Outlier eigenvalues and eigenvectors of generalized Wigner matrices with finite-rank perturbations

TL;DR

This paper analyzes the behavior of outlier eigenvalues and eigenvectors in generalized Wigner matrices with finite-rank perturbations, deriving their asymptotic distributions and eigenvector alignments.

Contribution

It provides new central limit theorems for outlier eigenvalues and eigenvectors in perturbed Wigner matrices, including Gaussian fluctuation results and eigenvector convergence.

Findings

Outlier eigenvalues follow a multivariate Gaussian distribution.

Eigenvectors exhibit asymptotic alignment with the perturbation's eigenvectors.

Eigenvector fluctuations are Gaussian around the origin for non-aligned components.

Abstract

A generalized Wigner matrix perturbed by a finite-rank deterministic matrix is considered. The fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors, are studied. Under certain assumptions on the perturbation and the matrix structure, we derive the first-order behavior of these eigenvalues and show that they are well separated from the bulk. The fluctuations of these eigenvalues are shown to follow a multivariate Gaussian distribution, and the asymptotic behavior of the associated eigenvectors is also studied. We prove central limit theorems that describe the asymptotic alignment of these eigenvectors with the perturbation's eigenvectors, as well as their Gaussian fluctuations around the origin for non-aligned components. Furthermore, we discuss the convergence of the eigenvector process in a Sobolev space framework.