Random Cayley Graphs II: Cutoff and Geometry for Abelian Groups

Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log \lvert G \rvert$. A conjecture of Aldous and Diaconis asserts, for $k \gg \log \lvert G \rvert$, that the random walk on this graph exhibits cutoff at a time which is a function only of $\lvert G \rvert$ and $k$. The conjecture is verified for all Abelian groups. We extend the conjecture to all $1 \ll \log k \ll \log \lvert G \rvert$, and prove it for a large class of Abelian groups: the cutoff time is the time at which the entropy of simple random walk on $\mathbb Z^k$ is $\log \lvert G \rvert$. We also show that the graph distance from the identity for all but $o(\lvert G \rvert)$ of the elements of $G$ lies in $[M - o(M), M + o(M)]$, where $M$ is the minimal radius of a ball in $\mathbb Z^k$ of cardinality $\lvert G \rvert$. In the spirit of the Aldous--Diaconis conjecture, this $M$ depends only on $\lvert G \rvert$ and $k$.