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

TL;DR

This work analyzes a triple mixed-spin ($\left(\tfrac{1}{2},1,\tfrac{3}{2}\right)$) Ising model on a Cayley tree with period-3 spin arrangement, using splitting Gibbs measures to derive boundary-law compatibility and reduce translation-invariant cases to a scalar fixed-point map $x=f(x,\theta,k)$ with $\theta=\exp(\beta J/2)$. It proves that $s_k(\theta)=f'(1,\theta,k)-1>0$ yields at least three positive fixed points, hence at least three TISGMs and a phase transition driven by inhomogeneity; for $k=2$ plus and minus TISGMs are extremal and the disordered phase can be non-extremal under a Kesten–Stigum (KS) criterion. The KS analysis constructs a 3-step channel $PQR$ yielding an effective $H$ with second eigenvalue $\lambda_{0,k}(\theta)$ and non-extremality when $g_k(\theta)=k^{3}\lambda_{0,k}(\theta)^{2}-1>0$, with regions expanding in $\theta$ as $k$ grows. The paper also provides thermodynamic expressions in terms of boundary laws and discusses open problems for higher $k$ and more general periodic patterns. Overall, it connects deterministic inhomogeneity to rich phase structure and reconstruction phenomena on tree-like graphs.

Abstract

We consider an Ising model on the Cayley tree $Γ_k$ 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 $θ=\exp(βJ/2)$, we further reduce the translation-invariant problem to a one-dimensional scalar fixed-point equation $x=f(x,θ,k)$ for a rational map $f$. We show that $f$ is strictly increasing and obtain an explicit sufficient condition for phase coexistence: if $s_k(θ)=f'(1,θ,k)-1>0$, then $x=f(x,θ,k)$ 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 $f(\cdot,θ,2)$, 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.

Figures (5)

• Figure 1: Some generations of a second-order Cayley tree with three different spin values assigned to alternating levels.
• Figure 2: Plots of $s_2(\theta)$ for $\theta \in (0,2)$ (left) and $s_2(1/\theta)$ for $\theta \in (0,1)$ (right). The change of variables $\theta \mapsto 1/\theta$ allows us to visualize and confirm the monotonicity of $s_2(\theta)$ on $(1,\infty)$.
• Figure 3: Plot of $s_3(\theta)=f'(1,\theta,3)-1$. The region where $s_3(\theta)>0$ corresponds to the existence of at least three translation-invariant splitting Gibbs measures.
• Figure 4: Graphs of $s_5(\theta)$ for $\theta \in (0,1.5)$ (left) and $s_5(1/\theta)$ for $\theta \in (0,1)$ (right).
• Figure 5: The plots of $g_k(\theta)$ for $\theta \in (0,5)$ and $k=2,3,4,5$ are shown. It is observed that the interval of $\theta$ for which $g_k(\theta)>0$ increases as $k$ becomes larger.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Lemma 4
• proof
• Remark 1
• Theorem 5
• proof
• Proposition 6