TY - JOUR

T1 - Concentration on the Boolean hypercube via pathwise stochastic analysis

AU - Eldan, Ronen

AU - Gross, Renan

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/12

Y1 - 2022/12

N2 - We develop a new technique for proving concentration inequalities which relate the variance and influences of Boolean functions. Using this technique, we 1.Settle a conjecture of Talagrand (Combinatorica 17(2):275–285, 1997), proving that (Formula presented.) where hf(x) is the number of edges at x along which f changes its value, μ(x) is the uniform measure on { - 1 , 1 } n, and Inf i(f) is the influence of the i-th coordinate.2.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, Var(f)≤C∑i=1nInfi(f)1+log(1/Infi(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.3.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.

AB - We develop a new technique for proving concentration inequalities which relate the variance and influences of Boolean functions. Using this technique, we 1.Settle a conjecture of Talagrand (Combinatorica 17(2):275–285, 1997), proving that (Formula presented.) where hf(x) is the number of edges at x along which f changes its value, μ(x) is the uniform measure on { - 1 , 1 } n, and Inf i(f) is the influence of the i-th coordinate.2.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, Var(f)≤C∑i=1nInfi(f)1+log(1/Infi(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.3.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.

KW - Boolean analysis

KW - Concentration

KW - Isoperimetric inequality

KW - Pathwise analysis

UR - http://www.scopus.com/inward/record.url?scp=85133450173&partnerID=8YFLogxK

U2 - 10.1007/s00222-022-01135-8

DO - 10.1007/s00222-022-01135-8

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AN - SCOPUS:85133450173

SN - 0020-9910

VL - 230

SP - 935

EP - 994

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -