Djalil CHAFAÏ (شافعي جليل)

Djalil Chafai's articles on arXiv

[1]  http://arxiv.org/abs/2412.15969v2 [ html pdf ]

On cutoff via rigidity for high dimensional curved diffusions

Djalil Chafaï, Max Fathi

We consider overdamped Lanvegin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein-Uhlenbeck process as well as non Gaussian and non product extensions with convex interaction, such as the Dyson process from random matrix theory. We show that a cutoff phenomenon or abrupt convergence to equilibrium occurs in high-dimension, at a critical time equal to the logarithm of the dimension divided by twice the spectral gap. This cutoff holds for Wasserstein distance, total variation, relative entropy, and Fisher information. A key observation is a relation to a spectral rigidity, linked to the presence of a Gaussian factor. A novelty is the extensive usage of functional inequalities, even for short-time regularization, and the reduction to Wasserstein. The proofs are short and conceptual. Since the product condition is satisfied, an Lp cutoff hold for all p. We moreover discuss a natural extension to Riemannian manifolds, a link with logarithmic gradient estimates in short-time for the heat kernel, and ask about stability by perturbation. Finally, beyond rigidity but still for diffusions, a discussion around the recent progresses on the product condition for nonnegatively curved diffusions leads us to introduce a new curvature product condition.

[2]  http://arxiv.org/abs/2503.11770v1 [ html pdf ]

On the cutoff phenomenon for fast diffusion and porous medium equations

Djalil Chafaï, Max Fathi, Nikita Simonov

The cutoff phenomenon, conceptualized at the origin for finite Markov chains, states that for a parametric family of evolution equations, started from a point, the distance towards a long time equilibrium may become more and more abrupt for certain choices of initial conditions, when the parameter tends to infinity. This threshold phenomenon can be seen as a critical competition between trend to equilibrium and worst initial condition. In this note, we investigate this phenomenon beyond stochastic processes, in the context of the analysis of nonlinear partial differential equations, by proving cutoff for the fast diffusion and porous medium Fokker-Planck equations on the Euclidean space, when the dimension tends to infinity. We formulate the phenomenon using quadratic Wasserstein distance, as well as using specific relative entropy and Fisher information. Our high dimensional asymptotic analysis uses the exact solvability of the model involving Barenblatt profiles. It includes the Ornstein-Uhlenbeck dynamics as a special linear case.

[3]  http://arxiv.org/abs/2405.00120v1 [ html pdf ]

Riesz Energy with a Radial External Field: When is the Equilibrium Support a Sphere?

Djalil Chafaï, Ryan W. Matzke, Edward B. Saff, Minh Quan H. Vu, Robert S. Womersley

We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.

[4]  http://arxiv.org/abs/2212.06090v2 [ html pdf ]

Monotonicity of the logarithmic energy for random matrices

Djalil Chafaï, Benjamin Dadoun, Pierre Youssef

Journal ref: RMTA volume 13, Issue 02, Article 2450008, (2024)

DOI: 10.1142/S2010326324500084

It is well-known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko--Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas of the logarithmic energy of the mentioned models which can be of independent interest.

[5]  http://arxiv.org/abs/2206.04956v3 [ html pdf ]

Threshold condensation to singular support for a Riesz equilibrium problem

Djalil Chafaï, Edward B. Saff, Robert S. Womersley

Journal ref: Anal. Math. Phys. 13, No. 1, Paper No. 19, 30 p. (2023)

DOI: 10.1007/s13324-023-00779-w

We compute the equilibrium measure in dimension d=s+4 associated to a Riesz s-kernel interaction with an external field given by a power of the Euclidean norm. Our study reveals that the equilibrium measure can be a mixture of a continuous part and a singular part. Depending on the value of the power, a threshold phenomenon occurs and consists of a dimension reduction or condensation on the singular part. In particular, in the logarithmic case s=0 (d=4), there is condensation on a sphere of special radius when the power of the external field becomes quadratic. This contrasts with the case d=s+3 studied previously, which showed that the equilibrium measure is fully dimensional and supported on a ball. Our approach makes use, among other tools, of the Frostman or Euler-Lagrange variational characterization, the Funk-Hecke formula, the Gegenbauer orthogonal polynomials, and hypergeometric special functions.

[6]  http://arxiv.org/abs/2108.00534v6 [ html pdf ]

On the solution of a Riesz equilibrium problem and integral identities for special functions

Djalil Chafaï, Edward B. Saff, Robert S. Womersley

Journal ref: J. Math. Anal. Appl. 515 (2022) 126367

DOI: 10.1016/j.jmaa.2022.126367

The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. It seems that these identities are not found in the existing tables for series and integrals, and are not recognized by advanced mathematical software. Among other ingredients, our proofs involve the Euler-Lagrange variational characterization, the Funk-Hecke formula, the Weyl regularity lemma, the maximum principle, and special properties of hypergeometric functions.

