We show that, for sufficiently well approximable frequencies, several spectral characteristics of the Almost Mathieu operator can be as poor as at all possible in the class of all discrete Schroedinger operators. For example, the modulus of continuity of the integrated density of states may be no better than logarithmic. Other characteristics to be discussed are homogeneity, the Parreau-Widom property, and (for the critical AMO) the Hausdorff content of the spectrum. Based on joint work with A. Avila, Y. Last, and Q. Zhou

We study discrete magnetic random Schrödinger operators on the square and honeycomb lattice, both with single-site potentials in weak magnetic fields under weak disorder. We show that there is, in the case of the honeycomb lattice, both strong dynamical localization and delocalization close to the conical point. We obtain similar results for the discrete random Schrödinger operator on the Z2-lattice close to the bottom and top of its spectrum. As part of this analysis, we give a rigorous derivation of the quantum hall effect for both models derived from the density of states for which we obtain an asymptotic expansion in the disorder parameter. The expansion implies (leading order in the disorder parameter) universality of the integrated density of states. Finally, we show that on the hexagonal lattice the Dirac cones occur for any rational magnetic flux. We use this observation to study the self-similarity of the Hall conductivity and transport properties of the random operator close to any rational magnetic flux.

Joint work with Rui Han.