Weakly periodic Gibbs measures for the Ising model on the Cayley tree of order $k=2$
Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 2, pp. 210-224 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove the existence of weakly periodic Gibbs measures for the Ising model on the Cayley tree of order $k=2$ with respect to a normal divisor of index 4.
Keywords: Cayley tree, Gibbs measure, Ising model, weakly periodic measure.
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M. M. Rahmatullaev; Zh. D. Dekhkonov. Weakly periodic Gibbs measures for the Ising model on the Cayley tree of order $k=2$. Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 2, pp. 210-224. http://geodesic.mathdoc.fr/item/TMF_2021_206_2_a5/

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

[2] 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

[3] F. Spitzer, “Markov random fields on an infinite tree”, Ann. Probab., 3:3 (1975), 387–398 | DOI | MR

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

[5] N. N. Ganikhodzhaev, U. A. Rozikov, “Opisanie periodicheskikh krainikh gibbsovskikh mer nekotorykh reshetochnykh modelei na dereve Keli”, TMF, 111:1 (1997), 109–117 | DOI | DOI | MR | Zbl

[6] 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

[7] U. A. Rozikov, “Postroenie neschetnogo chisla predelnykh gibbsovskikh mer neodnorodnoi modeli Izinga”, TMF, 118:1 (1999), 95–104 | DOI | MR | Zbl

[8] U. A. Rozikov, Yu. M. Suhov, “Gibbs measures for SOS models on a Cayley tree”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9:3 (2006), 471–488 | DOI

[9] J. B. Martin, U. A. Rozikov, Yu. M. Suhov, “A three state hard-core model on a Cayley tree”, J. Nonlinear Math. Phys., 12:3 (2005), 432–448 | DOI | MR

[10] U. A. Rozikov, “Opisanie predelnykh gibbsovskikh mer dlya $\lambda$-modelei na reshetkakh Bete”, Sib. matem. zhurn., 39:2 (1998), 427–435 | DOI | MR | Zbl

[11] F. M. Mukhamedov, U. A. Rozikov, “On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras”, J. Statist. Phys., 114:3–4 (2004), 825–848 | DOI | MR

[12] D. Gandolfo, J. Ruiz, S. Shlosman, “A manifold of pure Gibbs states of the Ising model on a Cayley tree”, J. Statist. Phys., 148:6 (2012), 999–1005 | DOI | MR

[13] U. A. Rozikov, Sh. A. Shoyusupov, “Mery Gibbsa dlya modeli SOS s chetyrmya sostoyaniyami na dereve Keli”, TMF, 149:1 (2006), 18–31 | DOI | DOI | MR | Zbl

[14] U. A. Rozikov, M. M. Rakhmatullaev, “Opisanie slabo periodicheskikh mer Gibbsa modeli Izinga na dereve Keli”, TMF, 156:2 (2008), 292–302 | DOI | DOI | MR | Zbl

[15] “Slabo periodicheskie osnovnye sostoyaniya i mery Gibbsa dlya modeli Izinga s konkuriruyuschimi vzaimodeistviyami na dereve Keli”, TMF, 160:3 (2009), 507–516 | DOI | DOI | MR | Zbl

[16] U. A. Rozikov, M. M. Rakhmatullaev, “O slabo periodicheskikh gibbsovskikh merakh modeli Izinga na dereve Keli”, Dokl. AN RUz, 4 (2008), 12–15

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

[18] M. M. Rakhmatullaev, “O slabo periodicheskikh merakh Gibbsa modeli Izinga s vneshnim polem na dereve Keli”, TMF, 183:3 (2015), 434–440 | DOI | MR

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

[20] M. M. Rahmatullaev, “On new weakly periodic Gibbs measures of the Ising model on the Cayley tree of order $\le 5$”, J. Phys.: Conf. Ser., 697 (2016), 012020, 7 pp. | DOI

[21] Ya. G. Sinai, Teoriya fazovykh perekhodov (strogie rezultaty), Nauka, M., 1980 | MR | MR | Zbl

[22] C. Preston, Gibbs States on Countable Sets, Cambridge Univ. Press, Cambridge, 1974 | MR

[23] V. A. Malyshev, R. A. Minlos, Gibbsovskie sluchainye polya. Metod klasternykh razlozhenii, Nauka, M., 1985 | DOI | MR | Zbl

[24] D. Ryuel, Statisticheskaya mekhanika, Mir, M., 1971 | MR | MR | Zbl | Zbl

[25] L. J. de Jongh, A. R. Miedema, “Experiments on simple magnetic model systems”, Adv. Phys., 23:1 (1974), 1–260 | DOI

[26] R. Feinman, Statisticheskaya mekhanika, Mir, M., 1978 | Zbl

[27] P. W. Kasteleyn, “Phase transitions”, Fundamental Problems in Statistical Mechanics II, Proceedings of 2nd NUFFIC International Summer Course (Noordwijk, The Netherlands, 1967), ed. E. G. D. Cohen, North-Holland, Amsterdam, 1968, 30–70 | Zbl

[28] B. Zimm, P. Doty, K. Iso, “Determination of the parameters for helix formations in poly-$\gamma$-benzil-L-glutamate”, Proc. Nat. Acad. Sci. USA, 45 (1959), 1601–1607 | DOI

[29] K. Khuang, Statisticheskaya mekhanika, Mir, M., 1966 | MR