Trees and matchings from point processes
Probability
2007-05-23 v1
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 \geq 4 our result answers a question posed by Ferrari, Landim and Thorisson. 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.
Keywords
Cite
@article{arxiv.math/0211455,
title = {Trees and matchings from point processes},
author = {Alexander E. Holroyd and Yuval Peres},
journal= {arXiv preprint arXiv:math/0211455},
year = {2007}
}