# Products of random walks on finite groups with moderate growth

## TL;DR

This paper investigates the cutoff phenomena in products of random walks on finite groups with moderate growth, establishing conditions for cutoffs and illustrating with examples like Heisenberg groups and finite cycles.

## Contribution

It introduces new comparison techniques linking total variation and Hellinger distance cutoffs, and provides criteria for the existence of cutoffs in these group products.

## Key findings

- Identified total variation cutoff for discrete-time lazy random walks
- Established equivalence of cutoff conditions including Peres' product condition
- Applied results to Heisenberg groups and finite cycle products

## Abstract

In this article, we consider products of random walks on finite groups with moderate growth and discuss their cutoffs in the total variation. Based on several comparison techniques, we are able to identify the total variation cutoff of discrete time lazy random walks with the Hellinger distance cutoff of continuous time random walks. Along with the cutoff criterion for Laplace transforms, we derive a series of equivalent conditions on the existence of cutoffs, including the existence of pre-cutoffs, Peres' product condition and a formula generated by the graph diameters. For illustration, we consider products of Heisenberg groups and randomized products of finite cycles.