A note on outlier eigenvectors for sparse non-Hermitian perturbations

TL;DR

This paper analyzes the behavior of outlier eigenvectors in sparse non-Hermitian random matrices with finite-rank perturbations, providing asymptotic formulas for eigenvector overlaps and generalizing previous results.

Contribution

It extends existing results to the finite-rank case for sparse non-Hermitian matrices, offering new asymptotic formulas for outlier eigenvector overlaps.

Findings

Squared projection of eigenvector converges to 1-|u|^{-2} for outliers with |u|>1

Generalizes previous theorems to finite-rank perturbations

Provides a resolvent reduction technique for asymptotic analysis

Abstract

We consider a sparse i.i.d.\ non-Hermitian random matrix model $X_n$ (with sparsity parameter $K_n$) and a deterministic finite-rank perturbation $E_n$. Assuming biorthogonality for $E_n$ and a growth condition on $K_n$, we outline a finite-rank resolvent reduction leading to asymptotics for the overlap between an outlier eigenvector of $Y_n:=X_n+E_n$ and the corresponding spike eigenspace. In particular, for an outlier spike $\mu$ with $|\mu|>1$, the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to $1-|\mu|^{-2}$. Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5.