Acta Mathematica

The Fourier spectrum of critical percolation

Christophe Garban, Gábor Pete, and Oded Schramm

Full-text: Open access


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.


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).

Article information

Acta Math., Volume 205, Number 1 (2010), 19-104.

Received: 6 May 2008
Revised: 12 November 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2010 © Institut Mittag-Leffler


Garban, Christophe; Pete, Gábor; Schramm, Oded. The Fourier spectrum of critical percolation. Acta Math. 205 (2010), no. 1, 19--104. doi:10.1007/s11511-010-0051-x.

Export citation


  • Aizenman, M., Duplantier, B. & Aharony, A., Path-crossing exponents and the external perimeter in 2D percolation. Phys. Rev. Let., 83 (1999), 1359–1362.
  • Benjamini, I., Häggström, O., Peres, Y. & Steif, J. E., Which properties of a random sequence are dynamically sensitive? Ann. Probab., 31 (2003), 1–34.
  • 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).
  • Benjamini, I. & Schramm, O., Exceptional planes of percolation. Probab. Theory Related Fields, 111 (1998), 551–564.
  • van den Berg, J., Meester, R. & White, D. G., Dynamic Boolean models. Stochastic Process. Appl., 69 (1997), 247–257.
  • Bernstein, E. & Vazirani, U., Quantum complexity theory. SIAM J. Comput., 26 (1997), 1411–1473.
  • Broman, E. I. & Steif, J. E., Dynamical stability of percolation for some interacting particle systems and ε-movability. Ann. Probab., 34 (2006), 539–576.
  • Friedgut, E. & Kalai, G., Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc., 124 (1996), 2993–3002.
  • Garban, C., Pete, G. & Schramm, O., Pivotal, cluster and interface measures for critical planar percolation. Preprint, 2010. arXiv:1008.1378v1 [math.PR].
  • — The scaling limits of dynamical and near-critical percolation. In preparation.
  • Grimmett, G., Percolation. Grundlehren der Mathematischen Wissenschaften, 321. Springer, Berlin–Heidelberg, 1999.
  • Hammond, A., Pete, G. & Schramm, O., Local time for dynamical percolation, and the incipient infinite cluster. In preparation.
  • Hoffman, C., Recurrence of simple random walk on ℤ2 is dynamically sensitive. ALEA Lat. Am. J. Probab. Math. Stat., 1 (2006), 35–45.
  • Häggström, O. & Pemantle, R., On near-critical and dynamical percolation in the tree case. Random Structures Algorithms, 15 (1999), 311–318.
  • Häggström, O., Peres, Y. & Steif, J.E., Dynamical percolation. Ann. Inst. Henri Poincaré Probab. Statist., 33 (1997), 497–528.
  • Jonasson, J. & Steif, J.E., Dynamical models for circle covering: Brownian motion and Poisson updating . Ann. Probab., 36 (2008), 739–764.
  • 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.
  • 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.
  • Kesten, H., The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields, 73 (1986), 369–394.
  • — Scaling relations for 2D-percolation. Comm. Math. Phys., 109 (1987), 109–156.
  • 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.
  • Khoshnevisan, D., Dynamical percolation on general trees. Probab. Theory Related Fields, 140 (2008), 169–193.
  • 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.
  • Lawler, G. F., Schramm, O. & Werner, W., Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187 (2001), 275–308.
  • — One-arm exponent for critical 2D percolation. Electron. J. Probab., 7 (2002), 13 pp.
  • Liggett, T. M., Schonmann, R. H. & Stacey, A. M., Domination by product measures. Ann. Probab., 25 (1997), 71–95.
  • Linial, N., Mansour, Y. & Nisan, N., Constant depth circuits, Fourier transform, and learnability. J. Assoc. Comput. Mach., 40 (1993), 607–620.
  • Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.
  • Mörters, P. & Peres, Y., Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010.
  • Nolin, P., Near-critical percolation in two dimensions. Electron. J. Probab., 13 (2008), 1562–1623.
  • 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.
  • Peres, Y. & Steif, J.E., The number of infinite clusters in dynamical percolation. Probab. Theory Related Fields, 111 (1998), 141–165.
  • Reimer, D., Proof of the van den Berg–Kesten conjecture. Combin. Probab. Comput., 9 (2000), 27–32.
  • 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.
  • Schramm, O. & Smirnov, S., On the scaling limits of planar percolation. To appear in Ann. Probab
  • Schramm, O. & Steif, J.E., Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math., 171 (2010), 619–672.
  • 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.
  • — Towards conformal invariance of 2D lattice models, in International Congress of Mathematicians (Madrid, 2006). Vol. II, pp. 1421–1451. Eur. Math. Soc., Zürich, 2006.
  • Smirnov, S. & Werner, W., Critical exponents for two-dimensional percolation. Math. Res. Lett., 8 (2001), 729–744.
  • 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.
  • 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.