Elsevier

Advances in Mathematics

Volume 219, Issue 3, 20 October 2008, Pages 743-779
Advances in Mathematics

Universality limits in the bulk for varying measures

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Abstract

Universality limits are a central topic in the theory of random matrices. We establish universality limits in the bulk of the spectrum for varying measures, using the theory of entire functions of exponential type. In particular, we consider measures that are of the form Wn2n(x)dx in the region where universality is desired. Wn does not need to be analytic, nor possess more than one derivative—and then only in the region where universality is desired. We deduce universality in the bulk for a large class of weights of the form W2n(x)dx, for example, when W=eQ where Q is convex and Q satisfies a Lipschitz condition of some positive order. We also deduce universality for a class of fixed exponential weights on a real interval.

Keywords

Universality limits
Random matrices
Orthogonal polynomials potential theory with external fields

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Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353.