Abstract
In this paper, we study a two-state Hard-Core (HC) model with activity λ > 0 on a Cayley tree of order k ≥ 2. It is known that there are λcr, \({\lambda }_{{\text{cr}}}^{0}\), and \({\lambda }_{{\text{cr}}}{\prime}\) such that
• for λ ≤ λcr this model has a unique Gibbs measure μ*, which is translation invariant. The measure μ* is extreme for λ < \({\lambda }_{{\text{cr}}}^{0}\) and not extreme for λ > \({\lambda }_{{\text{cr}}}{\prime}\);
• for λ > λcr there exist exactly three 2-periodic Gibbs measures, one of which is μ*, the other two are not translation invariant and are always extreme.
The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k = 2, 3. Our proof is based on the tree reconstruction method.
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References
P. M. Blekher and N. N. Ganikhodzhaev, “On pure phases of the Ising model on the Bethe lattice,” Teor. Ver. i Ee Prim., 35, No. 2, 920–930 (1990).
P. M. Blekher, J. Ruiz, and V. A. Zagrebnov, “On the purity of the limiting Gibbs states for the Ising model on the Bethe lattice,” J. Stat. Phys., 79, No. 2, 473–482 (1995).
N. N. Ganikhodzhaev and U. A. Rozikov, “Description of periodic extreme Gibbs measures of some lattice models on the Cayley tree,” Teor. Mat. Fiz., 111, No. 1, 109–117 (1997).
H.-O. Georgii, Gibbs Measures and Phase Transitions [Russian translation], Mir, Moscow (1992).
H. Kesten and B. P. Stigum, “Additional limit theorem for indecomposable multidimensional Galton–Watson processes,” Ann. Math. Statist., 37, 1463–1481 (1966).
R. M. Khakimov, “Uniqueness of the weakly periodic Gibbs measure for the HC-model,” Mat. Zametki, 94, No. 5, 796–800 (2013).
R. M. Khakimov, “Weakly periodic Gibbs measures for the NC-model for a normal index divisor of four,” Ukr. Mat. Zh., 67, No. 10, 1409–1422 (2015).
R. M. Khakimov, “HC model on a Cayley tree: translation invariant Gibbs measures,” Vestn. NUUz, 2, No. 2, 245–251 (2017).
R. M. Khakimov, “Weakly periodic Gibbs measures for NC-models on a Cayley tree,” Sib. Mat. Zh., 59, No. 1, 185–196 (2018).
R. M. Khakimov and G. T. Madgoziyev, “Weakly periodic Gibbs measures for two and three state HC models on a Cayley tree,” Uzb. Math. J., 3, 116–131 (2018).
R. M. Khakimov and M. T. Makhammadaliev, “Condition of uniqueness and non-uniqueness of weakly periodic Gibbs measures for the NC-model,” ArXiv, 1910.11772v1 [math.ph] (2019).
C. Külske and U. A. Rozikov, “Extremality of translation-invariant phases for a three-state SOS-model on the binary tree,” J. Stat. Phys., 160, No. 3, 659–680 (2015).
C. Külske and U. A. Rozikov, “Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree,” Random Structures Algorithms, 50, No. 4, 636–678 (2017).
J. B. Martin, “Reconstruction thresholds on regular trees,” In: Discrete Random Walks, DRW’03. Proceedings of the Conference, Paris, France, September 1–5, 2003, MIMD, Paris, pp. 191–204 (2003).
F. Martinelli, A. Sinclair, and D. Weitz, “Fast mixing for independent sets, coloring and other models on trees,” Random Structures Algoritms, 31, 134–172 (2007).
A. E. Mazel and Yu. M. Suhov, “Random surfaces with two-sided constraints: an application of the theory of dominant ground states,” J. Stat. Phys., 64, 111–134 (1991).
E. Mossel, “Reconstruction on trees: beating the second eigenvalue,” Ann. Appl. Probab., 11, No. 1, 285–300 (2001).
E. Mossel and Y. Peres, “Information flow on trees,” Ann. Appl. Probab., 13, No. 3, 817–844 (2003).
C. J. Preston, Gibbs States on Countable Sets, Cambridge Univ. Press, Cambridge (1974).
U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore (2013).
U. A. Rozikov and R. M. Khakimov, “Uniqueness condition for the weakly periodic Gibbs measure in the hard-core model,” Teor. Mat. Fiz., 173, No. 1, 60–70 (2012).
U. A. Rozikov and R. M. Khakimov, “The extremality of the translation invariant Gibbs measure for the NC-model on the Cayley tree,” Byull. In-ta Mat., 2, 17–22 (2019).
U. A. Rozikov and M. M. Rakhmatullaev, “Description of the weakly periodic Gibbs measures of the Ising model on the Cayley tree,” Teor. Mat. Fiz., 156, No. 2, 292–302 (2008).
Ya. G. Sinay, Theory of Phase Transitions. Strict Results [in Russian], Nauka, Moscow (1980).
Yu. 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).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 68, No. 1, Science — Technology — Education — Mathematics — Medicine, 2022.
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Rozikov, U.A., Khakimov, R.M. & Makhammadaliev, M.T. Gibbs Periodic Measures for a Two-State HC-Model on a Cayley Tree. J Math Sci 278, 647–660 (2024). https://doi.org/10.1007/s10958-024-06946-z
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DOI: https://doi.org/10.1007/s10958-024-06946-z