Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

We apply universality limits to asymptotics of spacing of zeros fxkng of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result is lim n!1 xkn xk+1;n ~ Kn (xkn; xkn) = 1 under minimal hypotheses on the weight, with ~ Kn denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the di¤erentiated kernels K (r;s) n (x; x) = n 1 X k=0 p (r) k (x) p (s) k (x) : 1. Introduction and Results Let be a …nite positive Borel measure on the real line, with all …nite power moments. Then we may de…ne orthonormal polynomials pn (x) = nx n + :::; n > 0; n = 0; 1; 2; ::: satisfying the orthonormality conditions Z pnpmd = mn: The zeros of pn are denoted xnn < xn 1;n < xn 2;n < ::: < x1n: The universality limit of random matrix theory [4], [16] involves the reproducing kernel (1.1) Kn (x; y) = n 1 X