Before revision:  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 .
After revision:  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 and apply to all boundary conditions. We show that the spatial mixing conditions known as Variance Mixing and Entropy Mixing, introduced in the study of local Markov chains by Martinelli et al. ( 2004)implies constant spectral gap of the Swendsen-Wang dynamics .