A Noise Sensitivity Theorem for Schreier Graphs

TL;DR

This paper extends the concept of noise sensitivity to Schreier graphs, establishing a key theorem that links influences to noise sensitivity in this setting and applying it to exclusion sensitivity.

Contribution

It proves a noise sensitivity theorem for Schreier graphs, generalizing previous results and providing an alternative proof for existing exclusion sensitivity findings.

Findings

Benjamini-Kalai-Schramm theorem holds for Schreier graphs

Established connection between influences and noise sensitivity in this setting

Provided an alternative proof for exclusion sensitivity results

Abstract

During the past 15 years, several extensions of the concept of noise sensitivity first coined in~\cite{schramm2000}, has been studied. One such extension was studied in ??, where the definition of noise sensitivity was extended to noise corresponding to any sequence of irreducible and reversible Markov chains. In this paper we focus on the case where the Markov chain is a random walk on a Schreier graph, and show that the Benjamini-Kalai-Schramm theorem, connecting influences to noise sensitivity, holds in this setting. We then apply this result to give an alternative proof of one of the main results from a recent paper on exclusion sensitivity by Broman, Garban and Steif.