The Swendsen-Wang Dynamics on Trees

The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models with inverse temperature $\beta>0$. Due to the global nature of the updates of the Swendsen-Wang algorithm, the underlying Markov chain often mixes rapidly in the low temperature region (large $\beta$) where long-range correlations impair the convergence rate of local chains such as the Glauber dynamics. With few exceptions, tight bounds on the convergence rate of the Swendsen-Wang algorithm only hold for high temperature regions (small $\beta$) corresponding to the uniqueness or the decay of correlations region. We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete $d$-ary tree that extend to the non-uniqueness region. Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. (2004) implies constant spectral gap of the Swendsen-Wang dynamics. This implies that the relaxation time (i.e., the inverse of the spectral gap) is $O(1)$ for all boundary conditions in the uniqueness region or when $\beta<\beta_1$ where $\beta_1$ exceeds the uniqueness threshold for the Ising model and for the $q$-state Potts model when $q$ is small with respect to $d$. In addition, we prove $O(1)$ relaxation time for all $\beta$ for the monochromatic boundary condition. Our proof introduces a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders.