Splitting Gibbs measures for a periodic triple mixed-spin Ising model on a regular tree

We consider an Ising model on the regular tree of arbitrary order k ≥ 1 with three spin species ( 1 / 2,1,3 / 2 ) distributed periodically with period three along the generations. Since the local spin space depends on the generation modulo three, the model is not invariant under all tree automorphisms; throughout the paper, translation invariance means invariance of the boundary law on each of the three periodic sublattices, or equivalently invariance under color-preserving automorphisms of the rooted Cayley tree. Within the framework of splitting Gibbs measures, we derive the exact boundary-law compatibility equations and characterize the corresponding period-three translation-invariant splitting Gibbs measures by a finite system of algebraic relations. In the ferromagnetic regime J > 0 this system reduces to a scalar fixed-point equation x = f ( x , θ , k ) , where θ = exp ( β J / 2 ) ; the condition s k ( θ ) = f ′ ( 1 , θ , k ) − 1 > 0 gives at least three distinct such Gibbs measures. For the binary tree we identify the plus and minus weak limits with the extremal ordered fixed points and use a period-three Kesten–Stigum martingale argument for the effective three-step transfer operator to obtain a sufficient non-extremality region for the disordered phase. To address the thermodynamic interpretation, we also provide numerical results for the critical temperature, spontaneous ferromagnetic and ferrimagnetic magnetizations, sublattice magnetizations, and magnetic susceptibility.

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