Random walks in degenerate random environments

TL;DR

This paper investigates the long-term behavior of random walks in non-elliptic, i.i.d. random environments on integer lattices, establishing conditions for transience and velocity, and extending classical laws to more general settings.

Contribution

It introduces new monotonicity results for velocity in 2-valued environments and generalizes 0-1 laws for directional transience to non-elliptic cases.

Findings

Monotonicity of velocity for 2-valued environments

Existence of directional transience under certain conditions

Deterministic limiting velocity in 2D environments

Abstract

We study the asymptotic behaviour of random walks in i.i.d. random environments on $\Z^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience, and in 2-dimensions the existence of a deterministic limiting velocity.