Total duration and score.
Experts took on average 77 min to complete the quiz, novices 118 min, which is significantly slower (one-tailed t-test, $p<0.05$). Experts achieved on average 82% of the total points, novices 66%. According to the t-test the difference in total scores between groups is significant at $p<0.05$. An ANOVA reveals that there is a significant difference in total duration and scores within the novices group, but not within the experts group.
#### Per question analysis.
No question was incorrectly answered by everyone. Three questions (\#6, \#11, \#18) were correctly answered by everyone–they were appeared to be easiest to answer and did not require deep understanding of the code. In addition, seven questions (\#1, \#13, \#15, \#21, \#22, \#28, \#29) were correctly answered by all experts, but not all novices–here their NMT experience was useful for working with hyperparameters and peculiarities like special tokens. However, for only one question, regarding the differences in data processing between training and validation (\#16), the difference between average expert and novice score was significant (at $p < 0.05$). Six questions (\#9, \#18, \#21, \#25, \#31) show a significantly longer average duration for novices than experts. These questions concerned post-processing, initialization, batching, end conditions for training termination and plotting, and required detailed code inspection.
#### LME.
In order to analyze the dependence of scores and duration on particular questions and individual users, we performed a linear mixed effects (LME) analysis using the R library `lme4` [@LME4]. Participants and questions are treated as random effects (categorical), the level of expertise as fixed effect (binary). Duration and score per question are response variables.[^14] For both response variables the variability is higher depending on the question than on the user (6x higher for score, 2x higher for time). The intercepts of the fixed effects show that novices score on average 0.14 points less while taking 2.47 min longer on each question than experts. The impact of the fixed effect is significant at $p<0.05$.
Findings
--------
First of all, we observe that the design of the questions was engaging enough for the students because all participants invested at least 1h to complete the quiz voluntarily. The experts also reported having gained new insights into the code through the quiz. We found that there are significant differences between both groups: Most prominently, the novices needed more time to answer each question, but still succeeded in answering the majority of questions correctly. There are larger variances within the group of novices, because they had to develop individual strategies to explore the code and use the available resources (documentation, code search, IDE), while experts could in many cases rely on prior knowledge.
Conclusion
==========
We presented Joey NMT, a toolkit for sequence-to-sequence learning designed for NMT novices. It implements the most common NMT features and achieves performance comparable to more complex toolkits, while being minimalist in its design and code structure. In comparison to other toolkits, it is smaller in size and but more extensively documented. A user study on code accessibility confirmed that the code is comprehensibly written and structured. We hope that [Joey NMT ]{} will ease the burden for novices to get started with NMT, and can serve as a basis for teaching.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Sariya Karimova, Philipp Wiesenbach, Michael Staniek and Tsz Kin Lam for their feedback on the early stages of the code and for their bug fixes. We also thank the student and expert participants of the user study.
Supplemental Material
=====================
NMT Features {#app:features}
------------
Table \[tab:feature-table\] gives an overview over Joey NMT’s features compared with several popular NMT toolkits implemented in Python, such as Sockeye, Neural Monkey, fair-seq, Tensor2Tensor (T2T), XNMT and OpenNMT-py. Sockeye is based on MXNet, Neural Monkey and Tensor2Tensor on TensorFlow, XNMT on Dynet and fair-seq, OpenNMT-py and JoeyNMT on PyTorch. We filled the table to our best knowledge with information obtained from GitHub repositories, published papers and provided documentation.
Extra Results
-------------
#### WMT14.
WMT14 has been a popular benchmark to compare MT systems, even though different pre-/post-processing methods make comparisons noisy.[^15] We train a recurrent 1-layer (“shallow”) and 4-layer (“deep”) and a Transformer model on the same data as @LuongETAL:15. Training the shallow RNN model took about 5 days on one P40 GPU; the deep model took around 9 days, the Transformer 10 days for en-de and 12 days for en-fr. For comparative purposes we report (Moses-)tokenized and compound-splitted (only en-de) `multibleu` scores. Table \[tab:wmt14\] compares the [Joey NMT ]{}models against GNMT, @LuongETAL:15, OpenNMT-py, and the original Tensor2Tensor Transformer. Without checkpoint averaging and extensive hyperparameter tuning, [Joey NMT ]{}achieves results that come close to these systems.
#### IWSLT En-Vi.
We also compared our RNNs against Tensorflow NMT and XNMT on the IWSLT15 en-vi data set as pre-processed by Stanford. Table \[tab:iwslt-vi\] shows the results. The first three systems were trained on sentences of up to 50 tokens, while last two systems were trained on sentences of up to 110 tokens. Our BLEU scores were computed with SacreBLEU with version string BLEU+case.mixed+numrefs.1+smooth.exp+ tok.none+version.1.3.6. We use the original tokenization and data from <https://nlp.stanford.edu/projects/nmt>.
Crash Course on NMT and Pytorch Basics {#sec:course}
--------------------------------------
Prior to the study, the novices attended a three-day crash course (ca. 20h in total) where they were introduced to the concepts of feed-forward, recurrent and attentional neural networks, to PyTorch and the encoder-decoder model for sequence-to-sequence learning. In addition to lectures on the theory and background, they completed a subset of the PyTorch and RNN exercises of the Udacity course on Deep Learning[^16], so that they had all implemented and trained a feed-forward neural network for image classification and an LSTM for character-level language modeling. Solutions were discussed in class. In addition, they worked through the “The Annotated Encoder Decoder”[^17] [@bastings2018annotated] to get a grasp of the building blocks of a NMT implementation in PyTorch. Note that we did not teach [Joey NMT ]{}explicitly in class, but the students had to self-sufficiently work through a [Joey NMT ]{}tutorial[^18].
Quiz Interface
--------------
{width="\textwidth"} {width="\textwidth"}
Figure \[fig:questions\] shows the interface for two example questions, one as a free-text question, and one as a multiple-choice task.
Quiz Statistics {#sec:stats}
---------------
Figure \[fig:duration\] compares the total completion time for the quiz, Figure \[fig:points\] the total points between experts and novices.
![Total duration of quiz taken by experts and novices.[]{data-label="fig:duration"}](duration_corrected.pdf){width="0.3\columnwidth"}
![Percentage of points scored by experts and novices.[]{data-label="fig:points"}](points_perc.pdf){width="0.3\columnwidth"}
Quiz Questions {#sec:questions}
--------------
LMEM Details {#sec:lmem}
------------
[^1]: Exposition adapted from Michael Collins <https://youtu.be/jfwqRMdTmLo>
[^2]: We refer the reader to the additional technical description in <https://arxiv.org/abs/1907.12484>: Table 6 in Appendix A.1 compares Joey NMT’s features with several popular NMT toolkits and shows that [Joey NMT ]{}covers all features that those toolkits have in common.
[^3]: <https://joeynmt.readthedocs.io>
[^4]: Demo video: <https://youtu.be/PzWRWSIwSYc>
[^5]: Using <https://github.com/AlDanial/cloc>
[^6]:
[^7]: Note that the scores reported for other models reflect their state when evaluated in @hieber2018sockeye.
[^8]: Blog posts like @rush2018annotated and @bastings2018annotated also offer simple code, but they do not perform as well.
[^9]:
[^10]: See §\[sec:course\] in the supplemental material of <https://arxiv.org/abs/1907.12484> for details.
[^11]: [Joey NMT ]{}commit hash `0708d596`, prior to the Transformer implementation.
[^12]: <https://arxiv.org/abs/1907.12484> contains the full list of questions, complete statistics and details of the LME analysis.
[^13]: Time measurement is noisy, since full minutes are measured and students might take breaks at various points in time.
[^14]: Modeling expertise with higher granularity instead of the binary classification into expertise groups (individual variables for experience with PyTorch, NMT and background in deep learning) did not have a significant effect on the model, since the number of participants is relatively low.
[^15]: See <https://github.com/tensorflow/tensor2tensor/issues/317> for a discussion on post-processing for en-de.
[^16]: Parts 1-6 of the publicly available notebooks on <https://github.com/udacity/deep-learning-v2-pytorch/tree/master/intro-to-pytorch> and <https://github.com/udacity/deep-learning-v2-pytorch/tree/master/recurrent-neural-networks/char-rnn>, commit hash 9b6001a.
[^17]: <https://github.com/bastings/annotated_encoder_decoder>
[^18]: [https://\\joeynmt.readthedocs.io](https://\joeynmt.readthedocs.io)
|
---
abstract: 'Conventional wireless information transfer by modulating the amplitude, phase or frequency leads to an inevitable Rate-Energy (RE) trade-off in the presence of simultaneous Wireless Power Transfer (WPT). In echoing Varshney’s seminal concept of *jointly* transmitting both information and energy, we propose the so-called Generalised Precoded Spatial Modulation (GPSM) aided Integrated Wireless Information and Power Transfer (IWIPT) concept employing a power-split receiver. The principle of GPSM is that a particular subset of *Receive* Antennas (RA) is activated and the activation pattern itself conveys useful information. Hence, the novelty of our GPSM aided IWIPT concept is that RA pattern-based information transfer is used in addition to the conventional waveform-based information carried by the classic $M$-ary PSK/QAM modulation. Following the Radio Frequency (RF) to Direct Current (DC) power conversion invoked for WPT at the power-split receiver, the non-coherent detector simply compares the remaining received power accumulated by each legitimate RA pattern for the sake of identifying the most likely RA. This operation is then followed by down-conversion and conventional Base Band (BB) $M$-ary PSK/QAM detection. Both our analysis and simulations show that the RA pattern-based information transfer represented in the Spatial Domain (SD) exhibits a beneficial immunity to any potential power-conversion induced performance degradation and hence improves the overall RE trade-off when additionally the waveform-based information transfer is also taken into account. Moreover, we investigate the impact of realistic imperfect Channel State Information at the Transmitter (CSIT) as well as that of the antenna correlations encountered. Finally, the system’s asymptotic performance is characterised in the context of large-scale Multiple Input Multiple Output (MIMO) systems.'
author:
- |
Rong Zhang, Lie-Liang Yang and Lajos Hanzo\
Communications, Signal Processing and Control, School of ECS, University of Southampton, SO17 1BJ, UK\
Email: [rz,lly,lh]{}@ecs.soton.ac.uk, http://www-mobile.ecs.soton.ac.uk[^1]
bibliography:
- 'ref.bib'
- 'rong\_pub.bib'
title: Generalised Precoded Spatial Modulation for Integrated Wireless Information and Power Transfer
---
[^1]: [The financial support of the EPSRC under the India-UK Advanced Technology Centre (IU-ATC), that of the EU under the Concerto project as well as that of the European Research Council’s (ERC) Advanced Fellow Grant is gratefully acknowledged.]{}
|
---
abstract: 'Results from the first two years of data from the Taiwanese-American Occultation Survey (TAOS) are presented. Stars have been monitored photometrically at 4 Hz or 5 Hz to search for occultations by small ($\sim$3 km) Kuiper Belt Objects (KBOs). No statistically significant events were found, allowing us to present an upper bound to the size distribution of KBOs with diameters 0.5 km $<D<$ 28 km.'
author:
- 'Z.-W. Zhang, F. B. Bianco, M. J. Lehner, N. K. Coehlo, J.-H. Wang, S. Mondal, C. Alcock, T. Axelrod, Y.-I. Byun, W. P. Chen, K. H. Cook, R. Dave, I. de Pater, R. Porrata, D.-W. Kim, S.-K. King, T. Lee, H.-C. Lin, J. J. Lissauer, S. L. Marshall, P. Protopapas, J. A. Rice, M. E. Schwamb, S.-Y. Wang and C.-Y. Wen'
title: 'First Results From The Taiwanese-American Occultation Survey (TAOS)'
---
Introduction
============
The study of the Kuiper Belt has exploded since the discovery of 1992 QB1 by @1993Natur.362..730J. The brightness distribution of objects with $R$ magnitude brighter than $\sim$26 is relatively well-established by many surveys, most recently by @2008Icar..195..827F [and references therein] . The brightness distribution is adequately described by a simple cumulative luminosity function , where $R_0 \sim 23$ and $\alpha \sim 0.6$, for objects with magnitude $R<26$. There is clear evidence for a break to a shallower slope for fainter objects: the deepest survey, conducted using the Advanced Camera for Surveys on the *Hubble Space Telescope* [@2004AJ....128.1364B] extended to $R = 28.5$, and found a factor of $\sim$25 fewer objects than would be expected if the same distribution extended into this range.
The size distribution of Kuiper Belt Objects (KBOs) is believed to reflect a history of *agglomeration* during the planetary formation epoch, when relative velocities between particles were low and collisions typically resulted in particles sticking together, followed by *destructive collisions* when the relative velocities were increased by dynamical processes after the giant planets formed [@1996AJ....112.1203S; @1997Icar..125...50D; @1997AJ....114..841S; @1999AJ....118.1101K; @1999ApJ...526..465K; @2004AJ....128.1916K; @2005Icar..173..342P]. The slope of the distribution function for larger objects reflects the early phase of agglomeration, while the shallower distribution for smaller objects reflects a subsequent phase of destructive collisions. The location of the break moves to larger sizes with time, while the distribution for smaller objects is expected to evolve towards a steady state collisional cascade [@2004AJ....128.1916K; @2005Icar..173..342P]. Models for the spectrum of small bodies differ between @2005Icar..173..342P, who derived a double power-law distribution, and @2004AJ....128.1916K, whose simulations show more structure, depending on material properties.
Thus, the size spectrum encodes information about the history of planet formation and dynamics. However, the size spectrum for small KBOs is not constrained by the imaging surveys because the objects of interest are too faint for direct detection using presently available instruments. These small objects may, however, be detected indirectly when they pass between an observer and a distant star [@1976Natur.259..290B; @1992QJRAS..33...45D; @1992ASPC...34..171A; @1997MNRAS.289..783B; @2000Icar..147..530R; @2003ApJ...587L.125C; @2007AJ....134.1596N]. The challenge confronting any survey exploiting this technique is the combination of very low anticipated event rate and short duration of the events (typically $<$ 1 second).
Other groups are attempting similar occultation surveys. @2006AJ....132..819R reported three events in 10 star-hours of photometric data sampled at 45 Hz, which they modeled as objects at 15 AU, 140 AU, and 210 AU, respectively, placing the inferred objects outside the Kuiper Belt. @2008AJ....135.1039B reported results of 5 star-hours of data sampled at 40 Hz, during which no events were detected. @2006Natur.442..660C reported a surprisingly high rate of possible occultation events in *RXTE* x-ray observations of Sco X-1, but many of these events have since been attributed to instrumental effects [@2008ApJ...677.1241J; @2007MNRAS.378.1287C].
We report here the first results of the Taiwanese American Occultation Survey (TAOS). TAOS differs from the previously reported projects primarily in the extent of the photometric time series, a total of $1.53 \times 10^5$ star-hours, and in that data are collected simultaneously with three telescopes. Some compromises have been made in regard to signal-to-noise (SNR), which is typically lower than in previously reported surveys, and in cadence, which is 4 Hz or 5 Hz, in contrast to the higher rates mentioned above. The substantial increase in exposure more than compensates for the lower cadence and SNR, and we are able to probe significant ranges of the model space for small KBOs. We have also developed a statistical analysis technique which allows efficient use of the multi-telescope data to detect brief occultation events that would be statistically insignificant if observed with only one telescope.
Data and Analysis {#sec:analysis}
=================
TAOS has been collecting scientific data since 2005. Observations are normally carried out simultaneously with three 50 cm telescopes (A, B, and D, separated by distances of 6 meters and 60 meters; this system is described by @2008arXiv0802.0303L). Over 15 TB of raw images have been taken. We report here on the first two years of data taken simultaneously with all three telescopes. The data set comprises 156 *data runs*, where a data run is defined as a series of three-telescope observations of a given field for durations of $\sim$90 minutes. Thirty data runs with a 4 Hz sampling rate were taken before 2005 December 15, and 126 data runs were collected subsequently with a sampling rate of 5 Hz. Only fields with ecliptic latitudes $|b| < 10^\circ$ were analyzed. Over 93% of the data were collected in fields with $|b| < 3^\circ$, so the results of our analysis are relevant to the sum of the cold and excited KBO populations [@2004AJ....128.1364B]. No data run was included unless each star was sampled more than 10,000 times in each telescope. The angle from opposition in these data runs is distributed from $0^\circ$ to $90^\circ$. The number of stars (with $R< 13.5$, which typically gives a SNR $\ge 5$) monitored in the data runs ranges between 200 and 2000[^1].
The images were analyzed using an aperture photometry package [@kiwi] devised exclusively for TAOS images. Lightcurves were produced for each star by assembling the photometric information into a time series. A star in each data run has a lightcurve from each of the three telescopes. The data presented in this paper comprises 110,895 *lightcurve sets* (where a lightcurve set is defined as a set of three lightcurves, one for each telescope, for the same star in a single data run), containing $7.1 \times 10^9$ individual photometric measurements. The photometric data are not calibrated to a standard system. Changes in atmospheric transparency during a data run produce flux variations () that could undermine our occultation search algorithm. Such low-frequency trends in a lightcurve can be removed by a numerical high-pass filter that preserves information of a brief occultation event, typically with a duration less than 1–2 data points with the TAOS sampling rate. Our filter takes a time series of $ f_{i}$ measured at time $t_i$ to produce an intermediate series $g_{i} = f_{i} - \bar{f_{i}}$ where $\bar{f_{i}}$ is the running average of 33 data points centered on $t_{i}$. The series $g_{i}$ is then scaled by the local fluctuation, $h_{i} = g_{i} / \sigma(g_{i})$ with $\sigma(g_{i})$ being the standard deviation of $g_{i}$ of 151 data points centered at $t_{i}$. Both the mean and standard deviation are calculated using *three-sigma clipping*. This filtering proves effective to remove slow-varying trends in the lightcurve, while preserving high-frequency fluctuations that we aim to detect, as illustrated in .
We now confront the two central challenges in the search for extremely rare occultation events in these data: *(1) to search for events simultaneously in three parallel data streams,* and *(2) to determine the statistical significance of any rare events that are found.* The second is not straightforward because the statistical distribution of our photometric measurements is not known in advance; approximations based on Gaussian statistics are unreliable far from the mean. This motivates a non-parametric approach.
We thus found it useful to represent each data point by its *rank* in the filtered, rescaled lightcurve data $h_{i}$. That is, the rank of a data point ranges from $r=1$ (lowest $h$) to $r={N_\mathrm{P}}$ (highest $h$), for a data run comprising ${N_\mathrm{P}}$ photometric measurements taken with telescope A, B, or D. The *rank triples* $(r_{i}^A, r_{i}^B, r_{i}^D)$ form the basis of further analysis of the multi-telescope data. The statistical distribution of these ranks is known exactly, since each rank must occur exactly once in each time series. Thus, the probability that a given rank will occur at time $t_i$ is $P(r_i)=1/{N_\mathrm{P}}$. When the photometric data are uncorrelated, the probability that a particular rank triple will occur is simply $P(r_{i}^A, r_{i}^B, r_{i}^D)=1/{N_\mathrm{P}}^3$. This allows a straightforward test for correlation between the photometric data taken in the three telescopes: the rank triples should be distributed uniformly in a cube with sides of length ${N_\mathrm{P}}$. A non-uniform pattern in the cube, on the other hand, indicates correlation. shows an example of the rank series of one telescope against another; this indicates that the raw photometric data $f_{i}$ are strongly correlated, but the filtered data $h_{i}$ are not.
Given that the rank triples are uncorrelated, the ranks can be used to search for possible occultation events, as follows: *A true occultation event will exhibit anomalous, correlated low ranks in all three telescopes.* The rank triples thus allow an elegant test for the statistical significance of a candidate event. Consider the quantity $$\eta_i = -\ln(r_{i}^{A} r_{i}^{B} r_{i}^{D} / {N_\mathrm{P}}^3).$$ Since the ranks are uncorrelated (unless we have an occultation event), we can calculate the exact probability density function[^2] for $\eta$. This in turn allows us to compute the probability that a given triple of low ranks randomly occurred in an uncorrelated lightcurve set; this is our measure of statistical significance. An illustration of the power of this approach is shown in , where a simulated occultation event is readily recovered.[^3]
We thus screened all of our series of rank triples for events with low ranks in all three telescopes. In the analysis reported here, we considered only events for which $P(\eta \geq c) \leq 10^{-10}$, which leads to an expected value of $0.24$ false positive events in the entire data set of $2.4\times 10^9$ rank triples. *No statistically significant events emerged from this analysis.*[^4]
Efficiency Test and Event Rate
==============================
While the example event shown in is readily recovered, objects with smaller diameters or which do not directly cross the line of sight might not be so easily detected. An efficiency calculation is thus necessary for understanding the detection sensitivity of our data and analysis pipeline to different event parameters, notably the KBO size distribution. Our efficiency test started with implanting synthetic events into observed lightcurves, with the original noise. Each original lightcurve $f_{i}$ was modified by the implanted occultation, $ k_{i} = f_{i} - (1 -d_{i})\bar{f_{i}},$ where $d_{i}$ is the simulated event lightcurve (with baseline $d_i
\rightarrow 1$ far from the event, [@2007AJ....134.1596N]), and $\bar{f_{i}}$ is the average of the original series over a 33 point rolling window. Since we preserved the original noise in the modified lightcurves, the noise where the implanted occultation event takes place—for which the flux diminishes—would be slightly overestimated, hence our efficiency estimate is conservative.
