Universality of mesoscopic fluctuations for orthogonal polynomial ensembles

Abstract

We prove that the fluctuations of mesocopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a Central Limit Theorem for the modified Jacobi Unitary Ensembles on all mesosopic scales.