B. Dubertret et al., LONG-RANGE GEOMETRICAL CORRELATIONS IN 2-DIMENSIONAL FOAMS, Journal of physics. A, mathematical and general, 31(3), 1998, pp. 879-900
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.