Oded Schramm: from circle packing to SLE. (English) Zbl 1236.60004

Rohde, Steffen

Ann. Probab. 39, No. 5, 1621-1667 (2011).

The author focuses on some highlights of Oded Schramm’s work in circle packing and the Koebe conjecture as well as on Schramm–Loewner evolution (SLE).

Reviewer: Nicko G. Gamkrelidze (Moskva)

MSC:

60-03	History of probability theory
01A70	Biographies, obituaries, personalia, bibliographies
60D05	Geometric probability and stochastic geometry
52C26	Circle packings and discrete conformal geometry
30C35	General theory of conformal mappings
60J67	Stochastic (Schramm-)Loewner evolution (SLE)

Keywords:

circle packing; Schramm–Loewner evolution

Software:

CirclePack

References:

[1]	Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419-453. · Zbl 0944.60022 · doi:10.1215/S0012-7094-99-09914-3
[2]	Alberts, T. and Sheffield, S. (2008). Hausdorff dimension of the SLE curve intersected with the real line. Electron. J. Probab. 13 1166-1188. · Zbl 1192.60025
[3]	Andreev, E. M. (1970). Convex polyhedra in Lobačevskiĭ spaces. Mat. Sb. ( N.S. ) 81 445-478. · Zbl 0194.23202
[4]	Andreev, E. M. (1970). Convex polyhedra of finite volume in Lobačevskiĭ space. Mat. Sb. ( N.S. ) 83 256-260.
[5]	Angel, O. and Schramm, O. (2003). Uniform infinite planar triangulations. Comm. Math. Phys. 241 191-213. · Zbl 1098.60010 · doi:10.1007/s00220-003-0932-3
[6]	Bauer, M. and Bernard, D. (2003). Conformal field theories of stochastic Loewner evolutions. Comm. Math. Phys. 239 493-521. · Zbl 1046.81091 · doi:10.1007/s00220-003-0881-x
[7]	Bauer, M. and Bernard, D. SLE, CFT and zig-zag probabilities. Available at .
[8]	Bauer, M. and Bernard, D. (2006). 2D growth processes: SLE and Loewner chains. Phys. Rep. 432 115-221.
[9]	Bauer, M., Bernard, D. and Houdayer, J. (2005). Dipolar stochastic Loewner evolutions. J. Stat. Mech. Theory Exp. 3 P03001, 18 pp. (electronic).
[10]	Bauer, R. O. and Friedrich, R. M. (2004). Stochastic Loewner evolution in multiply connected domains. C. R. Math. Acad. Sci. Paris 339 579-584. · Zbl 1064.60017 · doi:10.1016/j.crma.2004.08.010
[11]	Bauer, R. O. and Friedrich, R. M. (2008). On chordal and bilateral SLE in multiply connected domains. Math. Z. 258 241-265. · Zbl 1130.30006 · doi:10.1007/s00209-006-0041-z
[12]	Beardon, A. F. and Stephenson, K. (1991). Circle packings in different geometries. Tohoku Math. J. (2) 43 27-36. · Zbl 0703.52007 · doi:10.2748/tmj/1178227533
[13]	Beardon, A. F. and Stephenson, K. (1991). The Schwarz-Pick lemma for circle packings. Illinois J. Math. 35 577-606. · Zbl 0753.30016
[14]	Beffara, V. (2008). The dimension of the SLE curves. Ann. Probab. 36 1421-1452. · Zbl 1165.60007 · doi:10.1214/07-AOP364
[15]	Benjamini, I. and Schramm, O. (1996). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219-1238. · Zbl 0862.60053
[16]	Benjamini, I. and Schramm, O. (1996). Harmonic functions on planar and almost planar graphs and manifolds, via circle packings. Invent. Math. 126 565-587. · Zbl 0868.31008 · doi:10.1007/s002220050109
[17]	Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic). · Zbl 1010.82021
[18]	Benjamini, I. and Schramm, O. (2009). KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys. 289 653-662. · Zbl 1170.83006 · doi:10.1007/s00220-009-0752-1
[19]	Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1-65. · Zbl 1016.60009
[20]	Bishop, C. J., Jones, P. W., Pemantle, R. and Peres, Y. (1997). The dimension of the Brownian frontier is greater than 1. J. Funct. Anal. 143 309-336. · Zbl 0870.60077 · doi:10.1006/jfan.1996.2928
