Let X N be an N → 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  that the empirical distribution of the eigenvalues of X N , once renormalized by √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 α-stable law. We prove that if we renormalize the eigenvalues by a constant a N of order N1α, the corresponding spectral distribution converges in expectation towards a law μ which only depends on α. We characterize μα 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics