The two-type Richardson model with unbounded initial configurations

Abstract

The two-type Richardson model describes the growth of two competing infections on ℤ^d and the main question is whether both infection types can simultaneously grow to occupy infinite parts of ℤ^d. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points x=(x₁, …, 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 $\mathcal{H}\backslash\{\mathbf{0}\}$ 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 x₁-axis, it is shown that the type 2 infection at the origin can also grow unboundedly when the infection types have the same intensity.