WIAS Preprint No. 955, (2004)

Mott law as lower bound for a random walk in a random environment


  • Faggionato, Alessandra
    ORCID: 0000-0002-6168-3517
  • Schulz-Baldes, H.
  • Spehner, D.

2010 Mathematics Subject Classification

  • 60D05 60K3 88C44


  • Disordered systems, Mott law, random walk in random environment, point processes, percolation




We consider a random walk on the support of a stationary simple point process on $RR^d$, $dgeq 2$ 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 localisation. 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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