The Time Constant and Critical Probabilities in Percolation Models

We consider a first-passage percolation (FPP) model on a Delaunay triangulation D of the plane. In this model each edge e of D is independently equipped with a nonnegative random variable τe, with distribution function F, which is interpreted as the time it takes to traverse the edge. Vahidi-Asl and Wierman [9] 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 μ(F)|x|, where μ(F) is a nonnegative finite constant. However the exact value of the time constant μ(F) still a fundamental problem in percolation theory. Here we prove that if F(0) < 1− p∗ c then μ(F) > 0, where p∗ c is a critical probability for bond percolation on the dual graph D∗.