Geodesic
Noise stability of functions with low influences: Invariance and optimality
Elchanan Mossel1 ; Ryan O’Donnell2 ; Krzysztof Oleszkiewicz3
1 University of California at Berkeley, Department of Statistics, 367 Evans Hall, Berkeley, CA 94720-3860, United States
2 Carnegie Mellon University, School of Computer Science, 5000 Forbes Avenue, Pittsburgh, PA 15213-3891, United States
3 Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
Annals of mathematics, Tome 171 (2010) no. 1, pp. 295-341.

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In this paper we study functions with low influences on product probability spaces. These are functions $f : \Omega_1 \times \cdots \times \Omega_n \to\mathbb{R}$ that have ${\rm E}[{\rm Var}_{\Omega_i}[f]]$ small compared to ${\rm Var}[f]$ for each $i$. The analysis of boolean functions $f: \{-1,1\}^n \to \{-1,1\}$ 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.
DOI : 10.4007/annals.2010.171.295

Elchanan Mossel 1 ; Ryan O’Donnell 2 ; Krzysztof Oleszkiewicz 3

1 University of California at Berkeley, Department of Statistics, 367 Evans Hall, Berkeley, CA 94720-3860, United States
2 Carnegie Mellon University, School of Computer Science, 5000 Forbes Avenue, Pittsburgh, PA 15213-3891, United States
3 Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland 
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Elchanan Mossel; Ryan O’Donnell; Krzysztof Oleszkiewicz. Noise stability of functions with low influences: Invariance and optimality. Annals of mathematics, Tome 171 (2010) no. 1, pp. 295-341. doi : 10.4007/annals.2010.171.295. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.295/

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