Abstract
We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. A. Pirogov and Ya. G. Sinai, “Phase diagrams of classical lattice systems,” Theor. Math. Phys., 25, 1185–1192 (1975).
S. A. Pirogov and Ya. G. Sinai, “Phase diagrams of classical lattice systems continuation,” Theor. Math. Phys., 26, 39–49 (1976).
Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results [in Russian], Nauka, Moscow (1980); English transl., Pergamon, Oxford (1982).
R. Kotecky and S. B. Shlosman, “First-order phase transition in large entropy lattice models,” Commun. Math. Phys., 83, 493–515 (1982).
A. Mazel, Y. Suhov, and I. Stuhl, “A classical WR model with q particle types,” J. Stat. Phys., 159, 1040–1086 (2015).
A. Mazel, Y. Suhov, I. Stuhl, and S. Zohren, “Dominance of most tolerant species in multi-type lattice Widom–Rowlinson models,” J. Stat. Mech., 8, P08010 (2014).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad, Press, London (1982).
P. M. Bleher and N. N. Ganikhodjaev, “On pure phases of the Ising model on the Bethe lattice,” Theor. Probab. Appl., 35, 216–227 (1990).
P. M. Bleher, J. Ruiz, and V. A. Zagrebnov, “On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice,” J. Stat. Phys., 79, 473–482 (1995).
N. N. Ganikhodjaev and U. A. Rozikov, “On Ising model with four competing interactions on Cayley tree,” Math. Phys. Anal. Geom., 12, 141–156 (2009).
C. Preston, Gibbs States on Countable Sets, Cambridge Univ. Press, Cambridge (1974).
U. A. Rozikov, “Structures of partition of the group representation of the Cayley tree into adjacent classes by finite index normal subgroups and their application for discription of periodic Gibbs distributions,” Theor. Math. Phys., 112, 929–933 (1997).
F. Spitzer, “Markov random fields on an infinite tree,” Ann. Probab., 3, 387–398 (1975).
Y. M. Suhov and U. A. Rozikov, “A hard-core model on a Cayley tree: An example of a loss network,” Queueing Syst., 46, 197–212 (2004).
S. Zachary, “Countable state space Markov random fields and Markov chains on trees,” Ann. Probab., 11, 894–903 (1983).
N. N. Ganikhodjaev, C. H. Pah, and M. R. B. Wahiddin, “Exact solution of an Ising model with completing interections on a Cayley tree,” J. Phys. A.: Math. Gen., 36, 4283–4289 (2003).
J. L. Monroe, “Phase diagrams of Ising models on Husime trees: II. Pair and multisite interaction systems,” J. Stat. Phys., 67, 1185–2000 (1992).
J. L. Monroe, “A new criterion for the location of phase transitions for spin system on a recursive lattice,” Phys. Lett. A, 188, 80–84 (1994).
N. N. Ganikhodzhaev, “Exact solution of the Ising model on the Cayley tree with competing ternary and binary interactions,” Theor. Math. Phys., 130, 419–424 (2002).
F. M. Mukhamedov and U. A. Rozikov, “On Gibbs measures of models with completing ternary and binary interactions and corresponding von Neumann algebras,” J. Stat. Phys., 114, 825–848 (2004).
N. N. Ganikhodjaev, C. H. Pah, and M. R. B. Wahiddin, “An Ising model with three competing interactions on a Cayley tree,” J. Math. Phys., 45, 3645–3658 (2004).
N. N. Ganikhodjaev and U. A. Rozikov, “The Potts model with countable set of spin values on a Cayley tree,” Lett. Math. Phys., 75, 99–109 (2006).
Yu. R. Dashjan and Yu. M. Suhov, “On the question of the Gibbs description of random processes with discrete time,” Sov. Math. Dokl., 19, 1122–1126 (1978).
Yu. Kh. Eshkabilov, F. H. Haydarov, and U. A. Rozikov, “Uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley tree,” Math. Phys. Anal. Geom., 16, 1–17 (2013).
Yu. Kh. Eshkabilov, F. H. Haydarov, and U. A. Rozikov, “Non-uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley tree,” J. Stat. Phys., 147, 779–794 (2012).
Yu. Kh. Eshkabilov, Sh. D. Nodirov, and F. H. Haydarov, “Positive fixed points of quadratic operators and Gibbs measures,” Positivity, 20, 929–943 (2016).
Yu. Kh. Eshkabilov and F. H. Haydarov, “On positive solutions of the homogenous Hammerstein integral equation,” Nanosyst.: Phys. Chem. Math., 6, 618–627 (2015).
B. Jahnel, C. Külske, and G. I. Botirov, “Phase transition and critical value of nearest-neighbor system with uncountable local state space on Cayley tree,” Math. Phys. Anal. Geom., 17, 323–331 (2014).
U. A. Rozikov and Yu. Kh. Eshkabilov, “On models with uncountable set of spin values on a Cayley tree: Integral equations,” Math. Phys. Anal. Geom., 13, 275–286 (2010).
U. A. Rozikov and F. H. Haydarov, “Periodic Gibbs measures for models with uncountable set of spin values on a Cayley tree,” Infin. Dimens. Anal. Quantum Probab. Relat. Top., 18, 1550006 (2015).
U. A. Rozikov, Gibbs measures on a Cayley trees, World Scientific, Singapore (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 191, No. 3, pp. 503–517, June, 2017.
Rights and permissions
About this article
Cite this article
Rozikov, U.A., Haydarov, F.H. Four competing interactions for models with an uncountable set of spin values on a Cayley tree. Theor Math Phys 191, 910–923 (2017). https://doi.org/10.1134/S0040577917060095
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577917060095