Conveners
Lecture 2: Limit shapes
- Philippe Di Francesco
Description
A natural question in physics regards the thermodynamic limit, of large size and small mesh, where one expects to capture only the essential properties of a model, independently of its details. When it comes to tilings of specific domains a whole theory of limit shapes was developed in mathematics by Kenyon, Okounkov, Sheffield and others. These are generalizations of the famous arctic circle theorem of Cohn, Kenyon, Propp for the domino tilings of the so-called Aztec diamond. From the physics point of view, tiling problems are free fermion models, namely they can be expressed in terms of ``fermionic" degrees of freedom, generally in the form non-intersecting paths. However Vertex models such as the 6 vertex model are models of interacting fermions, where paths are allowed to touch thus violating the fermionic condition. In this lecture, we explore limit shapes of 6 and 20 vertex models, establishing their arctic phenomena by using their integrability. We find some remarkable relations between limit shapes of equinumerous yet non-bijective problems explored in lecture 1.
