The diameter of random Cayley digraphs of given degree

We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to \mathrm{\infty }$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$\mathrm{.}Inparticular,if$f(n)$isasymptoticallylinearin$n$, the probability converges exponentially fast to 1.