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

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

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


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.

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  • [1] Aldous, D.J. (2009). Which connected spatial networks on random points have linear route-lengths? Preprint. Available at arXiv:0911.5296.
  • [2] Baccelli, F., Coupier, D. and Tran, V.C. (2013). Semi-infinite paths of the two-dimensional radial spanning tree. Adv. in Appl. Probab. 45 895–916.
  • [3] Baccelli, F., Tchoumatchenko, K. and Zuyev, S. (2000). Markov paths on the Poisson–Delaunay graph with applications to routing in mobile networks. Adv. in Appl. Probab. 32 1–18.
  • [4] Chenavier, N. and Devillers, O. (2015). Stretch factor of long paths in a planar Poisson–Delaunay triangulation. Preprint.
  • [5] Coupier, D. (2016). Sublinearity of the number of semi-infinite branches for geometric random trees. Preprint. Available at arXiv:1501.04804.
  • [6] Coupier, D. and Tran, V.C. (2013). The 2D-directed spanning forest is almost surely a tree. Random Structures Algorithms 42 59–72.
  • [7] Daley, D.J. and Last, G. (2005). Descending chains, the lilypond model, and mutual-nearest-neighbour matching. Adv. in Appl. Probab. 37 604–628.
  • [8] Dobkin, D.P., Friedman, S.J. and Supowit, K.J. (1990). Delaunay graphs are almost as good as complete graphs. Discrete Comput. Geom. 5 399–407.
  • [9] Hammersley, J.M. and Welsh, D.J.A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif. 61–110. New York: Springer.
  • [10] Hirsch, C., Neuhäuser, D. and Schmidt, V. (2016). Moderate deviations for shortest-path lengths on random segment processes. ESAIM Probab. Stat. 20 261–292.
  • [11] Howard, C.D. and Newman, C.M. (1997). Euclidean models of first-passage percolation. Probab. Theory Related Fields 108 153–170.
  • [12] Howard, C.D. and Newman, C.M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29 577–623.
  • [13] Jaromczyk, J.W. and Toussaint, G.T. (1992). Relative neighborhood graphs and their relatives. Proc. IEEE 80 1502–1517.
  • [14] Keil, J.M. and Gutwin, C.A. (1992). Classes of graphs which approximate the complete Euclidean graph. Discrete Comput. Geom. 7 13–28.
  • [15] Licea, C. and Newman, C.M. (1996). Geodesics in two-dimensional first-passage percolation. Ann. Probab. 24 399–410.
  • [16] Newman, C.M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) 1017–1023. Basel: Birkhäuser.
  • [17] Pimentel, L.P.R. (2011). Asymptotics for first-passage times on Delaunay triangulations. Combin. Probab. Comput. 20 435–453.
  • [18] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73–205.