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Gibbs measure for the HC-Blume-Capel model in the case of a "wand" type graph on a Cayley tree

Nosirjon M. Khatamov, Malika A. Kodirova

Abstract

In this paper, we investigate translation-invariant splitting Gibbs measures (TISGMs) for the HC-Blume-Capel model on a "wand" graph embedded in the Cayley tree of arbitrary order $k \geq 2$. It is known that there is the exact critical value $θ_{cr}$ such that for $θ\geq θ_{cr}$, there exists a unique TISGM, whereas for $θ< θ_{cr}$, precisely three TISGMs exist in the case of a "wand" graph for the given model. In the present paper, we completely solve the (non-)extremality problem for one of these measures for any order $k$ of Cayley tree

Gibbs measure for the HC-Blume-Capel model in the case of a "wand" type graph on a Cayley tree

Abstract

In this paper, we investigate translation-invariant splitting Gibbs measures (TISGMs) for the HC-Blume-Capel model on a "wand" graph embedded in the Cayley tree of arbitrary order . It is known that there is the exact critical value such that for , there exists a unique TISGM, whereas for , precisely three TISGMs exist in the case of a "wand" graph for the given model. In the present paper, we completely solve the (non-)extremality problem for one of these measures for any order of Cayley tree

Paper Structure

This paper contains 8 sections, 10 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conditions for Non-Extremality of the Measure $\mu_{0}$
  4. Conditions for the Extremality of the Measure $\mu_0$ for $k=3$
  5. Conclusion
  6. Data Availability Statement
  7. Conflicts of Interest
  8. Funding

Key Result

Theorem 1

27 Let $k \geq 2$. The probability distribution $\mu^{(n)}, n=1,2,\dots$ in f2 is consistent if and only if, for any $x \in V$, the following conditions hold: where $\theta = \exp(-J \beta), \beta = 1/T$, and $z_{i,x} = \exp(h_{i,x} - h_{0,x})$ for $i=-1,1$.

Theorems & Definitions (25)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Theorem 3
  • Remark 2
  • Theorem 4
  • proof
  • ...and 15 more