The Annals of Probability

Vertex-reinforced random walk on ℤ eventually gets stuck on five points

Pierre Tarrès

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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.

Article information

Ann. Probab., Volume 32, Number 3B (2004), 2650-2701.

First available in Project Euclid: 6 August 2004

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Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03] 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

Reinforced random walks urn model random perturbations of dynamical systems repulsive traps


Tarrès, Pierre. Vertex-reinforced random walk on ℤ eventually gets stuck on five points. Ann. Probab. 32 (2004), no. 3B, 2650--2701. doi:10.1214/009117907000000694.

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