Stretch Factor in a Planar Poisson-Delaunay Triangulation with a Large Intensity

Nicolas Chenavier 1 Olivier Devillers 2
2 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
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 non-trivial 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 infinity. 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 $35/3\pi^2$, giving an upper bound for the expected length of the smallest path.
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Advances in Applied Probability, Applied Probability Trust, 2018, 50 (1), pp.35-56. 〈10.1017/apr.2018.3〉
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Nicolas Chenavier, Olivier Devillers. Stretch Factor in a Planar Poisson-Delaunay Triangulation with a Large Intensity. Advances in Applied Probability, Applied Probability Trust, 2018, 50 (1), pp.35-56. 〈10.1017/apr.2018.3〉. 〈hal-01700778〉

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