Fixed Energy Universality for Generalized Wigner Matrices

Paul Bourgade, L. Erdos, H. T. Yau, J. Yin

Research output: Contribution to journalArticle

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 languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - 2015

Fingerprint

Universality
Energy
Homogenization Theory
Brownian movement
Hermitian matrix
Random Matrices
Averaging
Brownian motion
Statistics

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Fixed Energy Universality for Generalized Wigner Matrices. / Bourgade, Paul; Erdos, L.; Yau, H. T.; Yin, J.

In: Communications on Pure and Applied Mathematics, 2015.

Research output: Contribution to journalArticle

@article{a417d3483d814daeaaa57ba9c6ac3f5c,
title = "Fixed Energy Universality for Generalized Wigner Matrices",
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.",
author = "Paul Bourgade and L. Erdos and Yau, {H. T.} and J. Yin",
year = "2015",
doi = "10.1002/cpa.21624",
language = "English (US)",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",

}

TY - JOUR

T1 - Fixed Energy Universality for Generalized Wigner Matrices

AU - Bourgade, Paul

AU - Erdos, L.

AU - Yau, H. T.

AU - Yin, J.

PY - 2015

Y1 - 2015

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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84950327437&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84950327437&partnerID=8YFLogxK

U2 - 10.1002/cpa.21624

DO - 10.1002/cpa.21624

M3 - Article

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -