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- Title
Gibbs Periodic Measures for a Two-State HC-Model on a Cayley Tree.
- Authors
Rozikov, U. A.; Khakimov, R. M.; Makhammadaliev, M. T.
- Abstract
In this paper, we study a two-state Hard-Core (HC) model with activity λ > 0 on a Cayley tree of order k ≥ 2. It is known that there are λcr, λ cr 0 , and λ cr ′ such that • for λ ≤ λcr this model has a unique Gibbs measure μ*, which is translation invariant. The measure μ* is extreme for λ < λ cr 0 and not extreme for λ > λ cr ′ ; • for λ > λcr there exist exactly three 2-periodic Gibbs measures, one of which is μ*, the other two are not translation invariant and are always extreme. The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k = 2, 3. Our proof is based on the tree reconstruction method.
- Subjects
TREES; CAYLEY graphs; GIBBS sampling; MEASUREMENT
- Publication
Journal of Mathematical Sciences, 2024, Vol 278, Issue 4, p647
- ISSN
1072-3374
- Publication type
Article
- DOI
10.1007/s10958-024-06946-z