The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and non-universality of the fluctuations

Mireille Capitaine (1) , Catherine Donati-Martin (2) , Delphine Féral (1)

1 LSP - Laboratoire de Statistique et Probabilités
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We investigate the asymptotic spectrum of deformed Wigner matrices. The deformation is deterministic will all but finitely many eigenvalues equal to zero. We show that, as soon as the first largest or last smallest eigenvalues of the deformation are sufficiently far from 0, the corresponding eigenvalues of the deformed Wigner matrix almost surely exit the limiting semicircle compact support as the size of the matrix becomes large. In the particular case of a diagonal pertubation of rank 1, we prove that the fluctuations of the largest eigenvalue are not universal and depend on the particular distribution of the entries of the Wigner matrix.

Mots clés: random matrices; largest eigenvalues; non universality

Domaines: Probabilités [math.PR]

Mireille Capitaine, Catherine Donati-Martin, Delphine Féral. The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and non-universality of the fluctuations. 2007. ⟨hal-00150997⟩