Abstract
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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Communicated by B. Simon
This work was partially supported by Miller institute for Basic Research in Science, University of California Berkeley.
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Arous, G.B., Guionnet, A. The Spectrum of Heavy Tailed Random Matrices. Commun. Math. Phys. 278, 715–751 (2008). https://doi.org/10.1007/s00220-007-0389-x
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DOI: https://doi.org/10.1007/s00220-007-0389-x