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

TL;DR

This paper establishes an upper bound on the diffusion behavior of a random walk in a disordered environment, aligning with Mott law, which models electron hopping in disordered solids under strong localization.

Contribution

It proves an upper bound on the asymptotic diffusion matrix for a random walk in a random environment, complementing existing lower bounds and confirming Mott law predictions.

Findings

Upper bound on the diffusion matrix matches Mott law.

Model describes phonon-induced electron hopping in disordered solids.

Results support theoretical predictions of electron transport in localization regime.

Abstract

We consider a random walk on the support of an ergodic simple point process on R^d, d>1, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann-type factor. This is an effective model for the phonon-induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process we prove an upper bound of the asymptotic diffusion matrix of the random walk in agreement with Mott law. A lower bound in agreement with Mott law was proved in \cite{FSS}.