Trees and Matchings from Point Processes

@article{Holroyd2002TreesAM,
  title={Trees and Matchings from Point Processes},
  author={Alexander E. Holroyd and Yuval Peres},
  journal={Electronic Communications in Probability},
  year={2002},
  volume={8},
  pages={17-27}
}
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… Expand

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