Journal of Mathematical Physics, Analysis, Geometry 2019, vol. 15, No 2, pp. 239-255   https://doi.org/10.15407/mag15.02.239     ( to contents , go back )

### Translation-Invariant Gibbs Measures for the Blum-Kapel Model on a Cayley Tree

Nosir Khatamov

Namangan State University, 316 Uychi Str., 160119, Namangan, Uzbekistan
E-mail: nxatamov@mail.ru

Rustam Khakimov

Namangan State University, 316 Uychi Str., 160119, Namangan, Uzbekistan
E-mail: rustam-7102@rambler.ru

Received July 26, 2017, revised September 14, 2018.

Abstract

In the paper, translation-invariant Gibbs measures for the Blum-Kapel model on a Cayley tree of order $k$ are considered. An approximate critical temperature $T_{cr}$ is found such that for $T\geq T_{cr}$ there exists a unique translation-invariant Gibbs measure and for $0 < T < T_{cr}$ there are exactly threetranslation-invariant Gibbs measures. In addition, the problem of (not) extremality for the unique Gibbs measure is studied.

Mathematics Subject Classification 2000: 82B26, 60K35.
Key words: Cayley tree, configuration, Blum-Kapel model, Gibbs measure, translation-invariant measure, extremality of measure.