Deterministic walk in an excited random environment

TL;DR

This paper studies a deterministic walk in an excited environment modeled by stacks of cookies on each site, analyzing its behavior, moments, and large deviations, revealing complex dynamics depending on cookie consumption.

Contribution

It introduces a new model of deterministic walks in cookie environments and establishes monotonicity and large deviation results for the process.

Findings

Moments of the walk are sub-linear in time.

The walk can be infinite without looping when multiple cookies are present.

Monotonicity results lead to large deviation principles.

Abstract

Deterministic walk in an excited random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{\infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of cookies on each site of $\mathbb Z$. Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. If each site has exactly one cookie, then the walk ends in a loop if it ever visits the same site twice. If the number of cookies per site is increased to two, the walk can visit a site infinitely many times and still not end in a loop. Nevertheless the moments of $X_n$ are sub-linear in $n$ and we establish monotonicity results on the environment that imply large deviations.