Fluctuations in the Aztec diamonds via a space-like maximal surface in Minkowski 3-space

TL;DR

This paper describes the scaling limit of dimer fluctuations in Aztec diamonds using a space-like maximal surface in Minkowski space, connecting combinatorial models with geometric structures.

Contribution

It introduces a new geometric framework for understanding Aztec diamond fluctuations through maximal surfaces in Minkowski space, extending recent embedding techniques.

Findings

Scaling limit characterized by a space-like maximal surface

Connection between origami maps and Minkowski geometry

New geometric perspective on dimer fluctuations

Abstract

We provide a new description of the scaling limit of dimer fluctuations in homogeneous Aztec diamonds via the intrinsic conformal structure of a space-like maximal surface in the three-dimensional Minkowski space $\mathbb{R}^{2,1}$. This surface naturally appears as the limit of the graphs of origami maps associated to symmetric t-embeddings of Aztec diamonds, fitting the framework recently developed in arXiv:2109.06272.