Attraction time for strongly reinforced walks

We consider a class of strongly edge reinforced random walks, where the corresponding reinforcement weight function is non-decreasing. It is known by Limic and Tarrès (2006) 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 multiplicative constant) asymptotics if the underlying graph has two edges. Next we show some extensions in the setting of finite and bounded degree infinite graphs. A nice corollary is 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 ρ ≤ 1 + 1+ √ 5 2 . AMS 2000 Subject Classification. 60G50, 60J10, 60K35