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<p>We study the three-state Potts-SOS model on a Cayley tree of order two and investigate the relationship between boundary configurations and translation-invariant splitting Gibbs measures (TISGMs). Although all TISGMs for this model have been previously classified, an explicit description of boundary conditions generating these measures has not been established. Using a rigorous probabilistic framework based on the Markov functional approach and the Kolmogorov consistency condition, we analyze the nonlinear recursive equations associated with boundary laws. We prove that for each TISGM, there exists a nonempty class of boundary configurations such that the corresponding limiting Gibbs measure coincides exactly with that TISGM. Our results reveal how the dynamical system induced by the recursive operator determines domains of attraction of stable fixed points and clarify the mechanism through which microscopic boundary effects select macroscopic equilibrium states. This provides the first characterization of boundary conditions generating prescribed TISGMs for the three-state Potts-SOS model on a Cayley tree.</p>

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ON THE RELATIONSHIP BETWEEN CONFIGURATIONS AND LIMITING GIBBS MEASURES FOR THE POTTS-SOS MODEL ON THE CAYLEY TREE

  • 期刊论文

  • 发表日期：

    2026年4月28日

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  • Muhayyo A. Rasulova

摘要

We study the three-state Potts-SOS model on a Cayley tree of order two and investigate the relationship between boundary configurations and translation-invariant splitting Gibbs measures (TISGMs). Although all TISGMs for this model have been previously classified, an explicit description of boundary conditions generating these measures has not been established. Using a rigorous probabilistic framework based on the Markov functional approach and the Kolmogorov consistency condition, we analyze the nonlinear recursive equations associated with boundary laws. We prove that for each TISGM, there exists a nonempty class of boundary configurations such that the corresponding limiting Gibbs measure coincides exactly with that TISGM. Our results reveal how the dynamical system induced by the recursive operator determines domains of attraction of stable fixed points and clarify the mechanism through which microscopic boundary effects select macroscopic equilibrium states. This provides the first characterization of boundary conditions generating prescribed TISGMs for the three-state Potts-SOS model on a Cayley tree.