On Quantitative Noise Stability and Influences for Discrete and Continuous Models. (23rd March 2018)

• Record Type: Journal Article
• Authors: BOUYRIE, RAPHAËL
• Is Part Of: Combinatorics, probability and computing. Volume 27:Number 3(2018)
• Page Start: 334
• Page End: 357
• Publication Date: 2018-03-23
• Subjects: Primary 60C05, -- Secondary 05D40; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548318000044
• Languages: English
• ISSNs: 0963-5483

Abstract

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.