G. I. Botirov, U. A. Rozikov, “Potts model with competing interactions on the Cayley tree: The contour method”, TMF, 153:1 (2007), 86–97; Theoret. and Math. Phys., 153:1 (2007), 1423

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Teoreticheskaya i Matematicheskaya Fizika, 2007, Volume 153, Number 1, Pages 86–97
DOI: https://doi.org/10.4213/tmf6123 (Mi tmf6123)

This article is cited in 23 scientific papers (total in 23 papers)

Potts model with competing interactions on the Cayley tree: The contour method

G. I. Botirov^a, U. A. Rozikov^b

^a National University of Uzbekistan named after M. Ulugbek
^b Romanovskii Mathematical Institute of the National Academy of Sciences of Uzbekistan

Full-text PDF (451 kB) Citations (23)

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DOI: https://doi.org/10.4213/tmf6123

Abstract: We consider the Potts model with three spin values and with competing interactions of radius

$r=2$ on the Cayley tree of order

$k=2$ . We completely describe the ground states of this model and use the contour method on the tree to prove that this model has three Gibbs measures at sufficiently low temperatures.

Keywords: Cayley tree, configuration, contour method, ground state, Gibbs measure.

Received: 11.12.2006

English version:
Theoretical and Mathematical Physics, 2007, Volume 153, Issue 1, Pages 1423–1433
DOI: https://doi.org/10.1007/s11232-007-0125-x

Bibliographic databases:

Language: Russian

Citation: G. I. Botirov, U. A. Rozikov, “Potts model with competing interactions on the Cayley tree: The contour method”, TMF, 153:1 (2007), 86–97; Theoret. and Math. Phys., 153:1 (2007), 1423–1433

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This publication is cited in the following 23 articles:

Begzod Isakov, Olimkhon Akhmedov, “FUNKTsIONALNYE URAVNENIYa DLYa PREDELNYKh MER GIBBSA MODELI IZINGA-POTTSA NA DEREVE KELI”, VOGUMFT, 2024, no. 1(4), 90
M. M. Rahmatullaev, M. A. Rasulova, “Ground States and Gibbs Measures for the Potts-SOS Model with an External Field on the Cayley Tree”, Lobachevskii J Math, 45:1 (2024), 518
Muhayyo A. Rasulova, “Ground states for the potts model with an external field”, Reports on Mathematical Physics, 94:3 (2024), 325
G. I. Botirov, Z. E. Mustafoeva, “Gibbs measures for the Potts model with a countable set of spin values on a Cayley tree”, Theoret. and Math. Phys., 214:2 (2023), 273–281
M. M. Rahmatullaev, B. M. Isakov, “Ground states of Ising-Potts model on Cayley tree”, Ufa Math. J., 15:1 (2023), 43–55
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
G. Botirov, F. Haydarov, U. Qayumov, “Gibbs Measures of the Blume–Emery–Griffiths Model on the Cayley Tree”, Math Phys Anal Geom, 26:1 (2023)
Muzaffar M. Rahmatullaev, Muhayyo A. Rasulova, Javohir N. Asqarov, “Ground States and Gibbs Measures of Ising Model with Competing Interactions and an External Field on a Cayley Tree”, J Stat Phys, 190:7 (2023)
Muzaffar M. Rahmatullaev, Bunyod U. Abraev, “On ground states for the SOS model with competing interactions”, Zhurn. SFU. Ser. Matem. i fiz., 15:2 (2022), 162–175
F. M. Mukhamedov, M. M. Rahmatullaev, M. A. Rasulova, “Extremality of translation-invariant Gibbs measures for the $\lambda$-model on the binary Cayley tree”, Theoret. and Math. Phys., 210:3 (2022), 411–424
M. M. Rahmatullaev, M. A. Rasulova, “Periodic Ground States for the Potts Model with External Field and a Countable Set of Spin Values on the Cayley Tree”, Math. Notes, 112:1 (2022), 116–125
G. I. Botirov, U. U. Qayumov, “Ground states for the Potts model with competing interactions and a countable set of spin values on a Cayley tree”, Theoret. and Math. Phys., 209:2 (2021), 1633–1642
Rozikov U.A., “Gibbs Measures of Potts Model on Cayley Trees: a Survey and Applications”, Rev. Math. Phys., 33:10 (2021), 2130007
Mukhamedov F.M., Rakhmatullaev M.M., Rasulova M.A., “Weakly Periodic Ground States For the Lambda-Model”, Ukr. Math. J., 72:5 (2020), 771–784
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
Farrukh M. Mukhamedov, Muzaffar M. Rahmatullaev, M. A. Rasulova, “Slabo periodicheskie osnovnye sostoyaniya dlya λ-modeli”, Ukr. Mat. Zhurn., 72:5 (2020)
Mukhamedov F., Pah Ch.H., Rahimatullaev M., Jamil H., “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, eds. Rakhimov A., Ural B., Daoud J., Saburov K., Chowdhury M., IOP Publishing Ltd, 2018, UNSP 012021
G. I. Botirov, M. M. Rahmatullaev, Springer Proceedings in Mathematics & Statistics, 264, Algebra, Complex Analysis, and Pluripotential Theory, 2018, 59
M. A. Rasulova, M. M. Rahmatullaev, “Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree”, Siberian Adv. Math., 26:3 (2016), 215–229
N. M. Khatamov, “New classes of ground states for the Potts model with random competing interactions on a Cayley tree”, Theoret. and Math. Phys., 180:1 (2014), 827–834

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

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