Random walk in a stratified independent random environment

Brémont Julien

doi:10.1214/19-ecp252

Abstract

We study Markov chains on a lattice in a codimension-one stratified independent random environment, exploiting results established in [2]. First of all the random walk is transient in dimension at least three. Focusing on dimension two, both recurrence and transience can happen, but transience remains by far the most general situation. We identify the critical scale of the local drift along the strata corresponding to the frontier between the two regimes.

Highlights

We study Markov chains on a lattice in a codimension-one stratified independent random environment, exploiting results previously established in [2]
The random walk is first shown to be transient in dimension at least three
We provide sharp sufficient conditions for either recurrence or transience

Summary

Introduction

A very first and important question concerning the asymptotic behaviour of a Markov chain on a lattice in inhomogeneous environment is the question of the recurrence/transience. The purpose of the present article is to extend the applications of [2], studying in detail the previous general model in the important case when the transition laws are random and independent, with a quenched point of view, i.e. for almost-every realization of the environment. On such a model, let us mention a result by Kochler in his doctoral thesis [12], concerning the case when P(μn = δ1) = P(μn = δ−1) = 1/2, giving P(εn = ±1) = 1/2, with rn/(pn + qn) = c (a constant independent on n) and (pn/qn, μm)(n,m)∈Z2 independent and identically distributed. This gives an indication on what should happen for more general models of random walks in independent environments

Independent setting: presentation and results

Preliminaries