We introduce a family of stochastic processes on the integers, depending on a
parameter pâ[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 xâZ 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
p1âpââX(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 tâ[0,â). Here B(t) is a standard Brownian motion
and a,b<1 are constants depending on the marginals of the initial rotors on
N and âN 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, limsupX(t)=+â and liminfX(t)=ââ [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.Comment: 22 pages, 2 figures; Remark about the connection between our model
and excited random walks with Markovian cookie stacks added. References adde