Continuum percolation and Euclidean minimal spanning trees in high dimensions
Continuum percolation and Euclidean minimal spanning trees in high dimensions
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高维连续渗流和欧几里得最小生成树
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发表时间：
1996
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通讯作者：
M. Penrose
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作者：
M. Penrose
我们证明了连续体渗流在rd中，由连接到原点的点的平均数目y参数化为d ！1加y (cid:12)，则原点处的聚类点数分布收敛于泊松(y)子代分布的分支过程的子代总数分布。我们还证明了对于足够大的f(cid:12) d，在（cid:12）非占用区和空区存在的临界点是不同的。我们的结果解决了Avram和Bertsimas关于d维中n个独立均匀分布点上的欧几里得最小生成树长度增长率公式的猜想。1。
We prove that for continuum percolation in R d , parametrized by the mean number y of points connected to the origin, as d ! 1 with y (cid:12)xed 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 suf(cid:12)ciently large d the critical points for the existence of in(cid:12)nite 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 ! 1 .