Conformal invariance and universality of the dimer model

Abstract

This thesis is dedicated to the study of the conformal invariance and the universality of the dimer model on planar bipartite graphs. Kenyon [41, 42] has established the conformal invariance of the limiting distribution of the dimer height function in the case of Temperleyan discretizations, discrete domains on the square lattice with special boundary conditions. In the thesis, we extended Kenyon’s result for more general classes of approximations on the square lattice. Yet another direction of research in the dimer model is the universality (which means that the scaling limit is independent of the shape of the lattice) of the planar dimer model. We describe how to construct a circle pattern embedding of a dimer planar graph using its Kasteleyn weights. We also introduce the definition of discrete holomorphicity on such an embedding. We focus on understanding the link between these functions and actual continuous holomorphic functions to study holomorphic observables of the dimer model.