zbMATH — the first resource for mathematics

The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. (English) Zbl 1163.15026
The asymptotic behavior of the first extremal eigenvalues of some complex or real deformed Wigner matrices \((M_N)_N\) is investigated. Such matrix models can be seen as additive analogue of the spiked population models and are defined by a sequence \((M_N)_N\) given by \(M_N = W_N/ \sqrt{N} + A_N\), where \(W_N\) is a Wigner matrix such that the common distribution of its entries satisfies some conditions (random variables with a symmetric distribution satisfying a Poincaré inequality) and \(A_N\) is a deterministic matrix of finite rank. In this way, an analogue of the main result of J. Baik and J. W. Silverstein [J. Multivariate Anal. 97, 1382–1408 (2006; Zbl 1220.15011)] is established, namely that, once \(A_N\) has exactly \(k\) (fixed) eigenvalues far enough from zero, the \(k\) first eigenvalues of \(M_N\) jump almost surely outside the limiting semicircle support. This result is universal since the corresponding limits only involve the variance of the entries of \(W_N\).
On the other hand, at the level of the fluctuations, a striking phenomenon in the particular case where \(A_N\) is diagonal with a sole simple nonzero eigenvalue large enough is exhibited. In this case, the fluctuations of the largest eigenvalue of \(M_N\) are not universal and strongly depend on the particular law of the entries of \(W_N\). It is proved that the limiting distribution of the (properly rescaled) largest value of \(M_N\) is the convolution of the distribution of the entries of \(W_N\) with a Gaussian law. In particular, if the entries of \(W_N\) are not Gaussian, the fluctuations of the largest eigenvalue of \(M_N\) are not Gaussian.

15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60F05 Central limit and other weak theorems
Full Text: DOI arXiv
[1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses [ Panoramas and Syntheses ] 10 . Société Mathématique de France, Paris. With a preface by Dominique Bakry and Michel Ledoux. · Zbl 0982.46026
[2] Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611-677. With comments by G. J. Rodgers and Jack W. Silverstein; and a rejoinder by the author. · Zbl 0949.60077
[3] Bai, Z. D. and Silverstein, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316-345. · Zbl 0937.60017
[4] Bai, Z. D. and Silverstein, J. W. (1999). Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27 1536-1555. · Zbl 0964.60041
[5] Bai, Z. D. and Yao, J. (2005). On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11 1059-1092. · Zbl 1101.60012
[6] Bai, Z. D. and Yao, J. F. (2007). Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. H. Poincaré 44 447-474. · Zbl 1274.62129
[7] Bai, Z. D. and Yin, Y. Q. (1988). Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16 1729-1741. · Zbl 0677.60038
[8] Baik, J., Ben Arous, G. and Péché, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643-1697. · Zbl 1086.15022
[9] Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382-1408. · Zbl 1220.15011
[10] Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley Series in Probability and Mathematical Statistics . Wiley, New York. · Zbl 0822.60002
[11] Biroli, G., Bouchaud, J.-P. and Potters, M. (2007). On the top eigenvalue of heavy-tailed random matrices. Europhys. Lett. EPL 78 Art. 10001, 5. · Zbl 1244.82029
[12] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1-28. · Zbl 0924.46027
[13] Capitaine, M. and Donati-Martin, C. (2007). Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56 767-803. · Zbl 1162.15013
[14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications II , 2nd ed. Wiley, New York. · Zbl 0219.60003
[15] Féral, D. (2006). Grandes déviations et fluctuations des valeurs propres maximales de matrices aléatoires. Ph.D. thesis, Univ. Toulouse.
[16] Féral, D. and Péché, S. (2007). The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272 185-228. · Zbl 1136.82016
[17] Füredi, Z. and Komlós, J. (1981). The eigenvalues of random symmetric matrices. Combinatorica 1 233-241. · Zbl 0494.15010
[18] Haagerup, U. and Thorbjørnsen, S. (2005). A new application of random matrices: Ext( C red * ( F 2 )) is not a group. Ann. of Math. (2) 162 711-775. · Zbl 1103.46032
[19] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis . Cambridge Univ. Press, Cambridge. Corrected reprint of the 1985 original. · Zbl 0704.15002
[20] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. · Zbl 1016.62078
[21] Khorunzhy, A. M., Khoruzhenko, B. A. and Pastur, L. A. (1996). Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 5033-5060. · Zbl 0866.15014
[22] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89 . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002
[23] Maïda, M. (2007). Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles. Electron. J. Probab. 12 1131-1150 (electronic). · Zbl 1127.60022
[24] Mathias, R. (1993). The Hadamard operator norm of a circulant and applications. SIAM J. Matrix Anal. Appl. 14 1152-1167. · Zbl 0786.15030
[25] Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 1617-1642. · Zbl 1134.62029
[26] Péché, S. (2006). The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 127-173. · Zbl 1088.15025
[27] Ruzmaikina, A. (2006). Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Comm. Math. Phys. 261 277-296. · Zbl 1130.82313
[28] Saporta, G. (1990). Probabilités, analyse des données et statistique . Gulf Pub., Houston, TX. · Zbl 0703.62003
[29] Schultz, H. (2005). Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. Probab. Theory Related Fields 131 261-309. · Zbl 1085.46045
[30] Silverstein, J. W. and Choi, S.-I. (1995). Analysis of the limiting spectral distribution of large-dimensional random matrices. J. Multivariate Anal. 54 295-309. · Zbl 0872.60013
[31] Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697-733. · Zbl 1062.82502
[32] Tillmann, H.-G. (1953). Randverteilungen analytischer Funktionen und Distributionen. Math. Z. 59 61-83. · Zbl 0051.08901
[33] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151-174. · Zbl 0789.35152
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.