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

Geyang Wang, Alexander Barg, Navin Kashyap

TL;DR

The paper studies recoverable systems on the two-dimensional lattice, focusing on a local rule that exactly entails MIS configurations. It develops a Gibbs-potential framework with cross-shaped interactions and analyzes both high-temperature uniqueness and entropy growth, as well as a maximal hard-core MIS model exhibiting phase coexistence. High-temperature results rely on Dobrushin’s condition to show uniqueness and rapid mixing, while zero- and positive-temperature entropy bounds are derived via transfer-matrix methods and contour arguments. At low temperature, a complete Pirogov-Sinai analysis classifies ground states and demonstrates the emergence of multiple extremal Gibbs measures through a Peierls condition and contour control. The findings illuminate the balance between information storage capacity and thermodynamic phases in recoverable MIS-based systems, with broader implications for constrained systems and ground-state structure on lattices.

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.

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

TL;DR

The paper studies recoverable systems on the two-dimensional lattice, focusing on a local rule that exactly entails MIS configurations. It develops a Gibbs-potential framework with cross-shaped interactions and analyzes both high-temperature uniqueness and entropy growth, as well as a maximal hard-core MIS model exhibiting phase coexistence. High-temperature results rely on Dobrushin’s condition to show uniqueness and rapid mixing, while zero- and positive-temperature entropy bounds are derived via transfer-matrix methods and contour arguments. At low temperature, a complete Pirogov-Sinai analysis classifies ground states and demonstrates the emergence of multiple extremal Gibbs measures through a Peierls condition and contour control. The findings illuminate the balance between information storage capacity and thermodynamic phases in recoverable MIS-based systems, with broader implications for constrained systems and ground-state structure on lattices.

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.
Paper Structure (28 sections, 25 theorems, 110 equations, 6 figures)

This paper contains 28 sections, 25 theorems, 110 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Overview of the paper and summary of results
  3. Preliminaries
  4. High-temperature disordered regime
  5. Uniqueness of the Gibbs measure
  6. Information storage and error probability
  7. Mixing rate
  8. Error probability
  9. Entropy of the system
  10. Zero temperature
  11. Positive temperature
  12. A hard-core model for recoverable systems
  13. Maximal independent sets
  14. The hard-core model
  15. The maximal hard-core model
  16. ...and 13 more sections

Key Result

Lemma 1.1

A configuration $\omega\in \Omega$ forms an MIS if and only if $\omega\in{\EuScript X}_0$ (see Fig. fig:MIS for an illustration of this equivalence).

Figures (6)

  • Figure 1: The blank/dotted squares in the grid correspond to 0's and 1's, respectively. The 1's not on the boundary are surrounded by 4 zeros, and every 0 is adjacent to at least one 1; cf.\ref{['eq:recovery']}. The set of 1's forms a maximal independent set in the grid graph, or a maximal hardcore configuration in the lattice, shown by the nonoverlapping gray squares.
  • Figure 2: Plot of the bound \ref{['eq:entropy_bound']}. The horizontal axis corresponds to the entropy value $H_0 \approx 0.3012$ (the topological entropy), attained at zero temperature. The singularity occurs at the $\beta$ for which the maximum switches from one term in \ref{['eq:entropy_bound']} to the other.
  • Figure 3: Illustration of the contour elimination procedure.
  • Figure 5: Construction of periodic ground states.
  • Figure 6: Lattice tilings ${\EuScript L}_1$ and ${\EuScript L}_2$.
  • ...and 1 more figures

Theorems & Definitions (48)

  • Lemma 1.1
  • proof
  • Proposition 3.1
  • proof
  • Remark 1
  • Theorem 3.2
  • Definition 3.1: Mixing time
  • Theorem 3.3
  • Proposition 3.4
  • proof
  • ...and 38 more