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 Td, where d.3. Starting from an initial configuration of .healthy. and .infected. vertices, infected vertices infect their neighbors at Poisson rate ..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 if and only if .>1. We show that there exists a threshold .c.(1,.) such that if .>.c then in the above setting with positive probability, all vertices will become eventually infected forever, while if .<.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 Td.above .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 Td.