Eigenvector correlations in non-Hermitian random matrix ensembles

We analyse correlations of eigenvectors in Ginibre's and Girko's ensembles of Gaussian, non-Hermitian random N x N matrices J. We study the ensemble average of [L-alpha/L-beta] [R-beta/R-alpha], where [L-alpha\ and \R-beta] are the left and right eigenvectors of J. The case of Ginibre's ensemble, in which the real and imaginary parts of each element of J are independent random variables, is sufficiently symmetric to allow for an exact solution. In the more general case of Girko's ensemble, we rely on approximations which become exact in the limit of N --> infinity