Biased random walk in positive random conductances on $\mathbb{Z}^{d}$

TL;DR

This paper investigates the behavior of biased random walks in positive random conductances on integer lattices, establishing conditions for ballistic movement and characterizing sub-ballistic regimes.

Contribution

It extends previous results by proving the equivalence between finite mean conductances and ballistic behavior, and determines polynomial order of displacement in sub-ballistic cases.

Findings

Random walk is ballistic if conductances have finite mean.

In sub-ballistic regimes, the polynomial order of displacement is characterized.

Extends prior work from uniformly elliptic to more general conductance distributions.

Abstract

We study the biased random walk in positive random conductances on $\mathbb {Z}^d$. This walk is transient in the direction of the bias. Our main result is that the random walk is ballistic if, and only if, the conductances have finite mean. Moreover, in the sub-ballistic regime we find the polynomial order of the distance moved by the particle. This extends results obtained by Shen [Ann. Appl. Probab. 12 (2002) 477-510], who proved positivity of the speed in the uniformly elliptic setting.