I'll describe how to invert the Kasteleyn matrix for a few of the most interesting weight functions on the Aztec Diamond. This, in principle, allows the computations of correlation kernels for the associated tiling model. In some of the simpler cases, it is possible to compute asymptotics for this kernel rigorously; though this is essentially a combinatorics talk, I'll mention the sorts of asymptotics which can be derived. In more difficult cases (specifically 2-periodic weights) the asymptotics are not yet tractable, although they represent perhaps the most promising way in which one might study the transition between liquid and gaseous regimes in a dimer model. Joint work with Sunil Chhita and Kurt Johansson.