## Abstract

The non-linear invariance principle of Mossel, O'Donnell and Oleszkiewicz establishes that if f(x1, . . . , xn) is a multilinear low-degree polynomial with low influences then the distribution of f(B1, . . . , Bn) is close (in various senses) to the distribution of f(G1, . . . , Gn), where Bi ϵR {-1, 1} are independent Bernoulli random variables and Gi ∼N(0, 1) are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAXCUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (X1, . . . ,Xn), (Y1, . . . ,Yn) such that (i) Matching moments: Xi and Yi have matching first and second moments, (ii) Independence: The variables X1, . . . ,Xn are independent, as are Y1, . . . ,Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (X1, . . . ,Xn) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice [n] k which consists of all vectors (x1, . . . , xn) ϵ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdos-Ko-Rado theorems) and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, . . . ,Xn) is the uniform distribution on a slice [n] pn and (Y1, . . . ,Yn) consists either of n independent Ber(p) random variables, or of n independent N(p, p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler- Safra structural theorem, and a stability version of the t-intersecting Erdos-Ko-Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

Original language | American English |
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Title of host publication | 31st Conference on Computational Complexity, CCC 2016 |

Editors | Ran Raz |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 15:1-15:10 |

ISBN (Electronic) | 9783959770088 |

DOIs | |

State | Published - 1 May 2016 |

Event | 31st Conference on Computational Complexity, CCC 2016 - Tokyo, Japan Duration: 29 May 2016 → 1 Jun 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 50 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st Conference on Computational Complexity, CCC 2016 |
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Country/Territory | Japan |

City | Tokyo |

Period | 29/05/16 → 1/06/16 |

### Bibliographical note

Funding Information:National Science Foundation under agreement No. DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors, and do not necessarily reflect the views of the National Science Foundation. The bulk of the work on this paper was done while at the Institute for Advanced Study, Princeton, NJ. E.M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K.W. would like to acknowledge the support of NSF grant CCF 1117079.

Publisher Copyright:

© Yuval Filmus, Guy Kindler, Elchanan Mossel, and Karl Wimmer.

## Keywords

- Analysis of boolean functions
- Invariance principle
- Johnson association scheme
- The slice