Periodic Gibbs measures for the Potts–SOS model on a Cayley tree

M. A. Rasulova

Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 1, pp. 134-141

Résumé

We describe periodic Gibbs measures for the Potts–SOS model on a Cayley tree of order $k\ge1$, i.e. a characterization of such measures with respect to any normal subgroup of finite index of the group representation of the Cayley tree.

Détail

Keywords: Cayley tree, Potts–SOS model, periodic Gibbs measure.
Mots-clés : configuration

Bibliographie

[1] Ya. G. Sinai, Teoriya fazovykh perekhodov. Strogie rezultaty, Nauka, M., 1980 | MR | MR | Zbl

[2] Kh.-O. Georgi, Gibbsovskie mery i fazovye perekhody, Mir, M., 1992 | MR | Zbl

[3] S. Zachary, “Countable state space Markov random fields and Markov chains on trees”, Ann. Probab., 11:4 (1983), 894–903 | DOI | MR | Zbl

[4] K. Preston, Gibbsovskie sostoyaniya na schetnykh mnozhestvakh, Mir, M., 1977 | MR

[5] S. Zachary, “Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice”, Stochastic Process. Appl., 20:2 (1985), 247–256 | DOI | MR

[6] M. M. Rakhmatullaev, “O novykh slabo periodicheskikh gibbsovskikh merakh modeli Izinga na dereve Keli”, Izv. vuzov. Matem., 2015, no. 11, 54–63 | DOI | MR

[7] M. M. Rakhmatullaev, “Novye gibbsovskie mery modeli Izinga na dereve Keli”, Uzb. matem. zhurn., 2 (2009), 144–152

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

[9] 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

[10] P. M. Bleher, “Extremity of the disordered phase in the Ising model on the Bethe lattice”, Commun. Math. Phys., 128:2 (1990), 411–419 | DOI | MR

[11] 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. Stat. Phys., 79:1–2 (1995), 473–482 ; “On the phase diagram of the random field Ising model on the Bethe lattice”, 93:1–2 (1998), 33–78 | DOI | MR | DOI | MR

[12] D. Ioffe, “On the extremality of the disordered state for the Ising model on the Bethe lattice”, Lett. Math. Phys., 3:2 (1996), 137–143 | DOI | MR

[13] D. Ioffe, “Extremality of the disordered state for the Ising model on general trees”, Trees (Versailles, June 14–16, 1995), Progress in Probability, 40, eds. B. Chauvin, S. Cohen, A. Rouault, Birkhäuser, Basel, 1996, 3–14 | DOI | MR

[14] P. M. Bleher, J. Ruiz, R. H. Schonmann, S. B. Shlosman, V. A. Zagrebnov, “Rigidity of the critical phases on a Cayley tree”, Mosc. Math. J., 1:3 (2001), 345–363 | DOI | MR

[15] C. Külske, U. A. Rozikov, R. M. Khakimov, “Description of all translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree”, J. Stat. Phys., 156:1 (2014), 189–200, arXiv: 1310.6220 | DOI | MR | Zbl

[16] U. A. Rozikov, R. M. Khakimov, “Periodicheskie mery Gibbsa dlya modeli Pottsa na dereve Keli”, TMF, 175:2 (2013), 300–312 | DOI | DOI | MR | Zbl

[17] R. M. Khakimov, “O suschestvovanii periodicheskikh mer Gibbsa dlya modeli Pottsa na dereve Keli”, Uzb. matem. zhurn., 3 (2014), 134–142

[18] M. M. Rakhmatullaev, “Suschestvovanie slabo periodicheskikh mer Gibbsa dlya modeli Pottsa na dereve Keli”, TMF, 180:3 (2014), 307–317 | DOI | DOI | MR

[19] U. A. Rozikov, Y. M. Suhov, “Gibbs measures of SOS models on a Cayley tree”, Infin. Dimen. Anal. Quant. Probab. Relat. Top., 9:3 (2006), 471–488 | DOI | MR

[20] C. Kulske, U. A. Rozikov, “Extremality of translation-invariant phases for a three-state SOS-model on the binary tree”, J. Stat. Phys., 160:3 (2015), 659–680 | DOI | MR

[21] H. Saygili, “Gibbs measures for the Potts–SOS model with three states of spin values”, Asian J. Current Res., 1:3 (2017), 114–121

[22] U. A. Rozikov, “Struktury razbienii na klassy smezhnosti gruppovogo predstavleniya dereva Keli po normalnym delitelyam konechnogo indeksa i ikh primeneniya dlya opisaniya periodicheskikh raspredelenii Gibbsa”, TMF, 112:1 (1997), 170–175 | DOI | DOI | MR | Zbl