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) no. 1, pp. 107-133.

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.

Classification : 60B20, 15A18, 60F05
Mots clés : random matrices, deformed Wigner matrices, extremal eigenvalues, fluctuations, localized eigenvectors, universality

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