Three interactions of holes in two dimensional dimer systems

Abstract

Consider the unit triangular lattice in the plane with origin $O$, drawn so that one of the sets of lattice lines is vertical. Let $l$ and $l^{\prime }$ denote respectively the vertical and horizontal lines that intersect $O$. Suppose the plane contains a pair of triangular holes of side length two, distributed symmetrically with respect to $l$ and $l^{\prime }$, and oriented so that both holes point toward $O$. Unit rhombus tilings of three different regions of the plane are considered, namely: tilings of the entire plane; tilings of the half plane that lies to the left of $l$ (where $l$ is considered a free boundary, so unit rhombi are allowed to protrude half-way across it); and tilings of the half plane that lies just below the fixed boundary $l^{\prime }$. Asymptotic expressions for the interactions of the triangular holes in these three different regions are obtained, providing further evidence for Ciucu's ongoing program that seeks to draw parallels between gaps in dimer systems on the hexagonal lattice and electrostatic phenomena.

arXiv comment38 pages, 5 figures. In most recent version: updated enumeration formulas (and proofs) for horizontally symmetric tilings to include the case k = n; corrected typos