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 Probability Surveys > Vol. 8 (2011) open journal systems 

Conformally invariant scaling limits in planar critical percolation

Nike Sun, Stanford University

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLEĸ), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE6. No prior knowledge is assumed beyond some general complex analysis and probability theory.

AMS 2000 subject classifications: Primary 60K35, 30C35; secondary 60J65.

Keywords: Conformally invariant scaling limits, percolation, Schramm-Loewner evolutions, preharmonicity, preholomorphicity, percolation exploration path.

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Sun, Nike, Conformally invariant scaling limits in planar critical percolation, Probability Surveys, 8, (2011), 155-209 (electronic). DOI: 10.1214/11-PS180.


[1]    Ahlfors, L. V. (1978). Complex analysis, Third ed. McGraw-Hill Book Co., New York. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics. MR510197 (80c:30001)

[2]    Ahlfors, L. V. (2010). Conformal invariants. AMS Chelsea Publishing, Providence, RI. Topics in geometric function theory, Reprint of the 1973 original, With a foreword by Peter Duren, F. W. Gehring and Brad Osgood. MR2730573

[3]    Aizenman, M. (1997). On the number of incipient spanning clusters. Nuclear Phys. B 485, 3, 551–582. http://dx.doi.org/10.1016/S0550-3213(96)00626-8. MR1431856 (98j:82030)

[4]    Aizenman, M. (1998). Scaling limit for the incipient spanning clusters. In Mathematics of multiscale materials (Minneapolis, MN, 1995–1996). IMA Vol. Math. Appl., Vol. 99. Springer, New York, 1–24. MR1635999 (99g:82034)

[5]    Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99, 3, 419–453. http://dx.doi.org/10.1215/S0012-7094-99-09914-3. MR1712629 (2000i:60012)

[6]    Aizenman, M., Duplantier, B., and Aharony, A. (1999). Path-crossing exponents and the external perimeter in 2D percolation. Phys. Rev. Lett. 83, 7 (Aug), 1359–1362.

[7]     Aizenman, M., Kesten, H., and Newman, C. M. (1987). Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111, 4, 505–531. http://projecteuclid.org/getRecord?id=euclid.cmp/1104159720.
MR901151 (89b:82060)

[8]    Alon, N., Benjamini, I., and Stacey, A. (2004). Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32, 3A, 1727–1745. http://dx.doi.org/10.1214/009117904000000414. MR2073175 (2005f:05149)

[9]    Beffara, V. (2004). Hausdorff dimensions for SLE6. Ann. Probab. 32, 3B, 2606–2629. http://dx.doi.org/10.1214/009117904000000072. MR2078552 (2005k:60295)

[10]    Beffara, V. (2007). Cardy’s formula on the triangular lattice, the easy way. In Universality and renormalization. Fields Inst. Commun., Vol. 50. Amer. Math. Soc., Providence, RI, 39–45. MR2310300 (2008i:82030)

[11]    Beffara, V. (2008a). The dimension of the SLE curves. Ann. Probab. 36, 4, 1421–1452. http://dx.doi.org/10.1214/07-AOP364.
MR2435854 (2009e:60026)

[12]    Beffara, V. (2008b). Is critical 2D percolation universal? In In and out of equilibrium. 2. Progr. Probab., Vol. 60. Birkhäuser, Basel, 31–58. http://dx.doi.org/10.1007/978-3-7643-8786-0_3. MR2477376 (2010h:60262)

[13]    Beffara, V. and Duminil-Copin, H. (2011). Planar percolation with a glimpse of Schramm-Loewner evolution. ArXiv preprint.

[14]    Benjamini, I. and Kozma, G. (2011). -actions and uniqueness of percolation. ArXiv preprint.

