On Phase Transition and Percolation in the Beach Model

Abstract: The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certainpercolation event. The Potts model extends the Ising model tomore than two spin states, and we go on to study thecorresponding extension of the beach model. Usingrandom-cluster model methods we obtain some results on where inthe parameter space this model exhibits phase transition.Finally we study the beach model on regular trees. Criticalvalues are estimated with iterative numerical methods. Indifferent parameter regions we will see indications of bothfirst and second order phase transition.