Random matrices with correlated elements: a model for disorder with interactions.

The complicated interactions in the presence of disorder lead to a correlated randomization of states. The Hamiltonian as a result behaves like a multiparametric random matrix with correlated elements. We show that the eigenvalue correlations of these matrices can be described by the single parametric Brownian ensembles. The analogy helps us to reveal many important features of the level statistics in interacting systems, e.g., a critical point behavior different from that of noninteracting systems, the possibility of extended states even in one dimension, and a universal formulation of level correlations.