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October 1996 The random minimal spanning tree in high dimensions
Mathew D. Penrose
Ann. Probab. 24(4): 1903-1925 (October 1996). DOI: 10.1214/aop/1041903210


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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Mathew D. Penrose. "The random minimal spanning tree in high dimensions." Ann. Probab. 24 (4) 1903 - 1925, October 1996.


Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0866.60021
MathSciNet: MR1415233
Digital Object Identifier: 10.1214/aop/1041903210

Primary: 05C05 , 60D05 , 90C27

Keywords: continuum percolation , geometric probability , Invasion percolation , Minimal spanning tree , vertex degrees

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 4 • October 1996
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