The interaction of a gap with a free boundary in a two dimensional dimer system

Let $\mathrm{\ell }$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let $\text{Cal}H$ be the half plane to the left of $\mathrm{\ell }$. We consider lozenge tilings of $\text{Cal}H$ that have a triangular gap of side-length two and in which $\mathrm{\ell }$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\mathrm{\ell }$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$, $r\to \mathrm{\infty }$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.