### Abstract

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 Erdõs-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} →0 polynomially fast in the number of variables.

Original language | English (US) |
---|---|

Pages (from-to) | 3239-3278 |

Number of pages | 40 |

Journal | Annals of Probability |

Volume | 43 |

Issue number | 6 |

DOIs | |

State | Published - 2015 |

### Fingerprint

### Keywords

- Noise sensitivity of Boolean functions
- Random graphs

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*43*(6), 3239-3278. https://doi.org/10.1214/14-AOP959

**Strong noise sensitivity and random graphs.** / Lubetzky, Eyal; Steif, Jeffrey E.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 43, no. 6, pp. 3239-3278. https://doi.org/10.1214/14-AOP959

}

TY - JOUR

T1 - Strong noise sensitivity and random graphs

AU - Lubetzky, Eyal

AU - Steif, Jeffrey E.

PY - 2015

Y1 - 2015

N2 - 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 pc. 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 Erdõs-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 pc →0 polynomially fast in the number of variables.

AB - 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 pc. 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 Erdõs-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 pc →0 polynomially fast in the number of variables.

KW - Noise sensitivity of Boolean functions

KW - Random graphs

UR - http://www.scopus.com/inward/record.url?scp=84950973718&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84950973718&partnerID=8YFLogxK

U2 - 10.1214/14-AOP959

DO - 10.1214/14-AOP959

M3 - Article

VL - 43

SP - 3239

EP - 3278

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 6

ER -