A new approach to universality limits involving orthogonal polynomials

Abstract

We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure on OE 1; 1. Assume that is a regular measure, and is absolutely continuous in an open interval containing some point x. Assume moreover, that 0 is positive and continuous at x. Then universality for holds at x. If the hypothesis holds for x in a compact subset of . 1; 1/, universality holds uniformly for such x. Indeed, this follows from universality for the classical Legendre weight. We also establish universality in an L p sense under weaker assumptions