Concentration on the Boolean hypercube via pathwise stochastic analysis

Abstract

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 $$\begin{aligned} \int _{\left\{ -1,1\right\} ^{n}}\sqrt{h_{f}\left( x\right) }d\mu \left( x\right) \ge C\cdot \mathrm {Var}\left( f\right) \cdot \left( \log \left( \frac{1}{\sum \mathrm {Inf}_{i}^{2}\left( f\right) }\right) \right) ^{1/2}, \end{aligned}$$ where \(h_{f}\left( x\right) \) is the number of edges at x along which f changes its value, \(\mu \left( x\right) \) is the uniform measure on \(\left\{ -1,1\right\} ^{n}\), and \(\mathrm {Inf}_{i}\left( f\right) \) 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, \(\mathrm {Var}\left( f\right) \le C\sum _{i=1}^{n}\frac{\mathrm {Inf}_{i}\left( f\right) }{1+\log \left( 1/\mathrm {Inf}_{i}\left( f\right) \right) }\). 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.