Transience, Recurrence and the Speed of a Random Walk in a Site-Based Feedback Environment

TL;DR

This paper analyzes a self-interacting random walk in a dynamic environment, establishing conditions for transience and recurrence based on environment parameters, and providing explicit formulas for the walk's speed.

Contribution

It introduces a sharp criterion for transience and recurrence in a site-based feedback environment, including explicit speed formulas and behavior characterization in critical and noncritical cases.

Findings

Walk is transient to +∞ if α > 1/2; to -∞ if α < 1/2.

In the critical case α=1/2, behavior depends on initial environment.

Walk has positive speed in noncritical cases, with explicit formulas available.

Abstract

We study a random walk on $\mathbb{Z}$ which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, $p$ and $q$. $R$ consecutive right jumps from a site in the $q$-mode are required to switch it to the $p$-mode, and $L$ consecutive left jumps from a site in the $p$-mode are required to switch it to the $q$-mode. From a site in the $p$-mode the walk jumps right with probability $p$ and left with probability $1-p$, while from a site in the $q$-mode these probabilities are $q$ and $1-q$. We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters $\alpha = \alpha(p,q,R,L)$. For $\alpha > 1/2$ the walk is transient to $+\infty$ for any initial environment, whereas for $\alpha < 1/2$ the walk is transient to $-\infty$ for any initial environment. In the critical case, $\alpha = 1/2$, the situation is more complicated and the behavior of the walk depends on the initial environment. Nevertheless, we are able to give a characterization of transience/recurrence in many instances, including when either $R=1$ or $L=1$ and when $R=L=2$. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed.