Inventiones mathematicae

, Volume 185, Issue 1, pp 75–119 | Cite as

Universality of random matrices and local relaxation flow

  • László ErdősEmail author
  • Benjamin Schlein
  • Horng-Tzer Yau


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.

Mathematics Subject Classification (2000)

15A52 82B44 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • László Erdős
    • 1
    Email author
  • Benjamin Schlein
    • 2
  • Horng-Tzer Yau
    • 3
  1. 1.Institute of MathematicsUniversity of MunichMunichGermany
  2. 2.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK
  3. 3.Harvard UniversityCambridgeUSA

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