A note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

Sandrine Dallaporta (Institut de Mathématiques de Toulouse)
Van H. Vu (Department of Mathematics, Rutgers)

Abstract


The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd?s, Yau and Yin.

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

Pages: 214-322

Publication Date: June 22, 2011

DOI: 10.1214/ECP.v16-1634

References

  1. G. Anderson, A. Guionnet, O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118 (2010). Math. Review 2760897
  2. O. Costin, J. L. Lebowitz. Gaussian Fluctuations in Random Matrices. Phys. Rev. Lett. 75 (1995), 69--72.
  3. L. Erd?s, H-T. Yau, J. Yin. Rigidity of Eigenvalues of Generalized Wigner Matrices. arXiv:1007.4652 (2010).
  4. P. Forrester, E. Rains. Inter-Relationships between Orthogonal, Unitary and Symplectic Matrix Ensembles. Cambridge University Press, Cambridge, United Kingdom (2001), 171--208. Math. Review 1842786
  5. J. Gustavsson. Gaussian Fluctuations of Eigenvalues in the GUE. Ann. I. PoincarÈ 41 (2005), 151--178. Math. Review 2124079
  6. S. O'Rourke. Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices. to appear in J. Stat. Phys., arXiv:0909.2677 (2009).
  7. A. Soshnikov. Gaussian Fluctuation for the Number of Particles in Airy, Bessel, Sine, and other Determinantal Random Point Fields. J. Statist. Phys. 100 (2000), 3-4, 491--522. Math. Review 1788476
  8. Z. Su, Gaussian Fluctuations in Complex Sample Covariance Matrices. Electronic Journal of Probability 11 (2006), 1284--1320. Math. Review 2268545
  9. T. Tao and V. Vu. Random Matrices: Universality of Local Eigenvalues Statistics. to appear in Acta Math., arXiv:0906.0510 (2009).
  10. T. Tao and V. Vu. Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge. Comm. Math. Phys. 298 (2010), 2, 549--572. Math. Review 2669449
  11. T. Tao and V. Vu. Random Covariance Matrices: Universality of Local Statistics of Eigenvalues. arXiv:0912.0966 (2010).
Bookmark and Share


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