On the tractability of sampling from the Potts model at low temperatures via random-cluster dynamics

TL;DR

The paper investigates sampling from the $q$-state ferromagnetic Potts model at low temperature on general graphs via Edwards–Sokal random-cluster dynamics, identifying a geometric-percolation criterion—the existence of a strongly supercritical edge-percolation phase—that dictates fast mixing. It proves fast mixing for two broad graph families: (i) graphs with at most stretched-exponential growth and (ii) locally treelike graphs, provided the percolation condition holds; the results hinge on a novel low-temperature disagreement percolation framework and burn-in with boundary-wiring techniques. The authors also construct explicit graphs where SW and FK dynamics mix exponentially slowly at arbitrarily low temperatures, demonstrating the necessity of the percolation condition and delineating the tractable regime. The work blends new probabilistic tools (disagreement percolation at low temperature), censoring arguments, and tree-based spatial mixing to yield near-optimal mixing-time bounds, including $O(n^2\log n)$ and $O(n\log n)$ rates in applicable settings. Overall, the paper advances understanding of when non-local Markov chains for Potts-model sampling are efficient, linking geometric graph properties to algorithmic tractability with potential implications for broader Gibbs-sampling tasks.

Abstract

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 may be 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 random-cluster dynamics. These are non-local Markov chains that have long been believed to converge rapidly to equilibrium at low temperatures in many graphs. However, the hardness of the sampling problem likely indicates that this is not even the case for all bounded degree graphs. Our results demonstrate that a 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 large linear-sized component, and the graph complement of that component is comprised of only small components. Specifically, we prove that such a condition implies fast mixing of the random-cluster Glauber and Swendsen--Wang 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.

Figures (3)

• Figure 3.1: For a vertex $v$ (dark green) and edge $e$ (purple), the sets $\mathcal{C}_v^{\ne 1}(\omega\setminus e)$ (edges in red) and $\mathsf{CE}_v^{\ne 1}(\omega)$ (edges highlighted in green) are shown in three different cases. Left: $v$ is not part of the giant (blue edges) in $\omega$ (blue and black edges). Middle: $v$ is part of the giant component but not of its $2$-connected core. Right: $v$ in the $2$-connected core of the giant.
• Figure 3.2: Three steps of the construction of the witness are shown. The ball $B_{R/2}$ is the highlighted region. For each $i$, the edges of the finite-connectivity cluster of $w_i$ (green) to $f_i$ (purple) in $Z_{i}$ (blue and black edges) are depicted in red. Note that the configuration changes from left to right, depicting the evolution of the dynamics (backwards in time).
• Figure 5.1: Two steps of the revealing process used in Lemma \ref{['coupling-with-separating-set']}. Left: the configurations on the sub-tree of $w_i\in \mathcal{P}_{i-1}$ are revealed, and in the configuration with $\xi$-boundary conditions, $w_i$ is not wired down to $\xi$. Thus, its parent $w_i'$ is added to $\mathcal{P}_i$. In the next step, this vertex is $w_{i+1}$, and when the remainder of its sub-tree is revealed to indeed include a wiring to $\xi$, the vertex is removed from $\mathcal{P}_i$ but its parent is no longer added.

Theorems & Definitions (67)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Definition 2.1