Asymptotics of random discrete interfaces, Richard W. Kenyon, Université Paris-Sud and Princeton University

We discuss joint work with Andrei Okounkov. We study a model of random interfaces arising in the dimer model (domino tiling model). These are two-dimensional interfaces in R^3, 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 "scaling limit" (limit when the mesh tends to zero). Specifically, there is a "law of large numbers" which says that at small mesh size a typical surface lies very close to its mean value. The mean value surface for given boundary values satisfies a variational principle (minimizing a certain energy functional), leading to a nonlinear PDE. Remarkably, its solutions can be parametrized by analytic functions (in a similar way as soap bubbles) but here one can see facets appearing in the limit shapes.