Random Walks in Dirichlet Random Environments on $\mathbb{Z}$ with Bounded Jumps

TL;DR

This paper studies one-dimensional random walks with bounded jumps in Dirichlet-distributed environments, providing a detailed characterization of recurrence, transience, and ballisticity based on specific parameters.

Contribution

It introduces a generalized model with bounded jumps, characterizes recurrence and transience, and identifies parameters governing ballistic behavior.

Findings

Walk is right-transient iff >0

Walk is ballistic iff >1 and >1

Parameters  and  determine trapping and traversal effects

Abstract

We examine a class of random walks in random environments on $\mathbb{Z}$ with bounded jumps, a generalization of the classic one-dimensional model. The environments we study have i.i.d. transition probability vectors drawn from Dirichlet distributions. For this model, we characterize recurrence and transience, and in the transient case we characterize ballisticity. For ballisticity, we give two parameters, $\kappa_0$ and $\kappa_1$. The parameter $\kappa_0$ governs finite trapping effects, and $\kappa_1$ governs repeated traversals of arbitrarily large regions of the graph. We show that the walk is right-transient if and only if $\kappa_1>0$, and in that case it is ballistic if and only if $\min(\kappa_0,\kappa_1)>1$.