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

Tunneling behavior of Ising and Potts models in the low-temperature regime

We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature $\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$, in which this multi-spin system has $q$ stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the $q$-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as $\beta \to \infty$ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case $q=2$, our results characterize the tunneling behavior of the Ising model on grid graphs.Comment: 13 figure

Sampling from the low temperature Potts model through a Markov chain on flows

In this paper we consider the algorithmic problem of sampling from the Potts model and computing its partition function at low temperatures. Instead of directly working with spin configurations, we consider the equivalent problem of sampling flows. We show, using path coupling, that a simple and natural Markov chain on the set of flows is rapidly mixing. As a result we find a $\delta$-approximate sampling algorithm for the Potts model at low enough temperatures, whose running time is bounded by $O(m^2\log(m\delta^{-1}))$ for graphs $G$ with $m$ edges.Comment: Slightly revised version based on referee comments. No significant changes. Accepted in Random Structures and Algorithm