An interacting particle system modelling competing growth on the Z 2 lattice is defined as follows. Each x ∈ Z 2 is in one of the states {0, 1, 2}. 1's and 2's remain in their states forever, while a 0 flips to a 1 (resp. a 2) at a rate equal to the number of its neighbours which are in state 1 (resp. 2). This is a generalization of the well known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's 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 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.
A twotype version of the Richardson model was introduced in REF , with two types competing to invade the sites of the Z d -lattice.
First passage percolation and a model for competing spatial growth
arXiv: Probability
Mathematics