A note on outlier eigenvectors for sparse non-Hermitian perturbations

Miltiadis Galanis; Michail Louvaris

paper

A note on outlier eigenvectors for sparse non-Hermitian perturbations

arXiv:2603.03972

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 $I^¼$ with $∣ I^¼∣ > 1$ , the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to $1 - ∣ I^¼ ∣^{- 2}$ . Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5.

10 pages