Dynamics for the Random Cluster Model

Abstract

The random cluster model is a statistical model on random graph structures. It can also be thought of as a unification of the percolation model, the Ising model and the q-state Potts models. To simulate such models, the state of the art procedure is given by the Markov chain Monte Carlo (MCMC) method. Here, Markov chains that converge to the probability distribution of the model are used to generate samples. In this thesis, we study the convergence speed of Markov chains for the random cluster model and related models on the planar rectangular grid. We review the methods that have led to optimal mixing results in the subcritical phase of the model, and show that in a certain sense they can be transferred to the supercritical phase. We introduce a new model closely related to the random cluster model and show that many results transfer to this case. For the Swendsen-Wang Markov chain in the supercritical phase, we introduce a coupling of Markov chains that shows optimal mixing behaviour. This is verified through extensive numerical simulations.