Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality in the bulk of the spectrum for complex sample covariance matrices

Sandrine Péché

Full-text: Open access

Abstract

We consider complex sample covariance matrices MN = (1/N)YY* where Y is a N × p random matrix with i.i.d. entries Yij, 1 ≤ iN, 1 ≤ jp, with distribution F. Under some regularity and decay assumptions on F, we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit where N → ∞ and limN→∞p/N = γ for any real number γ ∈ (0, ∞).

Résumé

On considère des matrices de covariance empirique complexes MN = (1/N)YY* où Y est une matrice de taille N × p dont les coefficients Yij, 1 ≤ iN, 1≤jp, sont des variables aléatoires i.i.d. de loi F. Sous certaines hypothèses de régularité et de décroissance sur F, on montre l’universalité de certaines statistiques locales de valeurs propres au milieu du spectre quand N → ∞ et limN→∞p/N = γ pour tout réel γ ∈ (0, ∞).

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 1 (2012), 80-106.

Dates
First available in Project Euclid: 23 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1327328015

Digital Object Identifier
doi:10.1214/11-AIHP442

Mathematical Reviews number (MathSciNet)
MR2919199

Zentralblatt MATH identifier
1238.60010

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60B10: Convergence of probability measures 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Random matrix Bulk universality Sample covariance matrices

Citation

Péché, Sandrine. Universality in the bulk of the spectrum for complex sample covariance matrices. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 1, 80--106. doi:10.1214/11-AIHP442. https://projecteuclid.org/euclid.aihp/1327328015


Export citation

References

  • [1] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.
  • [2] Z. D. Bai. Convergence rate of expected spectral distributions of large random matrices. II Sample covariance matrices. Ann. Probab. 21 (1993) 649–672.
  • [3] Z. D. Bai, B. Miao and J. Tsay. Remarks on the convergence rate of the spectral distributions of Wigner matrices. J. Theoret. Probab. 12 (1999) 301–311.
  • [4] G. Ben Arous and S. Péché. Universality of local eigenvalue statistics for some sample covariance matrices. Comm. Pure Appl. Math. LVIII (2005) 1–42.
  • [5] E. Brézin and S. Hikami. Spectral form factor in random matrix theory. Phys. Rev. E 55 (1997) 4067–4083.
  • [6] E. Brézin and S. Hikami. Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479 (1996) 697–706.
  • [7] P. Deift. Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics 3. American Mathematical Society, Providence, RI, 1999.
  • [8] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides and X. Zhou. Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (1999) 1335–1425.
  • [9] P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides and X. Zhou. Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (1999) 1491–1552.
  • [10] F. J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191–1198.
  • [11] L. Erdős, B. Schlein and H.-T. Yau. Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287 (2009) 641–655.
  • [12] L. Erdős, B. Schlein and H.-T. Yau. Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 (2009) 815–852.
  • [13] L. Erdős, B. Schlein and H.-T. Yau. Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Notices 2010 (2010) 436–479.
  • [14] L. Erdős, S. Péché, J. Ramirez, B. Schlein and H.-T. Yau. Bulk universality for Wigner matrices. Comm. Pure Appl. Math. 63 (2010) 895–925.
  • [15] L. Erdős, J. Ramirez, B. Schlein, T. Tao, V. Vu and H.-T. Yau. Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Research Letters 17 (2010) 667–674.
  • [16] A. Guionnet and O. Zeitouni. Concentration of the spectral measure for large random matrices. Electron. Comm. Probab. 5 (2000) 119–136.
  • [17] K. Johansson. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215 (2001) 683–705.
  • [18] V. A. Marčenko and L. Pastur. The distribution of eigenvalues for some sets of random matrices. Math. Sb. 72 (1967) 507–536.
  • [19] M. L. Mehta. Random Matrices. Academic Press, New York, 1991.
  • [20] J. W. Silverstein. Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. J. Multivariate Anal. 55 (1995) 331–339.
  • [21] F. Olver. Asymptotics and Special Functions. Computer Science and Applied Mathematics. Academic Press, New York–London, 1974.
  • [22] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics. Preprint. Available at arxiv:0906.0510.
  • [23] T. Tao and V. Vu. Random covariance matrices: Universality of local statistics of eigenvalues. Preprint. Available at arXiv:0912.0966.