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

Yi Han

TL;DR

This work studies finite rank perturbations of large non-Hermitian i.i.d. random matrices under minimal moment assumptions. It shows that, under merely a second moment, outlier eigenvalues of $n^{-1/2}A_n+C_n$ converge to the perturbation’s outliers for bounded-rank (and in sparse regimes, bounded-nonzero) perturbations, extending prior fourth-moment requirements. The paper also analyzes products of i.i.d. matrices, proving the spectral radius tends to 1 under second moment conditions and establishing outlier behavior for both additive and product perturbations via a linearization technique and direct characteristic-function calculations, yielding convergences to Gaussian analytic limits. The results unify and extend known BBP-type behavior to non-Hermitian settings, sparse regimes, and product models, with a coherent framework based on trace methods and reverse-characteristic-function limits.

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.

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

TL;DR

This work studies finite rank perturbations of large non-Hermitian i.i.d. random matrices under minimal moment assumptions. It shows that, under merely a second moment, outlier eigenvalues of converge to the perturbation’s outliers for bounded-rank (and in sparse regimes, bounded-nonzero) perturbations, extending prior fourth-moment requirements. The paper also analyzes products of i.i.d. matrices, proving the spectral radius tends to 1 under second moment conditions and establishing outlier behavior for both additive and product perturbations via a linearization technique and direct characteristic-function calculations, yielding convergences to Gaussian analytic limits. The results unify and extend known BBP-type behavior to non-Hermitian settings, sparse regimes, and product models, with a coherent framework based on trace methods and reverse-characteristic-function limits.

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 entries or Bernoulli entries with , the same result holds for perturbation matrices with a bounded number of nonzero elements.
Paper Structure (12 sections, 18 theorems, 107 equations)

This paper contains 12 sections, 18 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Spectral radius of product heavy-tailed i.i.d. matrices
  3. Finite rank perturbation: finite second moment
  4. Finite rank perturbation: sparse i.i.d. matrices
  5. Finite rank perturbation for product of i.i.d. matrices
  6. Spectral radius of product matrix with second moment
  7. Perturbation for an i.i.d. matrix with only finite second moment
  8. Illustrative example: a diagonal matrix of rank one
  9. The general case of finitely many nonzero elements
  10. Infinitesimally shifting the mean
  11. Perturbation of sparse i.i.d. matrices
  12. Perturbation for the product of i.i.d. matrices

Key Result

Theorem 1.1

Fix an integer $m\in\mathbb{N}_+$. Consider $A^1,\cdots,A^m$$m$ independent $n\times n$ random matrices with entry $(A^k)_{ij}=a^k_{ij}$, where $(a^k_{i,j})_{1\leq i,j\leq n,k=1,\cdots,m}$ are independent identically distributed random variables satisfying Then for any $\epsilon>0$, with probability $1-o(1)$, all eigenvalues of the product matrix lie in $(1+\epsilon)\mathbb{D}$, i.e. where $\rh

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 26 more