Noise stability of functions with low influences: invariance and optimality
Elchanan Mossel, Ryan O'Donnell, Krzysztof Oleszkiewicz
TL;DR
This work develops an invariance principle for low-influence, bounded-degree multilinear polynomials on product spaces, enabling transfer of noise-stability results from Gaussian space to general finite product spaces with explicit error bounds. By coupling this principle with hypercontractivity and smoothing techniques, the authors prove two central conjectures: Majority Is Stablest and It Ain't Over Till It's Over, with broad consequences for social choice theory and hardness of approximation. The framework unifies analysis across discrete and Gaussian settings, yielding precise bounds and revealing the central role of the Majority function in extremal low-influence problems, while also providing counterexamples to certain low-level weight conjectures. Overall, the paper advances a powerful, broadly applicable method for understanding stability and influences in complex product spaces, with significant implications for PCPs, computational hardness, and collective decision-making models.
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.