A comment on the Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices

László Erdős (LMU-University of Munich)
Horng-Tzer Yau (Harvard University)


Recently we proved that  the  eigenvalue correlation functions of a general class of random matrices converge,  weakly with respect to the energy, to  the corresponding ones of  Gaussian matrices. Tao and Vu gave a proof that for the special case of Hermitian Wigner matrices the convergence  can be strengthened to vague  convergence at any  fixed energy in the bulk. In this article we comment on this result in the context of the universality conjectures of Mehta. We show that this theorem is  an immediate corollary of our earlier results. Indeed,  a more general form of this theorem also follows directly from our previous work.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-5

Publication Date: April 10, 2012

DOI: 10.1214/EJP.v17-1779


  • Brézin, E.; Hikami, S. Correlations of nearby levels induced by a random potential. Nuclear Phys. B 479 (1996), no. 3, 697--706. MR1418841
  • Erdös, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral Statistics of Erdös-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues. ARXIV1103.3869
  • Erdős, László; Péché, Sandrine; Ramírez, José A.; Schlein, Benjamin; Yau, Horng-Tzer. Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 (2010), no. 7, 895--925. MR2662426
  • Erdős, László; Ramírez, José; Schlein, Benjamin; Tao, Terence; Vu, Van; Yau, Horng-Tzer. Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 17 (2010), no. 4, 667--674. MR2661171
  • Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 (2009), no. 2, 641--655. MR2481753
  • Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Universality of random matrices and local relaxation flow. Invent. Math. 185 (2011), no. 1, 75--119. MR2810797
  • Erdös, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. ph Annales Inst. H. Poincaré (B), Probability and Statistics. 48, no. 1, (2012), 1--46.
  • Erdös, L., Yau, H.-T.: Universality of local spectral statistics of random matrices. To appear in Bull. of Amer. Math. Soc. ARXIV1106.4986
  • Erdös, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. To appear in Prob. Theor. Rel. Fields. Preprint arXiv:1001.3453
  • Erdős, László; Yau, Horng-Tzer; Yin, Jun. Universality for generalized Wigner matrices with Bernoulli distribution. J. Comb. 2 (2011), no. 1, 15--81. MR2847916
  • Erdös, L., Yau, H.-T., Yin, J.: Rigidity of Eigenvalues of Generalized Wigner Matrices. To appear in Adv. Math. ARXIV1007.4652
  • Johansson, Kurt. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 (2001), no. 3, 683--705. MR1810949
  • Mehta, Madan Lal. Random matrices. Second edition. Academic Press, Inc., Boston, MA, 1991. xviii+562 pp. ISBN: 0-12-488051-7 MR1083764
  • Tao, Terence; Vu, Van. Random matrices: universality of local eigenvalue statistics. Acta Math. 206 (2011), no. 1, 127--204. MR2784665
  • Tao, Terence; Vu, Van. The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 16 (2011), no. 77, 2104--2121. MR2851058

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.