Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T12:06:14.875Z Has data issue: false hasContentIssue false
  • English
  • Français

On Quantitative Noise Stability and Influences for Discrete and Continuous Models

Part of: Extremal combinatorics Combinatorial probability

Published online by Cambridge University Press:  23 March 2018

RAPHAËL BOUYRIE
RAPHAËL BOUYRIE*
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219 du CNRS Université de Toulouse, 31062, Toulouse, France (e-mail: raphael.bouyrie@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

Keller and Kindler recently established a quantitative version of the famous Benjamini–Kalai–Schramm theorem on the noise sensitivity of Boolean functions. Their result was extended to the continuous Gaussian setting by Keller, Mossel and Sen by means of a Central Limit Theorem argument. In this work we present a unified approach to these results, in both discrete and continuous settings. The proof relies on semigroup decompositions together with a suitable cut-off argument, allowing for the efficient use of the classical hypercontractivity tool behind these results. It extends to further models of interest such as families of log-concave measures and Cayley and Schreier graphs. In particular we obtain a quantitative version of the Benjamini–Kalai–Schramm theorem for the slices of the Boolean cube.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 3 , May 2018 , pp. 334 - 357
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000) Sur les Inégalités de Sobolev Logarithmiques, Vol. 10 of Panoramas et Synthèses, Société Mathématique de France.Google Scholar
[2] Bakry, D. (1994) L'hypercontractivité et son utilisation en théorie des semigroupes. In Lectures on Probability Theory: École d'Été de Probabilités de Saint-Flour XXII, Vol. 1581 of Lecture Notes in Mathematics, Springer, pp. 1–114.Google Scholar
[3] Bakry, D. and Émery, M. (1985) Diffusions hypercontractives. In Séminaire de Probabilités XIX, Vol. 1123 of Lecture Notes in Mathematics, Springer, pp. 177–206.Google Scholar
[4] Bakry, D., Gentil, I. and Ledoux, M., M. (2014) Analysis and Geometry of Markov Diffusion Operators, Vol. 348 of Grundlehren der Mathematischen Wissenschaften, Springer.Google Scholar
[5] Beckner, W. (1975) Inequalities in Fourier analysis. Ann. of Math. 102 159–182.Google Scholar
[6] Benjamini, I., Kalai, G. and Schramm, O. (1999) Noise sensitivity of Boolean functions and applications to percolation. Publ. Math. Inst. Hautes Etudes Sci. 90 5–43.Google Scholar
[7] Bonami, A. (1970) Étude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier (Grenoble) 20 335–402.Google Scholar
[8] Cordero-Erausquin, D. and Ledoux, M. (2012) Hypercontractive measures, Talagrand's inequality, and influences. In Geometric Aspects of Functional Analysis: Israel Seminar 2006–2010, Vol. 2050 of Lecture Notes in Mathematics, Springer, pp. 169–189.CrossRefGoogle Scholar
[9] Diaconis, P. and Saloff-Coste, L. (1996) Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.Google Scholar
[10] Forsström, M. P. (2015) A noise sensitivity theorem for Schreier graphs. arXiv:1501.01828Google Scholar
[11] Garban, C. and Steif, J. E. (2014) Lectures on Noise Sensitivity and Percolation, Cambridge University Press.Google Scholar
[12] Gross, L. (1975) Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061–1083.Google Scholar
[13] Kalai, G. and Safra, S. (2006) Threshold phenomena and influence: Perspectives from mathematics, computer science, and economics. In Computational Complexity and Statistical Physics (Percus, A. G. et al., eds), Oxford University Press, pp. 25–60.Google Scholar
[14] Keller, N. and Kindler, G. (2013) Quantitative relationship between noise sensitivity and influences. Combinatorica 33 45–71.Google Scholar
[15] Keller, N., Mossel, E. and Sen, A. (2012) Geometric influences. Ann. Probab. 40 1135–1166.Google Scholar
[16] Keller, N., Mossel, E. and Sen, A. (2014) Geometric influences II: Correlation inequalities and noise sensitivity. Ann. Inst. H. Poincaré 50 1121–1139.Google Scholar
[17] Ledoux, M. (2000) The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. 9 305–366.Google Scholar
[18] Lee, T. Y. and Yau, H. T. (1998) Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26 1855–1873.Google Scholar
[19] Mazet, O. (1997) Classification des semi-groupes de diffusion sur ℝ associés à une famille de polynômes orthogonaux. In Séminaire de Probabilités XXXI, Vol. 1655 of Lecture Notes in Mathematics, Springer, pp. 40–53.CrossRefGoogle Scholar
[20] Nelson, E. (1973) The free Markov field. J. Funct. Anal. 12 211–227.Google Scholar
[21] O'Donnell, R. and Wimmer, K. (2009) KKL, Kruskal–Katona, and monotone nets. SIAM J. Comput. 42 2375–2399 (50th Annual IEEE Symposium on Foundations of Computer Science: FOCS 2009).Google Scholar
[22] Talagrand, M. (1994) On Russo's approximate zero-one law. Ann. Probab. 22 1576–1587.CrossRefGoogle Scholar
[23] Talagrand, M. (1996) How much are increasing sets positively correlated? Combinatorica 16 243–258.CrossRefGoogle Scholar
[24] Talagrand, M. (1997) On boundary and influences. Combinatorica 17 275–285.Google Scholar