Gibbs measure for mixed spins and mixed types model

TL;DR

This work analyzes a mixed $(2,q)$-Ising-Potts system on Cayley trees, formulating a Hamiltonian with Ising-Potts interactions and deriving a nonlinear recurrence that guarantees a splitting Gibbs measure. In the translation-invariant regime, the authors identify fixed-point structures on invariant sets, establish a phase-transition threshold via a closed-form expression for $\theta_c$, and prove the existence of multiple translation-invariant Gibbs measures, including at least 3 TIGMs for $k\ge2$ and at least 8 TIGMs for the $(2,3)$ case on the binary tree. The methodology hinges on a fixed-point operator $W$ acting on auxiliary variables $z_{i,j}$ and the Kolmogorov extension to construct the global Gibbs measures. These results extend classic Ising and Potts analyses to mixed-type interactions on trees and reveal robust phase-transition phenomena in the translation-invariant setting.

Abstract

In the present paper, we study the $(2,q)$-Ising-Potts model on the Cayley tree. We have derived a recurrence equation that shows the existence of a splitting Gibbs measure for this model. Furthermore, we have proven that for the $(2,q)$-Ising-Potts model on the Cayley tree of order $k\geq2$, there are at least 3 translation-invariant splitting Gibbs measures. We also prove that for the $(2,3)$-Ising-Potts model on the Cayley tree, specifically the binary tree, under certain conditions, there are at least 8 translation-invariant splitting Gibbs measures.

Figures (10)

• Figure 1: $a=1.5$, $x\in(0, 0.5)$
• Figure 2: $a=1.5$, $x\in(0.5, 3)$
• Figure 3: $a=3$, $x\in(0, 0.5)$
• Figure 4: $a=3$, $x\in(0.5, 3)$
• Figure 5: $a=5$, $x\in(0, 0.7)$

Theorems & Definitions (15)

• Remark 1
• Theorem 1
• proof
• Remark 2
• Theorem 2
• Theorem 3
• proof
• Remark 3
• Remark 4
• Remark 5