Mathematics Colloquium, Fall 2009
SPECIAL LECTURE
Yuliy Baryshnikov
Bell Laboratories
Algebraic curves and frozen regions in planar tilings
There are many examples of "Arctic circle" type phenomena in random tilings of planar regions such as domino tilings of Aztek diamonds or diabolo tilings of fortresses: regions with high correlations between tiling orientations are separated from those with nontrivial entropy (per area) by algebraic curves.
Our (with Robin Pemantle) recent results on asymptotics of multivariate generating function allow to derive the shapes of these boundaries easily as the duals to certain curves given by hyperbolic polynomials.