Geometric influences II: Correlation inequalities and noise sensitivity
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, p. 1121-1139

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand's lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand's bound for all discrete probability spaces and to reestablish analogues of the BKS theorem for biased two-point product spaces.

Dans un papier récent, nous avons présenté une nouvelle définition de l'influence dans des produits d'espaces de fonctions continues et montré que des résultats analogues aux résultats les plus importants sur les influences discrètes, comme le théorème KKL, sont valables pour la nouvelle définition dans des espaces gaussiens. Dans cet article, nous prouvons des analogues gaussiens de deux des applications principales des influences : la borne inférieure de Talagrand sur la corrélation de sous-ensembles croissants du cube discret et le théorème de Benjamini-Kalai-Schramm (BKS) sur la sensibilité au bruit. Ensuite nous utilisons les résultats gaussiens pour obtenir des analogues de la borne de Talagrand pour tous les espaces de probabilités discrets et pour retrouver l'analogue du théorème BKS pour des espaces produits biaisés à deux points.

DOI : https://doi.org/10.1214/13-AIHP557
Classification:  60C05,  05D40
Keywords: influences, geometric influences, noise sensitivity, correlation between increasing sets, Talagrand's bound, gaussian measure, isoperimetric inequality
@article{AIHPB_2014__50_4_1121_0,
     author = {Keller, Nathan and Mossel, Elchanan and Sen, Arnab},
     title = {Geometric influences II: Correlation inequalities and noise sensitivity},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1121-1139},
     doi = {10.1214/13-AIHP557},
     zbl = {1302.60023},
     mrnumber = {3269987},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1121_0}
}
Keller, Nathan; Mossel, Elchanan; Sen, Arnab. Geometric influences II: Correlation inequalities and noise sensitivity. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1121-1139. doi : 10.1214/13-AIHP557. http://www.numdam.org/item/AIHPB_2014__50_4_1121_0/

[1] D. Ahlberg, E. I. Broman, S. Griffith and R. Morris. Noise sensitivity in continuum percolation. Israel J. Math. To appear, 2014. Available at http://arxiv.org/abs/1108.0310. | MR 3265306 | Zbl pre06361320

[2] W. Beckner. Inequalities in Fourier analysis. Ann. Math. (2) 102 (1975) 159-182. | MR 385456 | Zbl 0338.42017

[3] I. Benjamini, G. Kalai and O. Schramm. Noise sensitivity of boolean functions and applications to percolation. Publ. Math. Inst. Hautes Études Sci. 90 (1999) 5-43. | Numdam | MR 1813223 | Zbl 0986.60002

[4] A. Bonami. Etude des coefficients Fourier des fonctiones de L p (G). Ann. Inst. Fourier 20 (1970) 335-402. | Numdam | MR 283496 | Zbl 0195.42501

[5] C. Borell. Positivity improving operators and hypercontractivity. Math. Z. 180 (2) (1982) 225-234. | MR 661699 | Zbl 0472.47015

[6] C. Borell. Geometric bounds on the Ornstein-Uhlenbeck velocity process. Probab. Theory Related Fields 70 (1) (1985) 1-13. | MR 795785 | Zbl 0537.60084

[7] J. Bourgain, J. Kahn, G. Kalai, Y. Katznelson and N. Linial. The influence of variables in product spaces. Israel J. Math. 77 (1992) 55-64. | MR 1194785 | Zbl 0771.60002

[8] S. Chatterjee. Chaos, concentration, and multiple valleys, 2008. Available at http://arxiv.org/abs/0810.4221.

