Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point

Abstract

We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice \({\mathbb{Z}^{d}}\)—heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L d-1. This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)2.