A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

AbstractThe 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 λc=1/(k−1)×(k/(k−1))k. Then: (i) for λ≤λc, the Gibbs measure is unique (and coincides with the above measure μ*), (ii) for λ>λc, 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>λc. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For $$\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $$ , measure μ*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λeand λo, for even and odd sites. We discuss open problems and state several related conjectures.