[7]  http://arxiv.org/abs/2107.14452v3 [ html pdf ]

Universal cutoff for Dyson Ornstein Uhlenbeck process

Jeanne Boursier, Djalil Chafaï, Cyril Labbé

Journal ref: Probab. Theory Related Fields 185 (2023), no. 1-2, 449-512. 60 (82)

DOI: 10.1007/s00440-022-01158-5

We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.

[8]  http://arxiv.org/abs/2108.10653v2 [ html pdf ]

Aspects of Coulomb gases

Djalil Chafaï

Journal ref: Quelques aspects de la m\'ecanique statistique, C\'edric Boutillier, B\'eatrice de Tili\`ere, Kilian Raschel (editors), Panoramas et synth\`eses 59, pp XXii + 230, 2023, Soci\'et\'e Math\'ematique de France

Coulomb gases are special probability distributions, related to potential theory, that appear at many places in pure and applied mathematics and physics. In these short expository notes, we focus on some models, ideas, and structures. We present briefly selected mathematical aspects, mostly related to exact solvability, and to first and second order global asymptotics. A particular attention is devoted to two-dimensional exactly solvable models of random matrix theory such as the Ginibre model. Thematically, these notes lie between probability theory, mathematical analysis, and statistical physics, and aim to be very accessible. They form a contribution to a volume of the "Panoramas et Synth\`eses" series around the workshop "\'Etats de la recherche en m\'ecanique statistique", organized by Soci\'et\'e Math\'ematique de France, held at Institut Henri Poincar\'e, Paris, in the fall of 2018 (http://statmech2018.sciencesconf.org).

[9]  http://arxiv.org/abs/2012.05602v2 [ html pdf ]

Convergence of the spectral radius of a random matrix through its characteristic polynomial

Charles Bordenave, Djalil Chafaï, David García-Zelada

Journal ref: Probab. Theory Related Fields 182 (2022), no. 3-4, 1163-1181

DOI: 10.1007/s00440-021-01079-9

Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in probability. This result can also be seen as the convergence of the support in the circular law theorem under optimal moment conditions. In the proof we establish the convergence in law of the reciprocal characteristic polynomial to a random analytic function outside the unit disc, related to a hyperbolic Gaussian analytic function. The proof is short and differs from the usual approaches for the spectral radius. It relies on a tightness argument and a joint central limit phenomenon for traces of fixed powers.

[10]  http://arxiv.org/abs/2012.04633v2 [ html pdf ]

At the edge of a one-dimensional jellium

Djalil Chafaï, David García-Zelada, Paul Jung

Journal ref: Bernoulli 2022, Vol. 28, No. 3, 1784-1809

DOI: 10.3150/21-BEJ1397

We consider a one-dimensional classical Wigner jellium, not necessarily charge neutral, for which the electrons are allowed to exist beyond the support of the background charge. The model can be seen as a one-dimensional Coulomb gas in which the external field is generated by a smeared background on an interval. It is a true one-dimensional Coulomb gas and not a one-dimensional log-gas. The system exists if and only if the total background charge is greater than the number of electrons minus one. For various backgrounds, we show convergence to point processes, at the edge of the support of the background. In particular, this provides asymptotic analysis of the fluctuations of the right-most particle. Our analysis reveals that these fluctuations are not universal, in the sense that depending on the background, the tails range anywhere from exponential to Gaussian-like behavior, including for instance Tracy-Widom-like behavior. We also obtain a Renyi-type probabilistic representation for the order statistics of the particle system beyond the support of the background.

[11]  http://arxiv.org/abs/1907.05803v3 [ html pdf ]

Coulomb gases under constraint: some theoretical and numerical results

Djalil Chafaï, Grégoire Ferré, Gabriel Stoltz

Journal ref: SIAM J. Math. Anal., 53(1), 181-220. (2021)

DOI: 10.1137/19M1296859

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an I-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.

