  We analyze domino tilings of the two-periodic Aztec diamond by means of
matrix valued orthogonal polynomials that we obtain from a reformulation of the
Aztec diamond as a non-intersecting path model with periodic transition
matrices. In a more general framework we express the correlation kernel for the
underlying determinantal point process as a double contour integral that
contains the reproducing kernel of matrix valued orthogonal polynomials. We use
the Riemann-Hilbert problem to simplify this formula for the case of the
two-periodic Aztec diamond.
  In the large size limit we recover the three phases of the model known as
solid, liquid and gas. We describe fine asymptotics for the gas phase and at
the cusp points of the liquid-gas boundary, thereby complementing and extending
results of Chhita and Johansson.
