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Continuum percolation and Euclidean minimal spanning trees in high dimensions. (English) Zbl 0855.60096
Summary: 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 F. Avram and D. Bertsimas [ibid. 2, No. 1, 113-130 (1992; Zbl 0755.60011)] 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 \(n \to \infty\).

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
82B43 Percolation
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