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

Charlie Dworaczek Guera, Ronan Memin, Michel Pain

Abstract

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 the central limit theorem for linear statistics with general test-functions and compute the subleading correction to the free energy. Our strategy relies on establishing an optimal local law in the spirit of [Bourgade, Mody, Pain 22']. Our results allow us to give a large $N$ expansion up to $o(N)$ of the log-volume of the unit balls of $N\times N$ self-adjoint matrices for the $p$-Schatten norms and to give a consistency check of the KLS conjecture. For the latter, we consider the functions $f(X)=\mathrm{Tr}\left(X^r\right)^q$ and the uniform distributions on these same Schatten balls for $N$ large enough. 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.

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

Abstract

In this paper, we are interested in the -ensembles (or 1D log-gas) with Freud weights, namely with a potential of the form with . Since this potential is not of class when , most of the literature does not apply. In this singular setting, we prove the central limit theorem for linear statistics with general test-functions and compute the subleading correction to the free energy. Our strategy relies on establishing an optimal local law in the spirit of [Bourgade, Mody, Pain 22']. Our results allow us to give a large expansion up to of the log-volume of the unit balls of self-adjoint matrices for the -Schatten norms and to give a consistency check of the KLS conjecture. For the latter, we consider the functions and the uniform distributions on these same Schatten balls for large enough. While the case , , was proven in [Dadoun, Fradelizi, Guédon, Zitt 23'], we address in the present paper the case , and an even integer. The proofs are based on a link between the moments of norms of uniform laws on -Schatten balls and the -ensembles with Freud weights.

Paper Structure

This paper contains 36 sections, 45 theorems, 313 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Main results
  4. About beta-ensembles
  5. About the KLS conjecture for p-Schatten balls
  6. Connection with the literature
  7. Central limit theorems in $\beta$-ensembles.
  8. Transition for bulk statistics at $p=1$.
  9. Multicriticality and quantum chromodynamics.
  10. Definitions and strategy
  11. The equilibrium measure.
  12. Strategy.
  13. Organization of the paper
  14. Notation
  15. Loop equations and local laws
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $p\geq 2$, and $f\in \mathcal{C}^3(\mathbb{R})$ with at most exponential growth, i.e. there exists $C>0$ such that $\mathopen{}\mathclose{\left\lvert f(x)\right\rvert \leq Ce^{C\mathopen{}}\mathclose{\left\lvert x\right\rvert}}$ for any $x \in \mathbb{R}$. Then, the following convergence holds i where the mean and the variance of the limiting Gaussian distribution are given by: where the oper

Theorems & Definitions (90)

  • Theorem 1.1: CLT
  • Remark 1.2
  • Theorem 1.3: Next-order expansion of the free energy
  • Remark 1.4
  • Theorem 1.5: Optimal local law
  • Remark 1.6
  • Conjecture 1.7: KLS conjecture
  • Lemma 1.8: Weyl's integration formula, raymond1984volumeguedon2007concentration
  • Corollary 1.9: Volume of $p$-Schatten balls
  • Theorem 1.10
  • ...and 80 more