Scaling limits of random structures in two dimensions often posses some conformal invariance properties, which can be analysed by different methods of geometric analysis. Random tilings are typical examples, with inviting questions on the geometry of their ordered and disordered (or frozen and liquid) limit regions.
Such liquid regions carry a natural complex structure, which can be described by a quasilinear Beltrami equation with very specific properties. On the other hand, the frozen boundary can be described by a specific (but very degenerate) free boundary problem.
In this presentation, based on joint work with E.Duse, I.Prause and X.Zhong, I show how these methods lead to an understanding and classifying the geometry of the limiting boundaries for different random tilings and other dimer models.
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