[9] D. Cordero-Erausquin and M. Ledoux. Hypercontractive measures, Talagrand's inequality, and influences. Preprint, 2011. Available at http://arxiv.org/abs/1105.4533. | Zbl 1280.60018

[10] C. M. Fortuin, P. W. Kasteleyn and J. Ginibre. Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 (1971) 89-103. | MR 309498 | Zbl 0346.06011

[11] P. Frankl. The shifting technique in extremal set theory. In Surveys in Combinatorics 81-110. C. W. Whitehead (Ed). Cambridge Univ. Press, Cambridge, 1987. | MR 905277 | Zbl 0633.05038

[12] G. R. Grimmett and B. Graham. Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34 (2006) 1726-1745. | MR 2271479 | Zbl 1115.60099

[13] T. E. Harris. A lower bound for the critical probability in a certain percolation process. Math. Proc. Cambridge Philos. Soc. 56 (1960) 13-20. | MR 115221 | Zbl 0122.36403

[14] H. Hatami. Decision trees and influence of variables over product probability spaces. Combin. Probab. Comput. 18 (2009) 357-369. | MR 2501432 | Zbl 1193.60007

[15] J. Kahn, G. Kalai and N. Linial. The influence of variables on boolean functions. In Proc. 29th Ann. Symp. on Foundations of Comp. Sci. 68-80. Computer Society Press, 1988.

[16] G. Kalai and M. Safra. Threshold phenomena and influence. In Computational Complexity and Statistical Physics 25-60. A. G. Percus, G. Istrate and C. Moore (Eds). Oxford Univ. Press, New York, 2006. | Zbl 1156.82317

[17] N. Keller. Influences of variables on boolean functions. Ph.D. thesis, Hebrew Univ. Jerusalem, 2009.

[18] N. Keller. On the influences of variables on boolean functions in product spaces. Combin. Probab. Comput. 20 (1) (2011) 83-102. | MR 2745679 | Zbl 1204.94120

[19] N. Keller. A simple reduction from the biased measure on the discrete cube to the uniform measure. European J. Combin. 33 1943-1957. Available at http://arxiv.org/abs/1001.1167. | MR 2950492 | Zbl 1248.28005

[20] N. Keller and G. Kindler. A quantitative relation between influences and noise sensitivity. Combinatorica 33 45-71. Available at http://arxiv.org/abs/1003.1839. | MR 3070086 | Zbl 1299.05308

[21] N. Keller, E. Mossel and A. Sen. Geometric influences. Ann. Probab. 40 (3) (2012) 1135-1166. | MR 2962089 | Zbl 1255.60015

[22] G. Kindler and R. O'Donnell. Gaussian noise sensitivity and Fourier tails. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity 137-147. IEEE Computer Society, Washington, DC, 2012. Available at http://www.cs.cmu.edu/~odonnell/papers/gaussian-noise-sensitivity.pdf. | MR 3026322

[23] D. J. Kleitman. Families of non-disjoint subsets. J. Combin. Theory 1 (1966) 153-155. | MR 193020 | Zbl 0141.00801

[24] M. Ledoux. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9 (2) (2000) 305-366. | Numdam | MR 1813804 | Zbl 0980.60097

[25] E. Mossel, R. O'Donnell and K. Oleszkiewicz. Noise stability of functions with low influences: Invariance and optimality. Ann. Math. (2) 171 (1) (2010) 295-341. | MR 2630040 | Zbl 1201.60031

[26] E. Mossel, R. O'Donnell, O. Regev, J. E. Steif and B. Sudakov. Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality. Israel J. Math. 154 (2006) 299-336. | MR 2254545 | Zbl 1140.60007

[27] R. O'Donnell. Some topics in analysis of boolean functions. In Proceedings of the 40th Annual ACM Sympsium on the Theory of Computing 569-578. ACM, New York, 2008. | MR 2582688 | Zbl 1231.94096

[28] M. Talagrand. On Russo's approximate zero-one law. Ann. Probab. 22 (1994) 1576-1587. | Zbl 0819.28002

[29] M. Talagrand. How much are increasing sets positively correlated? Combinatorica 16 (2) (1996) 243-258. | MR 1401897 | Zbl 0861.05008

[30] P. Wolff. Hypercontractivity of simple random variables. Studia Math. 180 (3) (2007) 219-236. | MR 2314078 | Zbl 1133.60011