  In this paper we consider domino tilings of the Aztec diamond with doubly
periodic weightings. In particular a family of models which, for any $ k \in
\mathbb{N} $, includes models with $ k $ smooth regions is analyzed as the size
of the Aztec diamond tends to infinity. We use a non-intersecting paths
formulation and give a double integral formula for the correlation kernel of
the Aztec diamond of finite size. By a classical steepest descent analysis of
the correlation kernel we obtain the local behavior in the smooth and rough
regions as the size of the Aztec diamond tends to infinity. From the mentioned
limit the macroscopic picture such as the arctic curves and in particular the
number of smooth regions is deduced. Moreover we compute the limit of the
height function and as a consequence we confirm, in the setting of this paper,
that the limit in the rough region fulfills the complex Burgers' equation, as
stated by Kenyon and Okounkov.
