Fast counting and sampling for ferromagnetic two-spin systems

2026-07-06

AI Summary

This study addresses the lack of efficient sampling and partition function approximation algorithms for ferromagnetic two-state systems within certain parameter regimes. By constructing weighted subgraph and random-cluster-like models, the authors establish novel equivalences between these combinatorial structures and the target physical system. Leveraging these connections, they propose the first efficient sampling algorithm tailored to this regime and design a partition function approximation algorithm running in nearly quadratic time on bounded-degree graphs and in polynomial time on general graphs. This represents a significant improvement over the method of Guo et al. (2020). The work integrates techniques from graph theory, statistical physics, and randomized approximation to deliver an enhanced computational framework for analyzing such systems.

Abstract

We introduce two new models equivalent to ferromagnetic two-spin systems: a weighted subgraph model and a random cluster type model. Using these new connections, we obtain an efficient sampling algorithm and a new randomised algorithm that efficiently approximates the partition function of ferromagnetic two-spin systems in certain parameter regimes. No efficient sampling algorithms are known before in this regime, and our new estimation algorithm runs in near-quadratic time for bounded degree graphs and in polynomial time for general graphs, improving upon the previous algorithm of Guo, Liu, and Lu (2020).

Problem

• ferromagnetic two-spin systems
• sampling
• partition function
• efficient algorithms
• randomised approximation

Innovation

• ferromagnetic two-spin systems
• weighted subgraph model
• random cluster model
• efficient sampling
• partition function approximation