Annals of Probability

Stuck walks: A conjecture of Erschler, Tóth and Werner

Daniel Kious

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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.

Article information

Ann. Probab., Volume 44, Number 2 (2016), 883-923.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G20: Generalized stochastic processes 60G42: Martingales with discrete parameter

Stuck walks reinforced random walks localization Rubin time-line construction martingale


Kious, Daniel. Stuck walks: A conjecture of Erschler, Tóth and Werner. Ann. Probab. 44 (2016), no. 2, 883--923. doi:10.1214/14-AOP991.

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