Excited random walks with Markovian cookie stacks

TL;DR

This paper studies a one-dimensional excited random walk with Markovian cookie stacks, revealing different behaviors in critical and non-critical regimes, including transience, recurrence, ballisticity, and limit laws, generalizing previous periodic models.

Contribution

It introduces a Markovian environment for excited random walks, providing new criteria for recurrence, transience, and limit laws, extending the understanding beyond periodic cookie stacks.

Findings

Non-critical regime: walk is transient with non-zero speed and obeys CLT.

Critical regime: diverse behaviors with specific recurrence/transience conditions.

Results generalize and improve upon previous models with periodic cookie stacks.

Abstract

We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $\omega_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection $(\omega_x(j))_{x\in\mathbb{Z},n\ge 0}$ is sometimes called the "cookie environment" due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probabilities according to the "strength" of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie "strengths" within the stack $(\omega_x(j))_{j\ge 0}$ at site $x$ follow a finite state Markov chain. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the original random walk time. The model admits two different regimes, critical or non-critical, depending on whether the expected probability to jump to the right (or left) under the invariant measure for the Markov chain is equal to $1/2$ or not. We show that in the non-critical regime the walk is always transient, has non-zero linear speed, and satisfies the classical central limit theorem. The critical regime allows for a much more diverse behavior. We give necessary and sufficient conditions for recurrence/transience and ballisticity of the walk in the critical regime as well as a complete characterization of limit laws under the averaged measure in the transient case. The setting considered in this paper generalizes the previously studied model with periodic cookie stacks. Our results on ballisticity and limit theorems are new even for the periodic model.