Concentration on the Boolean hypercube via pathwise stochastic analysis

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?ⁿgh_fg gxg g?dμ≥ C·g gfg g?·g glogg g g g g1g'_i²g gf1/2, where h_f(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=1ⁿ_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.