The Squish Map and the SL_2 Double Dimer Model

• Leigh Foster (University of Waterloo)

A plane partition, whose 3D Young diagram is made of unit cubes, can be approximated by a "coarser" plane partition, made of cubes of side length 2. We relate this coursening operation to the squish map, which is a measure-preserving map between the dimer model on the honeycomb graph, and the SL_2 double dimer model on a coarser honeycomb graph. This allows us to re-use existing computations of the 2-periodic single-dimer partition function (and, in principle, the correlation functions) in a portion of the parameter space of the harder double-dimer model. The other direction of the map allows for some interesting conjectures in plane partition enumeration, when some of the generating function variables are specialized to roots of unity.

The dimer model and geometric dynamics

• Paul Melotti (Université Paris-Saclay)

Several objects from discrete geometry may be seen as dynamical systems: an example is a discrete version of holomorphic functions via Shramm's orthogonal circle patterns, for which "analytic continuation" gives rise to discrete time dynamics. We will see that this evolution may as well be seen as a birational map, and that its iterates are encoded by the partition function of a model from statistical mechanics, the dimer model. This correspondence allow us to answer some questions about the initial dynamics, such as the study of its singularities. Joint works with Niklas Affolter and Béatrice de Tilière.