The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model

Abstract

The two-type Richardson model describes the growth of two competing infections on $\mathbb{Z}^d$. At time 0 two disjoint finite sets $\xi_1,\xi_2\subset \mathbb{Z}^d$ are infected with type 1 and type 2 infection respectively. An uninfected site then becomes type 1 (2) infected at a rate proportional to the number of type 1 (2) infected nearest neighbors and once infected it remains so forever. The main result in this paper is, loosely speaking, that the choice of the initial sets $\xi_1$ and $\xi_2$ is irrelevant in deciding whether the event of mutual unbounded growth for the two infection types has positive probability or not.