[21]	Bobenko, A. I., Hoffmann, T. and Springborn, B. A. (2006). Minimal surfaces from circle patterns: Geometry from combinatorics. Ann. of Math. (2) 164 231-264. · Zbl 1122.53003 · doi:10.4007/annals.2006.164.231
[22]	Bonk, M. and Kleiner, B. (2002). Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150 127-183. · Zbl 1037.53023 · doi:10.1007/s00222-002-0233-z
[23]	Bonk, M. (2006). Quasiconformal geometry of fractals. In International Congress of Mathematicians II 1349-1373. Eur. Math. Soc., Zürich. · Zbl 1102.30016
[24]	Brandt, M. (1980). Ein Abbildungssatz für endlich-vielfach zusammenhängende Gebiete. Bull. Soc. Sci. Lett. Łódź 30 12. · Zbl 0463.30006
[25]	Brooks, R. L., Smith, C. A. B., Stone, A. H. and Tutte, W. T. (1940). The dissection of rectangles into squares. Duke Math. J. 7 312-340. · Zbl 0024.16501 · doi:10.1215/S0012-7094-40-00718-9
[26]	Burdzy, K. (1989). Cut points on Brownian paths. Ann. Probab. 17 1012-1036. · Zbl 0691.60069 · doi:10.1214/aop/1176991254
[27]	Camia, F. and Newman, C. M. (2007). Critical percolation exploration path and SLE 6 : A proof of convergence. Probab. Theory Related Fields 139 473-519. · Zbl 1126.82007 · doi:10.1007/s00440-006-0049-7
[28]	Cannon, J. W. (1994). The combinatorial Riemann mapping theorem. Acta Math. 173 155-234. · Zbl 0832.30012 · doi:10.1007/BF02398434
[29]	Cannon, J. W., Floyd, W. J. and Parry, W. R. (1994). Squaring rectangles: The finite Riemann mapping theorem. In The Mathematical Legacy of Wilhelm Magnus : Groups , Geometry and Special Functions ( Brooklyn , NY , 1992). Contemp. Math. 169 133-212. Amer. Math. Soc., Providence, RI. · Zbl 0818.20043
[30]	Cardy, J. L. (1992). Critical percolation in finite geometries. J. Phys. A 25 L201-L206. · Zbl 0965.82501 · doi:10.1088/0305-4470/25/4/009
[31]	Cardy, J. (2005). SLE for theoretical physicists. Ann. Physics 318 81-118. · Zbl 1073.81068 · doi:10.1016/j.aop.2005.04.001
[32]	Carleson, L. and Makarov, N. (2001). Aggregation in the plane and Loewner’s equation. Comm. Math. Phys. 216 583-607. · Zbl 1042.82039 · doi:10.1007/s002200000340
[33]	Courant, R., Manel, B. and Shiffman, M. (1940). A general theorem on conformal mapping of multiply connected domains. Proc. Natl. Acad. Sci. USA 26 503-507. · Zbl 0063.00988 · doi:10.1073/pnas.26.8.503
[34]	de Verdière, C. (1989). Empilements de cercles: Convergence d’une méthode de point fixe. Forum Math. I 395-402. · Zbl 0685.52012 · doi:10.1515/form.1989.1.395
[35]	de Verdière, C. (1991). Une principe variationnel pour les empilements de cercles. Invent. Math. 104 655-669. · Zbl 0745.52010 · doi:10.1007/BF01245096
[36]	Doyle, P., He, Z.-X. and Rodin, B. (1994). Second derivatives of circle packings and conformal mappings. Discrete Comput. Geom. 11 35-49. · Zbl 0794.30006 · doi:10.1007/BF02573993
[37]	Dubédat, J. (2003). SLE and triangles. Electron. Comm. Probab. 8 28-42 (electronic). · Zbl 1061.60104
[38]	Dubédat, J. (2006). Excursion decompositions for SLE and Watts’ crossing formula. Probab. Theory Related Fields 134 453-488. · Zbl 1112.60032 · doi:10.1007/s00440-005-0446-3
[39]	Dubédat, J. (2009). Duality of Schramm-Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4) 42 697-724. · Zbl 1205.60147
[40]	Dubédat, J. (2009). SLE and the free field: Partition functions and couplings. J. Amer. Math. Soc. 22 995-1054. · Zbl 1204.60079 · doi:10.1090/S0894-0347-09-00636-5
[41]	Duffin, R. J. (1962). The extremal length of a network. J. Math. Anal. Appl. 5 200-215. · Zbl 0107.43604 · doi:10.1016/S0022-247X(62)80004-3
[42]	Duplantier, B. (2006). Conformal random geometry. In Les Houches 2005 Lecture Notes : Mathematical Statistical Physics 101-217. Elsevier, Amsterdam. · Zbl 1370.60013
