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On Quantitative Noise Stability and Influences for Discrete andContinuous Models

Part of: Combinatorial probability Extremal combinatorics

Published online by Cambridge University Press:  23 March 2018

RAPHAËL BOUYRIE
RAPHAËL BOUYRIE*
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219 du CNRS Université de Toulouse, 31062, Toulouse, France (e-mail: raphael.bouyrie@gmail.com)
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Abstract

Keller and Kindler recently established a quantitative version of the famousBenjamini–Kalai–Schramm theorem on the noise sensitivity of Boolean functions.Their result was extended to the continuous Gaussian setting by Keller, Mosseland Sen by means of a Central Limit Theorem argument. In this work we present aunified approach to these results, in both discrete and continuous settings. Theproof relies on semigroup decompositions together with a suitable cut-offargument, allowing for the efficient use of the classical hypercontractivitytool behind these results. It extends to further models of interest such asfamilies of log-concave measures and Cayley and Schreier graphs. In particularwe obtain a quantitative version of the Benjamini–Kalai–Schramm theorem for theslices of the Boolean cube.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05D40: Probabilistic methods

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 3 , May 2018 , pp. 334 - 357
Copyright
Copyright © Cambridge University Press 2018 

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