[15]    Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (1999a). Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27, 3, 1347–1356. http://dx.doi.org/10.1214/aop/1022677450. MR1733151 (2000k:60197)

[16]    Benjamini, I., Lyons, R., Peres, Y., and Schramm, O. (1999b). Group-invariant percolation on graphs. Geom. Funct. Anal. 9, 1, 29–66. http://dx.doi.org/10.1007/s000390050080. MR1675890 (99m:60149)

[17]    Benjamini, I. and Schramm, O. (1998). Conformal invariance of Voronoi percolation. Comm. Math. Phys. 197, 1, 75–107. http://dx.doi.org/10.1007/s002200050443. MR1646475 (99i:60172)

[18]    Billingsley, P. (1999). Convergence of probability measures, Second ed. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York. A Wiley-Interscience Publication, http://dx.doi.org/10.1002/9780470316962. MR1700749 (2000e:60008)

[19]    Binder, I., Chayes, L., and Lei, H. K. (2010a). On convergence to SLE6 I: conformal invariance for certain models of the bond-triangular type. J. Stat. Phys. 141, 2, 359–390. http://dx.doi.org/10.1007/s10955-010-0052-3. MR2726646

[20]    Binder, I., Chayes, L., and Lei, H. K. (2010b). On convergence to SLE6 II: discrete approximations and extraction of Cardy’s formula for general domains. J. Stat. Phys. 141, 2, 391–408. http://dx.doi.org/10.1007/s10955-010-0053-2. MR2726647

[21]    Bollobás, B. (2001). Random graphs, Second ed. Cambridge Studies in Advanced Mathematics, Vol. 73. Cambridge University Press, Cambridge. MR1864966 (2002j:05132)

[22]    Bollobás, B. and Kohayakawa, Y. (1994). Percolation in high dimensions. European J. Combin. 15, 2, 113–125. http://dx.doi.org/10.1006/eujc.1994.1014. MR1261058 (95c:60092)

[23]    Bollobás, B. and Riordan, O. (2006). Percolation. Cambridge University Press, New York. MR2283880 (2008c:82037)

[24]    Borgs, C., Chayes, J. T., Kesten, H., and Spencer, J. (2001). The birth of the infinite cluster: finite-size scaling in percolation. Comm. Math. Phys. 224, 1, 153–204. Dedicated to Joel L. Lebowitz, http://dx.doi.org/10.1007/s002200100521. MR1868996 (2002k:60199)

[25]    Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G., and Spencer, J. (2005a). Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Structures Algorithms 27, 2, 137–184. http://dx.doi.org/10.1002/rsa.20051. MR2155704 (2006e:05156)

[26]    Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G., and Spencer, J. (2005b). Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab. 33, 5, 1886–1944. http://dx.doi.org/10.1214/009117905000000260. MR2165583 (2006j:60009)

[27]    Borgs, C., Chayes, J. T., van der Hofstad, R., Slade, G., and Spencer, J. (2006). Random subgraphs of finite graphs. III. The phase transition for the n-cube. Combinatorica 26, 4, 395–410. http://dx.doi.org/10.1007/s00493-006-0022-1. MR2260845 (2007k:05194)

[28]    Broadbent, S. R. and Hammersley, J. M. (1957). Percolation processes. I. Crystals and mazes. Proc. Cambridge Philos. Soc. 53, 629–641. MR0091567 (19,989e)

[29]    Burton, R. M. and Keane, M. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121, 3, 501–505. http://projecteuclid.org/getRecord?id=euclid.cmp/1104178143. MR990777 (90g:60090)

[30]    Camia, F. and Newman, C. M. (2006). Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys. 268, 1, 1–38. http://dx.doi.org/10.1007/s00220-006-0086-1. MR2249794 (2007m:82032)

[31]    Camia, F. and Newman, C. M. (2007). Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields 139, 3-4, 473–519. http://dx.doi.org/10.1007/s00440-006-0049-7. MR2322705 (2008k:82040)

[32]    Carathéodory, C. (1913). Über die Begrenzung einfach zusammenhängender Gebiete. Math. Ann. 73, 3, 323–370.  http://dx.doi.org/10.1007/BF01456699. MR1511737

[33]    Cardy, J. (2001). Conformal invariance and percolation. ArXiv preprint.

