The spectrum of heavy-tailed random matrices

TL;DR

This paper extends the understanding of eigenvalue distributions of symmetric random matrices to cases where entries follow heavy-tailed stable laws, showing convergence to a new heavy-tailed spectral distribution depending on the stability parameter.

Contribution

It introduces a new spectral distribution for heavy-tailed stable law entries and characterizes its properties, expanding classical results beyond finite variance cases.

Findings

Eigenvalues, when properly scaled, converge to a heavy-tailed distribution depending on alpha.

The limiting spectral measure is absolutely continuous except possibly on a set of capacity zero.

The distribution depends only on the stability parameter alpha.

Abstract

Let $X_N$ be an $N\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\sqrt{N}$, converges almost surely and in expectation to the so-called semicircular distribution as $N$ goes to infinity. In this paper we study the same question when $P$ is in the domain of attraction of an $\alpha$-stable law. We prove that if we renormalize the eigenvalues by a constant $a_N$ of order $N^{\frac{1}{\alpha}}$, the corresponding spectral distribution converges in expectation towards a law $\mu_\alpha$ which only depends on $\alpha$. We characterize $\mu_\alpha$ and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero.