najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2026-17-1-5-16najo-1682Research ArticleMATHEMATICSМАТЕМАТИКАPhase transition and thermodynamic properties of the Hard-Core-Potts modelФазовый переход и термодинамические свойства модели Поттса с жестким ядромhttps://orcid.org/0000-0003-4127-174XХакимовР. М.KhakimovR. M.

Rustamjon M.Khakimov

Universitystreet, 100174, Tashkent

rustam7102@rambler.ruhttps://orcid.org/0009-0001-6388-1322МахаммадалиевМ. Т.MakhammadalievM. T.

Muhtorjon T. Makhammadaliev

Boburshoxstreet,161,160107, Namangan

https://orcid.org/0009-0008-3052-1807МуталлиевН. Н.MutallievN. N.

Nodirbek N. Mutalliev

I.Karimovstreet,12,160105, Namangan

nodirbekmutalliyev95@gmail.comInstitute of mathematics named after V. I. RomanovskyUzbekistanNamangan State UniversityUzbekistanNamangan State Technical UniversityUzbekistan202604032026171516Copyright © Khakimov R.M., Makhammadaliev M.T., Mutalliev N.N., 20262026Хакимов Р.М., Махаммадалиев М.Т., Муталлиев Н.Н.Khakimov R.M., Makhammadaliev M.T., Mutalliev N.N.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/1682

We investigate translation-invariant Gibbs measures of the Hard-Core–Potts (HC–Potts) model on the Cayley tree. The model combines Potts-type ferromagnetic interactions with a hard-core exclusion rule, leading to a nontrivial interplay between magnetic ordering and occupancy constraints. For the hinge-type fertile graph, we analyze the cases k = 2 and k = 3, and determine explicit critical values of θ that mark the transition from uniqueness to multiplicity of Gibbs measures. The model exhibits up to five translationinvariant phases depending on the interaction strength. Thermodynamic quantities such as magnetization and quadrupolar moment are computed, revealing ordered phases at low temperatures and a paramagnetic phase at high temperatures.

Мы исследуем трансляционно-инвариантные меры Гиббса модели Поттса с жестким ядром (HC–Potts) на дереве Кэли. Модель сочетает ферромагнитные взаимодействия типа Поттса с правилом исключения жесткого ядра, что приводит к нетривиальному взаимодействию между магнитным упорядочением и ограничениями на заселенность. Для фертильного графа шарнирного типа мы анализируем случаи k = 2 и k = 3 и определяем явные критические значения θ, которые отмечают переход от единственности к множественности мер Гиббса. Модель демонстрирует до пяти трансляционно-инвариантных фаз в зависимости от силы взаимодействия. Вычисляются термодинамические величины, такие как намагниченность и квадрупольный момент, выявляющие упорядоченные фазы при низких температурах и парамагнитную фазу при высоких температурах.

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The authors declare that there are no conflicts of interest present.