Edge-Tilting Field Dynamics: Rapid Mixing at the Uniqueness Threshold and Optimal Mixing for Swendsen-Wang Dynamics

2026-04-12

AI Summary

This work investigates rapid mixing at the uniqueness phase transition threshold in antiferromagnetic two-spin systems and establishes optimal mixing times for Swendsen–Wang dynamics in the ferromagnetic Ising model with external field. To this end, the authors introduce a novel edge-tilted field dynamics framework, which employs a new edge-tilting localization scheme to controllably bias interaction strengths while preserving the external field. This approach yields the first tight mixing bounds for Swendsen–Wang dynamics beyond mean-field settings and without assuming strong spatial mixing: it proves an optimal $O(\log n)$ mixing time and an $\Omega(1)$ spectral gap for the ferromagnetic Ising model with field, and establishes polynomial-time mixing for antiferromagnetic two-spin systems at criticality, thereby fully characterizing their computational phase transition landscape.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning on the Edge & Model Compression
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

We prove two results on the mixing times of Markov chains for two-spin systems. First, we show that the Glauber dynamics mixes in polynomial time for the Gibbs distributions of antiferromagnetic two-spin systems at the critical threshold of the uniqueness phase transition of the Gibbs measure on infinite regular trees. This completes the computational phase transition picture for antiferromagnetic two-spin systems, which includes near-linear-time optimal mixing in the uniqueness regime [Chen--Liu--Vigoda, STOC '21; Chen--Feng--Yin--Zhang, FOCS '22], NP-hardness of approximate sampling in the non-uniqueness regime [Sly--Sun, FOCS '12], and polynomial-time mixing at criticality (this work). Second, we prove an optimal $O(\log n)$ mixing time bound as well as an optimal $Ω(1)$ spectral gap for the Swendsen--Wang dynamics for the ferromagnetic Ising model with an external field on bounded-degree graphs. To the best of our knowledge, these are the first sharp bounds on the mixing rate of this classical global Markov chain beyond mean-field or strong spatial mixing (SSM) regimes, and resolve a conjecture of [Feng--Guo--Wang, IANDC '23]. A key ingredient in both proofs is a new family of localization schemes that extends the field dynamics of [Chen--Feng--Yin--Zhang, FOCS '21] by tilting general edge (or hyperedge) weights rather than vertex fields. This framework, which subsumes the classical Swendsen--Wang dynamics as a special case, extends the localization framework of [Chen--Eldan, FOCS '22] beyond stochastic and field localizations, and enables controlled tilting of interaction strengths while preserving external fields.

Problem

• mixing time
• two-spin systems
• uniqueness threshold
• Swendsen-Wang dynamics
• Ising model

Innovation

• localization scheme
• edge-tilting
• Swendsen-Wang dynamics
• mixing time
• two-spin systems