The time constant and critical probabilities in percolation models

Abstract

We consider a first-passage percolation model on a Delaunay triangulation of the plane. In this model each edge is independently equipped with a nonnegative random variable, with distribution function F, which is interpreted as the time it takes to traverse the edge. Vahidi-Asl and Wierman have shown that, under a suitable moment condition on F, the minimum time taken to reach a point x from the origin 0 is asymptotically \mu(F)|x|, where \mu(F) is a nonnegative finite constant. However the exact value of the time constant \mu(F) still a fundamental problem in percolation theory. Here we prove that if F(0)<1-p_c^* then \mu(F)>0, where p_c^* is a critical probability for bond percolation on the dual graph (Voronoi tessellation).