The Annals of Applied Probability

The radial spanning tree of a Poisson point process

Francois Baccelli and Charles Bordenave

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Abstract

We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root.

We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy.

We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than k generations away from the origin.

This class of spanning trees has applications in many fields and, in particular, in communications.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 1 (2007), 305-359.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1171377186

Digital Object Identifier
doi:10.1214/105051606000000826

Mathematical Reviews number (MathSciNet)
MR2292589

Zentralblatt MATH identifier
1136.60007

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C05: Trees
Secondary: 90C27: Combinatorial optimization 60G55: Point processes

Keywords
Spanning trees Poisson point process nearest neighbor graph directed spanning tree asymptotic shape

Citation

Baccelli, Francois; Bordenave, Charles. The radial spanning tree of a Poisson point process. Ann. Appl. Probab. 17 (2007), no. 1, 305--359. doi:10.1214/105051606000000826. https://projecteuclid.org/euclid.aoap/1171377186


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