## Probability Surveys

### Planar percolation with a glimpse of Schramm–Loewner evolution

#### Abstract

In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy–Smirnov formula. This theorem, together with the introduction of Schramm–Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $\theta(p)$ for site percolation on the triangular lattice behaves like $(p-p_{c})^{5/36+o(1)}$ as $p\searrow p_{c}=1/2$.

#### Article information

Source
Probab. Surveys, Volume 10 (2013), 1-50.

Dates
First available in Project Euclid: 20 September 2013

https://projecteuclid.org/euclid.ps/1379686423

Digital Object Identifier
doi:10.1214/11-PS186

Mathematical Reviews number (MathSciNet)
MR3161674

Zentralblatt MATH identifier
1283.60118

#### Citation

Beffara, Vincent; Duminil-Copin, Hugo. Planar percolation with a glimpse of Schramm–Loewner evolution. Probab. Surveys 10 (2013), 1--50. doi:10.1214/11-PS186. https://projecteuclid.org/euclid.ps/1379686423

#### References

• [AB99] Aizenman, M. and Burchard, A., Hölder regularity and dimension bounds for random curves, Duke Math. J. 99 (1999), no. 3, 419–453.
• [BB07] Berger, N. and Biskup, M., Quenched invariance principle for simple random walk on percolation clusters, Probab. Theory Related Fields 137 (2007), no. 1, 83–120.
• [BCL10] Binder, I., Chayes, L., and Lei, H. K., On convergence to $\mathrm{SLE}_{6}$ I: Conformal invariance for certain models of the bond-triangular type, J. Stat. Phys. 141 (2010), no. 2, 359–390.
• [BCL12] Binder, I., Chayes, L., and Lei, H. K., On the rate of convergence for critical crossing probabilities, preprint, arXiv:1210.1917, 2012.
• [BDC12] Beffara, V. and Duminil-Copin, H., The self-dual point of the two-dimensional random-cluster model is critical for $q\ge1$, Probab. Theory Related Fields 153 (2012), 511–542.
• [Bef04] Beffara, V., Hausdorff dimensions for $\mathrm{SLE}_{6}$, Ann. Probab. 32 (2004), no. 3B, 2606–2629.
• [Bef07] Beffara, V., Cardy’s formula on the triangular lattice, the easy way, Universality and Renormalization (I. Binder and D. Kreimer, eds.), Fields Institute Communications, vol. 50, The Fields Institute, 2007, pp. 39–45.
• [Bef08a] Beffara, V., The dimension of the SLE curves, Ann. Probab. 36 (2008), no. 4, 1421–1452.
• [Bef08b] Beffara, V., Is critical 2D percolation universal? In and Out of Equilibrium 2, Progress in Probability, vol. 60, Birkhäuser, 2008, pp. 31–58.
• [BH57] Broadbent, S. R. and Hammersley, J. M., Percolation processes I. Crystals and mazes, Math. Proc. Cambridge Philos. Soc. 53 (1957), no. 3, 629–641.
• [BKK$92] Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y., and Linial, N., The influence of variables in product spaces, Israel J. Math. 77 (1992), no. 1–2, 55–64. • [BN11] Beffara, V. and Nolin, P., On monochromatic arm exponents for 2D critical percolation, Ann. Probab. 39 (2011), 1286–1304. • [BPZ84a] Belavin, A. A., Polyakov, A. M., and Zamolodchikov, A. B., Infinite conformal cymmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), no. 2, 333–380. • [BPZ84b] Belavin, A. A., Polyakov, A. M., and Zamolodchikov, A. B., Infinite conformal symmetry of critical fluctuations in two dimensions, J. Stat. Phys. 34 (1984), no. 5–6, 763–774. • [BR06a] Bollobás, B. and Riordan, O., The critical probability for random Voronoi percolation in the plane is$1/2$, Probab. Theory Related Fields 136 (2006), no. 3, 417–468. • [BR06b] Bollobás, B. and Riordan, O., Percolation, Cambridge University Press, New York, 2006. • [BR06c] Bollobás, B. and Riordan, O., A short proof of the Harris–Kesten theorem, Bull. Lond. Math. Soc. 38 (2006), no. 3, 470. • [Car92] Cardy, J. L., Critical percolation in finite geometries, J. Phys. A 25 (1992), no. 4, L201–L206. • [CN06] Camia, F. and Newman, C. M., Two-dimensional critical percolation: The full scaling limit, Comm. Math. Phys. 