Long-range geometrical correlations in two-dimensional foams

The statistical properties of two-dimensional, space-filling random cellular structures (foams, or their dual, random triangulations) in statistical equilibrium are obtained by maximum entropy inference and topological simulations. We show by maximum entropy inference that for a broad class of foams (shell-structured, including three-sided cell inclusions), all two-cell topological correlators A(j)(k,n) (average number of pairs of k-cell and n-cell at a topological distance j) are linear in n and k, the numbers of neighbours of the cells. This generalizes a correlation known for neighbouring cells (j = 1) which implies the linearity of Aboav's relation (between the total number of neighbours of the cells adjacent to a n-neighboured cell and n). Our results, verified by simulations, also build up Gauss's theorem for cellular structures. Any additional restriction in exploring local cell configurations, besides the constraints of filling space at random, will manifest itself through a deviation from linearity of the correlators A(j)(k,n) and the Aboav relation. Notably, foams made of Feynman diagrams have additional, context-dependent restrictions and their Aboav relation is slightly curved. It is essential that the local random variable n denotes the number of neighbours of the cell and not that of its sides, whenever the two are different.