A Three State Hard-Core Model on a Cayley Tree

Abstract

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 σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ 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, ∀ λ > 0 and k ≥ 1, there exists a unique translationinvariant splitting Gibbs measure µ * .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.