## Electronic Communications in Probability

### Trees and Matchings from Point Processes

#### Abstract

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.

#### Article information

Source
Electron. Commun. Probab., Volume 8 (2003), paper no. 3, 17-27.

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

https://projecteuclid.org/euclid.ecp/1463608887

Digital Object Identifier
doi:10.1214/ECP.v8-1066

Mathematical Reviews number (MathSciNet)
MR1961286

Zentralblatt MATH identifier
1060.60048

Rights

#### Citation

Holroyd, Alexander; Peres, Yuval. Trees and Matchings from Point Processes. Electron. Commun. Probab. 8 (2003), paper no. 3, 17--27. doi:10.1214/ECP.v8-1066. https://projecteuclid.org/euclid.ecp/1463608887

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