  We analyze a random lozenge tiling model of a large regular hexagon, whose
underlying weight structure is periodic of period $2$ in both the horizontal
and vertical directions. This is a determinantal point process whose
correlation kernel is expressed in terms of non-Hermitian matrix valued
orthogonal polynomials. This model belongs to a class of models for which the
existing techniques for studying asymptotics cannot be applied. The novel part
of our method consists of establishing a connection between matrix valued and
scalar valued orthogonal polynomials. This allows to simplify the double
contour formula for the kernel obtained by Duits and Kuijlaars by reducing the
size of a Riemann-Hilbert problem. The proof relies on the fact that the matrix
valued weight possesses eigenvalues that live on an underlying Riemann surface
$\mathcal{M}$ of genus $0$. We consider this connection of independent
interest; it is natural to expect that similar ideas can be used for other
matrix valued orthogonal polynomials, as long as the corresponding Riemann
surface $\mathcal{M}$ is of genus $0$.
  The rest of the method consists of two parts, and mainly follows the lines of
a previous work of Charlier, Duits, Kuijlaars and Lenells. First, we perform a
Deift-Zhou steepest descent analysis to obtain asymptotics for the scalar
valued orthogonal polynomials. The main difficulty is the study of an
equilibrium problem in the complex plane. Second, the asymptotics for the
orthogonal polynomials are substituted in the double contour integral and the
latter is analyzed using the saddle point method.
  Our main results are the limiting densities of the lozenges in the disordered
flower-shaped region. However, we stress that the method allows in principle to
rigorously compute other meaningful probabilistic quantities in the model.
