I've searched the web now for ages to try and find a paper on the asymptotic distribution of the area of the union of randomly placed discs on the plane. Ideally, I would be looking for the discs to be placed in a unit square, but if anyone knows of other work that has been done, that would be great. I can calculate the expected area, and I have found a number of papers on the topic of 'coverage of wireless sensor networks' which give some useful ideas. Nevertheless, I was wondering whether this problem has ever been studied for its own sake.
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1$\begingroup$ Related: mathoverflow.net/questions/103432/… and math.stackexchange.com/questions/176383/… $\endgroup$– user6096Commented Jun 23, 2014 at 20:26
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If by "the area of ... circles" you mean the area of the union of disks, then there is work in the statistical literature, e.g.:
Zikmundová, Markéta, Kateřina Staňková Helisová, and Viktor Beneš. "Spatio-temporal model for a random set given by a union of interacting discs." Methodology and Computing in Applied Probability 14.3 (2012): 883-894.
Møller, Jesper, and Kateřina Helisová. "Likelihood inference for unions of interacting discs." Scandinavian Journal of Statistics 37.3 (2010): 365-381.
Møller, Jesper, and Kateřina Helisová. "Power diagrams and interaction processes for unions of discs." Advances in Applied Probability (2008): 321-347.
Fig. 1 from 3rd paper above.
answered Jun 23, 2014 at 10:56
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1$\begingroup$ Yes i did mean the union. Thank you very much this is helpful, sometimes the problem i have is not knowing exactly what to search for. $\endgroup$ Commented Jun 23, 2014 at 11:58