Continuum percolation with steps in an annulus
成果类型：
Article
署名作者：
Balister, P; Bollobás, B; Walters, M
署名单位：
University of Memphis
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/105051604000000891
发表日期：
2004
页码：
1869-1879
关键词：
摘要：
Let A be the annulus in R-2 centered at the origin with inner and outer radii r(1 - epsilon) and r, respectively. Place points {x(i)} in R-2 according to a Poisson process with intensity 1 and let G(A) be the random graph with vertex set {x(i)} and edges x(i)x(j) whenever x(i) - x(j) is an element of A. We show that if the area of A is large, then G(A) almost surely has an infinite component. Moreover, if we fix epsilon, increase r and let n(c) = n(c)(epsilon) be the area of A when this infinite component appears, then n(c) --> 1 as epsilon --> 0. This is in contrast to the case of a square annulus where we show that n(c) is bounded away from 1.