Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2721-2751.

Coalescence of Euclidean geodesics on the Poisson–Delaunay triangulation

David Coupier and Christian Hirsch

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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.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2721-2751.

Dates
Received: October 2016
Revised: March 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051223

Digital Object Identifier
doi:10.3150/17-BEJ943

Mathematical Reviews number (MathSciNet)
MR3779700

Zentralblatt MATH identifier
06853263

Keywords
Burton–Keane argument coalescence Delaunay triangulation first-passage percolation Poisson point process relative neighborhood graph sublinearity

Citation

Coupier, David; Hirsch, Christian. Coalescence of Euclidean geodesics on the Poisson–Delaunay triangulation. Bernoulli 24 (2018), no. 4A, 2721--2751. doi:10.3150/17-BEJ943. https://projecteuclid.org/euclid.bj/1522051223


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