Cluster Algorithms for the Ising Model

Abstract

We describe some algorithms that output random spin configurations of the Ising model whose distribution is close to (or exactly) the Gibbs distribution using Markov chains. Therefore we first introduce the Ising model and the related random cluster model and show that sampling on both state spaces is equivalent. Nevertheless simulation on the random cluster model seems to be easier, because since the Gibbs distribution at low temperature is multimodal, the random cluster distribution is (almost) unimodal. So we hope that simulation on the random cluster model is tractable, in contrast to simulation of the Ising model. Next we describe some algorithms on both models algorithmically and show that they have the right stationary distribution. Additionally we introduce two methods that allow us to generate exactly distributed states. The methods are the Propp-Wilson (or Coupling-From-The-Past) algorithm and the Bounding chain technique. Then we introduce techniques to evaluate integrals on the Ising model. As a consequence of the introduced algorithms we will see that we could make our estimators unbiased and so we save a lot of computing time. Finally we show some results of simulations that were done with the self-made Matlab programs that are available on my homepage.