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

Sandrine Péché

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 1, page 80-106
  • ISSN: 0246-0203

Abstract

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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 ≤ i ≤ N, 1 ≤ j ≤ p, 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, ∞).

How to cite

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Péché, Sandrine. "Universality in the bulk of the spectrum for complex sample covariance matrices." Annales de l'I.H.P. Probabilités et statistiques 48.1 (2012): 80-106. <http://eudml.org/doc/271975>.

@article{Péché2012,
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 ≤ i ≤ N, 1 ≤ j ≤ p, 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, ∞).},
author = {Péché, Sandrine},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random matrix; bulk universality; sample covariance matrices},
language = {eng},
number = {1},
pages = {80-106},
publisher = {Gauthier-Villars},
title = {Universality in the bulk of the spectrum for complex sample covariance matrices},
url = {http://eudml.org/doc/271975},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Péché, Sandrine
TI - Universality in the bulk of the spectrum for complex sample covariance matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 1
SP - 80
EP - 106
AB - 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 ≤ i ≤ N, 1 ≤ j ≤ p, 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, ∞).
LA - eng
KW - random matrix; bulk universality; sample covariance matrices
UR - http://eudml.org/doc/271975
ER -

References

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  1. [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. Zbl0643.33001MR167642
  2. [2] Z. D. Bai. Convergence rate of expected spectral distributions of large random matrices. II Sample covariance matrices. Ann. Probab. 21 (1993) 649–672. Zbl0779.60025MR1217560
  3. [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. Zbl0928.60007MR1684746
  4. [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. Zbl1075.62014MR2162782
  5. [5] E. Brézin and S. Hikami. Spectral form factor in random matrix theory. Phys. Rev. E55 (1997) 4067–4083. MR1449379
  6. [6] E. Brézin and S. Hikami. Correlations of nearby levels induced by a random potential. Nucl. Phys. B479 (1996) 697–706. Zbl0925.82117MR1418841
  7. [7] P. Deift. Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics 3. American Mathematical Society, Providence, RI, 1999. Zbl0997.47033MR1677884
  8. [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. Zbl0944.42013MR1702716
  9. [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. Zbl1026.42024MR1711036
  10. [10] F. J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys.3 (1962) 1191–1198. Zbl0111.32703MR148397
  11. [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. Zbl1186.60005MR2481753
  12. [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. Zbl1175.15028MR2537522
  13. [13] L. Erdős, B. Schlein and H.-T. Yau. Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Notices2010 (2010) 436–479. Zbl1204.15043MR2587574
  14. [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. Zbl1216.15025MR2662426
  15. [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 Letters17 (2010) 667–674. Zbl1277.15027MR2661171
  16. [16] A. Guionnet and O. Zeitouni. Concentration of the spectral measure for large random matrices. Electron. Comm. Probab.5 (2000) 119–136. Zbl0969.15010MR1781846
  17. [17] K. Johansson. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys.215 (2001) 683–705. Zbl0978.15020MR1810949
  18. [18] V. A. Marčenko and L. Pastur. The distribution of eigenvalues for some sets of random matrices. Math. Sb.72 (1967) 507–536. Zbl0152.16101MR208649
  19. [19] M. L. Mehta. Random Matrices. Academic Press, New York, 1991. Zbl0780.60014MR1083764
  20. [20] J. W. Silverstein. Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. J. Multivariate Anal.55 (1995) 331–339. Zbl0851.62015MR1370408
  21. [21] F. Olver. Asymptotics and Special Functions. Computer Science and Applied Mathematics. Academic Press, New York–London, 1974. Zbl0303.41035MR435697
  22. [22] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics. Preprint. Available at arxiv:0906.0510. Zbl1217.15043MR2784665
  23. [23] T. Tao and V. Vu. Random covariance matrices: Universality of local statistics of eigenvalues. Preprint. Available at arXiv:0912.0966. Zbl1247.15036MR2962092

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