# 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}
}
• Published 2002
• Mathematics
• Electronic Communications in Probability
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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#### References

SHOWING 1-10 OF 16 REFERENCES
Poisson trees, succession lines and coalescing random walks
• Mathematics, Physics
• 2004
Abstract We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S⊂ R d . The algorithmExpand
Group-invariant Percolation on Graphs
• Mathematics
• 1999
Abstract. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processesExpand
Asymptotics for Euclidean minimal spanning trees on random points
• Mathematics
• 1992
SummaryAsymptotic results for the Euclidean minimal spanning tree onn random vertices inRd can be obtained from consideration of a limiting infinite forest whose vertices form a Poisson process inExpand
Percolation and minimal spanning forests in infinite graphs
The structure of a spanning forest that generalizes the minimal spanning tree is considered for infinite graphs with a value f(b) attached to each bond b. Of particular interest are stationary randomExpand
Infinite clusters in dependent automorphism invariant percolation on trees
We study dependent bond percolation on the homogeneous tree T n of order n ≥ 2 under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clustersExpand
Nearest neighbor and hard sphere models in continuum percolation
• Computer Science
• Random Struct. Algorithms
• 1996
It is shown that if percolation occurs, then there is exactly one infinite cluster and some general properties of percolations models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). Expand
Amenability, Kazhdan’s property and percolation for trees, groups and equivalence relations
• Mathematics
• 1991
We prove amenability for a broad class of equivalence relations which have trees associated to the equivalence classes. The proof makes crucial use of percolation on trees. We also discuss relatedExpand
Nonamenable products are not treeable
• Mathematics
• 2000
LetX andY be infinite graphs such that the automorphism group ofX is nonamenable and the automorphism group ofY has an infinite orbit. We prove that there is no automorphism-invariant measure on theExpand
KAZHDAN GROUPS, COCYCLES AND TREES
• Mathematics
• 1990
We study cocycles of Kazhdan group actions with values in groups acting on trees. In particular we show that all cocycles of finite measure preserving actions of Kazhdan groups taking values in aExpand
Probability theory and combinatorial optimization
Preface 1. First View of Problems and Methods. A first example. Long common subsequences Subadditivity and expected values Azuma's inequality and a first application A second example. TheExpand