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.

翻译：从$q$态铁磁Potts模型中进行采样是统计物理学、概率论和理论计算机科学中的一个基本问题。在一般图上，该问题在计算上是困难的，且这种困难性在任意低温下都成立。与此同时，近年来，大量进展表明在多种特定图族中存在低温采样算法。本文旨在理解一般图在低温下能够从$q$态铁磁Potts模型中进行多项式时间采样的最小结构性质。我们从广泛使用的Swendsen-Wang动力学及其密切相关的随机团簇动力学的角度研究该问题。我们的结果表明，这些动力学收敛速度的快或慢所依赖的关键图属性是图上的独立边渗流是否具有强超临界相。我们这里所指的“强超临界相”意味着：在较大的$p<1$时，存在唯一的线性规模巨连通分支，且该巨连通分支的补集仅由小分支构成。具体而言，我们证明，该条件蕴含Swendsen-Wang动力学和随机团簇动力学在两类有界度图族上的快速混合：(a) 体积增长至多为亚指数型的图，以及(b) 局部树状图。另一方面，我们表明，即使在这些图族中，若边渗流条件不成立，这些马尔可夫链在任意低温下都可能以指数速度缓慢收敛。在此过程中，我们发展了分析非局部马尔可夫链的新工具，包括一个在长程关联存在时限制分歧传播速度的框架，以及对带随机边界条件的树上空间混合性质的理解。