Open Access
July 1999 Vertex-Reinforced Random Walk on Z Has Finite Range
Robin Pemantle, Stanislav Volkov
Ann. Probab. 27(3): 1368-1388 (July 1999). DOI: 10.1214/aop/1022677452


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.


Download Citation

Robin Pemantle. Stanislav Volkov. "Vertex-Reinforced Random Walk on Z Has Finite Range." Ann. Probab. 27 (3) 1368 - 1388, July 1999.


Published: July 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0960.60041
MathSciNet: MR1733153
Digital Object Identifier: 10.1214/aop/1022677452

Primary: 60G17
Secondary: 60J20

Keywords: Bernard Friedman’s urn , Reinforced random walk , urn model , Vertex-reinforced random walk , VRRW

Rights: Copyright © 1999 Institute of Mathematical Statistics

Vol.27 • No. 3 • July 1999
Back to Top