Noise stability on the Boolean hypercube via a renormalized Brownian motion

TL;DR

This paper introduces a new noise model on the Boolean hypercube using renormalized Brownian motion, providing novel proofs and bounds for noise stability theorems and related conjectures.

Contribution

It presents a new stochastic approach to analyze noise stability, improving bounds in the Majority is Stablest theorem and confirming a variant of Courtade and Kumar's conjecture.

Findings

Improved polynomial dependence on maximal influence in noise stability bounds

New proof of the Majority is Stablest theorem using the renormalized Brownian motion

Validation of a variant of Courtade and Kumar's conjecture with the new noise model

Abstract

We consider a variant of the classical notion of noise on the Boolean hypercube which gives rise to a new approach to inequalities regarding noise stability. We use this approach to give a new proof of the Majority is Stablest theorem by Mossel, O'Donnell, and Oleszkiewicz, improving the dependence of the bound on the maximal influence of the function from logarithmic to polynomial. We also show that a variant of the conjecture by Courtade and Kumar regarding the most informative Boolean function, where the classical noise is replaced by our notion, holds true. Our approach is based on a stochastic construction that we call the renormalized Brownian motion, which facilitates the use of inequalities in Gaussian space in the analysis of Boolean functions.