Cutoff for the Swendsen-wang dynamics on the lattice

We study the Swendsen-Wang dynamics for the q-state Potts model on the lattice. Introduced as an alternative algorithm of the classical singlesite Glauber dynamics, the Swendsen-Wang dynamics is a nonlocal Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work, we establish cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at sufficiently high temperatures, proving that it exhibits a sharp transition from "unmixed" to "well mixed." In particular, we show that at high enough temperatures the Swendsen-Wang dynamics on the torus (Z/nZ)^d has cutoff at time (-log(1-γ)^-1 log n, where γ (β) is the spectral gap of the infinitevolume dynamics.