Abstract
We consider an infinite system of functional equations for the Potts model with competing interactions of radius \(r=2\) and countable spin values \(\Phi=\{0,1,\ldots,\}\) on the Cayley tree of order \(k=2\). We reduce the problem to the description of the solutions of some infinite system of equations for any \(k=2\) and any fixed probability measure \(\nu\) with \(\nu(i)>0\) on the set of all nonnegative integer numbers. We also give a description of the class of measures \(\nu\) on \(\Phi\) such that the infinite system of equations has unique solution \(\{a^i,\,i=1,2,\ldots\}\), \(a\in(0,1)\), with respect to each element of this class.
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Acknowledgments
We thank the referees for the helpful suggestions.
Funding
This work was supported by the fundamental project (F-FA-2021-425) of the Ministry of Innovative Development of the Republic of Uzbekistan.
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 214, pp. 318–328 https://doi.org/10.4213/tmf10353.
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Botirov, G.I., Mustafoyeva, Z.E. Gibbs measures for the Potts model with a countable set of spin values on a Cayley tree. Theor Math Phys 214, 273–281 (2023). https://doi.org/10.1134/S0040577923020113
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DOI: https://doi.org/10.1134/S0040577923020113