The Riemann-Hilbert approach to the Aztec Diamond

The Aztec diamond is a random tiling model whereby a diamond-shaped region is randomly tiled with 1x2 and 2x1 dominos. While simple, this model serves as an important toy problem for the emergence of various phenomenon that have been observed in physical systems of interest, like the six-vertex model or the Ising model. The novelty considered in the present proposal is the particular choice of the model's parameters (or weights); namely we take these weights to be themselves random. The motivation for this choice lies in the fact that random weights are expected to model physical systems with impurities. A topic of great practical importance, as impurities are not only ubiquitous in all condensed matter systems, but also influence key physical properties like conductance or magnetism. The method of choice for the present proposal is primarily the Riemann-Hilbert approach. This is a powerful complex analytic technique to study various problems in asymptotic analysis, and has been used in recent years with great success in the study of a variety of models of the Aztec diamond. However, so far the case of random weights was not considered using this approach. It should be emphasized that the choice of random weights presented here will naturally lead to a random Riemann-Hilbert problem. Such problems appeared only recently in the analysis of soliton gases, but so far not in the context of random tilings. Thus, one of the aims of the present proposal is to further develop the theory of random Riemann-Hilbert problems in this new context, thereby extending the reach of the Riemann-Hilbert method.