@inproceedings{e71c4fc30c8441319c5d7ef074fc82db,

title = "Concentration on the Boolean hypercube via pathwise stochastic analysis",

abstract = "We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions.Second, we strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, (f)≤ C∑ i=1 n i (f)/1+log(1/ i (f)). We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn-Kalai-Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary.Lastly, we improve a quantitative relation between influences and noise stability given by Keller and Kindler.Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.",

keywords = "Boolean analysis, Concentration, Isoperimetric inequality, Pathwise analysis",

author = "Ronen Eldan and Renan Gross",

note = "Publisher Copyright: {\textcopyright} 2020 ACM.; STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing ; Conference date: 22-06-2020 Through 26-06-2020",

year = "2020",

month = jun,

day = "8",

doi = "https://doi.org/10.1145/3357713.3384230",

language = "الإنجليزيّة",

isbn = "9781450369794",

series = "Annual ACM SIGACT Symposium on Theory of Computing",

pages = "208--221",

editor = "Konstantin Makarychev and Yury Makarychev and Madhur Tulsiani and Gautam Kamath and Julia Chuzhoy",

booktitle = "STOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on theory of computing",

}