Central limit theorem for eigenvectors of heavy tailed matrices

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 $1$).