Noise sensitivity of Boolean functions and applications to percolation

Itai Benjamini*, Gil Kalai, Oded Schramm

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

163 Scopus citations

Abstract

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an n+1 by n grid. A configuration is a function that assigns to every edge the value 0 or 1. Let ω be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges ℓ with ω(ℓ)=1. By duality, the probability for having a crossing is 1/2. Fix an e{open} ∈ (0, 1). For each edge ℓ, let ω′(ℓ)=ω(ℓ) with probability 1 - e{open}, and ω′(ℓ)=1 - ω(ℓ) with probability e{open}, independently of the other edges. Let p(τ) be the probability for having a crossing in ω, conditioned on ω′ = τ. Then for all n sufficiently large, P{τ : |p(τ) - 1/2| > e{open}}<e{open}.

Original languageEnglish
Pages (from-to)5-43
Number of pages39
JournalPublications Mathematiques de l'Institut des Hautes Etudes Scientifiques
Volume90
Issue number1
DOIs
StatePublished - Dec 1999

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