Mesoscopic Perturbations of Large Random Matrices

TL;DR

This paper studies the eigenvalues and eigenvectors of large random matrices with small-rank perturbations, extending finite rank results to cases where the perturbation rank grows with the matrix size.

Contribution

It extends the analysis of eigenvalue outliers and eigenvector projections to perturbations with rank up to o(N), generalizing finite rank perturbation results.

Findings

Proves rigidity of outliers in large random matrices with growing perturbation rank.

Derives empirical distribution of outliers under additive and multiplicative models.

Provides approximate eigenvectors related to the perturbing matrix.

Abstract

We consider the eigenvalues and eigenvectors of small rank perturbations of random $N\times N$ matrices. We allow the rank of perturbation $M$ increases with $N$, and the only assumption is $M=o(N)$. In both additive and multiplicative perturbation models, we prove rigidity results for the outliers of the perturbed random matrices. Based on the rigidity results we derive the empirical distribution of outliers of the perturbed random matrices. We also compute the appropriate projection of eigenvectors corresponding to the outliers of the perturbed random matrices, which are approximate eigenvectors of the perturbing matrix. Our results can be regarded as the extension of the finite rank perturbation case to the full generality up to $M=o(N)$.