Topological correlations in cellular structures and planar graph theory
Claude Godrèche, Ivan K. Kostov, I. Yekutieli · Physical Review Letters · 1992
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.