Periodic Gibbs measures for the Potts model on the Cayley tree

Rozikov, U. A.; Khakimov, R. M.

doi:10.1007/s11232-013-0055-8

Cited by 30 publications

(14 citation statements)

References 12 publications

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“…In Section 3, we study periodic and weakly periodic ground states. Note that our results extend known ones [16,17]. Hence, the proposed model opens new perspectives in the theory of statistical mechanics over trees.…”

Section: Introductionsupporting

confidence: 83%

Periodic and Weakly Periodic Ground States for the λ-Model on Cayley Tree

Mukhamedov

Pah

Rahimatullaev

et al. 2017

J. Phys.: Conf. Ser.

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Section: Introductionsupporting

confidence: 83%

Periodic and Weakly Periodic Ground States for the λ-Model on Cayley Tree

Mukhamedov

Pah

Rahimatullaev

et al. 2017

J. Phys.: Conf. Ser.

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“…Moreover, (20) holds in this interval. Consequently, if r > 3θ 2 then the first equation of ( 11) has a positive real solution, if r ≤ 3θ 2 then the first equation of (11) cannot have a positive solution, i.e., any positive real pair (x, y), which is solution of the first equation of (11), does not satisfy r ≤ 3θ 2 .…”

Section: Translation-invariant Gibbs Measuresmentioning

confidence: 86%

“…In [19] all translation-invariant splitting Gibbs measures (TISGMs) for the Potts model on the CT are described. In [20], [21] periodic Gibbs measures, in [22]- [24] weakly periodic Gibbs measures for the Potts model are studied.…”

Section: Introductionmentioning

confidence: 99%

Extremality of translation-invariant Gibbs measures for the Potts-SOS model on the Cayley tree

Rahmatullaev,

Rasulova

2021

Preprint

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“…. , a k+1 , i.e., let a 2 i = e (see [22]). There exists a one-to-one correspondence between the set V of vertices of the Cayley tree of order k and the group G k [20].…”

Section: Preliminariesmentioning

confidence: 99%

On ground states and phase transition for λ-model with the competing Potts interactions on Cayley trees

Mukhamedov

Pah

Jamil

et al. 2020

Physica A: Statistical Mechanics and its Applications

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In this paper, we consider the λ-model with nearest neighbor interactions and with competing Potts interactions on the Cayley tree of order-two. We notice that if λfunction is taken as a Potts interaction function, then this model contains as a particular case of Potts model with competing interactions on Cayley tree. In this paper, we first describe all ground states of the model. We point out that the Potts model with considered interactions was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model in more general setting. Furthermore, we find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition.

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Periodic Gibbs measures for the Potts model on the Cayley tree

Cited by 30 publications

References 12 publications

Periodic and Weakly Periodic Ground States for the λ-Model on Cayley Tree

Periodic and Weakly Periodic Ground States for the λ-Model on Cayley Tree

Extremality of translation-invariant Gibbs measures for the Potts-SOS model on the Cayley tree

On ground states and phase transition for λ-model with the competing Potts interactions on Cayley trees

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