Extremal eigenvalues and eigenvectors of deformed Wigner matrices

Abstract

We consider random matrices of the form H = W + lambda V, lambda is an element of R+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i. i. d. entries that are independent of W. We assume subexponential decay of the distribution of the matrix entries of W and we choose lambda similar to 1, so that the eigenvalues of W and lambda V are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is lambda(+) is an element of R+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for lambda > lambda(+). In particular, the largest eigenvalue of H has a Weibull distribution in the limit N -> infinity if lambda > lambda(+). Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for lambda > lambda(+), while they are completely delocalized for lambda < lambda(+). Similar results hold for the lowest eigenvalues.