On the tractability of sampling from the Potts model at low temperatures via Swendsen--Wang dynamics

Sampling from the $q$-state ferromagnetic Potts model is a fundamental question in statistical physics, probability theory, and theoretical computer science. On general graphs, this problem is computationally hard, and this hardness holds at arbitrarily low temperatures. At the same time, in recent years, there has been significant progress showing the existence of low-temperature sampling algorithms in various specific families of graphs. Our aim in this paper is to understand the minimal structural properties of general graphs that enable polynomial-time sampling from the $q$-state ferromagnetic Potts model at low temperatures. We study this problem from the perspective of the widely-used Swendsen--Wang dynamics and the closely related random-cluster dynamics. Our results demonstrate that the key graph property behind fast or slow convergence time for these dynamics is whether the independent edge-percolation on the graph admits a strongly supercritical phase. By this, we mean that at large $p<1$, it has a unique giant component of linear size, and the complement of that giant component is comprised of only small components. Specifically, we prove that such a condition implies fast mixing of the Swendsen--Wang and random-cluster dynamics on two general families of bounded-degree graphs: (a) graphs of at most stretched-exponential volume growth and (b) locally treelike graphs. In the other direction, we show that, even among graphs in those families, these Markov chains can converge exponentially slowly at arbitrarily low temperatures if the edge-percolation condition does not hold. In the process, we develop new tools for the analysis of non-local Markov chains, including a framework to bound the speed of disagreement propagation in the presence of long-range correlations, and an understanding of spatial mixing properties on trees with random boundary conditions.