The Spectrum of Heavy Tailed Random Matrices
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چکیده
Take XN to be a symmetric matrix with real independent (modulo the symmetry constraint) equidistributed entries with law P and denote (λ1, · · · , λN) its eigenvalues. Then, Wigner [14] has shown that, if ∫ xdP (x) is finite, N−1 ∑N i=1 δλi/ √ N converges in expectation towards the semi-circle distribution. In this paper, we consider the case where P has a heavy tail and belong to the domain of attraction of an α-stable law for α ∈]0, 2[. We show the convergence of N−1 ∑N i=1 λi/N 1 α towards a law μα. We characterize and study μα, showing in particular that it is a symmetric measure with heavy tail.
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تاریخ انتشار 2007