The Annals of Probability

The random minimal spanning tree in high dimensions

Mathew D. Penrose

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For the minimal spanning tree on n independent uniform points in the d-dimensional unit cube, the proportionate number of points of degree k is known to converge to a limit $\alpha_{k,d}$ as $n \to \infty$. We show that $\alpha_{k,d}$ converges to a limit $\alpha_k$ as $d \to \infty$ for each k. The limit $\alpha_k$ arose in earlier work by Aldous, as the asymptotic proportionate number of vertices of degree k in the minimum-weight spanning tree on k vertices, when the edge weights are taken to be independent, identically distributed random variables. We give a graphical alternative to Aldous's characterization of the $\alpha_k$.

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Ann. Probab., Volume 24, Number 4 (1996), 1903-1925.

First available in Project Euclid: 6 January 2003

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Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C05: Trees 90C27: Combinatorial optimization

Geometric probability minimal spanning tree vertex degrees continuum percolation invasion percolation


Penrose, Mathew D. The random minimal spanning tree in high dimensions. Ann. Probab. 24 (1996), no. 4, 1903--1925. doi:10.1214/aop/1041903210.

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