Abstract
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.
Funding Statement
This research was supported by JSPS Grant-in-Aid for Scientific Research (C) 19K03540, JSPS Grant-in-Aid for Scientific Research (A) 17H01093, a JSPS Postdoctoral Fellowship for Research in Japan, Grant-in-Aid for JSPS Fellows 19F19814, and the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
Acknowledgments
The authors would like to thank Takashi Kumagai for his contributions in the early part of the discussions that led to this article. They also thank a referee for their very careful reading of an earlier version of the paper and pointing out an important error in the argument.
Citation
David A. Croydon. Ryoki Fukushima. Stefan Junk. "Anomalous scaling regime for one-dimensional Mott variable-range hopping." Ann. Appl. Probab. 33 (5) 4044 - 4090, October 2023. https://doi.org/10.1214/22-AAP1915
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