The two-type Richardson model with unbounded initial configurations

Abstract

The two-type Richardson model describes the growth of two competing infections on ${\mathbb{Z}}^d$ and the main question is whether both infection types can simultaneously grow to occupy infinite parts of ${\mathbb{Z}}^d$. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points $x=(x_1,...,x_d)$ in the hyperplane $\mathcal{H}=\{x\in {\mathbb{Z}}^d:x_1=0\}$ is considered. It is shown that, starting from a configuration where all points in $\mathcal{H} {\mathbf{0}\}$ are type 1 infected and the origin $0$ is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only if it has a strictly larger intensity than the type 1 infection. If, instead, the initial type 1 infection is restricted to the negative $x_1$-axis, it is shown that the type 2 infection at the origin can also grow unboundedly when the infection types have the same intensity.

Published in at http://dx.doi.org/10.1214/07-AAP440 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)