Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices

Florent Benaych-Georges (1, 2) , Alice Guionnet (3) , Mylène Maïda (4)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
4 LM-Orsay - Laboratoire de Mathématiques d'Orsay

Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X_n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X_n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.

Mots clés: Random matrices; largest eigenvalue; Limit theorems; Tracy-Widom law

Domaines: Probabilités [math.PR]

Florent Benaych-Georges, Alice Guionnet, Mylène Maïda. Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. 2010. ⟨hal-00505497v5⟩