We assumed spherical KBOs at a fixed geocentric distance of $\Delta=43$ AU. (Given our sampling rate, varying the KBO distance within the Kuiper Belt has little effect on our simulated lightcurves.) The event epoch $t_0$ was chosen randomly and uniformly within the duration $E$ of the lightcurve set. The angular size $\theta_*$ of each star, necessary for the simulated lightcurve calculation, was estimated using stellar color and apparent magnitude taken from the USNO-B [@2003AJ....125..984M] and 2MASS [@2006AJ....131.1163S] catalogs. The impact parameter of each event was chosen, again randomly and uniformly, between 0 and $H/2$, where $H$ is the *event cross section* [@2007AJ....134.1596N], $$H = 2\left[(\sqrt{3}F)^{3 \over 2} + (D / 2)^{3 \over 2}
\right]^{{2 \over 3}} + \theta_*\Delta.$$ Here $D$ is the diameter of the occulting object, and $F$ is the *Fresnel scale*, $F = \sqrt{\lambda \Delta/2}$, where $\lambda =
600$ nm is the median wavelength in the TAOS filter. The relative velocity between the Earth and KBO ${v_\mathrm{rel}}$, necessary for the conversion of the occultation diffraction pattern to a temporally sampled lightcurve, is calculated based on the angle from opposition during each data run.
To adequately cover a wide range of parameter space, two efficiency runs were completed in which we implanted each lightcurve set with exactly one simulated occultation event. For each event, the diameter of the KBO was chosen randomly according to a probability, or weighting factor, $w_D$. In the first run, objects of diameters $D
=$ 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 km were added with $w_D = 1/6$ for each diameter. In the second run, objects of diameters $D =$ 1, 2, 3, 5, 10 and 30 km were added with weights $w_D = \{100, 100, 100, 30, 5,
1\} / 336$. The modified lightcurves $k_i$ were reprocessed using the same procedure described in . The recovered events and the event parameters were then used to calculate the number of expected occultation event in our survey. That is, we calculated the quantity $${\Omega_\mathrm{e}}(D) =
{w_D^{-1}}\sum_j\left[E_j~v_{\mathrm{rel}_j}~ H_j(D) ~/~ \Delta^2\right],$$ where the sum is over all lightcurve sets where a simulated event is successfully recovered in the reanalysis (). Essentially ${\Omega_\mathrm{e}}(D)$ is the *effective solid angle* of our survey, insofar as TAOS can be considered equivalent to a survey that is capable of counting every KBO of diameter $D$ in a solid angle ${\Omega_\mathrm{e}}$ with 100% efficiency.
The expected number of detected events by KBOs with sizes ranging from $D_1$ to $D_2$ can then be written as $${N_\mathrm{exp}}= \int\limits_{D_2}^{D_1}{dn \over dD}{\Omega_\mathrm{e}}(D) dD,
\label{eq:nexp}$$ where ${dn}/{dD}$ is the differential surface number density of KBOs. The integrand of contains two factors: the *model-dependent* size distribution ${dn}/{dD}$, and the *model-independent* effective solid angle ${\Omega_\mathrm{e}}(D)$, which describes the sensitivity of the survey to objects of diameter $D$. Given the observed number of events and the value of ${\Omega_\mathrm{e}}(D)$ resulting from the efficiency calculation, we can place model-dependent limits on the the population of KBOs. Based on the absence of detections in this data set, any model with a size distribution such that ${N_\mathrm{exp}}~\geq~3.0$ is inconsistent with our data at the 95% confidence level.
Note that there are an infinite number of models that satisfy the above requirement. We thus make the reasonable choice of a power-law size distribution ${dn}/{dD} = n_\mathrm{B}(D/28~\mathrm{km})^{-q}$, where $n_\mathrm{B}$ is chosen such that the cumulative size distribution is continuous at 28 km with the results of @2004AJ....128.1364B. We integrate from $D_2 =
28$ km down to our detection limit of $D_1 = 0.5$ km, and solve with ${N_\mathrm{exp}}= 3$, to find $q = 4.60$. Our null detection thus eliminates any power law size distribution with $q > 4.60$ at the 95% c.l., setting a stringent upper limit (see ) to the number density of KBOs.
Conclusion
==========
We have surveyed the sky for occultations by small KBOs using the three telescope TAOS system. We have demonstrated that a dedicated occultation survey using an array of small telescopes, an innovative statistical analysis of multi-telescope data, and a large number of star-hours, can be used as a powerful probe of small objects in the Kuiper Belt, and we are thus able to place the strongest upper bound to date on the number of KBOs with $0.5~\mathrm{km} < D < 28$ km. We continue to operate TAOS, soon with an additional telescope, and will report more sensitive survey results in the future.
Work at NCU was supported by the grant NSC 96-2112-M-008-024-MY3. Work at the CfA was supported in part by the NSF under grant AST-0501681 and by NASA under grant NNG04G113G. Work at ASIAA was supported in part by the thematic research program AS-88-TP-A02. Work at UCB was supported by the NSF under grant DMS-0405777. Work at Yonsei was supported by the KRCF grant to Korea Astronomy and Space Science Institute. Work at LLNL was performed under the auspices of the U.S. DOE in part under Contract W-7405-Eng-48 and Contract DE-AC52-07NA27344. Work at SLAC was performed under U.S. DOE contract DE-AC02-76SF00515. Work at NASA Ames was funded by NASA/P.G.&G.
[28]{} natexlab\#1[\#1]{}
, T. S. [et al.]{} 1992, in Robotic Telescopes in the 1990s, Vol. 34, 171–181
, M. E. 1976, , 259, 290
, G. M. [et al.]{} 2004, AJ, 128, 1364
, S. J., [Kavelaars]{}, J. J., & [Welch]{}, D. L. 2008, , 135, 1039
, M. J. I. & [Webster]{}, R. L. 1997, , 289, 783
, H.-K. [et al.]{} 2006, , 442, 660
—. 2007, , 378, 1287
, W. P. [et al.]{} 2007, in IAU Symposium, Vol. 236, IAU Symposium, ed. G. B. [Valsecchi]{} & D. [Vokrouhlick[ý]{}]{}, 65–68
, A. & [Farmer]{}, A. J. 2003, , 587, L125
, D. R. & [Farinella]{}, P. 1997, Icarus, 125, 50
, F. J. 1992, , 33, 45
, W. C. [et al.]{} 2008, Icarus, 195, 827
, D. & [Luu]{}, J. 1993, Nature, 362, 730
, T. A., [Levine]{}, A. M., [Morgan]{}, E. H., & [Rappaport]{}, S. 2008, , 677, 1241
, S. J. & [Bromley]{}, B. C. 2004, , 128, 1916
, S. J. & [Luu]{}, J. X. 1999, , 118, 1101
—. 1999, , 526, 465
, M. J. [et al.]{} 2008, ArXiv e-prints, 802.0303
—. 2008, in preparation
, D. G. [et al.]{} 2003, , 125, 984
, T. C. [et al.]{} 2007, , 134, 1596
, M. & [Sari]{}, R. 2005, Icarus, 173, 342
, F. & [Moncuquet]{}, M. 2000, Icarus, 147, 530
, F. [et al.]{} 2006, , 132, 819
, M. F. [et al.]{} 2006, , 131, 1163
, S. A. 1996, , 112, 1203
, S. A. & [Colwell]{}, J. E. 1997, , 114, 841
Zhang, Z.-W. [et al.]{} 2008, in preparation
[^1]: Information on the TAOS fields is available at <http://taos.asiaa.sinica.edu.tw/taosfield/>.
[^2]: For small $\eta$, this distribution can be approximated by a $\Gamma$ distribution of the form $P(\eta) =
\eta^{{N_\mathrm{T}}-1}\,e^{-\eta}/({N_\mathrm{T}}- 1)!$, where ${N_\mathrm{T}}= 3$ is the number of telescopes.
[^3]: Details of our statistical methodology will be described in @stat.
[^4]: A candidate event was reported in @2007IAUS..236...65C. This event had a significance of $3.7\times 10^{-10}$, which did not pass our cut on $\eta$. We expect to have $\sim$1 false positive event at that significance level or higher.
|
---
abstract: 'Hot corino chemistry and warm carbon chain chemistry (WCCC) are driven by gas-grain interactions in star-forming cores: radical-radical recombination reactions to form complex organic molecules (COMs) in the ice mantle, sublimation of CH$_4$ and COMs, and their subsequent gas-phase reactions. These chemical features are expected to depend on the composition of ice mantle which is set in the prestellar phase. We calculated the gas-grain chemical reaction network considering a layered ice-mantle structure in star-forming cores, to investigate how the hot corino chemistry and WCCC depend on the physical condition of the static phase before the onset of gravitational collapse. We found that WCCC becomes more active, if the temperature is lower, or the visual extinction is lower in the static phase, or the static phase is longer. Dependence of hot corino chemistry on the static-phase condition is more complex. While CH$_3$OH is less abundant in the models with warmer static phase, some COMs are formed efficiently in those warm models, since there are various formation paths of COMs. If the visual extinction is lower, photolysis makes COMs less abundant in the static phase. Once the collapse starts and visual extinction increases, however, COMs can be formed efficiently. Duration of the static phase does not largely affect COM abundances. Chemical diversity between prototypical hot corinos and hybrid sources, in which both COMs and carbon chains are reasonably abundant, can be explained by the variation of prestellar conditions. Deficiency of gaseous COMs in prototypical WCCC sources is, however, hard to reproduce within our models.'
author:
- Yuri Aikawa
- Kenji Furuya
- Satoshi Yamamoto
- Nami Sakai
bibliography:
- 'reference.bib'
title: 'Chemical Variation among Protostellar Cores: Dependence on Prestellar Core Conditions'
---
Introduction {#sec:intro}
============
Various molecular emission lines have been detected in the central regions of low-mass protostellar cores. Emission lines of complex organic molecules (COMs), which are defined as organic molecules with 6 atoms or more, are detected in the hot ($\gtrsim 100$ K) central ($\lesssim 100$ au) region in several protostellar cores, e.g. IRAS 16293-2422B and NGC 1333-IRAS4A [e.g. @vandishoeck95; @cazaux03; @bottinelli04a; @bottinelli04b; @ceccarelli07; @caux11; @taquet15; @jorgensen16; @oya16; @lopez17]. Towards IRAS 04368+2557 in L1527 and IRAS 15398-3359, on the other hand, unsaturated carbon chains (e.g. C$_2$H and C$_4$H) are abundantly detected in the vicinity of protostar [@sakai08; @sakai09a]. In this work we reserve the term carbon chains for such significantly unsaturated carbon chains to discriminate them from hydrocarbons, which are molecules made of C and H in general.
In theoretical models, the formation of both carbon chains and COMs is explained, at least qualitatively, by the gas-grain chemistry in the sequence of star formation. In cold ($\sim 10$ K) prestellar cores, molecules with heavy-element such as CO are frozen onto grains. Hydrogen atoms can thermally migrate and hydrogenate CO and other adsorbed atoms and molecules on grain surfaces. While the successive hydrogenation of CO produces CH$_3$OH, the intermediate radicals such as HCO can also react with each other, if they are in neighboring adsorption sites, to form more complex organic molecules. Eventually, star-formation starts and dust temperature gradually rises, which enhances thermal diffusion of adsorbed species and reactions among them to produce various COMs. Icy molecules are sublimated according to their volatility, and reactions proceed in the gas-phase as well. When the dust temperature exceeds $\sim 100$ K, the dominant constituent of ice, H$_2$O, sublimates together with COMs [@rodgers03; @garrod06; @aikawa08; @herbst09; @aikawa12; @garrodCR13; @taquet14; @chuang16; @cuppen18; @lu18]. The low-mass protostars with bright emission of COMs at the central hot region ($R \lesssim 100$) are called hot corinos. The unsaturated carbon chains, on the other hand, are formed by the gas-phase reactions triggered by the sublimation of CH$_4$ at $\sim 25-30$ K; it is called warm carbon chains chemistry (WCCC) [@sakai08; @aikawa08; @hassel08; @sakai13]. Protostars are called WCCC source, if the emission lines of unsaturated carbon chains become bright inward of $\sim$ a few $10^3$ au suggesting their abundance jump.
In theory, both WCCC and COM formation proceed in a star forming core. Indeed, both carbon chains and COMs are recently detected in B335 and L483 [@imai16; @oya17], which are called [*hybrid*]{} (of hot corino and WCCC). The emission of carbon chains is more extended ($r \sim$ a few 100 $- 10^3$ au ) than that of COMs ($r \lesssim 100$ au), which supports the theoretical model. It should be noted, however, that the prototypical hot corino sources are deficient in carbon chains, while COMs are deficient in the first two WCCC sources. The column density ratio of C$_2$H/CH$_3$OH is $\sim 0.1$ in prototypical hot corinos such as IRAS 16293 and NGC 1333 IRAS 4A, while it is $\sim 4$ in L1527 [@higuchi18; @bouvier20 and references therein]. [@higuchi18] observed multiple lines of C$_2$H, c-C$_3$H$_2$ and CH$_3$OH towards 36 Class 0/I protostars in Perseus molecular cloud complex using IRAM 30 m and NRO 45 m telescopes to derive the chemical composition averaged over the 1,000 au scale. The column density ratio of C$_2$H/CH$_3$OH varies by two orders of magnitude among sources. One caution in such single dish observations is a possible contamination of the emission of a parental molecular cloud [@bouvier20]. Follow-up observations are necessary to disentangle the molecular cloud and protostellar core and to confirm the abundance jump of COMs and unsaturated carbon chains, which characterizes the hot corino and WCCC. In the ALMA observation of IRAS 16293 B, for example, C$_2$H emission is detected, but its spatial distribution is different from that of C$_3$H$_2$ [@murillo18]. It is indeed unexpected in WCCC, and thus suggests the chemistry in IRAS 162293 B would be different from that of prototypical WCCC.
What is the origin of such chemical variation among protostellar cores? Since the sublimation temperature of COMs are higher than that of CH$_4$, WCCC cores could be deficient in COMs, if the central protostar is not bright enough. But it cannot explain the deficiency of carbon chains (or lack of WCCC) around hot corinos. [@higuchi18] showed that the column density ratio of C$_2$H/CH$_3$OH in protostellar cores does not correlate with the evolutionary indicator, such as bolometric temperature, or with luminosity. Alternatively, the chemical composition of protostellar cores could reflect the ice composition set in the prestellar phase. [@graninger16] observed C$_4$H and CH$_3$OH towards 16 deeply embedded low-mass protostars using IRAM 30 m telescope, and found that the gaseous C$_4$H/CH$_3$OH abundance ratio tentatively correlates with the CH$_4$/CH$_3$OH ice abundance ratio determined by Spitzer c2d survey [@boogert08; @oberg08]. The ice composition, in turn, would be determined by the physical conditions. For example, if a core is located near the periphery of a molecular cloud, the penetrating UV radiation keeps carbon atoms abundant, which activates the formation of hydrocarbons. Then the ratio of CH$_4$/CH$_3$OH in the ice mantle could be relatively high compared with that in cores in the high visual extinction. While the correlation of ice abundances with UV radiation is hard to investigate observationally, the correlation between gaseous molecules with UV radiation field is found in low-mass star-forming cores. [@lindberg15] showed that in the protostellar core R CrA IRS7B, COMs are under-abundant, while CN emission is strong, which is an indicator of PDR chemistry. [@higuchi18] found that the protostars located near cloud edges or in isolated clouds tend to have a high C$_2$H/CH$_3$OH ratio. In a prototypical prestellar core, L1544, [@spezzano16] found that c-C$_3$H$_2$ emission peaks close to the southern part of the core, where the surrounding molecular cloud has a sharp edge, while CH$_3$OH mainly traces the northern part of the core. The gas in north and south would eventually fall to the central region to set the chemistry in protostellar phase. Since a protoplanetary disk is being formed in the central region of a protostellar core, the chemical composition of a core might be inherited by a forming disk and eventually by a planetary system.
Several theoretical work investigated the dependence of molecular abundances on core models. [@garrod06] showed that grain-surface reactions become more important for COM formation in models with slower warming-up, while the peak abundance of each molecular species show complex dependence on the warm-up timescale. [@aikawa12] showed that for WCCC to be active, the gas density around the CH$_4$ sublimation zone should be low enough for C$^+$ to be abundant. [@sakai13] pointed out that CH$_4$ should be more abundant than $10^{-7}$ relative to hydrogen to be the major reactant with C$^+$ (competing against C$^+$ + OH) to trigger WCCC. But there are few theoretical work which investigates both WCCC and hot corino chemistry systematically. In this paper, we study the dependence of COMs and warm carbon-chain abundances in protostellar cores on the physical conditions in prestellar phase via numerical calculations. Specifically, we vary the temperature, visual extinction of the ambient clouds, and duration of static phase before gravitational collapse. The rest of the paper is organized as follows. Our chemical and physical models are described in §2. Results of model calculations are presented in §3. We compare our results with previous work and observations in §4. Our conclusions are summarized in §5.
Model
=====
Model of a Star-forming Core
----------------------------
Structure and evolution of molecular clouds are summarized as follows [@andre14 and references therein]. Molecular clouds consist of filaments. Herschel observations revealed that gravitationally bound prestellar cores and protostars are primarily found in filaments with the H$_2$ column densities of $\gtrsim 7\times 10^{21}$ cm$^{-2}$. The radial density profile of the filaments show a flat plateau with FWHM $\sim 0.1$ pc. The mean gas density in the star-forming filaments is thus $n$(H$_2$)$\gtrsim 2\times 10^4$ cm$^{-3}$. Statistical observations of cloud cores suggest the lifetime of a core with $n$(H$_2$)$\sim 10^4$ cm$^{-3}$ to be $10^6$ yr, which coincides with the sound-crossing time scale of a filament. The typical column density of filaments with cores roughly coincides with the threshold value of gravitational instability, which indicates that the filaments fragment to form self-gravitating cores. Eventually the core collapses to form protostars.
In the present work, we solve the rate equations of gas-grain chemical reaction network in a fluid parcel that reaches the $T\gtrsim 100$ K region in a protostellar envelope. Considering the observational overview described above, we divide our model to two phases: the prestellar phase with density of $\sim 10^4$ cm$^{-3}$ and the collapse phase (Figure \[fig\_schem\]).
For the latter phase, we adopt the 1D (spherical) radiation hydrodynamic model of low-mass star formation by [@masunaga98] and [@masunaga00]. We assume the spherical collapse, although flattened structure will form in the vicinity of the protostars [e.g. @terebey84]. The total mass and initial radius of the core are 3.852 $M_{\odot}$ and $R=4\times 10^4$ au, respectively, The initial number density of hydrogen nuclei is $\sim 6\times 10^4$ cm$^{-3}$ (i.e. $n$(H$_2$)$\sim 3\times 10^4$ cm$^{-3}$) at the core center. After the onset of gravitational collapse, the central density increases with time, and the protostar is formed in $2.5\times 10^5$ yr. After the birth of the protostar, the model further follows the evolution for $9.3 \times 10^4$ yr, during which the protostar grows by mass accretion from the envelope. At each evolutionary stage, the model gives the total luminosity of the core and the radial distribution of density, temperature, and infall velocity self-consistently. At the final time step, the total luminosity of the core is 24 $L_{\odot}$ and the temperature is higher than 100 K inside $R \sim 100$ au. While the temperatures of gas and multiple components of dust material are calculated separately in the original model of [@masunaga98] and [@masunaga00], we assume that the dust temperature is equal to the gas temperature, for simplicity. More detailed description of our core model can be found in [@masunaga98; @masunaga00; @aikawa08; @aikawa12].
In our previous work [@aikawa08; @aikawa12], we solved the rate equations in multiple infalling fluid parcels to derive the radial distributions of molecular abundances. In the present work, on the other hand, we follow the temporal variation of molecular abundances in one infalling fluid parcel that is initially located at $R=1.01\times 10^4$ au and reaches $R=30.6$ au at the final time step, and plot the abundances as a function of the temporal radial position of the fluid parcel. The green line in Figure \[phys\_fid\] (b) shows the time after the onset of collapse as a function of the location of the infalling fluid parcel. The figure also depicts the density, temperature, and visual extinction of the fluid parcel. As we will show in section §3, the $R-$abundance plot is reasonably similar to the true radial distribution of molecules around WCCC and hot corino regions at the final time step. If we were to calculate the chemistry in multiple fluid parcels to derive the radial distributions with a similar spatial resolution as the $R-$ abundance plot, we would need $\sim 100$ fluid parcels for each model, which is computationally expensive. We decided to use the $R-$abundance plot as a proxy of the radial distribution, considering the similarity between them.
For the prestellar phase, we assume that the initial core stays static for $t_{\rm sta}=1\times 10^6$ yr. We thus call this phase as static phase in the following. This notation, static phase versus collapse phase, is clearer, since a protostar is not formed immediately after the onset of collapse; i.e. the early collapse phase is actually a prestellar phase. In the fluid parcel we follow, which is located at $R=1.01\times 10^4$ au, the density and temperature are $n_{\rm H}=2.28\times 10^4$ cm$^{-3}$ and $T_{\rm init}=10$ K (Figure \[fig\_schem\] and Figure \[phys\_fid\] a). The gas column density from the core outer boundary ($R=4\times 10^4$ au) to the location of the fluid parcel corresponds to the visual extinction of $A_{\rm v}=1.51$ mag. Outside the core, we assume the ambient gas of $A_{\rm v}^{\rm amb}=3$ mag. The interstellar UV radiation field is thus attenuated by the total visual extinction of 4.51 mag (Figure \[phys\_fid\] a). These parameters are chosen to be consistent with the physical conditions in filaments and for a simple and smooth transition to the infalling stage. Ideally, we need to also consider the formation of a filament and its fragmentation to form prestellar cores. We will include the filament formation to our model in §3.5.
