Round Aztec windows, a dual of the Aztec diamond theorem and a curious symmetry of the correlation of diagonal slits

TL;DR

This paper introduces new regions related to Aztec diamonds with holes, proving simple product formulas for their domino tilings, and explores symmetries in the correlation of holes, including a dual of the Aztec diamond theorem.

Contribution

It establishes a dual of the Aztec diamond theorem and uncovers symmetries in the correlation of diagonal slits, extending tiling enumeration to new geometries.

Findings

Proved product formulas for domino tilings of Aztec rectangles with holes.

Discovered a dual of the Aztec diamond theorem for toroidal graphs.

Identified a symmetry in the correlation of diagonal slits.

Abstract

Fairly shortly after the publication of the Aztec diamond theorem of Elkies, Kuperberg, Larsen and Propp in 1992, interest arose in finding the number of domino tilings of an Aztec diamond with an ``Aztec window,'' i.e.\ a hole in the shape of a smaller Aztec diamond at its center. Several intriguing patterns were discovered for the number of tilings of such regions, but the numbers themselves were not ``round'' -- they didn't seem to be given by a simple product formula. In this paper we consider a very closely related shape of holes (namely, odd Aztec rectangles), and prove that a large variety of regions obtained from Aztec rectangles by making such holes in them possess the sought-after property that the number of their domino tilings is given by a simple product formula. We find the same to be true for certain symmetric cruciform regions. We also consider graphs obtained from a toroidal Aztec diamond by making such holes in them, and prove a simple formula that governs the way the number of their perfect matchings changes under a natural evolution of the holes. This yields in particular a natural dual of the Aztec diamond theorem. Some implications for the correlation of such holes are also presented, including an unexpected symmetry for the correlation of diagonal slits on the square grid.