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 modifications of any graph, e.g. the square lattice, which is known to have a critical point for either model. En route we prove that the Widom-Rowlinson model does behave monotonically on the ordinary Cayley tree, and we obtain an explicit description of its critical behavior there. We conclude with some results about monotonicity of the phase transition phenomenon relative to graph structure.