To fixate or not to fixate in two-type annihilating branching random walks

Abstract

We study a model of competition between two types evolving as branching random walks on ${\mathbb{Z}}^{\mathit{d}}$. The two types are represented by red and blue balls, respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of ${\mathbb{Z}}^{\mathit{d}}$ contain one ball each which are independently coloured red with probability p and blue otherwise. We address the question of fixation, referring to the sites and eventually settling for a given colour or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for $\mathit{p}\ne 1/2$ and that every site will change colour infinitely often almost surely for the balanced initial condition $\mathit{p}=1/2$.