Hidden Potts-Observed SOS Models on a Two-Layer Structure

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.

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