Attraction time for strongly reinforced walks
Codina Cotar and Vlada Limic
Source: Ann. Appl. Probab.
Volume 19, Number 5
(2009), 1972-2007.
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 
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Primary Subjects: 60G50, 60J10, 60K35
Keywords: Attracting edge; reinforced walk; strong reinforcement; time of attraction
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1255699550
Digital Object Identifier: doi:10.1214/08-AAP564
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