Extremal eigenvalues and eigenvectors of deformed Wigner matrices

We consider random matrices of the form $$H = W + \lambda V, \lambda \in {\mathbb {R}}^+$$H=W+λV,λ∈R+, where $$W$$W is a real symmetric or complex Hermitian Wigner matrix of size $$N$$N and $$V$$V is a real bounded diagonal random matrix of size $$N$$N with i.i.d. entries that are independent of $$W$$W. We assume subexponential decay of the distribution of the matrix entries of $$W$$W and we choose $$\lambda \sim 1$$λ∼1, so that the eigenvalues of $$W$$W and $$\lambda V$$λV are typically of the same order. Further, we assume that the density of the entries of $$V$$V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is $$\lambda _+\in {\mathbb {R}}^+$$λ+∈R+ such that the largest eigenvalues of $$H$$H are in the limit of large $$N$$N determined by the order statistics of $$V$$V for $$\lambda >\lambda _+$$λ>λ+. In particular, the largest eigenvalue of $$H$$H has a Weibull distribution in the limit $$N\rightarrow \infty $$N→∞ if $$\lambda >\lambda _+$$λ>λ+. Moreover, for $$N$$N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for $$\lambda >\lambda _+$$λ>λ+, while they are completely delocalized for $$\lambda <\lambda _+$$λ