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Mott Law as Lower Bound for a Random Walk in a Random Environment

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Abstract

We consider a random walk on the support of an ergodic stationary simple point process on ℝd,d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.

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Authors and Affiliations

  1. Weierstrass Institut für Angewandte Analysis und Stochastic, 10117, Berlin, Germany

    A. Faggionato

  2. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    H. Schulz-Baldes

  3. Fachbereich Physik, Universität Duisburg-Essen, 45117, Essen, Germany

    D. Spehner

Authors
  1. A. Faggionato

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  2. H. Schulz-Baldes

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  3. D. Spehner

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Communicated by M. Aizenman

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Faggionato, A., Schulz-Baldes, H. & Spehner, D. Mott Law as Lower Bound for a Random Walk in a Random Environment.Commun. Math. Phys.263, 21–64 (2006). https://doi.org/10.1007/s00220-005-1492-5

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