Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices

TL;DR

This paper analyzes the extremal eigenvalues and eigenvectors of deformed Wigner matrices, showing phase transitions in eigenvalue distribution and eigenvector localization depending on the deformation parameter.

Contribution

It establishes the existence of a critical deformation strength where the largest eigenvalues follow a Weibull distribution and eigenvectors transition from delocalized to localized.

Findings

Largest eigenvalues follow Weibull distribution for strong deformation.

Eigenvectors become partially localized beyond a critical threshold.

Eigenvalues near the edges are determined by the order statistics of the diagonal entries.

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 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.