268 (2006), no. 1, 1–38. • [CN07] Camia, F. and Newman, C. M., Critical percolation exploration path and$\mathrm{SLE}_{6}$: A proof of convergence, Probab. Theory Related Fields 139 (2007), no. 3–4, 473–519. • [DCST13] Duminil-Copin, H., Sidoravicius, V., and Tassion, V., Absence of percolation for critical Bernoulli percolation on planar slabs, in preparation, 2013. • [DNS12] Damron, M., Newman, C. M., and Sidoravicius, V., Absence of site percolation at criticality in$\mathbb{Z}^{2}\times\{0,1\}$, 2012, arXiv:1211.4138. • [DSB04] Desolneux, A., Sapoval, B., and Baldassarri, A., Self-organized percolation power laws with and without fractal geometry in the etching of random solids, Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot: Multifractals, Probability and Statistical Mechanics, Applications (M. L. Lapidus and M. van Frankenhuijsen, eds.), Proceedings of Symposia in Pure Mathematics, vol. 72, AMS, 2004.. • [FK96] Friedgut, E. and Kalai, G., Every monotone graph property has a sharp threshold, Proceedings of the American Math. Society 124 (1996), no. 10, 2993–3002. • [FKG71] Fortuin, C. M., Kasteleyn, P. W., and Ginibre, J., Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), 89–103. • [Fri04] Friedgut, E., Influences in product spaces: KKL and BKKKL revisited, Combinatorics, Probability and Computing 13 (2004), no. 1, 17–29. • [Gar11] Garban, C., Oded Schramm’s contributions to noise sensitivity, Ann. Probab. 39 (2011), no. 5, 1702–1767. • [Geo88] Georgii, H.-O., Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. • [GM11a] Grimmett, G. R. and Manolescu, I., Inhomogeneous bond percolation on square, triangular, and hexagonal lattices, 2011, to appear in Ann. Probab., arXiv:1105.5535. • [GM11b] Grimmett, G. R. and Manolescu, I., Universality for bond percolation in two dimensions, 2011, to appear in Ann. Probab., arXiv:1108.2784. • [GM12] Grimmett, G. R. and Manolescu, I., Bond percolation on isoradial graphs, 2012, to appear in Prob. Theory Related Fields, arXiv:1204.0505. • [GPS10] Garban, C., Pete, G., and Schramm, O., The Fourier spectrum of critical percolation, Acta Math. 205 (2010), no. 1, 19–104. • [Gri99] Grimmett, G. R., Percolation, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math. Sciences], vol. 321, Springer-Verlag, Berlin, 1999. • [Gri06] Grimmett, G. R., The Random-Cluster model, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math. Sciences], vol. 333, Springer-Verlag, Berlin, 2006. • [Gri10] Grimmett, G. R., Probability on Graphs, Institute of Math. Statistics Textbooks, vol. 1, Cambridge University Press, Cambridge, 2010. • [Har60] Harris, T. E., A lower bound for the critical probability in a certain percolation process, Math. Proc. Cambridge Philos. Soc. 56 (1960), no. 1, 13–20. • [HS94] Hara, T. and Slade, G., Mean-field behaviour and the lace expansion, NATO ASI Series C, Math. and Physical Sciences 420 (1994), 87–122. • [Kes80] Kesten, H., The critical probability of bond percolation on the square lattice equals$1/2\$, Comm. Math. Phys. 74 (1980), no. 1, 41–59.
• [Kes82] Kesten, H., Percolation Theory for Mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston, Mass., 1982.
• [Kes86] Kesten, H., The incipient infinite cluster in two-dimensional percolation, Probab. Theory Related Fields 73 (1986), no. 3, 369–394.
• [Kes87] Kesten, H., Scaling relations for 2D percolation, Comm. Math. Phys. 109 (1987), no. 1, 109–156.
• [KKL88] Kahn, J., Kalai, G., and Linial, N., The influence of variables on Boolean functions, Proceedings of 29th Symposium on the Foundations of Computer Science, Computer Science Press, 1988, pp. 68– 80.
• [KN09] Kozma, G. and Nachmias, A., The Alexander–Orbach conjecture holds in high dimensions, Invent. Math. 178 (2009), no. 3, 635–654.
• [KS06] Kalai, G. and Safra, S., Threshold Phenomena and Influence, Oxford University Press, 2006.
