Abstract
The uniqueness of the translation-invariant extreme Gibbs measure for the antiferromagnetic Potts model with an external field and the existence of an uncountable number of extreme Gibbs measures for the Ising model with an external field on the Cayley tree are proved. The classes of normal subgroups of finite index of the Cayley tree group representation are constructed. The periodic extreme Gibbs measures, which are invariant with respect to subgroups of index 2, are constructed for the Ising model with zero external field. From these measures, the existence of an uncountable number of nonperiodic extreme Gibbs measures for the antiferromagnatic Ising model follows.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Baxter,Exactly Solved Models in Statistical Mechanics, Academic Press, London-New York-Paris (1982).
P. M. Blekher and N. N. Ganikhodzhaev,Teor. Veroyatn. Primenen.,35, 920 (1990).
N. N. Ganikhodzhaev,Dokl. Akad. Nauk Resp. Uzbekistan, Nos. 6–7, 4 (1992).
N. N. Ganikhodzhaev,Theor. Math. Phys.,85, 1125 (1990).
C. J. Preston,Gibbs States on Countable Sets, Cambridge Univ., Oxford (1974).
P. M. Bleher,Commun. Math. Phys.,128, 411 (1990).
H. O. Georgii,Gibbs Measures and Phase Transitions, Walter de Gruyter, Berlin-New York (1988).
F. Spitzer,Ann. Prob.,3, 387 (1975).
Y. Higuchi,Publ. RIMS Kyoto Univ.,3, 335 (1977).
N. N. Ganikhodzhaev,Dokl. Akad. Nauk Resp. Uzbekistan, No. 5, 3 (1994).
A. G. Kurosh,Gruppentheorie, Akademic Verlag, Berlin (1953).
M. I. Kargapolov and Yu. I. Merzlyakov,Foundations of Group Theory [in Russian], Nauka, Moscow (1982).
S. A. Pirogov and Ya. G. Sinai,Theor. Math. Phys.,25, 1185 (1975);26, 61 (1976).
Ya. G. Sinai,Theory of Phase Transitions: Rigorous Results, Pergamon, Oxford-New York-London-Paris (1982).
Author information
Authors and Affiliations
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 109–117, April, 1997.
Rights and permissions
About this article
Cite this article
Ganikhodzhaev, N.N., Rozikov, U.A. Description of periodic extreme gibbs measures of some lattice models on the Cayley tree. Theor Math Phys 111, 480–486 (1997). https://doi.org/10.1007/BF02634202
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02634202