Gaussian statistics for left and right eigenvectors of complex non-Hermitian matrices

Abstract

We consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+\epsilon}$. We show that arbitrary constant rank projections of these eigenvectors are Gaussian and jointly independent.