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Extremality of Gibbs Measures for the \(HC\)-Blume–Capel Model on the Cayley Tree

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Abstract

In this paper, we consider translation-invariant Gibbs measures (TIGMs) for the \(HC\)-Blume–Capel model in case of “wands” with chemical potential with parameters \((\theta,\eta)\) on the Cayley tree. It is proved that, for \(\eta\le\theta^{3}\), there is a unique TIGM and, for \(\eta>\theta^{3}\), there are exactly three TIGMs in the case of “wands” with chemical potential for the model under consideration. In addition, the problem of the (non)extremality of these measures is studied.

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Acknowledgments

The author wishes to express gratitude to Professors U. A. Rozikov and R. M. Khakimov for helpful advice.

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Authors and Affiliations

  1. National University of Uzbekistan, Tashkent, 100174, Uzbekistan

    N. M. Khatamov

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  1. N. M. Khatamov
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Correspondence to N. M. Khatamov.

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Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 762–777 https://doi.org/10.4213/mzm13217.

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Khatamov, N.M. Extremality of Gibbs Measures for the \(HC\)-Blume–Capel Model on the Cayley Tree. Math Notes 111, 768–781 (2022). https://doi.org/10.1134/S000143462205011X

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