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 $-1,0,1$ on a Cayley tree of order $k\geq 2$ and we are interested in the Gibbs measures of the models. The models depend on 3 parameters: the order $k$ of the tree, $\theta$ describing the strength of the (ferromagnetic or antiferromagnetic) interaction, and $\lambda$ describing the intensity for particles. The Hard-Core Widom-Rowlinson model corresponds to the case $\theta=0$. 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 (TISGM) 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. Also some periodic and non-periodic SGMs are constructed in the ferromagnetic model.