Transience/Recurrence and the speed of a one-dimensional random walk in a "have your cookie and eat it" environment

TL;DR

This paper studies a one-dimensional random walk with a 'cookie' environment where the probability to jump right is initially biased but becomes fair after the first left jump at each site, analyzing its transience, recurrence, and speed.

Contribution

It introduces a model with a dynamic transition mechanism and characterizes its transience, recurrence, and speed in deterministic and ergodic environments.

Findings

Characterizes conditions for transience and recurrence.

Analyzes the speed of the walk in deterministic environments.

Provides insights into the impact of environment on walk behavior.

Abstract

Consider a simple random walk on the integers with the following transition mechanism. At each site $x$, the probability of jumping to the right is $\omega(x)\in[\frac12,1)$, until the first time the process jumps to the left from site $x$, from which time onward the probability of jumping to the right is $\frac12$. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments $\{\omega(x)\}_{x\in Z}$. In deterministic environments, we also study the speed of the process.