A Hard-Core Model on a Cayley Tree: An Example of a Loss Network View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2004-01

AUTHORS

Y. Suhov, U.A. Rozikov

ABSTRACT

The paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ∀λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ*. Define λc=1/(k−1)×(k/(k−1))k. Then: (i) for λ≤λc, the Gibbs measure is unique (and coincides with the above measure μ*), (ii) for λ>λc, in addition to μ*, there exist two distinct translation-periodic measures, μ+and μ−, taken to each other by the unit space shift. Measures μ+and μ−are extreme ∀λ>λc. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $$ \end{document}, measure μ*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λeand λo, for even and odd sites. We discuss open problems and state several related conjectures. More... »

PAGES

197-212

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1023/b:ques.0000021149.43343.05

DOI

http://dx.doi.org/10.1023/b:ques.0000021149.43343.05

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1014082630


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