U. A. Rozikov, F. Kh. Khaidarov, “Four competing interactions for models with an uncountable set of spin values on a Cayley tree”, TMF, 191:3 (2017), 503–517; Theoret. and Math. Phys., 191:3 (2017), 910

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Teoreticheskaya i Matematicheskaya Fizika, 2017, Volume 191, Number 3, Pages 503–517
DOI: https://doi.org/10.4213/tmf9215 (Mi tmf9215)

This article is cited in 10 scientific papers (total in 10 papers)

Four competing interactions for models with an uncountable set of spin values on a Cayley tree

U. A. Rozikov^a, F. Kh. Khaidarov^b

^a Institute of Mathematics and Information Technologies, Tashkent, Uzbekistan
^b National University of Uzbekistan, Tashkent, Uzbekistan

Full-text PDF (493 kB) Citations (10)

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DOI: https://doi.org/10.4213/tmf9215

Abstract: We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set

[0, 1]

$[0,1]$ of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.

Keywords: Cayley tree, competing interaction, configuration, Gibbs measure, Ising model, Potts model, periodic Gibbs measure, phase transition.

Received: 23.04.2016

English version:
Theoretical and Mathematical Physics, 2017, Volume 191, Issue 3, Pages 910–923
DOI: https://doi.org/10.1134/S0040577917060095

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: U. A. Rozikov, F. Kh. Khaidarov, “Four competing interactions for models with an uncountable set of spin values on a Cayley tree”, TMF, 191:3 (2017), 503–517; Theoret. and Math. Phys., 191:3 (2017), 910–923

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tmf9215

https://doi.org/10.4213/tmf9215

https://www.mathnet.ru/eng/tmf/v191/i3/p503

This publication is cited in the following 10 articles:

I. M. Mavlonov, N. Kh. Khushvaktov, G. P. Arzikulov, F. Kh. Khaidarov, “On positive fixed points of operator of Hammerstein type with degenerate kernel and Gibbs measures”, Theoret. and Math. Phys., 220:3 (2024), 1580–1588
F. H. Haydarov, “On normal subgroups of the group representation of the Cayley tree”, Vladikavk. matem. zhurn., 25:4 (2023), 135–142
F. Kh. Khaidarov, R. A. Ilyasova, “On periodic Gibbs measures of the Ising model corresponding to new subgroups of the group representation of a Cayley tree”, Theoret. and Math. Phys., 210:2 (2022), 261–274
F. H. Khaidarov, “Existence and uniqueness of fixed points of an integral operator of Hammerstein type”, Theoret. and Math. Phys., 208:3 (2021), 1228–1238
F. H. Haydarov, “New condition on uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley tree”, Math. Phys. Anal. Geom., 24:4 (2021), 31
Yu. Kh. Eshkabilov, G. I. Botirov, F. H. Haydarov, “Phase transitions for models with a continuum set of spin values on a Bethe lattice”, Theoret. and Math. Phys., 205:1 (2020), 1372–1380
R. N. Ganikhodjaev, R. R. Kucharov, K. A. Aralova, “Positive fixed points of lyapunov operator”, Nanosyst.-Phys. Chem. Math., 11:4 (2020), 373–378
F. H. Haydarov, Sh. A. Akhtamaliyev, M. A. Nazirov, B. B. Qarshiyev, “Uniqueness of Gibbs measures for an Ising model with continuous spin values on a Cayley tree”, Rep. Math. Phys., 86:3 (2020), 293–302
F. H. Haydarov, “Fixed points of Lyapunov integral operators and Gibbs measures”, Positivity, 22:4 (2018), 1165–1172
Eshkabilov Yu.Kh., Haydarov F.H., “Lyapunov Operator l With Degenerate Kernel and Gibbs Measures”, Nanosyst.-Phys. Chem. Math., 8:5 (2017), 553–558

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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