TITLE: Probability of Generating a Connected Graph on the Unit Square
QUESTION [10 upvotes]: $N$ points are generated randomly on the unit square, with a uniform distribution. What is the probability that they form a connected graph, given that two points are connected iff the distance between them is less than or equal to $d\in(0,\sqrt{2})$?
This should obviously be some function of $N$ and $d$.

REPLY [3 votes]: Let's brute force it.  Start with code for Kruskal's algorithm, and then select a number of points (like 10), and find the maximal distance required by Kruskal over 10000 trials.  With 10 points, 20 points, and 30 points, I got figures like the following:

Here's code for the 30 point image.
Histogram[Table[Max[With[{pts = RandomReal[{0,1},{30,2}]},
EuclideanDistance[pts[[#]][[1]],pts[[#]][[2]]]&/@List@@@Kruskal[pts]]],{10000}]]

Looks like you could fit a distribution curve to those pretty easily.  But that's beyond the duties of a brute forcer, so I'll stop there.