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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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