Nonmonotonic behavior in hard-core and Widom-Rowlinson models

We give two examples of nonmonotonic behavior in symmetric systems exhibiting more than one critical point at which spontaneous symmetry breaking appears or disappears. The two systems are the hard-core model and the Widom-Rowlinson model, and both examples take place on a variation of the Cayley tree (Bethe lattice) devised by Schonmann and Tanaka. We obtain similar, though less constructive, examples of nonmonotonicity via certain local modifications of any graph, e.g., the square lattice, which is known to have a critical point for either model. En route we discuss the critical behavior of the Widom-Rowlinson model on the ordinary Cayley tree. Some results about monotonicity of the phase transition phenomenon relative to graph structure are also given.