Abstract
We consider a mixed-type model given by the three-state Ising–Potts model on a Cayley tree. A criterion for the existence of limit Gibbs measures for this model on an arbitrary-order Cayley tree is obtained. Translation-invariant Gibbs measures on a second-order Cayley tree are studied. The existence of a phase transition is proved: a range of parameter values is found in which there are one to seven Gibbs measures for the three-state Ising–Potts model.
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References
S. Katsura and M. Takizava, “Bethe lattice and Bethe approximation,” Progr. Theor. Phys., 51, 82–98 (1974).
C. J. Preston, Gibbs States on Countable Sets, (Cambridge Tracts in Mathematics, Vol. 68), Cambridge Univ. Press, Cambridge (2011).
U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore (2013).
N. N. Ganikhodzhaev and U. A. Rozikov, “Description of periodic extreme Gibbs measures of some lattice models on the Cayley tree,” Theoret. and Math. Phys., 111, 480–486 (1997).
U. A. Rozikov and M. M. Rakhmatullaev, “Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree,” Theoret. and Math. Phys., 156, 1218–1227 (2008).
U. A. Rozikov and M. M. Rakhmatullaev, “Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree,” Theoret. and Math. Phys., 160, 1292–1300 (2009).
M. M. Rahmatullaev and Zh. D. Dekhkonov, “Weakly periodic Gibbs measures for the Ising model on the Cayley tree of order \(k=2\),” Theoret. and Math. Phys., 206, 185–198 (2021).
C. Külske, U. A. Rozikov, and R. M. Khakimov, “Description of translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree,” J. Stat. Phys., 156, 189–200 (2014).
R. M. Khakimov and M. T. Makhammadaliev, “Translation Invariance of the periodic Gibbs measures for the Potts model on the Cayley tree,” Theoret. and Math. Phys., 199, 726–735 (2019).
R. M. Khakimov and F. Kh. Khaidarov, “Translation-invariant and periodic Gibbs measures for the Potts model on a Cayley tree,” Theoret. and Math. Phys., 189, 1651–1659 (2016).
M. M. Rakhmatullaev, “Weakly periodic Gibbs measures and ground states for the Potts model with competing interactions on the Cayley tree,” Theoret. and Math. Phys., 176, 1236–1251 (2013).
M. M. Rakhmatullaev, “The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree,” Theoret. and Math. Phys., 180, 1019–1029 (2014).
U. A. Rozikov, Gibbs measures in Biology and Physics: The Potts Model, World Sci., Singapore (2022).
H. Saygili, “Gibbs measures for the Potts–SOS model with three states of spin values,” Asian J. Current Research, 1, 114–121 (2016).
M. M. Rahmatullaev and M. M. Rasulova, “Extremality of translation-invariant Gibbs measures for the Potts–SOS model on the Cayley tree,” J. Stat. Mech. Theory Exp., 2021, 073201, 18 pp. (2021).
F. M. Mukhamedov, M. M. Rahmatullaev, and M. M. Rasulova, “Weakly periodic ground states for the \(\lambda\)-model,” Ukr. Math. J., 72, 771–784 (2020).
F. M. Mukhamedov, “Extremality of disordered phase of \(\lambda\)-model on a Cayley trees,” Algoritms, 15, 18, 12 pp. (2022).
M. M. Rahmatullaev and B. M. Isakov, “Ground states of Ising–Potts model on a Cayley tree,” Ufa Math. J., 15, 43–55 (2023).
M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1977).
N. Khatamov and R. Khakimov, “Translation-invariant Gibbs measures for the Blume–Capel model on a Cayley tree,” J. Math. Phys. Anal. Geom., 15, 239–255 (2019).
N. M. Khatamov, “Extremality of Gibbs Measures for the \(HC\)-Blume–Capel Model on the Cayley Tree,” Math. Notes, 111, 768–781 (2022).
U. A. Rozikov, “Description of limit Gibbs measures for \(\lambda\)-models on Bethe lattices,” Siberian Math. J., 39, 373–380 (1998).
Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results, (International Series in Natural Philosophy, Vol. 108), Pergamon Press, Oxford (1982).
K. T. Leung, I. A. C. Mok, and S. N. Seun, Polynomials and Equations, Hong Kong Univ. Press, Hong Kong (1992).
V. V. Prasolov, Polynomials, (Algorithms and Computation in Mathematics, Vol. 11), Springer, Berlin (2004).
A. G. Kurosh, Higher Algebra, Mir, Moscow (1975).
Acknowledgments
The authors express their deep gratitude to Professors U. A. Rozikov and N. N. Ganikhodzhaev for a useful advice on their research.
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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 219, pp. 597–609 https://doi.org/10.4213/tmf10669.
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Rakhmatullaev, M.M., Isakov, B.M. Translation-invariant Gibbs measures for the Ising–Potts model on a second-order Cayley tree. Theor Math Phys 219, 1048–1059 (2024). https://doi.org/10.1134/S0040577924060114
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DOI: https://doi.org/10.1134/S0040577924060114