TITLE: Mean minimum distance for N random points on a unit square (plane)
QUESTION [9 upvotes]: A previously posted question "mean minimum distance for N random points on a one-dimensional line" produced an elegant answer: for a line of length L, the expected minimum distance (between random points) = L/(N^2/1), or simply 1/(N^2-1) for a line of unit length. 
I am wondering what the corresponding value would be for the unit plane? I am not a mathematician, but my intuition suggests the answer is simply 1/(N -1), given N points on a unit square (plane). 
Am I correct?

REPLY [5 votes]: This is a continuation to Douglas Zare's answer. I believe his lower bound of $1/n\sqrt{2}$ is the correct asymptotic behaviour. To see that, notice that all in inequalities in the calculation of the lower bound tend to equalities as $n\to \infty$, with the possible exception of the first inequality.
But in fact, the first inequality also tend to an equality when $n\to \infty$. This is because the error term is the measure of the overlap between discs of radius $x$ around the $k$ points and for $x=O(1/n)$, which is the relevant order of magnitude, this overlap is negligible - it's $O(k^2/n^4)$ whereas the measure of the discs is $\Theta(k/n^2)$.