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 single-site 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 d2(.log(1..)).1logn, where .(.) is the spectral gap of the infinite-volume dynamics.