Skip to main content

The Fourier spectrum of critical percolation

Abstract

Consider the indicator function f of a 2-dimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension \({\frac{31}{36}}\) almost surely, and the corresponding dimension in the half-plane is \({\frac{5}{9}}\) . It is also proved that critical bond percolation on the square grid has exceptional times almost surely. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.

This is a preview of subscription content, access via your institution.

References

  1. Aizenman, M., Duplantier, B. & Aharony, A., Path-crossing exponents and the external perimeter in 2D percolation. Phys. Rev. Let., 83 (1999), 1359–1362.

    Article  Google Scholar 

  2. Benjamini, I., Häggström, O., Peres, Y. & Steif, J. E., Which properties of a random sequence are dynamically sensitive? Ann. Probab., 31 (2003), 1–34.

    MATH  Article  MathSciNet  Google Scholar 

  3. Benjamini, I., Kalai, G. & Schramm, O., Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math., 90 (1999), 5–43 (2001).

    Google Scholar 

  4. Benjamini, I. & Schramm, O., Exceptional planes of percolation. Probab. Theory Related Fields, 111 (1998), 551–564.

    MATH  Article  MathSciNet  Google Scholar 

  5. van den Berg, J., Meester, R. & White, D. G., Dynamic Boolean models. Stochastic Process. Appl., 69 (1997), 247–257.

    MATH  Article  MathSciNet  Google Scholar 

  6. Bernstein, E. & Vazirani, U., Quantum complexity theory. SIAM J. Comput., 26 (1997), 1411–1473.

    MATH  Article  MathSciNet  Google Scholar 

  7. Broman, E. I. & Steif, J. E., Dynamical stability of percolation for some interacting particle systems and ε-movability. Ann. Probab., 34 (2006), 539–576.

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  Article  MathSciNet  Google Scholar 

  9. Garban, C., Pete, G. & Schramm, O., Pivotal, cluster and interface measures for critical planar percolation. Preprint, 2010. arXiv:1008.1378v1 [math.PR].

  10. — The scaling limits of dynamical and near-critical percolation. In preparation.

  11. Grimmett, G., Percolation. Grundlehren der Mathematischen Wissenschaften, 321. Springer, Berlin–Heidelberg, 1999.

    MATH  Google Scholar 

  12. Hammond, A., Pete, G. & Schramm, O., Local time for dynamical percolation, and the incipient infinite cluster. In preparation.

  13. Hoffman, C., Recurrence of simple random walk on ℤ2 is dynamically sensitive. ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 35–45.

    MATH  MathSciNet  Google Scholar 

  14. Häggström, O. & Pemantle, R., On near-critical and dynamical percolation in the tree case. Random Structures Algorithms, 15 (1999), 311–318.

    MATH  Article  MathSciNet  Google Scholar 

  15. Häggström, O., Peres, Y. & Steif, J.E., Dynamical percolation. Ann. Inst. Henri Poincaré Probab. Statist., 33 (1997), 497–528.

    MATH  Article  Google Scholar 

  16. Jonasson, J. & Steif, J.E., Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab., 36 (2008), 739–764.

    MATH  Article  MathSciNet  Google Scholar 

  17. Kahn, J., Kalai, G. & Linial, N., The influence of variables on boolean functions, in 29th Annual Symposium on Foundations of Computer Science, pp. 68–80. IEEE Computer Society, Los Alamitos, CA, 1988.

    Google Scholar 

  18. Kalai, G. & Safra, S., Threshold phenomena and influence: perspectives from Mathematics, Computer Science, and Economics, in Computational Complexity and Statistical Physics, St. Fe Inst. Stud. Sci. Complex., pp. 25–60. Oxford Univ. Press, New York, 2006.

    Google Scholar 

  19. Kesten, H., The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields, 73 (1986), 369–394.

    MATH  Article  MathSciNet  Google Scholar 

  20. — Scaling relations for 2D-percolation. Comm. Math. Phys., 109 (1987), 109–156.

