I randomly scatter $N$ points on a bounded rectangular plane $P$ with dimensions $A \times B$. To be more specific, for $N$ iterations, I choose a real number $x \in [0, A]$ and a real number $y \in [0, B]$, and place my point at the coordinates $(x, y)$.


Let $R$ be the radius of some circle I place somewhere on the plane $P$ that encloses no points. What is the probability distribution for the maximum permissible size of $R$?


Note - I specified $P$ to be a rectangular plane but, please, let it be any shape you wish if you can answer my question.



The largest circle that encloses no points on a plane with points placed at $N$ random coordinates
The largest empty ball in a point set is known in the literature as the *dispersion*; see, for example, [this definition](http://planning.cs.uiuc.edu/node203.html). This was explored a bit in a previous MO question, "[Finding the most-isolated point in a high-dimensional cube](https://mathoverflow.net/questions/98682/)." Here is a paper that explores dispersion. This could serve to forward-search in Google Scholar. Or you could hope that Günter Rote, who is now participating in MO, will comment directly:



> 
> Rote, Tichy.
>  "Quasi-Monte-Carlo methods and the dispersion of point sequences"
>  *Mathematical and Computer Modelling* 23 (1996), 9-23.
>  ([ACM link](http://dl2.acm.org/citation.cfm?id=2275170))
> 
> 
> 


See also the recent paper on largest empty rectangles:



> 
> Dumitrescu, Jiang.
>  "On the largest empty axis-parallel box amidst $n$ points."
>  2012. ([PDF download](http://www.cs.uwm.edu/faculty/ad/tau2.pdf))
> 
> 
> 


