Three interactions of holes in two dimensional dimer systems

TL;DR

This paper analyzes the interactions of triangular holes in two-dimensional dimer systems on a triangular lattice, deriving asymptotic expressions that support parallels with electrostatic phenomena.

Contribution

It provides new asymptotic formulas for hole interactions in dimer systems on a triangular lattice, extending understanding of gap phenomena and electrostatic analogies.

Findings

Asymptotic expressions for hole interactions in full plane, half-plane with free boundary, and half-plane with fixed boundary.

Evidence supporting electrostatic analogy in dimer gap interactions.

Extension of Ciucu's program relating dimer gaps to electrostatics.

Abstract

Consider the unit triangular lattice in the plane with origin $O$, drawn so that one of the sets of lattice lines is vertical. Let $l$ and $l'$ denote respectively the vertical and horizontal lines that intersect $O$. Suppose the plane contains a pair of triangular holes of side length two, distributed symmetrically with respect to $l$ and $l'$, and oriented so that both holes point toward $O$. Unit rhombus tilings of three different regions of the plane are considered, namely: tilings of the entire plane; tilings of the half plane that lies to the left of $l$ (where $l$ is considered a free boundary, so unit rhombi are allowed to protrude half-way across it); and tilings of the half plane that lies just below the fixed boundary $l'$. Asymptotic expressions for the interactions of the triangular holes in these three different regions are obtained, providing further evidence for Ciucu's ongoing program that seeks to draw parallels between gaps in dimer systems on the hexagonal lattice and electrostatic phenomena.