[12]  http://arxiv.org/abs/1607.02741v3 [ html pdf ]

On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

Michel Bonnefont, Djalil Chafaï, Ronan Herry

Journal ref: Annales de la Facult\'e des Sciences de Toulouse. Math\'ematiques., Universit\'e Paul Sabatier, Cellule Mathdoc 2020, S\'erie 6 Tome 29 (2), pp. 335-355

DOI: 10.5802/afst.1633

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non commutative nature of the increments produces a new gradient which naturally involves a Brownian bridge on the Heisenberg group. This new inequality contains the optimal logarithmic Sobolev inequality for the Gaussian distribution in two dimensions. We compare this new inequality with the sub-elliptic logarithmic Sobolev inequality of Hong-Quan Li and with the more recent inequality of Fabrice Baudoin and Nicola Garofalo obtained using a generalized curvature criterion. Finally, we extend this inequality to the case of homogeneous Carnot groups of rank two.

[13]  http://arxiv.org/abs/1909.00613v2 [ html pdf ]

Macroscopic and edge behavior of a planar jellium

Djalil Chafaï, David García-Zelada, Paul Jung

Journal ref: J. Math. Phys. 61 (2020), no. 3, 033304, 18 pp

DOI: 10.1063/1.5126724

We consider a planar Coulomb gas in which the external potential is generated by a smeared uniform background of opposite-sign charge on a disc. This model can be seen as a two-dimensional Wigner jellium, not necessarily charge neutral, and with particles allowed to exist beyond the support of the smeared charge. The full space integrability condition requires low enough temperature or high enough total smeared charge. This condition does not allow at the same time, total charge neutrality and determinantal structure. The model shares similarities with both the complex Ginibre ensemble and the Forrester--Krishnapur spherical ensemble of random matrix theory. In particular, for a certain regime of temperature and total charge, the equilibrium measure is uniform on a disc as in the Ginibre ensemble, while the modulus of the farthest particle has heavy-tailed fluctuations as in the Forrester--Krishnapur spherical ensemble. We also touch on a higher temperature regime producing a crossover equilibrium measure, as well as a transition to Gumbel edge fluctuations. More results in the same spirit on edge fluctuations are explored by the second author together with Raphael Butez.

[14]  http://arxiv.org/abs/1805.00708v3 [ html pdf ]

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

Djalil Chafai, Joseph Lehec

Journal ref: Geometric aspects of functional analysis. Israel seminar (GAFA) 2017-2019. Volume 1. Lecture Notes in Mathematics 2256, 219-246 (2020) Springer

DOI: 10.1007/978-3-030-36020-7_10

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.

[15]  http://arxiv.org/abs/1806.05985v4 [ html pdf ]

Simulating Coulomb gases and log-gases with hybrid Monte Carlo algorithms

Djalil Chafaï, Grégoire Ferré

Journal ref: Journal of Statistical Physics 174(3):692-714 (2019)

DOI: 10.1007/s10955-018-2195-6

Coulomb and log-gases are exchangeable singular Boltzmann-Gibbs measures appearing in mathematical physics at many places, in particular in random matrix theory. We explore experimentally an efficient numerical method for simulating such gases. It is an instance of the Hybrid or Hamiltonian Monte Carlo algorithm, in other words a Metropolis-Hastings algorithm with proposals produced by a kinetic or underdamped Langevin dynamics. This algorithm has excellent numerical behavior despite the singular interaction, in particular when the number of particles gets large. It is more efficient than the well known overdamped version previously used for such problems, and allows new numerical explorations. It suggests for instance to conjecture a universality of the Gumbel fluctuation at the edge of beta Ginibre ensembles for all beta.

[16]  http://arxiv.org/abs/1607.05484v2 [ html pdf ]

On the spectral radius of a random matrix: an upper bound without fourth moment

Charles Bordenave, Pietro Caputo, Djalil Chafai, Konstantin Tikhomirov

Journal ref: The Annals of Probability 2018, Vol. 46, No. 4, 2268-2286

DOI: 10.1214/17-AOP1228

Consider a square matrix with independent and identically distributed entries of zero mean and unit variance. It is well known that if the entries have a finite fourth moment, then, in high dimension, with high probability, the spectral radius is close to the square root of the dimension. We conjecture that this holds true under the sole assumption of zero mean and unit variance, in other words that there are no outliers in the circular law. In this work we establish the conjecture in the case of symmetrically distributed entries with a finite moment of order larger than two. The proof uses the method of moments combined with a novel truncation technique for cycle weights that might be of independent interest.

[17]  http://arxiv.org/abs/1706.08776v3 [ html pdf ]

Dynamics of a planar Coulomb gas

François Bolley, Djalil Chafai, Joaquín Fontbona

Journal ref: Annals of Applied Probability 2018, Vol. 28, No. 5, 3152-3183

DOI: 10.1214/18-AAP1386

We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-posed for any inverse temperature and that Poincar{\'e} inequalities are available. Moreover the second moment dynamics turns out to be a nice Cox -- Ingersoll -- Ross process in which the dependency over the number of particles leads to identify two natural regimes related to the behavior of the noise and the speed of the dynamics.

