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Bulk universality for Wigner matrices |
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| Authors: | László Erdős Sandrine Péché José A. Ramírez Benjamin Schlein Horng‐Tzer Yau |
| Affiliation: | 1. University of Munich, Institute of Mathematics, Theresienstrasse 39, D‐80333 Munich, Germany;2. University of Grenoble 1, Institut Fourier, 100 rue des Maths, BP 74, 38402 St. Martin d' Hères, France;3. Universidad de Costa Rica, Department of Mathematics, San Jose 2060, Costa Rica;4. University of Cambridge, Department of Pure Mathematics, and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, United Kingdom;5. Harvard University, Department of Mathematics, Cambridge MA 02138 |
| 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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