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Universality of random matrices and local relaxation flow

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Abstract

Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we prove that the relaxation time to local equilibrium for the Dyson Brownian motion is bounded above by N −ζ for some ζ>0. The proof is based on an estimate of the entropy flow of the Dyson Brownian motion w.r.t. a “pseudo equilibrium measure”. As an application of this estimate, we prove that the eigenvalue spacing statistics in the bulk of the spectrum for N×N symmetric Wigner ensemble is the same as that of the Gaussian Orthogonal Ensemble (GOE) in the limit N→∞. The assumptions on the probability distribution of the matrix elements of the Wigner ensemble are a subexponential decay and some minor restriction on the support.

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Authors and Affiliations

  1. Institute of Mathematics, University of Munich, Theresienstr. 39, 80333, Munich, Germany

    László Erdős

  2. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WB, UK

    Benjamin Schlein

  3. Harvard University, Department of Mathematics, Cambridge, MA, 02138, USA

    Horng-Tzer Yau

Authors
  1. László Erdős
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  2. Benjamin Schlein
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  3. Horng-Tzer Yau
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Corresponding author

Correspondence to László Erdős.

Additional information

L. Erdős was partially supported by SFB-TR 12 Grant of the German Research Council.

H.-T. Yau was partially supported by NSF grants DMS-0757425, 0804279.

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Erdős, L., Schlein, B. & Yau, HT. Universality of random matrices and local relaxation flow. Invent. math. 185, 75–119 (2011). https://doi.org/10.1007/s00222-010-0302-7

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