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Queueing Systems

, Volume 81, Issue 1, pp 49–69 | Cite as

Gibbs measures for the fertile three-state hard-core models on a Cayley tree

  • U. A. Rozikov
  • R. M. Khakimov
Article

Abstract

We study translation-invariant splitting Gibbs measures (TISGMs, tree-indexed Markov chains) for the fertile three-state hard-core models with activity \(\lambda >0\) on the Cayley tree of order \(k\ge 1\). There are four such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. It is known that (i) for the wrench and pipe cases \(\forall \lambda >0\) and \(k\ge 1\), there exists a unique TISGM; (ii) for hinge (resp. wand) case at \(k=2\) if \(\lambda <\lambda _\mathrm{cr}=9/4\) (resp. \(\lambda <\lambda _\mathrm{cr}=1\)), there exists a unique TISGM, and for \(\lambda > 9/4\) (resp. \(\lambda >1\)), there exist three TISGMs. In this paper we generalize the result (ii) for any \(k\ge 2\), i.e., for hinge and wand cases we find the exact critical value \(\lambda _\mathrm{cr}(k)\) with properties mentioned in (ii). Moreover, we find some regions for the \(\lambda \) parameter ensuring that a given TISGM is extreme or non-extreme in the set of all Gibbs measures. For the Cayley tree of order two, we give explicit formulae and some numerical values.

Keywords

Fertile Hard-core model Critical temperature  Cayley tree  Gibbs measure Extreme measure Reconstruction problem 

Mathematics Subject Classification

82B26 (Primary) 60K35 (Secondary) 

Notes

Acknowledgments

U. A. Rozikov thanks the Université du Sud Toulon Var, the Centre de Physique Théorique for support of his many visits, and Aix-Marseille University Institute for Advanced Study IMéRA (Marseille, France) for support by a residency scheme. He also thanks the Ruhr-University Bochum (Germany) for financial support and hospitality. We thank both referees for careful reading of the manuscript and for useful suggestions.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of MathematicsTashkentUzbekistan

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