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March 2014 Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$
Anne-Laure Basdevant, Bruno Schapira, Arvind Singh
Ann. Probab. 42(2): 527-558 (March 2014). DOI: 10.1214/12-AOP811


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.


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Anne-Laure Basdevant. Bruno Schapira. Arvind Singh. "Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$." Ann. Probab. 42 (2) 527 - 558, March 2014.


Published: March 2014
First available in Project Euclid: 24 February 2014

zbMATH: 1297.60062
MathSciNet: MR3178466
Digital Object Identifier: 10.1214/12-AOP811

Primary: 60J17 , 60J20 , 60K35

Keywords: Localization , Reinforced random walk , urn model

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 2 • March 2014
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