Self-attracting self-avoiding walk

• 成果类型: Article
• 署名作者: Hammond, Alan; Helmuth, Tyler
• 署名单位: University of California System; University of California Berkeley; University of Bristol
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-00898-7
• 发表日期: 2019
• 页码: 677-719
• 关键词: lace expansion

摘要

This article is concerned with self-avoiding walks (SAW) on that are subject to a self-attraction. The attraction, which rewards instances of adjacent parallel edges, introduces difficulties that are not present in ordinary SAW. Ueltschi has shown how to overcome these difficulties for sufficiently regular infinite-range step distributions and weak self-attractions (Ueltschi in Probab Theory Relat Fields 124(2):189-203, 2002). This article considers the case of bounded step distributions. For weak self-attractions we show that the connective constant exists, and, in carry out a lace expansion analysis to prove the mean-field behaviour of the critical two-point function, hereby addressing a problem posed by den Hollander (Random Polymers, vol. 1974. Springer-Verlag, Berlin, 2009).