[18]  http://arxiv.org/abs/1610.00980v3 [ html pdf ]

Concentration for Coulomb gases and Coulomb transport inequalities

Djalil Chafai, Adrien Hardy, Mylène Maïda

Journal ref: Journal of Functional Analysis 275 (2018) 1447-1483

DOI: 10.1016/j.jfa.2018.06.004

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Ma{\"i}da and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.

[19]  http://arxiv.org/abs/1610.01836v2 [ html pdf ]

Spectrum of large random Markov chains: heavy-tailed weights on the oriented complete graph

Charles Bordenave, Pietro Caputo, Djalil Chafaï, Daniele Piras

Journal ref: Random Matrices: Theory and Applications, World Scientific, 6 (2), pp.1750006 (2017)

DOI: 10.1142/S201032631750006X

We consider the random Markov matrix obtained by assigning i.i.d. non-negative weights to each edge of the complete oriented graph. In this study, the weights have unbounded first moment and belong to the domain of attraction of an alpha-stable law. We prove that as the dimension tends to infinity, the empirical measure of the singular values tends to a probability measure which depends only on alpha, characterized as the expected value of the spectral measure at the root of a weighted random tree. The latter is a generalized two-stage version of the Poisson weighted infinite tree (PWIT) introduced by David Aldous. Under an additional smoothness assumption, we show that the empirical measure of the eigenvalues tends to a non-degenerate isotropic probability measure depending only on alpha and supported on the unit disc of the complex plane. We conjecture that the limiting support is actually formed by a strictly smaller disc.

[20]  http://arxiv.org/abs/1509.02231v1 [ html pdf ]

On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence

Djalil Chafaï, Konstantin Tikhomirov

Journal ref: Probab. Theory Relat. Fields 170 (2018) no. 3 pp. 847-889

DOI: 10.1007/s00440-017-0778-9

Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge, when the dimension and the sample size both tend to infinity, to the left and right edges of the Marchenko--Pastur distribution. The assumptions are related to tails of norms of orthogonal projections. They cover isotropic log-concave random vectors as well as random vectors with i.i.d. coordinates with almost optimal moment conditions. The method is a refinement of the rank one update approach used by Srivastava and Vershynin to produce non-asymptotic quantitative estimates. In other words we provide a new proof of the Bai and Yin theorem using basic tools from probability theory and linear algebra, together with a new extension of this theorem to random matrices with dependent entries.

[21]  http://arxiv.org/abs/1405.1003v4 [ html pdf ]

From Boltzmann to random matrices and beyond

Djalil Chafaï

Journal ref: Annales de la Facult\'e de Sciences de Toulouse, 2015, XXIV (4), pp.641-689

DOI: 10.5802/afst.1459

These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory of gases, various universes are visited, including Markov processes and their Helmholtz free energy, the Shannon monotonicity problem in the central limit theorem, the Voiculescu free probability theory and the free central limit theorem, random walks on regular trees, the circular law for the complex Ginibre ensemble of random matrices, and finally the asymptotic analysis of mean-field particle systems in arbitrary dimension, confined by an external field and experiencing singular pair repulsion. The text is written in an informal style driven by energy and entropy. It aims to be recreative and to provide to the curious readers entry points in the literature, and connections across boundaries.

[22]  http://arxiv.org/abs/1304.7569v4 [ html pdf ]

First order global asymptotics for confined particles with singular pair repulsion

Djalil Chafaï, Nathael Gozlan, Pierre-André Zitt

Journal ref: The Annals of Applied Probability 24, 6 (2014) 2371-2413

DOI: 10.1214/13-AAP980

We study a physical system of $N$ interacting particles in $\mathbb{R}^d$, $d\geq1$, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as $N$ tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension $d>2$, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as $N$ tends to infinity. In the more specific case of Coulomb interaction in dimension $d>2$, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.

[23]  http://arxiv.org/abs/1310.0727v2 [ html pdf ]

A note on the second order universality at the edge of Coulomb gases on the plane

Djalil Chafaï, Sandrine Péché

Journal ref: Journal of Statistical Physics, 2014, 156 (2), 368-383

DOI: 10.1007/s10955-014-1007-x

We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan and Rider, boils down to identities in law and to an instance of the Laplace method.

