Noise Sensitivity of Boolean Functions and Applications to Percolation

  • Itai BenjaminiEmail author
  • Gn. Kalai
  • Oded Schramm
Open Access
Part of the Selected Works in Probability and Statistics book series (SWPS)


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 ω (e) = 1. By duality, the probability for having a crossing is 1/2. Fix an ε ∈ (0, 1). For each edges e let ω' (e) = ω (e) with probability 1 − ε, and ω' (e) = 1 − ω (e) with probability ε. independently of the other edges. Let þ(τ) be the probability for having a crossing in ω, conditioned on ω' = τ. Then for all n sufficiently large, P{τ : |þ(τ) − 1/2| > ε} < ε.


Boolean Function Fourier Coefficient Uniform Measure Noise Sensitivity Polynomial Size 
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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.The Weizmann Institute of ScienceRehovotIsrael
  2. 2.The Hebrew UniversityGivat Ram, JerusalemIsrael

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