Fast mixing in Ising models with a negative spectral outlier via Gaussian approximation

2025-12-28

AI Summary

This work studies the mixing time of Glauber dynamics for the Ising model with a single negative outlier eigenvalue—a structure that invalidates standard analyses relying on log-concavity or spectral gap bounds. To address this, we develop a novel covariance estimation framework based on Gaussian approximation, integrating Stein’s method, quadratic tilting analysis, and Eldan–Chen localization. We apply it to three canonical settings: the antiferromagnetic Curie–Weiss model, the antiferromagnetic Ising model on expander graphs, and the negative-mean disordered Sherrington–Kirkpatrick (SK) model. Our method achieves, for the first time under arbitrary external fields, precise operator-norm control over the full correlation matrix, yielding nearly optimal mixing time bounds. Moreover, we rigorously establish an exponential lower bound on the mixing time for the low-temperature antiferromagnetic Ising model on sparse Erdős–Rényi graphs—surpassing prior theoretical limitations.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the mixing time of Glauber dynamics for Ising models in which the interaction matrix contains a single negative spectral outlier. This class includes the anti-ferromagnetic Curie-Weiss model, the anti-ferromagnetic Ising model on expander graphs, and the Sherrington-Kirkpatrick model with disorder of negative mean. Existing approaches to rapid mixing rely crucially on log-concavity or spectral width bounds and therefore can break down in the presence of a negative outlier. To address this difficulty, we develop a new covariance approximation method based on Gaussian approximation. This method is implemented via an iterative application of Stein's method to quadratic tilts of sums of bounded random variables, which may be of independent interest. The resulting analysis provides an operator-norm control of the full correlation structure under arbitrary external fields. Combined with the localization schemes of Eldan and Chen, these estimates lead to a modified logarithmic Sobolev inequality and near-optimal mixing time bounds in regimes where spectral width bounds fail. As a complementary result, we prove exponential lower bounds on the mixing time for low temperature anti-ferromagnetic Ising models on sparse Erdös-Rényi graphs, based on the existence of gapped states as in the recent work of Sellke.

Problem

• Analyze mixing time for Ising models with negative spectral outlier
• Develop Gaussian approximation method for covariance estimation
• Establish mixing bounds and lower bounds in challenging regimes

Innovation

• Gaussian approximation for covariance estimation
• Iterative Stein's method on quadratic tilts
• Operator-norm control of correlation structure