Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications

We give a simple proof of central limit theorem for linear statistics the circular $\beta $-ensembles which is valid at almost microscopic scales functions class $C^{3}$. Using coupling introduced by Valko and Virag [48], we deduce Sine$_{\beta }$ processes. also discuss connections between our result theory Gaussian Multiplicative Chaos. Based on results [37], show that exponential logarithm real (and imaginary) part characteristic polynomial $-ensembles, regularized small mesoscopic scale renormalized, converges to GMC measures in subcritical regime. This establishes leading order behavior extreme values consistent with predictions coming from log-correlated field theory.