Combinatorics Seminar

The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality.

The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig's theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes.

I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. We will see how various known results from total positivity provide answers to old and new questions regarding the Ising model. We will also explore the topology of the underlying spaces (joint work with S. Karp and T. Lam), and discuss how several other topics (such as electrical networks and the amplituhedron) fit into our story. Speaker(s): Pavel Galashin (M.I.T.)