Hidden Potts-Observed SOS Models on a Two-Layer Structure
DOI:
https://doi.org/10.56143/ujmcs.v2i3s.5Keywords:
Cayley tree; Potts model; SOS model; hidden Markov structure; Gibbs measure; probabilistic inference.Abstract
In this paper, we introduce a two-layer hidden Potts-observed SOS model on Cayley trees of arbitrary order. The model is constructed through a hidden Markov structure in which the observable SOS configuration is generated from an underlying latent Potts field by means of a local stochastic transformation. For the proposed model, we derive the compatibility conditions for Gibbs measures on Cayley trees of arbitrary order and investigate translation-invariant Gibbs states. In particular, explicit forms of translation-invariant Gibbs measures are obtained for Cayley trees of orders one and two, and conditions for their existence are established. Moreover, for these Gibbs measures, the hidden states are reconstructed and predicted from the observable layer. The obtained results provide a rigorous probabilistic framework for multistate hidden spin systems and extend hidden Markov methods to Potts-SOS interactions on hierarchical graphs. The proposed approach may be useful in the study of spatial stochastic systems, graphical models, and related inference problems involving latent structures.
References
[1] Abdullaev, L. U., Rozikov, U. A.: Gibbs Measures in Machine Learning. World Scientific, Singapore (2026). DOI: 10.1142/14350
[2] Akin, H. and Ulusoy, U.: A new approach to studying the thermodynamic properties of the q-state Potts model on a Cayley tree. Chaos, Solitons and Fractals (2023). DOI: 10.1016/j.chaos.2023.113811
[3] Baxter, R. J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London/New York (1982).
[4] Botirov, G. I., Rozikov, U. A.: On q-component models on the Cayley tree: the general case. J. Stat. Mech. (2006). DOI: 10.1088/17425468/2006/10/P10006
[5] Botirov, G. I., Rozikov, U. A.: Potts model with competing interactions on the Cayley tree: The contour method. Theor. Math. Phys. (2007). DOI: 10.1007/s11232-007-0125-x
[6] Ferreira, L. S., Jorge, L. N., DaSilva, C. J., Caparica, A. A.: An entropic approach to analyze phase transitions in the Potts model. Physica A: Statistical Mechanics and its Applications. 674, 130694 (2025). DOI: 10.1016/j.physa.2025.130694
[7] Herrera, F., Rozikov, U.A., Velasco, M.V.: Ising models with hidden Markov structure: applications to probabilistic inference in machine learning. J. Stat. Mech. (2025). DOI: 10.1088/1742-5468/ade5f8
[8] Kurbanova, D. R., Magomedov, M. A., Dzhamaludinov, M. R., Ramazanov, M. K., Murtazaev, A. K.: Phase diagram of the antiferromagnetic 3-state Potts model on a bcc lattice with competing interactions. Physica A: Statistical Mechanics and its Applications. 676, 130870 (2025). DOI: 10.1016/j.physa.2025.130870
[9] Minlos, R. A.: Introduction to mathematical statistical physics. University lecture series. 1 (2000).
[10] Mukhamedov, F., Akin, H.: Exact solution for the three-state asymmetric Potts model on a Cayley tree. Chaos, Solitons and Fractals (2026). DOI: 10.1016/j.chaos.2026.118118
[11] Ostilli, M.: Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists. Physica A: Statistical Mechanics and its Applications. 391(12), 3417–3423 (2012). DOI: 10.1016/j.physa.2012.01.038
[12] Pirogov,S.A.,Sinai,Ya.G.:Phasediagramsofclassicallatticesystems. Theoretical and Mathematical Physics (1975).DOI:10.1007/BF01040127
[13] Preston, C. J.: Gibbs states on countable sets. Cambridge University Press, London (2011). DOI: 10.1017/CBO9780511897122
[14] Rahmatullaev, M. M., Rasulova, M. A.: Extremality of translation-invariant Gibbs measures for the Potts-SOS model on the Cayley tree. J. Stat. Mech. (2021). DOI: 10.1088/1742-5468/ac08ff
[15] Rahmatullaev, M. M., Rasulova, M. A.: Ground States and Gibbs Measures for the Potts-SOS Model with an External Field on the Cayley Tree. Lobachevskii Journal of Mathematics (2024). DOI: 10.1134/S1995080224010451
[16] Rahmatullaev, M. M., Rasulova, M. A., Potts models featuring hidden Markov structure. Available at SSRN: https://ssrn.com/abstract=6819388 or http://dx.doi.org/10.2139/ssrn.6819388
[17] Rasulova, M. A.: Ground states for the Potts model with an external field. Reports on Mathematical Physics (2024). DOI: 10.1016/S0034 4877(24)00082-X
[18] Rasulova, M. A.: On the Relationship between Configurations and Limiting Gibbs Measures for the Potts-SOS Model on the Cayley Tree. Journal of Mathematical Sciences (2026). DOI: 10.1007/s10958-026-08419-x
[19] Rozikov, U.A.: A Contour method on Cayley trees. J. Stat. Phys. (2008). DOI: 10.1007/s10955-007-9455-1
[20] Rozikov, U. A.: Gibbs measures on Cayley trees. World Scientific, Singapore (2013). DOI: 10.1142/8841
[21] Rozikov, U. A.: Gibbs Measures in Biology and Physics: The Potts Model. World Scientific, Singapore (2023). DOI: 10.1142/12694
[22] Rozikov, U. A., Rakhmatullaev, M. M., Khakimov, R. M.: Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree. Theoretical and Mathematical Physics (2022). DOI: 10.1134/S004057792201010X
[23] Sinai, Ya. G.: Theory of phase transitions: Rigorous Results. Pergamon, Oxford (1982).
