The Williams Bjerknes Model on Regular Trees

We consider the Williams Bjerknes model, also known as the biased voter model on the $d$-regular tree $\bbT^d$, where $d \geq 3$. Starting from an initial configuration of "healthy" and "infected" vertices, infected vertices infect their neighbors at Poisson rate $\lambda \geq 1$, while healthy vertices heal their neighbors at Poisson rate 1. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability iff $\lambda > 1$. We show that there exists a threshold $\lambda_c \in (1, \infty)$ such that if $\lambda > \lambda_c$ then in the above setting with positive probability all vertices will become eventually infected forever, while if $\lambda < \lambda_c$, all vertices will become eventually healthy with probability 1. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on $\bbT^d$ -- above $\lambda_c$. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of $\bbT^d$.