[24]  http://arxiv.org/abs/1202.0644v3 [ html pdf ]

Spectrum of Markov generators on sparse random graphs

Charles Bordenave, Pietro Caputo, Djalil Chafai

Journal ref: Communications on Pure and Applied Mathematics 67, 4 (2014) 621-669

DOI: 10.1002/cpa.21496

We investigate the spectrum of the infinitesimal generator of the continuous time random walk on a randomly weighted oriented graph. This is the non-Hermitian random nxn matrix L defined by L(j,k)=X(j,k) if k<>j and L(j,j)=-sum(L(j,k),k<>j), where X(j,k) are i.i.d. random weights. Under mild assumptions on the law of the weights, we establish convergence as n tends to infinity of the empirical spectral distribution of L after centering and rescaling. In particular, our assumptions include sparse random graphs such as the oriented Erd\"os-R\'enyi graph where each edge is present independently with probability p(n)->0 as long as np(n) >> (log(n))^6. The limiting distribution is characterized as an additive Gaussian deformation of the standard circular law. In free probability terms, this coincides with the Brown measure of the free sum of the circular element and a normal operator with Gaussian spectral measure. The density of the limiting distribution is analyzed using a subordination formula. Furthermore, we study the convergence of the invariant measure of L to the uniform distribution and establish estimates on the extremal eigenvalues of L.

[25]  http://arxiv.org/abs/1402.3660v1 [ html pdf ]

Circular law for random matrices with exchangeable entries

Radosław Adamczak, Djalil Chafaï, Paweł Wolff

Journal ref: Random Structures & Algorithms Volume 48, Issue 3, pages 454-479, May 2016

DOI: 10.1002/rsa.20599

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non-Hermitian counterpart of a result of Chatterjee on the semi-circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on Hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest.

[26]  http://arxiv.org/abs/1011.2331v4 [ html pdf ]

Intertwining and commutation relations for birth-death processes

Djalil Chafaï, Aldéric Joulin

Journal ref: Bernoulli 2013, Vol. 19, No. 5A, 1855-1879

DOI: 10.3150/12-BEJ433

Given a birth-death process on $\mathbb {N}$ with semigroup $(P_t)_{t\geq0}$ and a discrete gradient ${\partial}_u$ depending on a positive weight $u$, we establish intertwining relations of the form ${\partial}_uP_t=Q_t\,{\partial}_u$, where $(Q_t)_{t\geq0}$ is the Feynman-Kac semigroup with potential $V_u$ of another birth-death process. We provide applications when $V_u$ is nonnegative and uniformly bounded from below, including Lipschitz contraction and Wasserstein curvature, various functional inequalities, and stochastic orderings. Our analysis is naturally connected to the previous works of Caputo-Dai Pra-Posta and of Chen on birth-death processes. The proofs are remarkably simple and rely on interpolation, commutation, and convexity.

[27]  http://arxiv.org/abs/1303.5838v1 [ html pdf ]

Circular law for random matrices with unconditional log-concave distribution

Radosław Adamczak, Djalil Chafai

Journal ref: Communications in Contemporary Mathematics Vol. 17, No. 04, 1550020 (2015)

DOI: 10.1142/S0219199715500200

We explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n \times n real matrix having as a random vector in R^{n^2} a log-concave isotropic unconditional law. In particular, the entries are uncorellated and have a symmetric law of zero mean and unit variance. This allows for some dependence and non equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries. Our main result states that as n goes to infinity, the empirical spectral distribution of n^{-1/2}A tends to the uniform law on the unit disc of the complex plane.

[28]  http://arxiv.org/abs/0903.3528v5 [ html pdf ]

Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph

Charles Bordenave, Pietro Caputo, Djalil Chafaï

Journal ref: Annals of Probability 2011, Vol. 39, No. 4, 1544-1590

DOI: 10.1214/10-AOP587

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an $\alpha$-stable law, $\alpha\in(0,2)$. When $1\leq\alpha<2$, we show that for a suitable regularly varying sequence $\kappa_n$ of index $1-1/\alpha$, the limiting spectral distribution $\mu_{\alpha}$ of $\kappa_nK$ coincides with the one of the random symmetric matrix of the un-normalized weights (L\'{e}vy matrix with i.i.d. entries). In contrast, when $0<\alpha<1$, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law $\widetilde{\mu}_{\alpha}$ supported on [-1,1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of $\mu_{\alpha}$ and $\widetilde{\mu}_{\alpha}$. We also study the limiting behavior of the invariant probability measure of K.

[29]  http://arxiv.org/abs/1109.3343v4 [ html pdf ]

Around the circular law

Charles Bordenave, Djalil Chafai

Journal ref: Probability Surveys 9 (2012) 1-89

DOI: 10.1214/11-PS183

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.

