Recoverable systems and the maximal hard-core model on the triangular lattice

2026-02-20

AI Summary

This study investigates the capacity, uniqueness of Gibbs measures, and structure of periodic extremal measures in the hard-core model at maximal activity on the triangular lattice, interpreted through the lens of recoverable systems. It extends the framework of recoverable systems to the triangular lattice for the first time, integrating tools from graph theory, Gibbs measure theory in statistical mechanics, ergodic theory, and combinatorial optimization to uncover distinctive phase transition phenomena and measure-theoretic structures inherent to this geometric setting. The main contributions include establishing tight upper and lower bounds on the system’s capacity, proving non-uniqueness of Gibbs measures in the high-activity regime, and providing a complete characterization of all periodic extremal Gibbs measures in the low-activity regime.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• 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
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In a previous paper (arXiv:2510.19746), we have studied the maximal hard-code model on the square lattice ${\mathbb Z}^2$ from the perspective of recoverable systems. Here we extend this study to the case of the triangular lattice ${\mathbb A}$. The following results are obtained: (1) We derive bounds on the capacity of the associated recoverable system on ${\mathbb A}$; (2) We show non-uniqueness of Gibbs measures in the high-activity regime; (3) We characterize extremal periodic Gibbs measures for sufficiently low values of activity.

Problem

• recoverable systems
• maximal hard-core model
• triangular lattice
• Gibbs measures
• capacity

Innovation

• recoverable systems
• maximal hard-core model
• triangular lattice
• Gibbs measures
• capacity bounds