Vertex-reinforced random walk on Z has finite range

A stochastic process called vertex-reinforced random walk (VRRW) is defined in Pemantle [Ann. Probab. 16 1229-1241]. We consider this process in the c ase where the underlying graph is an infinite chain (i.e., the one-dimensio nal integer lattice). We show that the range is almost surely finite, that at least five points are visited infinitely often almost surely and that wi th positive probability the range contains exactly five points. There are a lways 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 as ymptotically n(alpha) for a dense set of values alpha is an element of (0, 1). The power law analysis relies on analysis of a related urn model.