Mixing Time and Cutoff for a Random Walk on the Ring of Integers mod $n$

Abstract: We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive step' or a multiplicativejump'. When the probability of making a jump tends to zero as an appropriate power of $n$ we prove the existence of a total variation pre-cutoff for this walk. In addition, we show that the process obtained by subsampling our walk at jump times exhibits a true cutoff, with mixing time dependent on whether the step distribution has zero mean.