CLT for Circular beta-Ensembles at high temperature

We consider the macroscopic large N limit of Circular beta-Ensemble at high temperature, and its weighted version as well, in regime where inverse temperature scales β/N for some parameter β>0. More precisely, N→∞, equilibrium measure this particle system is described unique minimizer a functional which interpolates between relative entropy (β=0) logarithmic energy (β=∞). The purpose work to show that fluctuation empirical around converges towards Gaussian field whose covariance structure Lebesgue L2 Sobolev H1/2 (β=∞) norms. furthermore obtain rate convergence fluctuations W2 metric. Our proof uses normal approximation result Lambert et al. [24], Coulomb transport inequality Chafaï [8], spectral analysis operator associated with limiting structure.