Abstract
We study Gibbs measures for the HC model with a countable set \(\mathbb Z\) of spin values and a countable set of parameters (i.e., with the activity function \(\lambda_i>0\), \(i\in \mathbb Z\)) in the case of a “wand ”-type graph. In this case, analyzing a functional equation that ensures the consistency condition for finite-dimensional Gibbs measures, we obtain the following results. Exact values of the parameter \(\lambda_{\mathrm{cr}}\) are determined; it is shown that for \(0<\lambda\leq\lambda_{\mathrm{cr}}\), there exists exactly one translation-invariant nonprobabilistic Gibbs measure, and for \(\lambda>\lambda_{\mathrm{cr}}\), there exist precisely three such measures on a Cayley tree of order \(2\), \(3\), or \(4\). We obtain the uniqueness conditions for \(2\)-periodic nonprobabilistic Gibbs measures on a Cayley tree of an arbitrary order, as well as exact values of the parameter \(\lambda_{\mathrm{cr}}\); we also show that for \(\lambda\geq\lambda_{\mathrm{cr}}\), there exists precisely one such a measure, and for \(0<\lambda<\lambda_{\mathrm{cr}}\), there exist precisely three such measures on a Cayley tree of order \(2\) or \(3\).
Similar content being viewed by others
References
H.-O. Georgii, Gibbs Measures and Phase Transitions, (De Gruyter Studies in Mathematics, Vol. 9), Walter de Gruyter, Berlin (2011).
C. J. Preston, Gibbs States on Countable Sets, (Cambridge Tracts in Mathematics, Vol. 68), Cambridge Univ. Press, Cambridge (1974).
U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore (2013).
L. V. Bogachev and U. A. Rozikov, “On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field,” J. Stat. Mech. Theory Exp., 2019, 073205, 76 pp. (2019).
Y. K. 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); arXiv: 1202.2542.
S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems. A Concrete Mathematical Introduction, Cambridge Univ. Press, Cambridge (2018).
F. Henning, C. Külske, A. Le Ny, and U. A. Rozikov, “Gradient Gibbs measures for the SOS model with countable values on a Cayley tree,” Electron. J. Probab., 24, 106, 23 pp. (2019).
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).
N. N. Ganikhodjaev, “Limiting Gibbs measures of Potts model with countable set of spin values,” J. Math. Anal. Appl., 336, 693–703 (2007).
Z. Ye, “Models of gradient type with sub-quadratic actions,” J. Math. Phys., 60, 073304, 26 pp. (2019).
F. Henning and C. Külske, “Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees,” Ann. Appl. Probab., 31, 2284–2310 (2021).
S. Buchholz, “Phase transitions for a class of gradient fields,” Probab. Theory Related Fields, 179, 969–1022 (2021).
F. Henning, Ruhr-Universität, Bochum (2021).
C. Külske and P. Schriever, “Gradient Gibbs measures and fuzzy transformations on trees,” Markov Process. Relat. Fields, 23, 553–590 (2017).
F. Henning and C. Külske, “Existence of gradient Gibbs measures on regular trees which are not translation invariant,” arXiv: 2102.11899.
G. R. Brightwell, O. Häggström, and P. Winkler, “Non monotonic behavior in hard-core and Widom–Rowlinson models,” J. Statist. Phys., 94, 415–435 (1999).
F. P. Kelly, “Stochastic models of computer communication systems,” J. Roy. Statist. Soc. B, 47, 379–395 (1985).
A. E. Mazel’ and Yu. M. Suhov, “Random surfaces with two-sided constraints: An application of the theory of dominant ground states,” J. Statist. Phys., 64, 111–134 (1991).
R. M. Khakimov, M. T. Makhammadaliev, “Uniqueness and nonuniqueness conditions for weakly periodic Gibbs measures for the hard-core model,” Theoret. and Math. Phys., 204, 1059–1078 (2020).
G. R. Brightwell and P. Winkler, “Graph homomorphisms and phase transitions,” J. Combin. Theor. Ser. B, 77, 221–262 (1999).
N. Ganikhodjaev, F. Mukhamedov, and J. F. F. Mendes, “On the three state Potts model with competing interactions on the Bethe lattice,” J. Stat. Mech., 2006, P08012, 29 pp. (2006).
Acknowledgments
The authors thank the referee for the useful remarks and suggestions.
Funding
The work was supported by the Ministry of Innovation Development of the Republic of Uzbekistan (fundamental project No. F-FA-2021-425).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflicts of interest.
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 212, pp. 429–447 https://doi.org/10.4213/tmf10302.
Rights and permissions
About this article
Cite this article
Khakimov, R.M., Makhammadaliev, M.T. Nonprobability Gibbs measures for the HC model with a countable set of spin values for a “wand”-type graph on a Cayley tree. Theor Math Phys 212, 1259–1275 (2022). https://doi.org/10.1134/S0040577922090082
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577922090082