ATTRACTION TIME FOR STRONGLY REINFORCED WALKS

• 成果类型: Article
• 署名作者: Cotar, Codina; Limic, Vlada
• 署名单位: Technical University of Berlin; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP564
• 发表日期: 2009
• 页码: 1972-2007
• 关键词: edge; ant

摘要

We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarres [Ann. Probab. (2007), to appear], that the attracting edge emerges with probability I 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 omega(k) = k(rho), rho > 1, then (universally over finite graphs) the expected time to attraction is infinite if and only if rho <= 1 + 1+root 5/2.