@article{ECP1066,
author = {Alexander Holroyd and Yuval Peres},
title = {Trees and Matchings from Point Processes},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {8},
year = {2003},
keywords = {Poisson process, point process, random tree, random matching, minimal spanning forest.},
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 greater than or equal to 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.},
pages = {no. 3, 17-27},
issn = {1083-589X},
doi = {10.1214/ECP.v8-1066},
url = {http://ecp.ejpecp.org/article/view/1066}}