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Central limit theorem for eigenvectors of heavy tailed matrices

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1.	Title	Title of document	Central limit theorem for eigenvectors of heavy tailed matrices
2.	Creator	Author's name, affiliation, country	Florent Benaych-Georges; Université Paris Descartes; France
2.	Creator	Author's name, affiliation, country	Alice Guionnet; CNRS & MIT; United States
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Random matrices, heavy tailed random variables, eigenvectors, central limit theorem
3.	Subject	Subject classification	15A52; 60F05
4.	Description	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 $1$).
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	Simons Foundation ; NSF award DMS-1307704
7.	Date	(YYYY-MM-DD)	2014-06-23
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/3093
10.	Identifier	Digital Object Identifier	10.1214/EJP.v19-3093
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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