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Geometric properties of Poisson matchings
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  • Published: 16 March 2010

Geometric properties of Poisson matchings

  • Alexander E. Holroyd1,2 

Probability Theory and Related Fields volume 150, pages 511–527 (2011)Cite this article

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  • 12 Citations

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Abstract

Suppose that red and blue points occur as independent Poisson processes of equal intensity in \({\mathbb {R}^d}\), and that the red points are matched to the blue points via straight edges in a translation-invariant way. We address several closely related properties of such matchings. We prove that there exist matchings that locally minimize total edge length in d = 1 and d ≥ 3, but not in the strip \({\mathbb {R}\times[0,1]}\). We prove that there exist matchings in which every bounded set intersects only finitely many edges in d ≥ 2, but not in d = 1 or in the strip. It is unknown whether there exists a matching with no crossings in d = 2, but we prove positive answers to various relaxations of this question. Several open problems are presented.

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Author information

Authors and Affiliations

  1. Microsoft Research, 1 Microsoft Way, Redmond, WA, 98052, USA

    Alexander E. Holroyd

  2. Department of Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Alexander E. Holroyd

Authors
  1. Alexander E. Holroyd
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Corresponding author

Correspondence to Alexander E. Holroyd.

Additional information

Dedicated to Oded Schramm, 10 December 1961–1 September 2008.

Funded in part by Microsoft and NSERC.

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Cite this article

Holroyd, A.E. Geometric properties of Poisson matchings. Probab. Theory Relat. Fields 150, 511–527 (2011). https://doi.org/10.1007/s00440-010-0282-y

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  • Received: 09 September 2009

  • Revised: 23 February 2010

  • Published: 16 March 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0282-y

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Keywords

  • Poisson process
  • Point process
  • Matching

Mathematics Subject Classification (2000)

  • 60D05
  • 60G55
  • 05C70
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