### 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) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - 2015 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21624

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

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21624

}

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 -