The Annals of Probability

Strong noise sensitivity and random graphs

Eyal Lubetzky and Jeffrey E. Steif

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The noise sensitivity of a Boolean function describes its likelihood to flip under small perturbations of its input. Introduced in the seminal work of Benjamini, Kalai and Schramm [Inst. Hautes Études Sci. Publ. Math. 90 (1999) 5–43], it was there shown to be governed by the first level of Fourier coefficients in the central case of monotone functions at a constant critical probability $p_{c}$.

Here we study noise sensitivity and a natural stronger version of it, addressing the effect of noise given a specific witness in the original input. Our main context is the Erdős–Rényi random graph, where already the property of containing a given graph is sufficiently rich to separate these notions. In particular, our analysis implies (strong) noise sensitivity in settings where the BKS criterion involving the first Fourier level does not apply, for example, when $p_{c}\to0$ polynomially fast in the number of variables.

Article information

Ann. Probab., Volume 43, Number 6 (2015), 3239-3278.

Received: August 2013
Revised: May 2014
First available in Project Euclid: 11 December 2015

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Zentralblatt MATH identifier

Primary: 82C43: Time-dependent percolation [See also 60K35] 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Noise sensitivity of Boolean functions random graphs


Lubetzky, Eyal; Steif, Jeffrey E. Strong noise sensitivity and random graphs. Ann. Probab. 43 (2015), no. 6, 3239--3278. doi:10.1214/14-AOP959.

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