Paper detail

A triangular gap of size two in a sea of dimers in a $90^\circ$ angle with mixed boundary conditions, and a heat flow conjecture for the general case

We consider a triangular gap of side two in a $90^\circ$ angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. The image in the side with constrained boundary has the same orientation as the original gap, while the image in the side with free boundary has the opposite orientation. This, together with the parallel between the correlation of gaps in dimer packings and electrostatics we developed in previous work, provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we phrase as a conjecture. While the electrostatic interpretation is equivalent to a steady state heat flow interpretation in the bulk, it turns out that the latter view is more natural in the context of the interaction of the gaps with the boundary. The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).