Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces

Abstract

This is the second paper in the series devoted to the study of the dimer model on t-embeddings of planar bipartite graphs. We introduce the notion of perfect t-embeddings and assume that the graphs of the associated origami maps converge to a Lorentz-minimal surface ${\mathrm{S}}_{\xi }$ as $\delta \to 0$. In this setup we prove (under very mild technical assumptions) that the gradients of the height correlation functions converge to those of the Gaussian Free Field defined in the intrinsic metric of the surface ${\mathrm{S}}_{\xi }$. We also formulate several open questions motivated by our work.

arXiv comment39 pages, 3 figures

References in corpus (8)

• Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
• Fluctuations of particle systems determined by Schur generating functions
• Asymptotic domino statistics in the Aztec diamond
• Dimers and Circle patterns
• The Coleman correspondence at the free fermion point
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Cited by in corpus (3)

• Lozenge tilings and the Gaussian free field on a cylinder
• Fluctuations in the Aztec diamonds via a space-like maximal surface in Minkowski 3-space
• Möbius invariant Y-systems (cluster structures) for Miquel dynamics