Nonmonotonic behavior in hard-core and Widom-Rowlinson models

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