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

TL;DR

This work connects the scaling limit of dimer fluctuations on homogeneous Aztec diamonds to the intrinsic conformal structure of a space-like maximal surface in $\mathbb{R}^{2,1}$ by using symmetric t-embeddings and origami maps. The authors establish a recursive construction of $\mathcal{T}_n$ and $\mathcal{O}_n$ for reduced Aztec diamonds, show these objects satisfy a discrete wave equation on a growing cone with explicit boundary data, and demonstrate that, in the limit, the origami/embedding data converge to a space-like maximal surface $\mathrm{S}_\diamondsuit$ whose conformal structure governs the Gaussian fluctuations in the liquid region. They provide analytic connections via harmonic measures and a conformal parameter $\zeta$, and validate the picture through numerical simulations that visualize convergence to the Lorentzian surface. These results substantiate the CLR2CLR1 framework in a classical setting and offer a concrete Aztec-diamond instance where a space-like maximal surface describes dimer fluctuations, with rigorous aspects further developed in related work (e.g., $\text{BNR}$).

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.

Figures (6)

• Figure 1: Local change of a planar graph carrying the bipartite dimer model and of the edge weights under the urban renewal move.
• Figure 2: Aztec diamond $A_{n+1}$ and its reduction $A'_{n+1}$ for $n=3$. Double edges of $A'_{n+1}$ have weight $2$. The labeling $(j,k)$ of faces of $A'_{n+1}$ is shown, $|j|+|k|\!<\!n$.
• Figure 3: A sequence of updates leading from $A'_4$ to $A'_5$. Top row: graphs carrying the dimer model; double, thick and thin edges have weight $2$, $1$ and $\frac{1}{2}$, respectively. Bottom row: the same moves lead from the t-embedding $\mathcal{T}_3$ to $\mathcal{T}_4$; the last move (contraction of vertices of degree $2$) removes dashed edges. Positions of urban renewals/central moves are indicated by white squares.
• Figure 4: Reduced Aztec diamonds $A'_2$, $A'_3$, $A'_4$ and their symmetric t-embeddings $\mathcal{T}_1$, $\mathcal{T}_2$, $\mathcal{T}_3$. Bottom-right: illustration of the origami maps $\mathcal{O}_3$ and $\mathcal{O}_2$; the colors of faces inherit those from the north-east parts of $\mathcal{T}_3$ and $\mathcal{T}_2$. As discussed below (see Remark \ref{["rem:O'def"]} and Eq. \ref{['eq:f_0=o(1)']}), the images of the origami maps $\mathcal{O}_n$ are asymptotically one-dimensional for large $n$.
• Figure 5: Left: (symmetric) t-embedding of the Aztec diamond of size $27$. Right: t-embedding of the Aztec diamond of size $102$, the edges connecting $\mathcal{T}_{101}(j_1,k_1)$ and $\mathcal{T}_{101}(j_2,k_2)$ are colored red if both $(j_{1,2}^2+k_{1,2}^2)\le 0.49\cdot 10^4$, blue if both $(j_{1,2}^2+k_{1,2}^2)\ge 0.5\cdot 10^4$, and are not shown otherwise.

Theorems & Definitions (15)

• Remark 1.1
• Remark 1.2
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Definition 2.6