Sine kernel asymptotics for a class of singular measures

TL;DR

This paper constructs a family of purely singular measures on the real line that still exhibit universal sine-kernel asymptotics in the bulk, revealing new insights into spectral measures and universality.

Contribution

It introduces a class of singular measures with sine-kernel asymptotics, characterized by sparse perturbations of Chebyshev polynomial recursion coefficients.

Findings

Convergence of Christoffel-Darboux kernel to sine kernel for sparse perturbations

Construction of purely singular measures with universal asymptotics

Extension of sine-kernel universality to singular measures

Abstract

We construct a family of measures on $\bbR$ that are purely singular with respect to Lebesgue measure, and yet exhibit universal sine-kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients: these are sparse perturbations of the recursion coefficients corresponding to Chebyshev polynomials of the second kind. We prove convergence of the renormalized Christoffel-Darboux kernel to the sine kernel for any sufficiently sparse decaying perturbation.