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 for 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 $N\to\infty$ 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.