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Stretch factor in a planar Poisson–Delaunay triangulation with a large intensity

Part of: Graph theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  20 March 2018

Nicolas Chenavier  and
Olivier Devillers
Nicolas Chenavier*
Affiliation:
Université du Littoral Côte d'Opale
Olivier Devillers*
Affiliation:
INRIA, CNRS, Université de Lorraine
*
* Postal address: LMPA Joseph Liouville, Université du Littoral Côte d'Opale, 50 rue Ferdinand Buisson, BP 699, 62228 Calais Cedex, France. Email address: nicolas.chenavier@univ-littoral.fr
** Postal address: Université de Lorraine, 615 rue du Jardin Botanique, B.P. 101, 54602 Villers-lès-Nancy Cedex, France. Email address: olivier.devillers@inria.fr
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Abstract

Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

Keywords

Delaunay triangulationPoisson point processstretch factor

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 05C80: Random graphs
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 35 - 56
Copyright
Copyright © Applied Probability Trust 2018 

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