Higher Order Voronoi Diagram of a Poisson Point Process: What do we know?

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This question is looking for probabilistic results of the Voronoi diagrams of 2-D space when the points are distributed by a homogeneous Poisson point process. The results can be the distribution of the surface area of the cells, distribution of the number of edges, etc.

Specifically 2nd or higher order Voronoi diagrams (2nd order meaning the area close to two points among the Poisson points). For first order Voronoi diagrams with Poisson point processes here are some results. Does anyone know of results for higher order case please?

Thank you.