Asymptotic shapes of crystalline surfaces

This is joint work with Andrei Okounkov. We study a simple model of crystalline surfaces in R^3. These come from limits of discrete interfaces in the dimer model (domino tiling model), and can be viewed as a higher-dimensional generalization of the simple random walk, where the domain is (part of) Z^2 instead of Z. We are interested in the behavior of these interfaces in the scaling limit (limit when the mesh tends to zero): the limit surfaces minimize a certain surface tension functional which arises from purely entropic considerations. Remarkably, the limit surfaces, which are solutions of a nonlinear PDE, can be parametrized by analytic functions and may contain facets in certain rational directions.

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