An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0, 1, 2}. 1s and 2s remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. 1s and 2s may be thought of as two types of infection, and 0s as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.
