Mathematics – Probability
Scientific paper
2006-12-02
Annals of Applied Probability 2009, Vol. 19, No. 5, 1972-2007
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/08-AAP564 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/08-AAP564
We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarr\`{e}s [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 $w(k)=k^{\rho}$, $\rho>1$, then (universally over finite graphs) the expected time to attraction is infinite if and only if $\rho\leq1+\frac{1+\sqrt{5}}{2}$.
Cotar Codina
Limic Vlada
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