In order to investigate the dependence of WCCC and COM formation on the physical parameters of static phase, we vary the initial temperature ($T_{\rm init}=10, 15, 20,$ or 25 K), the visual extinction of the ambient gas ($A_{\rm v}^{\rm amb}=1, 3,$ or 5 mag) and duration of the static core phase ($t_{\rm sta}=3\times 10^5, 1\times 10^6,$ or $3\times 10^6$ yr) (Table \[tab:param\]). In the collapse phase of these models, the temporal variation of the temperature is assumed to be the same as the fiducial model, while the visual extinction is modified in accordance with $A_{\rm v}^{\rm amb}$. Namely, even in the model with high $T_{\rm init}$, the temperature of the fluid parcel decreases below 10 K in the early phase of collapse, shielded from the interstellar radiation field. As we will see in §3, the ice composition is significantly changed during this period, which dilutes the effect of warm static phase in the models of $T_{\rm init}>10$ K. In order to further investigate the effect of thermal history, we also calculated models in which the minimum temperature is set to be $T_{\rm min}=10, 15, 20$ or 25 K; the temperature of the fluid parcel is fixed at $T_{\min}$ in the static phase and early collapse phase, and start rising when it exceeds $T_{\rm min}$ in the original model. The dotted line in Figure \[phys\_fid\] (b), for example, depicts the temporal temperature variation in the model of $T_{\rm min}=10$ K. Warm temperatures in the static and early collapse phase would be possible, if the core is in e.g., cluster-forming regions.
[llc]{} total mass of the core & & 3.852 \[$M_{\odot}$\]\
core radius & & $4\times 10^4$ \[au\]\
initial position of the infalling fluid parcel & & $1.01 \times 10^4$ \[au\]\
final position of the infalling fluid parcel & & 30.6 \[au\]\
varied parameters: & &\
temperature in the static phase & $T_{\rm init}$ & [**10**]{}, 15, 20, 25 \[K\]\
minimum temperature & $T_{\rm min}$ & 10, 15, 20, 25 \[K\]\
visual extinction of the ambient gas & $A_{\rm v}^{\rm amb}$ & 1, [**3**]{}, 5 \[mag\]\
duration of static phase & $t_{\rm sta}$ & $3\times 10^5$, [$1\times 10^6$]{}, $3\times 10^6$ \[yr\]\
Chemistry
---------
Our reaction network model is based on [@garrod13] and includes the following updates. The gas-phase reaction of H$_2$CO + OH has two production paths: HCOOH + H and H$_2$O + HCO. In our previous model [@aikawa12], the former was one of the major formation paths of HCOOH in the gas phase. According to quantum calculations and laboratory experiments , however, the former branch has the activation barrier of $\sim 2500$ K and thus negligible compared with the latter branch, which is barrierless. We thus deleted the former path in the present work, while the rate coefficient of the latter path is set to be $1.0\times 10^{-11}(T/300.0 {\rm K)}^{-0.6}$ cm$^3$ s$^{-1}$ [see also @ruaud15]. While HCOOH consists of 5 atoms, we consider it as COM in the present work, since it is often observed in hot corinos [e.g. @remijan06; @imai16]. The gas-phase reaction rate coefficient of NH$_2$ + H$_2$CO $\rightarrow$ NH$_2$CHO + H is set to be $2.6 \times 10^{-12} (T/300.0 {\rm K})^{-2.1} \exp(-26.9 {\rm K}/ T)$ cm$^3$ s$^{-1}$ following [@barone15]. On grain surfaces, the reaction of H + NH$_2$CO results in hydrogen abstraction (H$_2$ + HNCO) rather than hydrogenation (NH$_2$CHO) [@noble15]. The grain-surface reaction of HCO + CH$_3$ has three possible production paths, CH$_4$ + CO, CH$_3$CHO, and CH$_3$OCH. We neglected the latter two paths following [@enrique-romero16], who showed that only the first path is plausible considering the orientation of the reactants on amorphous water ice (see §4.2 for further discussions). We also added the gas-phase reactions of H$_2$ with C$_2$H, C$_4$H and C$_3$H$_2$, which result in a hydrogen addition to the carbon chains (i.e. C$_x$H$_y$ + H$_2 \rightarrow$ C$_x$H$_{y+1}$ + H ); the barrier is set to be 998 K, 950 K, and 1740 K, respectively, referring the KIDA data base for astrochemistry (kida.obs.u-bordeaux1.fr).
The simplest model of gas-grain chemistry consists of two phases, gas phase and ice phase, and thus is called two-phase model [@hase92]. While the ice mantle consists of $\sim 10^2$ monolayers of molecules and atoms in cold dense prestellar cores, such layering structure is not considered in the two-phase model. For example, all CO molecules in the ice mantle have the equal probability of reaction and the equal probability of desorption. It is obviously a simplification. [@hase93] then proposed the three-phase model that consists of gas-phase, ice surface phase, and ice mantle phase. In the original three-phase model, the ice mantle phase is assumed to be chemically inert, and chemical reactions are considered only in the ice surface phase. In more recent three-phase models, the chemical reactions are often considered in the ice mantle phase as well, but with slower rates than in the ice surface phase. The basic assumption of the three-phase model is that the ice mantle phase has a uniform chemical composition, which is not true. Infrared absorption bands of interstellar ices suggest that ice mantle is made of polar (water-rich) component and apolar (CO- and CO$_2$-rich) component [e.g. @gibb04]. In order to take into account such inhomogeneity in the ice mantle, we adopt a seven-phase model, that consists of the gas phase, the ice surface phase, and 5 phases of ice mantle. We assume that the top 4 monolayers are the ice surface phase [@vasyunin13], while each mantle phase consists of a few tens of monolayers at a maximum. Since the term [*phase*]{} is used to discriminate gas and ice, i.e. the gas phase versus ice phase, and also to specify the evolutionary stage of a star-forming core, i.e. the static phase and infalling phase, in the present work, we use the term [*layer*]{} to specify different phases of ice hereinafter. In summary, ices in our model consists of six layers: the ice surface layer and 5 mantle layers.
We assume the Langmuir-Hinshelwood mechanism for reactions in the ice surface layer and mantle layers; species can diffuse by thermal hopping and react with each other when they meet. No quantum tunneling is assumed in the migration, even for H atoms, while we consider the tunneling effect for reaction probability of reactions with activation barrier. The barrier of thermal migration of atoms and molecules are set to be 40 % of the adsorption energy $E_{\rm ads}$ in the surface layer, while it is 80 % of $E_{\rm ads}$ in the ice mantle layers [e.g. @ruaud16]. The set of adsorption energies is adopted from [@garrod13]. We consider two-body reactions in each ice layer, but do not allow the reactions between species in different layers, assuming the vertical migration of species in ice mantles is limited. Swapping between the six ice layers is not considered. Only the molecules in the surface layer are subject to desorption, while the species in the deeper mantle layers are transported to the upper layers following the net loss of ices in the surface layer. Detailed explanation on the formulation and coding is given in [@furuya17]. The molecules in the ice mantle are subject to photolysis with extinction in the pre/protostellar core and the outer layers of ice mantles.
As elemental abundances, we adopt the so-called low-metal values [Table 1 of @aikawa01]. In our fiducial model, the species are assumed to be initially in the form of atoms or atomic ions except for hydrogen, which is entirely in its molecular form, in the static core. Later in section 3, we also calculate models in which the initial molecular abundance of the static phase is obtained via the 1-D shock model of molecular cloud (i.e. filament) formation [@bergin04; @furuya15].
Result
======
Fiducial Model
--------------
Figure \[dist\_fid\] (a)-(d) shows the temporal variation of molecular abundances in the static prestellar core phase. The solid lines depict the molecular abundances of gaseous molecules relative to hydrogen nuclei, while the dashed lines show those of icy molecules, i.e. the sum of ices in the surface layer and mantle layers. As we start with atoms and ions, C$^+$ is gradually converted to atomic carbon and then to CO in the gas phase, while oxygen is hydrogenated to form water on grain surfaces (Figure \[dist\_fid\] a). After $10^5$ yrs, the ice abundances of CO, CO$_2$, and CH$_3$OH relative to water are in reasonable agreement with observation towards low-mass protostars and background stars (i.e. CO/H$_2$O$\sim 0.3$, CO$_2$/H$_2$O $\sim 0.3-0.4$, and CH$_3$OH/H$_2$O $\sim$ a few $10^{-2}$) [@oberg11]. Hydrocarbons, including carbon chains, are efficiently formed from C$^+$ and atomic C both in the gas phase and ice before carbon is fully converted to CO (Figure \[dist\_fid\] d). Hydrocarbons and carbon chains also react with N atoms and O atoms to form CH$_3$CN and CH$_3$CHO in the gas phase (Figure \[dist\_fid\] b). CH$_3$CHO, for example, is mainly formed via O + C$_2$H$_5$ at $t\sim 10^4$ yr in our model.
Molecular evolution in the collapse phase is shown as a function of the radial position of the infalling fluid parcel in Figure \[dist\_fid\] (e)-(h). The sublimation temperatures of the most volatile C-bearing molecules, CO and CH$_4$, are $\sim 20-30$ K, depending of gas density, which is reached when the fluid parcel is at $R=2000-3000$ au. Since the ice mantle is multi-layered and made of mixture of ices in our model, only a fraction of CO and CH$_4$ is sublimated. The majority of icy molecules are trapped in water ice until water is sublimated at $\sim 100$ K. This entrapment is in agreement with laboratory experiments, at least qualitatively, while the actual fraction of trapped volatiles depends on the ice thickness and the mixture ratio in experiments, and parametrization of diffusion and swapping efficiency in numerical simulations [@collings04; @fayolle11].
Figure \[dist\_fid\] (c) and (g) show the abundances of several icy radical species in the static phase and collapse phase, respectively. Radicals are efficiently formed, trapped, and stored in ice in the cold phase [@lu18]. Compared with the two-phase model, various radicals are abundant in the multi-layered ice mantle model for the following reasons. Firstly, icy species, including radicals, are trapped beneath the surface layer, and not directly subject to desorption to the gas phase. Radicals beneath the surface layer do not react with newly adsorbed atoms and molecules from the gas phase, either. In the two-phase model, on the other hand, all icy radicals can react with newly adsorbed species, e,g. H atoms, to form saturated molecules. Lastly, the thermal diffusion and thus the reactions (e.g. radical-radical reactions) are slower in ice mantle layers than in the surface layer. When the temperature rises, however, the stored radicals start to react with each other to form COMs. CH$_3$OCH$_3$, for example, is mainly formed in the ice mantle via the reaction of CH$_3$ + CH$_3$O, when the temperature of the fluid parcel is $\sim 20$ K (Figure \[dist\_fid\] f). The CH$_3$OCH$_3$ abundance in the final time step in the present model ($9.7 \times 10^{-9}$) is much higher than that in the two-phase model of [@aikawa12] ($1.2\times 10^{-10}$) adopting the same collapse model. The gas-phase reactions of sublimated molecules also contribute to the formation of COMs, e.g. C$_2$H$_5$ + O $\rightarrow$ CH$_3$CHO + H and NH$_2$ + H$_2$CO $\rightarrow$ NH$_2$CHO + H at $R\sim 90$ au.
When the fluid parcel reaches $R\sim 2000$ au, the temperature rises to $\sim 25$ K, which is the sublimation temperature of CH$_4$. Sublimation of CH$_4$ triggers the WCCC; e.g. CH$_4$ reacts with C$^+$ to produce C$_2$H$_3^+$, which then recombine to form C$_2$H. The carbon chain also extends via the reactions with C atoms (e.g. C + CH$_3$ $\rightarrow$ C$_2$H$_2$ + H). It should be noted that CH$_4$ does not fully sublimate at its own sublimation temperature ($\sim 25$ K), which suppresses the WCCC compared with the two-phase model of [@aikawa08]. Closely looking at Figure \[dist\_fid\] (f) and (h), we also note that WCCC enhances the abundance of some COMs, as well. For example, the NH$_2$CHO abundance increases around the WCCC region ($R\sim 1900$ au); CH$_3$ reacts with O atoms in the gas phase to form H$_2$CO, which then reacts with NH$_2$ to form NH$_2$CHO. Later, the temperature rises to the sublimation temperature of H$_2$CO ($\sim 40$ K) when the fluid parcel reaches $\sim 750$ au. Then the gas-phase reactions initiated by the sublimation of H$_2$CO further enhance the abundance of NH$_2$CHO.
The crosses in Figure \[dist\_fid\] (e), (f), and (h) depict the radial distribution of gaseous molecular abundances at the final time step; we calculated the gas-grain chemistry in the fluid parcels that reach the radius of 62.4 au, 125 au, 250 au, 500 au, 1000 au, 2000 au, 4000 au and 8000 au, respectively. The $R$-abundance plot of the infalling fluid parcel (i.e. solid lines) is quite similar to this true radial distribution inside the radius of $\sim 10^3$ au. It is reasonable, because the infall timescale inside $10^3$ au is $\lesssim 5\times 10^3$ yr, which is much shorter than the age of the protostar $9\times 10^4$ yr in our model. At $R\gtrsim 10^3$ au, the difference between the $R-$abundance plot and the true radial distribution becomes more apparent. In the WCCC region ($r\sim$ a few $10^3$ au), however, the $R-$abundance plot of CH$_4$ and hydrocarbons are in reasonable agreement with the radial distribution. Thus we can use the $R-$abundance plot to investigate the hot corino chemistry and WCCC.
![ [*Left column*]{}: The molecular abundances relative to hydrogen nuclei as a function of time in the static phase in our fiducial model. The solid lines show the gas-phase abundances, while the dotted lines depict the ice abundances. [*Right column*]{}: The molecular abundances in the infalling fluid parcel in the collapse phase as a function of the radial location of the fluid parcel (see text) in our fiducial model. The arrows in panel (e) depict the radius where the thermal desorption timescale is equal to adsorption timescale for CO (black), CH$_4$ (orange) and CH$_3$OH (green). The crosses in panel show the radial distributions of gaseous molecules at the final time step in our model. \[dist\_fid\]](Fig1_chem.pdf)
Dependence of COM Abundances on $T_{\rm init}$ and $T_{\rm min}$
----------------------------------------------------------------
We calculate the molecular evolution in the fluid parcels with the temperature in the static phase $T_{\rm init}$ of 15 K, 20 K, and 25 K. The gaseous abundances of COMs in the final time step (i.e. $R=30.6$ au) are shown in Figure \[Tinit\] (a) as a function of $T_{\rm init}$. A naive expectation is that the higher $T_{\rm init}$ makes the freeze out and grain-surface hydrogenation less efficient, resulting in lower abundances of icy molecules and COMs. Indeed, the abundances of CH$_3$OH and CH$_3$CHO are lower in the model with $T_{\rm init}=25$ K than in the fiducial model (i.e. $T_{\rm init}=10$ K). But the abundances of some COMs, such as CH$_3$CN, CH$_3$OCH$_3$ and HCOOH, are higher with $T_{\rm init}=25$ K than in the fiducial model.
In order to investigate the chemistry with high $T_{\rm init}$, we plot the temporal variation of molecular abundances in the model with $T_{\rm init}=25$ K; Figure \[warm\_COMs\] (a, b) shows the temporal variation in the static phase, while Figure \[warm\_COMs\](c, d) is the $R-$abundance plot in the infalling phase. In the static phase, the abundances of H$_2$O ice and CO ice are indeed lower than in our fiducial model; at the end of static phase ($t=10^6$ yr), their abundances are $1\times 10^{-5}$ (H$_2$O) and $3\times 10^{-6}$ (CO). CO$_2$ ice, on the other hand, is abundantly formed via CO + OH on grain surfaces; e.g. its abundance reaches $\sim 7\times 10^{-5}$ at $t=10^6$ yr. The warm temperature makes the CO freeze-out less efficient, but enhances the thermal diffusion of species with moderately high binding energy. In numerical models of gas-grain chemistry, molecules tend to be converted to and accumulate as stable icy species whose sublimation temperature is higher than the current temperature [e.g. @aikawa97; @furuya14]. Icy abundances of COMs in the static phase are also higher than those in our fiducial model. While CH$_3$OH is mainly formed via hydrogenation of CO in our fiducial model ($T_{\rm init}=10$ K), it is formed via the grain-surface association of CH$_3$ + O $\rightarrow$ CH$_3$O and subsequent hydrogenation in the $T_{\rm init}=25$ K model. CH$_3$ and O atoms, in turn, are formed via photodissociation of larger hydrocarbons and CO$_2$, respectively. The grain-surface reaction of CH$_3$ with CH$_3$O forms CH$_3$OCH$_3$. In other words, formation of COMs starts from carbon chains and hydrocarbons, which are abundantly formed in both the gas and ice phases before carbon is fully converted to CO. Once CO becomes the dominant carbon reservoir ($t\sim 10^5$ yr), COM abundances temporally decline, as the destruction (e.g. photodissociation) dominates over the formation.
The model results are in line with recent laboratory experiments showing that CH$_3$OH can be formed in CH$_4$ ice mixture. [@qasim18] performed Temperature Programmed Desorption (TPD) of the mixed ice of CH$_4$, O$_2$ and H atoms to find that CH$_3$OH is produced. The analysis of the experimental data with different ice mixtures indicates that CH$_3$OH is formed via CH$_4$ + OH $\rightarrow$ CH$_3$ + H$_2$O and CH$_3$ + OH $\rightarrow$ CH$_3$OH. The latter reaction is one of the main formation paths of CH$_3$OH in the collapse phase of $T_{\rm init}=25$ K model. In the static phase of our model with $T_{\rm init}=25$ K, CH$_4$ ice abundance is not high, since $T_{\rm init}$ coincides with its sublimation temperature. CH$_3$ is thus formed from larger hydrocarbons, and then quickly reacts with atoms and radicals in the ice mantle.
Figure \[warm\_COMs\] (c, d) is the $R-$abundance plot in the infalling phase of the model with $T_{\rm init}=25$ K. It should be noted that the temperature of the fluid parcel falls to $\lesssim 10$ K at the onset of collapse and the low temperature $\lesssim 20$ K continues for $\sim 3.1\times 10^5$ yr (§2.1). During this period, CH$_3$OH ice is abundantly formed via CO ice hydrogenation, while the thermal diffusion of species with heavy elements and thus radical-radical reactions are quenched. In Figure \[Tinit\] (a), we can compare the abundances of gaseous CH$_3$OH at the final timestep (i.e. $R=30.6$ au) (solid cyan line) and CH$_3$OH ice at the end of static phase (dash-dotted cyan line); the former is much higher than the latter at $T_{\rm init}\gtrsim 15$ K mainly due to the CH$_3$OH formation during this temporal cold phase. Radicals are also abundantly stored in the ice mantle then. It makes the dependence of the final COM abundances on $T_{\rm init}$ less significant than that on $T_{\rm min}$ (see below). Yet, chemical signature of warm $T_{\rm init}$ remains in some layers in the ice mantle. For example, CH$_3$CHO is less abundant in the model with $T_{\rm init}=25$ K, since its precursor CH$_3$ in the ice mantle is less abundant. On the other hand, HCOOH is abundantly formed via OH + HCO in the upper layers of bulk ice mantle at $R\sim 1\times 10^3$ au in the model of $T_{\rm init}=25$ K. CO ice is not abundant in those layers, and HCO is formed via reaction of OH + H$_2$CO, while H$_2$CO is formed via photodissociation or H-abstraction of CH$_3$OH. At the same radius in the fiducial model, OH reacts with CO, which is relatively abundant in all the ice mantle layers.
While it is reasonable that the core temperature decreases as the gas density increases in the early collapse phase, we also calculated models with a fixed minimum temperature. The model with $T_{\rm min}=25$ K, for example, is similar to the model with $T_{\rm init}=25$ K, but the temperature is kept at 25 K even in the early collapse phase (i.e. $t=1\times 10^6 - 1.33\times 10^6$ yr). Figure \[Tinit\] (b) shows the abundances of gaseous COMs at the final time step, and Figure \[warm\_COMs\] (e, f) shows the $R-$abundance plot of radicals and COMs in the collapse phase of $T_{\rm min}=25$ K model. Note that their abundances in the static phase are the same as Figure \[warm\_COMs\] (a, b). Although the icy abundances of CH$_3$OH and some radicals increase temporally after the onset of collapse, the increment is smaller than in the model with $T_{\rm init}=25$ K (Figure \[warm\_COMs\] c, d). COM abundances depend more sensitively on $T_{\rm min}$ than on $T_{\rm init}$; e.g. CH$_3$OCH$_3$, which is formed via CH$_3$O + CH$_3$ in the ice mantle, is least abundant with $T_{\rm min}=20$ K, and becomes more abundant with $T_{\rm min}=25$ K. The reactant, CH$_3$O is mainly formed via photodissociation of CH$_3$OH in the low $T_{\rm min}$ model ($<20 K$), while it is formed from hydrocarbon (CH$_3$ + O) in the model of $T_{\rm min}=25$ K. Either path is not efficient in the $T_{\rm min}=20$ K model.
Dependence of WCCC on $T_{\rm init}$ and $T_{\rm min}$
------------------------------------------------------
In Figure \[Tinit\_WCCC\] (a)(b), we plot the minimum and maximum abundances of CH$_4$, C$_2$H, C$_3$H$_2$, and C$_4$H around the CH$_4$ sublimation region, which we set $R=1455-2850$ au, in models with $T_{\rm init}=10, 15, 20,$ and 25 K (panel a) and $T_{\rm min}=10, 15, 20,$ and 25 K (panel b). The minimum and maximum values are connected by a vertical solid line, if the abundance increases with decreasing radius in the CH$_4$ sublimation region, which indicates that WCCC is active. A dotted vertical line, on the other hand, is used if the abundance decreases inwards. We can see that CH$_4$ abundance decreases with increasing $T_{\rm init}$ and $T_{\rm min}$; it is natural, since both the grain surface formation and freeze-out of CH$_4$ become less efficient at warmer temperatures. The carbon chain abundances also decrease with $T_{\rm init}$ and $T_{\rm min}$, and the abundances of C$_4$H and C$_3$H$_2$ do not increase inwards when $T_{\rm init} \ge 20$ K and $T_{\rm min}\ge 20$ K. As an example, $R-$abundance plot of CH$_4$ and carbon chains in the model with $T_{\rm min}=20$ K is depicted in Figure \[Tinit\_WCCC\] (c).
