Central limit theorem for eigenvectors of heavy tailed matrices

• Main Authors: Benaych-Georges, Florent (Author), Guionnet, Alice (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Institute of Mathematical Statistics, 2014-09-15T15:31:57Z.
• Subjects: Article
• Online Access: Get fulltext

 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).