[30]  http://arxiv.org/abs/1006.1713v5 [ html pdf ]

Spectrum of non-Hermitian heavy tailed random matrices

Charles Bordenave, Pietro Caputo, Djalil Chafai

Journal ref: Communications in Mathematical Physics 307, 2 (2011) 513-560

DOI: 10.1007/s00220-011-1331-9

Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}| is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X_{jk}, we prove that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability measure mu_alpha on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1} (X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu_alpha. In contrast with the Hermitian case, we find that mu_alpha is not heavy tailed.

[31]  http://arxiv.org/abs/1104.2198v1 [ html pdf ]

Central limit theorems for additive functionals of ergodic Markov diffusions processes

Patrick Cattiaux, Djalil Chafai, Arnaud Guillin

Journal ref: ALEA, Lat. Am. J. Probab. Math. Stat. 9 (2), 337-382 (2012)

We revisit functional central limit theorems for additive functionals of ergodic Markov diffusion processes. Translated in the language of partial differential equations of evolution, they appear as diffusion limits in the asymptotic analysis of Fokker-Planck type equations. We focus on the square integrable framework, and we provide tractable conditions on the infinitesimal generator, including degenerate or anomalously slow diffusions. We take advantage on recent developments in the study of the trend to the equilibrium of ergodic diffusions. We discuss examples and formulate open problems.

[32]  http://arxiv.org/abs/0808.1502v3 [ html pdf ]

Circular Law Theorem for Random Markov Matrices

Charles Bordenave, Pietro Caputo, Djalil Chafai

Journal ref: Probability Theory and Related Fields 152, 3-4 (2012) 751-779

DOI: 10.1007/s00440-010-0336-1

Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded density, mean m, and finite positive variance sigma^2. Let M be the nxn random Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its sum. In particular, when X11 follows an exponential law, then M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Our main result states that with probability one, the counting probability measure of the complex spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law on the centered disk of radius sigma/m. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.

[33]  http://arxiv.org/abs/0811.1097v2 [ html pdf ]

Spectrum of large random reversible Markov chains: two examples

Charles Bordenave, Pietro Caputo, Djalil Chafai

Journal ref: ALEA Latin American Journal of Probability and Mathematical Statistics 7, 41-64 (2010)

We take on a Random Matrix theory viewpoint to study the spectrum of certain reversible Markov chains in random environment. As the number of states tends to infinity, we consider the global behavior of the spectrum, and the local behavior at the edge, including the so called spectral gap. Results are obtained for two simple models with distinct limiting features. The first model is built on the complete graph while the second is a birth-and-death dynamics. Both models give rise to random matrices with non independent entries.

[34]  http://arxiv.org/abs/0709.0036v3 [ html pdf ]

Circular law for non-central random matrices

Djalil Chafai

Journal ref: Journal of Theoretical Probability 23, 4 (2010) 945-950

DOI: 10.1007/s10959-010-0285-8

Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let $\la_{n,1},...,\la_{n,n}$ be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution $\frac{1}{n}(\de_{\la_{n,1}}+...+\de_{\la_{n,n}})$ converges weakly as $n\to\infty$ to the uniform law over the unit disc $\{z\in\dC;|z|\leq1\}$. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\leq j,k\leq n}$ provided that $\mathrm{Tr}(MM^*)=O(n^2)$ and $\mathrm{rank}(M)=O(n^\al)$ with $\al<1$. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems.

[35]  http://arxiv.org/abs/0805.0987v3 [ html pdf ]

On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

Djalil Chafai, Florent Malrieu

Journal ref: Annales de l'Institut Henri Poincar\'e (B) Probabilit\'es et Statistiques 46, 72-96 (2010)

DOI: 10.1214/08-AIHP309

Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. Additionally, our analysis of Sobolev type inequalities for two-component mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\'e constant of a two-component mixture may remain bounded as the mixture proportion goes to 0 or 1 while the logarithmic Sobolev constant may surprisingly blow up. This counter-intuitive result is not reducible to support disconnections, and appears as a reminiscence of the variance-entropy comparison on the two-point space. As far as mixtures are concerned, the logarithmic Sobolev inequality is less stable than the Poincar\'e inequality and the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-component mixtures. This work leads to many open questions.

[36]  http://arxiv.org/abs/0811.2180v3 [ html pdf ]

On the long time behavior of the TCP window size process

Djalil Chafai, Florent Malrieu, Katy Paroux

Journal ref: Stochastic Processes and their Applications 120, 8 (2010) 1518-1534

DOI: 10.1016/j.spa.2010.03.019

The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in $[0,\infty)$, is ergodic and irreversible. It belongs to the Additive Increase Multiplicative Decrease class of processes. The sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. Several aspects of this process have already been investigated in the literature. In the present paper, we mainly get quantitative estimates for the convergence to equilibrium, in terms of the $W_1$ Wasserstein coupling distance, for the process and also for its embedded chain.

