THE GROWTH OF THE INFINITE LONG-RANGE PERCOLATION CLUSTER

• 成果类型: Article
• 署名作者: Trapman, Pieter
• 署名单位: Stockholm University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP517
• 发表日期: 2010
• 页码: 1583-1608
• 关键词: chemical distance; models; epidemics; diameter; forest; graph

摘要

We consider long-range percolation on Z(d), where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 - exp[-lambda(r)] is an element of (0, 1) and the presence or absence of different edges are independent Here, lambda(r) is a strictly positive, nonincreasing, regularly varying function We investigate the asymptotic growth of the size of the k-ball around the origin, vertical bar B(k)vertical bar, that is, the number of vertices that ale within graphdistance k of the origin, for K -> infinity, for different lambda(r) We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonmereasimi regularly varying lambda(r) exist, for which, respectively vertical bar B(k)vertical bar(1/k) -> infinity almost surely, there exist 1 < a(1) < a(2) < infinity such that lim(k ->infinity)P(a(1) < vertical bar B(k)vertical bar(1/k) < a(2)) = 1. vertical bar B(k)vertical bar(1/k) -> 1 almost surely. This result can be applied to spatial SIR epidemics. In particular. regimes are identified for which the basic reproduction number. R(0), which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.