Geodesic
On periodic Gibbs measures of the Ising model corresponding to new subgroups of the group representation of a Cayley tree
Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 2, pp. 302-316 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Cayley tree, group representation of Cayley tree, subgroups, Ising model, Gibbs measure. 
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F. Kh. Khaidarov; R. A. Ilyasova. On periodic Gibbs measures of the Ising model corresponding to new subgroups of the group representation of a Cayley tree. Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 2, pp. 302-316. http://geodesic.mathdoc.fr/item/TMF_2022_210_2_a6/

[1] D. E. Cohen, R. C. Lyndon, “Free bases for normal subgroups of free groups”, Trans. Amer. Math. Soc., 108:3 (1963), 526–537 | DOI | MR

[2] D. S. Malik, J. N. Mordeson, M. K. Sen, Fundamentals of Abstract Algebra, McGraw–Hill, New York, 1997

[3] U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore, 2013 | DOI | MR

[4] S. Friedli, Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge Univ. Press, Cambridge, 2017 | DOI | MR

[5] S. A. Albeverio, Yu. G. Kondrat'ev, T. Pasurek, M. Röckner, “Euclidean Gibbs states of quantum crystals”, Mosc. Math. J., 1:3 (2001), 307–313 | DOI | MR | Zbl

[6] S. Albeverio, Y. Kondratiev, Y. Kozitsky, “Classical limits of Euclidean Gibbs states for quantum lattice models”, Lett. Math. Phys., 48:3 (1999), 221–233 | DOI | MR

[7] M. A. Rasulova, M. M. Rakhmatullaev, “Periodicheskie i slabo periodicheskie osnovnye sostoyaniya dlya modeli Pottsa s konkuriruyuschimi vzaimodeistviyami na dereve Keli”, Matem. tr., 18:2 (2015), 112–132 | DOI | DOI | MR

[8] F. H. Haydarov, Sh. A. Akhtamaliyev, M. A. Nazirov, B. B. Qarshiyev, “Uniqueness of Gibbs measures for an Ising model with continuous spin values on a Cayley tree”, Rep. Math. Phys., 86:3 (2020), 293–302 | DOI | MR

[9] U. A. Rozikov, F. Kh. Khaidarov, “Modeli s chetyrmya konkuriruyuschimi vzaimodeistviyami i s neschetnym mnozhestvom znachenii spina na dereve Keli”, TMF, 191:3 (2017), 503–517 | DOI | DOI | MR

[10] U. A. Rozikov, 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., 18:1 (2015), 1550006, 22 pp., arXiv: 1509.04883 | DOI | MR

[11] F. H. Haydarov, “New normal subgroups for the group representation of the Cayley tree”, Lobachevskii J. Math., 39:2 (2018), 213–217 | DOI | MR

[12] U. A. Rozikov, F. H. Haydarov, Invariance property on group representations of the Cayley tree and its applications, arXiv: 1910.13733

[13] U. A. Rozikov, F. H. Haydarov, “Normal subgroups of finite index for the group represantation of the Cayley tree”, TWMS J. Pure Appl. Math., 5:2 (2014), 234–240 | MR

[14] R. B. Ash, C. D. Doléans-Dade, Probability and Measure Theory, Harcourt Science and Technology Company, Burlington, MA, 2000 | MR

[15] P. M. Blekher, N. N. Ganikhodzhaev, “O chistykh fazakh modeli Izinga na reshetkakh Bete”, Teoriya veroyatn. i ee primen., 35:2 (1990), 220–230 | DOI | MR | Zbl

[16] P. M. Bleher, J. Ruiz, V. A. Zagrebnov, “On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice”, J. Statist. Phys., 79:1–2 (1995), 473–482 | DOI | MR