Topological correlations in cellular structures and planar graph theory

We study the topological properties of the ensemble of cells with trivalent vertices in the plane. The problem is equivalent to counting planar Feynman diagrams with a cubic interaction. This ensemble is also the equilibrium state of a topological model of cellular structure, obtained by applying repetitively a topological flip transformation to any initial configuration of cells with trivalent vertices. We give analytical expressions for two-cell correlations; in particular, we give the analytical form of the average number m(n) of edges of cells adjacent to an n-sided cell. These results are confirmed by simulations.