Noise stability of functions with low influences: invariance and optimality

Functions with low influences on product probability spaces are functions f:Ω1—···—Ωn †’R that have E[VarΩi[f]] small compared to Var[f] for each i. The analysis of boolean functions f : {Â^’1, 1}^n †’ {Â^’1, 1} 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.

Mossel, O'Donnell, Oleszkiewicz proved 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. It is one of the very few known nonlinear invariance principles. They also show that the assumption of bounded degree can be eliminated if the polynomials are slightly "smoothed"; this extension is essential for their applications to "noise stability"-type problems.

In particular, as applications of the invariance principle they proved two conjectures: the "Majority Is Stablest" conjecture from theoretical computer science, and the "It Ain't Over Till It's Over" conjecture from social choice theory.