Fast and Slow Mixing of the Kawasaki Dynamics on Bounded-Degree Graphs

2024-05-10

International Workshop and International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

AI Summary

This paper studies the worst-case mixing time of Kawasaki dynamics for the fixed-magnetization Ising model on bounded-degree graphs. For graphs with maximum degree Δ, it establishes the exact phase transition threshold distinguishing fast from slow mixing: within the tree uniqueness regime, polynomial mixing holds for all magnetizations; strikingly, even in parameter regimes previously believed to admit efficient sampling, Kawasaki dynamics can exhibit exponential slow mixing—refuting prior conjectures. Methodologically, the work integrates spectral independence analysis, characterization of metastable multimodality for the external-field Ising model on random regular graphs, and refined techniques for proving sharp mixing-time thresholds. Key contributions include: (i) a spectral independence framework for the fixed-magnetization Ising model; (ii) necessary and sufficient critical conditions governing the mixing behavior of Kawasaki dynamics; and (iii) a fundamental separation between computational tractability (e.g., existence of efficient algorithms) and dynamical efficiency (i.e., rapid mixing), revealing an intrinsic inconsistency between the two.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

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 study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree $Delta$. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree $Delta$ graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs.

Problem

• Analyzing mixing time of Kawasaki dynamics for Ising models
• Determining parameter ranges for fast versus slow mixing dynamics
• Investigating metastable states in Ising models on bounded-degree graphs

Innovation

• Studying Kawasaki dynamics mixing time on graphs
• Showing spectral independence in fixed-magnetization Ising model
• Proving sharp threshold for multiple metastable states