Perturbing transient Random Walk in a Random Environment with cookies of maximal strength

TL;DR

This paper studies a left-transient random walk in a random environment on Z, which is influenced by cookies of maximal strength, and establishes criteria for its recurrence and transience using branching process techniques.

Contribution

It introduces a novel analysis of a random walk disturbed by maximal strength cookies, providing new criteria for recurrence and transience.

Findings

Criteria for recurrence and transience are derived.

Branching processes in random environments are used for analysis.

The impact of cookies on the walk's behavior is quantified.

Abstract

We consider a left-transient random walk in a random environment on Z that will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.