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Fixed Energy Universality for Generalized Wigner Matrices
Author(s) -
Bourgade Paul,
Erdős Laszlo,
Yau HorngTzer,
Yin Jun
Publication year - 2016
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.21624
Subject(s) - universality (dynamical systems) , hermitian matrix , mathematics , mesoscopic physics , conjecture , mathematical physics , random matrix , brownian motion , homogenization (climate) , pure mathematics , statistical physics , quantum mechanics , eigenvalues and eigenvectors , physics , statistics , biodiversity , ecology , biology
We prove the Wigner‐Dyson‐Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.© 2016 Wiley Periodicals, Inc.