Mesoscopic Universality for Circular Orthogonal Polynomial Ensembles

Abstract

We study mesoscopic fluctuations of orthogonal polynomial ensembles on the unit circle. We show that asymptotics of such fluctuations are stable under decaying perturbations of the recurrence coefficients, where the appropriate decay rate depends on the scale considered. By directly proving Gaussian limits for certain constant coefficient ensembles, we obtain mesoscopic scale Gaussian limits for a large class of orthogonal polynomial ensembles on the unit circle. As a corollary we prove mesocopic central limit theorems (for all mesoscopic scales) for the $\beta=2$ circular Jacobi ensembles with real parameter $\delta>-1/2$.