Interpolating between random walk and rotor walk

We introduce a family of stochastic processes on the integers, depending on a parameter (Formula presented.) and interpolating between the deterministic rotor walk ((Formula presented.)) and the simple random walk ((Formula presented.)). This p-rotor walk is not a Markov chain but it has a local Markov property: for each (Formula presented.) the sequence of successive exits from (Formula presented.) 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 (Formula presented.), where (Formula presented.) is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation 1 (Formula presented.) for all (Formula presented.). Here (Formula presented.) is a standard Brownian motion and (Formula presented.) are constants depending on the marginals of the initial rotors on (Formula presented.) and (Formula presented.) respectively. Chaumont and Doney have shown that Equation 1 has a pathwise unique solution (Formula presented.), and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, (Formula presented.) and (Formula presented.). 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 (Formula presented.).