CLT for fluctuations of $\beta$-ensembles with general potential

TL;DR

This paper establishes a central limit theorem for linear statistics of one-dimensional log-gases ($eta$-ensembles) with general potentials, including complex multi-cut and critical cases, extending previous results and providing convergence rates.

Contribution

It introduces a new method based on variable change that handles more general $eta$-ensemble situations, including critical and multi-cut cases, surpassing prior limitations.

Findings

Proves CLT for $eta$-ensembles with general potentials.

Handles multi-cut and critical cases for the first time.

Provides convergence rates and free energy expansions.

Abstract

We prove a central limit theorem for the linear statistics of one-dimensional log-gases, or $\beta$-ensembles. We use a method based on a change of variables which allows to treat fairly general situations, including multi-cut and, for the first time, critical cases, and generalizes the previously known results of Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our approach also allows to retrieve a rate of convergence as well as previously known expansions of the free energy to arbitrary order.