Random covariance matrices: Universality of local statistics of eigenvalues

Abstract

We study the eigenvalues of the covariance matrix 1/n M^∗M of a large rectangular matrix M = M_n,p = (ζ_ij)_{1≤i≤p;1≤j≤n} whose entries are i.i.d. random variables of mean zero, variance one, and having finite C₀th moment for some sufficiently large constant C₀.

The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549–572])). We can use this theorem together with existing results to establish universality of local statistics of eigenvalues under mild conditions.

As a byproduct of our arguments, we also extend our previous results on random Hermitian matrices to the case in which the entries have finite C₀th moment rather than exponential decay.