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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Universality of local spectral statistics of random matrices

Authors: László Erdős and Horng-Tzer Yau
Journal: Bull. Amer. Math. Soc. 49 (2012), 377-414
MSC (2010): Primary 15B52, 82B44
Published electronically: January 30, 2012
MathSciNet review: 2917064
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Abstract: 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 $ \beta = 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. For $ \beta \notin \{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 $ \beta > 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.

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Additional Information

László Erdős
Affiliation: Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany

Horng-Tzer Yau
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Keywords: Random matrix, local semicircle law, Tracy-Widom distribution, Dyson Brownian motion
Received by editor(s): June 24, 2011
Received by editor(s) in revised form: December 28, 2011
Published electronically: January 30, 2012
Additional Notes: The first author was partially supported by SFB-TR 12 Grant of the German Research Council
The second author was partially supported by NSF grants DMS-0757425, 0804279
Article copyright: © Copyright 2012 American Mathematical Society