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Bulk universality for Wigner matrices
Author(s) -
Erdős László,
Péché Sandrine,
Ramírez José A.,
Schlein Benjamin,
Yau HorngTzer
Publication year - 2010
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20317
Subject(s) - mathematics , universality (dynamical systems) , hermitian matrix , random matrix , eigenvalues and eigenvectors , brownian motion , mathematical physics , density matrix , probability density function , sine , pure mathematics , mathematical analysis , quantum , quantum mechanics , statistics , geometry , physics
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 ∈ C 6 ( \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.