Voronoi tilings hidden in crystals - The case of maximal abelian coverings

TL;DR

This paper reveals that the crystal structure derived from a graph's maximal abelian covering contains a hidden Voronoi tiling, linking discrete geometric analysis with tropical geometry and invariant theory.

Contribution

It demonstrates that the Voronoi tiling is embedded within the crystal structure, connecting it to the tropical Abel-Jacobi map and extending previous geometric invariant theory results.

Findings

Voronoi tiling is embedded in the crystal structure

Crystal does not intrude into Voronoi cell interiors

Connection established with tropical Abel-Jacobi map

Abstract

Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadri as a by-product of Geometric Invariant Theory, he shows that the Voronoi tiling (also known as the Wigner-Seitz tiling) is hidden in the crystal, that is, the crystal does not intrude the interiors of the top-dimensional Voronoi cells. The result turns out to be closely related to the tropical Abel-Jacobi map of the associated compact tropical curve.