Abstract:

The Wegner orbital model is a version of the Anderson model in which instead of independent random variables on the diagonal one has independent random N X N blocks. The main special case is when the blocks on the diagonal are sampled from the Gaussian Orthogonal Ensemble. Physically, N represents the number of internal degrees of freedom (orbitals). One is interested in the spectral properties of such operators when N is large.

We shall present a couple of joint results obtained in collaboration with Ron Peled, Jeff Schenker and Sasha Sodin. First, we establish an N-independent uniform bound on the density of states (a Wegner estimate) and a corresponding Minami estimate. Second, we prove Anderson localization when the coupling is smaller than c/\sqrt{N}. It is believed that for coupling greater than C/\sqrt{N} one has delocalization in dimension 3 and above.

An important ingredient in the proof is the study of the deformed Gaussian ensembles, performed in collaboration with the same coauthors and Michael Aizenman.