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Abstract The Swendsen–Wang algorithm is a sophisticated, widely‐used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. This chain has proved difficult to analyze, due in part to its global nature. We present optimal bounds on the convergence rate of the Swendsen–Wang algorithm for the complete ‐ary tree. Our bounds extend to the non‐uniqueness region and apply to all boundary conditions. We show that the spatial mixing conditions known asvariance mixingandentropy mixingimply spectral gap and mixing time, respectively, for the Swendsen–Wang dynamics on the ‐ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish mixing for the Swendsen–Wang dynamics forallboundary conditions throughout (and beyond) the tree uniqueness region. Our proofs feature a novel spectral view of the variance mixing condition and utilize recent work on block factorization of entropy. 

Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool for sampling from high-dimensional distributions, a notable example is the equilibirum distribution of graphical models. The Glauber dynamics, also known as the Gibbs sampler, is the simplest example of an MCMC algorithm; the transitions of the chain update the configuration at a randomly chosen coordinate at each step. Several works have studied distributed versions of the Glauber dynamics and we extend these efforts to a more general family of Markov chains. An important combinatorial problem in the study of MCMC algorithms is random colorings. Given a graph G of maximum degree Δ and an integer k ≥ Δ+1, the goal is to generate a random proper vertex k-coloring of G. Jerrum (1995) proved that the Glauber dynamics has O(nlog{n}) mixing time when k > 2Δ. Fischer and Ghaffari (2018), and independently Feng, Hayes, and Yin (2018), presented a parallel and distributed version of the Glauber dynamics which converges in O(log{n}) rounds for k > (2+ε)Δ for any ε > 0. We improve this result to k > (11/6-δ)Δ for a fixed δ > 0. This matches the state of the art for randomly sampling colorings of general graphs in the sequential setting. Whereas previous works focused on distributed variants of the Glauber dynamics, our work presents a parallel and distributed version of the more general flip dynamics presented by Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle (2019)), which recolors local maximal two-colored components in each step.