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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
Article

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.

Mathematics Subject Classification (2000)

15A52 82B44 

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References

  1. 1.
    Bakry, D., Émery, M.: Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84. Lecture Notes in Mathematics, vol. 1123, pp. 177–206. Springer, Berlin (1985) CrossRefGoogle Scholar
  2. 2.
    Ben Arous, G., Péché, S.: Universality of local eigenvalue statistics for some sample covariance matrices. Commun. Pure Appl. Math. LVIII, 1–42 (2005) Google Scholar
  3. 3.
    Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bobkov, S.G., Götze, F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163(1), 1–28 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brézin, E., Hikami, S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479, 697–706 (1996) zbMATHCrossRefGoogle Scholar
  6. 6.
    Brézin, E., Hikami, S.: Spectral form factor in a random matrix theory. Phys. Rev. E 55, 4067–4083 (1997) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Erdős, L., Schlein, B., Yau, H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37(3), 815–852 (2009) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Erdős, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. 2010(3), 436–479 (2010) Google Scholar
  12. 12.
    Erdős, L., Ramirez, J., Schlein, B., Yau, H.-T.: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15(18), 526–604 (2010) MathSciNetGoogle Scholar
  13. 13.
    Erdős, L., Péché, S., Ramírez, J., Schlein, B., Yau, H.-T.: Bulk universality for Wigner matrices. Commun. Pure Appl. Math. 63, 895–925 (2010) Google Scholar
  14. 14.
    Erdős, L., Ramírez, J., Schlein, B., Tao, T., Vu, V., Yau, H.-T.: Bulk universality for Wigner hermitian matrices with subexponential decay. Math. Res. Lett. 17(4), 667–674 (2010). arXiv:0906.4400 MathSciNetGoogle Scholar
  15. 15.
    Erdős, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. arXiv:0911.3687
  16. 16.
    Erdős, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. arXiv:1001.3453
  17. 17.
    Erdős, L., Yau, H.-T., Yin, J.: Universality for generalized Wigner matrices with Bernoulli distribution. arXiv:1003.3813
  18. 18.
    Erdős, L., Yau, H.-T., Yin, J.: Rigidity of eigenvalues of generalized Wigner matrices. arXiv:1007.4652
  19. 19.
    Guionnet, A.: Large Random Matrices: Lectures on Macroscopic Asymptotics. École d’Et́é de Probabilités de Saint-Flour, vol. XXXVI. Springer, Berlin (2006) Google Scholar
  20. 20.
    Gustavsson, J.: Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. H. Poincaré, Probab. Stat. 41(2), 151–178 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hanson, D.L., Wright, F.T.: A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42(3), 1079–1083 (1971) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence (2001) zbMATHGoogle Scholar
  24. 24.
    Lu, S.-L., Yau, H.-T.: Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics. Commun. Math. Phys. 156, 399–433 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mehta, M.L.: Random Matrices. Academic Press, New York (1991) zbMATHGoogle Scholar
  26. 26.
    Pastur, L., Shcherbina, M.: Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130(2), 205–250 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sinai, Y., Soshnikov, A.: A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge. Funct. Anal. Appl. 32(2), 114–131 (1998) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207(3), 697–733 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Tao, T., Vu, V.: Random matrices: universality of the local eigenvalue statistics. arXiv:0906.0510
  30. 30.
    Vu, V.: Spectral norm of random matrices. Combinatorica 27(6), 721–736 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Wright, F.T.: A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probab. 1(6), 1068–1070 (1973) zbMATHCrossRefGoogle Scholar
  32. 32.
    Yau, H.T.: Relative entropy and the hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22, 63–80 (1991) MathSciNetzbMATHCrossRefGoogle Scholar

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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