Trees and matchings from point processes

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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.

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Probability, 60G55; 60K35

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http://salesiana.dossiersoluciones.com/handle/123456789/65832

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0211455
 http://arxiv.org/abs/math/0211455

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arXiv