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Consider the unit triangular lattice in the plane with origin $O$, drawn sothat one of the sets of lattice lines is vertical. Let $l$ and $l'$ denoterespectively the vertical and horizontal lines that intersect $O$. Suppose theplane contains a pair of triangular holes of side length two, distributedsymmetrically with respect to $l$ and $l'$, and oriented so that both holespoint toward $O$. Unit rhombus tilings of three different regions of the planeare considered, namely: tilings of the entire plane; tilings of the half planethat lies to the left of $l$ (where $l$ is considered a free boundary, so unitrhombi are allowed to protrude half-way across it); and tilings of the halfplane that lies just below the fixed boundary $l'$. Asymptotic expressions forthe interactions of the triangular holes in these three different regions areobtained, providing further evidence for Ciucu's ongoing program that seeks todraw parallels between gaps in dimer systems on the hexagonal lattice andelectrostatic phenomena.

Three interactions of holes in two dimensional dimer systems