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Mixing time and cutoff for a random walk on the ring of integers mod \(n\). (English) Zbl 1429.60008
Summary: 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.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60B10 Convergence of probability measures
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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