## Abstract

The noise sensitivity of a Boolean function measures how susceptible the value of f on a typical input x to a slight perturbation of the bits of x: it is the probability f(x) and f(y) are different when x is a uniformly chosen n-bit Boolean string, and y is formed by flipping each bit of x with small probability ϵ. The noise sensitivity of a function is a key concept with applications to combinatorics, complexity theory, learning theory, percolation theory and more. In this paper, we investigate noise sensitivity on the p-biased hypercube, extending the theory for polynomially small p. Specifically, we give sufficient conditions for monotone functions with large groups of symmetries to be noise sensitive (which in some cases are also necessary). As an application, we show that the 2-SAT function is noise sensitive around its critical probability. En route, we study biased versions of the invariance principle for monotone functions and give p-biased versions of Bourgain's tail theorem and the Majority is Stablest theorem, showing that in this case the correct analog of ''small low degree influences'' is lack of correlation with constant width DNF formulas.

Original language | English |
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Title of host publication | Proceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019 |

Publisher | IEEE Computer Society |

Pages | 1205-1226 |

Number of pages | 22 |

ISBN (Electronic) | 9781728149523 |

DOIs | |

State | Published - Nov 2019 |

Externally published | Yes |

Event | 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 - Baltimore, United States Duration: 9 Nov 2019 → 12 Nov 2019 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2019-November |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 |
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Country/Territory | United States |

City | Baltimore |

Period | 9/11/19 → 12/11/19 |

### Bibliographical note

Publisher Copyright:© 2019 IEEE.

### Funding

ACKNOWLEDGMENTS We would like to thank Itai Benjamini for raising the question of the noise sensitivity of the 2-SAT function that motivated much of this work. We would also like to thank Gadi Kozma and Avi Wigderson for helpful conversations. Author D.M was supported by NSF grant CCF-1412958 and Rothschild Fellowship.

Funders | Funder number |
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National Science Foundation | CCF-1412958 |

## Keywords

- Analysis of Boolean Functions
- Graph Properties
- Noise Sensitivity