School of Mathematics Colloquium
Thursday, March 5, 2009 - 11:00
1 hour (actually 50 minutes)
This is joint work with Andrei Okounkov. The ``honeycomb dimer model'' is a natural model of discrete random surfaces in R^3. It is possible to write down a ``Law of Large Numbers" for such surfaces which describes the typical shape of a random surface when the mesh size tends to zero. Surprisingly, one can parameterize these limit shapes in a very simple way using analytic functions, somewhat reminiscent of the Weierstrass parameterization of minimal surfaces. This is even more surprising since the limit shapes tend to be facetted, that is, only piecewise analytic. There is a large family of boundary conditions for which we can obtain exact solutions to the limit shape problem using algebraic geometry techniques. This family includes the (well-known) solution to the limit shape of a ``boxed plane partition'' and has many generalizations.