Interactions between interleaving holes in a sea of unit rhombi

TL;DR

This paper explores how the interactions between collinear triangular holes in a sea of rhombi can be approximated using electrostatic energy models, revealing deep connections between tiling enumeration and physical analogies.

Contribution

It introduces a novel electrostatic energy approximation for hole interactions and relates boundary effects to image charge methods in tiling configurations.

Findings

Interaction approximated by exponential of electrostatic energy

Boundary effects modeled by image charge method

Connections established between tiling enumeration and electrostatics

Abstract

Consider a family of collinear, equilateral triangular holes of any even side length lying within a sea of unit rhombi. The results presented below show that as the distance between the holes grows large, the interaction between them may be approximated, up to a multiplicative constant, by taking the exponential of the negative of the electrostatic energy of the system obtained by viewing the holes as a set of point charges, each with a signed magnitude given by a certain statistic. Furthermore it is shown that the interaction between a family of left pointing collinear triangular holes and a free boundary may be approximated (again up to some multiplicative constant) by taking the exponential of the negative of the electrostatic energy of the system obtained by considering the holes as a set of point charges and the boundary a straight equipotential conductor. These two differing systems of point charges can be related via the method of image charges, a well-known physical law that also surfaces in the following mathematical analysis of enumeration formulas that count tilings of certain regions of the plane by unit rhombi.