Dimers and M-curves: Limit Shapes from Riemann Surfaces
Explore a comprehensive lecture on dimer models and M-curves presented by Alexander Bobenko from Technische Universität Berlin at IPAM's Statistical Mechanics and Discrete Geometry Workshop. Delve into a general approach to dimer models analogous to Krichever's scheme in integrable systems theory, leading to models on doubly periodic bipartite graphs with quasiperiodic positive weights. Discover how this generalization from Harnack curves to M-curves reveals transparent algebro-geometric structures, with the Ronkin function and surface tension expressed as integrals of meromorphic differentials. Examine explicit representations for limit shapes in terms of Abelian integrals, and understand the connection to discrete conformal mappings and hyperbolic polyhedra. Learn about the application of Schottky uniformization of Riemann surfaces in computing weights and dimer configurations, with computational results aligning with theoretical predictions.