Periodic Gibbs measures for three-state hard-core models in the case Wand

TL;DR

This paper investigates three-state hard-core models on Cayley trees, identifying critical activity values for non-uniqueness of two-periodic Gibbs measures and analyzing their extremality, with a focus on the wand model.

Contribution

It provides exact critical activity values for the wand model on Cayley trees and examines the extremality of two-periodic Gibbs measures, advancing understanding of phase transitions in these models.

Findings

Critical activity values for non-uniqueness of Gibbs measures identified.

Existence of multiple two-periodic Gibbs measures established.

Extremality of Gibbs measures analyzed for specific Cayley tree orders.

Abstract

We consider fertile three-state Hard-Core (HC) models with the activity parameter $\lambda>0$ on a Cayley tree. It is known that there exist four types of such models: wrench, wand, hinge, and pipe. These models arise as simple examples of loss networks with nearest-neighbor exclusion. In the case wand on a Cayley tree of order $k\geq2$, exact critical values $\lambda>0$ are found for which two-periodic Gibbs measures are not unique. Moreover, we study the extremality of the existing two-periodic Gibbs measures on a Cayley tree of order two.