Nonmonotonic Coexistence Regions for the Two-type Richardson Model on Graphs

Journal article, 2006

In the two-type Richardson model on a graph G=(V,E), each vertex is at a given time in state 0,1 or 2. A 0 flips to a 1 (resp. 2) at rate λ1 (λ2) times the number of neighboring 1's (2's), while 1's and 2's never flip. When G is infinite, the main question is whether, starting from a single 1 and a single 2, with positive probability we will see both types of infection reach infinitely many sites. This has previously been studied on the d-dimensional cubic lattice Z2, d≥2, where the conjecture (on which a good deal of progress has been made) is that such coexistence has positive probability if and only if λ1 =λ2. In the present paper examples are given of other graphs where the set of points in the parameter space which admit such coexistence has a more surprising form. In particular, there exist graphs exhibiting coexistence at some value of (λ1 /λ2) and non-coexistence when this ratio is brought closer to 1.