A proof of finite crystallization via stratification

TL;DR

This paper introduces a new stratification technique to prove finite crystallization in two-dimensional square lattices, linking energy minimization to isoperimetric inequalities and establishing a fluctuation estimate for minimizers.

Contribution

The paper presents a novel stratification method that simplifies the proof of crystallization and extends understanding of energy minimizers in particle systems.

Findings

Proves finite crystallization in the square lattice for specific particle systems.

Establishes an $n^{3/4}$-law fluctuation estimate for energy minimizers.

Reduces crystallization proof to a slicing argument using isoperimetric inequalities.

Abstract

We devise a new technique to prove two-dimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring two-body short-ranged particle interactions and three-body angular potentials favoring bond-angles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality in $\ell^1$. As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the $n^{3/4}$-law.