The effect of microscopic gap displacement on the correlation of gaps in dimer systems

TL;DR

This paper demonstrates that the local correlation changes in dimer systems caused by microscopic gap displacements are governed by an electric field analogy, extending previous electrostatic interpretations of dimer correlations.

Contribution

It introduces a new field ${old T}$ that describes the effect of gap displacements, linking microscopic perturbations to an electrostatic field analogy in dimer systems.

Findings

Correlation changes are governed by a field analogous to electric fields.

The new field ${old T}$ captures the instantaneous pull on gaps.

The fields ${old T}$ and ${old F}$ differ but both relate to electrostatics.

Abstract

In earlier work we showed that in the bulk, the correlation of gaps in dimer systems on the hexagonal lattice is governed, in the fine mesh limit, by Coulomb's law for 2D electrostatics. We also proved that the scaling limit of the discrete field ${\bold F}$ of average tile orientations is, up to a multiplicative constant, the electric field produced by a 2D system of charges corresponding to the gaps. In this paper we show that in the bulk, the relative change $T_{\al,\be}$ in correlation caused by displacing a hole by a fixed vector $(\al,\be)$ is, in the fine mesh limit, the projection on $(\al,\be)$ of a new field ${\bold T}$, which is also equal up to a multiplicative constant to the electric field of the corresponding system of charges. We also discuss the differences between the fields ${\bold T}$ and ${\bold F}$ and present conjectures for their fine mesh limits in the more general case of a dimer system with boundary. The new field ${\bold T}$ can be viewed as capturing the instantaneous pull on each gap in the surrounding fluctuating sea of dimers. From the point of view of the parallel to physics, the electrostatic force emerges then as an entropic force.