A hard-core model on a Cayley tree: An example of a loss network

The paper is about a nearest-neighbor hard-core model, with fugacity λ > 0, on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on 'splitting' Gibbs measures generating a Markov chain along each path on the tree. In this model, ∀λ > 0 and k ≥ 1, there exists a unique translation-invariant splitting Gibbs measure μ*. Define λ_c = 1/(k - 1) × (k/(k-1)^k. Then: (i) for λ ≥ λ_c, the Gibbs measure is unique (and coincides with the above measure μ*), (ii) for λ > λ_c, in addition to μ*, there exist two distinct translation-periodic measures, μ₊ and μ_-, taken to each other by the unit space shift. Measures μ₊ and μ_- are extreme ∀λ > λ_c. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For λ > 1/(√k - 1) × (√k/√k - 1))^k, measure μ* is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λ_e and λ_o, for even and odd sites. We discuss open problems and state several related conjectures.