Cluster Algorithms in Monte Carlo Simulations.

In this thesis, a special kind of Monte Carlo method--Swendsen and Wang's cluster algorithm--is discussed in detail. This algorithm is applied to 2-dimensional Ising, 3-state, and 4-state Potts models, 3-dimensional Ising model and mean-field Curie-Weiss Ising model. Numerical results are presented for the dynamic properties of the algorithm. It has been shown that the cluster algorithm leads to a significant reduction of critical slowing-down, compared to local algorithms, but not to its total elimination. Using a Rayleigh-Ritz variational argument, a rigorous lower bound on the dynamic critical exponents of the Swendsen -Wang and multi-level Swendsen-Wang algorithms is derived. This theoretical work is a valuable guide to further numerical experiments.