## Advances in Applied Probability

### Semi-infinite paths of the two-dimensional radial spanning tree

#### Abstract

We study semi-infinite paths of the radial spanning tree (RST) of a Poisson point process in the plane. We first show that the expectation of the number of intersection points between semi-infinite paths and the sphere with radius r grows sublinearly with r. Then we prove that in each (deterministic) direction there exists, with probability 1, a unique semi-infinite path, framed by an infinite number of other semi-infinite paths of close asymptotic directions. The set of (random) directions in which there is more than one semi-infinite path is dense in [0, 2π). It corresponds to possible asymptotic directions of competition interfaces. We show that the RST can be decomposed into at most five infinite subtrees directly connected to the root. The interfaces separating these subtrees are studied and simulations are provided.

#### Article information

Source
Adv. in Appl. Probab., Volume 45, Number 4 (2013), 895-916.

Dates
First available in Project Euclid: 12 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1386857849

Digital Object Identifier
doi:10.1239/aap/1386857849

Mathematical Reviews number (MathSciNet)
MR3161288

Zentralblatt MATH identifier
1287.60016

#### Citation

Baccelli, François; Coupier, David; Tran, Viet Chi. Semi-infinite paths of the two-dimensional radial spanning tree. Adv. in Appl. Probab. 45 (2013), no. 4, 895--916. doi:10.1239/aap/1386857849. https://projecteuclid.org/euclid.aap/1386857849

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