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

  • François Baccelli (a1), David Coupier (a2) and Viet Chi Tran (a2)

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.

Corresponding author
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.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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