Quenched limits for sub-ballistic random walks in random conductances: high and low dimensions.

Abstract: Consider a random walk amongst elliptic conductances with a deterministic directional bias. Fribergh proved that the walk is ballistic iff the mean of a conductance is finite. In the infinite mean case, under proper regularity conditions, Fribergh and Kious showed the convergence of the rescaled process towards Fractional Kinetics, in the annealed setting. I will explain how to get a quenched limit by exploiting the classical strategy of Bolthausen and Sznitman. I will also highlight the difference between the high dimensional case (d \ge 5) and the lower dimensional ones (d = 2,3,4). This is based on joint work with A. Fribergh and T. Lions and ongoing work with T. Lions and U. De Ambroggio.