  Random domino tilings of the Aztec diamond shape exhibit interesting features
and some of the statistical properties seen in random matrix theory. As a
statistical mechanical model it can be thought of as a dimer model or as a
certain random surface. We consider the Aztec diamond with a two-periodic
weighting which exhibits all three possible phases that occur in these types of
models, often referred to as solid, liquid and gas. To analyze this model, we
use entries of the inverse Kasteleyn matrix which give the probability of any
configuration of dominoes. A formula for these entries, for this particular
model, was derived by Chhita and Young (2014). In this paper, we find a major
simplification of this formula expressing entries of the inverse Kasteleyn
matrix by double contour integrals which makes it possible to investigate their
asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is
simpler, we use this formula to show that the entries of the inverse Kasteleyn
matrix converge to the known entries of the full-plane inverse Kasteleyn
matrices for the different phases. We also study the detailed asymptotics of
the inverse Kasteleyn matrix at both the "liquid-solid" and "liquid-gas"
boundaries, and find the extended Airy kernel in the next order asymptotics.
Finally we provide a potential candidate for a combinatorial description of the
liquid-gas boundary.
