Interpolating between random walk and rotor walk

Abstract

We introduce a family of stochastic processes on the integers, depending on a parameter $p \in [0,1]$ and interpolating between the deterministic rotor walk (p=0) and the simple random walk (p=1/2). This p-rotor walk is not a Markov chain but it has a local Markov property: for each $x \in \mathbb{Z}$ the sequence of successive exits from $x$ is a Markov chain. The main result of this paper identifies the scaling limit of the p-rotor walk with two-sided i.i.d. initial rotors. The limiting process takes the form $\sqrt{\frac{1-p}{p}} X(t)$, where $X$ is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation \begin{equation} X(t) = \mathcal{B}(t) + a \sup_{s\leq t} X(s) + b \inf_{s\leq t} X(s) \end{equation} for all $t \in [0,\infty)$. Here $\mathcal{B}(t)$ is a standard Brownian motion and $a,b<1$ are constants depending on the marginals of the initial rotors on $\mathbb{N}$ and $-\mathbb{N}$ respectively. Chaumont and Doney [CD99] have shown that the above equation has a pathwise unique solution $X(t)$, and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, $\limsup X(t) = +\infty$ and $\liminf X(t) = -\infty$ [CDH00]. This last result, together with the main result of this paper, implies that the p-rotor walk is recurrent for any two-sided i.i.d. initial rotors and any $0<p<1$.