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May 1996 Continuum percolation and Euclidean minimal spanning trees in high dimensions
Mathew D. Penrose
Ann. Appl. Probab. 6(2): 528-544 (May 1996). DOI: 10.1214/aoap/1034968142

Abstract

We prove that for continuum percolation in $\mathbb{R}^d$, parametrized by the mean number y of points connected to the origin, as $d \to \infty$ 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 $n \to \infty$.

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Mathew D. Penrose. "Continuum percolation and Euclidean minimal spanning trees in high dimensions." Ann. Appl. Probab. 6 (2) 528 - 544, May 1996. https://doi.org/10.1214/aoap/1034968142

Information

Published: May 1996
First available in Project Euclid: 18 October 2002

zbMATH: 0855.60096
MathSciNet: MR1398056
Digital Object Identifier: 10.1214/aoap/1034968142

Subjects:
Primary: 60D05 , 60K35
Secondary: 60J80 , 82B43

Keywords: branching process , continuum percolation , geometric probability , high dimensions , minimal spanning tree constant , Phase transitions , Poisson process

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.6 • No. 2 • May 1996
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