| 3 ABSTRACT: (2007) | |||||||||||||||
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| A stochastic process called Vertex-Reinforced Random Walk (VRRW) is dened in Pemantle (1988a). We consider this process in the case where the underlying graph is an in nite chain (i.e., the one-dimensional integer lattice). We show that the range is almost surely nite, that at least 5 points are visited innitely often almost surely, and that with positive probability the range contains exactly 5 points. There are always points visited in nitely 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 n for a dense set of values 2 (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 | |||||||||||||||
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