[43]	Duplantier, B. and Kwon, K. (1988). Conformal invariance and intersection of random walks. Phys. Rev. Lett. 61 2514-2517.
[44]	Duplantier, B. and Sheffield, S. Liouville Quantum Gravity and KPZ. Available at .
[45]	Duren, P. L. (1983). Univalent Functions. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 259 . Springer, New York. · Zbl 0514.30001
[46]	Garban, C., Pete, G. and Schramm, O. The scaling limit of the Minimal Spanning Tree-A preliminary report. Available at . · Zbl 1219.60084
[47]	Garban, C., Rohde, S. and Schramm, O. (2010). Continuity of the SLE trace in simply connected domains. Israel J. Math. · Zbl 1261.60079
[48]	Garnett, J. B. and Marshall, D. E. (2005). Harmonic Measure. New Mathematical Monographs 2 . Cambridge Univ. Press, Cambridge. · Zbl 1077.31001
[49]	Gruzberg, I. A. and Kadanoff, L. P. (2004). The Loewner equation: Maps and shapes. J. Stat. Phys. 114 1183-1198. · Zbl 1072.81052 · doi:10.1023/B:JOSS.0000013973.40984.3b
[50]	Harrington, A. N. (1982). Conformal mappings onto domains with arbitrarily specified boundary shapes. J. Anal. Math. 41 39-53. · Zbl 0528.30006 · doi:10.1007/BF02803393
[51]	Hastings, M. and Levitov, L. (1998). Laplacian growth as one-dimensional turbulence. Phys. D 116 244-252. · Zbl 0962.76542 · doi:10.1016/S0167-2789(97)00244-3
[52]	He, Z.-X. (1991). An estimate for hexagonal circle packings. J. Differential Geom. 33 395-412. · Zbl 0743.52005
[53]	He, Z.-X. and Rodin, B. (1993). Convergence of circle packings of finite valence to Riemann mappings. Comm. Anal. Geom. 1 31-41. · Zbl 0777.30003
[54]	He, Z.-X. and Schramm, O. (1993). Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2) 137 369-406. · Zbl 0777.30002 · doi:10.2307/2946541
[55]	He, Z.-X. and Schramm, O. (1994). Rigidity of circle domains whose boundary has \sigma -finite linear measure. Invent. Math. 115 297-310. · Zbl 0809.30006 · doi:10.1007/BF01231761
[56]	He, Z.-X. and Schramm, O. (1995). Hyperbolic and parabolic packings. Discrete Comput. Geom. 14 123-149. · Zbl 0830.52010 · doi:10.1007/BF02570699
[57]	He, Z. X. and Schramm, O. (1991). A hyperbolicity criterion for circle packings.
[58]	He, Z.-X. and Schramm, O. (1995). Koebe uniformization for “almost circle domains”. Amer. J. Math. 117 653-667. · Zbl 0830.30006 · doi:10.2307/2375085
[59]	He, Z.-X. and Schramm, O. (1995). The inverse Riemann mapping theorem for relative circle domains. Pacific J. Math. 171 157-165. · Zbl 0860.30005
[60]	He, Z.-X. and Schramm, O. (1996). On the convergence of circle packings to the Riemann map. Invent. Math. 125 285-305. · Zbl 0868.30010 · doi:10.1007/s002220050076
[61]	He, Z.-X. and Schramm, O. (1998). The C \infty -convergence of hexagonal disk packings to the Riemann map. Acta Math. 180 219-245. · Zbl 0913.30004 · doi:10.1007/BF02392900
[62]	Heinonen, J. and Koskela, P. (1995). Definitions of quasiconformality. Invent. Math. 120 61-79. · Zbl 0832.30013 · doi:10.1007/BF01241122
[63]	Kager, W. and Nienhuis, B. (2004). A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115 1149-1229. · Zbl 1157.82327 · doi:10.1023/B:JOSS.0000028058.87266.be
[64]	Kennedy, T. (2004). Conformal invariance and stochastic Loewner evolution predictions for the 2D self-avoiding walk-Monte Carlo tests. J. Stat. Phys. 114 51-78. · Zbl 1060.82022 · doi:10.1023/B:JOSS.0000003104.35024.f9
[65]	Kenyon, R. (2000). Conformal invariance of domino tiling. Ann. Probab. 28 759-795. · Zbl 1043.52014 · doi:10.1214/aop/1019160260
[66]	Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239-286. · Zbl 0982.05013 · doi:10.1007/BF02392811
[67]	Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128-1137. · Zbl 1034.82021 · doi:10.1214/aop/1015345599
