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Bulk universality for Wigner matrices

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2010

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Wiley-Blackwell
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Erdős, László, Sandrine Péché, José A. Ramírez, Benjamin Schlein, and Horng-Tzer Yau. 2010. “Bulk Universality for Wigner Matrices.” Communications on Pure and Applied Mathematics. doi:10.1002/cpa.20317.

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Abstract

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C6( \input amssym $\Bbb R$) with at most polynomially growing derivatives and ν(x) ≥ Ce−C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.

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Wigner random matrix, Dyson sine kernel

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