Universality of local spectral statistics of random matrices
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- by László Erdős and Horng-Tzer Yau PDF
- Bull. Amer. Math. Soc. 49 (2012), 377-414 Request permission
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.References
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Additional Information
- László Erdős
- Affiliation: Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany
- MR Author ID: 343945
- Email: lerdos@math.lmu.de
- Horng-Tzer Yau
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 237212
- Email: htyau@math.harvard.edu
- 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 - © Copyright 2012 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 49 (2012), 377-414
- MSC (2010): Primary 15B52, 82B44
- DOI: https://doi.org/10.1090/S0273-0979-2012-01372-1
- MathSciNet review: 2917064