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Phase Transitions, Non-Extremality (Reconstruction), and Markov Entropy Rate for the Mixed Spin-$(s,\tfrac12)$ Ising Model on a Cayley Tree of Order Three

Hasan Akin

Abstract

We investigate the mixed spin-$(s,\tfrac12)$ Ising model on a Cayley tree of order three ($k=3$), extending the approach of \cite{Akin2024}. For the representative case $s=5$, the associated recursion leads to an 11-dimensional dynamical system, and phase-transition regions are examined via the local stability of the disordered (symmetric) fixed point, detected through the condition $|λ_{\max}|\ge 1$ for the Jacobian matrix. To study extremality (non-reconstruction) of the disordered phase, we represent translation-invariant splitting Gibbs measures by tree-indexed Markov chains and compute the relevant Dobrushin coefficients. At the symmetric fixed point we obtain explicit transition kernels and the induced two-step kernel on the spin-$\tfrac12$ layer; its second eigenvalue $λ_2$ yields a spectral reconstruction test consistent with the Kesten--Stigum condition $3|λ_2|^2>1$. In addition, we introduce the Markov entropy rate as a computable thermodynamic/information-theoretic observable and derive closed-form expressions for this entropy rate at the symmetric fixed point for an arbitrary spin $s$. Numerical illustrations for $s=1,2,\ldots,5$ are provided to compare the entropy-rate behavior with the spectral criteria. Our results connect naturally to themes in information theory and statistical physics and are also relevant to reconstruction problems on trees that appear in biology/phylogenetics.

Phase Transitions, Non-Extremality (Reconstruction), and Markov Entropy Rate for the Mixed Spin-$(s,\tfrac12)$ Ising Model on a Cayley Tree of Order Three

Abstract

We investigate the mixed spin- Ising model on a Cayley tree of order three (), extending the approach of \cite{Akin2024}. For the representative case , the associated recursion leads to an 11-dimensional dynamical system, and phase-transition regions are examined via the local stability of the disordered (symmetric) fixed point, detected through the condition for the Jacobian matrix. To study extremality (non-reconstruction) of the disordered phase, we represent translation-invariant splitting Gibbs measures by tree-indexed Markov chains and compute the relevant Dobrushin coefficients. At the symmetric fixed point we obtain explicit transition kernels and the induced two-step kernel on the spin- layer; its second eigenvalue yields a spectral reconstruction test consistent with the Kesten--Stigum condition . In addition, we introduce the Markov entropy rate as a computable thermodynamic/information-theoretic observable and derive closed-form expressions for this entropy rate at the symmetric fixed point for an arbitrary spin . Numerical illustrations for are provided to compare the entropy-rate behavior with the spectral criteria. Our results connect naturally to themes in information theory and statistical physics and are also relevant to reconstruction problems on trees that appear in biology/phylogenetics.
Paper Structure (53 sections, 12 theorems, 262 equations, 16 figures, 3 tables, 2 algorithms)

This paper contains 53 sections, 12 theorems, 262 equations, 16 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed-spin configurations and finite-volume Gibbs measures
  4. Compatibility and splitting Gibbs measures
  5. Construction of the TISGMs for arbitrary k
  6. Construction and Stability Analysis of the 11-Dimensional Dynamical System for the Mixed Spin-(5,1/2) Ising Model on a CT of Order 3
  7. Stability Analysis via the Jacobian Matrix
  8. Stability of the one-dimensional dynamical system
  9. Consistency with the Jacobian analysis
  10. Numerical results
  11. Stability analysis of other fixed points
  12. Extremality and Non-Extremality of the Disordered Phase in the MSIM via Tree-Indexed Markov Chains
  13. Construction of the stochastic matrices corresponding to the fixed point $\ell_0^{(5)}$ for $k=3$ and $s=5$
  14. The extremality criterion (non-reconstruction) for a TISGM
  15. Extremality of the disordered phase associated with $(2,1/2)$-MSIM on the CT of order three
  16. ...and 38 more sections

Key Result

Theorem 1

Let $n = 1,2,\ldots$ and $\varphi = e^{\frac{\beta J}{2}}$. The family of finite-volume Gibbs measures $\{\mu_n^{\pmb{h}}\}$ defined by Gibbs1 is compatible (i.e., satisfies compatile1) if and only if, for every $x \in V$, the following conditions are satisfied: (i) For $i \in \{-s, -(s-1), \ldots, (ii) For $i \in \{1, 2, \ldots, s-1, s\}$, (iii) For the spin-$\tfrac{1}{2}$ component,

Figures (16)

  • Figure 1: (Color online) Plots of the one-dimensional map $F(Z)$ (solid curve) and the diagonal $y = Z$ (dashed line) at the two critical parameter values: $\varphi = 0.901258081777163$ (left) and $\varphi = 1.10956009185308$ (right). At these values, $|F'(1)| = 1$, indicating a change in the stability of the fixed point $Z = 1$. For $0 < \varphi < 0.901258081777163$ and $\varphi > 1.10956009185308$, the fixed point is repelling; for $0.901258081777163 < \varphi < 1.10956009185308$, it is attracting.
  • Figure 2: (Color online) Intersection points between $\lambda_{\max}(\varphi)$ and the line $y = 1$. In the region where $\lambda_{\max}(\varphi) > 1$, the fixed point $Z = 1$ is repelling, corresponding to the phase transition region of the model.
  • Figure 3: (Color online) Phase diagram showing the stable fixed points of the one-dimensional rational map defined in \ref{['eq:1D-Stability1a']} as a function of $\varphi$. The repelling regime of the $Z = 1$ fixed point (shaded/darker region) aligns with the phase transition boundaries identified in the previous figures.
  • Figure 4: (Color online) Fixed points of the rational function defined in \ref{['Zi_eq2']} for $\varphi = 1.12434$. One of the fixed points is approximately $Z = 4$, and an analysis of its stability shows that this fixed point is attractive.
  • Figure 5: (Left) A three-branch Cayley tree growing upward from the root. (Right) A three-branch Cayley tree of depth three: the root has three children (level 1), each of which has three children (level 2, 9 nodes in total), and each of those has three children (level 3, 27 nodes in total).
  • ...and 11 more figures

Theorems & Definitions (20)

  • Theorem 1: Akin2024
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Remark 4.1
  • Definition 7: KuelskeRozikov2017
  • Lemma 8
  • ...and 10 more