Universality limits in the bulk for varying measures

Eli Levin; Doron S. Lubinsky

doi:10.1016/j.aim.2008.06.010

Universality limits in the bulk for varying measures

Eli Levin, Doron S. Lubinsky

Mathematics discipline

Research output: Contribution to journal › Article › peer-review

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 W_n^{2 n} (x) d x in the region where universality is desired. W_n 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 W^{2 n} (x) d x, for example, when W = e^{- Q} 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.

Original language	English
Pages (from-to)	743-779
Number of pages	37
Journal	Advances in Mathematics
Volume	219
Issue number	3
DOIs	https://doi.org/10.1016/j.aim.2008.06.010
State	Published - 20 Oct 2008

Bibliographical note

Funding Information:
Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353. Corresponding author. E-mail addresses: [email protected] (E. Levin), [email protected] (D.S. Lubinsky).

Keywords

Orthogonal polynomials potential theory with external fields
Random matrices
Universality limits

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10.1016/j.aim.2008.06.010

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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 Wn2 n (x) d x 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 W2 n (x) d x, for example, when W = e- Q 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.",

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Universality limits in the bulk for varying measures

Abstract

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Keywords

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