Mesoscopic Edge Universality of Orthogonal Polynomial Ensembles

TL;DR

This paper establishes a universal central limit theorem for mesoscopic fluctuations at the edges of orthogonal polynomial ensembles, using resolvent estimates for Jacobi matrices with slowly varying coefficients.

Contribution

It introduces a new resolvent estimate for Jacobi matrices with slowly varying entries, enabling a CLT for mesoscopic edge fluctuations in various ensembles.

Findings

Universal CLT for mesoscopic edge fluctuations.

Applicable to Jacobi, Laguerre, Gaussian ensembles, and discrete tilings.

Resolves limitations of the Combes-Thomas estimate at edges.

Abstract

In this paper, we study the mesoscopic fluctuations at edges of orthogonal polynomial ensembles with both continuous and discrete measures. Our main result is a Central limit Theorem (CLT) for linear statistics at mesoscopic scales. We show that if the recurrence coefficients for the associated orthogonal polynomials are slowly varying, a universal CLT holds. Our primary tool is the resolvent for the truncated Jacobi matrices associated with the orthogonal polynomials. While the Combes-Thomas estimate has been successful in obtaining bulk mesoscopic fluctuations in the literature, it is too rough at the edges. Instead, we prove an estimate for the resolvent of Jacobi matrices with slowly varying entries. Particular examples to which our CLT applies are Jacobi, Laguerre and Gaussian unitary ensembles as well as discrete ensembles from random tilings.