Existence of stable Voronoi facets as a fundamental signature of the crystalline solid phase

Prove that, in crystalline solids, there exist Voronoi cell facets that persist with probability one up to the melting temperature despite thermal fluctuations, and demonstrate that their existence constitutes a fundamental signature of the crystalline solid phase that enables quantification of the Gibbs measure across the solid phase.

Background

From ab-initio molecular dynamics on silicon, the authors observe that exactly four large Voronoi facets endure up to the melting line, while the distribution of facets and their areas changes abruptly at the solid–liquid transition.

Based on this observation, they conjecture that the presence of such persistent facets is a universal hallmark of crystalline solids and is key to defining and quantifying the Gibbs measure over the entire solid phase.