The Annals of Probability

Vertex-Reinforced Random Walk on Z Has Finite Range

Robin Pemantle and Stanislav Volkov

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A stochastic process called vertex-reinforced random walk (VRRW) is defined in Pemantle [Ann. Probab. 16 1229–1241] . 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 five points are visited infinitely often almost surely and that with positive probability the range contains exactly five 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 $n ^[\alpha}$ for a dense set of values $\alpha \in (0,1)$. The power law analysis relies on analysis of a related urn model.

Article information

Ann. Probab., Volume 27, Number 3 (1999), 1368-1388.

First available in Project Euclid: 29 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 60J20

Vertex-reinforced random walk Reinforced random walk VRRW Urn model Bernard Friedman’s urn


Pemantle, Robin; Volkov, Stanislav. Vertex-Reinforced Random Walk on Z Has Finite Range. Ann. Probab. 27 (1999), no. 3, 1368--1388. doi:10.1214/aop/1022677452.

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