Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree
U. A. RozikovNational University of Uzbekistan named after Mirzo Ulugbek, Tashkent, UzbekistanM. M. RakhmatullaevV. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, UzbekistanР. М. ХакимовV. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan
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Annotatsiya
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.
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