[34]    Cardy, J. L. (1992). Critical percolation in finite geometries. J. Phys. A 25, 4, L201–L206. http://stacks.iop.org/0305-4470/25/L201. MR1151081 (92m:82048)

[35]    Chayes, L. and Lei, H. K. (2007). Cardy’s formula for certain models of the bond-triangular type. Rev. Math. Phys. 19, 5, 511–565. http://dx.doi.org/10.1142/S0129055X0700305X. MR2337476 (2008j:82019)

[36]    Chelkak, D. and Smirnov, S. (2009). Universality in the 2D Ising model and conformal invariance of fermionic observables. ArXiv preprint.

[37]    Chelkak, D. and Smirnov, S. (2010). Conformal invariance of the 2D Ising model at criticality. In preparation.

[38]    Ding, J., Kim, J. H., Lubetzky, E., and Peres, Y. (2011). Anatomy of a young giant component in the random graph. Random Structures Algorithms 39, 2, 139–178. http://dx.doi.org/10.1002/rsa.20342.

[39]    Dubédat, J. (2005). SLE(κ,ρ) martingales and duality. Ann. Probab. 33, 1, 223–243. http://dx.doi.org/10.1214/009117904000000793. MR2118865 (2005j:60180)

[40]    Dubédat, J. (2006). Excursion decompositions for SLE and Watts’ crossing formula. Probab. Theory Related Fields 134, 3, 453–488. http://dx.doi.org/10.1007/s00440-005-0446-3. MR2226888 (2007d:60019)

[41]    Dubédat, J. (2009). Duality of Schramm-Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4) 42, 5, 697–724. MR2571956 (2011g:60151)

[42]    Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist 39, 1563–1572. MR0230338 (37 #5900)

[43]    Duminil-Copin, H. and Smirnov, S. (2010). The connective constant of the honeycomb lattice equals ∘ ---√--
  2+   2. ArXiv preprint.

[44]    Duminil-Copin, H. and Smirnov, S. (2011). Conformal invariance of lattice models. ArXiv preprint.

[45]    Epstein, D. B. A. (1981). Prime ends. Proc. London Math. Soc. (3) 42, 3, 385–414. http://dx.doi.org/10.1112/plms/s3-42.3.385. MR614728 (83c:30025)

[46]    Fortuin, C. M., Kasteleyn, P. W., and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22, 89–103. MR0309498 (46 #8607)

[47]    Grimmett, G. (1999). Percolation, Second ed. Grundlehren der Mathematischen Wissenschaften, Vol. 321. Springer-Verlag, Berlin. MR1707339 (2001a:60114)

[48]    Grimmett, G. (2010). Probability on graphs. Institute of Mathematical Statistics Textbooks, Vol. 1. Cambridge University Press, Cambridge. Random processes on graphs and lattices. MR2723356

[49]    Grimmett, G. R. and Stacey, A. M. (1998). Critical probabilities for site and bond percolation models. Ann. Probab. 26, 4, 1788–1812. http://dx.doi.org/10.1214/aop/1022855883. MR1675079 (2000d:60160)

[50]    Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128, 2, 333–391. http://projecteuclid.org/getRecord?id=euclid.cmp/1104180434. MR1043524 (91a:82037)

[51]    Hara, T. and Slade, G. (1994). Mean-field behaviour and the lace expansion. In Probability and phase transition (Cambridge, 1993). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 420. Kluwer Acad. Publ., Dordrecht, 87–122. MR1283177 (95d:82033)

[52]    Hara, T. and Slade, G. (1995). The self-avoiding-walk and percolation critical points in high dimensions. Combin. Probab. Comput. 4, 3, 197–215. http://dx.doi.org/10.1017/S0963548300001607. MR1356575 (96i:82081)

[53]    Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56, 13–20. MR0115221 (22 #6023)

[54]    Hongler, C. and Smirnov, S. (2010). The energy density in the planar Ising model. ArXiv preprint.

