Rapid mixing of Swendsen-Wang dynamics in two dimensions

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.Comment: Ph.D. thesis, 66 page

Rapid mixing of Swendsen-Wang and single-bond dynamics in two dimensions

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model on arbitrary graphs with m edges is bounded above by 16 m log m times the spectral gap of the single-bond (or heat-bath) dynamics. This and the corresponding lower bound imply that rapid mixing of these two dynamics is equivalent. Using the known lower bound on the spectral gap of the Swendsen-Wang dynamics for the two dimensional square lattice $Z_L^2$ of side length L at high temperatures and a result for the single-bond dynamics on dual graphs, we obtain rapid mixing of both dynamics on $\Z_L^2$ at all non-critical temperatures. In particular this implies, as far as we know, the first proof of rapid mixing of a classical Markov chain for the Ising model on $\Z_L^2$ at all temperatures.Comment: 20 page