This article is cited in 11 papers

Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree

U. A. Rozikov^abc, M. M. Rahmatullaev^ad, R. M. Khakimov<sup>ad

a</sup> V. I. Romanovskiy Institute of Mathematcs of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan
^b Akfa University, Tashkent, Uzbekistan
^c National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan
^d Namangam State University, Namangam, Uzbekistan

Abstract: We study the Potts model in translation-invariant and periodic external fields on the Cayley tree of order $k\geq 2$. For the Potts model in a translation-invariant external field for $k\geq 2$, the nonuniqueness of the translation-invariant and periodic Gibbs measure is shown. It is proved that for the Potts model in an external field that is not translation-invariant, translation-invariant Gibbs measures do not exist on the Cayley tree of order $k\geq 2$. Periodic Gibbs measures are also studied for the Potts model in a periodic external field. We prove that under certain conditions, the number of such measures can be at least three.

Keywords: Cayley tree, Gibbs measure, Potts model, periodic external field, translation-invariant external field.

Received: 05.08.2021
Revised: 24.09.2021

DOI: 10.4213/tmf10158

 English version: Theoretical and Mathematical Physics, 2022, 210:1, 135–153

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