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

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

2009-12-10

arxiv.org/abs/0912.2023v1

Commun. Math. Phys. 302 (2011), 253-289

Mathematics

Combinatorics

34 pages, AmS-TeX

Scientific paper

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let $\Cal H$ be the half plane to the left of $\ell$. We consider lozenge tilings of $\Cal H$ that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$, $r\to\infty$, 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.

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