---
_id: '2765'
abstract:
- lang: eng
text: 'This is a study of the universality of spectral statistics for large random
matrices. Considered are N×N symmetric, Hermitian, or quaternion self-dual random
matrices with independent identically distributed entries (Wigner matrices), where
the probability distribution of each matrix element is given by a measure v with
zero expectation and with subexponential decay. The main result is that the correlation
functions of the local eigenvalue statistics in the bulk of the spectrum coincide
with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble
(GUE), and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit
as N → ∞. This approach is based on a study of the Dyson Brownian motion via a
related new dynamics, the local relaxation flow. As a main input, it is established
that the density of the eigenvalues converges to the Wigner semicircle law, and
this holds even down to the smallest possible scale. Moreover, it is shown that
the eigenvectors are completely delocalized. These results hold even without the
condition that the matrix elements are identically distributed: only independence
is used. In fact, for the matrix elements of the Green function strong estimates
are given that imply that the local statistics of any two ensembles in the bulk
are identical if the first four moments of the matrix elements match. Universality
at the spectral edges requires matching only two moments. A Wigner-type estimate
is also proved, and it is shown that the eigenvalues repel each other on arbitrarily
small scales.'
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
citation:
ama: 'Erdös L. Universality of Wigner random matrices: A survey of recent results.
*Russian Mathematical Surveys*. 2011;66(3):507-626. doi:10.1070/RM2011v066n03ABEH004749'
apa: 'Erdös, L. (2011). Universality of Wigner random matrices: A survey of recent
results. *Russian Mathematical Surveys*. IOP Publishing Ltd. https://doi.org/10.1070/RM2011v066n03ABEH004749'
chicago: 'Erdös, László. “Universality of Wigner Random Matrices: A Survey of Recent
Results.” *Russian Mathematical Surveys*. IOP Publishing Ltd., 2011. https://doi.org/10.1070/RM2011v066n03ABEH004749.'
ieee: 'L. Erdös, “Universality of Wigner random matrices: A survey of recent results,”
*Russian Mathematical Surveys*, vol. 66, no. 3. IOP Publishing Ltd., pp.
507–626, 2011.'
ista: 'Erdös L. 2011. Universality of Wigner random matrices: A survey of recent
results. Russian Mathematical Surveys. 66(3), 507–626.'
mla: 'Erdös, László. “Universality of Wigner Random Matrices: A Survey of Recent
Results.” *Russian Mathematical Surveys*, vol. 66, no. 3, IOP Publishing
Ltd., 2011, pp. 507–626, doi:10.1070/RM2011v066n03ABEH004749.'
short: L. Erdös, Russian Mathematical Surveys 66 (2011) 507–626.
date_created: 2018-12-11T11:59:29Z
date_published: 2011-01-01T00:00:00Z
date_updated: 2019-04-26T07:22:20Z
day: '01'
doi: 10.1070/RM2011v066n03ABEH004749
extern: 1
intvolume: ' 66'
issue: '3'
main_file_link:
- open_access: '0'
url: http://arxiv.org/abs/1004.0861
month: '01'
page: 507 - 626
publication: Russian Mathematical Surveys
publication_status: published
publisher: IOP Publishing Ltd.
publist_id: '4125'
quality_controlled: 0
status: public
title: 'Universality of Wigner random matrices: A survey of recent results'
type: review
volume: 66
year: '2011'
...