arXiv subjects

Ronan Memin

Publications and source records attributed to Ronan Memin.

6 recordsLinked to original sources

arXiv2025

CLT for $\beta$-ensembles with Freud weights, application to the KLS conjecture in Schatten balls

In this paper, we are interested in the $\beta$-ensembles (or 1D log-gas) with Freud weights, namely with a potential of the form $|x|^{p}$ with $p \geq 2$. Since this potential is not of class $\mathcal{C}^{3}$ when $p \in (2,3]$, most of the literature does not apply. In this singular setting, we prove a central limit theorem for linear statistics with general test-functions. Our strategy relies on establishing an optimal local law in the spirit of [Bourgade, Mody, Pain 22'. Our results allow us to give a consistency check of the KLS conjecture for the uniform distributions on $p$-Schatten balls and the functions $f(X)=\mathrm{Tr}\left(X^r\right)^q$. While the case $p>3$, $q=1$, $r=2$ was proven in [Dadoun, Fradelizi, Gu\'edon, Zitt 23'], we address in the present paper the case $p\geq2$, $q\geq1$ and $r\geq2$ an even integer. The proofs are based on a link between the moments of norms of uniform laws on $p$-Schatten balls and the $\beta$-ensembles with Freud weights.

math.PR↗

arXiv2025

Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature

We consider a model of a gas of $N$ confined particles subject to a two-body repulsive interaction, namely the one-dimensional log or Riesz gas. We are interested in the so-called \textit{high temperature} regime, \textit{i.e.} when the inverse temperature is given by $\beta_N=2\alpha/N$ for some $\alpha>0$. We establish, in the log case, a large deviation (LD) principle and moderate deviations estimates for the largest particle $x_\mathrm{max}$ when appropriately rescaled. Our result is in the continuity of [Ben Arous Dembo Guionnet 01', Pakzad 20'] where such estimates were shown for the largest particle of the $\beta$-ensemble at fixed $\beta_N=\beta>0$ and $\beta_N\gg N^{-1}$ respectively. We show that the corresponding rate function is the same as in the case of iid particles. We also provide LD estimates in the Riesz case. Additionally, we consider related models of symmetric tridiagonal random matrices with independent entries having Gaussian tails; for which we establish the LD principle for the top eigenvalue. In a certain specialization of the entries, we recover the result for the largest particle of the log-gas. We show that LD are created by a few entries taking abnormally large values.

math.PR↗

arXiv2023

CLT for $\beta$ ensembles at high-temperature, and for integrable systems: a transfer operator approach

In this paper, we prove a polynomial Central Limit Theorem for several integrable models, and for the $\beta$-ensembles at high-temperature with polynomial potential. Furthermore, we connect the mean values, the variances and the correlations of the moments of the Lax matrices of these integrable systems with the ones of the $\beta$-ensembles. Moreover, we show that the local functions' space-correlations decay exponentially fast for the considered integrable systems. For these models, we also established a Berry--Esseen type bound.

math.PR↗

arXiv2023

CLT for real beta-ensembles at high temperature

We establish a central limit theorem for the fluctuations of the linear statistics in the $\beta$-ensemble of dimension $N$ at a temperature proportional to $N$ and with confining smooth potential. In this regime, the particles do not accumulate in a compact set as in the fixed $\beta>0$ case which results in an equilibrium measure supported on the whole real line. The space of test functions for which the CLT holds includes bounded $C^2$ functions. The method that we use is based on a change of variables in the partition function introduced in Johansson [1998] and allows to deduce the convergence of the Laplace transform of the recentred linear statistics towards the Laplace transform of the normal distribution. It is obtained by the inversion of the master operator, which is the main contribution of the present paper, by following the scheme developed in Hardy, Lambert [2019] in the compact case. In the high-temperature regime, the master operator contains an additional differential term due to entropic effects which makes it an unbounded operator. The techniques used in this article involve Schr\"odinger operators theory as well as concentration of measure.

math.PR↗

arXiv2022

Large Deviations for Ablowitz-Ladik lattice, and the Schur flow

We consider the Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, and the Schur flow. We derive large deviations principles for the distribution of the empirical measures of the equilibrium measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limit in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble respectively.

math.PR↗

arXiv2021

Large deviations for generalized Gibbs ensembles of the classical Toda chain

We prove large deviation principles for the distribution of the empirical measure of the eigenvalues of Lax matrices following the Generalized Gibbs ensembles of the classical Toda chain introduced in [10]. We deduce the almost sure convergence of this empirical measure towards a limit which we describe in terms of the limiting empirical measure of beta-ensembles. Our results apply to general smooth potentials.

math.PR↗