A three state hard-core model on a Cayley tree

We consider a nearest-neighbor hard-core model, with three states, on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state sigma(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate lambda at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on 'splitting' Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, for all lambda > 0 and k >= 1, there exists a unique translation-invariant splitting Gibbs measure mu(*). We also study periodic splitting Gibbs measures and show that the above model admits only translation-invariant and periodic with period two (chess-board) Gibbs measures. We discuss some open problems and state several related conjectures.