[@sakai13] argued that the CH$_4$ abundance relative to H$_2$ should be higher than several $10^{-7}$ for the WCCC to be active, i.e. for CH$_4$ to be the major reactant for C$^+$ competing with OH and other molecules. In the sublimation region of CH$_4$ in our fiducial model, the reaction of C$^+$ + CH$_4$ $\rightarrow$ C$_2$H$_3^+$ + H competes with C$^+$ + H$_2$ $\rightarrow$ CH$_2^+$ and C$^+$ + NH$_3 \rightarrow$ H$_2$CN$^+$ + H, although the rates are higher for the latter two. The reaction of C + CH$_3 \rightarrow$ C$_2$H$_2$ + H also contributes the WCCC, where a fraction of CH$_4$ contributes to form C atom and CH$_3$, as well. While the model details are different between the present work and [@sakai13], we can see in Figure \[Tinit\_WCCC\] that the gaseous CH$_4$ abundance of $10^{-7}$ would roughly be a condition for WCCC. The abundances of C$_2$H, C$_4$H, and C$_3$H$_2$ correlate with the sublimated abundance of CH$_4$.
In Figure \[Tinit\_WCCC\] (a)(b), the dash-dotted orange lines depict the CH$_4$ ice abundance at the end of the static phase, while the (non-vertical) solid orange lines depict the gaseous CH$_4$ abundance at the final timestep ($R=30.6$ au). Except for the models with $T_{\rm init} > 20$ K, the solid line overlaps with the dash-dotted line, which means that the total abundance of CH$_4$ is mostly determined by the CH$_4$ ice abundance in the static phase. In the models with $T_{\rm init} = 25$ K, the CH$_4$ abundance increases on the grain surfaces in the early collapse phase. We also note that the (non-vertical) solid lines are well above the upper values of the orange vertical solid lines, which means that only a fraction of icy CH$_4$ sublimates in the WCCC region, except in the model of $T_{\rm init}\ge 20$ K. The trapping and layering of molecules in ice determine what fraction of CH$_4$ can sublimates at $T\sim 25$ K and contributes to the WCCC.
Dependence on UV extinction by ambient clouds
---------------------------------------------
In the fiducial model, we assumed that the initial prestellar core is embedded in ambient gas, which attenuate the interstellar UV radiation field by $A_{\rm v}^{\rm amb}=3$ mag (Figure \[fig\_schem\]). We calculated two additional models with $A_{\rm v}^{\rm amb}=1$ mag and 5 mag; i.e. the visual extinction in the static phase is 2.51 mag and 6.51 mag in the fluid parcel. Figure \[Av\_COM\_WCCC\] shows the gaseous COM abundances at the final time step (panel a) and the minimum and maximum abundances of CH$_4$, C$_2$H, C$_3$H$_2$, and C$_4$H around the CH$_4$ sublimation region (panel b) as a function of $A_{\rm v}^{\rm amb}$. A naive expectation is that in the prestellar core with lower $A_{\rm v}^{\rm amb}$, CH$_3$OH would be less abundant, while CH$_4$ would be more abundant, which results in less abundant COMs and more active WCCC in the protostellar phase. While CH$_4$ abundance is indeed high and WCCC is more active in the models with lower $A_{\rm v}^{\rm amb}$ (Figure \[Av\_COM\_WCCC\] b), the dependence of COM abundances on $A_{\rm v}^{\rm amb}$ is not so simple.
Figure \[Av\_COM\_WCCC\] (c)-(e) shows the $R-$abundance plot of the infalling fluid parcel in the model with $A_{\rm v}^{\rm amb}=1$ mag. In the static phase, photolysis makes CH$_3$OH and other COMs less abundant both in the gas phase and ice mantle in the model with $A_{\rm v}^{\rm amb}=1$ mag than in the fiducial model. Their abundances, however, become as high as in the fiducial model, once the visual extinction gets high enough in the collapse phase. Even though CO ice abundance is lower than in the fiducial model, CH$_3$OH can be formed via CH$_3$ + OH. In the model with $A_{\rm v}^{\rm amb}=5$ mag, HCOOH is less abundant, while HCOOCH$_3$ is more abundant than in the fiducial model. In the fiducial model, HCOOH is formed around $R\sim 3\times 10^3$ au in the ice mantle via CO + OH $\rightarrow$ COOH and COOH + H $\rightarrow$ HCOOH. The abundance of OH radical in the ice mantle is lower in the model with $A_{\rm v}^{\rm amb}=5$ mag. HCOOCH$_3$ is formed via HCO + CH$_3$O in the ice mantle at $R\sim 1\times 10^3$ au in the model with $A_{\rm v}^{\rm amb}=5$ mag. Both of the reactants are formed in the reaction network of CH$_3$OH (i.e. photo-dissociation of CH$_3$OH and CH$_3$O + H$_2$CO $\rightarrow$ CH$_3$OH + HCO), which is more abundant in the ice mantle of $A_{\rm v}^{\rm amb}=5$ mag than in the fiducial model.
Dependence on the duration of static phase
------------------------------------------
In the classical pseudo-time dependent models of molecular clouds, carbon is initially in the form of C$^+$, which is converted to C atom and then to CO. Carbon chains and CH$_4$ are formed from C atoms and thus reach the maximum abundance in the gas phase before carbon is fully converted to CO [e.g. @suzuki92]. Then one would naively expect that if the star formation sets in earlier, the ice abundance ratios of CO/CH$_4$ and thus CH$_3$OH/CH$_4$ would be lower, which is favorable for WCCC rather than hot corinos.
In order to test this expectation, we run models with various $t_{\rm sta}$; Figure \[precol\_COM\_WCCC\] shows the gaseous COM abundances at the final time step (panel a) and the minimum and maximum abundances of CH$_4$, C$_2$H, C$_3$H$_2$ and C$_4$H around the CH$_4$ sublimation region (panel b). While the dependence of COM abundances on $t_{\rm sta}$ is rather weak, the abundance jump of carbon chains around CH$_4$ sublimation region is larger (i.e. WCCC is more active) in models with longer $t_{\rm sta}$. This counter-intuitive dependence is caused by the stability of CH$_4$ in the ice. Referring to Figure \[dist\_fid\], we note that after the conversion of C atom to CO and CO ice, CH$_4$ ice abundance does not decrease, while gaseous CH$_4$ (and other carbon chains in the gas phase) does decrease. Actually, at $t\gtrsim 10^6$ yr in the static phase, CO is gradually converted to CH$_4$ via CO + He$^+$ $\rightarrow $ C$^+$ + O + He and subsequent reactions with H$_2$. The conversion of CO to CH$_4$ is also found in previous work, e.g. [@hassel08], and is also responsible for the slight decrease of CH$_3$OH with $t_{\rm sta}$. We also note that in the models with $t_{\rm sta}=3\times 10^5$ yr, the abundances of C$_3$H$_2$ and C$_4$H decrease inwards around the CH$_4$ sublimation radius. Their abundances at the outermost radius ($\sim 10^4$ au) are higher than those in our fiducial model by about one order of magnitude. In other words, the remnant carbon chains from the prestellar phase dominates over production via WCCC.
So far, we assumed that species are in the form of atoms or atomic ions except for hydrogen, which is in H$_2$, at the start of static phase with $n_{\rm H}\sim 2\times 10^4$ cm$^{-3}$ and $A_{\rm v}\gtrsim 2$ mag. Such an initial abundance is often assumed, e.g. in pseudo-time dependent models and collapsing core models, because H$_2$ is self-shielded to be firstly formed, which is essential for the gas-phase two-body reactions to proceed. But it is obviously a simplified initial condition. Observations indicate that molecular formation both in the gas and ice phases starts in lower densities [e.g. @whittet09; @snow06]. Theoretical work shows that molecular clouds are formed via converging flows of HI gas [e.g. @inoue12], and the ice formation in the post-shock gas of converging flow reproduces important features of interstellar ices: D/H ratio of water ice and inhomogeneous distribution of polar (i.e. water) and apolar species within the ice mantle [@furuya15]. Since we aim to investigate the effect of ice composition on protostellar chemistry, we calculated another set of model, in which the initial molecular abundance of the static phase is set by solving the gas-grain chemistry in the converging flow. We solve the molecular evolution in the post-shock gas in the same 1D steady-state shock model as [@furuya15] [@bergin04; @hassel10], and adopt the molecular abundance when $A_{\rm v}$ reaches 1.2 mag. This choice of visual extinction is rather arbitrary, but it roughly corresponds to the time when both gas-phase CO and water ice abundances reach the abundance of $10^{-4}$; CO abundance starts to decline due to the conversion to CH$_4$ at larger $A_{\rm v}$. The gas and ice are then put in the prestellar core, which is kept static for $t_{\rm sta}=3\times 10^5$ yr, $1\times 10^6$ yr or $3\times 10^6$ yr.
Figure \[precol\_COM\_WCCC\] ($c, d$) shows the gaseous COM abundances and the minimum and maximum abundances of CH$_4$ and carbon chains as in Figure \[precol\_COM\_WCCC\] ($a, b$), but with the initial abundance set in the converging flow. We can see that the COM abundances, activity of WCCC, and their dependence on $t_{\rm sta}$ are similar to those in panels ($a$, $b$). We note that the abundance of CH$_3$OCH$_3$ is higher in the model with converging flow. It is most abundantly formed in the inner-most layer of the ice mantle, the molecular abundance of which reflects the early evolution. A major characteristics of ice mantle composition set by the converging flow is that the inner-most layer is CO-poor, which results in relatively high abundance of OH. In our fiducial model, CO ice is more uniformly distributed in bulk ice layers, and reacts with OH to form CO$_2$, when the thermal diffusion becomes efficient. CH$_3$OCH$_3$ is formed via CH$_3$ + CH$_3$O. In the OH-rich ice layer, CH$_3$ reacts with OH to reform CH$_3$OH, which is photodissociated to CH$_3$O. In the fiducial model, on the other hand, CH$_3$ reacts mainly with NH$_2$ to form CH$_3$NH$_2$. Thus CH$_3$CN is more abundant in our fiducial model, while CH$_3$OCH$_3$ is more abundant with the ice mantle set by the converging flow.
a grid of models
----------------
So far we varied one of the parameters of temperature, the visual extinction of ambient gas ($A_{\rm v}^{\rm amb}$), or the duration of the static phase ($t_{\rm sta}$). COMs abundances and WCCC activity in the protostellar phase are found to be most sensitive to the minimum temperature among these parameters. In this subsection, we investigate how this sensitivity depends on $A_{\rm v}^{\rm amb}$ and $t_{\rm sta}$.
Figure \[two\_param\_COM\] shows the COM abundances as a function of $T_{\rm min}$ in models with $A_{\rm v}=1$ mag, 3 mag, and 5 mag and $t_{\rm sta}=3\times 10^5$ yr, $1\times 10^6$ yr, and $3\times 10^6$ yr. We can see that the dependence of COM abundances on $T_{\rm min}$ is more significant in models with lower $A_{\rm v}^{\rm amb}$. In these models, photodissociation of molecules, including COMs, is more effective. While the photodissociation produces radicals, which can recombine to reform COMs, the rates of their diffusion in ice mantle and sublimation (i.e. loss to the gas phase) are very sensitive to temperatures. Photodissociation thus enhances the dependence of COM abundances on temperatures. The dependence of some COM abundances on $T_{\rm min}$ is also stronger in models with longer static phase. In the middle row (Av=3mag) in Figure \[two\_param\_COM\], for example, the CH$_3$CN abundance varies more than two orders of magnitudes in the models with $t_{\rm sta}=3 \times 10^6$ yr, while the variation is within an order of magnitude in the models with $t_{\rm sta}=3 \times 10^5$ yr. When a gas-grain reaction network is kept at a constant temperature, specific molecules and radicals accumulate in ices: e.g. species that can freeze-out and/or species produced by the recombination of radical that can thermally diffuse. If the duration of the constant temperature is longer, the accumulation of those specific species becomes more significant, which results in the higher dependence of protostellar core chemistry on $T_{\rm min}$. Comparing the solid and dash-dotted cyan lines, we note that a significant amount of CH$_3$OH is formed after the onset of collapse in models with low $A_{\rm v}^{\rm amb}$ ($\lesssim 3$ mag). The abundance ratio of COMs to gaseous CH$_3$OH, as well as that to CH$_3$OH ice at the end of static phase, vary significantly with $T_{\rm min}$.
The WCCC activity in the same set of models are shown in Figure \[two\_param\_WCCC\]. The WCCC is activated when the sublimated CH$_4$ abundance is higher than $\sim 10^{-7}$. In models with $A_{\rm v}^{\rm amb}= 3$ mag and 5 mag, the CH$_4$ ice abundance at the end of static phase (and the gaseous CH$_4$ at the final timestep) does not sensitively depend on $t_{\rm sta}$. The sublimated CH$_4$ abundance around WCCC region (i.e. $T\sim 25$ K) is, however, higher in models with larger $t_{\rm sta}$. Since the conversion of CO to CH$_4$ becomes efficient later in the static phase, the CH$_4$ ice in the surface layer, which is subject to the immediate sublimation at $\sim 25$ K, is more abundant in models with larger $t_{\rm sta}$. WCCC is thus more active in models with longer $t_{\rm sta}$. In the models with $A_{\rm v}^{\rm amb}=1$ mag and $T_{\rm min}\gtrsim 20$ K, CH$_4$ ice abundance is very low at the end of static phase, and thus WCCC is not active. In these models, a large fraction of CH$_4$ is formed after the onset of collapse by the gas-phase reactions starting from C$^+$ + H$_2$ $\rightarrow$ CH$_2^+$. WCCC is activated in models with $T_{\rm min}\lesssim 15$ K, even if the ambient visual extinction is low (i.e. $A_{\rm v}^{\rm amb}=1$ mag).
Discussion
==========
comparison with previous work
-----------------------------
The abundances of carbon chains and COMs in our fiducial model are different from those in our previous work [@aikawa08; @aikawa12], since we adopt the multi-layered ice mantle model, rather than the two-phase model as described in §3.1. Entrapment of CH$_4$ suppresses the WCCC, while the entrapment of radicals enhances the formation of COMs [@lu18].
[@acharyya18] investigated the hot corino chemistry in Large Magellanic Cloud and Small Magellanic Cloud by calculating the two-phase model of gas-grain chemistry in the cold collapse stage and warm-up stage. The dependence of peak molecular abundances on the temperature in the collapse stage is investigated, as well. The peak abundances of COMs are basically lower in models with higher temperature in the collapse phase, which is qualitatively consistent with our results on $T_{\rm min}$ (§3.2). The COM abundances in our model with $T_{\rm min}=25$ K, however, tends to be much higher than those in [@acharyya18]. It would be mostly due to the effect of multi-layered ice mantle model, which can keep various radicals even at warm temperatures. Our grid of models also suggest that the COM abundances depend more sensitively on $T_{\rm min}$ when $A_{\rm v}^{\rm amb}$ is lower. It is in line with the recent observations, which show large abundance variations of COMs among cores in LMC and SMC [@shimonishi16; @sewilo18; @shimonishi18; @shimonishi20], since the visual extinction is lower in those low-metalicity galaxies compared with that in our Galaxy.
[@vidal19] calculated the three-phase model in 110 models of star-forming core of [@vaytet17]. They statistically analyzed the correlation of molecular abundances and physical model parameters, and found that CH$_3$CN abundance correlates positively with the initial core temperature. Since they plot the final molecular abundance in all the fluid parcels of 110 models as a function of the initial temperature, CH$_3$CN abundance is scattered over one order of magnitude or more at each initial temperature bin. The dependence of the CH$_3$CN abundance (i.e. the maximum or mean value) on the initial temperature is similar to the dependence of CH$_3$CN on $T_{\rm init}$ in our model (Figure \[Tinit\]); the dependence is weak at $T_{\rm init} =10-20$ K, while the abundance is about one-order of magnitude higher at $T_{\rm init}=25$ K. They concluded that the positive correlation is caused by enhanced diffusion of CN and CH$_3$ in the ice mantle. We speculate that this explanation is too simplified; other COM abundances would show a positive correlation with the initial core temperature, if the diffusion rate is the key. As we discussed in §3, the formation paths of COMs vary with the initial temperature; new formation paths open in the model with $T_{\rm init}=25$ K.
We adopted the chemical reaction network of [@garrod13], which investigated formation of COMs, especially glycine, in warm-up models mimicking the star-forming core. Their model consists of the cold collapse phase and warm-up phase. In the former, the initial gas density increases from $n_{\rm H}=3 \times 10^3$ cm$^{-3}$ to $1\times 10^7$ cm$^{-3}$ in $\sim 10^6$ yr. The dust temperature and visual extinction $A_{\rm v}$ are initially 16 K and 2 mag, respectively. The temperature decreases as $A_{\rm v}$ increases, reaching the minimum value of 8 K. In the warm-up phase, temperature increases from 8 K to 400 K, while the density is kept constant ($1\times 10^7$ cm$^{-3}$). Three models are calculated with the warm-up timescale of $7.12 \times 10^4$ yr (fast), $2.85\times 10^5$ yr (medium), and $1.43 \times 10^6$ yr (slow). Comparing the timescale of temperature rise from 20 K to 100 K, our model is similar to the fast or medium model of [@garrod13], in which the peak abundances of gaseous CH$_3$OCH$_3$, CH$_3$CN, and CH$_3$CHO are $(3-5) \times 10^{-8}$, $(2-5) \times 10^{-9}$, and $(3-9)\times 10^{-9}$, respectively. Despite the differences in the physical model and chemical model (the three-phase model with swapping versus the multi-layered ice mantle model without swapping), these COM abundances are similar to our results. Our abundances of NH$_2$CHO and HCOOH at the final timestep are smaller than those in [@garrod13] by more than an order of magnitude, since we deleted some reactions relevant for their formation (§2.2) (see also §4.2).
uncertainties in the reaction network of COMs
---------------------------------------------
Besides the treatment of grain surface chemistry (e.g. two-phase, three-phase, and multi-phase model) and physical model of core formation and evolution, there are uncertainties in chemical reaction network. One of the major uncertainties in the hot corino chemistry is the branching ratio of radical reactions in the ice, as it is difficult to directly measure in laboratory experiments. Quantum chemical calculations have been useful to estimate such branching ratios, but are often not straightforward, since the interaction with the grain surface and neighboring icy species need to be included [e.g. @kayanuma19]. While the main aim of the present work is to investigate how the WCCC and hot corino chemistry as a whole depend on the physical conditions in prestellar pase, rather than the dependence of each COM species, it is useful to check the effect of uncertainties in reaction network on our model results.
Specifically we modified the branching ratio of two grain surface reactions: NH$_2$ + H$_2$CO and HCO + CH$_3$. In the gas phase, [@barone15] found that the activation barrier of NH$_2$ + H$_2$CO $\rightarrow$ NH$_2$CHO + H is very low (26.9 K), which we adopt in our fiducial model. For the reaction of NH$_2$ + H$_2$CO in ice, on the other hand, we assumed the products to be NH$_3$ + HCO with the activation barrier of 2360 K, as assumed in the original network of [@garrod13] to be conservative (see also discussions in [@fedoseev16]). In order to check the effect of uncertainty of this reaction, here we assume that NH$_2$CHO and H are formed without activation barrier in ice. As for the icy reaction of HCO + CH$_3$, we assume the products to be CH$_4$ + CO in our fiducial model referring to [@enrique-romero16]. Recently [@enrique-romero20] re-investigated this reaction with the broken-symmetry approach to find that the formation of CH$_3$CHO proceeds without barrier, while the direct H transfer to form CO + CH$_4$ can be a competitive channel. We thus assume 1:1 branching ratio for the product channels of CH$_3$CHO and CO + CH$_4$.
Figure \[mod\_COM\] shows the COM abundances at the final timestep ($R=30.6$ au) in the models with modified branching ratios with $T_{\rm min}$ of 10 K, 15 K, 20 K, and 25 K. Compared with Figure \[Tinit\] (b), the abundance of CH$_3$CHO is enhanced by a factor of 3.7 in the model with $T_{\rm min}=10$ K, while that of NH$_2$CHO is enhanced by a factor of $5.3-23$ in the models with $T_{\rm min} \le 20$ K. It is as expected, since we modified the branching ratios to be favorable for the formation of these molecules. It also suggests that other reaction paths dominate in their formation at higher temperatures. In order to discuss the dependence of each COM abundance on prestellar temperatures, it is essential to refine the chemical reaction network by the laboratory experiments and quantum chemical calculations.
![Gas phase abundances of COMs at the final time step ($R=30.6$ au) in models with the minimum temperature of 10 K, 15 K, 20 K, and 25 K with the modified branching ratios of NH$_2$ + H$_2$CO and HCO + CH$_3$. The dash-dotted cyan lines depict the CH$_3$OH ice abundance at the end of the static phase. \[mod\_COM\]](mod_COM.pdf)
comparison with observations
----------------------------
We found that WCCC, i.e. the formation of carbon chains around the CH$_4$ sublimation region, is more active and their abundances are higher in models with lower $T_{\rm init}$ and $T_{\rm min}$, lower $A_{\rm v}^{\rm amb}$, and longer $t_{\rm sta}$. The abundance of C$_4$H in L1527 and IRAS 15398, which are prototypical WCCC sources, is $\gtrsim 10^{-9}$, while it is $\sim 10^{-11}$ in a prototypical hot corino IRAS 16293 [@sakai09a]. This range of C$_4$H abundance is covered by the models in the present work. The molecular D/H ratio could be a key to discriminate which parameter, $T_{\rm init}$ ($T_{\rm min}$), $A_{\rm v}^{\rm amb}$, or $t_{\rm sta}$, is responsible for the variation of carbon chain abundances in protostellar cores; low $T_{\rm init}$ and long $t_{\rm sta}$ would enhance D/H ratio, while low D/H ratio is expected for low $A_{\rm v}^{\rm amb}$. In L1527, the column density ratio of c-C$_3$HD/c-C$_3$H$_2$ is 4.4 % [@yoshida19] (see also [@sakai09b]), while the ratio is observed to be 14 % towards IRAS 16293 [@majumdar17], which may suggest that the variation is caused by the visual extinction.
