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