  Consider the random Cayley graph of a finite Abelian group $G$ with respect
to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$.
Draw a vertex $U \sim \operatorname{Unif}(G)$.
  We show that the graph distance $\operatorname{dist}(\mathsf{id},U)$ from the
identity to $U$ concentrates at a particular value $M$, which is the minimal
radius of a ball in $\mathbb Z^k$ of cardinality at least $|G|$, under mild
conditions. In other words, the distance from the identity for all but $o(|G|)$
of the elements of $G$ lies in the interval $[M - o(M), M + o(M)]$. In the
regime $k \gtrsim \log |G|$, we show that the diameter of the graph is also
asymptotically $M$. In the spirit of a conjecture of Aldous and Diaconis
(1985), this $M$ depends only on $k$ and $|G|$, not on the algebraic structure
of $G$.
  Write $d(G)$ for the minimal size of a generating subset of $G$. We prove
that the order of the spectral gap is $|G|^{-2/k}$ when $k - d(G) \asymp k$ and
$|G|$ lies in a density-$1$ subset of $\mathbb N$ or when $k - 2 d(G) \asymp
k$. This extends, for Abelian groups, a celebrated result of Alon and Roichman
(1994).
  The aforementioned results all hold with high probability over the random
Cayley graph.
