The spectrum of heavy-tailed random matrices

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.