A new approach to universality limits involving orthogonal polynomials


We show how localization and smoothing techniques can be used to establish universality in the bulk of the spectrum for a fixed positive measure $\mu $ on $[-1,1] $. Assume that $\mu $ is a regular measure, and is absolutely continuous in an open interval containing some point $x$. Assume moreover, that $\mu ^{\prime }$ is positive and continuous at $x$. Then universality for $\mu $ holds at $x$. If the hypothesis holds for $x$ in a compact subset of $\left( -1,1\right) $, 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 on $\mu .$