2002/11/29 by Alexander E. Holroyd, Yuval Peres, Holroyd, Alexander E. +1
Mathematics · #60G55 #60K35 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G55 #msc:60K35
paper · pdf · doi:10.48550/arxiv.math/0211455
arxiv created 2002/11/29 · arxiv updated 2009/11/30
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 ≥ 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.