The speed of biased random walks among dynamical random conductances

TL;DR

This paper investigates the behavior of biased random walks in dynamic environments, establishing positive speed under certain conditions, deriving asymptotic formulas, and revealing complex monotonicity properties and counterexamples.

Contribution

It provides new theoretical results on the speed of biased random walks in dynamical conductances, including explicit formulas and counterexamples.

Findings

Walks have positive speed for all biases > 0 under bounded conductances.

Explicit asymptotic formula for speed as bias tends to infinity.

Speed can decrease with increasing bias, even with bounded conductances.

Abstract

We study biased variable-speed random walks in dynamical random conductances. Assuming that the conductances are upper-bounded, we prove that the walk has strictly positive speed for every bias $\lambda>0$. We then give an explicit asymptotic formula for the speed for $\lambda \to + \infty$, and prove two monotonicity properties for the speed. Finally, we provide an example showing that, even for conductances that are bounded and bounded away from zero, the speed can be asymptotically decreasing in the bias.