[68]	Koebe, P. (1908). Ueber die Uniformisierung beliebiger analytischer Kurven (Dritte Mitteilung). Nachr. Ges. Wiss. Gött. 337-358. · JFM 39.0489.02
[69]	Koebe, P. (1936). Kontaktprobleme der konformen abbildung. Ber. Sächs. Akad. Wiss. 88 141-164. · Zbl 0017.21701
[70]	Kufarev, P. P. (1947). A remark on integrals of Löwner’s equation. Doklady Akad. Nauk SSSR ( N.S. ) 57 655-656. · Zbl 0029.03702
[71]	Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47 655-693. · Zbl 0445.60058 · doi:10.1215/S0012-7094-80-04741-9
[72]	Lawler, G. F. (1998). Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions. In Random Walks ( Budapest , 1998). Bolyai Society Mathematical Studies 9 219-258. János Bolyai Math. Soc., Budapest. · Zbl 0955.60076
[73]	Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114 . Amer. Math. Soc., Providence, RI. · Zbl 1074.60002
[74]	Lawler, G. F., Schramm, O. and Werner, W. (2001). The dimension of the planar Brownian frontier is 4/3. Math. Res. Lett. 8 401-411. · Zbl 1114.60316 · doi:10.4310/MRL.2001.v8.n4.a1
[75]	Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237-273. · Zbl 1005.60097 · doi:10.1007/BF02392618
[76]	Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275-308. · Zbl 0993.60083 · doi:10.1007/BF02392619
[77]	Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 13 pp. (electronic). · Zbl 1015.60091
[78]	Lawler, G. F., Schramm, O. and Werner, W. (2002). Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38 109-123. · Zbl 1006.60075 · doi:10.1016/S0246-0203(01)01089-5
[79]	Lawler, G. F., Schramm, O. and Werner, W. (2002). Sharp estimates for Brownian non-intersection probabilities. In In and Out of Equilibrium ( Mambucaba , 2000). Progress in Probability 51 113-131. Birkhäuser, Boston, MA. · Zbl 1011.60062
[80]	Lawler, G. F., Schramm, O. and Werner, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 179-201. · Zbl 1024.60033 · doi:10.1007/BF02392842
[81]	Lawler, G., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917-955 (electronic). · Zbl 1030.60096 · doi:10.1090/S0894-0347-03-00430-2
[82]	Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939-995. · Zbl 1126.82011 · doi:10.1214/aop/1079021469
[83]	Lawler, G. F., Schramm, O. and Werner, W. (2004). On the scaling limit of planar self-avoiding walk. In Fractal Geometry and Applications : A Jubilee of Benoît Mandelbrot , Part 2. Proceedings of Symposia in Pure Mathematics 72 339-364. Amer. Math. Soc., Providence, RI. · Zbl 1069.60089
[84]	Lawler, G. F. and Werner, W. (1999). Intersection exponents for planar Brownian motion. Ann. Probab. 27 1601-1642. · Zbl 0965.60071 · doi:10.1214/aop/1022677543
[85]	Lawler, G. F. and Werner, W. (2000). Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. ( JEMS ) 2 291-328. · Zbl 1098.60081 · doi:10.1007/s100970000024
[86]	Lawler, G. F. and Werner, W. (2004). The Brownian loop soup. Probab. Theory Related Fields 128 565-588. · Zbl 1049.60072 · doi:10.1007/s00440-006-0319-6
[87]	Le Gall, J.-F. (2007). The topological structure of scaling limits of large planar maps. Invent. Math. 169 621-670. · Zbl 1132.60013 · doi:10.1007/s00222-007-0059-9
[88]	Le Gall, J.-F. and Paulin, F. (2008). Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. 18 893-918. · Zbl 1166.60006 · doi:10.1007/s00039-008-0671-x
[89]	Löwner, K. (1923). Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89 103-121. · JFM 49.0714.01
[90]	Lyons, R. (1997). A bird’s-eye view of uniform spanning trees and forests. In Microsurveys in Discrete Probability ( Princeton , NJ , 1997). DIMACS Series in Discrete Mathematics and Theoretical Computer Science 41 135-162. Amer. Math. Soc., Providence, RI. · Zbl 0909.60016
