## Bernoulli

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

### Mixing time and cutoff for a random walk on the ring of integers mod $n$

Michael Bate and Stephen Connor

#### 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**

Received: December 2015

Revised: February 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.3150/16-BEJ832

**Mathematical Reviews number (MathSciNet)**

MR3706784

**Zentralblatt MATH identifier**

06778355

**Keywords**

cutoff phenomenon group representation theory mixing time pre-cutoff random number generation random walk

#### 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