Longest edge length of the greedy spanner on uniformly distributed pointsets in $[0,1]^d$
In our paper Distribution-Sensitive Construction of the Greedy Spanner (accepted for ESA 2014) we prove the following (combining Theorem 4 and Lemma 6):


  There exists $c_t$ dependent only on $t$ such that for every $c &gt; 0$, if $P$ is a set of points uniformly and independently distributed at random in a $\sqrt{n} \times \sqrt{n}$ square and $n$ is large enough, then with probability at least $1 - n^c$, the greedy $t$-spanner on $P$ does not have edges longer than $c \cdot c_t \log n$.


Our bound on $c_t$ is fairly large, but we have experimental evidence that the 'right' bound is $\frac{1}{\sqrt[4]{t-1}} \cdot \frac{\log n}{\log \log n}$.