Attracting edge and strongly edge reinforced walks

• 成果类型: Article
• 署名作者: Limic, Vlada; Tarres, Pierre
• 署名单位: Aix-Marseille Universite; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000001097
• 发表日期: 2007
• 页码: 1783-1806
• 关键词: points

摘要

The goal is to show that an edge-reinforced random walk on a graph of bounded degree, with reinforcement weight function W taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting edge at all large times. The statement of the main theorem is very close to settling a conjecture of Sellke [Technical Report 94-26 (1994) Purdue Univ.]. An important corollary of this main result says that if W is reciprocally summable and nondecreasing, the attracting edge exists on any graph of bounded degree, with probability 1. Another corollary is the main theorem of Limic [Ann. Probab. 31 (2003) 1615-1654], where the class of weights was restricted to reciprocally summable powers. The proof uses martingale and other techniques developed by the authors in separate studies of edge- and vertex-reinforced walks [Ann. Probab. 31 (2003) 1615-1654, Ann. Prol ab. 32 (2004) 2650-270 1] and of nonconvergence properties of stochastic algorithms toward unstable equilibrium points of the associated deterministic dynamics [C R. Acad. Sci. Ser I Math. 330 (2000) 125-130].