LONG-RANGE GEOMETRICAL CORRELATIONS IN 2-DIMENSIONAL FOAMS

The statistical properties of two-dimensional, space-filling random ce llular structures (foams, or their dual, random triangulations) in sta tistical equilibrium are obtained by maximum entropy inference and top ological simulations. We show by maximum entropy inference that for a broad class of foams (shell-structured, including three-sided cell inc lusions), all two-cell topological correlators A(j)(k,n) (average numb er of pairs of k-cell and n-cell at a topological distance j) are line ar in n and k, the numbers of neighbours of the cells. This generalize s 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, ve rified by simulations, also build up Gauss's theorem for cellular stru ctures. Any additional restriction in exploring local cell configurati ons, 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 i s slightly curved. It is essential that the local random variable n de notes the number of neighbours of the cell and not that of its sides, whenever the two are different.