Abstract
We consider the d-dimensional Poisson–Voronoi tessellation and investigate the applicabilityof heuristic methods developed recently for two dimensions. Letpn(d) be the probabilitythat a cell have n neighbors (be ‘n-faced’) and mn(d) the average facedness of a cell adjacent to ann-faced cell. We obtain the leading order terms of the asymptotic large-n expansionsfor pn(d) and mn(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 MonteCarlo work remains inconclusive. However, simulations upgraded to the level ofpresent-day computer capacity will in principle be able to confirm (or invalidate) ourtheory.
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