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Semi-Infinite Paths of the Two-Dimensional Radial Spanning Tree

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  04 January 2016

François Baccelli ,
David Coupier  and
Viet Chi Tran
François Baccelli*
Affiliation:
INRIA
David Coupier*
Affiliation:
Université Lille1
Viet Chi Tran*
Affiliation:
Université Lille1
*
∗ Postal address: Research group on Network Theory and Communications (TREC), INRIA-ENS, 75214 Paris, France.
∗∗ Postal address: Laboratoire Paul Painlevé, Université Lille 1, Cité Scientifique, 59655 Villeneuve d'Ascq Cedex, France.
∗∗ Postal address: Laboratoire Paul Painlevé, Université Lille 1, Cité Scientifique, 59655 Villeneuve d'Ascq Cedex, France.
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Abstract

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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.

Keywords

Stochastic geometryrandom treesemi-infinite pathasymptotic directioncompetition interface

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 895 - 916
Copyright
© Applied Probability Trust 

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