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

TL;DR

This paper investigates the structural graph properties that determine the efficiency of sampling from the Potts model at low temperatures, revealing that certain percolation phases enable polynomial-time algorithms while others lead to slow convergence.

Contribution

It identifies the key graph property—strongly supercritical percolation—that influences the rapid mixing of random-cluster dynamics at low temperatures.

Findings

Strongly supercritical percolation implies fast mixing on certain graph families.

Absence of this percolation property can cause exponential slowdowns.

Results apply to graphs with bounded degree, including those with exponential growth and locally treelike structures.

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.