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

ABSTRACT

We consider fertile three-state Hard-Core (HC) models 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 concept of alternative Gibbs measure is introduced and in the case ‘wand’ translational invariance conditions for alternative Gibbs measures are found. Also, we show the existence of alternative Gibbs measures which are not translation-invariant.

KEYWORDS:

• Cayley tree
• configuration
• Hard-Core model
• Gibbs measure
• translation-invariant measure
• Alternating Gibbs measure

PACS:

• Primary 82B05
• 82B20
• Secondary 60K35

Acknowledgments

We thank the referees for the careful reading of the manuscript and especially for a number of suggestions that have improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).