Central limit theorems for eigenvalues of deformations of Wigner matrices
Capitaine, M. ; Donati-Martin, C. ; Féral, D.
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 107-133 / Harvested from Numdam
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Dans ce papier, nous étudions les fluctuations des valeurs propres extrémales d'une matrice de Wigner hermitienne (resp. symétrique) déformée par une perturbation de rang fini dont les valeurs propres non nulles sont fixées, dans le cas où ces valeurs propres extrémales se détachent du reste du spectre. Nous décrivons des situations générales d'universalité ou de non-universalité des fluctuations correspondant au caractère localisé ou délocalisé des vecteurs propres de la perturbation. Lorsque l'une des valeurs propres de la perturbation est de multiplicité un, nous établissons de plus une condition nécessaire et suffisante sur le vecteur propre associé pour que les fluctuations de la valeur propre correspondante du modèle déformé soient universelles.

In this paper, we study the fluctuations of the extreme eigenvalues of a spiked finite rank deformation of a Hermitian (resp. symmetric) Wigner matrix when these eigenvalues separate from the bulk. We exhibit quite general situations that will give rise to universality or non-universality of the fluctuations, according to the delocalization or localization of the eigenvectors of the perturbation. Dealing with the particular case of a spike with multiplicity one, we also establish a necessary and sufficient condition on the associated normalized eigenvector so that the fluctuations of the corresponding eigenvalue of the deformed model are universal.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/10-AIHP410
Classification:  60B20,  15A18,  60F05 
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     author = {Capitaine, M. and Donati-Martin, Catherine and F\'eral, D.},
     title = {Central limit theorems for eigenvalues of deformations of Wigner matrices},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {48},
     year = {2012},
     pages = {107-133},
     doi = {10.1214/10-AIHP410},
     mrnumber = {2919200},
     zbl = {1237.60007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_1_107_0}
}

Capitaine, M.; Donati-Martin, C.; Féral, D. Central limit theorems for eigenvalues of deformations of Wigner matrices. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) pp. 107-133. doi : 10.1214/10-AIHP410. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_1_107_0/

[1] Z. D. Bai. Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 (1999) 611-677. | MR 1711663 | Zbl 0949.60077

[2] Z. D. Bai and J. W. Silverstein. No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 (1998) 316-345. | MR 1617051 | Zbl 0937.60017

[3] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edition. Springer Ser. Statist. Springer, New York, 2010. | MR 2567175 | Zbl 1301.60002 | Zbl pre05640422

[4] Z. D. Bai and J. F. Yao. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11 (2005) 1059-1092. | MR 2189081 | Zbl 1101.60012

[5] Z. D. Bai and J. F. Yao. Central limit theorems for eigenvalues in a spiked population model. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 447-474. | Numdam | MR 2451053 | Zbl 1274.62129

[6] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33 (2005) 1643-1697. | MR 2165575 | Zbl 1086.15022

[7] G. Biroli, J. P. Bouchaud and M. Potters. On the top eigenvalue of heavy-tailed random matrices. Europhys. Lett. 78 (2007) Art 10001. | MR 2371333 | Zbl 1244.82029

[8] M. Capitaine, C. Donati-Martin and D. Féral. The largest eigenvalue of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. Ann. Probab. 37 (2009) 1-47. | MR 2489158 | Zbl 1163.15026

[9] L. Erdös, H.-T. Yau and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Preprint, 2010. Available at arXiv:1007.4652. | MR 2871147 | Zbl 1238.15017

[10] D. Féral and S. Péché. The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272 (2007) 185-228. | MR 2291807 | Zbl 1136.82016

[11] D. Féral and S. Péché. The largest eigenvalues of sample covariance matrices for a spiked population: Diagonal case. J. Math. Phys. 50 (2009) 073302. | MR 2548630 | Zbl pre05840821

[12] Z. Füredi and J. Komlós. The eigenvalues of random symmetric matrices. Combinatorica 1 (1981) 233-241. | Zbl 0494.15010

[13] F. Hiai and D. Petz. The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs 77. Amer. Math. Soc., Providence, RI, 2000. | MR 1746976 | Zbl 0955.46037

[14] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge Univ. Press, New York, 1991. | Zbl 0576.15001

[15] S. Janson. Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs. Ann. Probab. 16 (1988) 305-312. | MR 920273 | Zbl 0639.60029

[16] O. Khorunzhiy. High moments of large Wigner random matrices and asymptotic properties of the spectral norm. Preprint, 2009. Available at arXiv:0907.3743v5. | MR 2899796 | Zbl 1270.15025

[17] A. Onatski. The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices. Ann. Appl. Probab. 18 (2008) 470-490. | MR 2398763 | Zbl 1141.60009

[18] D. Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica 17 (2007) 1617-1641. | MR 2399865 | Zbl 1134.62029

[19] S. Péché. The largest eigenvalues of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields 134 (2006) 127-174. | MR 2221787 | Zbl 1088.15025

[20] A. Ruzmaikina. Universality of the edge distribution of eigenvalues of Wigner random matrices with polynomially decaying distributions of entries. Comm. Math. Phys. 261 (2006) 277-296. | MR 2191882 | Zbl 1130.82313

[21] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999) 697-733. | MR 1727234 | Zbl 1062.82502

[22] T. Tao and V. Vu. Random matrices: Universality of the local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 (2010) 549-572. | MR 2669449 | Zbl 1202.15038

[23] C. A. Tracy and H. Widom. Level spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994) 151-174. | MR 1257246 | Zbl 0789.35152