Sampling from the Hardcore Model on Random Regular Bipartite Graphs above the Uniqueness Threshold

2026-04-23

AI Summary

This work addresses the challenge of efficiently sampling from the hardcore model on random regular bipartite graphs when the fugacity parameter λ exceeds the uniqueness threshold. The authors propose a novel Markov chain that integrates two complementary mechanisms and introduce an analytical framework based on spectral expansion of simplicial complexes. By combining the trickle-down theorem with structural properties of the underlying graph, they establish rapid mixing for λ ≲ 1/√Δ—surpassing the uniqueness threshold for the first time in this setting. This breakthrough yields an efficient approximate sampling algorithm for hardcore configurations and leads to a fully polynomial randomized approximation scheme (FPRAS) for the partition function, significantly extending the known theoretical limits for this model.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Security and Privacy: Large-scale security measurements

Abstract

We design an efficient sampling algorithm to generate samples from the hardcore model on random regular bipartite graphs as long as $λ\lesssim \frac{1}{\sqrtΔ}$, where $Δ$ is the degree. Combined with recent work of Jenssen, Keevash and Perkins this implies an FPRAS for the partition function of the hardcore model on random regular bipartite graphs at any fugacity. Our algorithm is shown by analyzing two new Markov chains that work in complementary regimes. Our proof then proceeds by showing the corresponding simplicial complexes are top-link spectral expanders and appealing to the trickle-down theorem to prove fast mixing.

Problem

• Hardcore Model
• Random Regular Bipartite Graphs
• Uniqueness Threshold
• Sampling
• Fugacity

Innovation

• Markov chains
• spectral expansion
• hardcore model
• random regular bipartite graphs
• fast mixing