Central limit theorem for eigenvectors of heavy tailed matrices
Name
Benaych-Georges-2014-Central limit theorem.pdf
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429.29 KB
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Author(s)
Benaych-Georges, Florent
Guionnet, Alice
Date Issued
June 2014
Journal
Electronic Journal of Probability
Publisher
Institute of Mathematical Statistics
Citation
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).
Version
Final published version
Abstract
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).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1214/EJP.v19-3093
