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

Abstract

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.

34 pages, AmS-TeX

References in corpus (1)

• The 'Cheerios effect'

Cited by in corpus (4)

• The free boundary Schur process and applications I
• Lozenge tilings with free boundaries
• Off-diagonally symmetric domino tilings of the Aztec diamond of odd order
• Hitting Matrix and Domino Tiling with Diagonal Impurities