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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. Rozikova, R. M. Khakimovb

a Institute of Mathematics, Tashkent, Uzbekistan
b Namangan State University, Namangan, Uzbekistan
References:
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
Citation in format AMSBIB
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Linking options:
  • https://www.mathnet.ru/eng/tmf8423
  • https://doi.org/10.4213/tmf8423
  • https://www.mathnet.ru/eng/tmf/v175/i2/p300
  • This publication is cited in the following 25 articles:
    1. B. Z. Tozhiboev, R. M. Khakimov, “Mery Gibbsa dlya HC-modeli v sluchae grafa tipa “klyuch” na dereve Keli”, Matem. zametki, 117:4 (2025), 575–590  mathnet  crossref
    2. 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  crossref
    3. 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  mathnet  crossref  crossref  mathscinet  adsnasa
    4. 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  mathnet  crossref
    5. 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  mathnet  crossref  crossref  mathscinet  adsnasa
    6. 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  mathnet  crossref
    7. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    8. J. D. Dekhkonov, “On (k0)-translation-invariant and (k0)-periodic Gibbs measures for Potts model on Cayley tree”, Ufa Math. J., 14:4 (2022), 42–55  mathnet  crossref  mathscinet
    9. 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  crossref  mathscinet  isi
    10. 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  crossref  mathscinet  isi
    11. 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  crossref
    12. U. A. Rozikov, “Gibbs measures of Potts model on Cayley trees: A survey and applications”, Rev. Math. Phys., 33:10 (2021)  crossref
    13. 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  crossref  mathscinet  isi
    14. M. A. Rasulova, “Periodic Gibbs measures for the Potts–SOS model on a Cayley tree”, Theoret. and Math. Phys., 199:1 (2019), 586–592  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. F. M. Mukhamedov, Ch. Pah, H. Jamil, “Ground states and phase transition of the λ model on the Cayley tree”, Theoret. and Math. Phys., 194:2 (2018), 260–273  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    17. F. Mukhamedov, Ch. H. Pah, M. Rahimatullaev, H. Jamil, “Periodic and weakly periodic ground states for the λ-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  crossref  isi  scopus
    18. 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  crossref  mathscinet  zmath  isi  scopus
    19. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    20. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
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