Noise sensitivity of Boolean functions and applications to percolation

  • Itai Benjamini
  • Gil Kalai
  • Oded Schramm

DOI: 10.1007/BF02698830

Cite this article as:
Benjamini, I., Kalai, G. & Schramm, O. Publications Mathématiques de L’Institut des Hautes Scientifiques (1999) 90: 5. doi:10.1007/BF02698830


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 ann+1 byn 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 ɛ ∈ (0, 1). For each edge ℓ, let ω′(ℓ)=ω(ℓ) with probability 1 − ɛ, and ω′(ℓ)=1 − ω(ℓ) with probability ɛ, independently of the other edges. Letp(τ) be the probability for having a crossing in ω, conditioned on ω′ = τ. Then for alln sufficiently large,P{τ : |p(τ) − 1/2| > ɛ}<ɛ.

Copyright information

© Publications Mathematiques de L’I.H.E.S. 1999

Authors and Affiliations

  • Itai Benjamini
    • 1
  • Gil Kalai
    • 2
  • Oded Schramm
    • 3
  1. 1.The Weizmann Institute of ScienceRehovotIsrael
  2. 2.The Hebrew University, Givat RamJerusalemIsrael
  3. 3.The Wiezmann Institute of ScienceRehovotIsrael

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