najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-2-164-175najo-12Research ArticleMATHEMATICSМАТЕМАТИКАTranslation-invariant p-adic quasi Gibbs measures for the Potts model with an external field on the Cayley treeТрансляционно-инвариантная p-адическая квазигиббсовская мера для модели Поттса с внешним полем на дереве Кэлиhttps://orcid.org/0000-0003-2987-7714РахматуллаевМ. М.RakhmatullaevM. M.

Muzaffar M. Rakhmatullaev

University str. 9, 100174,Tashkent

100000, Tashkent, Uzbekistan

mrahmatullaev@rambler.ruhttps://orcid.org/0009-0004-0060-2180СамиджоноваН. Д.SamijonovaN. D.

Nurkhon D. Samijonova –

161, Boburshox str, 160107, Namangan

nurxonsamijonova@gmail.comV. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences; New Uzbekistan UniversityUzbekistanNamangan State UniversityUzbekistan202519052025162164175Copyright © Rakhmatullaev M.M., Samijonova N.D., 20252025Рахматуллаев М.М., Самиджонова Н.Д.Rakhmatullaev M.M., Samijonova N.D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/12

The study is focused on investigation of p-adic Gibbs measures for the q-state Potts model with an external field and determination of the conditions for the existence of a phase transition. In this work, we derive a functional equation that satisfies the compatibility condition for p-adic quasi-Gibbs measures on a Cayley tree of order k ≥ 2. Furthermore, we prove that if |q|p = 1 there exists a unique p-adic Gibbs measure for this model. Additionally, for the Potts model on a binary tree, we identify three p-adic quasi-Gibbs measures under specific circumstances: one bounded and two unbounded, which implies a phase transition.

Целью настоящего исследования являются изучение p‑адических квазигиббсовских мер для модели Поттса с q-состояниями и с внешним полем и определить условия при которых существует фазовый переход. В данной работе получено функциональное уравнение, удовлетворяющее условию согласованности для p-адических квазигиббсовских мер на дереве Кэли порядка k ≥ 2. Кроме того, мы доказано, что |q|p = 1, то для данной модели существует единственная p‑адическая квазимера Гиббса. Более того, для модели Поттса на дереве Кэли порядка два найдены три p-адические квазигиббсовские меры, при которых одна из них ограниченная, а две неограниченные, что дает существование фазового перехода.

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