[55]    Ikhlef, Y. and Rajabpour, M. A. (2011). Discrete holomorphic parafermions in the Ashkin-Teller model and SLE. J. Phys. A 44, 4, 042001, 11. http://dx.doi.org/10.1088/1751-8113/44/4/042001. MR2754707

[56]    Kager, W. and Nienhuis, B. (2004). A guide to stochastic Löwner evolution and its applications. J. Statist. Phys. 115, 5-6, 1149–1229. http://dx.doi.org/10.1023/B:JOSS.0000028058.87266.be. MR2065722 (2005f:82037)

[57]    Kakutani, S. (1944). Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo 20, 706–714. MR0014647 (7,315b)

[58]    Karatzas, I. and Shreve, S. E. (1991). Brownian motion and stochastic calculus, Second ed. Graduate Texts in Mathematics, Vol. 113. Springer-Verlag, New York. MR1121940 (92h:60127)

[59]    Kemppainen, A. and Smirnov, S. (2011). Conformal invariance in random cluster models. III. Full scaling limit. In preparation.

[60]    Kenyon, R. (2000). Conformal invariance of domino tiling. Ann. Probab. 28, 2, 759–795. http://dx.doi.org/10.1214/aop/1019160260.
MR1782431 (2002e:52022)

[61]    Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29, 3, 1128–1137. http://dx.doi.org/10.1214/aop/1015345599.
MR1872739 (2002k:82039)

[62]    Kenyon, R. (2011). Conformal invariance of loops in the double-dimer model. ArXiv preprint.

[63]    Kenyon, R. W. and Wilson, D. B. (2011). Boundary partitions in trees and dimers. Trans. Amer. Math. Soc. 363, 3, 1325–1364. http://dx.doi.org/10.1090/S0002-9947-2010-04964-5. MR2737268 (2011i:60023)

[64]    Kesten, H. (1980). The critical probability of bond percolation on the square lattice equals 1 2. Comm. Math. Phys. 74, 1, 41–59. http://projecteuclid.org/getRecord?id=euclid.cmp/1103907931. MR575895 (82c:60179)

[65]    Kesten, H. (1982). Percolation theory for mathematicians. Progress in Probability and Statistics, Vol. 2. Birkhäuser Boston, Mass. MR692943 (84i:60145)

[66]    Kesten, H. (1990). Asymptotics in high dimensions for percolation. In Disorder in physical systems. Oxford Sci. Publ. Oxford Univ. Press, New York, 219–240. MR1064563 (91k:60114)

[67]    Kesten, H., Sidoravicius, V., and Zhang, Y. (1998). Almost all words are seen in critical site percolation on the triangular lattice. Electron. J. Probab. 3, no. 10, 75 pp. (electronic).  http://www.math.washington.edu/ejpecp/EjpVol3/paper10.abs.html. MR1637089 (99j:60155)

[68]    Lalley, S., Lawler, G., and Narayanan, H. (2009). Geometric interpretation of half-plane capacity. Electron. Commun. Probab. 14, 566–571. MR2576752 (2011b:60332)

[69]    Langlands, R., Pouliot, P., and Saint-Aubin, Y. (1994). Conformal invariance in two-dimensional percolation. Bull. Amer. Math. Soc. (N.S.) 30, 1, 1–61. http://dx.doi.org/10.1090/S0273-0979-1994-00456-2. MR1230963 (94e:82056)

[70]    Lawler, G. (2009). Schramm-Loewner evolution (SLE). In Statistical mechanics. IAS/Park City Math. Ser., Vol. 16. Amer. Math. Soc., Providence, RI, 231–295. MR2523461 (2011d:60244)

[71]    Lawler, G., Schramm, O., and Werner, W. (2003). Conformal restriction: the chordal case. J. Amer. Math. Soc. 16, 4, 917–955 (electronic). http://dx.doi.org/10.1090/S0894-0347-03-00430-2. MR1992830 (2004g:60130)

[72]    Lawler, G. F. (2005). Conformally invariant processes in the plane. Mathematical Surveys and Monographs, Vol. 114. American Mathematical Society, Providence, RI. MR2129588 (2006i:60003)

[73]    Lawler, G. F., Schramm, O., and Werner, W. (2001a). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187, 2, 237–273. http://dx.doi.org/10.1007/BF02392618.
MR1879850 (2002m:60159a)

[74]    Lawler, G. F., Schramm, O., and Werner, W. (2001b). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187, 2, 275–308. http://dx.doi.org/10.1007/BF02392619.
MR1879851 (2002m:60159b)

