Colorings, Potts Models, Height Representations, and Entropic Forces

We will discuss the three-color model on the square lattice, and the four-color model on the triangular lattice, from a physicists' point of view (the so-called antiferromagnetic Potts models). Both of these have a height representation which allows us to idealize them, at large length scales, as being described by an elastic surface. In the latter case the height is two-dimensional, leading to a four-dimensional surface. I will review how such a representation gives rise to power-law correlations in the system, and how defects or vortices of opposite type attract each other with an entropic force --- a force which is driven by the fact that there are more ways for the surrounding lattice to be colored when the defects are closer together.