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 β_c(q), Swendsen. Wang dynamics has tMIX ≥ exp(c √ 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 (β_s,β_S) around β_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 √ 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 β ∈ (β_s,β_S), and the same bound holds for the related MCMC samplers for the mean-field FK model when q > 2.