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 ℤ^d 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∈ℤ^d, there is a strictly positive probability that {z∈ℤ^d;d(y,z)<d(x,z)} and {z∈ℤ^d;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.