Trees and Matchings from Point Processes
Alexander E. Holroyd, University of California, Berkeley
Yuval Peres, University of California, Berkeley
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.
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