Random Geometric graphs (graphs where n points are placed at random in the unit square, and two nodes are connected with probability 1 if $r \leq r^*$) are known to percolate iff:
$$\pi r^2 = \frac{\log{n} + c\left(n\right)}{n}$$
This implies that the diameter of the graph is $\Theta\left(\sqrt{\frac{\pi n}{\log{n}}}\right)$.
Is it known what the distribution of expected path length between any two nodes is for this graph is, assuming we are in the supercritical regime, especially for fixed $n$?