[37]  http://arxiv.org/abs/0709.4678v3 [ html pdf ]

The Dirichlet Markov Ensemble

Djalil Chafai

Journal ref: Journal of Multivariate Analysis 101, 555-567 (2010)

DOI: 10.1016/j.jmva.2009.10.013

We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if $\bM$ is such a random matrix, then the empirical distribution built from the singular values of$\sqrt{n} \bM$ tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of $\sqrt{n} \bM$ tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of $\bM$ is of order $1-1/\sqrt{n}$ when $n$ is large.

[38]  http://arxiv.org/abs/0811.0600v2 [ html pdf ]

Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model

Patrick Cattiaux, Djalil Chafai, Sébastien Motsch

Journal ref: Asymptotic Analysis 67, 17-31 (2010)

DOI: 10.3233/ASY-2009-0969

The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al in Mathematical Biology for the modelling of fish motion. It involves a nonlinear pathwise functional of a non-elliptic hypo-elliptic diffusion. This diffusion solves a kinetic Fokker-Planck equation based on an Ornstein-Uhlenbeck Gaussian process. The long time "diffusive" behavior of this model was recently studied by Degond & Motsch using partial differential equations techniques. This model is however intrinsically probabilistic. In the present paper, we show how the long time diffusive behavior of this model can be essentially recovered and extended by using appropriate tools from stochastic analysis. The approach can be adapted to many other kinetic "probabilistic" models.

[39]  http://arxiv.org/abs/0805.1971v2 [ html pdf ]

Confidence regions for the multinomial parameter with small sample size

Djalil Chafai, Didier Concordet

Journal ref: Journal of the American Statistical Association 104, 1071-1079 (2009)

DOI: 10.1198/jasa.2009.tm08152

Consider the observation of n iid realizations of an experiment with d>1 possible outcomes, which corresponds to a single observation of a multinomial distribution M(n,p) where p is an unknown discrete distribution on {1,...,d}. In many applications, the construction of a confidence region for p when n is small is crucial. This concrete challenging problem has a long history. It is well known that the confidence regions built from asymptotic statistics do not have good coverage when n is small. On the other hand, most available methods providing non-asymptotic regions with controlled coverage are limited to the binomial case d=2. In the present work, we propose a new method valid for any d>1. This method provides confidence regions with controlled coverage and small volume, and consists of the inversion of the "covering collection"' associated with level-sets of the likelihood. The behavior when d/n tends to infinity remains an interesting open problem beyond the scope of this work.

[40]  http://arxiv.org/abs/0709.0111v2 [ html pdf ]

A new method for the estimation of variance matrix with prescribed zeros in nonlinear mixed effects models

Djalil Chafai, Didier Concordet

Journal ref: Statistics and Computing 19, 2 (2009) 129-138

DOI: 10.1007/s11222-008-9076-9

We propose a new method for the Maximum Likelihood Estimator (MLE) of nonlinear mixed effects models when the variance matrix of Gaussian random effects has a prescribed pattern of zeros (PPZ). The method consists in coupling the recently developed Iterative Conditional Fitting (ICF) algorithm with the Expectation Maximization (EM) algorithm. It provides positive definite estimates for any sample size, and does not rely on any structural assumption on the PPZ. It can be easily adapted to many versions of EM.

[41]  http://arxiv.org/abs/0710.3139v5 [ html pdf ]

On gradient bounds for the heat kernel on the Heisenberg group

Dominique Bakry, Fabrice Baudoin, Michel Bonnefont, Djalil Chafai

Journal ref: Journal of Functional Analysis 255, 8 (2008) 1905-1938

DOI: 10.1016/j.jfa.2008.09.002

It is known that the couple formed by the two dimensional Brownian motion and its L\'evy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.

[42]  http://arxiv.org/abs/math/0412193v3 [ html pdf ]

Explicit formulas for a continuous stochastic maturation model. Application to anticancer drug pharmacokinetics/pharmacodynamics

Djalil Chafai, Didier Concordet

Journal ref: Stochastic Models 24, 3 (2008) 376-400

DOI: 10.1080/15326340802232244

We present a continuous time model of maturation and survival, obtained as the limit of a compartmental evolution model when the number of compartments tends to infinity. We establish in particular an explicit formula for the law of the system output under inhomogeneous killing and when the input follows a time-inhomogeneous Poisson process. This approach allows the discussion of identifiability issues which are of difficult access for finite compartmental models. The article ends up with an example of application for anticancer drug pharmacokinetics/pharmacodynamics.

