U. A. Rozikov, R. M. Khakimov, “Periodic Gibbs measures for the Potts model on the Cayley tree”, TMF, 175:2 (2013), 300–312; Theoret. and Math. Phys., 175:2 (2013), 699

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Teoreticheskaya i Matematicheskaya Fizika, 2013, Volume 175, Number 2, Pages 300–312
DOI: https://doi.org/10.4213/tmf8423 (Mi tmf8423)

This article is cited in 25 scientific papers (total in 25 papers)

Periodic Gibbs measures for the Potts model on the Cayley tree

U. A. Rozikov^a, R. M. Khakimov^b

^a Institute of Mathematics, Tashkent, Uzbekistan
^b Namangan State University, Namangan, Uzbekistan

Full-text PDF (444 kB) Citations (25)

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DOI: https://doi.org/10.4213/tmf8423

Abstract: We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures.

Keywords: Cayley tree, configuration, Potts model, Gibbs measure, periodic measure, translation-invariant measure.

Received: 08.10.2012
Revised: 26.12.2012

English version:
Theoretical and Mathematical Physics, 2013, Volume 175, Issue 2, Pages 699–709
DOI: https://doi.org/10.1007/s11232-013-0055-8

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: U. A. Rozikov, R. M. Khakimov, “Periodic Gibbs measures for the Potts model on the Cayley tree”, TMF, 175:2 (2013), 300–312; Theoret. and Math. Phys., 175:2 (2013), 699–709

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https://doi.org/10.4213/tmf8423

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This publication is cited in the following 25 articles:

B. Z. Tozhiboev, R. M. Khakimov, “Mery Gibbsa dlya $\mathrm{HC}$ -modeli v sluchae grafa tipa “klyuch” na dereve Keli”, Matem. zametki, 117:4 (2025), 575–590
Rustamjon Khakimov, Kamola Umirzakova, “Periodic Gibbs Measures for Three-State Hard-Core Models in the Case Wand”, Z. mat. fiz. anal. geom., 20:1 (2024), 66
R. M. Khakimov, B. Z. Tozhiboev, “Gibbs measures for fertile models with hard-core interactions and four states”, Theoret. and Math. Phys., 219:2 (2024), 823–838
M. T. Makhammadaliev, “Pure phases of the ferromagnetic Potts model with $q$ states on the Cayley tree of order three”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 34:4 (2024), 499–517
M. M. Rahmatullaev, Zh. D. Dekhkonov, “Potts model on a Cayley tree: a new class of Gibbs measures”, Theoret. and Math. Phys., 215:1 (2023), 586–597
M. M. Rakhmatullaev, M. A. Rasulova, “Opisanie slabo periodicheskikh osnovnykh sostoyanii dlya modeli Pottsa s vneshnim polem i schetnym mnozhestvom znachenii spina na dereve Keli”, Vladikavk. matem. zhurn., 25:4 (2023), 103–119
U. A. Rozikov, M. M. Rahmatullaev, R. M. Khakimov, “Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree”, Theoret. and Math. Phys., 210:1 (2022), 135–153
J. D. Dekhkonov, “On $(k_0)$ -translation-invariant and $(k_0)$ -periodic Gibbs measures for Potts model on Cayley tree”, Ufa Math. J., 14:4 (2022), 42–55
Rahmatullaev M.M. Rafikov F.K. Azamov Sh.Kh., “On the Constructive Description of Gibbs Measures For the Potts Model on a Cayley Tree”, Ukr. Math. J., 73:7 (2021), 1092–1106
Rahmatullaev M.M. Rasulova M.A., “Extremality of Translation-Invariant Gibbs Measures For the Potts-Sos Model on the Cayley Tree”, J. Stat. Mech.-Theory Exp., 2021:7 (2021), 073201
M. Rahmatullaev, F. K. Rafikov, Sh. Kh. Azamov, “Pro konstruktivnі opisi mіr Gіbbsa dlya modelі Pottsa na derevі Kelі”, Ukr. Mat. Zhurn., 73:7 (2021), 938
U. A. Rozikov, “Gibbs measures of Potts model on Cayley trees: A survey and applications”, Rev. Math. Phys., 33:10 (2021)
Mukhamedov F. Pah Ch.H. Jamil H. Rahmatullaev M., “On Ground States and Phase Transition For Lambda-Model With the Competing Potts Interactions on Cayley Trees”, Physica A, 549 (2020), 124184
M. A. Rasulova, “Periodic Gibbs measures for the Potts–SOS model on a Cayley tree”, Theoret. and Math. Phys., 199:1 (2019), 586–592
R. M. Khakimov, M. T. Makhammadaliev, “Translation Invariance of the periodic Gibbs measures for the Potts model on the Cayley tree”, Theoret. and Math. Phys., 199:2 (2019), 726–735
F. M. Mukhamedov, Ch. Pah, H. Jamil, “Ground states and phase transition of the $\lambda$ model on the Cayley tree”, Theoret. and Math. Phys., 194:2 (2018), 260–273
F. Mukhamedov, Ch. H. Pah, M. Rahimatullaev, H. Jamil, “Periodic and weakly periodic ground states for the $\lambda$ -model on Cayley tree”, 4th International Conference on Mathematical Applications in Engineering 2017 (ICMAE'17), Journal of Physics Conference Series, 949, ed. A. Rakhimov, B. Ural, J. Daoud, K. Saburov, M. Chowdhury, IOP Publishing Ltd, 2018, UNSP 012021
L. Ahmad Mohd Ali Khameini Liao, M. Saburov, “Periodic $p$ -adic Gibbs measures of $q$ -state Potts model on Cayley trees I: the chaos implies the vastness of the set of $p$ -adic Gibbs measures”, J. Stat. Phys., 171:6 (2018), 1000–1034
U. A. Rozikov, R. M. Khakimov, F. Kh. Khaidarov, “Extremality of the translation-invariant Gibbs measures for the Potts model on the Cayley tree”, Theoret. and Math. Phys., 196:1 (2018), 1043–1058
U. A. Rozikov, M. M. Rakhmatullaev, “Free energies of the Potts model on a Cayley tree”, Theoret. and Math. Phys., 190:1 (2017), 98–108

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

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