Noise sensitivity of random walks on groups

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.