Heuristic theory for many-facedd-dimensional Poisson–Voronoi cells
H. J. Hilhorst · Journal of Statistical Mechanics Theory and Experiment · 2009
We consider the d-dimensional Poisson-Voronoi tessellation and investigate the applicability of heuristic methods developed recently for two dimensions.Let p n (d) be the probability that a cell have n neighbors (be 'nfaced') and m n (d) the average facedness of a cell adjacent to an n-faced cell.We obtain the leading order terms of the asymptotic large-n expansions for p n (d) and m n (3).It appears that, just as in dimension 2, the Poisson-Voronoi tessellation violates Aboav's 'linear law' also in dimension 3. A comparison of this statement against existing Monte Carlo work remains inconclusive.However, simulations upgraded to the level of present-day computer capacity will in principle be able to confirm (or invalidate) our theory.