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Asymptotics for Euclidean minimal spanning trees on random points
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  • Published: June 1992

Asymptotics for Euclidean minimal spanning trees on random points

  • David Aldous1 &
  • J. Michael Steele2 

Probability Theory and Related Fields volume 92, pages 247–258 (1992)Cite this article

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  • 69 Citations

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Summary

Asymptotic results for the Euclidean minimal spanning tree onn random vertices inR d can be obtained from consideration of a limiting infinite forest whose vertices form a Poisson process in allR d. In particular we prove a conjecture of Robert Bland: the sum of thed'th powers of the edge-lengths of the minimal spanning tree of a random sample ofn points from the uniform distribution in the unit cube ofR d tends to a constant asn→∞.

Whether the limit forest is in fact a single tree is a hard open problem, relating to continuum percolation.

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Author information

Authors and Affiliations

  1. Department of Statistics, University of California, 94720, Berkeley, CA, USA

    David Aldous

  2. Department of Statistics, Wharton School, University of Pennsylvania, 19104, Philadelphia, PA, USA

    J. Michael Steele

Authors
  1. David Aldous
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  2. J. Michael Steele
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Additional information

Research supported by N.S.F. Grants MCS87-11426 and MCS 90-01710

Research supported in part by N.S.F. Grant DMS88-12868, A.F.O.S.R. Grant 89-0301, ARO Grant DAAL03-89-G-0092 and NSA Grant MDA-904-H-2034

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Aldous, D., Steele, J.M. Asymptotics for Euclidean minimal spanning trees on random points. Probab. Th. Rel. Fields 92, 247–258 (1992). https://doi.org/10.1007/BF01194923

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  • Received: 01 March 1991

  • Revised: 03 September 1991

  • Issue Date: June 1992

  • DOI: https://doi.org/10.1007/BF01194923

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Keywords

  • Random Sample
  • Uniform Distribution
  • Stochastic Process
  • Probability Theory
  • Open Problem
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