The spectrum of heavy tailed random matrices

Let X-N be an N x 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 [14] that the empirical distribution of the eigenvalues of X-N , once renormalized by root 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-1/alpha, the corresponding spectral distribution converges in expectation towards a law 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.