  21. Kesten, H., Sidoravicius, V. & Zhang, Y., Almost all words are seen in critical site percolation on the triangular lattice. Electron. J. Probab., 3 (1998), 75 pp.

    MathSciNet  Google Scholar 

  22. Khoshnevisan, D., Dynamical percolation on general trees. Probab. Theory Related Fields, 140 (2008), 169–193.

    MATH  Article  MathSciNet  Google Scholar 

  23. Khoshnevisan, D., Levin, D. A. & Méndez-Hernández, P. J., Exceptional times and invariance for dynamical random walks. Probab. Theory Related Fields, 134 (2006), 383–416.

    MATH  Article  MathSciNet  Google Scholar 

  24. Lawler, G. F., Schramm, O. & Werner, W., Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187 (2001), 275–308.

    MATH  Article  MathSciNet  Google Scholar 

  25. — One-arm exponent for critical 2D percolation. Electron. J. Probab., 7 (2002), 13 pp.

  26. Liggett, T. M., Schonmann, R. H. & Stacey, A. M., Domination by product measures. Ann. Probab., 25 (1997), 71–95.

    MATH  Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  28. Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.

    MATH  Google Scholar 

  29. Mörters, P. & Peres, Y., Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010.

    MATH  Google Scholar 

  30. Nolin, P., Near-critical percolation in two dimensions. Electron. J. Probab., 13 (2008), 1562–1623.

    MATH  MathSciNet  Google Scholar 

  31. Peres, Y., Schramm, O. & Steif, J. E., Dynamical sensitivity of the infinite cluster in critical percolation. Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 491–514.

    MATH  Article  MathSciNet  Google Scholar 

  32. Peres, Y. & Steif, J.E., The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields, 111 (1998), 141–165.

    MATH  Article  MathSciNet  Google Scholar 

  33. Reimer, D., Proof of the van den Berg–Kesten conjecture. Combin. Probab. Comput., 9 (2000), 27–32.

    MATH  Article  MathSciNet  Google Scholar 

  34. Schramm, O., Conformally invariant scaling limits: an overview and a collection of problems, in International Congress of Mathematicians (Madrid, 2006). Vol. I, pp. 513–543. Eur. Math. Soc., Zürich, 2007.

  35. Schramm, O. & Smirnov, S., On the scaling limits of planar percolation. To appear in Ann. Probab

  36. Schramm, O. & Steif, J.E., Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math., 171 (2010), 619–672.

    MATH  Article  MathSciNet  Google Scholar 

  37. Smirnov, S., Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 239–244.

    MATH  Google Scholar 

  38. — Towards conformal invariance of 2D lattice models, in International Congress of Mathematicians (Madrid, 2006). Vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich, 2006.

  39. Smirnov, S. & Werner, W., Critical exponents for two-dimensional percolation. Math. Res. Lett., 8 (2001), 729–744.

    MATH  MathSciNet  Google Scholar 

  40. Tsirelson, B., Scaling limit, noise, stability, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1840, pp. 1–106. Springer, Berlin–Heidelberg, 2004.

    Google Scholar 

  41. Werner, W., Lectures on two-dimensional critical percolation, in Statistical Mechanics, IAS/Park City Math. Ser., 16, pp. 297–360. Amer. Math. Soc., Providence, RI, 2009.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gábor Pete.

Additional information

CG was partially supported by the ANR under the grant ANR-06-BLAN-0058. GP was partially supported by the Hungarian National Foundation for Scientific Research, grant T049398, and by an NSERC Discovery Grant. For large parts of the work, all three authors were at Microsoft Research.

Deceased on September 1st, 2008 (Oded Schramm).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Garban, C., Pete, G. & Schramm, O. The Fourier spectrum of critical percolation. Acta Math 205, 19–104 (2010). https://doi.org/10.1007/s11511-010-0051-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11511-010-0051-x

Keywords

  • Boundary Component
  • Fourier Spectrum
  • Triangular Lattice
  • Triangular Grid
  • Noise Sensitivity