Mario Ullrich

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.

The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative

We prove upper bounds on the order of convergence of Frolovs 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

Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube

\mathbf{B}^s_{p,\theta}

Comparison of Swendsen‐Wang and heat‐bath dynamics

and Triebel--Lizorkin spaces

A Monte Carlo Method for Integration of Multivariate Smooth Functions

\mathbf{F}^s_{p,2}

On weak tractability of the Clenshaw–Curtis Smolyak algorithm

and our results treat the whole range of admissible parameters

Complexity of oscillatory integration for univariate Sobolev spaces

(s\geq 1/p)

A lower bound for the dispersion on the torus

. In particular, we obtain upper bounds for the difficult the case of small smoothness which is given for Triebel--Lizorkin spaces

On “Upper Error Bounds for Quadrature Formulas on Function Classes” by K.K. Frolov

\mathbf{F}^s_{p,2}

An upper bound on the minimal dispersion

in case