Coexistence in two-type first-passage percolation models

submitted 18 dec 2003We study the problem of coexistence in a two-type competition model governedby first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation.Actually, we prove for a large class of ergodic stationary passage times that for distinct points $x,y\in\Zd$, there is a strictly positive probability that$\{z\in\Zd;d(y,z)d(x,z)\}$ 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{ä}ggstr{ö}m and Pemantle forindependent exponential times on the square lattice