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Abstract: Let $X N$ be an $N\ts N$ random symmetric matrix with independentequidistributed entries. If the law $P$ of the entries has a finite secondmoment, it was shown by Wigner \cite{wigner} that the empirical distribution ofthe eigenvalues of $X N$, once renormalized by $\sqrt{N}$, converges almostsurely 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 thedomain of attraction of an $\alpha$-stable law. We prove that if we renormalizethe eigenvalues by a constant $a N$ of order $N^{\frac{1}{\alpha}}$, thecorresponding spectral distribution converges in expectation towards a law$\mu \alpha$ which only depends on $\alpha$. We characterize $\mu \alpha$ andstudy some of its properties; it is a heavy-tailed probability measure which isabsolutely continuous with respect to Lebesgue measure except possibly on acompact set of capacity zero.



Author: Gerard Ben Arous, Alice Guionnet

Source: https://arxiv.org/



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