Tight bounds for mixing of the Swendsen-Wang algorithm at the Potts transition point
The authors study the mixing properties of the Swendsen-Wang and heat-bath stochastic algorithms for the \(q\)-state Potts model on \(\mathbb{Z}^{d}, d\geq 2\), providing upper and lower bounds for the variational mixing time of such processes when the system is confined in a box of size \(L\) with periodic boundary conditions. In particular, estimating the inverse gap of the generators of these Markov processes, they prove that, in both cases, the upper bound of the variational mixing time is exponential in \(\beta L^{d-1}\), for any positive \(\beta\). Providing an upper bound for the conductance of such Markov chains, for \(L\) and \(q\) sufficiently large, the authors prove the lower bound of the variational mixing time for the heat-bath dynamics to be also exponential in \(\beta L^{d-1}\), for any positive \(\beta\) larger than or equal to the critical inverse temperature of the model, while, in the case of Swendsen-Wang dynamics, they prove the same bound to hold at criticality. To compute such lower bounds, the authors make use of the FK representation to describe of the configuration space of the system in terms of contours and therefore of Pirogov-Sinai theory.
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