Noise stability of functions with low influences: invariance and optimality

TL;DR

This paper establishes an invariance principle for low-influence functions on product spaces, proving two major conjectures in computer science and social choice theory with explicit error bounds.

Contribution

It introduces a simple, non-linear invariance principle for multilinear polynomials with low influences, extending to unbounded degree via smoothing, and applies it to key conjectures.

Findings

Proved the 'Majority Is Stablest' conjecture.

Proved the 'It Ain't Over Till It's Over' conjecture.

Provided explicit error bounds for the invariance principle.

Abstract

In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly ``smoothed''; this extension is essential for our applications to ``noise stability''-type problems. In particular, as applications of the invariance principle we prove two conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer science, which was the original motivation for this work, and the ``It Ain't Over Till It's Over'' conjecture from social choice theory.