ASYMPTOTIC BEHAVIOR OF THE GYRATION RADIUS FOR LONG-RANGE SELF-AVOIDING WALK AND LONG-RANGE ORIENTED PERCOLATION

• 成果类型: Article
• 署名作者: Chen, Lung-Chi; Sakai, Akira
• 署名单位: Fu Jen Catholic University; Hokkaido University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP557
• 发表日期: 2011
• 页码: 507-548
• 关键词: limit distribution

摘要

We consider random walk and self-avoiding walk whose 1-step distribution is given by D, and oriented percolation whose bond-occupation probability is proportional to D. Suppose that D(x) decays as vertical bar x vertical bar(-d-alpha) with alpha > 0. For random walk in any dimension d and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension d(c) equivalent to 2(alpha boolean AND 2), we prove large-t asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length t or the average spatial size of an oriented percolation cluster at time t. This proves the conjecture for long-range self-avoiding walk in [Ann. Inst. H. Poincare Probab. Statist. (2010), to appear] and for long-range oriented percolation in [Probab. Theory Related Fields 142 (2008) 151-188] and [Probab. Theory Related Fields 145 (2009) 435-458].