• [KS12] Kemppainen, A. and Smirnov, S., Random curves, scaling limits and Loewner evolutions, preprint, 2012, arXiv:1212.6215.
• [Lan99] Lang, S., Complex Analysis, vol. 103, Springer Verlag, 1999.
• [Law05] Lawler, G. F., Conformally Invariant Processes in the Plane, Math. Surveys and Monographs, vol. 114, American Math. Society, Providence, RI, 2005.
• [LPSA94] Langlands, R., Pouliot, P., and Saint-Aubin, Y., Conformal invariance in two-dimensional percolation, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 1, 1–61.
• [LSW01a] Lawler, G. F., Schramm, O., and Werner, W., Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237–273.
• [LSW01b] Lawler, G. F., Schramm, O., and Werner, W., Values of Brownian intersection exponents. II. Plane exponents, Acta Math. 187 (2001), no. 2, 275–308.
• [LSW02] Lawler, G. F., Schramm, O., and Werner, W., One-arm exponent for critical 2D percolation, Electron. J. Probab. 7 (2002), 13 pp. (electronic).
• [LSW04] Lawler, G. F., Schramm, O., and Werner, W., Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), no. 1B, 939–995.
• [MNW12] Mendelson, D., Nachmias, A., and Watson, S. S., Rate of convergence for Cardy’s formula, preprint, arXiv:1210.4201, 2012.
• [MP07] Mathieu, P. and Piatnitski, A., Quenched invariance principles for random walks on percolation clusters, Proc. R Soc. A: Math., Physical and Engineering Science 463 (2007), 2287–2307.
• [Nol08] Nolin, P., Near-critical percolation in two dimensions, Electron. J. Probab. 13 (2008), 1562–1623.
• [Rei00] Reimer, D., Proof of the van den Berg–Kesten conjecture, Combinatorics, Probability and Computing 9 (2000), no. 1, 27–32.
• [RS05] Rohde, S. and Schramm, O., Basic properties of SLE, Ann. Math. (2) 161 (2005), no. 2, 883–924.
• [Rus78] Russo, L., A note on percolation, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43 (1978), no. 1, 39–48.
• [Sch00] Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288.
• [Smi01] Smirnov, S., Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244.
• [Smi06] Smirnov, S., Towards Conformal Invariance of 2D Lattice Models, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421–1451.
• [Smi10] Smirnov, S., Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model, Ann. Math. (2) 172 (2010), no. 2, 1435–1467.
• [SRG85] Sapoval, B., Rosso, M., and Gouyet, J.-F., The fractal nature of a diffusion front and the relation to percolation, Journal de Physique Lettres 46 (1985), no. 4, 149–156.
• [SS10] Schramm, O. and Steif, J., Quantitative noise sensitivity and exceptional times for percolation, Ann. Math. 171 (2010), no. 2, 619–672.
• [Sun11] Sun, N., Conformally invariant scaling limits in planar critical percolation, Probability Surveys 8 (2011), 155–209.
• [SW78] Seymour, P. D. and D. J. A. Welsh, Percolation probabilities on the square lattice, Ann. Discrete Math. 3 (1978), 227–245.
• [SW01] Smirnov, S. and Werner, W., Critical exponents for two-dimensional percolation, Math. Res. Lett. 8 (2001), no. 5–6, 729–744.
• [SW12] Sheffield, S. and Werner, W., Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Ann. Math. 176 (2012), 1827–1917.
• [vdBK85] van den Berg, J. and Kesten, H., Inequalities with applications to percolation and reliability, Journal of applied probability (1985), 556–569.
• [Wer04] Werner, W., Random planar curves and Schramm–Loewner evolutions, Lecture Notes in Math. 1840 (2004), 107–195.
• [Wer05] Werner, W., Conformal restriction and related questions, Probability Surveys 2 (2005), 145–190.
• [Wer09a] Werner, W., Lectures on two-dimensional critical percolation, Statistical mechanics, IAS/Park City Math. Ser., vol. 16, Amer. Math. Soc., 2009, pp. 297–360.
• [Wer09b] Werner, W., Percolation et modèle d’Ising, Cours Spécialisés [Specialized Courses], vol. 16, Société Mathématique de France, Paris, 2009.