Extreme eigenvalues of sparse, heavy tailed random matrices

TL;DR

This paper investigates the behavior of the largest eigenvalues of sparse, heavy-tailed random matrices, revealing phase transitions depending on tail heaviness and sparsity, and extends previous Hermitian matrix results.

Contribution

It provides new results on the asymptotic distribution of largest eigenvalues for sparse, heavy-tailed matrices, removing previous restrictions on sparsity.

Findings

Largest eigenvalues are Poissonian for certain tail indices and sparsity levels.

Eigenvalues converge to a constant when tail index exceeds a threshold.

Extends prior Hermitian matrix results to more general sparse matrices.

Abstract

We study the statistics of the largest eigenvalues of $p \times p$ sample covariance matrices $\Sigma_{p,n} = M_{p,n}M_{p,n}^{*}$ when the entries of the $p \times n$ matrix $M_{p,n}$ are sparse and have a distribution with tail $t^{-\alpha}$, $\alpha>0$. On average the number of nonzero entries of $M_{p,n}$ is of order $n^{\mu+1}$, $0 \leq \mu \leq 1$. We prove that in the large $n$ limit, the largest eigenvalues are Poissonian if $\alpha<2(1+\mu^{{-1}})$ and converge to a constant in the case $\alpha>2(1+\mu^{{-1}})$. We also extend the results of Benaych-Georges and Peche [7] in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.