Splitting Gibbs Measures for a Periodic Triple Mixed-Spin Ising Model on a Cayley Tree

Abstract

We consider an Ising model on the Cayley tree $\Gamma_k$ of arbitrary order $k\ge1$ with three spin species of values $(\tfrac12,1,\tfrac32)$ distributed deterministically with period three along the generations. Within the framework of splitting Gibbs measures, we derive the exact boundary-law compatibility equations and characterize translation-invariant splitting Gibbs measures (TISGMs) via a finite system of algebraic relations. In the ferromagnetic regime $J>0$, writing $\theta=\exp(\beta J/2)$, we further reduce the translation-invariant problem to a one-dimensional scalar fixed-point equation $x=f(x,\theta,k)$ for a rational map $f$. We show that $f$ is strictly increasing and obtain an explicit sufficient condition for phase coexistence: if $s_k(\theta)=f'(1,\theta,k)-1>0$, then $x=f(x,\theta,k)$ admits at least three distinct positive solutions, yielding at least three distinct TISGMs and hence a phase transition driven by the periodic inhomogeneity of the spin structure. For the binary tree $k=2$ we exploit attractiveness to construct plus and minus Gibbs measures as weak limits with extremal boundary conditions, prove that they are TISGMs corresponding to the minimal and maximal fixed points of $f(\cdot,\theta,2)$, and show that they are the minimal and maximal Gibbs measures in the natural stochastic order. Finally, we construct the tree-indexed Markov chain associated with a TISGM and apply the Kesten--Stigum criterion to the disordered TISGM, identifying nonempty parameter regions where this measure is non-extremal and reconstruction occurs.