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 Probability Surveys > Vol. 10 (2013) open journal systems 

Planar percolation with a glimpse of Schramm–Loewner evolution

Vincent Beffara, UMPA - ENS Lyon - CNRS
Hugo Duminil-Copin, Université de Genève

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 \(p\searrow p_c=1/2\).

Keywords: Site percolation, critical phenomenon, conformal invariance.

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Beffara, Vincent, Duminil-Copin, Hugo, Planar percolation with a glimpse of Schramm–Loewner evolution, Probability Surveys, 10, (2013), 1-50 (electronic). DOI: 10.1214/11-PS186.


[AB99]    Aizenman, M. and Burchard, A., Hölder regularity and dimension bounds for random curves, Duke Math. J. 99 (1999), no. 3, 419–453. MR1712629

[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. MR2278453

[BCL10]    Binder, I., Chayes, L., and Lei, H. K., On convergence to SLE6 I: Conformal invariance for certain models of the bond-triangular type, J. Stat. Phys. 141 (2010), no. 2, 359–390. MR2726646

[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 1, Probab. Theory Related Fields 153 (2012), 511–542. MR2948685

[Bef04]    Beffara, V., Hausdorff dimensions for SLE6, Ann. Probab. 32 (2004), no. 3B, 2606–2629. MR2078552

[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. MR2435854

[Bef08b]    Beffara, V., Is critical 2D percolation universal? In and Out of Equilibrium 2, Progress in Probability, vol. 60, Birkhäuser, 2008, pp. 31–58. MR2477376

[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. MR0091567

[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. MR1194785

[BN11]    Beffara, V. and Nolin, P., On monochromatic arm exponents for 2D critical percolation, Ann. Probab. 39 (2011), 1286–1304. MR2857240

[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. MR0757857

[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. MR0751712

[BR06a]    Bollobás, B. and Riordan, O., The critical probability for random Voronoi percolation in the plane is 12, Probab. Theory Related Fields 136 (2006), no. 3, 417–468. MR2257131

[BR06b]    Bollobás, B. and Riordan, O., Percolation, Cambridge University Press, New York, 2006. MR2283880

[BR06c]    Bollobás, B. and Riordan, O., A short proof of the Harris–Kesten theorem, Bull. Lond. Math. Soc. 38 (2006), no. 3, 470. MR2239042

[Car92]    Cardy, J. L., Critical percolation in finite geometries, J. Phys. A 25 (1992), no. 4, L201–L206. MR1151081

[CN06]    Camia, F. and Newman, C. M., Two-dimensional critical percolation: The full scaling limit, Comm. Math. Phys. 268 (2006), no. 1, 1–38. MR2249794

[CN07]    Camia, F. and Newman, C. M., Critical percolation exploration path and SLE6: A proof of convergence, Probab. Theory Related Fields 139 (2007), no. 3–4, 473–519. MR2322705

[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 2 × {0,1}, 2012, arXiv:1211.4138. MR3059543

[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, arXiv:cond-mat/0302072. MR2112129

[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. MR1371123

[FKG71]    Fortuin, C. M., Kasteleyn, P. W., and Ginibre, J., Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), 89–103. MR0309498

[Fri04]    Friedgut, E., Influences in product spaces: KKL and BKKKL revisited, Combinatorics, Probability and Computing 13 (2004), no. 1, 17–29. MR2034300

[Gar11]    Garban, C., Oded Schramm’s contributions to noise sensitivity, Ann. Probab. 39 (2011), no. 5, 1702–1767. MR2884872

[Geo88]    Georgii, H.-O., Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. MR0956646

[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. MR2736153

[Gri99]    Grimmett, G. R., Percolation, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math. Sciences], vol. 321, Springer-Verlag, Berlin, 1999. MR1707339

[Gri06]    Grimmett, G. R., The Random-Cluster model, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math. Sciences], vol. 333, Springer-Verlag, Berlin, 2006. MR2243761

[Gri10]    Grimmett, G. R., Probability on Graphs, Institute of Math. Statistics Textbooks, vol. 1, Cambridge University Press, Cambridge, 2010. MR2723356

[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. MR0115221

[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. MR1283177

[Kes80]    Kesten, H., The critical probability of bond percolation on the square lattice equals 12, Comm. Math. Phys. 74 (1980), no. 1, 41–59. MR0575895

[Kes82]    Kesten, H., Percolation Theory for Mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston, Mass., 1982. MR0692943

[Kes86]    Kesten, H., The incipient infinite cluster in two-dimensional percolation, Probab. Theory Related Fields 73 (1986), no. 3, 369–394. MR0859839

[Kes87]    Kesten, H., Scaling relations for 2D percolation, Comm. Math. Phys. 109 (1987), no. 1, 109–156. MR0879034

[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. MR2551766

[KS06]    Kalai, G. and Safra, S., Threshold Phenomena and Influence, Oxford University Press, 2006. MR2208732

[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. MR1659317

[Law05]    Lawler, G. F., Conformally Invariant Processes in the Plane, Math. Surveys and Monographs, vol. 114, American Math. Society, Providence, RI, 2005. MR2129588

[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. MR1230963

[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. MR1879850

[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. MR1879851

[LSW02]    Lawler, G. F., Schramm, O., and Werner, W., One-arm exponent for critical 2D percolation, Electron. J. Probab. 7 (2002), 13 pp. (electronic). MR1887622

[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. MR2044671

[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. MR2345229

[Nol08]    Nolin, P., Near-critical percolation in two dimensions, Electron. J. Probab. 13 (2008), 1562–1623. MR2438816

[Rei00]    Reimer, D., Proof of the van den Berg–Kesten conjecture, Combinatorics, Probability and Computing 9 (2000), no. 1, 27–32. MR1751301

[RS05]    Rohde, S. and Schramm, O., Basic properties of SLE, Ann. Math. (2) 161 (2005), no. 2, 883–924. MR2153402

[Rus78]    Russo, L., A note on percolation, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43 (1978), no. 1, 39–48. MR0488383

[Sch00]    Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288. MR1776084

[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. MR1851632

[Smi06]    Smirnov, S., Towards Conformal Invariance of 2D Lattice Models, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1421–1451. MR2275653

[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. MR2680496

[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. MR2630053

[Sun11]    Sun, N., Conformally invariant scaling limits in planar critical percolation, Probability Surveys 8 (2011), 155–209. MR2846901

[SW78]    Seymour, P. D. and D. J. A. Welsh, Percolation probabilities on the square lattice, Ann. Discrete Math. 3 (1978), 227–245. MR0494572

[SW01]    Smirnov, S. and Werner, W., Critical exponents for two-dimensional percolation, Math. Res. Lett. 8 (2001), no. 5–6, 729–744. MR1879816

[SW12]    Sheffield, S. and Werner, W., Conformal loop ensembles: the Markovian characterization and the loop-soup construction, Ann. Math. 176 (2012), 1827–1917. MR2979861

[vdBK85]    van den Berg, J. and Kesten, H., Inequalities with applications to percolation and reliability, Journal of applied probability (1985), 556–569. MR0799280

[Wer04]    Werner, W., Random planar curves and Schramm–Loewner evolutions, Lecture Notes in Math. 1840 (2004), 107–195. MR2079672

[Wer05]    Werner, W., Conformal restriction and related questions, Probability Surveys 2 (2005), 145–190. MR2178043

[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. MR2523462

[Wer09b]    Werner, W., Percolation et modèle d’Ising, Cours Spécialisés [Specialized Courses], vol. 16, Société Mathématique de France, Paris, 2009.

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