We introduce a family of stochastic processes on the integers, depending on a parameter p ∈ [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 ∈ Z the sequence of successive exits from x is a Markov chain. The main result of this paper identifies the sc...

A self-modifying random walk on Q is deened from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which is supported on a subset of R having Hausdorr dimension less than 1. By generating function techniques we then calcul...