Macroscopically separated gaps in dimer coverings of Aztec rectangles

TL;DR

This paper analyzes the interactions of defect clusters in Aztec rectangles, deriving asymptotic formulas for defect correlations influenced by boundaries, revealing novel interaction behaviors including Casimir-like forces.

Contribution

It extends previous work by providing formulas for defect interactions at arbitrary boundary points and explores complex defect configurations with new interaction insights.

Findings

Defect clusters interact via Casimir-like forces.

Neutral doublets interact independently of each other.

Long strings of monomers exhibit specific interaction patterns.

Abstract

In this paper we determine the interaction of diagonal defect clusters in regions of an Aztec rectangle that scale to arbitrary points on its symmetry axis (in earlier work we treated the case when this point was the center of the scaled Aztec rectangle). We use the resulting formulas to determine the asymptotics of the correlation of defects that are macroscopically separated from one another and feel the influence of the boundary. In several of the treated situations this seems not to be accomplishable by previous methods. Our applications include the case of two long neutral strings, which turn out to interact by an analog of the Casimir force, two families of neutral doublets that turn out to interact completely independently of one another, a neutral doublet and a very long neutral string, a general collection of macroscopically separated monomer and separation defects, and the case of long strings consisting of consecutive monomers.