Fixed Energy Universality for Generalized Wigner Matrices

Paul Bourgade; Laszlo Erdős; Horng Tzer Yau; Jun Yin

doi:10.1002/cpa.21624

Fixed Energy Universality for Generalized Wigner Matrices

Paul Bourgade, Laszlo Erdős, Horng Tzer Yau, Jun Yin

Mathematics

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

Abstract

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.

Original language	English (US)
Pages (from-to)	1815-1881
Number of pages	67
Journal	Communications on Pure and Applied Mathematics
Volume	69
Issue number	10
DOIs	https://doi.org/10.1002/cpa.21624
State	Published - Oct 1 2016

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1002/cpa.21624

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abstract = "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.",

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N2 - 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.

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Fixed Energy Universality for Generalized Wigner Matrices

Abstract

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