Coexistence in two-type first-passage percolation models

Abstract. We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Zd or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x, y ∈ Zd, there is a strictly positive probability that {z ∈ Zd; d(y, z) < d(x, z)} and {z ∈ Zd; d(y, z)> d(x, z)} are both infinite sets. We also show that there is a strictly positive probability that the graph of time-minimizing path from the origin in first-passage percolation has at least two topological ends. This generalizes results obtained by Häggström and Pemantle for independent exponential times on the square lattice. 1