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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