Mixing and cut-off in cycle walks

Given a sequence $({\mathfrak{X}}_i,{\mathcal{K}}_i{)}_{i=1}^{\mathrm{\infty }}$ of Markov chains, the cut-off phenomenon describes a period of transition to stationarity which is asymptotically lower order than the mixing time. We study mixing times and the cut-off phenomenon in the total variation metric in the case of random walk on the groups $\mathbb{Z}\mathrm{/}p\mathbb{Z}$, $p$ prime, with driving measure uniform on a symmetric generating set $A_p\subset \mathbb{Z}\mathrm{/}p\mathbb{Z}$.