# Noise sensitivity of random walks on groups

## TL;DR

This paper investigates how small perturbations in the steps of random walks on groups affect their outputs, introducing precise notions of noise sensitivity and exploring their implications and examples.

## Contribution

It defines two types of noise sensitivity for random walks on groups, links these properties to the Liouville property, and provides examples and open questions.

## Key findings

- Groups with noise sensitivity are necessarily Liouville.
- Homomorphisms to free abelian groups obstruct $ ext{l}^1$-noise sensitivity.
- Examples of noise sensitive random walks are constructed.

## Abstract

A random walk on a group is noise sensitive if resampling every step independantly with a small probability results in an almost independant output. We precisely define two notions: $\ell^1$-noise sensitivity and entropy noise sensitivity. Groups with one of these properties are necessarily Liouville. Homomorphisms to free abelian groups provide an obstruction to $\ell^1$-noise sensitivity. We also provide examples of $\ell^1$ and entropy noise sensitive random walks. Noise sensitivity raises many open questions which are described at the end of the paper.