Splitting Gibbs Measures for a Periodic Triple Mixed-Spin Ising Model on a Cayley Tree
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866912901858590720 |
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| author | Mukhamedov, Farrukh Rahmatullaev, Muzaffar Karshiboev, Obid |
| author_facet | Mukhamedov, Farrukh Rahmatullaev, Muzaffar Karshiboev, Obid |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12369 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Splitting Gibbs Measures for a Periodic Triple Mixed-Spin Ising Model on a Cayley Tree Mukhamedov, Farrukh Rahmatullaev, Muzaffar Karshiboev, Obid Probability Mathematical Physics 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. |
| title | Splitting Gibbs Measures for a Periodic Triple Mixed-Spin Ising Model on a Cayley Tree |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2602.12369 |