On the speed of Random Walks among Random Conductances

TL;DR

This paper investigates how the moments of conductance distributions influence the speed of random walks in random environments, establishing conditions for zero speed and providing counterexamples.

Contribution

It identifies moment conditions on conductances that guarantee zero speed and constructs examples where the speed is non-zero or undefined despite finite moments.

Findings

If E[log^α(ω_e)]<∞ for some α>1, the walk has zero speed.

Existence of non-zero speed walks with finite moments of conductances.

Examples of walks with undefined limiting speed despite finite moments.

Abstract

We consider random walk among random conductances where the conductance environment is shift invariant and ergodic. We study which moment conditions of the conductances guarantee speed zero of the random walk. We show that if there exists \alpha>1 such that E[log^\alpha({\omega}_e)]<\infty, then the random walk has speed zero. On the other hand, for each \alpha>1 we provide examples of random walks with non-zero speed and random walks for which the limiting speed does not exist that have E[log^\alpha({\omega}_e)]<\infty.