Large random tilings of a hexagon display the fascinating phenomenon of distinct asymptotic phases (frozen and rough, also referred to as solid and liquid), separated by a well-defined Arctic curve. In a weighted tiling model with periodically varying weights, a third phase (smooth, or gaseous) emerges, in which correlations between tiles decay at an exponential rate. After a general introduction, I will discuss an approach towards a rigorous analysis of a three-periodic hexagon tiling model. This involves matrix-valued orthogonal polynomials, Riemann-Hilbert problems, and steepest descent analysis on a Harnack curve.
|