The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations

Abstract

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices (M_N)_N defined by $M_{N}=W_{N}/\sqrt{N}+A_{N}$ where W_N is an N×N Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix A_N is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of A_N are sufficiently far from zero, the corresponding eigenvalues of M_N almost surely exit the limiting semicircle compact support as the size N becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of W_N. On the other hand, when A_N is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of W_N.