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March 2016 Stuck walks: A conjecture of Erschler, Tóth and Werner
Daniel Kious
Ann. Probab. 44(2): 883-923 (March 2016). DOI: 10.1214/14-AOP991


In this paper, we work on a class of self-interacting nearest neighbor random walks, introduced in [Probab. Theory Related Fields 154 (2012) 149–163], for which there is competition between repulsion of neighboring edges and attraction of next-to-neighboring edges. Erschler, Tóth and Werner proved in [Probab. Theory Related Fields 154 (2012) 149–163] that, for any $L\ge1$, if the parameter $\alpha$ belongs to a certain interval $(\alpha_{L+1},\alpha_{L})$, then such random walks localize on $L+2$ sites with positive probability. They also conjectured that this is the almost sure behavior. We prove this conjecture partially, stating that the walk localizes on $L+2$ or $L+3$ sites almost surely, under the same assumptions. We also prove that, if $\alpha\in(1,+\infty)=(\alpha_{2},\alpha_{1})$, then the walk localizes a.s. on $3$ sites.


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Daniel Kious. "Stuck walks: A conjecture of Erschler, Tóth and Werner." Ann. Probab. 44 (2) 883 - 923, March 2016.


Received: 1 September 2013; Revised: 1 November 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1344.60096
MathSciNet: MR3474462
Digital Object Identifier: 10.1214/14-AOP991

Primary: 60K35
Secondary: 60G20 , 60G42

Keywords: Localization , martingale , reinforced random walks , Rubin , Stuck walks , time-line construction

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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