Random Walks on Dynamical Random Environments with Non-Uniform Mixing

TL;DR

This paper investigates random walks in dynamical environments in one dimension, establishing laws of large numbers and concentration inequalities under mild mixing conditions, applicable to models like the contact process and East model.

Contribution

It introduces new results on random walks in non-uniformly mixing environments, including LLN and concentration bounds, expanding understanding beyond uniform mixing assumptions.

Findings

Law of Large Numbers for the walk's position

Concentration inequality around asymptotic speed

Positive speed for the distinguished zero in the East model

Abstract

In this paper we study random walks on dynamical random environments in $1 + 1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a concentration inequality around the asymptotic speed. The mixing hypothesis imposes a polynomial decay rate of covariances on the environment with sufficiently high exponent but does not impose uniform mixing. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the East model. For the East model we also obtain that the distinguished zero satisfies a Law of Large Numbers with strictly positive speed.