SELF-AVOIDING WALK ON NONUNIMODULAR TRANSITIVE GRAPHS

• 成果类型: Article
• 署名作者: Hutchcroft, Tom
• 署名单位: University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1322
• 发表日期: 2019
• 页码: 2801-2829
• 关键词: connective constants; percolation

摘要

We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length n is comparable to the nth power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product T-k x Z(d) of a k-regular tree (k >= 3) with Z(d), for which these results were previously only known for large k.