Exact solution for the three-state asymmetric Potts model on a Cayley tree

In this study, we investigate the Gibbs measures of an asymmetric Potts model on a Cayley tree with asymmetric branching couplings (J1≠J2), revealing new critical phenomena and phase transitions that extend beyond the classical, symmetric case. The Gibbs distributions are derived based on the Kolmogorov consistency condition, and the recursive equations governing the system are formulated. We analyze the solutions of these recurrence relations by identifying their fixed points and exploring their behavior under specific symmetry assumptions. We analyze the extremality conditions of the disordered phases corresponding to translation-invariant Gibbs measures using the eigenvalues of the stochastic matrices constructed using the tree-indexed Markov chain approach. Furthermore, we compute the free energy of the model as a function of the compatible boundary conditions (CBC) and derive the associated thermodynamic quantities through its derivatives. The convexity properties of the system are examined in the configuration space {1,2,3}Vn, providing further insights into the structure of the Gibbs measures. Finally, to study the dynamical features of the model — such as chaoticity, periodicity, and bifurcations — we calculate the Lyapunov exponent, offering a deeper understanding of the system’s complex behavior. Furthermore, the dynamical evolution toward the thermodynamic limit is characterized through a 2D stability analysis based on the Oseledets theorem. By employing trajectory embedding, we identify global basin restructuring at the phase transition thresholds and confirm the structural stability of the model, rigorously proving the absence of chaotic attractors within the physical parameter space.