The dependence of COM abundances on the static-phase conditions are more complex, since there are various formation paths of COMs, and since their efficiency depends on the composition of each layer of ice mantle. Among the parameters investigated, $T_{\rm min}$ is the most effective. While CH$_3$OH basically decreases with $T_{\rm min}$, CH$_3$OCH$_3$, for example, is least abundant in the model with $T_{\rm min}=20$ K, and is more abundant in the model with $T_{\rm min}=25$ K, in which COMs can be formed from large hydrocarbons. It should be noted that the observations often evaluate the relative COM abundances to CH$_3$OH. In our models, the COM abundances relative to CH$_3$OH tend to increase with $T_{\rm min}$ and $T_{\rm init}$ as CH$_3$OH decreases, although either CH$_3$OH or other COMs would not be detected if their abundances are too low. Recently, [@oya19] observed Class I protostellar source Elias 29 to find both COMs and carbon chains are deficient; the abundances of C$_2$H and c-C$_3$H$_2$ are $\lesssim 10^{-11}$ and those of HCOOCH$_3$ and CH$_3$OCH$_3$ are $\lesssim 10^{-9}$. Considering the uncertainties of the upper limits, these abundances are consistent in our model with $T_{\rm min}= 20$ K.
In our models with higher $T_{\rm init}$, higher $A_{\rm v}^{\rm amb}$, or shorter $t_{\rm sta}$, WCCC is less active, while COMs can be abundantly formed; these models can explain the deficiency of carbon chains in hot corinos. The deficiency of COMs in WCCC sources is, on the other hand, hard to reproduce by simply varying the static-phase conditions; COM abundances do not monotonically decrease with a specific parameter, and COMs are abundantly formed in models with active WCCC. While COM abundances are low in the model with $T_{\rm min}=20$ K, WCCC is not active, either. Alternatively, the deficiency of COM emissions could be due to temperature distributions in the WCCC sources. The sublimation temperature of COMs is typically 100 K. The size of the hot corino, i.e. the radius of the COM sublimation region, is typically $\lesssim 100$ au. If the central protostar is less luminous, the sublimation region could be significantly smaller; the COM emission lines could be weakened by the beam dilution and high dust opacity, since the (column) density is higher at smaller radii. Indeed, the luminosity of the prototypical hot corino sources (e.g. 9.1 $L_{\odot}$ in NGC 1333 IRAS4A and 22 $L_{\odot}$ in IRAS 16293-2422 ) tend to be higher than that of WCCC sources; e.g. 1.9 $L_{\odot}$ and 1.8 $L_{\odot}$ for L1527 and IRAS 15398, respectively [@froebrich05; @crimier10; @kristensen12; @karska13; @jorgensen13]. One notable exception is B335; while its luminosity is as low as 0.72 $L_{\odot}$, COMs emission is detected within a few 10 au at the core center, where the fractional abundances of COMs are comparable to those in the prototypical hot corino IRAS 16293 [@imai16]. A possible explanation would be that B335 experienced temporal outburst, which sublimated COMs, but is currently back to its quiescent phase. COMs are still in the gas phase, if the time after the outburst is less than the re-freeze-out timescale, i.e. $\sim 10^3 (10^7$ cm$^{-3}$/$n_{\rm H}$) yrs.
We note that the kinetic structure inside a few hundreds au is as relevant as temperature distributions. Once CH$_3$OH and other COMs are sublimated to the gas-phase, they are destroyed by gas-phase reactions within several $10^4$ yrs [@charnley92; @nomura09; @taquet16]. The spatial extent of gaseous COMs is thus estimated to be the product of destruction timescale and radial velocity of the gas. When the $T\sim 100$ K region is located in the envelope, we can naively expect constant gaseous abundance of COMs inside this region, since infall is faster than COM destruction in the gas phase (Figure \[discussion\] a). If the $T\sim 100$ K region is inside the rotationally-supported disk, on the other hand, the spatial extent of gas-phase COMs would be very narrow, limited by the competition between gas-phase destruction of COMs and slow radial migration of gas in the disk (Figure \[discussion\] b). While the spatially resolved observation of hot corino is challenging, [@imai19] recently resolved the COM emission in B335, and showed that the variation between velocity gradients of COM emissions are well explained by the model of infalling and rotating gas, rather than Keplerian motion. Observations with high spatial resolution (e.g. $\sim 0.1$ ) or in lower frequency band, in which the dust opacity is lower, are desirable to investigate the physical structure such as temperature and density distributions, and the radius of the forming disk.
![Schematic view of CH$_3$OH gas distribution in (a) a hot corino source with the $T\sim 100$ K region located in the infalling envelope and (b) a protostellar core with the $T\sim 100$ K region located in the rotationally-supported disk. \[discussion\]](discussion.pdf)
Summary
=======
We investigated the dependence of WCCC and hot corino chemistry on the physical parameters in the static phase before the onset of collapse: the initial and minimum temperatures ($T_{\rm init}$ and $T_{\rm min}$), visual extinction of ambient gas ($A_{\rm v}^{\rm amb}$), and the duration of the static phase ($t_{\rm sta}$). Our findings are as follows.
- [Among the parameters, $T_{\rm min}$ is the most effective on COM abundances. CH$_3$OH and some other COMs tend to decrease with increasing $T_{\rm min}$, since freeze-out of molecules and hydrogenation on grain surfaces are less efficient at warm temperatures. But molecules with higher sublimation temperatures can still be adsorbed onto grains, and there are various formation paths of COMs, some of which become efficient at warm temperature. The abundance of CH$_3$OCH$_3$, for example, is higher in the model with $T_{\rm min}=25$ K than that with $T_{\rm min}=20$ K. Dependence of COM abundances on $T_{\rm init}$ is weaker, since various molecules, including CO, can be frozen and hydrogenated on grain surfaces during the cold phase right after the onset of collapse. ]{}
- [The gaseous CH$_3$OH abundance in the central hot region ($\ge 100$ K) monotonically decreases with increasing $T_{\rm init}$ and $T_{\rm min}$, except for the models with low visual extinction ($A_{\rm v}^{\rm amb}=1$ mag). Dependence of other COM abundances on $T_{\rm init}$ and $T_{\rm min}$ is not monotonic, as described above. The relative abundance of COMs to CH$_3$OH, which is often derived and discussed in the observational studies, could then be higher in cloud cores with higher $T_{\rm init}$ and $T_{\rm min}$, ]{}
- [WCCC is less active and carbon-chain species are less abundant in models with higher $T_{\rm init}$ or $T_{\rm min}$, since both the grain-surface formation and freeze-out of CH$_4$ become less effective at warm temperatures. Warm temperature also enhances the conversion of CO to CO$_2$ on grain surfaces, which reduces gaseous CO and the production of C$^+$ via CO + He$^+$ $\rightarrow$ C$^+$ + O + He. ]{}
- [While CH$_4$ and carbon chains are more abundant in the models with lower $A_{\rm v}^{\rm amb}$, the dependence of COM abundances on $A_{\rm v}^{\rm amb}$ is more complex. Even though photolysis makes CH$_3$OH and other COMs less abundant in the static phase of lower $A_{\rm v}^{\rm amb}$ model, they can be formed via various reactions in ice mantle, e.g. CH$_3$ + OH $\rightarrow$ CH$_3$OH, once the collapse starts and $A_{\rm v}$ increases.]{}
- [When the duration of static phase $t_{\rm sta}$ is varied from $3\times 10^5$ yr to $3\times 10^6$ yr, the COM abundances in the protostellar phase vary less than an order of magnitude. WCCC, on the other hand, is more active in the model with longer $t_{\rm sta}$, since CH$_4$ ice is stable and accumulate during the static phase. In the model with short $t_{\rm sta}$, the remnant of carbon chains from the prestellar phase dominates over those formed via WCCC. ]{}
- [We also calculated additional models in which the initial molecular abundances are set by considering the cloud formation via converging flow. The COM abundances and WCCC activities in the protostellar phase are basically similar to those in the fiducial model. A notable difference is that the ice mantle has a larger chemical gradient, e.g. the inner most ice layer is deficient in CO ice. It enhances the abundances of OH radical and CH$_3$OCH$_3$ in the ice mantle in the protostellar phase compared with the fiducial model.]{}
- [We calculated a grid of models to investigate how the dependence of COMs and WCCC on $T_{\rm min}$ vary with $A_{\rm v}^{\rm amb}$ and $t_{\rm sta}$. Variation of COM abundances with $T_{\rm min}$ is enhanced in models with low $A_{\rm v}^{\rm amb}$. The models with low $A_{\rm v}^{\rm amb}$ could also be relevant to the recent observations of hot cores in LMC and SMC, where significant variations are found in COM emission. The dependence of some COM abundances on $T_{\rm min}$ is also stronger in models with longer $t_{\rm sta}$. In the models with longer $t_{\rm sta}$, a larger fraction of CH$_4$ ice is in surface layers of ice mantle, which sublimates and activates WCCC at $T\sim 25$ K. ]{}
- [Our models show that the variation of $T_{\rm init}$ (or $T_{\rm min}$), $A_{\rm v}^{\rm amb}$, and $t_{\rm sta}$ can explain the chemical diversity between prototypical hot corinos, in which carbon-chains are deficient, and hybrid sources, toward which both COM and carbon chains are abundant. A relatively low D/H ratio of carbon chain observed in a prototypical WCCC source L1527 may indicate that $A_{\rm v}^{\rm amb}$ is the key parameter. Deficiency of COMs in prototypical WCCC sources is, however, hard to reproduce within our models; i.e. models with active WCCC have relatively abundant COMs. A possible explanation for the deficiency would be the small size of COM sublimation region (i.e. $\gtrsim 100$ K) and/or the kinetic structure there. Brightness of the COM lines would be suppressed, if the COM sublimation region is significantly smaller than the beam size or within the rotationally-supported disk, in which the radial extent of gaseous COMs are limited by the competition between destruction in the gas-phase and slow radial accretion of gas. In case the COM sublimation region is small, the emission lines could also be hidden by the high dust opacity. Observations with high spatial resolution are desirable to investigate the physical structure in the central regions of protostellar cores such as temperature and density distributions, and the radius of the forming disk.]{}
We thank R. T. Garrod for helpful discussions and sharing the chemical reaction network model. We thank the anonymous referee for his/her constructive comments. This work is supported by Grant-in-Aid for Scientific Research (S) 18H05222, Grant-in-Aid for Young Scientists (B) 17K14245, and NAOJ ALMA Scientific Research Grant Numbers 2019-13B.
|
---
abstract: 'Following the recent progress on the calculation of three-point correlators with two “heavy” (with large quantum numbers) and one “light” states at strong coupling, we compute the logarithmic divergent terms of leading bosonic quantum corrections to correlation functions with “heavy” operators corresponding to simple string solutions in $AdS_5\times S^5$. The “light” operator is chosen to be the dilaton. An important relation connecting the corrections to both the dimensions of “heavy” states, and the structure constants is recovered.'
author:
- |
D. Arnaudov${}^{\star}$ and R. C. Rashkov${}^{\dagger,\star}$[^1] \
\
${}^{\star}$ Department of Physics, Sofia University,\
5 J. Bourchier Blvd, 1164 Sofia, Bulgaria \
\
${}^{\dagger}$ Institute for Theoretical Physics,\
Vienna University of Technology,\
Wiedner Hauptstr. 8-10, 1040 Vienna, Austria
title: 'Quadratic corrections to three-point functions'
---
Introduction
============
One of the most active fields of research in theoretical physics in recent years has been the correspondence between the large $N$ limit of gauge theories and string theory, and particularly the AdS/CFT correspondence [@Maldacena]. Many impressive results from the duality between type IIB string theory on $AdS_5\times S^5$ and ${\cal N}=4$ super Yang-Mills theory [@Maldacena; @GKP; @Witten] have been obtained, but much more lies beyond our knowledge. One of the problems that lack proper understanding is the calculation of three-point functions of string states (dual to operators with large quantum numbers in the gauge theory) at strong coupling ($\sqrt{\lambda}\gg1$). Although the problem remains unsolved in general, recently there has been significant progress in the semiclassical calculation of two-, three-, and four-point correlators with two “heavy” states [@Janik:2010gc]–[@Lee:2011]. Extending these studies, we consider the bosonic quadratic fluctuations[^2] of correlation functions of two “heavy” operators and the dilaton, utilizing the methods for calculation of three-point correlators suggested in [@Janik:2010gc; @Costa:2010].
The paper is organized as follows. To explain the method, in the next section we give a short review of and extend the procedure for computing semiclassically three-point correlators with dilaton “light” operator in the case of $\axs$. Next, we proceed with the calculation of quadratic fluctuations of correlation functions for some simple solutions. We conclude with a brief discussion on the results.
Calculation of three-point correlators
======================================
We consider $\axs$ background with Poincare coordinates $(z,x)$ in $AdS_5$, so that the boundary is a four-dimensional Minkowski space with coordinates $x$. As was shown in [@Costa:2010] the partition function assumes the form $$\tilde{Z}(x_i,x_f,\Phi_0)\approx\int DX\,D\gamma\,D\Phi\,e^{i\left(S_P[X,\gamma,\Phi]+S_{SUGRA}[\Phi]\right)}\,.
\label{stringgenfun}$$ We consider fluctuations around given string solution $\bar{X}^{\mu},\,\mu=0,\dots,9$. Examining the relevant geodesic equation $\ddot{\lambda}^{\mu}+\Gamma^{\mu}_{\nu\rho}\dot{\lambda}^{\nu}\dot{\lambda}^{\rho}=0$ with $\lambda^{\mu}(0)=\bar{X}^{\mu}$, we define $\xi^{\mu}=\dot{\lambda}^{\mu}(0)$. Up to quadratic fluctuations (second order in $\xi$) the partition function should be modified to [$$\tilde{Z}(x_i,x_f,\Phi_0)\approx\int D\Phi\,D\xi\,e^{i(S_P[\bar{X},\bar{s},\Phi,\xi]+S_{SUGRA}[\Phi])}\,,$$]{} where the classical solution $\bar{X}$ to the equations of motion with suitable boundary conditions corresponds to an operator ${\cal O}_A$ with large quantum numbers in the dual gauge theory. We will confine ourselves to a solution which is point-particle in AdS [$$z=z(\tau)=R/\cosh\,\kappa\tau\,,\qquad
x=x(\tau)=R\tanh\kappa\tau+x_0\,.
\label{xzparticle}$$]{} As was pointed out in [@Janik:2010gc] [$$\label{kappabc}
\kappa\approx\frac{2}{s}\log{\frac{x_f}{\varepsilon}}\,,\qquad R\approx x_0\approx\frac{x_f}{2}\,,$$]{} where $\varepsilon$ is an ultraviolet regulator and $\kappa$ is defined through $t=\kappa\tau$. In addition, $\bar{s}$ is the saddle-point value of the modular parameter $s$ on the worldsheet cylinder, whose minimization of area gives the two-point function. $\Phi$ denotes the supergravity fields, one of which is the dilaton $\phi$. It sources the operator ${\cal D}_\phi\equiv{\cal L}$, which has scaling dimension $\Delta=4$ in the leading semiclassical approximation, near the boundary. The Lagrangian $\cal L$ of the ${\cal N}=4$ SYM theory generates a deformation of the ’t Hooft coupling $\lambda$ [@Costa:2010]. The bosonic Polyakov action to leading order in the coupling $g=\frac{\sqrt{\lambda}}{4\pi}$ is [@Foerste][^3] [$$\begin{aligned}
S_P[\bar{X},\bar{s},\Phi,\xi]&=S_P^{(0)}[\bar{X},\bar{s},\Phi]+S_P^{(2)}[\bar{X},\bar{s},\Phi,\xi]\,,\\ \nonumber
S_P^{(0)}[\bar{X},\bar{s},\Phi]&=-g\!\int_{-\bar{s}/2}^{\bar{s}/2}\!\!d\tau\!\int\!d\sigma\,e^{\phi/2}\,\eta^{\alpha\beta}
\partial_\alpha\bar{X}^A\partial_\beta\bar{X}^Bg_{AB}\,,\\
S_P^{(2)}[\bar{X},\bar{s},\Phi,\xi]&=-g\!\int_{-\bar{s}/2}^{\bar{s}/2}\!\!d\tau\!\int\!d\sigma\,e^{\phi/2}\,\eta^{\alpha\beta}\!
\left(D_\alpha\xi^AD_\beta\xi^Bg_{AB}+\partial_\alpha\bar{X}^A\partial_\beta\bar{X}^B\xi^C\xi^DR_{ACBD}\right),
\nonumber
\end{aligned}$$]{} where $g_{AB}$ is the background metric, $D_\alpha\xi^A=\partial_\alpha\xi^A+\Gamma^A_{BC}\xi^B\partial_\alpha\bar{X}^C$, and the Riemann tensor is defined as [$${R^A}_{BCD}\equiv\partial_C\Gamma^A_{BD}-\partial_D\Gamma^A_{BC}+\Gamma^E_{BD}\Gamma^A_{EC}-\Gamma^E_{BC}\Gamma^A_{ED}\,.$$]{} We want to calculate the correction to the partition function [$$\tilde{Z}^{(2)}=\int D\xi\,e^{iS_P^{(2)}[\bar{X},\bar{s},\Phi,\xi]}\,.$$]{} Since $S_P^{(2)}$ is quadratic in $\xi$, $\tilde{Z}^{(2)}$ is equal to the determinant of an operator ${\cal O}$. In order to find the determinant we consider the heat kernel technique [@Foerste], which utilizes the powerful method of $\zeta$-function regularization. The correction to the partition function can be expressed as the formal sum ($t$ is an auxiliary parameter) [$$\log\tilde{Z}^{(2)}=\frac12\int\frac{dt}{t}e^{-{\cal O}t}=\frac12\int_{\t\varepsilon}^\infty\frac{dt}{t}\sum_{n=-2}^\infty a_nt^{\frac{n}{2}-1},$$]{} where $\t\varepsilon$ is an ultraviolet cutoff with dimension of mass. We are only interested in the logarithmic divergent part of the quadratic corrections. Therefore we concentrate on $n=2$, and get [$$\label{Z2}
\tilde{Z}^{(2)}\sim\t\varepsilon^{-a_2/2}\,,\quad
a_2=a_2[\bar{X},\bar{s},\Phi]=-\frac{1}{4\pi}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\int d\sigma\, e^{\phi/2}\eta^{\alpha\beta}\partial_\alpha\bar{X}^A\partial_\beta\bar{X}^BR^C_{\phantom{C}ACB}\,,$$]{} where $a_2$ is the relevant Seeley coefficient.
Let us start with calculating the logarithmic divergent quadratic correction to the two-point function. The detailed analysis in [@Janik:2010gc] shows that there is a subtlety in obtaining the string propagator, so that the classical solution for the cylinder coincides with the classical state. Therefore we have to work with $$\tilde{S}_P=S_P-\int_{-s/2}^{s/2}d\tau\int d\sigma\,\Pi^A\dot{X}_A\,,
\label{NewAction}$$ which is minus the integral of the Hamiltonian. It was shown in [@Bak:2011] that, strictly speaking, we have to use the Routhian instead of minus the Hamiltonian, but they coincide for our considerations. To obtain the correction to the two-point function we should examine the quadratic fluctuation of $\tilde{S}_P$, i.e., we should calculate the corresponding $\tilde{Z}^{(2)}$. It can be shown straightforwardly that again one can use just by substituting $\eta^{\alpha\beta}$ with $\delta^{\alpha\beta}$. Following [@Janik:2010gc; @Costa:2010], we get for the quantum correction of the two-point function (and scaling dimension) of two “heavy” operators [$$\begin{aligned}
\nonumber
\langle{\cal O}_A(0){\cal O}^*_A(x_f)\rangle&\sim\left(\frac{\varepsilon}{x_f}\right)^{2\Delta_A}\!\!=\tilde{Z}(0,x_f,\Phi_0=0)\\ \label{2point}
&=\int D\xi\,e^{i\tilde{S}_P[\bar{X},\bar{s},\Phi=0,\xi]}=\left(\frac{\varepsilon}{x_f}\right)^{2\Delta_A^{(0)}}\!\!\t\varepsilon^{-\tilde{a}_2/2}\,,\\
\Delta_A&=\Delta_A^{(0)}+\Delta_A^{(2)}\,,\qquad\Delta_A^{(2)}=-\frac{\tilde{a}_2\log\t\varepsilon}{4\log\frac{\varepsilon}{x_f}}\,,
\nonumber
\end{aligned}$$]{} where we have used , and having defined the “modified” Seeley coefficient [$$\tilde{a}_2=\tilde{a}_2[\bar{X},\bar{s},\Phi=0]=-\frac{1}{4\pi}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\int d\sigma\, \delta^{\alpha\beta}\partial_\alpha\bar{X}^A\partial_\beta\bar{X}^BR^C_{\phantom{C}ACB}\,.
\label{tildea2}$$]{} The three-point correlation function at strong coupling of two “heavy” operators and one dilaton can be obtained by functional differentiation of the partition function with respect to the dilaton field [$$\langle{\cal O}_A(0){\cal O}_A^*(x_f){\cal D}_\phi(y)\rangle\approx\frac{I_\phi[\bar{X},\bar{s};y]}{x_f^{2\Delta_A}}\,.$$]{} With a slight abuse of notation, we get for the logarithmic divergent part [$$\begin{aligned}
I_\phi[\bar{X},\bar{s};y]&=I_\phi^{(0)}[\bar{X},\bar{s};y]+I_\phi^{(2)}[\bar{X},\bar{s};y]\,,\\
I_\phi^{(0)}[\bar{X},\bar{s};y]&=i\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\int d\sigma\left.
