Domino shuffling on Novak half-hexagons and Aztec half-diamonds

TL;DR

This paper investigates the relationship between Aztec Diamond graphs and Novak's Half-Hexagons, establishing domino shuffling algorithms and proving an arctic parabola theorem for Half-Hexagons based on the Arctic Circle theorem.

Contribution

It introduces a new family of graphs called Half-Hexagons, linking their domino shuffling algorithms to those of Aztec Diamonds, and proves a new arctic parabola theorem.

Findings

Established domino shuffling algorithms for Half-Hexagons

Connected the arctic parabola to the Arctic Circle theorem

Proved the arctic parabola theorem for Half-Hexagons

Abstract

We explore the connections between the well-studied Aztec Diamond graphs and a new family of graphs called the Half-Hexagons, discovered by Jonathan Novak. In particular, both families of graphs have very simple domino shuffling algorithms, which turn out to be intimately related. This connection allows us to prove an "arctic parabola" theorem for the Half-Hexagons as a corollary of the Arctic Circle theorem for the Aztec Diamond.