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Large deviations from the circular law

Gérard Ben ArousOfer Zeitouni — 2010

ESAIM: Probability and Statistics

We prove a full large deviations principle, in the scale N, for the empirical measure of the eigenvalues of an N x N (non self-adjoint) matrix composed of i.i.d. zero mean random variables with variance N. The (good) rate function which governs this rate function possesses as unique minimizer the circular law, providing an alternative proof of convergence to the latter. The techniques are related to recent work by Ben Arous and Guionnet, who treat the self-adjoint case. A crucial role is...

Poisson convergence for the largest eigenvalues of heavy tailed random matrices

Antonio AuffingerGérard Ben ArousSandrine Péché — 2009

Annales de l'I.H.P. Probabilités et statistiques

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in ( (2004) 82–91), we prove that, in the absence of the fourth moment, the asymptotic behavior of the top eigenvalues is determined by the behavior of the largest entries of the matrix.

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