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

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 $[-2,2]$.