Invariance principle for Mott variable range hopping and other walks on point processes

TL;DR

This paper proves that a class of random walks on Poisson point processes with energy-dependent jump rates exhibit diffusive behavior converging to Brownian motion, extending results to various point processes including lattices.

Contribution

It establishes an invariance principle for Mott variable range hopping and similar walks on point processes, showing convergence to Brownian motion under broad conditions.

Findings

Random walk converges to Brownian motion with positive definite diffusion matrix.

Results hold for Poisson point processes and diluted lattices.

Diffusive behavior is independent of the environment.

Abstract

We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the A-power of the jump length and depend on the energy marks via a Boltzmann--like factor. The case A=1 corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with arbitrary start point, converges to a Brownian motion whose diffusion matrix is positive definite, and independent of the environment. Finally, we extend the above result to other point processes including diluted lattices.