Nearest neighbors and Voronoi regions in certain point processes

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Abstract
We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that Poisson (λ= 1). An asymptotic Poisson (λ= 1) distribution also holds for many of the other models. On the other hand, we find that . Related results concern the (random) volume, , of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that while

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