Recent Advances in the Dimer Model and Its Applications

This lecture discusses recent developments in the dimer model, focusing on its applications in probability theory and conformal field theory. The instructor begins by defining the dimer model using a planar bipartite graph, explaining the concept of dimer configurations as perfect matchings. The notion of height functions, introduced by Thurston, is presented as a key analytical tool for understanding the model. The lecture explores the large-scale behavior of the height function, emphasizing the effects of boundary conditions and the significance of Kasteleyn's solution in computing partition functions. The instructor highlights the importance of Temperleyan boundary conditions and their role in establishing conformal invariance. The discussion extends to Riemann surfaces, where the dimer model's behavior is analyzed in relation to punctures and conical singularities. The lecture concludes with conjectures regarding the Sine-Gordon field and its connection to the dimer model, illustrating the rich interplay between combinatorial structures and probabilistic phenomena.