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 non-monotonicity via certain local modiications 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.