Eigenvalue correlations for generalized Gaussian ensembles

We study the statistical properties of the eigenvalues of a generalized Gaussian ensemble of Hermitian matrices described by a set of the variances of the matrix elements and non-invariant under an unitary transformation. We find that the eigenvalue distribution in this case can be mapped on to the corresponding distribution in Dysonʹs Brownian ensembles, with a function of variances in the former playing the role of “time” in the latter. The pre-existing information about the spectral correlations of Brownian ensembles can therefore be used to obtain the same for generalized Gaussian ensembles.