Markov random fields and percolation on general graphs

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 fi ve models: bond percolation, site percolation, the Ising model, the Widom-R owlinson model and the beach model. Some, but not all, of these implication s hold without the bounded degree assumption. We finally give two examples of (random) unbounded degree graphs in which phase transition in all five m odels can be established: supercritical Galton-Watson trees, and Poisson-Vo ronoi tessellations of R-d for d greater than or equal to 2.