Crystallization in hard-core lattice models occurs at low and high activity
A general framework for crystallization in maximal hard-core models
Information Theory
Summary
This paper studies how particles arranged on a repeating grid cannot be too close together, a rule called the hard-core model. The researchers look at a version called maximal hard-core models, where particles fill as much space as possible without touching. They show that these systems can form stable patterns, called crystals, not only when particles are very active but also when activity is low. The authors develop a general mathematical framework that explains this behavior and apply it to many examples, clarifying when these patterns appear.
What this means in practice
- •For statistical physics modelers: Identify parameter ranges for phase coexistence in complex lattice-based particle systems using the unified assumptions provided.
- •For materials simulation engineers: Predict crystallization behavior in models of densely packed particles on grids, aiding design of structured materials.
A theory result. No direct application yet.
Authors
Alexander Barg, Qidong He, Geyang Wang
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.