- Volume 24, Number 4A (2018), 2721-2751.
Coalescence of Euclidean geodesics on the Poisson–Delaunay triangulation
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.
Bernoulli, Volume 24, Number 4A (2018), 2721-2751.
Received: October 2016
Revised: March 2017
First available in Project Euclid: 26 March 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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