Equilibrium crystal shapes for two-dimensional Potts models

Low temperature Potts model is the simplest statistical mechanical model of q co-existing phases. In this talk we shall explain how to prove that in two dimensions Potts equilibrium crystal shapes are always smooth and strictly convex. In other words, in two dimensions Potts models do not undergo roughening transition. Since the models in question (except for the Ising q=2 case) are not exactly soluble, the proof relies on an intrinsic probabilistic analysis of random phase separation lines. The main step of the latter is to develop finite scale renormalization procedures which enable a coding of the interface distribution in terms of Ruelle operator for full shifts over countable alphabets.

Joint work with Massimo Camapanino and Yvan Velenik