3-state hybrid Potts-SOS model with different coupling constants and its phase transition phenomenon

We studyP a hybrid Potts-SOS model with spin states {1, 2, 3} on a Cayley tree of order two, governed by a newly proposed Hamiltonian. The Gibbs measures are constructed through the Kolmogorov consistency condition, which guarantees the compatibility of finite-volume distributions and yields a unique measure for the infinite system. We derive recurrence equations for the partial partition functions and analyze their fixed points. A stability analysis is performed to identify phase transition conditions. The analysis provides an explicit characterization of Gibbs measures in terms of fixed-point solutions and determines critical parameter regimes for phase transitions. This work deepens the theoretical understanding of Gibbs measures and their relationship to fixed points in statistical mechanics and mathematical physics, particularly for tree-like lattices.