The Annals of Probability

Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$

Anne-Laure Basdevant, Bruno Schapira, and Arvind Singh

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Abstract

We characterize nondecreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on $4$ sites. A phase transition appears for weights of order $n\log\log n$: for weights growing faster than this rate, the VRRW localizes almost surely on, at most, $4$ sites, whereas for weights growing slower, the VRRW cannot localize on less than $5$ sites. When $w$ is of order $n\log\log n$, the VRRW localizes almost surely on either $4$ or $5$ sites, both events happening with positive probability.

Article information

Source
Ann. Probab. Volume 42, Number 2 (2014), 527-558.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1393251295

Digital Object Identifier
doi:10.1214/12-AOP811

Mathematical Reviews number (MathSciNet)
MR3178466

Zentralblatt MATH identifier
1297.60062

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J17 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 walk localization urn model

Citation

Basdevant, Anne-Laure; Schapira, Bruno; Singh, Arvind. Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$. Ann. Probab. 42 (2014), no. 2, 527--558. doi:10.1214/12-AOP811. https://projecteuclid.org/euclid.aop/1393251295


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References

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