Central limit theorem for eigenvectors of heavy tailed matrices

作者：Benaych Georges Florent^*; Guionnet Alice

来源：Electronic Journal of Probability, 2014, 19(0): 1-27.

DOI：10.1214/EJP.v19-3093

• Random matrices
• heavy tailed random variables
• eigenvectors
• central limit theorem

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 Erdos-Renyi graphs. We denote by U - [u(ij)] the eigenvectors matrix (corresponding to increasing eigenvalues) and prove that the bivariate process
B-s,t(n) := 1/root n Sigma(1 <= i <= ns 1 <= j <= nt) (vertical bar u(ij)vertical bar(2) - 1/n) (0 <= s, t <= 1)
converges in law to a non trivial Gaussian process. An interesting part of this result is the 1/root n rescaling, proving that from this point of view, the eigenvectors matrix U behaves more like a permutation matrix (as it was proved in [17] that for U a permutation matrix, 1/root n is the right scaling) than like a Haar-distributed orthogonal or unitary matrix (as it was proved in [18, 5] that for U such a matrix, the right scaling is 1).

• 出版日期2014-6-23
• 单位MIT