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2014 Central limit theorem for eigenvectors of heavy tailed matrices
Florent Benaych-Georges, Alice Guionnet
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Electron. J. Probab. 19: 1-27 (2014). DOI: 10.1214/EJP.v19-3093

Abstract

We consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of $\alpha$-stable laws, or adjacencymatrices of Erdos-Renyi graphs. We denote by $U=[u_{ij}]$ the eigenvectors matrix (corresponding to increasing eigenvalues) and prove that the bivariate process $$B^n_{s,t}:=n^{-1/2}\sum_{1\le i\le ns, 1\le j\le nt}(|u_{ij}|^2 -n^{-1}),$$ indexed by $s,t\in [0,1]$, 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 $1$).

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Florent Benaych-Georges. Alice Guionnet. "Central limit theorem for eigenvectors of heavy tailed matrices." Electron. J. Probab. 19 1 - 27, 2014. https://doi.org/10.1214/EJP.v19-3093

Information

Accepted: 23 June 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1293.15021
MathSciNet: MR3227063
Digital Object Identifier: 10.1214/EJP.v19-3093

Subjects:
Primary: 15A52
Secondary: 60F05

Keywords: central limit theorem , eigenvectors , Heavy tailed random variables , random matrices

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