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July 2004 Vertex-reinforced random walk on ℤ eventually gets stuck on five points
Pierre Tarrès
Ann. Probab. 32(3B): 2650-2701 (July 2004). DOI: 10.1214/009117907000000694


Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the one-dimensional integer lattice ℤ. They proved that the range is almost surely finite and that with positive probability the range contains exactly five points. They conjectured that this second event holds with probability 1. The proof of this conjecture is the main purpose of this paper.


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Pierre Tarrès. "Vertex-reinforced random walk on ℤ eventually gets stuck on five points." Ann. Probab. 32 (3B) 2650 - 2701, July 2004.


Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1068.60072
MathSciNet: MR2078554
Digital Object Identifier: 10.1214/009117907000000694

Primary: 60G17
Secondary: 34F05 , 60J20

Keywords: random perturbations of dynamical systems , reinforced random walks , repulsive traps , urn model

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3B • July 2004
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