This Ph.D. thesis presents recent developments in the theory of random Schrodinger operators. Differently from what is often studied in the subject, our main results consider potentials which are not independent at distinct sites but, rather, display some form of long range correlation. These are natural objects to investigate if one wishes to understand the long term behavior of a single particle which evolves in a disordered environment but also interacts with different members of this environment (other particles, spins, etc). In chapter 2 it is shown that, within the Hartree-Fock approximation for the disordered Hubbard Hamiltonian, weakly interacting Fermions at positive temperature exhibit localization, suitably defined as exponential decay of eigenfunction correlators. Our result holds in any dimension in the regime of large disorder and at any disorder in the one dimensional case. As a consequence of our methods, we are able to show Holder continuity of the integrated density of states with respect to energy, disorder and interaction using known techniques. This is based on joint work with Jeffrey Schenker. Chapter 3 is based on joint work with Jeffrey Schenker and Rajinder Mavi. There we present simple, physically motivated, examples where small geometric changes on a two-dimensional graph G, combined with high disorder, have a significant impact on the spectral and dynamical properties of the random Schrodinger operator -AG+V[omega] obtained by adding a random potential to the graph's adjacency operator. Differently from the standard Anderson model, the random potential will be constant along vertical line, hence the models exhibit long range correlations. Moreover, one of the models presented here is a natural example where the transient and recurrent components of the absolutely continuous spectrum, introduced by Avron and Simon in [9] coexist and allow us to capture a sharp phase transition present in the system.
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