CONTINUUM PERCOLATION AND EUCLIDEAN MINIMAL SPANNING TREES IN HIGH DIMENSIONS
成果类型：
Article
署名作者：
Penrose, Mathew D.
署名单位：
Durham University
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
1996
页码：
528-544
关键词：
摘要：
We prove that for continuum percolation in R-d, parametrized by the mean number y of points connected to the origin, as d -> infinity with y fixed the distribution of the number of points in the cluster at the origin converges to that of the total number of progeny of a branching process with a Poisson(y) offspring distribution. We also prove that for sufficiently large d the critical points for the existence of infinite occupied and vacant regions are distinct. Our results resolve conjectures made by Avram and Bertsimas in connection with their formula for the growth rate of the length of the Euclidean minimal spanning tree on n independent uniformly distributed points in d dimensions as n -> infinity.