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Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree

Rustamjon Khakimov, Muhtorjon Makhammadaliev, Kamola Umrzakova

TL;DR

The paper investigates fertile three-state Hard-Core models on the Cayley tree, focusing on the wand graph and introducing alternative Gibbs measures (AGMs) in addition to translation-invariant Gibbs measures (TIGMs). It derives TI consistency equations and identifies a TI phase transition with a critical value $\lambda_{cr}^{(1)}=\frac{2^{k}}{(k-1)k^{k}}$ for wand, revealing multiple TIGMs beyond the threshold. By imposing nonuniform boundary data, it constructs AGMs via a 4-variable boundary-map on invariant sets, proving explicit non-TI AGM existence in many parameter regimes and detailing several $\lambda$-thresholds for various configurations. The work also relates AGMs to known periodic and weakly periodic Gibbs measures, showing that TI and periodic measures are encompassed within the AGM framework. Overall, it advances understanding of the full Gibbs-measure landscape for multi-state HC models on trees and demonstrates rich non-TI behavior arising from alternative boundary data.

Abstract

We consider fertile three-state Hard-Core (HC) models with the activity parameter $λ>0$ on a Cayley tree. It is known that there exist four types of such models: "wrench"\,, "wand"\,, "hinge"\, and "pipe"\,. In cases "wand"\, and "hinge"\ on a Cayley tree of arbitrary order a complete description of translation-invariant Gibbs measures is obtained. The conception of alternative Gibbs measure is introduced and in the case "wand"\, translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant

Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree

TL;DR

The paper investigates fertile three-state Hard-Core models on the Cayley tree, focusing on the wand graph and introducing alternative Gibbs measures (AGMs) in addition to translation-invariant Gibbs measures (TIGMs). It derives TI consistency equations and identifies a TI phase transition with a critical value for wand, revealing multiple TIGMs beyond the threshold. By imposing nonuniform boundary data, it constructs AGMs via a 4-variable boundary-map on invariant sets, proving explicit non-TI AGM existence in many parameter regimes and detailing several -thresholds for various configurations. The work also relates AGMs to known periodic and weakly periodic Gibbs measures, showing that TI and periodic measures are encompassed within the AGM framework. Overall, it advances understanding of the full Gibbs-measure landscape for multi-state HC models on trees and demonstrates rich non-TI behavior arising from alternative boundary data.

Abstract

We consider fertile three-state Hard-Core (HC) models with the activity parameter on a Cayley tree. It is known that there exist four types of such models: "wrench"\,, "wand"\,, "hinge"\, and "pipe"\,. In cases "wand"\, and "hinge"\ on a Cayley tree of arbitrary order a complete description of translation-invariant Gibbs measures is obtained. The conception of alternative Gibbs measure is introduced and in the case "wand"\, translational invariance conditions for alternative Gibbs measures are found. Also, we show that the existence of alternative Gibbs measures which are not translation-invariant
Paper Structure (8 sections, 16 theorems, 88 equations, 5 figures)

This paper contains 8 sections, 16 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Translation-invariant Gibbs measures
  3. Alternative Gibbs measures in the case $G=\textit{Wand}$
  4. Alternative Gibbs measures on $I_3$
  5. Alternative Gibbs measures on $I_4$
  6. Relation of the Alternative Gibbs measures to known ones
  7. data availability statement
  8. conflicts of interest

Key Result

Theorem 1

Ro Probability measures $\mu^{(n)}$, $n=1,2,\ldots$, given by the formula (e1), are consistent if and only if for any $x\in V$ the following equation holds: where $z'_{i,x}=\lambda z_{i,x}/z_{0,x}, \ \ i=1,2$.

Figures (5)

  • Figure 1: Fertile graphs $G=\textit{wand}$ (on the left) and $G=\textit{hinge}$ (on the right).
  • Figure 2: Graph of the function $\lambda(t)$ for $k=4$.
  • Figure 3: In this figure the values of function $z_x$ on the vertices of the Cayley tree of order 5 are shown in the case $m=3$ and $r=1$.
  • Figure 4: Graph of the function $\lambda_3(z)$.
  • Figure 5: In this figure the values of function $z_x$ on the vertices of the Cayley tree of order 4 are shown.

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Definition 4
  • Definition 5
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 28 more