A note on outlier eigenvectors for sparse non-Hermitian perturbations
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arXiv
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| Format: | Preprint |
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2026
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| author | Galanis, Miltiadis Louvaris, Michail |
| author_facet | Galanis, Miltiadis Louvaris, Michail |
| contents | 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 $μ$ with $|μ|>1$, the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to $1-|μ|^{-2}$. Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03972 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on outlier eigenvectors for sparse non-Hermitian perturbations Galanis, Miltiadis Louvaris, Michail Probability Statistics Theory 60B20, 15B52 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 $μ$ with $|μ|>1$, the squared projection of the associated (right) eigenvector onto the spike eigenspace converges in probability to $1-|μ|^{-2}$. Our result generalizes Theorem 1.6 of [HLN26] to general finite rank case solving Open Problem 5. |
| title | A note on outlier eigenvectors for sparse non-Hermitian perturbations |
| topic | Probability Statistics Theory 60B20, 15B52 |
| url | https://arxiv.org/abs/2603.03972 |