In recent years important progress has been made in the understanding of random tilings of large Aztec diamonds with doubly periodic weights. Due to the double periodicity a new phase appears that has not been observed in tiling models with uniform weights. One of the challenges is to find expressions of for the correlation functions that are amenable for asymptotic studies. In the case of the uniform weight the model is an example of a Schur process and, consequently, such expressions for the correlation functions are known and well-studied in that case. In a joint work with Tomas Berggren we studied a more general integrable structure that includes certain doubly periodic weightings planar domains, such as the Aztec diamond. A key feature is a dynamical system hiding in the background. In case of a periodic orbit, explicit double integrals for the correlation function can be found, paving the way for an asymptotic study using saddle point methods.
