Extremal eigenvalues and eigenvectors of deformed Wigner matrices

We consider random matrices of the form (H = W + lambda V, lambda in {mathbb {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 sim 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 _+in {mathbb {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 (Nrightarrow infty ) 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 . Similar results hold for the lowest eigenvalues.