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November 2018 Coalescence of Euclidean geodesics on the Poisson–Delaunay triangulation
David Coupier, Christian Hirsch
Bernoulli 24(4A): 2721-2751 (November 2018). DOI: 10.3150/17-BEJ943

Abstract

Let us consider Euclidean first-passage percolation on the Poisson–Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an argument of Burton–Keane type and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesics starting at a given vertex and leaving a disk of a certain radius grows at most sublinearly in the radius.

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David Coupier. Christian Hirsch. "Coalescence of Euclidean geodesics on the Poisson–Delaunay triangulation." Bernoulli 24 (4A) 2721 - 2751, November 2018. https://doi.org/10.3150/17-BEJ943

Information

Received: 1 October 2016; Revised: 1 March 2017; Published: November 2018
First available in Project Euclid: 26 March 2018

zbMATH: 06853263
MathSciNet: MR3779700
Digital Object Identifier: 10.3150/17-BEJ943

Keywords: Burton–Keane argument , Coalescence , Delaunay triangulation , First-passage percolation , Poisson point process , relative neighborhood graph , sublinearity

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4A • November 2018
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