[75]    Lawler, G. F., Schramm, O., and Werner, W. (2002a). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7, no. 2, 13 pp. (electronic). http://www.math.washington.edu/ejpecp/EjpVol7/paper2.abs.html. MR1887622 (2002k:60204)

[76]    Lawler, G. F., Schramm, O., and Werner, W. (2002b). Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38, 1, 109–123. http://dx.doi.org/10.1016/S0246-0203(01)01089-5. MR1899232 (2003d:60163)

[77]     Lawler, G. F., Schramm, O., and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32, 1B, 939–995. http://dx.doi.org/10.1214/aop/1079021469. MR2044671 (2005f:82043)

[78]    Lawler, G. F. and Sheffield, S. (2009). The natural parametrization for the Schramm-Loewner evolution. ArXiv preprint.

[79]    Lawler, G. F. and Werner, W. (2000). Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. (JEMS) 2, 4, 291–328. http://dx.doi.org/10.1007/s100970000024. MR1796962 (2002g:60123)

[80]    Lévy, P. (1992). Processus stochastiques et mouvement brownien. Les Grands Classiques Gauthier-Villars. Éditions Jacques Gabay, Sceaux. Followed by a note by M. Loève, Reprint of the second (1965) edition. MR1188411 (93i:60003)

[81]    Lieb, E. H. and Loss, M. (2001). Analysis, Second ed. Graduate Studies in Mathematics, Vol. 14. American Mathematical Society, Providence, RI. MR1817225 (2001i:00001)

[82]    Marshall, D. E. and Rohde, S. (2005). The Loewner differential equation and slit mappings. J. Amer. Math. Soc. 18, 4, 763–778 (electronic). http://dx.doi.org/10.1090/S0894-0347-05-00492-3. MR2163382 (2006d:30022)

[83]    Mendelson, D., Nachmias, A., Sheffield, S., and Watson, S. (2011). Convergence rate in Cardy’s formula. In preparation.

[84]    Miller, J. (2010a). Fluctuations for the Ginzburg-Landau ϕ interface model on a bounded domain. ArXiv preprint.

[85]    Miller, J. (2010b). Universality for SLE(4). ArXiv preprint.

[86]    Miller, J. and Sheffield, S. (2011). CLE(4) and the Gaussian free field. In preparation.

[87]    Mörters, P. and Peres, Y. (2010). Brownian motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge. With an appendix by Oded Schramm and Wendelin Werner. MR2604525

[88]    Munkres, J. R. (2000). Topology, Second ed. Prentice Hall, Upper Saddle River, NJ.

[89]    Nachmias, A. (2009). Mean-field conditions for percolation on finite graphs. Geom. Funct. Anal. 19, 4, 1171–1194. http://dx.doi.org/10.1007/s00039-009-0032-4. MR2570320 (2011e:60228)

[90]    Pommerenke, C. (1992). Boundary behaviour of conformal maps. Grundlehren der Mathematischen Wissenschaften, Vol. 299. Springer-Verlag, Berlin. MR1217706 (95b:30008)

[91]    Reimer, D. (2000). Proof of the van den Berg-Kesten conjecture. Combin. Probab. Comput. 9, 1, 27–32. http://dx.doi.org/10.1017/S0963548399004113. MR1751301 (2001g:60017)

[92]    Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161, 2, 883–924. http://dx.doi.org/10.4007/annals.2005.161.883. MR2153402 (2006f:60093)

[93]    Russo, L. (1978). A note on percolation. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 43, 1, 39–48. MR0488383 (58 #7931)

[94]    Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288. http://dx.doi.org/10.1007/BF02803524. MR1776084 (2001m:60227)

[95]    Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6, 115–120 (electronic). MR1871700 (2002h:60227)

[96]    Schramm, O. (2007). Conformally invariant scaling limits: an overview and a collection of problems. In International Congress of Mathematicians. Vol. I. Eur. Math. Soc., Zürich, 513–543. http://dx.doi.org/10.4171/022-1/20. MR2334202 (2008j:60237)

[97]    Schramm, O. and Sheffield, S. (2005). Harmonic explorer and its convergence to SLE4. Ann. Probab. 33, 6, 2127–2148. http://dx.doi.org/10.1214/009117905000000477. MR2184093 (2006i:60013)

[98]    Schramm, O. and Sheffield, S. (2009). Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 1, 21–137. http://dx.doi.org/10.1007/s11511-009-0034-y. MR2486487 (2010f:60238)

[99]    Schramm, O. and Sheffield, S. (2010). A contour line of the continuum Gaussian free field. ArXiv preprint.

