A random tiling model for two dimensional electrostatics

TL;DR

This paper models the interaction of triangular holes in a dimer-covered hexagonal lattice, revealing that their joint correlation asymptotics mimic two-dimensional electrostatic potential energy, with a superposition of pairwise Coulomb interactions.

Contribution

It introduces a random tiling model that connects lattice dimer interactions with electrostatic potential energy, providing a novel physical interpretation.

Findings

Joint correlation asymptotics match electrostatic potential energy

Interactions follow Coulomb's law

Superposition principle applies to pairwise interactions

Abstract

We consider triangular holes on the hexagonal lattice and we study their interaction when the rest of the lattice is covered by dimers. More precisely, we analyze the joint correlation of these triangular holes in a ``sea'' of dimers. We determine the asymptotics of the joint correlation (for large separations between the holes) in the case when one of the holes has side 1, all remaining holes have side 2, and the holes are distributed symmetrically with respect to a symmetry axis. Our result has a striking physical interpretation. If we regard the holes as electrical charges, with charge equal to the difference between the number of down-pointing and up-pointing unit triangles in a hole, the logarithm of the joint correlation behaves exactly like the electrostatic potential energy of this two-dimensional electrostatic system: it is obtained by a Superposition Principle from the interaction of all pairs, and the pair interactions are according to Coulomb's law.