\frac{\delta S_{P}^{(0)}[\bar{X},\bar{s},\Phi]}{\delta\phi}\right|_{\Phi=0}K_\phi(\bar{X};y)\,,\\
I_\phi^{(2)}[\bar{X},\bar{s};y]&=-\frac{\log\t\varepsilon}{2}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\int d\sigma\left.
\frac{\delta a_2[\bar{X},\bar{s},\Phi]}{\delta\phi}\right|_{\Phi=0}K_\phi(\bar{X};y)\,,
\label{Iphi}
\end{aligned}$$]{} where the bulk-to-boundary propagator has the following form [@Freedman:1998] [$$K_\phi(\bar{X};y)=K_{\phi}(z(\tau),x(\tau);y)=\frac{6}{\pi^2}\!\left(\frac{z(\tau)}{z^2(\tau)+(x(\tau)-y)^2}\right)^4\!\!.$$]{}
Also, the following relation was discovered in [@Costa:2010] for the leading semiclassical approximation [$$\langle{\cal O}_A(0){\cal O}_A^*(x_f){\cal L}(y)\rangle\approx-\frac{g^2}{2\pi^2}\frac{\partial\Delta_A}{\partial g^2}\frac{x_f^{4-2\Delta_A}}{y^4(x_f-y)^4}\,,$$]{} where the conserved charges are assumed constant. As we shall see below, it holds even for the quadratic corrections [$$I_\phi^{(2)}[\bar{X},\bar{s};y]=-\frac{g^2}{2\pi^2}\frac{\partial\Delta_A^{(2)}}{\partial g^2}\frac{x_f^4}{y^4(x_f-y)^4}\,.
\label{aLAA}$$]{}
Quantum corrections to three-point functions
============================================
In this section we apply the methods described in the previous one to particular solutions.
Circular rotating string
------------------------
Let us consider the case of circular rotating string with two equal spins in the sphere [@Frolov:2003qc]. First, we fix the notation by writing down the explicit form of the metric [$$ds^2_{\axs}/R_{\rm str}^2=\frac{dz^2+dx^2}{z^2}+[d\gamma^2+\cos^2\gamma d\varphi_3^2+\sin^2\gamma(d\psi^2+\cos^2\psi d\varphi_1^2+\sin^2\psi d\varphi_2^2)]\,.$$]{} If we assume the point-particle solution in $AdS_5$ , the Polyakov action in conformal gauge can be written as [$$\begin{aligned}
\nonumber
S_P[X,s,\Phi]&=g\int_{-s/2}^{s/2}d\tau\int d\sigma\,e^{\phi/2}\,\{\kappa^2+\dot{\gamma}^2-{\gamma'}^2+\cos^2\gamma(\dot{\varphi_3}^2-{\varphi'_3}^2)\\
&+\sin^2\gamma[\dot{\psi}^2-{\psi'}^2+\cos^2\psi(\dot{\varphi_1}^2-{\varphi'_1}^2)+\sin^2\psi(\dot{\varphi_2}^2-{\varphi'_2}^2)]\}\,.
\end{aligned}$$]{} The solution has the following form [@Frolov:2003qc] [$$\gamma=\frac{\pi}{2}\,,\quad\psi=\sigma\,,\quad\varphi_1=\varphi_2=\omega\tau\,,\quad\varphi_3=0\,.$$]{} It is straightforward to find the conserved quantities and dispersion relation for this string configuration [$$J\equiv J_1=J_2=2\pi g\omega\,,\qquad E=2\sqrt{J^2+4\pi^2g^2}=\Delta_A^{(0)}.$$]{} The dual operator is of the type ${\cal O}_A\sim{\rm Tr}(X^{J_1}Z^{J_2})$.
Let us apply now the procedure outlined briefly in the previous section. For the logarithmic divergent part of the quadratic correction to the Polyakov action we obtain [$$a_2[\bar{X},\bar{s},\Phi]=\frac{1}{\pi}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\int_0^{2\pi}d\sigma\,e^{\phi/2}(\kappa^2-\omega^2+1)\,.$$]{} The modified correction takes the form [$$\tilde{a}_2[\bar{X},\bar{s},\Phi=0]=2\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\,(\omega^2-\kappa^2+1)\,.$$]{} The saddle point with respect to the modular parameter $s$ is given by [@Costa:2010] [$$\label{saddlecircular}
\bar{s}=\frac{2i}{\sqrt{1+\omega^2}}\log\frac{\varepsilon}{x_f}\,,$$]{} which along with implies the Virasoro constraint $\kappa=i\sqrt{1+\omega^2}$. Thus, the evaluation of $\tilde{Z}(0,x_f,\Phi_0=0)$ gives [$$\langle{\cal O}_A(0){\cal O}^*_A(x_f)\rangle\sim\left(\frac{\varepsilon}{x_f}\right)^{2\Delta_A}
=\left(\frac{\varepsilon}{x_f}\right)^{2\Delta_A^{(0)}-4i\sqrt{1+\omega^2}\log\t\varepsilon},\qquad\Delta_A=\Delta_A^{(0)}+\Delta_A^{(2)},$$]{} which leads to [$$\Delta_A^{(2)}=-\frac{i\sqrt{J^2+4\pi^2g^2}}{\pi g}\log\t\varepsilon\,.$$]{}
Now we turn to the derivation of the fluctuation of the three-point function. We evaluate at the saddle point [$$I_\phi^{(2)}[\bar{X},\bar{s};y]=\frac{3\omega^2\log\t\varepsilon}{\pi^2}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\left(\frac{z}{z^2+(x-y)^2}\right)^4
=-\frac{i\omega^2\log\t\varepsilon}{2\pi^2\sqrt{1+\omega^2}}\frac{x_f^4}{y^4(x_f-y)^4}\,.$$]{} Therefore, the final expression for the three-point correlator is [$$\langle{\cal O}_A(0){\cal O}^*_A(x_f){\cal L}(y)\rangle\approx\frac{I_\phi^{(0)}[\bar{X},\bar{s};y]}{x_f^{2\Delta_A}}
-\frac{iJ^2\log\t\varepsilon}{4\pi^3g\sqrt{J^2+4\pi^2g^2}}\frac{x_f^{4-2\Delta_A}}{y^4(x_f-y)^4}\,.
\label{3pointcir}$$]{} One can see immediately that holds, provided that $J$ is kept constant.
Giant magnon
------------
One very important for the AdS/CFT correspondence class of string solutions is the so called giant magnon. In this subsection we will consider the simplest solution of this type [@Hofman:2006xt]. We will use a different parametrization of the sphere [$$ds_{S^5}^2=d\theta^2+\sin^2\theta d\varphi^2+\cos^2\theta d\Omega^2_3\,.$$]{} We start with the ansatz suggested in [@Hofman:2006xt] [$$\cos\theta=\sin\frac{p}{2}\,{\rm sech}(\omega u)\,,\quad\tan(\varphi-\omega\tau)=\tan\frac{p}{2}\tanh(\omega u)\,,
\label{mag-ans}$$]{} where $u=\left(\sigma-\tau\cos\frac{p}{2}\right)\csc\frac{p}{2}$, and $p\in[0,2\pi)$ is the momentum of the magnon. The angular momentum of the string becomes [$$J=2g\int_{-L}^{L}d\sigma\,\sin^2\theta\,\dot{\varphi}=2g\omega\int_{-L}^{L}d\sigma\,\tanh^2(\omega u)\approx 4g\!\left(\omega L-\sin\frac{p}{2}\right)$$]{} in the limit of large $L$. The energy can be obtained to be [$$E=\Delta^{(0)}=J+4g\sin\frac{p}{2}\approx4g\omega L\,.$$]{}
In this case assumes the form [$$a_2[\bar{X},\bar{s},\Phi]=-\frac{2\omega^2}{\pi}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\int_{-L}^{L}d\sigma\,e^{\phi/2}\tanh^2(\omega u)\,,$$]{} where we have used that the saddle point is [@Costa:2010] [$$\label{saddlemagnon}
\bar{s}=\frac{2i}{\omega}\log\frac{\varepsilon}{x_f}\,,$$]{} which again due to implies the Virasoro constraint $\kappa=i\omega$. The modified correction takes the form [$$\tilde{a}_2[\bar{X},\bar{s},\Phi=0]=\frac{8i\omega L}{\pi}\log\frac{\varepsilon}{x_f}\,.$$]{} Therefore we find that the contribution of $\tilde{Z}(0,x_f,\Phi_0=0)$ is [$$\langle{\cal O}_A(0){\cal O}^*_A(x_f)\rangle\sim\left(\frac{\varepsilon}{x_f}\right)^{2\Delta_A}=\left(\frac{\varepsilon}{x_f}\right)^{2\Delta_A^{(0)}-\frac{4i\omega L}{\pi}\log\t\varepsilon}\ \ \Longrightarrow\ \
\Delta_A^{(2)}\approx-\frac{2i}{\pi}\!\left(\frac{J}{4g}+\sin\frac{p}{2}\right)\log\t\varepsilon\,,$$]{} where we have used that $\Delta_A=\Delta_A^{(0)}+\Delta_A^{(2)}$.
In order to obtain the correction to the three-point correlator, we evaluate at the saddle point $$\begin{aligned}
\nonumber
I_\phi^{(2)}[\bar{X},\bar{s};y]&=\frac{3\omega^2\log\t\varepsilon}{\pi^3}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\int_{-L}^{L}d\sigma\,\tanh^2(\omega u)\left(\frac{z}{z^2+(x-y)^2}\right)^4\\
&=\frac{3\omega J\log\t\varepsilon}{2\pi^3g}\int_{-\bar{s}/2}^{\bar{s}/2}d\tau\left(\frac{z}{z^2+(x-y)^2}\right)^4
=-\frac{iJ\log\t\varepsilon}{8\pi^3g}\frac{x_f^4}{y^4(x_f-y)^4}\,.\end{aligned}$$ For the three-point function we end up with the expression [$$\langle{\cal O}_A(0){\cal O}^*_A(x_f){\cal L}(y)\rangle\approx\frac{I_\phi^{(0)}[\bar{X},\bar{s};y]}{x_f^{2\Delta_A}}
-\frac{iJ\log\t\varepsilon}{8\pi^3g}\frac{x_f^{4-2\Delta_A}}{y^4(x_f-y)^4}\,.$$]{} Again it can be seen that holds (up to zeroth order of $L$), provided that $p$ and $J$ are kept independent of the coupling.
Conclusion
==========
One of the active fields in AdS/CFT correspondence recently has been the calculation of three-point correlators beyond the supergravity approximation [@Janik:2010gc; @Zarembo:2010; @Costa:2010; @Roiban:2010]. The authors consider string theory on $\axs$ and obtain three-point functions of two “heavy” operators and one supergravity state at strong coupling. Very recently three-point correlators of three BMN operators with large charges were computed [@Klose:2011], albeit in light-cone gauge.
In this paper we generalize the ideas for calculation of correlation functions in [@Janik:2010gc; @Costa:2010] by including the leading bosonic quadratic correction to the action. We apply the method in the cases of simple string solutions. The careful analysis in [@Costa:2010] states that the structure constants are given by $-g^2\partial\Delta_A/\partial g^2$. We are the first to check this relation for quadratic corrections to correlators, and we found perfect agreement.
Unfortunately, to the best of our knowledge, there are no known results from gauge theory side which can be used for comparison, although there has been progress in the computation of structure constants using the Bethe eigenstates of the underlying integrable spin chain in the weak coupling limit of the dual gauge theory [@Escobedo:2010; @Escobedo:2011]. The corrections to correlation functions we obtained here should be considered only as a first step. For example, it would be interesting to extend our results to other “light” operators[^4] and backgrounds, as in [@Klose:2011].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank H. Dimov for valuable discussions and careful reading of the paper. This work was supported in part by the Austrian Research Funds FWF P22000 and I192, and NSFB DO 02-257.
[99]{}
J. M. Maldacena, “The large $N$ limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. [**2**]{}, 231 (1998) \[arXiv:hep-th/9711200\].
S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,” Phys. Lett. B [**428**]{}, 105 (1998) \[arXiv:hep-th/9802109\].
E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. [**2**]{}, 253 (1998) \[arXiv:hep-th/9802150\].
R. A. Janik, P. Surowka, and A. Wereszczynski, “On correlation functions of operators dual to classical spinning string states,” JHEP [**1005**]{}, 030 (2010) \[arXiv:1002.4613\]. E. I. Buchbinder and A. A. Tseytlin, “On semiclassical approximation for correlators of closed string vertex operators in AdS/CFT,” JHEP [**1008**]{}, 057 (2010) \[arXiv:1005.4516\]. K. Zarembo, “Holographic three-point functions of semiclassical states,” JHEP [**1009**]{}, 030 (2010) \[arXiv:1008.1059\]. M. S. Costa, R. Monteiro, J. E. Santos, and D. Zoakos, “On three-point correlation functions in the gauge/gravity duality,” JHEP [**1011**]{}, 141 (2010) \[arXiv:1008.1070\]. R. Roiban and A. A. Tseytlin, “On semiclassical computation of 3-point functions of closed string vertex operators in $\axs$,” Phys. Rev. D [**82**]{}, 106011 (2010) \[arXiv:1008.4921\]. R. Hernández, “Three-point correlation functions from semiclassical circular strings,” J. Phys. A [**44**]{}, 085403 (2011) \[arXiv:1011.0408\]. S. Ryang, “Correlators of Vertex Operators for Circular Strings with Winding Numbers in $\axs$,” JHEP [**1101**]{}, 092 (2011) \[arXiv:1011.3573\]. G. Georgiou, “Two and three-point correlators of operators dual to folded string solutions at strong coupling,” JHEP [**1102**]{}, 046 (2011) \[arXiv:1011.5181\]. J. G. Russo and A. A. Tseytlin, “Large spin expansion of semiclassical 3-point correlators in $\axs$,” JHEP [**1102**]{}, 029 (2011) \[arXiv:1012.2760\]. C. Park and B. Lee, “Correlation functions of magnon and spike,” Phys. Rev. D [**83**]{}, 126004 (2011) \[arXiv:1012.3293\]. E. I. Buchbinder and A. A. Tseytlin, “Semiclassical four-point functions in $\axs$,” JHEP [**1102**]{}, 072 (2011) \[arXiv:1012.3740\]. D. Bak, B. Chen, and J. Wu, “Holographic Correlation Functions for Open Strings and Branes,” JHEP [**1106**]{}, 014 (2011) \[arXiv:1103.2024\]. A. Bissi, C. Kristjansen, D. Young, and K. Zoubos, “Holographic three-point functions of giant gravitons,” JHEP [**1106**]{}, 085 (2011) \[arXiv:1103.4079\]. D. Arnaudov, R. C. Rashkov, and T. Vetsov, “Three- and four-point correlators of operators dual to folded string solutions in $\axs$,” Int. J. Mod. Phys. A [**26**]{}, 3403 (2011) \[arXiv:1103.6145\]. R. Hernández, “Three-point correlators for giant magnons,” JHEP [**1105**]{}, 123 (2011) \[arXiv:1104.1160\]. X. Bai, B. Lee, and C. Park, “Correlation function of dyonic strings,” Phys. Rev. D [**84**]{}, 026009 (2011) \[arXiv:1104.1896\]. L. F. Alday and A. A. Tseytlin, “On strong-coupling correlation functions of circular Wilson loops and local operators,” J. Phys. A [**44**]{}, 395401 (2011) \[arXiv:1105.1537\]. C. Ahn and P. Bozhilov, “Three-point Correlation functions of Giant magnons with finite size,” Phys. Lett. B [**702**]{}, 286 (2011) \[arXiv:1105.3084\]. B. Lee and C. Park, “Finite size effect on the magnon’s correlation functions,” Phys. Rev. D [**84**]{}, 086005 (2011) \[arXiv:1105.3279\]. S. Förste, “Strings, Branes and Extra Dimensions,” Fortschr. Phys. [**50**]{}, 221 (2002) \[arXiv:hep-th/0110055\]. D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Correlation functions in the CFT(d)/AdS(d+1) correspondence,” Nucl. Phys. B [**546**]{}, 96 (1999) \[arXiv:hep-th/9804058\]. S. Frolov and A. A. Tseytlin, “Multi-spin string solutions in AdS(5) x S\*\*5,” Nucl. Phys. B [**668**]{}, 77 (2003) \[arXiv:hep-th/0304255\]. $\bullet$ S. Frolov and A. A. Tseytlin, “Semiclassical quantization of rotating superstring in AdS(5) x S(5),” JHEP [**0206**]{}, 007 (2002) \[arXiv:hep-th/0204226\]. D. M. Hofman and J. M. Maldacena, “Giant magnons,” J. Phys. A [**39**]{}, 13095 (2006) \[arXiv:hep-th/0604135\]. T. Klose and T. McLoughlin, “A light-cone approach to three-point functions in $AdS_5\times S^5$,” \[arXiv:1106.0495\]. J. Escobedo, N. Gromov, A. Sever, and P. Vieira, “Tailoring Three-Point Functions and Integrability,” JHEP [**1109**]{}, 028 (2011) \[arXiv:1012.2475\]. J. Escobedo, N. Gromov, A. Sever, and P. Vieira, “Tailoring Three-Point Functions and Integrability II. Weak/strong coupling match,” JHEP [**1109**]{}, 029 (2011) \[arXiv:1104.5501\].
[^1]: e-mail: rash@hep.itp.tuwien.ac.at.
[^2]: The logarithmic divergent terms of fermionic quadratic corrections are known to cancel exactly the bosonic ones so that conformal invariance is preserved. For this reason, we concentrate only on bosonic fluctuations.
[^3]: We work in conformal gauge.
[^4]: We note, however, that in the cases of other “light” operators a different approach should be used, probably involving vertex operators.
|
---
author:
- 'Rebecca E. Field'
date: 'March 19, 2001'
title: 'The Chow ring of the symmetric space $Gl(2n,{\Bbb C})/SO(2n,{\Bbb C})$'
---
> ABSTRACT. We show this Chow ring is ${\Bbb Z}\oplus {\Bbb Z}$. We do this by partitioning the space into $2n$ subvarieties each of which is fibered over $Gl(2n-2,{\Bbb C})/SO(2n-2,{\Bbb C})$.
Introduction
============
This paper is devoted to the following result.
$CH^*(Gl(2n)/SO(2n)) \cong {\Bbb Z}\oplus {\Bbb Z}y,$ where y is a codimension $n$ cycle.
Throughout this paper $Gl(n)$, $O(n)$, $SO(n)$, etc. denote the complex algebraic groups of these types.
Theorem 1 should be contrasted with the following result which will used in its proof.
$CH^0(Gl(2n)/O(2n)) \cong {\Bbb Z}$ and $CH^i(Gl(2n)/O(2n)) = 0$ for $i \geq 1$.
$Gl(2n)/O(2n) \cong Symm_{2n}({\Bbb C})$, where $Symm_{2n}({\Bbb C})$ is the space of symmetric, non-degenerate $2n \times 2n$ matrices over ${\Bbb C}$. $Symm_{2n}({\Bbb C})$ is an open subset of the vector space of all $2n \times 2n$ symmetric matrices over ${\Bbb C}$. By the fundamental exact sequence for Chow groups (Lemma 2 below), the Chow groups of any Zariski open subset of affine space vanish in codimensions higher than zero and are ${\Bbb Z}$ in codimension zero.
There is much work in the literature on the geometry of symmetric spaces of the form $G/K$ for $G$ an adjoint group and $K$ the fixed point set of an involution of $G$, see for example [@DeCP]. However, these results do not include a computation of the Chow ring. Moreover, the Chow ring of such a $G/K$ does not determine the Chow ring of the symmetric space $G/K^0$ where $K^0$ is the connected component of the identity in $K$. This can be significantly more complicated as Theorem 1 and Lemma 1 show.
In fact, there is currently no general theorem that computes the Chow ring of a reductive symmetric space $G/K$ in terms of Lie theoretic data. The most general result I know of is when $G/K$ is a group; the Chow ring $mod \hspace{.15cm} p$ is computed in Kac [@Kac]. For a history of similar problems and extensive references to previous results, see the survey article [@Mim].
In a subsequent paper, we will use Theorem 1 as the key step in the computation of the Chow ring of the classifying space $BSO(2n)$. The method of computing the Chow ring of $Gl(2n)/SO(2n)$ given in this paper can be extended to a computation of the cohomology of this symmetric space for all other cohomology theories.
[*Acknowledgements*]{}\
I would like to thank Brendan Hassett, J. Peter May, Burt Totaro, Madhav Nori, and especially Ian Grojnowski, without whom this paper would be impossible to read if it ever appeared at all.
Basic results
=============
For notation and conventions on Chow rings, we will refer to Fulton’s [*Intersection theory*]{} [@F] with one minor difference. The Chow ring that we will be using is the ring of algebraic cycles mod rational equivalence (denoted $CH^*(X)$) rather than the ring of cycles mod algebraic equivalence (denoted $A^*(X)$) as used by Fulton.
The basic result about Chow rings that we will need is the following.
Let $Y \subseteq X$ be a closed subvariety. Then
$$CH_*Y \to CH_*X \to CH_*(X-Y) \to 0$$ is exact.
Recall that $CH^iX=CH_{dimX-i}X.$
As in [@FMSS], given an action of an algebraic group $\Gamma$ on a variety $X$, one forms $CH^\Gamma_kX = Z^\Gamma_kX/R^\Gamma_kX$. Here $Z^\Gamma_kX$ is the free Abelian group generated by the $\Gamma$ stable closed subvarieties of $X$ and $R^\Gamma_k X$ is the subgroup generated by all divisors of eigenfunctions on $\Gamma$. A rational function $f$ on $X$ is an eigenfunction if $g\cdot f=\chi(g)f$ for all $g \in \Gamma$ and for some one dimensional character $\chi$ of $\Gamma$.
