Georgia Institute of Technology College of Sciences

We will present the solution to a statistical mechanics model on random lattices. More precisely, we consider the Potts model on the set of planar triangulations (embedded planar graph such that every face has degree 3). The partition function of this model is the generating function of vertex-colored triangulations counted according to the number of monochromatic edges and dichromatic edges. We characterize this partition function by a simple system of differential equations. Some special cases, such as properly 4-colored triangulations, lead to particularly simple equations waiting for a more direct combinatorial explanation. This is joint work with Mireille Bousquet-Melou.