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 (1999) found that this dynamics may in fact exhibit slow mixing: they showed that, for the Potts model with $q\geq 3$ colors on the complete graph on $n$ vertices at the critical point $\beta_c(q)$, Swendsen-Wang dynamics has $t_{\mathrm{mix}}\geq \exp(c\sqrt n)$. The same lower bound was extended to the critical window $(\beta_s,\beta_S)$ around $\beta_c$ by Galanis et al. (2015), as well as to the corresponding mean-field FK model by Blanca and Sinclair (2015). In both cases, an upper bound of $t_{\mathrm{mix}} \leq \exp(c' n)$ was known. Here we show that the mixing time is truly exponential in $n$: namely, $t_{\mathrm{mix}} \geq \exp (cn)$ for Swendsen-Wang dynamics when $q\geq 3$ and $\beta\in(\beta_s,\beta_S)$, and the same bound holds for the related MCMC samplers for the mean-field FK model when $q>2$.