Spectrum of Heavy-Tailed Elliptic Random Matrices

TL;DR

This paper extends the understanding of elliptic random matrices with heavy-tailed entries, showing their spectral distribution converges to a deterministic limit when entries are in the domain of attraction of an alpha-stable law, even without finite moments.

Contribution

It generalizes previous results by establishing spectral convergence for heavy-tailed elliptic matrices with no finite moments, using bounds on singular values and operator convergence.

Findings

Spectral measure converges to a deterministic limit for heavy-tailed entries.

Established bounds on the least singular value without moment assumptions.

Proved convergence of matrices to a Poisson Weighted Infinite Tree operator.

Abstract

An elliptic random matrix $X$ is a square matrix whose $(i,j)$-entry $X_{ij}$ is independent of the rest of the entries except possibly $X_{ji}$. Elliptic random matrices generalize Wigner matrices and non-Hermitian random matrices with independent entries. When the entries of an elliptic random matrix have mean zero and unit variance, the empirical spectral distribution is known to converge to the uniform distribution on the interior of an ellipse determined by the covariance of the mirrored entries. We consider elliptic random matrices whose entries fail to have two finite moments. Our main result shows that when the entries of an elliptic random matrix are in the domain of attraction of an $\alpha$-stable random variable, for $0<\alpha<2$, the empirical spectral measure converges, in probability, to a deterministic limit. This generalizes a result of Bordenave, Caputo, and Chafa\"i for heavy-tailed matrices with independent and identically distributed entries. The key elements of the proof are (i) a general bound on the least singular value of elliptic random matrices under no moment assumptions; and (ii) the convergence, in an appropriate sense, of the matrices to a random operator on the Poisson Weighted Infinite Tree.