Noise sensitivity of random walks on groups

Itaï Benjamini; Jérémie Brieussel

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arXiv · 1901.03617

Noise sensitivity of random walks on groups

Itaï Benjamini·Jérémie Brieussel

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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.

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Itaï Benjamini, Jérémie Brieussel. 2019-01-11. Noise sensitivity of random walks on groups. https://arxiv.org/abs/1901.03617

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