Central limit theorems for eigenvalues of deformations of Wigner matrices

C. Donati-Martin (1) , Mireille Capitaine (2) , D. Féral

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Domaines: Probabilités [math.PR]

C. Donati-Martin, Mireille Capitaine, D. Féral. Central limit theorems for eigenvalues of deformations of Wigner matrices. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2012, 48 (1), pp.107-133. ⟨10.1214/10-AIHP410⟩. ⟨hal-00686868⟩