The Effect of Microscopic Gap Displacement on the Correlation of Gaps in Dimer Systems

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 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 α , β in correlation caused by displacing a hole by a fixed vector ( α , β ) is, in the fine mesh limit, the projection on ( α , β ) of a new field 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 T and F and present conjectures for their fine mesh limits in the more general case of a dimer system with boundary. The new field 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.