[100]    Schramm, O. and Wilson, D. B. (2005). SLE coordinate changes. New York J. Math. 11, 659–669 (electronic). http://nyjm.albany.edu:8000/j/2005/11_659.html. MR2188260 (2007e:82019)

[101]    Seymour, P. D. and Welsh, D. J. A. (1978). Percolation probabilities on the square lattice. Ann. Discrete Math. 3, 227–245. Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977). MR0494572 (58 #13410)

[102]    Sheffield, S. (2009). Exploration trees and conformal loop ensembles. Duke Math. J. 147, 1, 79–129. http://dx.doi.org/10.1215/00127094-2009-007. MR2494457 (2010g:60184)

[103]    Sheffield, S. and Werner, W. (2010). Conformal loop ensembles: The Markovian characterization and the loop-soup construction. To appear, Ann. of Math.

[104]    Sheffield, S. and Wilson, D. B. (2010). Schramm’s proof of Watts’ formula. ArXiv preprint.

[105]    Smirnov, S. (2001). Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333, 3, 239–244. http://dx.doi.org/10.1016/S0764-4442(01)01991-7. MR1851632 (2002f:60193)

[106]    Smirnov, S. (2005). Critical percolation and conformal invariance. In XIVth International Congress on Mathematical Physics. World Sci. Publ., Hackensack, NJ, 99–112. MR2227824 (2007h:82030)

[107]    Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II. Eur. Math. Soc., Zürich, 1421–1451. MR2275653 (2008g:82026)

[108]    Smirnov, S. (2009). Critical percolation in the plane. ArXiv preprint.

[109]    Smirnov, S. (2010a). Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172, 2, 1435–1467. http://dx.doi.org/10.4007/annals.2010.172.1441. MR2680496

[110]    Smirnov, S. (2010b). Discrete complex analysis and probability. ArXiv preprint.

[111]    Smirnov, S. (2011). Conformal invariance in random cluster models. II. Scaling limit of the interface. In preparation.

[112]    Stauffer, D. and Aharony, A. (1992). Introduction to percolation theory, 2nd ed. Taylor & Francis, London.

[113]    Stein, E. M. and Shakarchi, R. (2003a). Complex analysis. Princeton Lectures in Analysis, II. Princeton University Press, Princeton, NJ. MR1976398 (2004d:30002)

[114]    Stein, E. M. and Shakarchi, R. (2003b). Fourier analysis. Princeton Lectures in Analysis, Vol. 1. Princeton University Press, Princeton, NJ. An introduction. MR1970295 (2004a:42001)

[115]    van den Berg, J. and Kesten, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22, 3, 556–569. MR799280 (87b:60027)

[116]    Werner, W. (2004). Random planar curves and Schramm-Loewner evolutions. In Lectures on probability theory and statistics. Lecture Notes in Math., Vol. 1840. Springer, Berlin, 107–195. MR2079672 (2005m:60020)

[117]    Werner, W. (2009). Lectures on two-dimensional critical percolation. In Statistical mechanics. IAS/Park City Math. Ser., Vol. 16. Amer. Math. Soc., Providence, RI, 297–360. MR2523462 (2011e:82042)

[118]    Wilson, D. B. (2011). XOR-Ising loops and the Gaussian free field. ArXiv preprint.

[119]     Zhan, D. (2008). Duality of chordal SLE. Invent. Math. 174, 2, 309–353. http://dx.doi.org/10.1007/s00222-008-0132-z. MR2439609 (2010f:60239)

[120]    Zhan, D. (2010). Duality of chordal SLE, II. Ann. Inst. Henri Poincaré Probab. Stat. 46, 3, 740–759. http://dx.doi.org/10.1214/09-AIHP340. MR2682265

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