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

TL;DR

This paper proves that the diffusion of a random walk in a disordered medium is bounded below by Mott's law, linking microscopic hopping dynamics to macroscopic conductivity in strongly localized electronic systems.

Contribution

It establishes a rigorous lower bound for the diffusion coefficient of a random walk in a random environment, matching Mott's law for variable range hopping conductivity.

Findings

Diffusion converges to a Brownian motion with a lower bound given by Mott's law.

The lower bound is proven using percolation theory estimates.

The model captures phonon-induced electron hopping in disordered solids.

Abstract

We consider a random walk on the support of a stationary simple point process on $R^d$, $d\geq 2$ 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.