Open Access
November 2015 Strong noise sensitivity and random graphs
Eyal Lubetzky, Jeffrey E. Steif
Ann. Probab. 43(6): 3239-3278 (November 2015). DOI: 10.1214/14-AOP959


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.


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Eyal Lubetzky. Jeffrey E. Steif. "Strong noise sensitivity and random graphs." Ann. Probab. 43 (6) 3239 - 3278, November 2015.


Received: 1 August 2013; Revised: 1 May 2014; Published: November 2015
First available in Project Euclid: 11 December 2015

zbMATH: 1339.82005
MathSciNet: MR3433581
Digital Object Identifier: 10.1214/14-AOP959

Primary: 60K35 , 82B43 , 82C43

Keywords: Noise sensitivity of Boolean functions , Random graphs

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 6 • November 2015
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