[91]	Mandelbrot, B. (1982). The beauty of fractals, Freeman. · Zbl 0504.28001
[92]	Marshall, D. E. and Rohde, S. (2005). The Loewner differential equation and slit mappings. J. Amer. Math. Soc. 18 763-778 (electronic). · Zbl 1078.30005 · doi:10.1090/S0894-0347-05-00492-3
[93]	Marshall, D. E. and Sundberg, C. (1989). Harmonic measure and radial projection. Trans. Amer. Math. Soc. 316 81-95. · Zbl 0692.30020 · doi:10.2307/2001274
[94]	Miller, G. and Thurston, W. (1990). Separators in two and three dimensions. In Proceedings of the 22 nd Annual ACM Symposium on Theory of Computing 300-309. ACM, New York.
[95]	Nolin, P. and Werner, W. (2009). Asymmetry of near-critical percolation interfaces. J. Amer. Math. Soc. 22 797-819. · Zbl 1218.60088 · doi:10.1090/S0894-0347-08-00619-X
[96]	Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559-1574. · Zbl 0758.60010 · doi:10.1214/aop/1176990223
[97]	Pommerenke, C. (1966). On the Loewner differential equation. Michigan Math. J. 13 435-443. · Zbl 0163.31801 · doi:10.1307/mmj/1028999601
[98]	Pommerenke, C. (1992). Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 299 . Springer, Berlin. · Zbl 0762.30001
[99]	Popova, N. V. (1949). Investigation of some integrals of the equation dw / dt = A /( w - \lambda ( t )). Novosibirsk Gos. Ped. Inst. Uchon. Zapiski 8 13-26.
[100]	Popova, N. V. (1954). Relations between Löwner’s equation and the equation dw / dt =1/( w - \lambda ( t )). Izv. Akad. Nauk BSSR Ser. Fiz.-Mat. Nauk 6 97-98.
[101]	Rhodes, R. and Vargas, V. (2008). KPZ formula for log-infinitely divisible multifractal random measures. Available at . · Zbl 1268.60070
[102]	Rodin, B. and Sullivan, D. (1987). The convergence of circle packings to the Riemann mapping. J. Differential Geom. 26 349-360. · Zbl 0694.30006
[103]	Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883-924. · Zbl 1081.60069 · doi:10.4007/annals.2005.161.883
[104]	Sachs, H. (1994). Coin graphs, polyhedra, and conformal mapping. Discrete Math. 134 133-138. · Zbl 0808.05043 · doi:10.1016/0012-365X(93)E0068-F
[105]	Schramm, O. (1990). Combinatorially prescribed packings and applications to conformal and quasiconformal maps. Ph.D. thesis, Princeton.
[106]	Schramm, O. (1991). Rigidity of infinite (circle) packings. J. Amer. Math. Soc. 4 127-149. · Zbl 0726.52008 · doi:10.2307/2939257
[107]	Schramm, O. (1993). How to cage an egg. Invent. Math. 107 543-560. · Zbl 0726.52003 · doi:10.1007/BF01231901
[108]	Schramm, O. (1991). Existence and uniqueness of packings with specified combinatorics. Israel J. Math. 73 321-341. · Zbl 0756.05039 · doi:10.1007/BF02773845
[109]	Schramm, O. (1993). Square tilings with prescribed combinatorics. Israel J. Math. 84 97-118. · Zbl 0788.05019 · doi:10.1007/BF02761693
[110]	Schramm, O. (1995). Transboundary extremal length. J. Anal. Math. 66 307-329. · Zbl 0842.30006 · doi:10.1007/BF02788827
[111]	Schramm, O. (1996). Conformal uniformization and packings. Israel J. Math. 93 399-428. · Zbl 0872.30005 · doi:10.1007/BF02761115
[112]	Schramm, O. (1997). Circle patterns with the combinatorics of the square grid. Duke Math. J. 86 347-389. · Zbl 1053.30525 · doi:10.1215/S0012-7094-97-08611-7
[113]	Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093 · doi:10.1007/BF02803524
[114]	Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115-120 (electronic). · Zbl 1008.60100
[115]	Schramm, O. (2007). Conformally invariant scaling limits: An overview and a collection of problems. In International Congress of Mathematicians I 513-543. Eur. Math. Soc., Zürich. · Zbl 1131.60088
