Concentration on the Boolean hypercube via pathwise stochastic analysis

12 Mar 2020
•
Eldan Ronen
•
Gross Renan

We develop a new technique for proving concentration inequalities which
relate between the variance and influences of Boolean functions. Using this
technique, we
1...Settle a conjecture of Talagrand [Tal97] proving that $$\int_{\left\{
-1,1\right\} ^{n}}\sqrt{h_{f}\left(x\right)}d\mu\geq
C\cdot\mathrm{var}\left(f\right)\cdot\left(\log\left(\frac{1}{\sum\mathrm{Inf}_{i}^{2}\left(f\right)}\right)\right)^{1/2},$$
where $h_{f}\left(x\right)$ is the number of edges at $x$ along which $f$
changes its value, 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
$f$, $\mathrm{var}\left(f\right)\leq
C\sum_{i=1}^{n}\frac{\mathrm{Inf}_{i}\left(f\right)}{1+\log\left(1/\mathrm{Inf}_{i}\left(f\right)\right)}$. 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.(read more)