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.

Abstract: We establish strong ballistic transport for a family of discrete quasiperiodic Schrodinger operators as a consequence of exponential dynamical localization for the dual family. The latter has been, essentially, shown by Jitomirskaya and Kruger in the one-frequency setting and by Ge--You--Zhou in the multi-frequency case. In both regimes, we obtain strong convergence of $\frac{1}{T}X(T)$ to the asymptotic velocity operator $Q$, which improves recent perturbative results by Zhao and provides the strongest known form of ballistic motion. In the one-frequency setting, this approach allows to treat Diophantine frequencies non-perturbatively and also consider the weakly Liouville case.

The proof is based on the duality method. Originally, localization for the dual model allows to obtain ballistic transport in expectation. Combined with dynamical localization bounds, the improved convergence allows to replace ``in expectation’’ by ``almost surely’'.