Interpolating between random walk and rotor walk.
RANDOM STRUCTURES & ALGORITHMS(2018)
摘要
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 xZ 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 1-ppX(t), where X is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation X(t)=B(t)+asupfor all t[0,). 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 -N respectively. Chaumont and Doney have shown that Equation 1 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, lim?sup?X(t)=+ and lim?inf?X(t)=-. 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.
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关键词
Abelian network,correlated random walk,locally Markov walk,martingale,perturbed Brownian motion,recurrence,rotor-router model,scaling limit
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