Perfect t-embeddings and the octahedron equation of the two-periodic Aztec diamond

TL;DR

This paper establishes a link between perfect t-embeddings and the octahedron equation in the two-periodic Aztec diamond, showing that embedding positions can be expressed via solutions to the octahedron equation with specific initial conditions.

Contribution

It demonstrates that the positions of t-embeddings and origami maps in the two-periodic Aztec diamond can be represented as sums of density functions from octahedron equation solutions, revealing a new mathematical connection.

Findings

Embedding positions are expressed as sums of octahedron equation solutions.

The origami map can be described using the same framework.

The results apply to the two-periodic Aztec diamond setting.

Abstract

This paper explores the connection between perfect t-embeddings and the octahedron equation in the setting of the two-periodic Aztec diamond. In particular, we show that the positions of both the t-embedding and the corresponding origami map can be expressed as sums of density functions arising from solutions to the octahedron equation with appropriate flat initial conditions.