Wenkui Liu: Limit shape and global fluctuations for inhomogeneous random tilings

Lozenge tiling in a hexagon intuitively is piling up unit boxes in the corner of a room. Further, one can randomly pick one configuration out of all possible tilings according to a certain weight. The uniform weight case is well-studied in the literature. Nevertheless, less is known for inhomogeneous weights. One example is the q-Hahn weight, where the more boxes, the more likely to appear. This random tiling is revealed as an orthonormal polynomial ensemble. We can study hypergeometric weights up to q-Racah with their recurrence relation. We show the almost sure convergence for the height function of the tiling, called the limit shape, in the literature. We also show that the limit shape is the minimiser of surface tension. Furthermore, we prove that the push-forward of the scaled height function converges to the Gaussian free field, which, heuristically speaking, is a Gaussian fluctuation with logarithm covariance and exploding variance.