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.
@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},
     publisher = {mathdoc},
     number = {11},
     year = {2015},
     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
PB  - mathdoc
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
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2015_11_a4/
%G ru
%F IVM_2015_11_a4
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/

[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