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On Quantitative Noise Stability and Influences for Discrete and Continuous Models

Part of: Extremal combinatorics Combinatorial probability

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: [email protected])
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Abstract

Keller and Kindler recently established a quantitative version of the famous Benjamini–Kalai–Schramm theorem on the noise sensitivity of Boolean functions. Their result was extended to the continuous Gaussian setting by Keller, Mossel and Sen by means of a Central Limit Theorem argument. In this work we present a unified approach to these results, in both discrete and continuous settings. The proof relies on semigroup decompositions together with a suitable cut-off argument, allowing for the efficient use of the classical hypercontractivity tool behind these results. It extends to further models of interest such as families of log-concave measures and Cayley and Schreier graphs. In particular we obtain a quantitative version of the Benjamini–Kalai–Schramm theorem for the slices of the Boolean cube.

MSC classification

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