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

Swendsen-Wang is faster than single-bond dynamics

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model is larger than the spectral gap of a single-bond dynamics, that updates only a single edge per step. For this we give a representation of the algorithms on the joint (Potts/random-cluster) model. Furthermore we obtain upper and lower bounds on the mixing time of the single-bond dynamics on the discrete $d$-dimensional torus of side length $L$ at the Potts transition temperature for $q$ large enough that are exponential in $L^{d-1}$, complementing a result of Borgs, Chayes and Tetali.Comment: 17 page

The role of Frolov's cubature formula for functions with bounded mixed derivative

We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov $\mathbf{B}^s_{p,\theta}$ and Triebel-Lizorkin spaces $\mathbf{F}^s_{p,\theta}$ and our results treat the whole range of admissible parameters $(s\geq 1/p)$. In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel-Lizorkin spaces $\mathbf{F}^s_{p,\theta}$ in case $1<\theta<p<\infty$ with $1/p<s\leq 1/\theta$. The presented upper bounds on the worst-case error show a completely different behavior compared to "large" smoothness $s>1/\theta$. In the latter case the presented upper bounds are optimal, i.e., they can not be improved by any other cubature formula. The optimality for "small" smoothness is open.Comment: 23 page

Structure and eigenvalues of heat-bath Markov chains

We prove that heat-bath chains (which we define in a general setting) have no negative eigenvalues. Two applications of this result are presented: one to single-site heat-bath chains for spin systems and one to a heat-bath Markov chain for sampling contingency tables. Some implications of our main result for the analysis of the mixing time of heat-bath Markov chains are discussed. We also prove an alternative characterisation of heat-bath chains, and consider possible generalisations.Comment: 15 pages. Minor edits to address referee's comment

On Weak Tractability of the Clenshaw-Curtis Smolyak Algorithm

We consider the problem of integration of d-variate analytic functions defined on the unit cube with directional derivatives of all orders bounded by 1. We prove that the Clenshaw Curtis Smolyak algorithm leads to weak tractability of the problem. This seems to be the first positive tractability result for the Smolyak algorithm for a normalized and unweighted problem. The space of integrands is not a tensor product space and therefore we have to develop a different proof technique. We use the polynomial exactness of the algorithm as well as an explicit bound on the operator norm of the algorithm.Comment: 18 page