Random Vornoi Diagrams (particular measures)

This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for.

I'm interested to know whether there are any known results about the following measures in 2-dimensional (preferably) random Vornoi diagrams. I think i should state also that i'm working in a circular disk so if the setting of any currently known results is different would the nature of the results change if i was to change the setting?

The measures are

a) The variance of the number of neighbours of each cell in the Vornoi Diagram. b) The variance of the length of the edges in the random Delaunay triangulation. c) The variance of the smallest, medium, and largest angles in the Delaunay triangulation.

I know it's perhaps not feasible to get information on the exact measures i am looking for, i'm therefore interested in anything related to the measures.

Thanks

2 Answers

Perhaps search from this first paper, which presents "a concise review of advances ... on planar Poisson-Voronoi tesselations":

Hilhorsta, H. J. "Statistical properties of planar Voronoi tessellations." The European Physical Journal B-Condensed Matter and Complex Systems 64.3 (2008): 437-441. (journal link)

And then work backwards through the references. For example, he computes "the average sidedness of a cell neighbouring an $n$-sided cell":

Hilhorst, Hendrik-Jan. "Asymptotic statistics of the $n$-sided planar Poisson–Voronoi cell: I. Exact results." Journal of Statistical Mechanics: Theory and Experiment 2005.09 (2005): P09005. (journal link)

And you can also work forward from 2008 via Google Scholar.

• Brilliant thanks for the great references! – Pavan Sangha Sep 15 '14 at 13:40

Page 104 of Moller's book (which I mentioned in my answer to your previous question) has the heading "On the distribution of the typical Poisson-Delaunay cell and related statistics".

It gives an exact distribution for the distance between the nucleus and vertices of the cell plus the joint distribution for the angles to the vertices as seen from the nucleus.

He then goes on to deduce the moments (e.g. variance) of some of these quantities.

Of course this is for the tesselation associated to a homogeneous Poisson point process on the entire plane. It may not be exactly what you're looking for... but I think you should at least take a look and try to understand what's done there. He also gives a reference to a 1992 paper of Rathie for some of the results and the planar case is discussed separately.