[43]  http://arxiv.org/abs/math/0507102v2 [ html pdf ]

On the strong consistency of asymptotic M-estimators

Djalil Chafai, Didier Concordet

Journal ref: Journal of Statistical Planning and Inference 137, 9 (2007) 2774-2783

DOI: 10.1016/j.jspi.2006.09.027

The aim of this article is to simplify Pfanzagl's proof of consistency for asymptotic maximum likelihood estimators, and to extend it to more general asymptotic M-estimators. The method relies on the existence of a sort of contraction of the parameter space which admits the true parameter as a fixed point. The proofs are short and elementary.

[44]  http://arxiv.org/abs/math/0510488v2 [ html pdf ]

Binomial-Poisson entropic inequalities and the M/M/$\infty$ queue

Djalil Chafai

Journal ref: ESAIM P&S 10 (2006) 317-339

DOI: 10.1051/ps:2006013

This article provides entropic inequalities for binomial-Poisson distributions, derived from the two point space. They appear as local inequalities of the M/M/$\infty$ queue. They describe in particular the exponential dissipation of $\Phi$-entropies along this process. This simple queueing process appears as a model of ``constant curvature'', and plays for the simple Poisson process the role played by the Ornstein-Uhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semi-group interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the Ornstein-Uhlenbeck process as a fluid limit of M/M/$\infty$ queues. Proofs are elementary and rely essentially on the development of a ``$\Phi$-calculus''.

[45]  http://arxiv.org/abs/math/0411516v1 [ html pdf ]

On nonparametric maximum likelihood for a class of stochastic inverse problems

Djalil Chafai, Jean-Michel Loubes

Journal ref: Statistics & Probability Letters / Statistics and Probability Letters 76, 12 (2006) 1225-1237

DOI: 10.1016/j.spl.2005.12.019

We establish the consistency of a nonparametric maximum likelihood estimator for a class of stochastic inverse problems. We proceed by embedding the framework into the general settings of early results of Pfanzagl related to mixtures.

[46]  http://arxiv.org/abs/math/0211108v1 [ html pdf ]

Glauber versus Kawasaki for spectral gap and logarithmic Sobolev inequalities of some unbounded conservative spin systems

Djalil Chafai

Journal ref: Markov Processes and Related Fields 9, 3 (2003) 341-362

Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau [LPY] on spectral gap and logarithmic Sobolev inequalities for unbounded conservative spin systems, we study uniform bounds in these inequalities for Glauber dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n), (x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or small perturbation of strictly convex) and, following [LPY], the case V is a bounded perturbation of a quadratic potential. By a simple path counting argument for the standard random walk, uniform bounds for the Glauber dynamics yields, in a transparent way, the classical L^{-2} decay for the Kawasaki dynamics on d-dimensional cubes of length L. The arguments of proofs however closely follow and make heavy use of the conservative approach and estimates of [LPY], relying in particular on the Lu-Yau martingale decomposition and clever partitionings of the conditional measure.

[47]  http://arxiv.org/abs/math/0211103v2 [ html pdf ]

Entropies, convexity, and functional inequalities

Djalil Chafai

Journal ref: Journal of Mathematics of Kyoto University, vol. 44 (2004), no. 2, 325--363

DOI: 10.1215/kjm/1250283556

Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies can be seen in particular as an inclusive interpolation between Poincare and Gross logarithmic Sobolev inequalities. In addition to the known material, extensions are provided and improvements are given for some aspects. Stability by tensor products, convolution, and bounded perturbations are addressed. We show that under simple convexity assumptions on Phi, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, Wiener measure (paths space of Brownian Motion on Riemannian Manifolds) and generic Poisson space (includes paths space of some pure jumps Levy processes and related infinitely divisible laws). Proofs are simple and relies essentially on convexity. We end up by a short parallel inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic Gases and Information Theories.

[48]  http://arxiv.org/abs/math/0102227v1 [ html pdf ]

Gaussian maximum of entropy and reversed log-Sobolev inequality

Djalil Chafai

Journal ref: S\'eminaire de Probabilit\'es XXXVI, 194--200, Lecture Notes in Mathematics 1801, ISBN 978-3-540-00072-3, 2003

DOI: 10.1007/978-3-540-36107-7_5

The aim of this note is to connect a reversed form of the Gross logarithmic Sobolev inequality with the Gaussian maximum of Shannon's entropy power. There is thus a complete parallel with the well-known link between logarithmic Sobolev inequalities and their information theoretic counterparts. We moreover provide an elementary proof of the reversed Gross inequality via a two-point inequality and the Central Limit Theorem.

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