Matthew Nicoletti: Perfect t-embeddings of the uniform hexagon

Speaker
Matthew Nicoletti, MIT

Abstract
We construct and study the asymptotic properties of “perfect t-embeddings” of uniformly weighted hexagon graphs. Hexagon graphs are subgraphs of the honeycomb lattice, and the corresponding dimer model is equivalent to the model of uniformly random lozenge tilings of the hexagon. We provide exact formulas describing the perfect t-embeddings of these graphs, and we use these to prove the convergence of naturally associated discrete surfaces (coming from the “origami maps”) to a maximal surface in Minkowski space carrying the conformal structure of the limiting Gaussian free field (GFF). The emergence of such a maximal surface is predicted to hold for a large class of dimer models by Chelkak, Laslier, and Russkikh. In addition, we check all conditions of a theorem of Chelkak, Laslier, and Russkikh which uses perfect t-embeddings to prove convergence of height fluctuations to the GFF, and thus we complete give a new proof, via t-embeddings, of convergence to the GFF. This is based on joint work with Marianna Russkikh and Tomas Berggren.