Noise sensitivity of random walks on groups

Itaï Benjamini (1) , Jérémie Brieussel (2)

1 Department of Computer Science and Applied Mathematics [Rehovot]
2 IMAG - Institut Montpelliérain Alexander Grothendieck

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.

Domaines: Mathématiques [math]

Itaï Benjamini, Jérémie Brieussel. Noise sensitivity of random walks on groups. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2023, 20 (2), pp.1139. ⟨10.30757/ALEA.v20-42⟩. ⟨hal-02048243⟩