Anomalous scaling regime for one-dimensional Mott variable-range hopping

We derive an anomalous, sub-diffusive scaling limit for a one-dimen-sional version of the Mott random walk. 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. We also discuss how, by incorporating a Bouchaud trap model element into the setting, it is possible to combine this “blocking” mechanism with one of “trapping”. Our proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces.