CLT for Circular beta-Ensembles at high temperature

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 β/N for some parameter β>0. More precisely, in the limit N→∞, the equilibrium measure of this particle system is described as the unique minimizer of a functional which interpolates between the relative entropy (β=0) and the weighted logarithmic energy (β=∞). 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 L2 (β=0) and the Sobolev H1/2 (β=∞) norms. We furthermore obtain a rate of convergence for the fluctuations in the W2 metric. Our proof uses the normal approximation result of Lambert et al. [24], the Coulomb transport inequality of Chafaï et al. [8], and a spectral analysis for the operator associated with the limiting covariance structure.