Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model.
type: Item
instance of: scholarly article
title: Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model.
zbMATH Open document ID: 1047.60099
DOI: 10.1016/S0304-4149(00)00042-9
author: Olle Häggström
author: Robin Pemantle
published in: Stochastic Processes and their Applications
publication date: 2004-09-22T00:00:00Z
review text: The Richardson model is a~particular case of interacting particle systems. The two-type Richardson model is a~system with the three states \(\{0,1,2\}\) and the evolution described by the following flip rates: 1's and 2's never flip, while 0 flips to 1 (resp. 2) at rate \(\lambda _1\) \((\lambda _2)\) times the number of nearest neighbors with value 1 (2). The state 1 (2) at site \(x\in {\mathbb Z}^{d}\) means occupation by a~particle of the first (second) type, and once the site being occupied, it cannot be changed. Starting with only one particle of each type, there are only three possible scenarios: (i) The set of 1's surrounds the (finite) set of 2's. (ii) The set of 2's surrounds the (finite) set of 1's. (iii) Both the sets of 1's and 2's grow indefinitely. The problem to be solved is whether the scenario (iii) can happen with positive probability. In a~previous paper the authors proved that for \(d=2\) and \(\lambda _1=\lambda _2\), (iii) really occurs with positive probability. In the present paper they conjecture that (iii) has zero probability whenever \(\lambda _1\neq \lambda _2\). As the main result of the paper they manage to prove the conjecture for \(\lambda _1=1\) and all but countably many values of \(\lambda _2\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 2102349
zbMATH Keywords: interacting particle system
zbMATH Keywords: Richardson model
zbMATH Keywords: shape theorem
reviewed by: Martin Janžura
MaRDI profile type: Publication
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Wikidata QID: Q126805868
description: scientific article; zbMATH DE number 2102349