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

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.

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.

DOI : 10.1214/13-AIHP557
Classification : 60C05, 05D40
Mots clés : 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},
     pages = {1121--1139},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     doi = {10.1214/13-AIHP557},
     mrnumber = {3269987},
     zbl = {1302.60023},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP557/}
}
TY  - JOUR
AU  - Keller, Nathan
AU  - Mossel, Elchanan
AU  - Sen, Arnab
TI  - Geometric influences II: Correlation inequalities and noise sensitivity
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 1121
EP  - 1139
VL  - 50
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP557/
DO  - 10.1214/13-AIHP557
LA  - en
ID  - AIHPB_2014__50_4_1121_0
ER  - 
%0 Journal Article
%A Keller, Nathan
%A Mossel, Elchanan
%A Sen, Arnab
%T Geometric influences II: Correlation inequalities and noise sensitivity
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 1121-1139
%V 50
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP557/
%R 10.1214/13-AIHP557
%G en
%F 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, Tome 50 (2014) no. 4, pp. 1121-1139. doi : 10.1214/13-AIHP557. http://archive.numdam.org/articles/10.1214/13-AIHP557/

[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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

Cité par Sources :