The Annals of Probability

Oded Schramm: From circle packing to SLE

Steffen Rohde

Full-text: Open access

Abstract

In this note, I will describe some highlights of Oded Schramm’s work in circle packings and the Koebe conjecture, as well as on SLE.

Article information

Source
Ann. Probab., Volume 39, Number 5 (2011), 1621-1667.

Dates
First available in Project Euclid: 18 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1318940778

Digital Object Identifier
doi:10.1214/10-AOP590

Mathematical Reviews number (MathSciNet)
MR2884870

Zentralblatt MATH identifier
1236.60004

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 52C26: Circle packings and discrete conformal geometry 30C35: General theory of conformal mappings

Keywords
Circle packing Schramm–Loewner evolution

Citation

Rohde, Steffen. Oded Schramm: From circle packing to SLE. Ann. Probab. 39 (2011), no. 5, 1621--1667. doi:10.1214/10-AOP590. https://projecteuclid.org/euclid.aop/1318940778


Export citation

References

  • [1] Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419–453.
  • [2] Alberts, T. and Sheffield, S. (2008). Hausdorff dimension of the SLE curve intersected with the real line. Electron. J. Probab. 13 1166–1188.
  • [3] Andreev, E. M. (1970). Convex polyhedra in Lobačevskiĭ spaces. Mat. Sb. (N.S.) 81 445–478.
  • [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.
  • [6] Bauer, M. and Bernard, D. (2003). Conformal field theories of stochastic Loewner evolutions. Comm. Math. Phys. 239 493–521.
  • [7] Bauer, M. and Bernard, D. SLE, CFT and zig-zag probabilities. Available at arXiv:math-ph/0401019v1.
  • [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.
  • [11] Bauer, R. O. and Friedrich, R. M. (2008). On chordal and bilateral SLE in multiply connected domains. Math. Z. 258 241–265.
  • [12] Beardon, A. F. and Stephenson, K. (1991). Circle packings in different geometries. Tohoku Math. J. (2) 43 27–36.
  • [13] Beardon, A. F. and Stephenson, K. (1991). The Schwarz–Pick lemma for circle packings. Illinois J. Math. 35 577–606.
  • [14] Beffara, V. (2008). The dimension of the SLE curves. Ann. Probab. 36 1421–1452.
  • [15] Benjamini, I. and Schramm, O. (1996). Random walks and harmonic functions on infinite planar graphs using square tilings. Ann. Probab. 24 1219–1238.
  • [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.
  • [17] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic).
  • [18] Benjamini, I. and Schramm, O. (2009). KPZ in one dimensional random geometry of multiplicative cascades. Comm. Math. Phys. 289 653–662.
  • [19] Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (2001). Uniform spanning forests. Ann. Probab. 29 1–65.
  • [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.
  • [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.
  • [22] Bonk, M. and Kleiner, B. (2002). Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150 127–183.
  • [23] Bonk, M. (2006). Quasiconformal geometry of fractals. In International Congress of Mathematicians II 1349–1373. Eur. Math. Soc., Zürich.
  • [24] Brandt, M. (1980). Ein Abbildungssatz für endlich-vielfach zusammenhängende Gebiete. Bull. Soc. Sci. Lett. Łódź 30 12.
  • [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.
  • [26] Burdzy, K. (1989). Cut points on Brownian paths. Ann. Probab. 17 1012–1036.
  • [27] Camia, F. and Newman, C. M. (2007). Critical percolation exploration path and SLE6: A proof of convergence. Probab. Theory Related Fields 139 473–519.
  • [28] Cannon, J. W. (1994). The combinatorial Riemann mapping theorem. Acta Math. 173 155–234.
  • [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.
  • [30] Cardy, J. L. (1992). Critical percolation in finite geometries. J. Phys. A 25 L201–L206.
  • [31] Cardy, J. (2005). SLE for theoretical physicists. Ann. Physics 318 81–118.
  • [32] Carleson, L. and Makarov, N. (2001). Aggregation in the plane and Loewner’s equation. Comm. Math. Phys. 216 583–607.
  • [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.
  • [34] de Verdière, C. (1989). Empilements de cercles: Convergence d’une méthode de point fixe. Forum Math. I 395–402.
  • [35] de Verdière, C. (1991). Une principe variationnel pour les empilements de cercles. Invent. Math. 104 655–669.
  • [36] Doyle, P., He, Z.-X. and Rodin, B. (1994). Second derivatives of circle packings and conformal mappings. Discrete Comput. Geom. 11 35–49.
  • [37] Dubédat, J. (2003). SLE and triangles. Electron. Comm. Probab. 8 28–42 (electronic).
  • [38] Dubédat, J. (2006). Excursion decompositions for SLE and Watts’ crossing formula. Probab. Theory Related Fields 134 453–488.
  • [39] Dubédat, J. (2009). Duality of Schramm–Loewner evolutions. Ann. Sci. Éc. Norm. Supér. (4) 42 697–724.
  • [40] Dubédat, J. (2009). SLE and the free field: Partition functions and couplings. J. Amer. Math. Soc. 22 995–1054.
  • [41] Duffin, R. J. (1962). The extremal length of a network. J. Math. Anal. Appl. 5 200–215.
  • [42] Duplantier, B. (2006). Conformal random geometry. In Les Houches 2005 Lecture Notes: Mathematical Statistical Physics 101–217. Elsevier, Amsterdam.
  • [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 arXiv:0808.1560v1.
  • [45] Duren, P. L. (1983). Univalent Functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 259. Springer, New York.
  • [46] Garban, C., Pete, G. and Schramm, O. The scaling limit of the Minimal Spanning Tree—A preliminary report. Available at arXiv:0909.3138.
