After a short introduction on deterministic random walks (called also rotor walks) and some related cluster growth models, I will introduce a family of stochastic processes on the integers, depending on a parameter p. These processes interpolate between the deterministic rotor walk (for p=0) and the simple random walk (for p=1/2), and they are not Markovian. For such processes, I will prove that the scaling limit is a one-sided perturbed Brownian motion, which is a linear combination of a Brownian motion and its running maximum.
Interpolating between rotor walk and random walk