On the speed of a one-dimensional random walk in a random environment perturbed by cookies of strength one

TL;DR

This paper investigates how cookies of strength one influence the speed of a one-dimensional random walk in a random environment, using branching processes to analyze the effects.

Contribution

It introduces a novel approach linking the random walk with branching processes in a random environment with migration to study speed.

Findings

Derived results on the walk's speed under cookie perturbations

Established conditions for positive or zero speed

Demonstrated the effectiveness of branching process correspondence

Abstract

We consider a random walk in an i.i.d. random environment on Z that is perturbed by cookies of strength 1. The number of cookies per site is assumed to be i.i.d. Results on the speed of the random walk are obtained. Our main tool is the correspondence in certain cases between the random walk and a branching process in a random environment with migration.