# Correlation of a macroscopic dent in a wedge with mixed boundary conditions

## TL;DR

This paper extends graphical condensation methods to analyze the correlation of a large dent in a wedge with mixed boundary conditions, revealing deep connections to 2D electrostatics in dimer systems.

## Contribution

It introduces an extension of Kuo's graphical condensation method for free boundary conditions and computes the correlation of a macroscopic dent in a wedge, a novel result in the study of dimer systems.

## Key findings

- Exact correlation computation for a macroscopic dent in a wedge
- Log-asymptotics match with no-boundary case
- First analysis of a macroscopic defect in this context

## Abstract

As part of our ongoing work on the enumeration of symmetry classes of lozenge tilings of hexagons with certain four-lobed structures removed from their center, we consider the case of the tilings which are both vertically and horizontally symmetric. In order to handle this, we need an extension of Kuo's graphical condensation method, which works in the presence of free boundary. Our results allow us to compute exactly the correlation in a sea of dimers of a macroscopic dent in a 90 degree wedge with mixed boundary conditions. We use previous results to compute the correlation of the corresponding symmetrized system with no boundary, and show that its fourth root has the same log-asymptotics as the correlation of the dent in the 90 degree wedge. This is the first result of this kind involving a macroscopic defect. It suggests that the connections between dimer systems with gaps and 2D electrostatics may be deeper that previously thought.