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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 [14] 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 α-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 α. 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.

The Spectrum of Heavy Tailed Random Matrices