Finite rank perturbation of non-Hermitian random matrices: heavy tail and sparse regimes

TL;DR

This paper investigates how finite rank perturbations affect the outlier eigenvalues of large non-Hermitian random matrices under minimal moment conditions, including sparse and heavy-tailed regimes.

Contribution

It extends existing results by proving outlier eigenvalue convergence under only second moment assumptions and for sparse perturbations.

Findings

Outlier eigenvalues converge to the perturbation's eigenvalues under second moment conditions.

Results hold for sparse matrices with a bounded number of nonzero entries.

The convergence is valid even in heavy-tailed and sparse regimes.

Abstract

We revisit the problem of perturbing a large, i.i.d. random matrix by a finite rank error. It is known that when elements of the i.i.d. matrix have finite fourth moment, then the outlier eigenvalues of the perturbed matrix are close to the outlier eigenvalues of the error, as long as the perturbation is relatively small. We first prove that under a merely second moment condition, for a large class of perturbation matrix with bounded rank and bounded operator norm, the outlier eigenvalues of perturbed matrix still converge to that of the perturbation. We then prove that for a matrix with i.i.d. Bernoulli $(d/n)$ entries or Bernoulli $(d_n/n)$ entries with $d_n=n^{o(1)}$, the same result holds for perturbation matrices with a bounded number of nonzero elements.