Consider the random Cayley graph of a finite, Abelian group $G = \oplus_{j=1}^d \mathbb{Z}_{m_j}$ with respect to $k$ generators chosen uniformly at random. We prove that the simple random walk on this graph exhibits abrupt convergence to equilibrium, known as cutoff, subject to $k \gg 1$, $\log k \ll \log |G|$ and mild conditions on $d$ and $\min_j m_j$ in terms of $|G|$ and $k$. In accordance with spirit of a conjecture of Aldous and Diaconis, the cutoff time is shown to be independent of the algebraic structure of the group; it occurs around the time that the entropy of the simple random walk on $\mathbb{Z}^k$ is $\log|G|$, independent of $d$ and $\{m_j\}_{j=1}^d$. Moreover, we prove a Gaussian profile of convergence to equilibrium inside the cutoff window. We also prove that the order of the spectral gap is $|G|^{-2/k}$ with high probability (as $|G|$ and $k$ diverge); this extends a celebrated result of Alon and Roichman. 查看全文>>