Abstract
For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T c,k =J/arctan (1/k) the limiting Gibbs measure is unique, and for T<T c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k}<k_{0}<k\) then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k 0≠k′0. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1}<T_{c,k}\) then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}<k_{0}<k}{\mathcal{G}}_{k,k_{0}})\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London/New York (1982)
Bleher, P.M., Ruiz, J., Schonmann, R.H., Shlosman, S., Zagrebnov, V.A.: Rigidity of the critical phases on a Cayley tree. Mosc. Math. J. 3, 345–362 (2001)
Bleher, P.M., Ganikhodjaev, N.N.: On pure phases of the Ising model on the Bethe lattice. Theory Probab. Appl. 35, 216–227 (1990)
Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Stat. Phys. 79, 473–482 (1995)
Bleher, P.M., Ruiz, J., Zagrebnov, V.A.: On the phase diagram of the random field Ising model on the Bethe lattice. J. Stat. Phys. 93, 33–78 (1998)
Ding, J., Lubetzky, E., Peres, Y.: Mixing time of critical Ising model on trees is polynomial in the height. Commun. Math. Phys. 295, 161–207 (2010)
Dobrushin, R.L.: Gibbs states describing coexistence of phases for a three-dimensional Ising model. Theory Probab. Appl. 17, 582–600 (1972)
Ganikhodjaev, N.N., Rozikov, U.A.: A description of periodic extremal Gibbs measures of some lattice models on the Cayley tree. Theor. Math. Phys. 111, 480–486 (1997)
Georgii, H.O.: Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin (1988)
Müller-Hartmann, E.: Theory of the Ising model on a Cayley tree. Z. Phys. B, Condens. Matter Quanta 27, 161–168 (1977)
Higuchi, Y.: Remarks on the limiting Gibbs states on a tree. Publ. RIMS Kyoto Univ. 13, 335–348 (1977)
Ioffe, D.: On the extremality of the disordered state for the Ising model on the Bethe lattice. Lett. Math. Phys. 37, 137–143 (1996)
Pemantle, R., Peres, Y.: The critical Ising model on trees, concave recursions and nonlinear capacity. Ann. Probab. 38, 184–206 (2010)
Preston, C.: Gibbs States on Countable Sets. Cambridge University Press, London (1974)
Rozikov, U.A., Rakhmatullaev, M.M.: Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree. Theor. Math. Phys. 160, 1291–1299 (2009)
Sinai, Ya.G.: Theory of Phase Transitions: Rigorous Results. Pergamon, Oxford (1982)
Spitzer, F.: Markov random fields on an infinite tree. Ann. Probab. 3, 387–398 (1975)
Suhov, Y.M., Rozikov, U.A.: A hard-core model on a Cayley tree: an example of a loss network. Queueing Syst. 46, 197–212 (2004)
Zachary, S.: Countable state space Markov random fields and Markov chains on trees. Ann. Probab. 11, 894–903 (1983)
Zachary, S.: Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice. Stoch. Process. Appl. 20, 247–256 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Akin, H., Rozikov, U.A. & Temir, S. A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree. J Stat Phys 142, 314–321 (2011). https://doi.org/10.1007/s10955-010-0106-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-0106-6