On four state Hard Core Models on the Cayley Tree

TL;DR

This paper investigates a four-state hard-core model on a Cayley tree, analyzing conditions for unique Gibbs measures and characterizing phase transitions for specific graph configurations.

Contribution

It provides new conditions for uniqueness of Gibbs measures and characterizes phase transition lines for various graph structures in the four-state hard-core model.

Findings

Conditions for unique Gibbs measure identified

Transition lines between regimes derived for specific graphs

Analysis of translation-invariant and periodic measures

Abstract

We consider a nearest-neighbor four state hard-core (HC) model on the homogeneous Cayley tree of order $k$. The Hamiltonian of the model is considered on a set of "admissible" configurations. Admissibility is specified through a graph with four vertices. We first exhibit conditions (on the graph and on the parameters) under which the model has a unique Gibbs measure. Next we turn to some specific cases. Namely, first we study, in the case of a particular graph (the diamond), translation-invariant and periodic Gibbs measures. We provide in both cases the equations of the transition lines separating uniqueness from non--uniqueness regimes. Finally the same is done for "fertile" graphs, the so--called stick, gun, and key (here only translation invariant states are taken into account).