CLT for Circular beta-Ensembles at High Temperature

Abstract

We consider the macroscopic large $N$ limit of the Circular beta-Ensemble at high temperature, and its weighted version as well, in the regime where the inverse temperature scales as $\beta/N$ for some parameter $\beta>0$. More precisely, in the limit $N\to\infty$, the equilibrium measure of this particle system is described as the unique minimizer of a functional which interpolates between the relative entropy ($\beta=0$) and the weighted logarithmic energy $(\beta=\infty$). The purpose of this work is to show that the fluctuation of the empirical measure around the equilibrium measure converges towards a Gaussian field whose covariance structure interpolates between the Lebesgue $L^2$ ($\beta=0$) and the Sobolev $H^{1/2}$ ($β=\infty$) norms. We furthermore obtain a rate of convergence for the fluctuations in the $W_2$ metric. Our proof uses the normal approximation result of Lambert, Ledoux, and Webb [2017], the Coulomb transport inequality of Chafaï, Hardy, and Maïda [2018], and a spectral analysis for the operator associated with the limiting covariance structure