Under suitable boundary conditions, scaling limits of random tilings present surprising geometric features: one observes definite deterministic and disordered (or frozen and liquid) limit configurations with interesting geometric properties.

In this talk we will show how with properties of the Beltrami equation it is possible to understand, for all dimer models and even beyond, the geometry of the boundary between the frozen/deterministic and liquid/random phase. It turns out that in this class the geometry frozen boundaries is universal, i.e. independent of the specific model.

The talk is based on a joint work with E. Duse, I. Prause and X. Zhong.