Efficient sampling and counting algorithms for the Potts model on $\mathbb Z^d$ at all temperatures

For $d \ge 2$ and all $q\geq q_{0}(d)$ we give an efficient algorithm to approximately sample from the $q$-state ferromagnetic Potts and random cluster models on the torus $(\mathbb Z / n \mathbb Z )^d$ for any inverse temperature $\beta\geq 0$. This stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly...More