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 updatesof 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  .
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 the global nature of its updates. We present optimal bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree . Our bounds 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. ( 2003), imply \Omega(1) spectral gap and O(\log{n the Swendsen-Wang dynamics on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish \Theta(\log{n all boundary conditions throughout the tree uniqueness region; in fact, our bounds hold beyond the uniqueness threshold for the Ising model , and for the q-state Potts model when q is small with respect to d. Our proofs feature a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders and utilize recent work on block factorization of entropy under spatial mixing conditions .
R
with inverse temperature \beta>0. Due to
. This chain has proved difficult to analyze, due in part to
R
the updatesof 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
its updates. We present optimal
R
that
. Our bounds
R
. Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced
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
R
2004)implies constant spectral gap of
2003), imply \Omega(1) spectral gap and O(\log{n
R
. This implies that the relaxation time (i. e., the inverse of the spectral gap) is O(1) for
on the d-ary tree. We also show that these bounds are asymptotically optimal. As a consequence, we establish \Theta(\log{n
R
in the uniqueness region or when \beta<\beta_1 where \beta_1 exceeds
throughout the tree uniqueness region; in fact, our bounds hold beyond
A
,
R
In addition, we prove O(1) relaxation time for all \beta for the monochromatic boundary condition. Our proof introduces
Our proofs feature
A
and utilize recent work on block factorization of entropy under spatial mixing conditions