Stability of Mesoscopic Fluctuations of Orthogonal Polynomial Ensembles Under Sparse Decaying Perturbations

TL;DR

This paper investigates how mesoscopic fluctuations of orthogonal polynomial ensembles on the real line remain stable under sparse, decaying perturbations of recurrence coefficients, establishing a mesoscopic CLT for certain singular measures.

Contribution

It proves the stability of mesoscopic fluctuations under sparse decaying perturbations and establishes a mesoscopic CLT for singular continuous measures.

Findings

Mesoscopic fluctuations are stable under sparse decaying perturbations.

A mesoscopic central limit theorem is proved for specific singular measures.

The stability result applies to orthogonal polynomial ensembles with perturbed recurrence coefficients.

Abstract

We study the stability of the mesoscopic fluctuations of certain orthogonal polynomial ensembles on the real line utilizing the recurrence relation of the associated orthogonal polynomials. We prove that under a sparse enough decaying perturbation of the recurrence coefficients the limiting distribution is stable. As a corollary we prove a mesoscopic central limit theorem (at any scale) for a family of singular continuous measures on $[-2,2]$.