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Владикавказский математический журнал

Vladikavkaz Mathematical Journal

Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН)

Hard-Core Model with Countable States: Translation-Invariant Gibbs Measures

Hard-core модель со счетным числом состояний: трансляционно-инвариантные меры Гиббса

10.46698/b9897-5479-1766-n

19364

06 2026

28 2 91 107

https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=19395&SECTION_ID=669

Махаммадалиев

М.

Makhammadaliev

mmtmuxtor93@mail.ru

Хакимов

Р.

Khakimov

rustam-7102@rambler.ru

Наманганский государственный университет, Узбекистан, 160107, Наманган, ул. Бабуршах, 161 Namangan State University, 161 Baburshah St., Namangan 160107, Uzbekistan

In this paper, we investigate the translation-invariant Gibbs measures (TIGM) for the Hard-Core (HC) lattice model with a countable set \(\mathbb{Z}\) of spin values on the Cayley tree of order \(k \geq 2\). This statistical mechanics model is uniquely characterized by an infinite set of parameters governed by a~positive activity function \(\lambda_{i}>0\), \(i \in \mathbb{Z}\). We explicitly derive the functional consistency equation and establish the exact normalizability conditions required for the boundary laws of the system. The analysis specifically focuses on the convergence of the activity series partitions, denoted by \(\sum_{j\in\mathbb{Z}_{0}}\lambda_{2j}=\Delta\) and \({\sum_{j\in\mathbb{Z}}\lambda_{2j+1}=\Psi}\). In the symmetric subcase where \(\Delta=\Psi\equiv\gamma\), we determine the precise critical value \(\gamma_{cr}(k)\), proving that a unique TIGM exists for \(\gamma \le \gamma_{cr}(k)\), while there exist exactly three distinct TIGMs when \(\gamma>\gamma_{cr}(k)\) on Cayley trees of orders \(k=2,3,4,5\). Furthermore, a generalized lower bound for the total number of TIGMs is formulated for all orders \(k \geq 2\). Finally, in the asymmetric subcase (\(\Delta \neq \Psi\)), we provide a comprehensive mathematical description and classification of all translation-invariant Gibbs measures on Cayley trees of orders \(k=1,2,3\).

В данной работе исследуются трансляционно-инвариантные меры Гиббса (ТИМГ) для решеточной модели Hard-Core (HC) со счетным множеством значений спина \(\mathbb{Z}\) на дереве Кэли порядка \(k \geq 2\). Эта модель статистической механики однозначно характеризуется бесконечным набором параметров, задаваемых положительной функцией активности \(\lambda_{i}>0\), \(i \in \mathbb{Z}\). Явно выведено функциональное уравнение согласованности и установлены точные условия нормализуемости, необходимые для граничных законов системы. Анализ подробно сосредоточен на сходимости разбиений рядов активности, обозначаемых как \(\sum_{j\in\mathbb{Z}_{0}}\lambda_{2j}=\Delta\) и \(\sum_{j\in\mathbb{Z}}\lambda_{2j+1}=\Psi\). В симметричном подслучае, когда \(\Delta=\Psi\equiv\gamma\), определено точное критическое значение \(\gamma_{cr}(k)\) и доказано, что при \(\gamma \le \gamma_{cr}(k)\) существует единственная ТИМГ, тогда как при \(\gamma>\gamma_{cr}(k)\) на деревьях Кэли порядков \(k=2,3,4,5\) возникают ровно три различные ТИМГ. Кроме того, сформулирована обобщённая нижняя граница для общего числа ТИМГ при всех порядках \(k \geq 2\). Наконец, в асимметричном подслучае (\(\Delta \neq \Psi\)) представлено полное математическое описание и классификация всех трансляционно-инвариантных мер Гиббса на деревьях Кэли порядков \(k=1,2,3\).

дерево Кэли Hard-Core модель конфигурация мера Гиббса трансляционно-инвариантные меры граничный закон.

Cayley tree Hard-Core model configuration Gibbs measure translation-invariant measures boundary law.

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