Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

• 成果类型: Article
• 署名作者: van der Hofstad, R; Sakai, A
• 署名单位: Eindhoven University of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0405-4
• 发表日期: 2005
• 页码: 438-470
• 关键词: field critical-behavior; super-brownian motion; oriented percolation; lace expansion; TREE

摘要

We consider self-avoiding walk and percolation in Z(d), oriented percolation in X-d x Z(+), and the contact process in Z(d), with pD(center dot) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y-x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p(c). We investigate the value of p(c) when d > 6 for percolation and d > 4 for the other models, and L >> 1. We prove in a unified way that p(c)=1+C(D)+O(L-2), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L-d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p(c)=1+cL(-d) +O(L-d-1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.