Abstract:
In this paper, we study one of the fertile Hard-Core ($\mathrm{HC}$)
models with four states on a Cayley tree.
It is known that there are three
types of fertile
$\mathrm{HC}$-models: “stick,” “key,” and “generalized
key.”
We consider the “key” case and, in this case, the uniqueness of a
translation-invariant Gibbs measure on the Cayley tree of orders four, five,
and six is proved.
We also find conditions for the nonuniqueness of such
measures on the Cayley tree of order seven.
In addition, periodic Gibbs
measures are investigated.
It is shown that, under some conditions, there exist
two-periodic Gibbs measures, different from the translation-invariant ones, on
the Cayley tree of order three.
Citation:
B. Z. Tozhiboev, R. M. Khakimov, “Gibbs measures for the $\mathrm{HC}$-model
in the case of a graph of type “key” on a Cayley tree”, Mat. Zametki, 117:4 (2025), 575–590; Math. Notes, 117:4 (2025), 629–642
\Bibitem{TozKha25}
\by B.~Z.~Tozhiboev, R.~M.~Khakimov
\paper Gibbs measures for the $\mathrm{HC}$-model
in the case of a graph of type ``key'' on a Cayley tree
\jour Mat. Zametki
\yr 2025
\vol 117
\issue 4
\pages 575--590
\mathnet{http://mi.mathnet.ru/mzm14176}
\crossref{https://doi.org/10.4213/mzm14176}
\mathscinet{https://mathscinet.ams.org/mathscinet-getitem?mr=4920229}
\transl
\jour Math. Notes
\yr 2025
\vol 117
\issue 4
\pages 629--642
\crossref{https://doi.org/10.1134/S0001434625030289}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-105008244515}
Citation in format AMSBIB
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