Extremal eigenvalues and eigenvectors of deformed Wigner matrices

Abstract

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 $$N\rightarrow \infty $$ 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 $$\lambda <\lambda _+$$ . Similar results hold for the lowest eigenvalues.