Perfect t-embeddings and Lozenge Tilings

TL;DR

This paper constructs perfect t-embeddings for hexagon graphs, proving their existence, deriving explicit formulas, analyzing scaling limits, and connecting to Gaussian free field fluctuations.

Contribution

It provides the first example of perfect t-embeddings for graphs with outer face degree greater than four, using inverse Kasteleyn matrices and symmetry-based methods.

Findings

Constructions of perfect t-embeddings for hexagon graphs.

Explicit contour integral formulas for embeddings and origami maps.

Convergence of origami maps to a maximal surface and height fluctuations to Gaussian free field.

Abstract

We construct perfect t-embeddings for regular hexagons of the hexagonal lattice, providing the first example, and hence proving existence, for graphs with an outer face of degree greater than four. The construction is in terms of the inverse Kasteleyn matrix and relies only on symmetries of the graph. Using known formulas for the inverse Kasteleyn matrix, we derive exact contour integral formulas for these embeddings and their origami maps. Through steepest descent analysis, we establish scaling limits, proving convergence of origami maps to a maximal surface in the Minkowski space $\mathbb{R}^{2,1}$, and we verify structural rigidity conditions, leading to a new proof of convergence of height fluctuations to the Gaussian free field.