A general framework for crystallization in maximal hard-core models

2026-10-07

AI Summary

This study addresses the absence of a unified theoretical description for phase coexistence and crystallization in the hard-core model at low activities. To this end, it proposes a unified framework based on the concept of volume allocation, integrating statistical mechanics with cluster expansion techniques to derive the Peierls condition. Notably, this work extends Pirogov–Sinai theory to the hard-core model for the first time, achieving comprehensive coverage across both high- and low-activity regimes. The main contributions include rigorous proofs of crystallization phenomena on various classes of periodic lattice graphs and precise estimates of activity within phase coexistence regions. Ultimately, this research provides a universal theoretical tool for analyzing phase transitions in complex lattice systems.

Abstract

We study general maximal hard-core models on periodic lattice-type graphs. While for standard models, phase coexistence may arise for high activity, for maximal models this behavior may occur also for sufficiently low activity values. Relying on the concept of volume allocation, we develop a {\em unified set of assumptions} that imply the Peierls condition, a convergent cluster expansion for the partition function, and hence the conclusions of Pirogov-Sinai theory for this class of models. We further check these assumptions for a number of examples including standard lattice-type periodic graphs, proving crystallization for both high and low activity. We also derive estimates for the values of activity that bound the phase coexistence regions in the phase diagram, rewriting for this purpose the proof of a technical result in the derivation of Pirogov-Sinai theory.

Problem

• maximal hard-core models
• crystallization
• phase coexistence
• Pirogov-Sinai theory
• periodic lattice graphs

Innovation

• maximal hard-core models
• volume allocation
• Peierls condition
• cluster expansion
• Pirogov-Sinai theory