The Williams Bjerknes Model on Regular Trees

TL;DR

This paper establishes a phase transition threshold for the Williams Bjerknes model on regular trees, showing that infection either persists or dies out depending on the infection rate relative to this critical value.

Contribution

The paper proves the existence of a critical infection rate for the Williams Bjerknes model on regular trees and provides a complete convergence theorem for the process and its dual.

Findings

Existence of a critical infection rate rac{}{-1} for the model.

Above the threshold, the entire tree becomes infected with positive probability.

Below the threshold, the infection dies out almost surely.

Abstract

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 $\bbT^d$.