Hard-core and soft-core Widom-Rowlinson models on Cayley trees

We consider both hard-core and soft-core Widom-Rowlinson models with spin values on a Cayley tree of order and we are interested in the Gibbs measures of the models. The models depend on three parameters: the order k of the tree, describing the strength of the (ferromagnetic or antiferromagnetic) interaction, and describing the intensity for particles. The hard-core Widom-Rowlinson model corresponds to the case . For the binary tree k = 2, and for k = 3 we prove that the ferromagnetic model has either one or three splitting Gibbs measures (tree-automorphism invariant Gibbs measures (TISGMs) which are tree-indexed Markov chains). We also give the exact form of the corresponding critical curves in parameter space. For higher values of k we give an explicit sufficient bound ensuring non-uniqueness which we conjecture to be the exact curve. Moreover, for the antiferromagnetic model we explicitly give two critical curves , , and prove that on these curves there are exactly two TISGMs; between these curves there are exactly three TISGMs; otherwise there exists a unique TISGM. Some periodic and non-periodic SGMs are also constructed in the ferromagnetic model.