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

TL;DR

This paper investigates the mixing times of a random walk on the integers mod n that combines additive steps and multiplicative jumps, establishing conditions for cutoff phenomena depending on jump probabilities and step distribution properties.

Contribution

It introduces a new model of a random walk with additive and multiplicative moves and proves the existence of pre-cutoff and cutoff phenomena under specific conditions.

Findings

Pre-cutoff occurs when jump probability tends to zero at a certain rate.

Subsampled process at jump times exhibits a true cutoff.

Mixing time depends on whether the step distribution has zero mean.

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.