Zeno's Walk: a Random Walk with Refinements

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 calculate the exponential rate of convergence of the process to its limit point, which may be taken as a bound for the convergence of the measure in the Wasserstein metric. The process may also be described as a random dynamical system, or as a random walk on the space of monotone piecewise linear functions, where moves are taken by successive compositions with a randomly chosen such function. 1. introduction Following a model introduced by Coppersmith and Diaconis 4], a fair amount of work (cf. Davis 5], Pemantle 8]) has been devoted to analyzing reinforced random walks, that is, random walks on graphs where the probability of moving to a given vertex or over a given edge is a function of the number of times that vertex or edge has been selected in the past. Often the reinforcement is supposed to be positive; that is, the more often an edge has been crossed in the past, the more likely it is to be chosen in the future. Among other results, Davis has shown that all positively reinforced random walks on the integers are recurrent to 0, as long as the weights increase moderately Allowing negative reinforcement complicates the analysis. One approach, discussed at greater length in section 6.4, is to redeene the negative reinforcement as a process of reenement, whereby each step modiies the graph itself, rather than some weights attached to the graph. As a method solving problems about reinforced random walk this has so far proved nearly useless. The natural negatively reinforced random walks are translated into fairly unnatural and intractable random walks with reenement. Zeno's walk, the topic to be examined in this paper, arose from the search for a process that would sit more congenially in this new context. It could be viewed as a particularly radical sort of negative reinforcement process, since an edge once crossed is likely never to be crossed again in its entirety { it is trapped instead on an innnity of intermediate points, crossing and …