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.

arXiv commentPaper streamlined. An error in the proof of Theorem 6.1 has been corrected