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.
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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\by U.~A.~Rozikov, R.~M.~Khakimov
\paper Periodic Gibbs measures for the~Potts model on the~Cayley tree
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\jour Theoret. and Math. Phys.
\yr 2013
\vol 175
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This publication is cited in the following 25 articles:
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
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 (k0)-translation-invariant and (k0)-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 λ 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 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
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