The Annals of Probability

The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations

Mireille Capitaine, Catherine Donati-Martin, and Delphine Féral

Full-text: Open access


In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices (MN)N defined by $M_{N}=W_{N}/\sqrt{N}+A_{N}$ where WN is an N×N Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix AN is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of AN are sufficiently far from zero, the corresponding eigenvalues of MN almost surely exit the limiting semicircle compact support as the size N becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of WN. On the other hand, when AN is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of WN.

Article information

Ann. Probab. Volume 37, Number 1 (2009), 1-47.

First available in Project Euclid: 17 February 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A52 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems 60F05: Central limit and other weak theorems

Deformed Wigner matrices asymptotic spectrum Stieltjes transform largest eigenvalues fluctuations nonuniversality


Capitaine, Mireille; Donati-Martin, Catherine; Féral, Delphine. The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. Ann. Probab. 37 (2009), no. 1, 1--47. doi:10.1214/08-AOP394.

Export citation


  • [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.
  • [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.
  • [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.
  • [4] Bai, Z. D. and Silverstein, J. W. (1999). Exact separation of eigenvalues of large-dimensional sample covariance matrices. Ann. Probab. 27 1536–1555.
  • [5] Bai, Z. D. and Yao, J. (2005). On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11 1059–1092.
  • [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.
  • [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.
  • [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.
  • [9] Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97 1382–1408.
  • [10] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. Wiley, New York.
  • [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.
  • [12] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28.
  • [13] Capitaine, M. and Donati-Martin, C. (2007). Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56 767–803.
  • [14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York.
  • [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.
  • [17] Füredi, Z. and Komlós, J. (1981). The eigenvalues of random symmetric matrices. Combinatorica 1 233–241.
  • [18] Haagerup, U. and Thorbjørnsen, S. (2005). A new application of random matrices: Ext(Cred*(F2)) is not a group. Ann. of Math. (2) 162 711–775.
  • [19] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge Univ. Press, Cambridge. Corrected reprint of the 1985 original.
  • [20] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295–327.
  • [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.
  • [22] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
  • [23] Maïda, M. (2007). Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles. Electron. J. Probab. 12 1131–1150 (electronic).
  • [24] Mathias, R. (1993). The Hadamard operator norm of a circulant and applications. SIAM J. Matrix Anal. Appl. 14 1152–1167.
  • [25] Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 1617–1642.
  • [26] Péché, S. (2006). The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 127–173.
  • [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.
  • [28] Saporta, G. (1990). Probabilités, analyse des données et statistique. Gulf Pub., Houston, TX.
  • [29] Schultz, H. (2005). Non-commutative polynomials of independent Gaussian random matrices. The real and symplectic cases. Probab. Theory Related Fields 131 261–309.
  • [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.
  • [31] Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697–733.
  • [32] Tillmann, H.-G. (1953). Randverteilungen analytischer Funktionen und Distributionen. Math. Z. 59 61–83.
  • [33] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.