We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive ‘step’ or a multiplicative ‘jump’. 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.
"Mixing time and cutoff for a random walk on the ring of integers mod $n$." Bernoulli 24 (2) 993 - 1009, May 2018. https://doi.org/10.3150/16-BEJ832