Central limit theorem for eigenvectors of heavy tailed matrices
Author(s)Benaych-Georges, Florent; Guionnet, Alice
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We consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of α-stable laws, or adjacencymatrices of Erdos-Renyi graphs. We denote by U=[uij] the eigenvectors matrix (corresponding to increasing eigenvalues) and prove that the bivariate process [formula] indexed by s,t∈[0,1], converges in law to a non trivial Gaussian process. An interesting part of this result is the n−1/2 rescaling, proving that from this point of view, the eigenvectors matrix U behaves more like a permutation matrix (as it was proved by Chapuy that for U a permutation matrix, n−1/2 is the right scaling) than like a Haar-distributed orthogonal or unitary matrix (as it was proved by Rouault and Donati-Martin that for U such a matrix, the right scaling is 1).
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Electronic Journal of Probability
Institute of Mathematical Statistics
Benaych-Georges, Florent, and Alice Guionnet. “Central Limit Theorem for Eigenvectors of Heavy Tailed Matrices.” Electronic Journal of Probability 19, no. 0 (January 2, 2014).
Final published version