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 . Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. ( 2003), imply \Omega(1) spectral gap and O(\log{n the Swendsen-Wang dynamics .