Communications in Mathematical Physics

, Volume 298, Issue 2, pp 549–572

Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge

Open Access
Article

Abstract

This is a continuation of our earlier paper (Tao and Vu, http://arxiv.org/abs/0908.1982v4[math.PR], 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of this paper are an extension of the results in Tao and Vu (http://arxiv.org/abs/0908.1982v4[math.PR], 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.

References

  1. 1.
    Anderson, G., Guionnet, A., Zeitouni, O.: An introduction to random matrices. To be published by Cambridge Univ. Press Google Scholar
  2. 2.
    Bai, Z.D., Silverstein, J.: Spectral analysis of large dimensional random matrices. Mathematics Monograph Series 2, Beijing: Science Press, 2006Google Scholar
  3. 3.
    Bai Z.D., Yin Y.Q.: Convergence to the semicircle law. Ann. Probab. 16, 863–875 (1988)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bai Z.D., Yin Y.Q.: Necessary and Sufficient Conditions for Almost Sure Convergence of the Largest Eigenvalue of a Wigner Matrix. Ann. Probab. 16, 1729–1741 (1988)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Deift, P.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, 3. New York University, Courant Institute of Mathematical Sciences, New York; Providence, RI: Amer. Math. Soc., 1999Google Scholar
  6. 6.
    Deift, P.: Universality for mathematical and physical systems. In: International Congress of Mathematicians Vol. I, Zürich: Eur. Math. Soc., 2007, pp. 125–152Google Scholar
  7. 7.
    Erdős L., Schlein B., Yau H.-T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Prob. 37(3), 815–852 (2009)CrossRefGoogle Scholar
  8. 8.
    Erdős L., Schlein B., Yau H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287(2), 641–655 (2009)CrossRefADSGoogle Scholar
  9. 9.
    Erdős, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Submitted, available at http://arxiv.org/abs/0811.2591v3[math.ph], 2009
  10. 10.
    Erdős, L., Schlein, B., Yau, H.-T.: Universality of Random Matrices and Local Relaxation Flow. http://arxiv.org/abs/0907.5605v3[math-ph], 2009
  11. 11.
    Erdős, L., Ramirez, J., Schlein, B., Yau, H.-T.: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. http://arxiv.org/abs/0905.2089v1[math-ph], 2009
  12. 12.
    Erdős, L., Ramirez, J., Schlein, B., Yau, H.-T.: Bulk universality for Wigner matrices. http://arxiv.org/abs/0905.4176v2[math-ph], 2009
  13. 13.
    Erdős, L., Ramirez, J., Schlein, B., Tao, T., Vu, V., Yau, H.-T.: Bulk universality for Wigner hermitian matrices with subexponential decay. http://arxiv.org/abs/0906.4400v1[math.PR], 2009
  14. 14.
    Forrester P.: The spectral edge of random matrix ensembles. Nucl. Phys. B 402, 709–728 (1993)MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Johansson K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001)MATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Johansson, K.: Universality for certain Hermitian Wigner matrices under weak moment conditions, preprintGoogle Scholar
  17. 17.
    Katz, N., Sarnak, P.: Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Publications, 45. Providence, RI: Amer. Math. Soc., 1999Google Scholar
  18. 18.
    Khorunzhiy, O.: High Moments of Large Wigner Random Matrices and Asymptotic Properties of the Spectral Norm. http://arxiv.org/abs/0907.3743v2[math.PR], 2009
  19. 19.
    Mehta M.L.: Random Matrices and the Statistical Theory of Energy Levels. Academic Press, New York (1967)MATHGoogle Scholar
  20. 20.
    Péché S., Soshnikov A.: On the lower bound of the spectral norm of symmetric random matrices with independent entries. Electron. Commun. Probab. 13, 280–290 (2008)MATHMathSciNetGoogle Scholar
  21. 21.
    Péché S., Soshnikov A.: Wigner random matrices with non-symmetrically distributed entries. J. Stat. Phys. 129(5–6), 857–884 (2007)MATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Ruzmaikina A.: Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Commun. Math. Phys. 261(2), 277–296 (2006)MATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Sinai Y., Soshnikov A.: Central limit theorem for traces of large symmetric matrices with independent matrix elements. Bol. Soc. Brazil. Mat. 29, 1–24 (1998)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Sinai Y., Soshnikov A.: A refinement of Wigners semicircle law in a neighborhood of the spectrum edge for random symmetric matrices. Func. Anal. Appl. 32, 114–131 (1998)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Soshnikov A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207(3), 697–733 (1999)MATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Soshnikov A.: Gaussian limit for determinantal random point fields. Ann. Probab. 30(1), 171–187 (2002)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Tao, T., Vu, V.: Random matrices: Universality of the local eigenvalue statistics. Submitted, available at http://arxiv.org/abs/0908.1982v4[math.PR], 2010
  28. 28.
    Tracy C., Widom H.: Level spacing distribution and Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)MATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Tracy C., Widom H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)MATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Department of MathematicsRutgersPiscatawayUSA

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