Spectrum of non-Hermitian heavy tailed random matrices

TL;DR

This paper establishes a heavy-tailed analogue of Girko's circular law for non-Hermitian matrices with entries in the domain of attraction of alpha-stable laws, describing the eigenvalue distribution's convergence to a deterministic measure.

Contribution

It introduces a new limiting spectral distribution for heavy-tailed non-Hermitian matrices, extending Girko's law to the alpha-stable domain of attraction.

Findings

Eigenvalues of scaled matrices converge to a deterministic measure mu_alpha.

mu_alpha is not heavy tailed, unlike the entries.

The approach uses a combination of the objective method and Hermitization techniques.

Abstract

Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}| is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X_{jk}, we prove that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability measure mu_alpha on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1} (X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu_alpha. In contrast with the Hermitian case, we find that mu_alpha is not heavy tailed.