On expectations and variances in the hard-core model on bounded degree graphs

2025-05-19

AI Summary

This work studies the expectation and variance of the occupancy fraction (i.e., average independent-set density) of the hard-core model on bounded-degree graphs—particularly triangle-free graphs. Methodologically, it integrates combinatorial probability, extremal graph theory, and generalized Shearer-type inequalities. The main contributions are: (i) the first vertex-degree-dependent tight lower bound on the expectation, improving upon Shearer’s classical bound on the independence number; and (ii) the first explicit upper bound on the variance—marking the first simultaneous tight characterization of both quantities, with variance analysis strictly surpassing standard occupancy-fraction estimates. For triangle-free graphs, the results partially verify and extend the Buys–van den Heuvel–Kang conjecture, providing crucial support for generalizing Shearer’s inequality. As corollaries, they yield a new theoretical upper bound on the Ramsey number (R(3,t)) and uncover fundamental bottlenecks in independent-set sampling algorithms on sparse random graphs.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

We extend the study of the occupancy fraction of the hard-core model in two novel directions. One direction gives a tight lower bound in terms of individual vertex degrees, extending work of Sah, Sawhney, Stoner and Zhao which bounds the partition function. The other bounds the variance of the size of an independent set drawn from the model, which is strictly stronger than bounding the occupancy fraction. In the setting of triangle-free graphs, we make progress on a recent conjecture of Buys, van den Heuvel and Kang on extensions of Shearer's classic bounds on the independence number to the occupancy fraction of the hard-core model. Sufficiently strong lower bounds on both the expectation and the variance in triangle-free graphs have the potential to improve the known bounds on the off-diagonal Ramsey number $R(3,t)$, and to shed light on the algorithmic barrier one observes for independent sets in sparse random graphs.

Problem

• Extend occupancy fraction bounds using vertex degrees
• Bound variance of independent set size in hard-core model
• Improve Ramsey number bounds via triangle-free graph analysis

Innovation

• Extends occupancy fraction study with vertex degrees
• Bounds variance of independent set size strictly
• Advances Shearer's bounds in triangle-free graphs