(Fulton, MacPherson, Sotille, Sturmfels)[@FMSS] If a connected solvable linear algebraic group $\Gamma$ acts on a scheme $X$, then the canonical homorphism $A^\Gamma_kX \rightarrow A_kX$ is an isomorphism.
These results allow us to restrict attention to $B$-stable cycles in our proof of Theorem 1. In fact the decomposition of $Gl(2n)/SO(2n)$ into $B$-orbits is not used essentially in the proof of the theorem, but it does provide a convenient notation for cycles.
We now recall some results about $Gl(2n)/SO(2n)$ and its partition into $B$-orbits. All of these statements are well known and their proofs are easy linear algebra. We let $Symm_k({\Bbb C})$ denote the subset of $Gl(k)$ consisting of symmetric matrices, and define the map $$f:Gl(k) \rightarrow Symm_k({\Bbb C})$$ by $f(g)= gg^t$. This map induces an isomorphism $$Gl(k)/O(k) \cong Symm_k({\Bbb C}),$$ and allows us to make the identification $$Gl(k)/SO(k) \cong
\{(q,\epsilon ) | q \in Gl(k)/O(k) \mbox { and } \epsilon^2 = det(q)\}.$$
With this identification, the obvious double cover $$\pi : Gl(2n)/SO(2n) \to Gl(2n)/O(2n)$$ takes a pair $(q, \epsilon)$ to $q.$ We will identify $Symm_k({\Bbb C})$ with the space of symmetric non-degenerate bilinear forms on ${\Bbb C}^{2n}$ by setting $q_A(v,w)=v^tAw$, so that $Gl(2n)$ acts on bilinear forms by $(gq)(v,w)=q(g^tv,g^tw)$.
Define the Borel subgroup $B$ to be the subgroup of upper triangular matrices, i.e. the stabilizer of the standard flag $F_1 \subset F_2 \subset \cdots \subset F_{2n},$ where $F_i=$$<e_1, e_2, \ldots , e_i>$. We have an isomorphism between $B$-orbits in $Gl(2n)/O(2n)$ and involutions in the Weyl group $S_{2n}$: $$B\backslash Gl(2n)/O(2n) \cong \{ \omega \in S_{2n} \mid \omega^2=1\},$$ defined by sending a quadratic form $q$ to the relative position between the standard flag $F$ and the flag $F^{\perp_q}$ of orthogonal complements with respect to $q$. It is well known and easy to see that this map is an isomorphism (this is a consequence of Gram-Schmidt orthonormalization).
To describe the $B$-orbits on $Gl(2n)/SO(2n)$, it is enough to describe the pullback of the orbits on $Gl(2n)/O(2n)$ through the double cover $\pi$. An inverse image $\pi^{-1}(O)$ is either a single orbit or a union of two disjoint orbits in the double cover $Gl(2n)/SO(2n)$.
The pullback of a $B$-orbit is the union of two disjoint orbits if and only if the permutation indexing the orbit is fixed point free.
Let $O$ be an orbit in $Gl(2n)/SO(2n)$ and $q \in O$. Then $\pi^{-1}(O)$ is a single $B$-orbit if and only if the stabilizer of $q$ in $B$ is disconnected; i.e. if and only if the stabilizer of $q$ in the set of diagonal matrices $T$ is disconnected. Choose $q$ such that $q(x,y)=x^t\omega y$ where $\omega \in S_{2n}$ is the permutation indexing $O$. Then $q(e_i,e_j)=\delta_{\omega(i),j}$. The component group of the stabilizer of $q$ in $T$ is $({\Bbb Z}/2{\Bbb Z})^l$, where $l$ is the number of fixed points of $\omega$ on $\{1,2,\ldots,2n\}$.
A $B$-orbit $O$ on $Gl(2n)/O(2n)$ is **fixed point free** if $\pi^{-1}(O)$ consists of two disjoint $B$-orbits.
The Chow ring of $Gl(2n)/SO(2n)$ is generated by elements corresponding to closures of fixed point free orbits in $Gl(2n)/O(2n)$ along with the codimension zero cycle.
The point of this paper is to prove that all but one of these codimension non-zero generators are rationally equivalent to zero.
As Lemma 1 states, the Chow ring of $Gl(2n)/O(2n)$ is trivial. Therefore, if $O$ is a $B$-orbit in $Gl(2n)/O(2n)$ whose codimension is larger than zero, then ${\pi}^{-1}(O)\sim 0$ in $CH^*(Gl(2n)/SO(2n))$.
By Definition 1, any orbit in $Gl(2n)/O(2n)$ which is not fixed point free lifts to a single orbit in $Gl(2n)/SO(2n)$ and is therefore rationally equivalent to zero.
On the other hand, if an orbit $O$ is fixed point free, then $\pi^{-1}(O)=O_+\amalg O_-$ where $O_+$ and $O_-$ are two disjoint copies of $O$ distinguished by the sign of $\epsilon$. Again, since the Chow ring of the base space is trivial, $0 \sim \pi^{-1}(O)\sim O_+ + O_-$, so $O_+ \sim -O_-$. Therefore, $CH^*(Gl(2n)/SO(2n))$ is generated by elements corresponding to closures of fixed point free orbits in $Gl(2n)/O(2n)$, along with the codimension zero cycle.
Proof of Theorem 1
==================
Define the graph of $B$-orbits to be a graph whose vertices are $B$-orbits and whose edges are codimension $1$ inclusion relations between orbit closures. This graph of $B$-orbits contains the full subgraph of fixed point free $B$-orbits. The proof of Theorem 1 was arrived at through a careful examination of the fixed point free graphs of $Gl(2n)/O(2n),$ for $n \leq 4$. These examples are reproduced in the appendix to make the proof easier to visualize.
The proof is by induction. We start by decomposing $Gl(2n)/SO(2n)$ into $2n$ disjoint subvarieties, each of which may be compared with $Gl(2n-2)/SO(2n-2)$. Our induction hypothesis will be $CH^*(Gl(2n-2)/SO(2n-2)) \cong {\Bbb Z}x_0 \oplus {\Bbb Z}y$, where $x_0$ is the codimension zero cycle and $y$ is in codimension $n-1$. We will then construct a map from all but one of the $2n$ disjoint subvarieties to $Gl(2n-2)/SO(2n-2)$. This map will be a trivial fibration. The remaining subvariety is easily dealt with using Proposition 1. This map will take fixed point free orbits of $Gl(2n)/SO(2n)$ to fixed point free orbits of $Gl(2n-2)/SO(2n-2)$, hence the relevance of the examples in the appendix. We will use this map and the induction step to show that each of the $2n$ subsets contributes at most ${\Bbb Z}\oplus {\Bbb Z}$ to $CH^*(Gl(2n)/SO(2n))$. From there, a second and third induction and the results of the previous section will show that only two of these disjoint subsets actually contribute to the Chow ring.
We will start the induction with the case $n=1$. Since there is only one fixed point free permutation in two letters, namely $(1 \hspace{.2cm} 2)$, the Chow ring $CH^*(Gl(2)/SO(2))$ has a single generator in addition to the trivial codimension zero cycle. Since this single fixed point free orbit is the largest fixed point free orbit, it cannot be the zero of any $B$-semi-invariant function. Therefore, $CH^*(Gl(2)/SO(2)) \cong {\Bbb Z}x_0 \oplus {\Bbb Z}y$, where $x_0$ is the codimension zero cycle and $y$ is this codimension $1$ cycle.
We now define a decomposition of $Gl(2n)/SO(2n)$ into $2n$ disjoint subspaces.
Define $X_i$ to be the subvariety of $Gl(2n)/SO(2n)$ consisting of pairs $(q,\epsilon)$ where $q(e_{2n},e_{2n})=q(e_{2n-1},e_{2n})=\cdots=q(e_{i+1},e_{2n})=0$ and $q(e_i,e_{2n})\neq 0$. The following properties of the varieties $X_i$ are immediate.
$$Gl(2n)/SO(2n)=X_1 \amalg X_2 \amalg \cdots
\amalg X_{2n}.$$ Also, $$\overline{X_i}=\coprod_{j\geq i}X_j.$$ The $X_i$ and the $\overline{X_i}$ are all quasi-projective varieties, and each $X_i$ is $B$-stable.
Notice that if $(q, \epsilon) \in X_{2n}$, then the $B$-orbit through $q$ is not fixed point free, and hence by Proposition 1, the only $B$-orbit in $X_{2n}$ that contributes to the Chow ring is the codimension zero open orbit.
For $i<2n$, we define a map $$f_i:X_i \longrightarrow Gl(2n-2)/SO(2n-2)$$ as follows. For $(q,\epsilon)\in X_i$, we know that $q(e_{i+1},e_{2n})=q(e_{i+2},e_{2n})=\cdots=q(e_{2n},e_{2n})=0$, while $q(e_i,e_{2n})\neq 0$. Therefore, the quadratic form $q$ is nondegenerate on $<e_i,e_{2n}>$, the subspace of ${\Bbb C}^{2n}$ generated by $e_i$ and $e_{2n}$. The orthogonal compliment $<e_i,e_{2n}>^\perp$ with respect to $q$ of this subspace is isomorphic to ${\Bbb C}^{2n-2}$.
Let $q'$ be the quadratic form on ${\Bbb C}^{2n-2}$ defined by restricting $q$ to $<e_i,e_{2n}>^\perp$ and let $\epsilon'=\epsilon/q(e_i,e_{2n})\sqrt{-1}$. As $det(q)=-q(e_i,e_{2n})^2det(q')$, we know $(\epsilon')^2=det(q').$ Define $f_i(q,\epsilon)=(q',\epsilon').$
This map takes $B$-orbits to $\overline{B}$-orbits, where $\overline{B}$ is the stabilizer in $Gl(2n-2)$ of the standard flag associated to the basis $$\overline{e_1},\ldots,\overline{e_{i-1}},\overline{e_{i+1}},\ldots,\overline{e_{2n-1}},$$ where $\overline{e_j}$ is equal to the orthogonal projection of $e_j$ onto $<e_i,e_{2n}>^\perp$.
$f_i$ is a trivial fibration with fibers isomorphic to ${\Bbb C}^*\times {\Bbb C}^{2n+i-2}.$
Fix a quadratic form $q'$ in $Gl(2n-2)/SO(2n-2)$. An element $q \in f^{-1}(q')$ is determined by the numbers $${q(e_j,e_{2n})\mbox{ for }j<i,
\mbox{ and } \atop q(e_j,e_i)\mbox{ for }j\neq i.}$$ These numbers may be chosen freely subject only to the constraint that $q(e_i,e_{2n})\neq 0.$
$CH^*(X_i)$ is a quotient of ${\Bbb Z}\oplus {\Bbb Z}$.
Since the map $f_i$ is a fibration with fibers isomorphic to an open subset of affine space, it induces a surjection of Chow rings ([@F] example 1.9.2). By our induction hypothesis $CH^*(Gl(2n-2)/SO(2n-2))\cong {\Bbb Z}\oplus {\Bbb Z}$.
More specifically, the Chow ring of $Gl(2n-2)/SO(2n-2)$ is generated by cycles $x_0$ and $y$, where $x_0$ is the codimension zero cycle and $y$ is in codimension $n-1$. Therefore, $CH^*(X_i)$ is generated by the pullbacks of these cycles.
In fact $CH^*(X_i)\cong {\Bbb Z}\oplus {\Bbb Z}$ as follows from the proof of Theorem 1 below. This can also easily be seen directly.
If $n>1$, then $CH^*(Gl(2n)/SO(2n))=CH^*(\overline{X_{2n}})$ is a quotient of ${\Bbb Z}\widetilde{x_0} \oplus {\Bbb Z}f_{2n-1}^*(y) \oplus {\Bbb Z}f_{2n-2}^*(y)
\oplus \cdots \oplus {\Bbb Z}f_1^*(y)$, where $\widetilde{x_0}$ is the codimension zero cycle in $\overline{X_{2n}}$.
We will show by induction on $i$ that $CH^*(\overline{X_i})$ is a quotient of ${\Bbb Z}f_{i}^*(y) \oplus {\Bbb Z}f_{i-1}^*(y)
\oplus \cdots \oplus {\Bbb Z}f_1^*(y).$
For the case $i=1$, since $\overline{X_1}=X_1$, Lemma 6 tells us that $CH^*(\overline{X_1})$ is a quotient of ${\Bbb Z}f^*_1(x_0) \oplus {\Bbb Z}f^*_1(y)$. The pullback of the codimension zero orbit $f^*_1(x_0)$ is indexed by the permutation $(1 \hspace{.2cm} 2n)$ so since $n>1$ it is not a fixed point free orbit in $Gl(2n)/SO(2n)$. Therefore, by Proposition 1, $f_1^*(x_0)$ is rationally equivalent to zero in $CH^*(Gl(2n)/SO(2n))$, and the subvariety $\overline{X_1}$ contributes at most $f^*_1(y)$ to the Chow ring of the whole symmetric space. This completes the $i=1$ case.
As stated in Lemma 2, we know that $$CH^*(\overline{X_{i-1}}) \to CH^*(\overline{X_i}) \to
CH^*(X_i) \to 0$$ is exact. Therefore, by induction, $CH^*(\overline{X_i})$ is a quotient of ${\Bbb Z}f_i^*(x_0) \oplus {\Bbb Z}f_i^*(y) \oplus {\Bbb Z}f_{i-1}^*(y) \oplus \cdots
\oplus {\Bbb Z}f_1^*(y)$. Again, we note that $f_i^*(x_0)$ is indexed by the permutation $(i \hspace{.2cm} 2n)$ so is not fixed point free. For $i<2n$, this orbit has codimension larger than zero, so by Proposition 1, this orbit is rationally equivalent to zero in $CH^*(Gl(2n)/SO(2n))$. Finally, if $i=2n$, as we noticed previously (immediately following Lemma 4), the subvariety $X_{2n}$ contributes only the codimension zero cycle $\widetilde{x_0}$.
Note that we have reduced the generators down to the largest fixed point free $B$-orbit in each of the $X_i$. These $2n-1$ $B$-orbits are visible in the appendix along the bottom right hand side of each graph.
We claim that $f_j^*(y)$ is rationally equivalent to zero in $CH^*(Gl(2n)/SO(2n))$ for $j<2n$. To this end, we inductively show that $CH^*(\overline{X_j})$ is a quotient of ${\Bbb Z}f_j^*(y)$ for $j < 2n$.
Define the function $g_j:\overline{X_j} \longrightarrow {\Bbb C}$ by $g_j(q)=q(e_j,e_{2n})$. This function is clearly non-zero on $X_j$ and is clearly zero on all of $\overline{X_{j-1}}$. To see that this zero is a simple one, we look at the function restricted to a line in $Gl(2n)/SO(2n)$. Let $q \in X_j, q'\in X_{j-1}$ and consider the line of quadratic forms $\{q'+aq\mid a \in {\Bbb C}\}$. Clearly the function $g_j$ has a simple zero along this line. This line meets $X_{j-1}$ only when $a=0$ and is otherwise completely contained in $X_j$. It is clear that the line intersects $\overline{X_{j-1}}$ transversally since $\overline{X_{j-1}}$ is an open subvariety of the vector space of quadratic forms for which $q(e_j,e_{2n})=0$. Therefore, the intersection is transversal and the zero is simple.
By our induction step, $CH^*(\overline{X_{j-1}})$ is a quotient of ${\Bbb Z}f_{j-1}^*(y)$, so $CH^*(\overline{X_j})$ is a quotient of ${\Bbb Z}f_{j-1}^*(y)\oplus {\Bbb Z}f_{j}^*(y)$. Let $\gamma$ be the $B$-orbit in $Gl(2n-2)/SO(2n-2)$ representing the cycle $y$. Restricting $g_j$ to the closure of $f_j^{-1}(\gamma)$ in $\overline{X_j}$, this argument shows that $f_{j-1}^*(y) \sim 0$ in $CH^*(\overline{X_j})$.
We have shown $f_j^*(y)\sim 0$ for $j<2n-1$. Consider $f_{2n-1}^*(y)$. This cycle is the closure of the largest fixed point free orbit, so any $B$-semi-invariant function vanishing along it must be defined on the closure of a non-fixed point free orbit. Therefore, any such function will produce a relation involving both copies of the orbit pulled back from $Gl(2n)/O(2n)$. Since we already know that the sum of these copies is rationally equivalent to zero, such a function can produce no new relations. The same argument shows that no multiple of $f_{2n-1}^*(y)$ is rationally equivalent to zero.
We have shown that $f_{j}^*(y) \sim 0$ for $1<j<2n-1$ and that no multiple of $f_{2n-1}^*(y)$ is rationally equivalent to zero. This combined with Proposition 2 proves Theorem 1.
Appendix
========
As mentioned in the proof, the graph of fixed point free $B$-orbits for $Gl(2)/SO(2)$ is a single point corresponding to the orbit $O_{(12)}$.
The graph of fixed point free $B$-orbits for $Gl(4)/SO(4)$ is: $$\spreaddiagramrows{-.3pc}
\xymatrix{
O_{(14)(23)} \ar[d]\\
O_{(13)(24)} \ar[d]\\
O_{(12)(34)}
}$$
where the codimensions of the orbits are (counting from the bottom) 2, 3, and 4 and the arrows represent inclusion of an orbit in the closure of the larger orbit. Since the graph for $Gl(2)/SO(2)$ is a single point, this is also its decomposition into subgraphs.
This leads to the induction step for the graph of fixed point free $B$-orbits for $Gl(6)/SO(6)$, namely: $$\spreaddiagramcolumns{-1pc}
\spreaddiagramrows{-.3pc}
\xymatrix{
&&O_{(16)(25)(34)} \morphism\dashed\tip\notip[1,-1] \ar[dr]&&\\
&O_{(15)(26)(34)}\morphism\dashed\tip\notip[1,-1]\ar[dr]&&O_{(16)(24)(35)}\morphism\dashed\tip\notip[1,-3]\morphism\dashed\tip\notip[1,-1]\ar[dr]&\\
O_{(15)(24)(36)}\morphism\dashed\tip\notip[1,0]\ar[drr]&&O_{(14)(26)(35)}\morphism\dashed\tip\notip[1,0]\ar[drr]&&O_{(16)(23)(45)}\morphism\dashed\tip\notip[1,-4]\ar[d]\\
O_{(15)(23)(46)}\morphism\dashed\tip\notip[1,0]\ar[drr]&&O_{(14)(25)(36)}\morphism\dashed\tip\notip[1,0]\morphism\dashed\tip\notip[1,-2]\ar[drr]&&
O_{(13)(26)(45)}\morphism\dashed\tip\notip[1,-2]\ar[d]\\
O_{(14)(23)(56)}\ar[dr]&&O_{(13)(25)(46)}\morphism\dashed\tip\notip[1,-1]\ar[dr]&&O_{(12)(36)(45)}\ar[dl]\\
&O_{(13)(24)(56)}\ar[dr]&&O_{(12)(35)(46)}\ar[dl]&\\
&&O_{(12)(34)(56)}&&
}
\vspace{-.1cm}$$
where the solid diagonal lines that are more or less parallel to each other represent inclusion relations within the $X_i$ and the dotted lines represent other inclusion relations that are not relevant to our proof Furthermore, orbits that appear on the same row have the same codimension.
The following is the graph of fixed point free $B$-orbits in $Gl(8)/O(8)$. Notice its seven subgraphs.
[FM2]{} C.DeConcini and C.Procesi. Complete symmetric varieties.II. Intersection theory. , 481–513. Advanced Studies in Pure Mathematics, 6, North-Holland, Amsterdam, 1985.
W.Fulton, R.MacPherson, F.Sottile, and B.Sturmfels. Intersection Theory on Spherical Varieties. , 4:181–193, 1995.
W.Fulton. . second edition, Springer-Verlag, 1998.
V.G.Kac. Torsion in cohomology of compact Lie groups and Chow rings of algebraic groups. , 80:69–79, 1985.
M.Mimura. Homotopy theory of Lie groups. , 951–991, [*North-Holland, Amsterdam*]{}, 1995.
[ DEPARTMENT OF MATHEMATICS, UNIVERSITY OF WISCONSIN, 480 LINCOLN DRIVE, MADISON, WI 53706-1388 USA]{}
[ `field@math.wisc.edu`]{}
|
---
abstract: 'The local complementation rule is applied for continuous-variable (CV) graph states in the paper, which is an elementary graph transformation rule and successive application of which generates the orbit of any graph states. The corresponding local Gaussian transformations of local complementation for four-mode unweighted graph states were found, which do not mirror the form of the local Clifford unitary of qubit exactly. This work is an important step to characterize the local Gaussian equivalence classes of CV graph states.'
author:
- 'Jing Zhang$^{\dagger }$'
title: 'Local complementation rule for continuous-variable four-mode unweighted graph states'
---
Entanglement lies at the heart of quantum mechanics and plays a crucial role in quantum information processing. Recently, special types of multipartite entangled states, the so-called the graph states [@one; @two], have moved into the center of interest. A graph quantum state is described by a mathematical graph, i.e. a set of vertices connected by edges. A vertex represents a physical system, e. g. a qubit (2-dimensional Hilbert space), qudit (d-dimensional Hilbert space), or CV (continuous Hilbert space). An edge between two vertices represents the physical interaction between the corresponding systems. An interesting feature is that many entanglement properties of graph states are closely related to their underlying graphs. They not only provide an efficient model to study multiparticle entanglement [@one], but also find applications in quantum error correction [@three; @four], multi-party quantum communication [@five] and most prominently, serve as the initial resource in one-way quantum computation [@six]. Considerable efforts have been stepped toward generating and characterizing cluster state with linear optics experimentally [@seven; @eight; @nine; @ten]. The principle feasibility of one-way quantum computing model has been experimentally demonstrated through photon cluster state successfully [@seven; @ten].
