Gibbs measures for the fertile three-state hard-core models on a Cayley tree
type: Item
instance of: scholarly article
title: Gibbs measures for the fertile three-state hard-core models on a Cayley tree
zbMATH Open document ID: 1369.82010
author: R. M. Khakimov
author: U. A. Rozikov
published in: Queueing Systems
publication date: 2015-09-30T00:00:00Z
review text: This paper studies translation-invariant splitting Gibbs measures (TISGMs) for the fertile three-state hard-core models with activity \(\lambda>0\) on the Cayley tree of order \(k \geq 1\). There are four such models: wrench, wand, hinge, and pipe, which arise as simple examples of loss networks with nearest-neighbor exclusion. Some rigorous results have been obtained before for these models concerning the uniqueness and non-uniqueness of TISGMs in different parameter regions. This paper generalizes the result for hinge and wand cases and finds the exact critical value \(\lambda_{\mathrm{cr}}(k)\) for any \(k\geq 2\) such that for \(\lambda<\lambda_{\mathrm{cr}}\) there exists a unique TISGM and for \(\lambda>\lambda_{\mathrm{cr}}\) there exist multiple TISGMs. The authors also investigate sufficient conditions for the activity parameter \(\lambda\) to see extremality and non-extremality of the TISGM. Explicit formulae and some numerical values are given for the Cayley tree of order two.
reviewed by: Mei Yin
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B27
zbMATH DE Number: 6487391
zbMATH Keywords: fertile
zbMATH Keywords: hard-core model
zbMATH Keywords: critical temperature
zbMATH Keywords: Cayley tree
zbMATH Keywords: Gibbs measure
zbMATH Keywords: extreme measure
zbMATH Keywords: reconstruction problem
MaRDI profile type: Publication
full work available at URL: s11134-015-9450-1
OpenAlex ID: W639522976
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DOI: 10.1007/S11134-015-9450-1
description: scientific article; zbMATH DE number 6487391