The two-type Richardson model with unbounded initial configurations

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 $\mathbf{0}$ 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.