Universality of level correlation function of sparse random matrices

The statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the Hubbard-Stratonovich transformation and the method is shown to be absolutely equivalent to the replica trick.

Abstract

The statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the Hubbard-Stratonovich transformation. When used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick. The model turns out to bear a close resemblance to the Anderson model on the Bethe lattice: it possesses a delocalization transition that occurs with an increase in the 'mean connectivity' parameter. In the delocalized phase the level-level correlation function proves to have a universal (Dyson) form with the full density of states replaced by the contribution from the infinite cluster.