Kurt Johansson: The two-periodic Aztec diamond

Abstract: Random uniform domino tilings of the region called the Aztec diamond exhibit interesting featureand statistical properties related to random matrix theory. For example there is a stochastic process describing therandom interface between a regularly tiled region (frozen phase) and an irregularly tiled region (liquid phase). In the limit of a large tiling this interface process converges to a limiting process called the Airy process which has been much studied recently in models of random growth. As a statistical mechanical model a tiling of an Aztec diamond can be thought of as a dimer model or as a certain random surface. The Aztec diamond with a two-periodic weighting exhibits all the three possible phases that can occur in these models usually called
liquid, solid and gas. This model is considerably harder to analyze than the standard one-periodic or uniform  model which only has two phases, liquid and solid. I will give an overview of some of the asymptotic results that have been obtained in joint work with Sunil Chhita. The tiles in the Aztec diamond form a determinantal point process. We are able to obtain a useful double contour integral formula for what is called the inverse Kasteleyn matrix, which then gives a formula for the
correlation kernel of the determinantal point process.