Open Access
October 2014 The Williams–Bjerknes model on regular trees
Oren Louidor, Ran Tessler, Alexander Vandenberg-Rodes
Ann. Appl. Probab. 24(5): 1889-1917 (October 2014). DOI: 10.1214/13-AAP966

Abstract

We consider the Williams–Bjerknes model, also known as the biased voter model on the d-regular tree Td, where d3. Starting from an initial configuration of “healthy” and “infected” vertices, infected vertices infect their neighbors at Poisson rate λ1, 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 λ>1. We show that there exists a threshold λc(1,) such that if λ>λc then in the above setting with positive probability, all vertices will become eventually infected forever, while if λ<λc, 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 λc. 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.

Citation

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Oren Louidor. Ran Tessler. Alexander Vandenberg-Rodes. "The Williams–Bjerknes model on regular trees." Ann. Appl. Probab. 24 (5) 1889 - 1917, October 2014. https://doi.org/10.1214/13-AAP966

Information

Published: October 2014
First available in Project Euclid: 26 June 2014

zbMATH: 1319.60181
MathSciNet: MR3226167
Digital Object Identifier: 10.1214/13-AAP966

Subjects:
Primary: 60K35
Secondary: 82C41

Keywords: biased voter , fixation , local survival , Williams–Bjerknes

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.24 • No. 5 • October 2014
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