Random Matrices with Correlated Elements: A Model for Disorder with Interactions

The complicated interactions in presence of disorder lead to a correlated randomization of states. The Hamiltonian as a result behaves like a multi-parametric 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 non-interacting systems, the possibility of extended states even in one dimension and a universal formulation of level correlations.Comment: 19 Pages, No Figures, Major Changes to Explain the Mathematical Detail

Diffusion in a Random Velocity Field: Spectral Properties of a Non-Hermitian Fokker-Planck Operator

We study spectral properties of the Fokker-Planck operator that describes particles diffusing in a quenched random velocity field. This random operator is non-Hermitian and has eigenvalues occupying a finite area in the complex plane. We calculate the eigenvalue density and averaged one-particle Green's function, for weak disorder and dimension d>2. We relate our results to the time-evolution of particle density, and compare them with numerical simulations.Comment: 4 pages, 2 figure