随机漫步对群体的噪声敏感性

Pub Date : 2023-01-01 DOI:10.30757/alea.v20-42

Itaï Benjamini, Jérémie Brieussel

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引用次数: 2

摘要

如果以小概率独立地对每一步重新采样会产生几乎独立的输出，则对组上的随机漫步是噪声敏感的。我们精确地定义了两个概念:$\ well ^1$-噪声灵敏度和熵噪声灵敏度。具有这些性质之一的群必然是刘维尔群。自由阿贝尔群的同态对$\ell^1$-噪声灵敏度有阻碍作用。我们还提供了$\ well ^1$和熵噪声敏感随机漫步的例子。

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Noise sensitivity of random walks on groups

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.

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