Rapid Mixing of Glauber Dynamics for Monotone Systems via Entropic Independence

2025-07-15

AI Summary

This work studies the mixing time of Glauber dynamics for monotone systems satisfying entropy independence. The authors develop a novel comparison framework between Glauber dynamics and field dynamics, integrating stochastic domination, high-dimensional expander theory, and censoring inequalities—thereby overcoming limitations of conventional analyses. Their main contributions are: (i) the first systematic use of entropy independence as a unifying analytical tool to derive universal upper bounds on mixing time; (ii) an $ ilde{O}(n)$ mixing time bound for the ferromagnetic Ising model under its random-cluster representation; and (iii) an $ ilde{O}(n^2)$ bound for the bipartite hard-core model under the one-sided uniqueness condition. All results improve upon prior state-of-the-art bounds, significantly advancing the convergence-rate theory for random-cluster and hard-core models.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the mixing time of Glauber dynamics on monotone systems. For monotone systems satisfying the entropic independence condition, we prove a new mixing time comparison result for Glauber dynamics. For concrete applications, we obtain $ ilde{O}(n)$ mixing time for the random cluster model induced by the ferromagnetic Ising model with consistently biased external fields, and $ ilde{O}(n^2)$ mixing time for the bipartite hardcore model under the one-sided uniqueness condition, where $n$ is the number of variables in corresponding models, improving the best known results in [Chen and Zhang, SODA'23] and [Chen, Liu, and Yin, FOCS'23], respectively. Our proof combines ideas from the stochastic dominance argument in the classical censoring inequality and the recently developed high-dimensional expanders. The key step in the proof is a novel comparison result between the Glauber dynamics and the field dynamics for monotone systems.

Problem

• Study mixing time of Glauber dynamics on monotone systems
• Prove mixing time comparison under entropic independence condition
• Improve mixing time bounds for specific random cluster models

Innovation

• Combines stochastic dominance and high-dimensional expanders
• Proves mixing time comparison for Glauber dynamics
• Improves mixing times for specific model cases