  Given a sequence $(\mathfrak{X}_i, \mathscr{K}_i)_{i=1}^\infty$ 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 $\mathbb{Z}/p\mathbb{Z}$, $p$ prime, with driving measure
uniform on a symmetric generating set $A_p \subset \mathbb{Z}/p\mathbb{Z}$.
