The Annals of Probability

On the scaling limits of planar percolation

Oded Schramm, Stanislav Smirnov, and Christophe Garban

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We prove Tsirelson’s conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.

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Ann. Probab., Volume 39, Number 5 (2011), 1768-1814.

First available in Project Euclid: 18 October 2011

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 82B43: Percolation [See also 60K35] 60G60: Random fields

Percolation noise scaling limit


Schramm, Oded; Smirnov, Stanislav; Garban, Christophe. On the scaling limits of planar percolation. Ann. Probab. 39 (2011), no. 5, 1768--1814. doi:10.1214/11-AOP659.

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  • [1] Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419–453.
  • [2] Astala, K., Prause, I. and Smirnov, S. (2010). Holomorphic motions and harmonic measure. Preprint.
  • [3] Benjamini, I., Kalai, G. and Schramm, O. (1999). Noise sensitivity of Boolean functions and applications to percolation. Publ. Math. Inst. Hautes Études Sci. 90 5–43.
  • [4] Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge Univ. Press, New York.
  • [5] Bollobás, B. and Riordan, O. (2010). Percolation on self-dual polygon configurations. Preprint. Available at arXiv:1001.4674.
  • [6] Camia, F., Fontes, L. R. G. and Newman, C. M. (2006). Two-dimensional scaling limits via marked nonsimple loops. Bull. Braz. Math. Soc. (N.S.) 37 537–559.
  • [7] Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: The full scaling limit. Comm. Math. Phys. 268 1–38.
  • [8] Cardy, J. L. (1992). Critical percolation in finite geometries. J. Phys. A 25 L201–L206.
  • [9] Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
  • [10] Federer, H. (1969). Geometric Measure Theory. Die Grundlehren der Mathematischen Wissenschaften 153. Springer, New York.
  • [11] Fell, J. M. G. (1962). A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc. 13 472–476.
  • [12] Garban, C., Pete, G. and Schramm, O. (2010). The scaling limit of the Minimal Spanning Tree—A preliminary report. In XVIth International Congress on Mathematical Physics, Prague, 2009 (P. Exner, ed.) 475–480. World Scientific, Singapore.
  • [13] Garban, C., Pete, G. and Schramm, O. (2010). The Fourier spectrum of critical percolation. Acta Math. 205 19–104.
  • [14] Garban, C., Pete, G. and Schramm, O. (2010). Pivotal, cluster and interface measures for critical planar percolation. Preprint. Available at arXiv:1008.1378.
  • [15] Garban, C., Rohde, S. and Schramm, O. (2011). Continuity of the SLE trace in simply connected domains. Israel J. Math. To appear. Available at arXiv:0810.4327.
  • [16] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [17] Kelley, J. L. (1975). General Topology. Graduate Texts in Mathematics 27. Springer, New York.
  • [18] Kemppainen, A. and Smirnov, S. (2010). Conformal invariance in random cluster models. III. Full scaling limit. Unpublished manuscript.
  • [19] Kenyon, R. (2009). Lectures on dimers. In Statistical Mechanics. IAS/Park City Mathematics Series 16 191–230. Amer. Math. Soc., Providence, RI.
  • [20] Kesten, H. (1980). The critical probability of bond percolation on the square lattice equals ½. Comm. Math. Phys. 74 41–59.
  • [21] Kesten, H. (1982). Percolation Theory for Mathematicians. Progress in Probability and Statistics 2. Birkhäuser, Boston, MA.
  • [22] Kesten, H. (1987). Scaling relations for 2D-percolation. Comm. Math. Phys. 109 109–156.
  • [23] Langlands, R., Pouliot, P. and Saint-Aubin, Y. (1994). Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30 1–61.
  • [24] Langlands, R. P. (2005). The renormalization fixed point as a mathematical object. In Twenty Years of Bialowieza: A Mathematical Anthology. World Sci. Monogr. Ser. Math. 8 185–216. World Sci. Publ., Hackensack, NJ.
  • [25] Langlands, R. P. and Lafortune, M. A. (1994). Finite models for percolation. In Representation Theory and Analysis on Homogeneous Spaces (New Brunswick, NJ, 1993). Contemporary Mathematics 177 227–246. Amer. Math. Soc., Providence, RI.
  • [26] Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 13 pp. (electronic).
  • [27] Mattila, P. (1995). Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics 44. Cambridge Univ. Press, Cambridge.
  • [28] Nienhuis, B. (1984). Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys. 34 731–761.
  • [29] Nolin, P. (2008). Near-critical percolation in two dimensions. Electron. J. Probab. 13 1562–1623.
  • [30] O’Donnell, R. and Servedio, R. A. (2007). Learning monotone decision trees in polynomial time. SIAM J. Comput. 37 827–844 (electronic).
  • [31] Rudin, W. (1973). Functional Analysis. McGraw-Hill, New York.
  • [32] Schramm, O. and Steif, J. E. (2010). Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. (2) 171 619–672.
  • [33] Sheffield, S. (2009). Exploration trees and conformal loop ensembles. Duke Math. J. 147 79–129.
  • [34] Smirnov, S. (2001). Critical percolation in the plane. Preprint. Available at arXiv:0909.4499.
  • [35] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Math. Acad. Sci. Paris Sér. I 333 239–244.
  • [36] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744.
  • [37] Steif, J. E. (2009). A survey of dynamical percolation. In Fractal Geometry and Stochastics IV. Proceedings of the 4th Conference, Greifswald, Germany, September 812, 2008. Progress in Probability 61 145–174. Birkhäuser, Basel.
  • [38] Tsirelson, B. (2004). Nonclassical stochastic flows and continuous products. Probab. Surv. 1 173–298 (electronic).
  • [39] Tsirelson, B. (2004). Scaling limit, noise, stability. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1840 1–106. Springer, Berlin.
  • [40] Tsirelson, B. (2005). Percolation, boundary, noise: An experiment. Preprint. Available at Arxiv:math/0506269.
  • [41] Tsirelson, B. S. and Vershik, A. M. (1998). Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations. Rev. Math. Phys. 10 81–145.
  • [42] Werner, W. (2009). Lectures on two-dimensional critical percolation. In Statistical Mechanics. IAS/Park City Mathematics Series 16 297–360. Amer. Math. Soc., Providence, RI.
  • [43] Wierman, J. C. (1981). Bond percolation on honeycomb and triangular lattices. Adv. in Appl. Probab. 13 298–313.