Coexistence in two-type first-passage percolation models

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Z or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times that for distinct points x, y ∈ Z, there is a strictly positive probability that {z ∈ Z; d(y, z) < d(x, z)} and {z ∈ Z; 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.