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TMF, 2016, Volume 186, Number 2, Pages 340–352 (Mi tmf8886)  

This article is cited in 1 scientific paper (total in 1 paper)

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

R. M. Khakimov

Institute for Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

Abstract: We study fertile hard-core models with the activity parameter $\lambda>0$ and four states on the Cayley tree. It is known that there are three types of such models. For each of these models, we prove the uniqueness of the translation-invariant Gibbs measure for any value of the parameter $\lambda$ on the Cayley tree of order three. Moreover, for one of the models, we obtain critical values of $\lambda$ at which the translation-invariant Gibbs measure is nonunique on the Cayley tree of order five. In this case, we verify a sufficient condition (the Kesten–Stigum condition) for a measure not to be extreme.

Keywords: Cayley tree, configuration, fertile graph, hard-core model, Gibbs measure, translation-invariant measure

DOI: https://doi.org/10.4213/tmf8886

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English version:
Theoretical and Mathematical Physics, 2016, 186:2, 294–305

Bibliographic databases:

Received: 16.02.2015

Citation: R. M. Khakimov, “Gibbs measures for fertile hard-core models on the Cayley tree”, TMF, 186:2 (2016), 340–352; Theoret. and Math. Phys., 186:2 (2016), 294–305

Citation in format AMSBIB
\Bibitem{Kha16}
\by R.~M.~Khakimov
\paper Gibbs measures for fertile hard-core models on the~Cayley tree
\jour TMF
\yr 2016
\vol 186
\issue 2
\pages 340--352
\mathnet{http://mi.mathnet.ru/tmf8886}
\crossref{https://doi.org/10.4213/tmf8886}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3462759}
\elib{http://elibrary.ru/item.asp?id=25707861}
\transl
\jour Theoret. and Math. Phys.
\yr 2016
\vol 186
\issue 2
\pages 294--305
\crossref{https://doi.org/10.1134/S0040577916020136}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000373359400012}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84962428121}


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  • http://mi.mathnet.ru/eng/tmf8886
  • https://doi.org/10.4213/tmf8886
  • http://mi.mathnet.ru/eng/tmf/v186/i2/p340

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. R. M. Khakimov, “Weakly periodic Gibbs measures for HC-models on Cayley trees”, Siberian Math. J., 59:1 (2018), 147–156  mathnet  crossref  crossref  isi  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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