The Swendsen–Wang dynamics on trees

The Swendsen–Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution ferromagnetic Ising and Potts models. This has proved difficult to analyze, due in part its global nature. We present optimal bounds on convergence rate of complete d $$ -ary tree. Our extend non-uniqueness region apply all boundary conditions. show that spatial mixing conditions known as variance entropy imply Ω ( 1 ) \Omega (1) spectral gap O log n O\left(\log n\right) time, respectively, dynamics also these are asymptotically optimal. As consequence, we establish Θ \Theta \left(\log throughout (and beyond) tree uniqueness region. proofs feature novel view condition utilize recent work block factorization entropy.