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.

Fertile HC models with three states on a cayley tree

“…These problems were solved in [4] in the case where G = wrench. We consider the cases G = hinge, G = pipe, and G = wand.…”

Gibbs measures for the SOS model with four states on a Cayley tree

“…The simplicity of the Cayley tree allows describing a rather wide class of GMs for models without "good" symmetries (see [9] and [15]) and with competing interactions (see, e.g., [16]). …”

Gibbs measures for the fertile three-state hard-core models on a Cayley tree

“…There are four such models: wrench, wand, hinge, and pipe [2]. It is known (see [16,19,21]) that (i) for the wrench and pipe cases ∀λ > 0 and k ≥ 1, there exists a unique TISGM; (ii) for hinge (resp. wand) case at k = 2 if λ < λ cr = 9/4 (resp.…”