Abstract

A stochastic process called Vertex-Reinforced Random Walk (VRRW) is defined in Pemantle (1988a). We consider this process in the case where the underlying graph is an infinite chain (i.e., the one-dimensional integer lattice). We show that the range is almost surely finite, that at least 5 points are visited infinitely often almost surely, and that with positive probability the range contains exactly 5 points. There are always points visited infinitely often but at a set of times of zero density, and we show that the number of visits to such a point to time n may be asymptotically nα for a dense set of values α ∈ (0,1). The power law analysis relies on analysis of a related urn model. Keywords: Vertex-reinforced random walk, Reinforced random walk, VRRW, Urn model, Bernard Friedman’s urn

Citations