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

TL;DR

This paper studies recoverable systems on a 2D lattice, analyzing Gibbs measures and phase coexistence, with a focus on the maximal hard-core model, revealing unique behaviors at different temperature regimes.

Contribution

It introduces a new interaction model for recoverable systems, classifies ground states at low activity, and proves phase coexistence using Pirogov-Sinai theory.

Findings

Uniqueness of Gibbs measure at high temperature

Existence of phase coexistence at high activity

Explicit classification of ground states at low activity

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.