{ "id": "1603.04107", "version": "v1", "published": "2016-03-14T01:38:18.000Z", "updated": "2016-03-14T01:38:18.000Z", "title": "Interpolating between random walk and rotor walk", "authors": [ "Wilfried Huss", "Lionel Levine", "Ecaterina Sava-Huss" ], "comment": "22 pages, 2 figures", "categories": [ "math.PR" ], "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