Analysis of Gibbs Measures and Stability of Dynamical System Linked to (1,1/2)-Mixed Ising Model on (m,k)-Ary Trees

This paper introduces a new (1,1/2) mixed spin Ising model (shortly, (1,1/2)-MSIM) having J1 and J2 competing interactions on (m, k)-ary trees. By constructing splitting Gibbs measures, we establish the presence of multiple Gibbs measures, which implies the occurrence of the phase transition for the (1, 1/2)-MSIM on the (m, k)-ary trees. Furthermore, the extremality of the two translation-invariant Gibbs measures is demonstrated for (1, k)-ary trees. Moreover, the extremality condition for the disordered phases is found and its non-extremality regimes are examined as well. It is well known that, to investigate lattice models on tree-like structures, conducting a stability analysis of the dynamical systems that represent the model and examining the behavior of the fixed points of these dynamical systems can provide significant insights into the model. We conducted a stability analysis around fixed points to investigate the behavior of the MSIM on the (m, k)-ary trees, also referred to as k-ary trees.