You may be interested in this paper, which takes a semi-empirical approach to come up with a closed-form approximate formula:

• Jarai, Neda: On the size-distribution of Poisson Voronoi cells https://arxiv.org/abs/cond-mat/0406116

At the time when it was written, Mathematica did not have VoronoiMesh, but one could "trick" it to quickly compute a Voronoi tessellation by using ListDensityPlot with InterpolationOrder -> 0 and extracting the polygons.

Thanks for sharing everybody! Maybe this can add some more information:

In the article "Statistical Distributions of Poisson Voronoi Cells in Two and Three Dimensions" by Masaharu Tanemura, mentioned by Marco Thiel, the distribution of the perimeter of the cells (and the number of edges etc.) is also analyzed. Here I adapted the code of Vitaly to perimeters and we see (as in the article) that now we have a normal distribution:

perimeters = Perimeter /@ vorInner;
hist = Histogram[perimeters, Automatic, "PDF", 
   PlotTheme -> "Detailed"];
dist = FindDistribution[perimeters]
Show[hist, Plot[PDF[dist, x], {x, 0, .1}]]

And for the number of edges, we find a BinomialDistribution with FindDistribution although the result look almost "normal" as we use EstimatedDitribution

numberOfEdges[poly_] := Length[First@poly]
edges = numberOfEdges /@ vorInner;
histE = Histogram[edges, Automatic, "PDF", PlotTheme -> "Detailed"];
distE = EstimatedDistribution[edges, 
  NormalDistribution[\[Mu], \[Sigma]]]
Show[histE, Plot[PDF[distE, x], {x, 0, 12}]]
distEB = FindDistribution[edges]
Plot[CDF[distEB, x], {x, 0, 12}, Filling -> Bottom]

Dear All,

that is indeed interesting. I also believe that the exact distribution is unknown but it can be approximated by a GammaDistribution, see:

https://arxiv.org/pdf/1207.0608.pdf

https://arxiv.org/pdf/1612.02375.pdf

https://www.sciencedirect.com/science/article/pii/002608008990030X

http://www.scipress.org/journals/forma/pdf/1804/18040221.pdf

http://www.eurecom.fr/~arvanita/PVT.pdf

Interestingly, if you run the code in @Vitaliy Kaurov's cod for 50000 points you obtain:

dis = FindDistribution[areas]
(*GammaDistribution[3.2718, 0.0000197063]*)

I managed to get it done for 500000 points. I think that the edge effects should be getting smaller; as the area increases faster than the circumference.

So I obtain:

dis = FindDistribution[areas, 5]

(*
{MixtureDistribution[{0.658342, 0.341658}, {MaxwellDistribution[2.92588*10^-6], GammaDistribution[7.1867, 1.37091*10^-6]}], 

ExtremeValueDistribution[4.80446*10^-6, 2.73831*10^-6], 

GammaDistribution[2.94963, 2.17794*10^-6], 

BetaDistribution[2.94962, 459143.], 

MixtureDistribution[{0.687855, 0.312145}, {MaxwellDistribution[2.82857*10^-6], 
   LogNormalDistribution[-11.5031, 0.28619]}]}*)

This happens when we plot all of them:

Plot[Evaluate[PDF[#, x] & /@ dis], {x, 0, 0.00003}, PlotRange -> All, LabelStyle -> Directive[Bold, 16]]

If you compare that with the histogram:

Show[Histogram[areas, 200, "PDF"], Plot[Evaluate[PDF[#, x] & /@ dis], {x, 0, 0.00003}, PlotRange -> All,LabelStyle -> Directive[Bold, 16]]]

I would say that the second (ExtremeValueDistribution) fits best:

but it does not get the small areas quite right.

Regarding the boundary effects, one might also (similar to what Sander suggests) use the torus as the geometry:

disfun[x_, y_] := Sqrt[Min[Abs[x[[1]] - y[[1]]], 1 - Abs[x[[1]] - y[[1]]]]^2 + Min[Abs[x[[2]] - y[[2]]], 1 - Abs[x[[2]] - y[[2]]]]^2]

M = 50; nf = Nearest[Rule @@@ Transpose[{RandomReal[1, {M, 2}], Range[M]}], DistanceFunction -> disfun]

DensityPlot[First[nf[{x, y}]], {x, 0, 1}, {y, 0, 1}, PlotPoints -> 100, ColorFunction -> "TemperatureMap"]

Cheers,

Marco

I don't find it all that surprising that the result is a Gamma distribution. The 1-d case is well known as it is the distribution of interarrival times of a Poisson process and is exponentially distributed. The exponential distribution is a special case of the Gamma so maybe you're onto something.

Thanks! Awesome idea about edge-effects! Did you estimate any analytical models beyond Gamma or it was just pure empirical thing?