Universality in the bulk of the spectrum for complex sample covariance matrices
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 1, p. 80-106

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, ∞).

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, ∞).

DOI : https://doi.org/10.1214/11-AIHP442
Classification:  60B20,  60B10,  60B12
Keywords: random matrix, bulk universality, sample covariance matrices
@article{AIHPB_2012__48_1_80_0,
     author = {P\'ech\'e, Sandrine},
     title = {Universality in the bulk of the spectrum for complex sample covariance matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {1},
     year = {2012},
     pages = {80-106},
     doi = {10.1214/11-AIHP442},
     zbl = {1238.60010},
     mrnumber = {2919199},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_1_80_0}
}
Péché, Sandrine. Universality in the bulk of the spectrum for complex sample covariance matrices. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 1, pp. 80-106. doi : 10.1214/11-AIHP442. http://www.numdam.org/item/AIHPB_2012__48_1_80_0/

[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. | MR 167642 | Zbl 0643.33001

[2] Z. D. Bai. Convergence rate of expected spectral distributions of large random matrices. II Sample covariance matrices. Ann. Probab. 21 (1993) 649-672. | MR 1217560 | Zbl 0779.60025

[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. | MR 1684746 | Zbl 0928.60007

[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. | MR 2162782 | Zbl 1075.62014

[5] E. Brézin and S. Hikami. Spectral form factor in random matrix theory. Phys. Rev. E 55 (1997) 4067-4083. | MR 1449379

[6] E. Brézin and S. Hikami. Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479 (1996) 697-706. | MR 1418841 | Zbl 0925.82117

[7] P. Deift. Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics 3. American Mathematical Society, Providence, RI, 1999. | MR 1677884 | Zbl 0997.47033

[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. | MR 1702716 | Zbl 0944.42013

[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. | MR 1711036 | Zbl 1026.42024

[10] F. J. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191-1198. | MR 148397 | Zbl 0111.32703

[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. | MR 2481753 | Zbl 1186.60005

[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. | MR 2537522 | Zbl 1175.15028

[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. | MR 2587574 | Zbl 1204.15043

[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. | MR 2662426 | Zbl 1216.15025

[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. | MR 2661171 | Zbl 1277.15027

[16] A. Guionnet and O. Zeitouni. Concentration of the spectral measure for large random matrices. Electron. Comm. Probab. 5 (2000) 119-136. | MR 1781846 | Zbl 0969.15010

[17] K. Johansson. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215 (2001) 683-705. | MR 1810949 | Zbl 0978.15020

[18] V. A. Marčenko and L. Pastur. The distribution of eigenvalues for some sets of random matrices. Math. Sb. 72 (1967) 507-536. | MR 208649 | Zbl 0152.16101

[19] M. L. Mehta. Random Matrices. Academic Press, New York, 1991. | MR 1083764 | Zbl 0780.60014

[20] J. W. Silverstein. Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices. J. Multivariate Anal. 55 (1995) 331-339. | MR 1370408 | Zbl 0851.62015

[21] F. Olver. Asymptotics and Special Functions. Computer Science and Applied Mathematics. Academic Press, New York-London, 1974. | MR 435697 | Zbl 0303.41035

[22] T. Tao and V. Vu. Random matrices: Universality of local eigenvalue statistics. Preprint. Available at arxiv:0906.0510. | MR 2784665 | Zbl 1217.15043

[23] T. Tao and V. Vu. Random covariance matrices: Universality of local statistics of eigenvalues. Preprint. Available at arXiv:0912.0966. | MR 2962092 | Zbl 1247.15036