Abstract
We give a full description of all index-\(4\) subgroups of the group representation of a Cayley tree. Also, we give new weakly periodic Gibbs measures of the Ising model corresponding to index-\(4\) subgroups of the group representation of the Cayley tree.
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D. E. Cohen and R. C. Lyndon, “Free bases for normal subgroups of free groups,” Trans. Amer. Math. Soc., 108, 526–537 (1963).
D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw–Hill, New York (1997).
U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore (2013).
S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge Univ. Press, Cambridge (2017).
S. A. Albeverio, Yu. G. Kondrat’ev, T. Pasurek, and M. Röckner, “Euclidean Gibbs states of quantum crystals,” Mosc. Math. J., 1, 307–313 (2001).
S. Albeverio, Y. Kondratiev, and Y. Kozitsky, “Classical limits of Euclidean Gibbs states for quantum lattice models,” Lett. Math. Phys., 48, 221–233 (1999).
M. A. Rasulova and M. M. Rahmatullaev, “Periodic and weakly periodic ground states for the Potts model with competing interactions on the Cayley tree,” Siberian Adv. Math., 26, 215–229 (2016).
F. H. Haydarov, Sh. A. Akhtamaliyev, M. A. Nazirov, and B. B. Qarshiyev, “Uniqueness of Gibbs measures for an Ising model with continuous spin values on a Cayley tree,” Rep. Math. Phys., 86, 293–302 (2020).
U. A. Rozikov and F. Kh. Khaidarov, “Four competing interactions for models with an uncountable set of spin values on a Cayley tree,” Theoret. and Math. Phys., 191, 910–923 (2017).
U. A. Rozikov and F. H. Haydarov, “Periodic Gibbs measures for models with uncountable set of spin values on a Cayley tree,” Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18, 1550006, 22 pp. (2015); arXiv: 1509.04883.
F. H. Haydarov, “New normal subgroups for the group representation of the Cayley tree,” Lobachevskii J. Math., 39, 213–217 (2018).
U. A. Rozikov and F. H. Haydarov, “Invariance property on group representations of the Cayley tree and its applications,” arXiv: 1910.13733.
U. A. Rozikov and F. H. Haydarov, “Normal subgroups of finite index for the group represantation of the Cayley tree,” TWMS J. Pure Appl. Math., 5, 234–240 (2014).
R. B. Ash and C. D. Doléans-Dade, Probability and Measure Theory, Harcourt Science and Technology Company, Burlington, MA (2000).
P. M. Blekher and N. N. Ganikhodzhaev, “On pure phases of the Ising model on the Bethe lattices,” Theory Probab. Appl., 35, 216–227 (1990).
P. M. Bleher, J. Ruiz, and V. A. Zagrebnov, “On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice,” J. Statist. Phys., 79, 473–482 (1995).
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We thank the referees for the careful reading of the manuscript and especially for a number of suggestions that have improved the paper.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 210, pp. 302-316 https://doi.org/10.4213/tmf10156.
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Haydarov, F.H., Ilyasova, R.A. On periodic Gibbs measures of the Ising model corresponding to new subgroups of the group representation of a Cayley tree. Theor Math Phys 210, 261–274 (2022). https://doi.org/10.1134/S0040577922020076
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DOI: https://doi.org/10.1134/S0040577922020076