[116]	Schramm, O. and Sheffield, S. (2005). Harmonic explorer and its convergence to SLE 4 . Ann. Probab. 33 2127-2148. · Zbl 1095.60007 · doi:10.1214/009117905000000477
[117]	Schramm, O. and Sheffield, S. (2009). Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 21-137. · Zbl 1210.60051 · doi:10.1007/s11511-009-0034-y
[118]	Schramm, O. and Sheffield, S. (2010). A contour line of the continuum Gaussian free field. Available at . · Zbl 1331.60090
[119]	Schramm, O., Sheffield, S. and Wilson, D. B. (2009). Conformal radii for conformal loop ensembles. Comm. Math. Phys. 288 43-53. · Zbl 1187.82044 · doi:10.1007/s00220-009-0731-6
[120]	Schramm, O. and Wilson, D. B. (2005). SLE coordinate changes. New York J. Math. 11 659-669 (electronic). · Zbl 1094.82007
[121]	Schramm, O. and Zhou, W. (2010). Boundary proximity of SLE. Probab. Theory Related Fields 146 435-450. · Zbl 1227.60101 · doi:10.1007/s00440-008-0195-1
[122]	Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521-541. · Zbl 1132.60072 · doi:10.1007/s00440-006-0050-1
[123]	Sheffield, S. (2009). Exploration trees and conformal loop ensembles. Duke Math. J. 147 79-129. · Zbl 1170.60008 · doi:10.1215/00127094-2009-007
[124]	Sheffield, S. and Wilson, D. (2010). Schramm’s proof of Watts’ formula. Available at . · Zbl 1238.60089
[125]	Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239-244. · Zbl 0985.60090 · doi:10.1016/S0764-4442(01)01991-7
[126]	Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians II 1421-1451. Eur. Math. Soc., Zürich. · Zbl 1112.82014
[127]	Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Ann. of Math. (2) 172 1435-1467. · Zbl 1200.82011 · doi:10.4007/annals.2010.172.1441
[128]	Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729-744. · Zbl 1009.60087 · doi:10.4310/MRL.2001.v8.n6.a4
[129]	Stephenson, K. (1996). A probabilistic proof of Thurston’s conjecture on circle packings. Rend. Sem. Mat. Fis. Milano 66 201-291 (1998). · Zbl 0997.60504 · doi:10.1007/BF02925361
[130]	Stephenson, K. (2005). Introduction to Circle Packing : The Theory of Discrete Analytic Functions . Cambridge Univ. Press, Cambridge. · Zbl 1074.52008
[131]	Strebel, K. (1951). Über das Kreisnormierungsproblem der konformen Abbildung. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 101 22. · Zbl 0044.08501
[132]	Thurston, W. (1980). The Geometry and Topology of 3 -Manifolds. Mathematical Notes 13 . Princeton Univ. Press, Princeton, NJ.
[133]	Virág, B. (2003). Brownian beads. Probab. Theory Related Fields 127 367-387. · Zbl 1035.60085 · doi:10.1007/s00440-003-0289-8
[134]	Werner, W. (2004). Random planar curves and Schramm-Loewner evolutions. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1840 107-195. Springer, Berlin. · Zbl 1057.60078
[135]	Werner, W. (2005). Conformal restriction and related questions. Probab. Surv. 2 145-190 (electronic). · Zbl 1189.60032 · doi:10.1214/154957805100000113
[136]	Werner, W. (2008). The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc. 21 137-169. · Zbl 1130.60016 · doi:10.1090/S0894-0347-07-00557-7
[137]	Witten, T. and Sander, L. (1981). Diffusion-limited aggregation, a kinetic phenomenon. Phys. Rev. Lett. 47 1400-1403.
[138]	Zhan, D. (2004). Stochastic Loewner evolution in doubly connected domains. Probab. Theory Related Fields 129 340-380. · Zbl 1054.60104 · doi:10.1007/s00440-004-0343-1
[139]	Zhan, D. (2008). Reversibility of chordal SLE. Ann. Probab. 36 1472-1494. · Zbl 1157.60051 · doi:10.1214/07-AOP366
[140]	Zhan, D. (2008). Duality of chordal SLE. Invent. Math. 174 309-353. · Zbl 1158.60047 · doi:10.1007/s00222-008-0132-z