Critical statistics for non-Hermitian matrices

Abstract

We introduce a generalized ensemble of nonhermitian matrices interpolating between the Gaussian Unitary Ensemble, the Ginibre ensemble and the Poisson ensemble. The joint eigenvalue distribution of this model is obtained by means of an extension of the Itzykson-Zuber formula to general complex matrices. Its correlation functions are studied both in the case of weak nonhermiticity and in the case of strong nonhermiticity. In the weak nonhermiticity limit we show that the spectral correlations in the bulk of the spectrum display critical statistics: the asymptotic linear behavior of the number variance is already approached for energy differences of the order of the eigenvalue spacing. To lowest order, its slope does not depend on the degree of nonhermiticity. Close the edge, the spectral correlations are similar to the Hermitian case. In the strong nonhermiticity limit the crossover behavior from the Ginibre ensemble to the Poisson ensemble first appears close to the surface of the spectrum. Our model may be relevant for the description of the spectral correlations of an open disordered system close to an Anderson transition.