Exponentially slow mixing in the mean-field Swendsen-Wang dynamics

Swendsen-Wang dynamics for the Potts model was proposed in the late 1980's as an alternative to single-site heat-bath dynamics, in which global updates allow this MCMC sampler to switch between metastable states and ideally mix faster. Gore and Jerrum (J. Stat. Phys. 97 (1999) 67-86) found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with q >= 3 colors on the complete graph on n vertices at the critical point beta(c) (q), Swendsen-Wang dynamics has t(mix) >= exp(c root n). Galanis et al. (In Proc. of the 19th International Workshop on Randomization and Computation (RANDOM 2015) (2015) 815-828) showed that t(mix) >= exp(cn(1/3)) throughout the critical window (beta(s) , beta(s)) around beta(c) , and Blanca and Sinclair (In Proc. of the 19th International Workshop on Randomization and Computation (RANDOM 2015) (2015) 528-543) established that t(mix) >= exp(c root n) in the critical window for the corresponding mean-field FK model, which implied the same bound for Swendsen-Wang via known comparison estimates. In both cases, an upper bound of t(mix )<= exp(c'n) was known. Here we show that the mixing time is truly exponential in n: namely, t(mix) >= exp(cn) for Swendsen-Wang dynamics when q >= 3 and beta is an element of (beta(s) , beta(s)), and the same bound holds for the related MCMC samplers for the mean-field FK model when q > 2.