×

Geometric influences. II: Correlation inequalities and noise sensitivity. (English. French summary) Zbl 1302.60023

Summary: In Part I [Ann. Probab. 40, No. 3, 1135–1166 (2012; Zbl 1255.60015)], 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 sequel, 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.

MSC:

60C05 Combinatorial probability
05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.)

Citations:

Zbl 1255.60015
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] D. Ahlberg, E. I. Broman, S. Griffith and R. Morris. Noise sensitivity in continuum percolation. Israel J. Math. To appear, 2014. Available at . · Zbl 1305.60100 · doi:10.1007/s11856-014-1038-y
[2] W. Beckner. Inequalities in Fourier analysis. Ann. Math. (2) 102 (1975) 159-182. · Zbl 0338.42017 · doi:10.2307/1970980
[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. · Zbl 0986.60002 · doi:10.1007/BF02698830
[4] A. Bonami. Etude des coefficients Fourier des fonctiones de \(L^{p}(G)\). Ann. Inst. Fourier 20 (1970) 335-402. · Zbl 0195.42501 · doi:10.5802/aif.357
[5] C. Borell. Positivity improving operators and hypercontractivity. Math. Z. 180 (2) (1982) 225-234. · Zbl 0472.47015 · doi:10.1007/BF01318906
[6] C. Borell. Geometric bounds on the Ornstein-Uhlenbeck velocity process. Probab. Theory Related Fields 70 (1) (1985) 1-13. · Zbl 0537.60084 · doi:10.1007/BF00532234
[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. · Zbl 0771.60002 · doi:10.1007/BF02808010
[8] S. Chatterjee. Chaos, concentration, and multiple valleys, 2008. Available at .
[9] D. Cordero-Erausquin and M. Ledoux. Hypercontractive measures, Talagrand’s inequality, and influences. Preprint, 2011. Available at . · Zbl 1280.60018 · doi:10.1007/978-3-642-29849-3_10
[10] C. M. Fortuin, P. W. Kasteleyn and J. Ginibre. Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 (1971) 89-103. · Zbl 0346.06011 · doi:10.1007/BF01651330
[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. · Zbl 0633.05038
[12] G. R. Grimmett and B. Graham. Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34 (2006) 1726-1745. · Zbl 1115.60099 · doi:10.1214/009117906000000278
[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. · Zbl 0122.36403 · doi:10.1017/S0305004100034241
[14] H. Hatami. Decision trees and influence of variables over product probability spaces. Combin. Probab. Comput. 18 (2009) 357-369. · Zbl 1193.60007 · doi:10.1017/S0963548309009833
[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. · Zbl 1204.94120 · doi:10.1017/S0963548310000234
[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 . · Zbl 1248.28005 · doi:10.1016/j.ejc.2012.06.003
[20] N. Keller and G. Kindler. A quantitative relation between influences and noise sensitivity. Combinatorica 33 45-71. Available at . · Zbl 1299.05308 · doi:10.1007/s00493-013-2719-2
[21] N. Keller, E. Mossel and A. Sen. Geometric influences. Ann. Probab. 40 (3) (2012) 1135-1166. · Zbl 1255.60015 · doi:10.1214/11-AOP643
[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 .
[23] D. J. Kleitman. Families of non-disjoint subsets. J. Combin. Theory 1 (1966) 153-155. · Zbl 0141.00801 · doi:10.1016/S0021-9800(66)80012-1
[24] M. Ledoux. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9 (2) (2000) 305-366. · Zbl 0980.60097 · doi:10.5802/afst.962
[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. · Zbl 1201.60031 · doi:10.4007/annals.2010.171.295
[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. · Zbl 1140.60007 · doi:10.1007/BF02773611
[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. · Zbl 1231.94096
[28] M. Talagrand. On Russo’s approximate zero-one law. Ann. Probab. 22 (1994) 1576-1587. · Zbl 0819.28002 · doi:10.1214/aop/1176988612
[29] M. Talagrand. How much are increasing sets positively correlated? Combinatorica 16 (2) (1996) 243-258. · Zbl 0861.05008 · doi:10.1007/BF01844850
[30] P. Wolff. Hypercontractivity of simple random variables. Studia Math. 180 (3) (2007) 219-236. · Zbl 1133.60011 · doi:10.4064/sm180-3-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.