If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? Equivalently, I want the probability that the area covered by the circles is $k\pi r(k)^{2}$.
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This is equivalent to asking what the distribution of minimal interpoint distance is. This is addressed in a number of places, in particular, in this article of Tanagawa, Mochizuki, Tanaka, 1992.
answered Oct 8, 2014 at 21:10
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$\begingroup$ Thanks, could this stuff be generalized if my points where say in a square or circle? $\endgroup$ Commented Oct 9, 2014 at 9:03
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$\begingroup$ @PavanSangha Yes, these sorts of questions have been done in convex polygons - generalization is usually not trivial, but not super hard (and for large $k,$ the results are usually more or less independent of the enclosing shape). $\endgroup$ Commented Oct 9, 2014 at 11:26
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$\begingroup$ Brilliant, do you have any links to papers? $\endgroup$ Commented Oct 13, 2014 at 11:27
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$\begingroup$ For large $n,$ this reduces to a local question, so the distance will be, roughly a function of the area of the region and the number of points. Just google "Poisson point process". $\endgroup$ Commented Oct 13, 2014 at 13:44
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$\begingroup$ Sorry could you elaborate a little more? $\endgroup$ Commented Oct 13, 2014 at 15:35