Universality Limits Involving Orthogonal Polynomials on the Unit Circle

Abstract

We establish universality limits for measures on the unit circle. Assume that μ is a regular measure on the unit circle in the sense of Stahl and Totik, and is absolutely continuous in an open arc containing some point z = e ^iθ. Assume, moreover, that μ′ is positive and continuous at z. then universality for μ holds at z, in the sense that the normalized reproducing kernel ~K_n(z, t) satisfies

$${\rm lim}_{n\to \infty}{1\over n}\tilde{K}_{n}(e^{i(\theta + 2\pi a/n)},e^{i(\theta + 2\pi b/n)}\bigg) = e^{i\pi(a-b)}{{\rm sin} \pi (b-a)\over \pi (b-a)}$$

, uniformly for a, b in compact subsets of the real line.