Concentration on the Boolean hypercube via pathwise stochastic analysis

Ronen Eldan, Renan Gross

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we first settle a conjecture of Talagrand, proving that g g gg-1,1g g¬ g?nghfg gxg g?dμ≥ C·g gfg g?·g glogg g g g g1g'i2g gf1/2, where hf(x) is the number of edges at x along which f changes its value, and i(f) is the influence of the i-th coordinate. 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)≤ Cg'i=1ni(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.

Original languageEnglish
Title of host publicationSTOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
EditorsKonstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
PublisherAssociation for Computing Machinery
Number of pages14
ISBN (Electronic)9781450369794
StatePublished - 8 Jun 2020
Externally publishedYes
Event52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States
Duration: 22 Jun 202026 Jun 2020

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
Country/TerritoryUnited States


  • Boolean analysis
  • Concentration
  • Isoperimetric inequality
  • Pathwise analysis


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