## The Annals of Applied Probability

### Continuum percolation and Euclidean minimal spanning trees in high dimensions

Mathew D. Penrose

#### 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 $n \to \infty$.

#### Article information

Source
Ann. Appl. Probab., Volume 6, Number 2 (1996), 528-544.

Dates
First available in Project Euclid: 18 October 2002

https://projecteuclid.org/euclid.aoap/1034968142

Digital Object Identifier
doi:10.1214/aoap/1034968142

Mathematical Reviews number (MathSciNet)
MR1398056

Zentralblatt MATH identifier
0855.60096

#### Citation

Penrose, Mathew D. Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Probab. 6 (1996), no. 2, 528--544. doi:10.1214/aoap/1034968142. https://projecteuclid.org/euclid.aoap/1034968142

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