The interaction of a gap with a free boundary in a two dimensional dimer system

TL;DR

This paper analyzes the asymptotic behavior of a triangular gap in a lozenge tiling near a free boundary, revealing a decay rate similar to electrostatic phenomena, and uses the method of images analogy.

Contribution

It establishes the asymptotic correlation function of a gap near a free boundary in a lozenge tiling, connecting combinatorics with electrostatic analogies.

Findings

Correlation function decays as 1/(4πr) for large r

Gap behavior parallels electrostatic image charge phenomena

Provides asymptotic analysis for lozenge tilings with free boundary

Abstract

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let $\Cal H$ be the half plane to the left of $\ell$. We consider lozenge tilings of $\Cal H$ that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$, $r\to\infty$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.