Skip to main content
SpringerLink
Log in

Noise sensitivity of Boolean functions and applications to percolation

  • Published:
Publications Mathématiques de l'Institut des Hautes Études Scientifiques Aims and scope Submit manuscript

Abstract

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given.

Consider, for example, bond percolation on ann+1 byn grid. A configuration is a function that assigns to every edge the value 0 or 1. Let ω be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges ℓ with ω(ℓ)=1. By duality, the probability for having a crossing is 1/2. Fix an ɛ ∈ (0, 1). For each edge ℓ, let ω′(ℓ)=ω(ℓ) with probability 1 − ɛ, and ω′(ℓ)=1 − ω(ℓ) with probability ɛ, independently of the other edges. Letp(τ) be the probability for having a crossing in ω, conditioned on ω′ = τ. Then for alln sufficiently large,P{τ : |p(τ) − 1/2| > ɛ}<ɛ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Ambjorn, B. Durhuus andT. Jonsson,Quantum Geometry, Cambridge University Press, Cambridge, 1997.

    Google Scholar 

  2. N. Alon andJ. Spencer,The Probabilistic Method, Wiley, New York (1992).

    MATH  Google Scholar 

  3. W. Beckner, Inequalities in Fourier analysis,Annals of Math. 102 (1975), 159–182.

    Article  MathSciNet  Google Scholar 

  4. M. Ben-or andN. Linial, Collective coin flipping, inRandomness and Computation (S. Micali, ed.), Academic Press, New York (1990), pp. 91–115. Earlier version: Collective coin flipping, robust voting games, and minima of Banzhaf value, Proc. 26th IEEE Symp. on the Foundation of Computer Science (1985), 408–416.

    Google Scholar 

  5. I. Benjamini andO. Schramm, Conformal invariance of Voronoi percolation,Commun. Math. Phys.,197 (1998), 75–107.

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Benjamini, G. Kalai andO. Schramm, Noise sensitivity, concentration of measure and first passage percolation, in preparation.

  7. A. Bonami, Etude des coefficients Fourier des fonctions de Lp(G),Ann. Inst. Fourier,20 (1970), 335–402.

    MATH  MathSciNet  Google Scholar 

  8. R. Boppana, Threshold functions and bounded depth monotone circuits,Proceedings of 16th Annual ACM Symposium on Theory of Computing (1984), 475–479.

  9. R. Boppana, The average sensitivity of bounded depth circuits,Inform. Process. Lett. 63 (1997), 257–261.

    Article  MathSciNet  Google Scholar 

  10. J. Bourgain, J. Kahn, G. Kalai, Y. Katznelson andN. Linial, The influence of variables in product spaces,Isr. J. Math. 77 (1992), 55–64.

    MATH  MathSciNet  Google Scholar 

  11. J. Bourgain andG. Kalai, Influences of variables and threshold intervals under group symmetries,Geom. Funct. Anal.,7 (1997), 438–461.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. Bruck, Harmonic analysis of polynomial threshold functions.SIAM J. Discrete Math. 3 (1990), 168–177.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Bruck andR. Smolensky, Polynomial threshold functions, AC0 functions, and spectral norms.SIAM J. Comput. 21 (1992), 33–42.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Bunde andS. Havlin (ed.s’),Fractals and Disordered Systems, Springer 1991.

  15. J. T. Chayes, L. Chayes, D. S. Fisher andT. Spencer, Finite-size scaling and correlation length for disordered systems,Phys. Rev. Lett. 57 (1986), 2999–3002.

    Article  MathSciNet  Google Scholar 

  16. E. Friedgut, Boolean functions with low average sensitivity,Combinatorica 18 (1998), 27–36.

    Article  MATH  MathSciNet  Google Scholar 

  17. E. Friedgut, Necessary and sufficient conditions for sharp thresholds of graphs properties and thek-sat problem,Jour. Amer. Math. Soc. 12 (1999), 1017–1054.

    Article  MATH  MathSciNet  Google Scholar 

  18. E. Friedgut andG. Kalai, Every monotone graph property has a sharp threshold,Proc. Amer. Math. Soc. 124 (1996), 2993–3002.

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Grimmett,Percolation, Springer-Verlag, Berlin (1989).

    MATH  Google Scholar 

  20. O. Haggstrom, Y. Peres andJ. E. Steif, Dynamical percolation,Ann. IHP 33 (1997), 497–528.

    MathSciNet  Google Scholar 

  21. J. Håstad, Almost optimal lower bounds for small depth circuits, inRandomness and Computation,5, ed. S. Micali, (1989), 143–170.

    Google Scholar 

  22. J. Håstad andM. Goldmann, On the power of small-depth threshold circuits,Computational Complexity,1 (1991), 113–129.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. Kahn, G. Kalai andN. Linial, The influence of variables on boolean functions,Proc. 29-th Ann. Symp. on Foundations of Comp. Sci., (1988), 68–80.

  24. H. Kesten, Scaling relations for 2D-percolation,Comm. Math. Phys. 109 (1987), 109–156.

    Article  MATH  MathSciNet  Google Scholar 

  25. H. Kesten andY. Zhang, Strict inequalites for some critical exponents in 2D-percolation.J. Statist. Phys. 46 (1987), 1031–1055.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. P. Langlands, P. Pouliot andY. Saint-aubin, Conformal invariance in two-dimensional percolation,Bull. Amer. Math. Soc. (N.S.) 30 (1994), 1–61.

    MATH  MathSciNet  Google Scholar 

  27. N. Linial, Y. Mansour andN. Nisan, Constant depth circuits, Fourier transform, and learnability,J. Assoc. Comput. Mach. 40 (1993), 607–620.

    MATH  MathSciNet  Google Scholar 

  28. V. V. Petrov,Limit theorems of probability theory, Oxford University Press, (1995).

  29. L. Russo, A note on percolation,ZW. 43 (1978), 39–48.

    Article  MATH  Google Scholar 

  30. P. Seymour andD. Welsh, Percolation probabilities on the square lattice. Advances in Graph Theory.Ann. Discrete Math. 3 (1978), 227–245.

    MATH  MathSciNet  Google Scholar 

  31. M. Talagrand, On Russo’s approximate zero-one law,Ann. of Prob. 22 (1994), 1576–1587.

    MATH  MathSciNet  Google Scholar 

  32. M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces,Publ. I.H.E.S.,81 (1995), 73–205.

    Article  MATH  MathSciNet  Google Scholar 

  33. M. Talagrand, How much are increasing sets positively correlated?Combinatorica 16 (1996), 243–258.

    Article  MATH  MathSciNet  Google Scholar 

  34. B. Tsirelson, Fourier-Walsh coefficients for a coalescing flow (discrete time), preprint, math.PR/9903068.

  35. B. Tsirelson, The Five noises, preprint.

  36. A. Yao, Circuits and local computation,Proceedings of 21st Annual ACM Symposium on Theory of Computing, (1989), 186–196.

Download references

Author information

Authors and Affiliations

  1. The Weizmann Institute of Science, 76100, Rehovot, Israel

    Itai Benjamini

  2. The Hebrew University, Givat Ram, 91904, Jerusalem, Israel

    Gil Kalai

  3. The Wiezmann Institute of Science, 76100, Rehovot, Israel

    Oded Schramm

Authors
  1. Itai Benjamini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gil Kalai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oded Schramm
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Benjamini, I., Kalai, G. & Schramm, O. Noise sensitivity of Boolean functions and applications to percolation. Publications Mathématiques de L’Institut des Hautes Scientifiques 90, 5–43 (1999). https://doi.org/10.1007/BF02698830

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02698830

Keywords

Search

Navigation