Sub-diffusive scaling regimes for one-dimensional Mott variable-range hopping

I will describe anomalous, sub-diffusive scaling limits for a one-dimensional version of the Mott random walk. The first setting considered nonetheless results in polynomial space-time scaling. In this case, the limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. I will outline how the proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces. The second setting considered concerns a regime that exhibits even more severe blocking (and sub-polynomial scaling). For this, I will describe how, for any fixed time, the appropriately-rescaled Mott random walk is situated between two
environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, I will give an asymptotic
description of the distribution of the Mott random walk between the barriers that contain it. This is joint work with Ryoki Fukushima (University of Tsukuba) and Stefan Junk (Tohoku University).