On Four State Hard Core Models on the Cayley Tree

D. Gandolfo, U.A. Rozikov, J. Ruiz

2016, v.22, №2, 359-378

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)

Keywords: Cayley tree, hard core interaction, Gibbs measures, splitting measures