Central limit theorem for eigenvectors of heavy tailed matrices. (English) Zbl 1293.15021

Benaych-Georges, Florent; Guionnet, Alice

Electron. J. Probab. 19, Paper No. 54, 27 p. (2014).

Summary: 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 adjacency matrices of Erdős-Rényi graphs. We denote by \(U=[u_{ij}]\) the eigenvectors matrix (corresponding to increasing eigenvalues) and prove that the bivariate process \[ B^n_{s,t}:=n^{-1/2}\sum_{1\leq i\leq ns, 1\leq j\leq nt}(|u_{ij}|^2 -n^{-1}), \] 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 G. Chapuy [in: 2007 Conference on analysis of algorithms, AofA 07. Papers from the 13th Conference held in Juan-les-Pins, France, 2007. Nancy: The Association Discrete Mathematics & Theoretical Science (DMTCS), 415–426 (2007; Zbl 1192.68456)] 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 C. Donati-Martin and A. Rouault [Random Matrices Theory Appl. 1, No. 1, 1150007, 24 p. (2012; Zbl 1247.15031)] that for \(U\) such a matrix, the right scaling is 1).

MSC:

15B52	Random matrices (algebraic aspects)
60B20	Random matrices (probabilistic aspects)
60F05	Central limit and other weak theorems
05C50	Graphs and linear algebra (matrices, eigenvalues, etc.)

Keywords:

random matrices; heavy tailed random variables; eigenvectors; central limit theorem; Erdős-Rényi graphs; Gaussian process

Citations:

Zbl 1192.68456; Zbl 1247.15031