Phase transitions and global bifurcations in asymmetric q-state Potts models on arbitrary Cayley trees

Mukhamedov and Akın (2026) recently investigated the asymmetric three-state Potts model on a second-order Cayley tree using the Kolmogorov consistency condition and established a rigorous framework for the existence of Gibbs measures and phase transitions. In this study, we extend their approach to an asymmetric q-state Potts model defined on a Cayley tree of arbitrary order k, employing the cavity method. We derive recursive relations for local magnetizations and analyze their fixed-point structure, which characterizes the set of Gibbs measures in the system. In particular, we consider a setting in which the coupling constants differ between odd and even generations, leading to alternating interaction patterns that significantly influence macroscopic behavior. We determine the conditions for the existence of multiple limiting Gibbs measures and identify the critical parameter regimes associated with phase coexistence and symmetry breaking. In addition, we examine how the order of the tree and the degree of nonhomogeneity affect the nature of the phase transitions. To further support the analysis, we compute Lyapunov exponents for the associated reduced dynamical systems. In particular, for the case q = 3, we investigate the corresponding two-dimensional mapping and analyze its stability structure using the Oseledets theorem (Oseledets, 1968). These results broaden the theoretical understanding of spin systems on nonamenable structures and provide a methodological basis for studying related asymmetric models in statistical physics and in probability theory.