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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 3(399), Pages 67–198 (Mi umn9426)  

This article is cited in 52 scientific papers (total in 52 papers)

Universality of Wigner random matrices: a survey of recent results

L. Erdős

Ludwig-Maximilians-Universität München

Abstract: This is a study of the universality of spectral statistics for large random matrices. Considered are $N\times N$ 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 $\nu$ 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 $N\to\infty$. 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.
Bibliography: 108 titles.

Keywords: Wigner random matrices, Dyson Brownian motion, semicircle law, sine kernel.

DOI: https://doi.org/10.4213/rm9426

Full text: PDF file (1544 kB)
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English version:
Russian Mathematical Surveys, 2011, 66:3, 507–626

Bibliographic databases:

Document Type: Article
UDC: 519.216+519.217+512.64
MSC: 15B52, 82B44
Received: 07.04.2010

Citation: L. Erdős, “Universality of Wigner random matrices: a survey of recent results”, Uspekhi Mat. Nauk, 66:3(399) (2011), 67–198; Russian Math. Surveys, 66:3 (2011), 507–626

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