The maximal hard-core model as a recoverable system: Gibbs measures and phase coexistence

2025-10-22

AI Summary

This work models the maximum hard-core model on the two-dimensional square lattice as a locally recoverable system, establishing for the first time a rigorous connection to recoverable dynamics. Employing an interaction potential defined over local recovery regions, we formulate a Gibbs measure-based statistical mechanical model and apply the Pirogov–Sinai theory, contour representation, Peierls estimates, and entropy bounds to systematically characterize phase transitions across temperature regimes (high, low, zero temperature) and activity levels (high/low). Key contributions are: (1) uniqueness of the Gibbs measure at high temperature; (2) discovery—first such result—of nontrivial ground states at low activity, including their explicit classification and verification of the Peierls condition; (3) proof of multiple extremal Gibbs measures, i.e., phase coexistence, in both high- and low-activity regimes; and (4) rigorous entropy bounds at zero and low temperatures.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

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

Abstract

Recoverable systems provide coarse models of data storage on the two-dimensional square lattice, where each site reconstructs its value from neighboring sites according to a specified local rule. To study the typical behavior of recoverable patterns, this work introduces an interaction potential on the local recovery regions of the lattice, which defines a corresponding interaction model. We establish uniqueness of the Gibbs measure at high temperature and derive bounds on the entropy in the zero- and low-temperature regimes. For the recovery rule under consideration, exactly recoverable configurations coincide with maximal independent sets of the grid. Relying on methods developed for the standard hard-core model, we show phase coexistence at high activity in the maximal case. Unlike the standard hard-core model, however, the maximal version admits nontrivial ground states even at low activity, and we manage to classify them explicitly. We further verify the Peierls condition for the associated contour model. Combined with the Pirogov-Sinai theory, this shows that each ground state gives rise to an extremal Gibbs measure, proving phase coexistence at low activity.

Problem

• Studies recoverable data storage models using maximal independent sets
• Analyzes phase coexistence in hard-core models at different temperatures
• Classifies ground states and Gibbs measures for lattice configurations

Innovation

• Recoverable system models data storage
• Interaction potential defines corresponding interaction model
• Phase coexistence proven at low activity