Asymptotics for First-Passage Times on Delaunay Triangulations

In this paper we study planar first-passage percolation (FPP) models, originally defined in the context of Z lattice by Hammersley and Welsh [6], on random Delaunay triangulations. The setup is as follows: to each edge e attach a positive random variable τe; the first-passage time T (v, v̄) between two vertexes v and v̄ is defined as the infimum of ∑ e∈γ τe over all paths γ connecting v to v̄. By using subadditivity, VahidiAsl and Wierman [19] showed that the rescaled first-passage time converges to a constant, called the time constant. We show a sufficient condition to ensure that the time constant is strictly positive and derive some upper bounds for fluctuations. Our proofs are based on renormalization ideas and on the method of bounded differences.