The two-type Richardson model with unbounded initial configurations

The two-type Richardson model describes the growth of two competing infections on .d and the main question is whether both infection types can simultaneously grow to occupy infinite parts of .d. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points x=(x1, ., xd) in the hyperplane H={x.Zd:x1=0} is considered. It is shown that, starting from a configuration where all points in H.{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.