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2003 Trees and Matchings from Point Processes
Alexander Holroyd, Yuval Peres
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Electron. Commun. Probab. 8: 17-27 (2003). DOI: 10.1214/ECP.v8-1066


A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the $d$-dimensional Poisson process has a one-ended tree as a factor graph. This implies that the Poisson points can be given an ordering isomorphic to the usual ordering of the integers in a deterministic isometry-invariant way. For $d \ge 4$ our result answers a question posed by Ferrari, Landim and Thorisson [7]. We prove also that any isometry-invariant ergodic point process of finite intensity in Euclidean or hyperbolic space has a perfect matching as a factor graph provided all the inter-point distances are distinct.


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Alexander Holroyd. Yuval Peres. "Trees and Matchings from Point Processes." Electron. Commun. Probab. 8 17 - 27, 2003.


Accepted: 3 March 2003; Published: 2003
First available in Project Euclid: 18 May 2016

zbMATH: 1060.60048
MathSciNet: MR1961286
Digital Object Identifier: 10.1214/ECP.v8-1066

Primary: 60G55
Secondary: 60K35

Keywords: Minimal spanning forest , point process , Poisson process , random matching , Random tree


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