Coalescence of Euclidean geodesics on the Poisson-Delaunay triangulation

TL;DR

This paper proves that semi-infinite geodesics in Euclidean first-passage percolation on Poisson-Delaunay triangulations almost surely coalesce, extending the approach to other geometric graphs and analyzing the growth of geodesics.

Contribution

It establishes almost sure coalescence of geodesics in Euclidean FPP on Poisson-Delaunay triangulations using a Burton-Keane argument and extends the method to other geometric graphs.

Findings

Almost sure coalescence of semi-infinite geodesics with same direction.

Extension of the approach to relative neighborhood graphs.

Expected number of geodesics grows sublinearly with radius.

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 adapted Burton-Keane argument 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.