Markov random fields and percolation on general graphs

LetGbe an infinite, locally finite, connected graph with bounded degree. We show thatGsupports 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 ℝdford≥ 2.