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

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 $[-2,2]$.