---
_id: '2775'
abstract:
- lang: eng
text: The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue
statistics of large random matrices exhibit universal behavior depending only
on the symmetry class of the matrix ensemble. For invariant matrix models, the
eigenvalue distributions are given by a log-gas with potential V and inverse temperature
β = 1, 2, 4, corresponding to the orthogonal, unitary and symplectic ensembles.
For β ∉ {1, 2, 4}, there is no natural random matrix ensemble behind this model,
but the statistical physics interpretation of the log-gas is still valid for all
β > 0. The universality conjecture for invariant ensembles asserts that the
local eigenvalue statistics are independent of V. In this article, we review our
recent solution to the universality conjecture for both invariant and non-invariant
ensembles. We will also demonstrate that the local ergodicity of the Dyson Brownian
motion is the intrinsic mechanism behind the universality. Furthermore, we review
the solution of Dyson's conjecture on the local relaxation time of the Dyson Brownian
motion. Related questions such as delocalization of eigenvectors and local version
of Wigner's semicircle law will also be discussed.
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
citation:
ama: Erdös L, Yau H. Universality of local spectral statistics of random matrices.
*Bulletin of the American Mathematical Society*. 2012;49(3):377-414. doi:10.1090/S0273-0979-2012-01372-1
apa: Erdös, L., & Yau, H. (2012). Universality of local spectral statistics
of random matrices. *Bulletin of the American Mathematical Society*. American
Mathematical Society. https://doi.org/10.1090/S0273-0979-2012-01372-1
chicago: Erdös, László, and Horng Yau. “Universality of Local Spectral Statistics
of Random Matrices.” *Bulletin of the American Mathematical Society*. American
Mathematical Society, 2012. https://doi.org/10.1090/S0273-0979-2012-01372-1.
ieee: L. Erdös and H. Yau, “Universality of local spectral statistics of random
matrices,” *Bulletin of the American Mathematical Society*, vol. 49, no.
3. American Mathematical Society, pp. 377–414, 2012.
ista: Erdös L, Yau H. 2012. Universality of local spectral statistics of random
matrices. Bulletin of the American Mathematical Society. 49(3), 377–414.
mla: Erdös, László, and Horng Yau. “Universality of Local Spectral Statistics of
Random Matrices.” *Bulletin of the American Mathematical Society*, vol. 49,
no. 3, American Mathematical Society, 2012, pp. 377–414, doi:10.1090/S0273-0979-2012-01372-1.
short: L. Erdös, H. Yau, Bulletin of the American Mathematical Society 49 (2012)
377–414.
date_created: 2018-12-11T11:59:32Z
date_published: 2012-01-30T00:00:00Z
date_updated: 2021-01-12T06:59:36Z
day: '30'
doi: 10.1090/S0273-0979-2012-01372-1
extern: 1
intvolume: ' 49'
issue: '3'
month: '01'
page: 377 - 414
publication: Bulletin of the American Mathematical Society
publication_status: published
publisher: American Mathematical Society
publist_id: '4115'
quality_controlled: 0
status: public
title: Universality of local spectral statistics of random matrices
type: journal_article
volume: 49
year: '2012'
...