The Spectrum of Heavy Tailed Random Matrices

Arous, Gérard; Guionnet, Alice
March 2008
Communications in Mathematical Physics;Mar2008, Vol. 278 Issue 3, p715
Academic Journal
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.


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