Excited deterministic walk in a random environment

Abstract

Excited deterministic walk in a 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 $x$ arbitrarily many times before getting stuck in a loop, which may or may not contain $x$. Nevertheless, we establish monotonicity results on the environment that imply large deviations.