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 $\text{Zd}$ or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times that for distinct points $x,y\in \text{Zd}$, there is a strictly positive probability that $\{z\in \text{Zd};d(y,z)<d(x,z)\}$ and $\{z\in \text{Zd};d(y,z)>d(x,z)\}$ 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 H{"a}ggstr{"o}m and Pemantle for independent exponential times on the square lattice.