Periodic Dimer Models

The goal of this project is large scale analysis of probabilistic systems known as dimer models. They often have simple descriptions; for example, a randomly chosen tiling of the chessboard with domino tiles is one of them. Yet, as the domain that is being tiled becomes large, such systems exhibit intricate behavior and serve as mathematical models for phase transitions seen in nature. More concretely, the experimentally observed phenomenon of roughening of crystals at temperatures close to absolute zero has similar phenomenology. Mathematical analysis of dimer models is challenging, and it requires a diverse set of sophisticated tools from different branches of mathematics including algebra, representation theory, and algebraic geometry. For a broad range of planar domains, these tools will ultimately make it possible to analyze the large scale behavior at a very fine resolution, offering key insights into a much wider range of probabilistic and physical systems. The proposal is dedicated to dimer models on growing planar graphs with periodically varying edge weights. The overarching goal is to describe limit shapes and bulk/edge/global fluctuations around them for a broad class of domains as the size of the domain grows. Establishing an explicit correspondence between periodic dimer models and Yang-Baxter solvable Solid-On-Solid models of statistical physics is essential, and this bridge will provide access to new algebraic structures that in turn will yield asymptotic results. Another key connection is to the theory of Riemann surfaces, whose geometry is essential to describing the limiting behavior of the models. The zero-temperature limit will also be analyzed, and it will be described in terms of tropical geometry. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.