arXiv subjects

Charlie Dworaczek Guera

Publications and source records attributed to Charlie Dworaczek Guera.

6 recordsLinked to original sources

arXiv2026

Mesoscopic transition for $β$-ensembles at intermediary temperature

This paper establishes a mesoscopic central limit theorem for linear statistics of $β$-ensembles or log-gas, as the dimension $N\to\infty$, in the temperature regime $1/N\llβ(N)\le 1$. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depends on $β(N)$ and the mesoscopic scale. We show that there is a transition at a critical $η\asymp 1/ Nβ(N)$ between a Random Matrix regime, where the limiting variance is given by the $\mathsf{H}^{1/2}$-norm and a Poisson regime where the limiting variance is given by the $L^2$-norm. We also describe the critical regime. The proof of the CLT relies on optimal local laws at intermediate temperatures and Stein's method for $β$-ensembles. In particular, in this regime, it is necessary to construct new correction terms to the classical equilibrium measure to obtain a suitable re-centring of linear statistics and describe their fluctuations. We also obtain a free energy expansion.

math.PR↗

arXiv2026

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 $β_N=2α/N$ for some $α>0$. 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 $β$-ensemble at fixed $β_N=β>0$ and $β_N\gg N^{-1}$ 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↗

arXiv2026

Asymptotics of the partition function for $β$-ensembles at high temperature

We consider the real $β$-ensemble (or 1D log-gas) of dimension $N$ in the high-temperature regime, \textit{i.e.} where the inverse temperature $β$ scales as $Nβ=2P$ with $P$ a fixed positive parameter. We establish the large-$N$ asymptotic expansion at all orders of the partition function: \begin{equation*} Z_N[V]=\int_{\mathbb{R}^N}\prod_{i<j}^{N}\left |x_i-x_j\right|^{\frac{2P}{N}}\cdot\prod_{i=1}^{N}e^{-V(x_i)} \mathrm{d}x_i \end{equation*} for $V(x)=x^2+ϕ(x)$ with $ϕ$ a bounded smooth function, and identify the first two terms of this expansion. In this regime, the energy no longer dominates the entropy, as in the fixed-$β$ case, but rather scales at the same order in $N$. Consequently, at large $N$, the system is macroscopically described by the so-called\textit{ thermal equilibrium measure} which is supported on the entire real line. Our proof relies on the loop equations method, previously applied in the fixed-$β$ setting in \cite{BoG1,BoG2}, and provides the first example in which this approach can be successfully implemented using the thermal equilibrium measure. This requires a detailed understanding of both the thermal equilibrium measure and the associated master operator, an unbounded differential operator, leading to several new analytical challenges. In this setting, we carry out a technically involved analysis to obtain precise estimates for the inverse of the master operator in suitable functional norms. In addition we establish, through subtle operator arguments, a crucial continuity property of the equilibrium density with respect to the potential dependence. These two results constitute the main novelties of the paper and allow us to exhibit a new class of multiple integrals for which such an expansion can be obtained, while providing a deeper understanding of the thermal equilibrium measure and its properties.

math.PR↗

arXiv2026

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

In this paper, we are interested in the $β$-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, Guédon, 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 $β$-ensembles with Freud weights.

math.PR↗

arXiv2024

CLT for real beta-ensembles at high temperature

We establish a central limit theorem for the fluctuations of the linear statistics in the $β$-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 $β>0$ 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 Schrödinger operators theory as well as concentration of measure.

math.PR↗

arXiv2024

On the equilibrium measure for the Lukyanov integral

In 2000, Lukyanov conjectured that a certain ratio of $N$-fold integrals should provide access, in the large-$N$ regime, to the ground state expectation value of the exponential of the Sinh-Gordon quantum field in 1+1 dimensions and finite volume $R$. This work aims at rigorously constructing the fundamental objects necessary to address the large-$N$ analysis of such integrals. More precisely, we construct and establish the main properties of the the equilibrium measure minimising a certain $N$-dependent energy functional that naturally arises in the study of the leading large-$N$ behaviour of the Lukyanov integral. Our construction allows us to heuristically advocate the leading term in the large-$N$ asymptotic behaviour of the mentioned ratio of Lukyanov integrals, hence supporting Lukyanov's prediction -- obtained by other means -- on the exponent $σ$ of the power-law $N^σ$ term of its asymptotic expansion as $N\rightarrow + \infty$.

math-ph↗