Central limit theorems for eigenvalues of deformations of Wigner matrices

M. Capitaine; C. Donati-Martin; D. Féral

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

  • Volume: 48, Issue: 1, page 107-133
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Capitaine, M., Donati-Martin, C., and Féral, D.. "Central limit theorems for eigenvalues of deformations of Wigner matrices." Annales de l'I.H.P. Probabilités et statistiques 48.1 (2012): 107-133. <http://eudml.org/doc/272000>.

@article{Capitaine2012,
abstract = {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.},
author = {Capitaine, M., Donati-Martin, C., Féral, D.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random matrices; deformed Wigner matrices; extremal eigenvalues; fluctuations; localized eigenvectors; universality},
language = {eng},
number = {1},
pages = {107-133},
publisher = {Gauthier-Villars},
title = {Central limit theorems for eigenvalues of deformations of Wigner matrices},
url = {http://eudml.org/doc/272000},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Capitaine, M.
AU - Donati-Martin, C.
AU - Féral, D.
TI - Central limit theorems for eigenvalues of deformations of Wigner matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 1
SP - 107
EP - 133
AB - 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.
LA - eng
KW - random matrices; deformed Wigner matrices; extremal eigenvalues; fluctuations; localized eigenvectors; universality
UR - http://eudml.org/doc/272000
ER -

References

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  1. [1] Z. D. Bai. Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica9 (1999) 611–677. Zbl0949.60077MR1711663
  2. [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. Zbl0937.60017MR1617051
  3. [3] Z. D. Bai and J. W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices, 2nd edition. Springer Ser. Statist. Springer, New York, 2010. Zbl1301.60002MR2567175
  4. [4] Z. D. Bai and J. F. Yao. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli11 (2005) 1059–1092. Zbl1101.60012MR2189081
  5. [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. Zbl1274.62129MR2451053
  6. [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. Zbl1086.15022MR2165575
  7. [7] G. Biroli, J. P. Bouchaud and M. Potters. On the top eigenvalue of heavy-tailed random matrices. Europhys. Lett. 78 (2007) Art 10001. Zbl1244.82029MR2371333
  8. [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. Zbl1163.15026MR2489158
  9. [9] L. Erdös, H.-T. Yau and J. Yin. Rigidity of eigenvalues of generalized Wigner matrices. Preprint, 2010. Available at arXiv:1007.4652. Zbl1238.15017MR2871147
  10. [10] D. Féral and S. Péché. The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys.272 (2007) 185–228. Zbl1136.82016MR2291807
  11. [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. Zbl05840821MR2548630
  12. [12] Z. Füredi and J. Komlós. The eigenvalues of random symmetric matrices. Combinatorica1 (1981) 233–241. Zbl0494.15010
  13. [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. Zbl0955.46037MR1746976
  14. [14] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge Univ. Press, New York, 1991. Zbl0576.15001
  15. [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. Zbl0639.60029MR920273
  16. [16] O. Khorunzhiy. High moments of large Wigner random matrices and asymptotic properties of the spectral norm. Preprint, 2009. Available at arXiv:0907.3743v5. Zbl1270.15025MR2899796
  17. [17] A. Onatski. The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices. Ann. Appl. Probab.18 (2008) 470–490. Zbl1141.60009MR2398763
  18. [18] D. Paul. Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statist. Sinica17 (2007) 1617–1641. Zbl1134.62029MR2399865
  19. [19] S. Péché. The largest eigenvalues of small rank perturbations of Hermitian random matrices. Probab. Theory Related Fields134 (2006) 127–174. Zbl1088.15025MR2221787
  20. [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. Zbl1130.82313MR2191882
  21. [21] A. Soshnikov. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys.207 (1999) 697–733. Zbl1062.82502MR1727234
  22. [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. Zbl1202.15038MR2669449
  23. [23] C. A. Tracy and H. Widom. Level spacing distributions and the Airy kernel. Comm. Math. Phys.159 (1994) 151–174. Zbl0789.35152MR1257246

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