New weakly periodic Gibbs measures of Ising model on Cayley tree
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2015), pp. 54-63

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the Ising model on the Cayley tree. For the Ising model, we find a new weakly periodic Gibbs measures corresponding to normal subgroups of index two in the group representation of a Cayley tree.
Keywords: Cayley tree, Gibbs measure, Ising model, weakly periodic measure.
M. M. Rakhmatullaev. New weakly periodic Gibbs measures of Ising model on Cayley tree. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2015), pp. 54-63. http://geodesic.mathdoc.fr/item/IVM_2015_11_a4/
@article{IVM_2015_11_a4,
     author = {M. M. Rakhmatullaev},
     title = {New weakly periodic {Gibbs} measures of {Ising} model on {Cayley} tree},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {54--63},
     year = {2015},
     number = {11},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2015_11_a4/}
}
TY  - JOUR
AU  - M. M. Rakhmatullaev
TI  - New weakly periodic Gibbs measures of Ising model on Cayley tree
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2015
SP  - 54
EP  - 63
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/IVM_2015_11_a4/
LA  - ru
ID  - IVM_2015_11_a4
ER  - 
%0 Journal Article
%A M. M. Rakhmatullaev
%T New weakly periodic Gibbs measures of Ising model on Cayley tree
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2015
%P 54-63
%N 11
%U http://geodesic.mathdoc.fr/item/IVM_2015_11_a4/
%G ru
%F IVM_2015_11_a4

[1] Rozikov U. A., Gibbs measures on Cayley trees, World Scientific Publ. Co. Pte. Ltd, Hackensack, NJ, 2013 | MR | Zbl

[2] Blekher P. M., Ganikhodzhaev N. N., “O chistykh fazakh modeli Izinga na reshetke Bete”, Teor. veroyat. i ee prim., 35:2 (1990), 220–230 | MR | Zbl

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

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

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

[6] Rozikov U. A., “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 | MR | Zbl

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

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

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

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

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

[12] Ganikhodzhaev N. N., “Gruppovoe predstavlenie i avtomorfizmy na dereve Keli”, Dokl. AN RUz, 4 (1994), 3–5

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

[14] Preston C., Gibbs states on countable sets, Cambridge Univ. Press, 1974 | MR | Zbl

[15] Malyshev V. A., Minlos R. A., Gibbsovskie sluchainye polya, Nauka, M., 1985 | MR

[16] Ryuel D., Statisticheskaya mekhanika, Mir, M., 1971

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

[18] Feiman R., Statisticheskaya mekhanika, Mir, M., 1978

[19] Kasteleyn P. W., “Phase transitions”, Fundamental Problems in Statistical Mechanics. II, Proc. of 2nd NUFFIC Int. Summer Course (Noordwijk, the Netherlands, 1967), New York, 1968

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

[21] Khuang K., Statisticheskaya mekhanika, Mir, M., 1966