Most of the concepts of quantum information and computation have been initially developed for discrete quantum variables, in particular two-level or spin-$\frac{1}{2}$ quantum variables (qubits). In parallel, quantum variables with a continuous spectrum, such as the position and momentum of a particle or amplitude and phase quadrature of an electromagnetic field, in informational or computational processes have attracted a lot of interest and appears to yield very promising perspectives concerning both experimental realizations and general theoretical insights [@eleven; @twelve], due to relative simplicity and high efficiency in the generation, manipulation, and detection of CV state. Although up to six-qubit single-photon cluster states have been created via postselection using nonlinear and linear optics, the deterministic, unconditional realization of optical cluster states would be based on continuous variables. CV cluster and graph states have been proposed [@thirteen], which can be generated by squeezed state and linear optics [@thirteen1; @thirteen2], and demonstrated experimentally for four-mode cluster state [@forteen]. The one-way CV quantum computation was also proposed with CV cluster state [@fifteen]. Moreover, the protocol of CV anyonic statistics implemented with CV graph states is proposed [@fifteen1].
One of the interesting issues on entanglement is how to define the equivalence of two entangled states. The transformations of qubit graph states under local Clifford operations were studied by Hein [@one] and Van den Nest [@sixteen]. They translate the action of local Clifford operations on qubit graph states into transformations on their associated graphs, that is, to derive transformations rules called the local complement rule, stated in purely graph theoretical terms, which completely characterize the evolution of graph states under local Clifford operations. The corresponding local Clifford unitary is a single and simple form. The successive application of this rule suffices to generate the complete orbit of any qubit graph state under local Clifford operations. In this paper, the local complement rule for CV four-mode unweighted graph state is applied and the corresponding local Clifford transformations (also called local Gaussian transformation for CV) for four-mode graph state were found. The local Gaussian equivalence classes of CV four-mode unweighted graph states can be obtained by this way. It was shown that the corresponding local Gaussian unitary can not exactly mirror that for qubit, which is not a single form compared with qubit. This result shows the complexity of CV quantum systems and stimulate the research on the local Gaussian equivalence of CV graph states. Although only focusing on the CV four-mode unweighted graph states, this work makes an important step in the direction of addressing the general question “What are the graph transformation rules that describe local unitary equivalence of any CV graph states?”.
The CV operations are reviewed firstly that follow the standard prescription given in Ref.[@seventeen]. The Pauli $X$ and $Z$ operators of qubit are generalized to the Weyl-Heisenberg group, which is the group of phase-space displacements. For CVs, this is a Lie group with generators $\hat{x}=(\hat{a}+\hat{a}^\dagger)/\sqrt{2}$ (quadrature-amplitude or position) and $\hat{p}=-i(\hat{a}-\hat{a}^\dagger)/\sqrt{2}$ (quadrature-phase or momentum) of the electromagnetic field as the CV system. These operators satisfy the canonical commutation relation $[\hat{x},\hat{p}]=i$ (with $\hbar=1$). In analogy to the qubit Pauli operators, the single mode Pauli operators are defined as $X(s)=exp[-is\hat{p}]$ and $Z(t)=exp[it\hat{x}]$ with $s,t\in
\mathbb{R}$. The Pauli operator $X(s)$ is a position-translation operator, which acts on the computational basis of position eigenstates $\{|q\rangle; q\in \mathbb{R}\}$ as $X(s)|q\rangle=|q+s\rangle$, whereas $Z$ is a momentum-translation operator, which acts on the momentum eigenstates as $Z(t)|p\rangle=|p+t\rangle$. These operators are non-commutative and obey the identity $ X(s)Z(t)=e^{-ist}Z(t)X(s)$. The Pauli operators for one mode can be used to construct a set of Pauli operators $\{X_{i}(s_{i}),Z_{i}(t_{i}); i=1,...,n\}$ for n-mode systems. This set generates the Pauli group $\mathcal{C}_{1} $. The clifford group $\mathcal{C}_{2} $ is the normalizer of the Pauli group, whose transformations acting by conjugating, preserve the Pauli group $\mathcal{C}_{1} $; i.e., a gate $\emph{U}$ is in the Clifford group if $\emph{UR}\emph{U}^{-1}\in\mathcal{C}_{1}$ for every $\emph{R}\in\mathcal{C}_{1}$. The clifford group $\mathcal{C}_{2} $ for CV is shown [@seventeen] to be the (semidirect) product of the Pauli group and linear symplectic group of all one-mode and two-mode squeezing transformations. Transformation between the position and momentum basis is given by the Fourier transform operator $F=exp[i(\pi/4)(\hat{x}^{2}+\hat{p}^{2})]$, with $F|q\rangle_{x}=|q\rangle_{p}$. The action $FRF^{-1}$ of the Fourier transform on the Pauli operators is $$\begin{aligned}
F:X(s)&\rightarrow& Z(s)
,\nonumber \\
Z(t)&\rightarrow& X(-t). \label{Four}\end{aligned}$$ This is the generalization of the Hadamard gate for qubits. The phase gate $ P(\eta)=exp[i(\eta/2)\hat{x}^{2}]$ with $\eta\in
\mathbb{R}$ is a squeezing operation for CV and the action on the Pauli operators is $$\begin{aligned}
P(\eta):X(s)&\rightarrow& e^{-is^{2}\eta/2}Z(s\eta)X(s)
,\nonumber \\
Z(t)&\rightarrow& Z(t), \label{phase}\end{aligned}$$ in analogy to the phase gate of qubit [@eighteen]. The controlled operation C-Z is generalized to controlled-$ Z (C_{Z})$. This gate $C_{Z}=exp[i\hat{x}_{1}\bigotimes\hat{x}_{2}]$ provides the basic interaction for two mode 1 and 2, and describes the quantum nondemolition (QND) interaction. This set $ \{X(s), F,
P(\eta), C-Z; s,\eta \in \mathbb{R}\}$ generates the Clifford group. Transformations in the Clifford group do not form a universal set of gates for CV quantum computation. However, Clifford group transformation (Gaussian transformations) together with any higher-order nonlinear transformation (non-Gaussian transformation) acting on a single-mode form a universal set of gates [@seventeen]. The local Gaussian group only was concerned here, which can be obtained by repeated application of Fourier and phase gates. In the following, another type of the phase gate will be used $ P_{X}(\eta)=FP(\eta)F^{-1}=exp[i(\eta/2)\hat{p}^{2}]$ and the action on the Pauli operators is $$\begin{aligned}
P_{X}(\eta):X(s)&\rightarrow& X(s)
,\nonumber \\
Z(t)&\rightarrow& e^{-it^{2}\eta/2}X(-t\eta)Z(t), \label{phase1}\end{aligned}$$ where $P_{X}(\eta)^{\dagger}=P_{X}(\eta)^{-1}=P_{X}(-\eta)$.
A graph quantum state is described by a mathematical graph $G=(V,E)$, i.e. a finite set of $n$ vertices $V$ connected by a set of edges $E$ [@ninteen]. An $\left\{ a,c\right\} $-path is a order list of vertices $%
a=a_1,a_2,\ldots ,a_{n-1},a_n=c$, such that for all $i$, $a_i$ and $a_{i+1}$ are adjacent. A connected graph is a graph that has an $\left\{ a,c\right\} $-path for any two $a,c\in V$. Otherwise it is referred to as disconnected. The neighborhood $N_a\subset V$ is defined as the set of vertices $b$ for which $\left\{ a,b\right\} \in E$. When a vertex a is deleted in a graph G, together with all edges incident with a, one obtains a new graph, denoted by $G-a$. For a subset of vertices $U\subset V$ of a graph $G=(V,E)$ let us denote with $G-U$ the graph that is obtained from $G$ by deleting the set $U$ of vertices and all edges which are incident with an element of $U$. Similarly, an subgraph $G[C]$ of a graph $G=(V,E)$, where $C\subset V$, is obtained by deleting all vertices and the incident edges that are not contained in $C$. The preparation procedure of CV graph states [@thirteen] can exactly mirror that for qubit graph states only using the Clifford operations: first, prepare each mode (or graph vertex) in a phase-squeezed state, approximating a zero-phase eigenstate (analog of Pauli-X eigenstates), then, apply a QND interaction (C-Z gate) to each pair of modes $(j,k)$ linked by an edge in the graph. All C-Z gates commute. Thus, the resulting CV graph state becomes, in the limit of infinite squeezing, $g_{a}=(\hat{p}_{a}-\sum_{b\in N_{a}}\hat{x}_{b})\rightarrow0$, where the modes $a\in V$ correspond to the vertices of the graph of $n$ modes, while the modes $b\in N_{a}$ are the nearest neighbors of mode $a$. This relation is as a simultaneous zero-eigenstate of the position-momentum linear combination operators. The stabilizers $G_{a}(\xi)=exp[-i \xi g_{a}]=X_{a}(\xi)\prod_{b\in
N_{a}}Z_{b}(\xi)$ with $\xi\in \mathbb{R}$ for CV graph states are analogous to $n$ independent stabilizers $G_{a}=X_{a}\prod_{b\in
N_{a}}Z_{b}$ for qubit graph states. Note that the CV graph states that is discussed here are unweighted since the QND interactions all have the same strength. For the CV weighted graph states generated by the different QND interaction strength, the stabilizers become $G_{a}(\xi)=X_{a}(\xi)\prod_{b\in N_{a}}Z_{b}(\Omega_{ab}\xi)$, where $\Omega_{ab}$ is the interaction strength between mode a and b. The CV weighted graph states are more complex, which is not considered in this paper.
![ The connected four-vertex graphs for an successive application of the local complementation. The rule is successively applied to the vertex, which is circle in the figure. \[Fig1\] ](fig1.eps){width="3in"}
The action of the local complement rule, can be described as: letting $G=(V,E)$ be a graph and $a\in V$ be a vertex, the local complement of $G$ for $a$, denoted by $\lambda_{a}(G)$, is obtained by complementing the subgraph of $G$ generated by the neighborhood $N_{a}$ of $a$ and leaving the rest of the graph unchanged. The successive application of this rule suffices to generate the complete orbit of any graph. Here, the corresponding local Gaussian unitary for CV four-mode graph state were examined. The corresponding four-mode graph state $|\lambda_{a}(G)\rangle$ by local complement of a graph $G$ at some vertex $a\in V$, is given by a local Gaussian unitary operation $$\begin{aligned}
|\lambda_{a}(G)\rangle=U_{\lambda_{a}}|G\rangle, \label{LC-graph}\end{aligned}$$ where $U_{\lambda_{a}}$ is local Gaussian operation. A form of the local Gaussian unitary comprising two types of phase gate is defined $$\begin{aligned}
U_{LG_{a}}=P_{Xa}(1)\prod_{b\in N_{a}}P_{b}(-1),\label{LC}\end{aligned}$$ which mirrors the form of qubit local Clifford operation for local complementation. Fig.1 depicts connected four-mode graphs by such a successive application of the local complement rule. The four independent stabilizers of the first graph state $|G^{(1)}\rangle$ are given by $$\begin{aligned}
G_{1}^{(1)}(\xi)&=&X_{1}(\xi)Z_{2}(\xi),\nonumber \\
G_{2}^{(1)}(\xi)&=&X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi),\nonumber
\\
G_{3}^{(1)}(\xi)&=&X_{3}(\xi)Z_{2}(\xi)Z_{4}(\xi),\nonumber
\\
G_{4}^{(1)}(\xi)&=&X_{4}(\xi)Z_{3}(\xi).\label{stab1}\end{aligned}$$ with $G_{i}^{(1)}(\xi)|G^{(1)}\rangle=|G^{(1)}\rangle$ in the limit of infinite squeezing, where $i=1,...,4$. Applying the local Gaussian unitary $U_{LG_{3}}$ to the vertex 3, I can compute the four independent stabilizers of the resulting graph state $|G^{(2)}\rangle$ by Eqs. (\[phase\],\[phase1\],\[LC-graph\],\[LC\]), for example calculating $G_{2}^{(2)}(\xi)$, $$\begin{aligned}
|G^{(2)}\rangle&=&|\lambda_{3}(G^{(1)})\rangle\nonumber \\
&=&U_{LG_{3}}G_{2}^{(1)}(\xi)|G^{(1)}\rangle\nonumber \\
&=&U_{LG_{3}}G_{2}^{(1)}(\xi)U_{LG_{3}}^{-1}U_{LG_{3}}|G^{(1)}\rangle \nonumber \\
&=&[e^{i\xi^{2}/2}Z_{2}(-\xi)X_{2}(\xi)]Z_{1}(\xi)\nonumber\\&&
[e^{-i\xi^{2}/2}X_{3}(-\xi)Z_{3}(\xi)]U_{LG_{3}}|G^{(1)}\rangle\nonumber \\
&=&X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)U_{LG_{3}}\nonumber\\&&
[Z_{2}(-\xi)X_{3}(-\xi)]|G^{(1)}\rangle\nonumber \\
&=&X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)U_{LG_{3}}\nonumber\\&&
[Z_{2}(-\xi)X_{3}(-\xi)]G_{3}^{(1)}(\xi)|G^{(1)}\rangle\nonumber \\
&=&X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)Z_{4}(\xi)|\lambda_{3}(G^{(1)})\rangle\nonumber \\
&=&G_{2}^{(2)}(\xi)|G^{(2)}\rangle\end{aligned}$$ to obtain $$\begin{aligned}
G_{1}^{(2)}(\xi)&=&X_{1}(\xi)Z_{2}(\xi),\nonumber \\
G_{2}^{(2)}(\xi)&=&X_{2}(\xi)Z_{1}(\xi)Z_{3}(\xi)Z_{4}(\xi),\nonumber
\\
G_{3}^{(2)}(\xi)&=&X_{3}(\xi)Z_{2}(\xi)Z_{4}(\xi),\nonumber
\\
G_{4}^{(2)}(\xi)&=&X_{4}(\xi)Z_{2}(\xi)Z_{3}(\xi),\end{aligned}$$ which exactly correspond to the stabilizers of No.2 graph state in Fig.1. The complete orbit of the first graph can be obtained by applying the local complement rule repeatedly to the vertices and the corresponding local Gaussian unitary is shown in the following forms: $No.1\stackrel{U_{LG_{3}}}{\longrightarrow
}No.2\stackrel{U_{LG_{3}}^{2}F^{2}_{1}U_{LG_{2}}^{\dagger}}{\longrightarrow
}No.3\stackrel{U_{LG_{3}}^{\dagger}}{\longrightarrow
}No.4\stackrel{U_{LG_{1}}}{\longrightarrow
}No.5\stackrel{U_{LG_{2}}^{2}F^{2}_{1}U_{LG_{3}}^{\dagger}}{\longrightarrow
}No.6\stackrel{U_{LG_{1}}^{\dagger}}{\longrightarrow
}No.7\stackrel{U_{LG_{3}}}{\longrightarrow
}No.8\stackrel{U_{LG_{4}}^{\dagger}}{\longrightarrow
}No.9\stackrel{U_{LG_{1}}}{\longrightarrow
}No.10\stackrel{U_{LG_{2}}^{\dagger}}{\longrightarrow }No.11$. Here the complete orbit means the local complement rule is applied on the graph until exhaust all possibilities.
![ The set of four-vertex graphs is equivalent to Fig.1 under local Gaussian transformation and graph isomorphisms. The graph No.7, which is repeated and placed in the dash-line box behind the No.10, is used for generating the graph No.11 directly. \[Fig2\] ](fig2.eps){width="3in"}
Notice the difference in the Gaussian operations of $2\rightarrow3$, and $5\rightarrow6$. In the qubit case, these would have been of identical form. This shows the added richness of CV graph states over their qubit counterparts. Note that Hein et al. [@one] classify the equivalence of the graph states by considering the local complementation and additional graph isomorphisms, which corresponds to the permutations of the vertices. Fig.2 shows another set of graphs, which are not equivalent to any graph in the equivalence class represented in Fig.1 only considering the local complementation. However, they belong to the same equivalence class when considering both, local Gaussian unitary and graph isomorphisms. The corresponding local Gaussian unitary in Fig.2 is shown in the following forms: $No.1\stackrel{U_{LG_{1}}}{\longrightarrow
}No.2\stackrel{U_{LG_{3}}^{2}F^{2}_{1}U_{LG_{2}}^{\dagger}}{\longrightarrow
}No.3\stackrel{U_{LG_{1}}^{\dagger}}{\longrightarrow
}No.4\stackrel{U_{LG_{2}}}{\longrightarrow
}No.5\stackrel{U_{LG_{2}}^{2}F^{2}_{3}U_{LG_{1}}^{\dagger}}{\longrightarrow
}No.6\stackrel{U_{LG_{4}}^{2}F^{2}_{2}U_{LG_{3}}^{\dagger}}{\longrightarrow
}No.7\stackrel{U_{LG_{4}}^{\dagger}}{\longrightarrow
}No.8\stackrel{U_{LG_{3}}}{\longrightarrow
}No.9\stackrel{U_{LG_{1}}^{\dagger}}{\longrightarrow }No.10;$ $No.7\stackrel{U_{LG_{2}}^{\dagger}}{\longrightarrow }No.11$.
![ The set of four-vertex graphs is not equivalent to Fig.1 and 2 under local Gaussian transformation and graph isomorphisms. The graph No.1, which is placed in the dash-line box, is used repeatedly by the local complementation. \[Fig3\] ](fig3.eps){width="3in"}
The set of graphs in Fig.3, usually called GHZ (Greenberger-Horne-Zeilinger) entangled states, is not equivalent with Fig.1 and 2 under local Gaussian transformation and graph isomorphisms. The local Gaussian unitary is applied to four-mode graph states in Fig.3, which is written above the arrows of the following diagram: $No.1\stackrel{U_{LG_{1}}^{\dagger}}{\longrightarrow
}No.2\stackrel{U_{LG_{1}}}{\longrightarrow
}No.1\stackrel{U_{LG_{2}}^{\dagger}}{\longrightarrow
}No.3\stackrel{U_{LG_{2}}}{\longrightarrow
}No.1\stackrel{U_{LG_{3}}^{\dagger}}{\longrightarrow
}No.4\stackrel{U_{LG_{3}}}{\longrightarrow
}No.1\stackrel{U_{LG_{4}}^{\dagger}}{\longrightarrow }No.5$. Fig. 4 lists the graphs with up to four vertices that are not equivalent under local Gaussian transformation and graph isomorphisms.
![ The connected graphs with up to four vertices are not equivalent under local Gaussian transformation and graph isomorphisms. \[Fig4\] ](fig4.eps){width="3in"}
In summary, the local complement rule was extended for CV graph states and the corresponding local Gaussian transformations of four-mode unweighted graph states were given. Thus the local Gaussian equivalence classes of CV four-mode unweighted graph states can be obtained. It was shown that the corresponding local Clifford unitary can not exactly mirror that for qubit and demonstrate the complexity of CV quantum systems. It is worth remarking that, whether the local complementation for any CV graph states can be implemented completely by the local Gaussian transformations and the general form of the corresponding local Gaussian unitary can be found, still need be further investigated. This work not only contribute to a deeper and more complete understanding of CV multipartite entanglement, but also stimulate the research on CV graph states theoretically and experimentally.
$^{\dagger} $Corresponding author’s email address: jzhang74@sxu.edu.cn, jzhang74@yahoo.com
**ACKNOWLEDGMENTS**
===================
J. Zhang thanks K. Peng and C. Xie for the helpful discussions. This research was supported in part by NSFC for Distinguished Young Scholars (Grant No. 10725416), National Basic Research Program of China (Grant No. 2006CB921101), NSFC (Grant No. 60678029), Program for the Top Young and Middle-aged Innovative Talents of Higher Learning Institutions of Shanxi and NSF of Shanxi Province (Grant No. 2006011003).
Reference
=========
[99]{} M. Hein, et al., Phys. Rev. A **69**, 062311 (2003).
H. J. Briegel and R. Raussendorf, Phys. Rev. Lett. **86**, 910 (2001).
D. Schlingemann, R. F. Werner, Phys. Rev. A **65**, 012308 (2002).
S. Yu, Q. Chen, and C. H. Oh, quant-ph:0709.1780.
R. Cleve, et al., Phys. Rev. Lett. **83**, 648 (1999).
R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. **86**, 5188 (2001); R. Raussendorf, et al., Phys. Rev. A **68**, 022312 (2003).
P. Walther et al., Nature (London) **434**, 169 (2005);
N. Kiesel et al., Phys. Rev. Lett. **95**, 210502 (2005);
C. Y. Lu et al., Nature Physics **3**, 91-95 (2007).
R. Prevedel et al., Nature (London) **445**, 65 (2007).
S. L. Braunstein and A. K. Pati, *Quantum Information with Continuous Variables* (Kluwer Academic, Dordrecht, 2003).
S. L. Braunstein, P. van Loock, Rev. Mod. Phys. **77**, 513 (2005).
J. Zhang, S. L. Braunstein, Phys. Rev. A **73**, 032318 (2006).
P. van Loock, C. Weedbrook and M. Gu, Phys. Rev. A **76**, 032321 (2007).
N. C. Menicucci et al.,Phys. Rev. A **76**, 010302 (2007).
X. Su, et al., Phys. Rev. Lett. **98**, 070502 (2007); M. Yukawa, et al., arXiv: 0804.0289
N. C. Menicucci et al., Phys. Rev. Lett. **97**, 110501 (2006); P. van Loock, J. Opt. Soc. Am. **B** **24**, 340 (2007).
J. Zhang, C. Xie, and K. Peng, quant-ph/0711.0820.
M. Van den Nest, J. Dehaene, and B. De Moor, Phys. Rev. A **69**, 022316 (2004).
S. D. Bartlett, et al., Phys. Rev. Lett. **88**, 097904 (2002).
D. Gottesman, et al., Phys. Rev. A **64**, 012310 (2001).
M. Hein, et al., quant-ph/0602096.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.