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October 2009 Attraction time for strongly reinforced walks
Codina Cotar, Vlada Limic
Ann. Appl. Probab. 19(5): 1972-2007 (October 2009). DOI: 10.1214/08-AAP564

Abstract

We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarrès [Ann. Probab. (2007), to appear], that the attracting edge emerges with probability 1 whenever the underlying graph is locally bounded. We study the asymptotic behavior of the tail distribution of the (random) time of attraction. In particular, we obtain exact (up to a multiplicative constant) asymptotics if the underlying graph has two edges. Next, we show some extensions in the setting of finite graphs, and infinite graphs with bounded degree. As a corollary, we obtain the fact that if the reinforcement weight has the form w(k)=kρ, ρ>1, then (universally over finite graphs) the expected time to attraction is infinite if and only if $\rho\leq1+\frac{1+\sqrt{5}}{2}$.

Citation

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Codina Cotar. Vlada Limic. "Attraction time for strongly reinforced walks." Ann. Appl. Probab. 19 (5) 1972 - 2007, October 2009. https://doi.org/10.1214/08-AAP564

Information

Published: October 2009
First available in Project Euclid: 16 October 2009

zbMATH: 1213.60087
MathSciNet: MR2569814
Digital Object Identifier: 10.1214/08-AAP564

Subjects:
Primary: 60G50 , 60J10 , 60K35

Keywords: Attracting edge , reinforced walk , strong reinforcement , time of attraction

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.19 • No. 5 • October 2009
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