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

We give a simple proof of a central limit theorem for linear statistics of the Circular beta-ensembles which is valid at almost arbitrary mesoscopic scale and for functions of class C^3. As a consequence, using a coupling introduced by Valko and Virag, we deduce a central limit theorem for the Sine beta processes. We also discuss the connection between our result and the theory of Gaussian Multiplicative Chaos. Based on the results of Lambert-Ostrovsky-Simm, we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the Circular beta-ensembles, regularized at a small mesoscopic scale and renormalized, converges to GMC measures in the subcritical regime. This implies that the leading order behavior for the extreme values of the logarithm of the characteristic polynomial is consistent with the predictions of log-correlated Gaussian fields.