A new approach to Markov chain Monte Carlo simulation was recently proposed by Propp and Wilson. This approach, unlike traditional ones, yields samples which have exactly the desired distribution. The Propp-Wilson algorithm requires this distribution to have a certain structure called monotonicity. In this paper an idea of Kendall is applied to show how the algorithm can be extended to the case where monotonicity is replaced by anti-monotonicity. As illustrating examples, simulations of the hard-core model and the random-cluster model are presented.

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. Keywords. phase transition, symmetry breaking, Widom-Rowlinson model, hard-core model, critical points, Gibbs measures, Bethe lattice, monotonicity 1 Intr...

Let G be an infinite, locally finite, connected graph with bounded degree. We show that G supports phase transition in all or none of the following five models: bond percolation, site percolation, the Ising model, the Widom--Rowlinson model and the beach model. Some, but not all, of these implications hold without the bounded degree assumption. We finally give two examples of (random) unbounded degree graphs in which phase transition in all five models can be established: supercritical Galton--Watson trees, and Poisson--Voronoi tessellations of R d for d 2. Keywords: Percolation, Ising model, Widom--Rowlinson model, beach model, Galton--Watson tree, Poisson--Voronoi tessellation. AMS Subject Classification: Primary 60K35, Secondary 82B20, 82B43 1 Introduction Over the last few decades, it has become increasingly clear that there are important connections between percolation theory on one hand, and the issue of Gibbs state multiplicity in Markov random fields on the other. Example...

We construct a coupling of two distinct Gibbs measures for Markov random fields with the same specifications, such that the existence of an infinite path of disagreements between the two configurations has probability 0. This shows that the independence assumption in the disagreement percolation method for proving Gibbsian uniqueness, cannot be dropped without being replaced by other conditions. A similar counterexample is given for couplings of Markov chains.