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 lambda > 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, For Alllambda > 0 and k greater than or equal to 1, there exists a unique translation-invariant splitting Gibbs measure mu*. Define lambda(c) = 1/(k - 1) x ( k/(k-1))(k). Then: (i) for lambda less than or equal to lambda(c), the Gibbs measure is unique (and coincides with the above measure mu*), (ii) for lambda > lambda(c), in addition to mu*, there exist two distinct translation-periodic measures, mu(+) and mu(-), taken to each other by the unit space shift. Measures mu(+) and mu(-) are extreme For Alllambda > lambda(c). We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For lambda > 1/(rootk - 1) x (rootk/rootk - 1))(k), measure mu* is not extreme (this result can be improved). Finally, we consider a model with two fugacities, lambda(e) and lambda(o), for even and odd sites. We discuss open problems and state several related conjectures.
cayley trees loss networks nearest-neighbor exclusion gibbs measures hard core translation invariant measures periodic measures splitting measures extreme gibbs measures