Perfect t-embeddings of doubly periodic Aztec diamonds

Abstract: We study the large-scale geometry of t-surfaces -- pairs of perfect t-embeddings and their associated origami maps -- arising from dimer models on Aztec diamonds with periodic edge weights. We prove that these t-surfaces converge to space-like maximal surfaces in the Minkowski space $\mathbb{R}^{2,2}$. We observe that the frozen and gas regions influence the geometry of the limiting surface in striking ways: all frozen regions collapse to four boundary points, regardless of the number of frozen regions, while each gas region collapses to a distinct light-like cusp in the interior of the surface. In the absence of gas regions, the limiting surface lies entirely within $\mathbb{R}^{2,1}$; in the general case, however, this is no longer true. The limiting surface is sensitive to the detailed structure of the model: both the positions of the cusps, and the placement of the boundary vertices, depend on the precise way the edge weights are distributed on the Aztec diamond. Nevertheless, we show that the global conformal structure remains robust and coincides with the Kenyon-Okounkov conformal structure. We further conjecture that the cusp locations encode the shift in the discrete Gaussian component that appears in the global fluctuations of the dimer model.