The Annals of Applied Probability

Continuum percolation and Euclidean minimal spanning trees in high dimensions

Mathew D. Penrose

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1034968142

Digital Object Identifier
doi:10.1214/aoap/1034968142

Mathematical Reviews number (MathSciNet)
MR1398056

Zentralblatt MATH identifier
0855.60096

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 82B43: Percolation [See also 60K35]

Keywords
Geometric probability continuum percolation phase transitions minimal spanning tree constant high dimensions Poisson process branching process

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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References

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