Large random tilings of a hexagon have the fascinating behavior of separation of phases (frozen and rough; also called solid and liquid) that are separated by a well-defined Arctic curve.

In a weighted tiling model with periodically varying weights a third phase (smooth; or gaseous) appears where correlations between tiles decay at an exponential rate.

After a general introduction, I will discuss a technique based on matrix valued orthogonal polynomials to analyse a particular case of the three-periodic lozenge tiling model.