## The Annals of Applied Probability

### The Williams–Bjerknes model on regular trees

#### Abstract

We consider the Williams–Bjerknes model, also known as the biased voter model on the $d$-regular tree $\mathbb{T} ^{d}$, where $d\geq3$. Starting from an initial configuration of “healthy” and “infected” vertices, infected vertices infect their neighbors at Poisson rate $\lambda\geq1$, 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 $\lambda>1$. 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 $\mathbb{T} ^{d}$—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 $\mathbb{T} ^{d}$.

#### Article information

Source
Ann. Appl. Probab., Volume 24, Number 5 (2014), 1889-1917.

Dates
First available in Project Euclid: 26 June 2014

https://projecteuclid.org/euclid.aoap/1403812365

Digital Object Identifier
doi:10.1214/13-AAP966

Mathematical Reviews number (MathSciNet)
MR3226167

Zentralblatt MATH identifier
1319.60181

#### Citation

Louidor, Oren; Tessler, Ran; Vandenberg-Rodes, Alexander. The Williams–Bjerknes model on regular trees. Ann. Appl. Probab. 24 (2014), no. 5, 1889--1917. doi:10.1214/13-AAP966. https://projecteuclid.org/euclid.aoap/1403812365

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