Random Walks in Random Environments with Rare Anomalies

Abstract

We study random walks in i.i.d. random environments on $\mathbb{Z}^d$ when there are two basic types of vertices, which we call "blue" and "red". Each color represents a different probability distribution on transition probability vectors. We introduce a method of studying these walks that compares the expected amount of time spent at a specific site on the event that the site is red with the expected amount of time spent there on the event that the site is blue. This method produces explicit bounds on the asymptotic velocity of the walk. We recover an early result of Kalikow, but with new bounds on the velocity. Next, we consider a "rare anomaly" model where the vast majority of sites are blue, and blue sites are uniformly elliptic, with some almost-sure bounds on the quenched drift. We show that if the red sites satisfy a certain uniform ellipticity assumption in two fixed, non-parallel directions, then even if red sites break the almost-sure bounds on the quenched drift, making red sites unlikely enough lets us obtain bounds on the asymptotic velocity of the walk arbitrarily close to the bounds on the quenced drift at blue sites. Significantly, the required proportion $p^*$ of blue sites to do this does not depend on the distribution of red sites, except through the uniform ellipticity assumption in two directions. Our proof is based on a coupling technique, where two walks run in environments that are the same everywhere except at one vertex. They decouple when they hit that vertex, and our proof is driven by bounds on how long it takes to recouple. We then demonstrate the importance of the i.i.d. assumption by providing a counterexample to the statement of the theorem with this assumption removed. We conclude with open questions.