A reversible nearest particle system on the homogeneous tree
type: Item
instance of: scholarly article
title: A reversible nearest particle system on the homogeneous tree
zbMATH Open document ID: 0922.60086
DOI: 10.1023/A:1021709013569
author: Amber L. Puha
published in: Journal of Theoretical Probability
publication date: 1999-10-03T00:00:00Z
review text: It is introduced a modified contact process on the homogeneous (connected) tree in which each vertex has \(d+1\) neighbors [for definitions and some results see \textit{T. M. Liggett}, ``Interacting particle systems'' (1985; Zbl 0559.60078), Ann. Probab. 24, No. 4, 1675-1710 (1996; Zbl 0871.60087) and \textit{D. Chen}, Acta Math. Sci. 14, No. 3, 348-353 (1994; Zbl 0814.60101)]. The modification is to the death rate: an occupied site becomes vacant at rate one if the number of its occupied neighbors is at most one. This modification leads to a growth model that is reversible, off empty set, provided the initial set of occupied sites is connected. Reversibility admits tools for studying the survival properties of the system not available in a nonreversible situation. Four potential phases are considered: extinction, weak survival, strong survival, and complete convergence. The main result of the paper is that there is exactly one phase transition on the binary tree. The value of the birth parameter at which the phase transitions occur is explicitly computed. In particular, survival and complete convergence hold if the birth parameter exceeds \(1/4\). Otherwise, the expected extinction time is finite.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 1275655
zbMATH Keywords: trees
zbMATH Keywords: growth models
zbMATH Keywords: phase transition
zbMATH Keywords: reversible
zbMATH Keywords: flows
reviewed by: Valentin Topchii
MaRDI profile type: Publication
full work available at URL: a:1021709013569
OpenAlex ID: W106217647
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description: scientific article; zbMATH DE number 1275655