The Annals of Probability

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

Pierre Tarrès

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Abstract

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

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

Dates
First available in Project Euclid: 6 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1091813627

Digital Object Identifier
doi:10.1214/009117907000000694

Mathematical Reviews number (MathSciNet)
MR2078554

Zentralblatt MATH identifier
1068.60072

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.aop/1091813627


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