Speaker: Marianna Russkikh (University of Notre Dame)

Title: From Dimers to Maximal Surfaces in Minkowski Space R^{2,2}

Abstract: 

We discuss a class of graph embeddings into Minkowski space R^{2,2}= C^{1,1}, called t-surfaces, which arise in the study of the planar dimer model. A t-surface consists of a (perfect) t-embedding together with its associated origami map. Perfect t-embeddings were recently introduced as a key tool for proving convergence of the gradient of the dimer height function to the gradient of the Gaussian Free Field in a canonically associated metric, under suitable technical assumptions. After introducing the notion of a t-embedding, we present examples in which explicit constructions of perfect t-embeddings are known, and in which the corresponding t-surfaces converge to space-like maximal surfaces in Minkowski space R^{2,1}. As a consequence, these constructions yield a new proof of convergence of the (gradient of the) fluctuations of the dimer height function to the (gradient of the) Gaussian Free Field in the conformal structure induced by the limiting maximal surface. In addition, I will discuss a class of dimer graphs for which we study t-surfaces without verifying all of the assumptions required to obtain a full proof of convergence of the height fluctuations. Instead, the focus will be on illustrating several interesting connections between t-surfaces and the spectral data of the corresponding dimer model.