## Bernoulli

• Bernoulli
• Volume 24, Number 2 (2018), 993-1009.

### 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 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.

#### Article information

Source
Bernoulli, Volume 24, Number 2 (2018), 993-1009.

Dates
Revised: February 2016
First available in Project Euclid: 21 September 2017

https://projecteuclid.org/euclid.bj/1505980886

Digital Object Identifier
doi:10.3150/16-BEJ832

Mathematical Reviews number (MathSciNet)
MR3706784

Zentralblatt MATH identifier
06778355

#### Citation

Bate, Michael; Connor, Stephen. Mixing time and cutoff for a random walk on the ring of integers mod $n$. Bernoulli 24 (2018), no. 2, 993--1009. doi:10.3150/16-BEJ832. https://projecteuclid.org/euclid.bj/1505980886

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