Mesoscopic Universality for Circular Orthogonal Polynomial Ensembles

TL;DR

This paper establishes mesoscopic fluctuation universality for orthogonal polynomial ensembles on the unit circle, demonstrating Gaussian limits and stability under perturbations across various scales.

Contribution

It proves mesoscopic fluctuation universality and Gaussian limits for a broad class of orthogonal polynomial ensembles on the circle, including circular Jacobi ensembles.

Findings

Gaussian limits for constant coefficient ensembles

Stability of fluctuations under decaying perturbations

Mesoscopic CLTs for circular Jacobi 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 $\delta>-1/2$.