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## Archive of Issues

Uzbekistan Namangan; Tashkent
Year
2022
Volume
32
Issue
2
Pages
256-277
 Section Mathematics Title Extremality of some Gibbs measures for the Blume–Capel HC-model on a Cayley tree Author(-s) Khatamov N.M.ab Affiliations Institute of Mathematics, National Academy of Sciences of Uzbekistana, Namangan State Universityb Abstract In this paper, we consider translation-invariant Gibbs measures (TIGM) for the Blume–Capel HC-model in the case of a “generalized wand” on a second-order Cayley tree. An approximate critical value of $\theta_{cr}$ is found such that for $\theta \geq\theta_{cr}$ there is only one TIGM, and for $0<\theta<\theta_{cr}$ there are exactly three TIGMs in the case of “generalized wand” for the model under consideration. In addition, the (non)extreme problem for these measures is studied. Keywords Cayley tree, configuration, Blume–Capel HC-model, Gibbs measure, translation-invariant measures, extreme measure UDC 517.98 MSC 82B05, 82B20, 60K35 DOI 10.35634/vm220207 Received 19 April 2022 Language Russian Citation Khatamov N.M. 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