Mott Law as Lower Bound for a Random Walk in a Random Environment |
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Authors: | A Faggionato H Schulz-Baldes D Spehner |
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Institution: | 1. Weierstrass Institut für Angewandte Analysis und Stochastic, 10117, Berlin, Germany 2. Institut für Mathematik, Technische Universit?t Berlin, 10623, Berlin, Germany 3. Fachbereich Physik, Universit?t Duisburg-Essen, 45117, Essen, Germany
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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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