  • [47] Garban, C., Rohde, S. and Schramm, O. (2010). Continuity of the SLE trace in simply connected domains. Israel J. Math. To appear.
  • [48] Garnett, J. B. and Marshall, D. E. (2005). Harmonic Measure. New Mathematical Monographs 2. Cambridge Univ. Press, Cambridge.
  • [49] Gruzberg, I. A. and Kadanoff, L. P. (2004). The Loewner equation: Maps and shapes. J. Stat. Phys. 114 1183–1198.
  • [50] Harrington, A. N. (1982). Conformal mappings onto domains with arbitrarily specified boundary shapes. J. Anal. Math. 41 39–53.
  • [51] Hastings, M. and Levitov, L. (1998). Laplacian growth as one-dimensional turbulence. Phys. D 116 244–252.
  • [52] He, Z.-X. (1991). An estimate for hexagonal circle packings. J. Differential Geom. 33 395–412.
  • [53] He, Z.-X. and Rodin, B. (1993). Convergence of circle packings of finite valence to Riemann mappings. Comm. Anal. Geom. 1 31–41.
  • [54] He, Z.-X. and Schramm, O. (1993). Fixed points, Koebe uniformization and circle packings. Ann. of Math. (2) 137 369–406.
  • [55] He, Z.-X. and Schramm, O. (1994). Rigidity of circle domains whose boundary has σ-finite linear measure. Invent. Math. 115 297–310.
  • [56] He, Z.-X. and Schramm, O. (1995). Hyperbolic and parabolic packings. Discrete Comput. Geom. 14 123–149.
  • [57] He, Z. X. and Schramm, O. (1991). A hyperbolicity criterion for circle packings. Preprint.
  • [58] He, Z.-X. and Schramm, O. (1995). Koebe uniformization for “almost circle domains”. Amer. J. Math. 117 653–667.
  • [59] He, Z.-X. and Schramm, O. (1995). The inverse Riemann mapping theorem for relative circle domains. Pacific J. Math. 171 157–165.
  • [60] He, Z.-X. and Schramm, O. (1996). On the convergence of circle packings to the Riemann map. Invent. Math. 125 285–305.
  • [61] He, Z.-X. and Schramm, O. (1998). The C-convergence of hexagonal disk packings to the Riemann map. Acta Math. 180 219–245.
  • [62] Heinonen, J. and Koskela, P. (1995). Definitions of quasiconformality. Invent. Math. 120 61–79.
  • [63] Kager, W. and Nienhuis, B. (2004). A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115 1149–1229.
  • [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.
  • [65] Kenyon, R. (2000). Conformal invariance of domino tiling. Ann. Probab. 28 759–795.
  • [66] Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math. 185 239–286.
  • [67] Kenyon, R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29 1128–1137.
  • [68] Koebe, P. (1908). Ueber die Uniformisierung beliebiger analytischer Kurven (Dritte Mitteilung). Nachr. Ges. Wiss. Gött. 337–358.
  • [69] Koebe, P. (1936). Kontaktprobleme der konformen abbildung. Ber. Sächs. Akad. Wiss. 88 141–164.
  • [70] Kufarev, P. P. (1947). A remark on integrals of Löwner’s equation. Doklady Akad. Nauk SSSR (N.S.) 57 655–656.
  • [71] Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47 655–693.
  • [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.
  • [73] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI.
  • [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.
  • [75] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237–273.
  • [76] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187 275–308.
  • [77] Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 13 pp. (electronic).
  • [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.
  • [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.
  • [80] Lawler, G. F., Schramm, O. and Werner, W. (2002). Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 179–201.
  • [81] Lawler, G., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917–955 (electronic).
  • [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.
  • [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.
  • [84] Lawler, G. F. and Werner, W. (1999). Intersection exponents for planar Brownian motion. Ann. Probab. 27 1601–1642.
  • [85] Lawler, G. F. and Werner, W. (2000). Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. (JEMS) 2 291–328.
  • [86] Lawler, G. F. and Werner, W. (2004). The Brownian loop soup. Probab. Theory Related Fields 128 565–588.
  • [87] Le Gall, J.-F. (2007). The topological structure of scaling limits of large planar maps. Invent. Math. 169 621–670.
  • [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.
  • [89] Löwner, K. (1923). Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89 103–121.
  • [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.
  • [91] Mandelbrot, B. (1982). The beauty of fractals, Freeman.
  • [92] Marshall, D. E. and Rohde, S. (2005). The Loewner differential equation and slit mappings. J. Amer. Math. Soc. 18 763–778 (electronic).
  • [93] Marshall, D. E. and Sundberg, C. (1989). Harmonic measure and radial projection. Trans. Amer. Math. Soc. 316 81–95.
  • [94] Miller, G. and Thurston, W. (1990). Separators in two and three dimensions. In Proceedings of the 22nd 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.
  • [96] Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 1559–1574.
  • [97] Pommerenke, C. (1966). On the Loewner differential equation. Michigan Math. J. 13 435–443.
  • [98] Pommerenke, C. (1992). Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 299. Springer, Berlin.
  • [99] Popova, N. V. (1949). Investigation of some integrals of the equation dw/dt=A/(wλ(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/(wλ(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 arXiv:0807.1036.
  • [102] Rodin, B. and Sullivan, D. (1987). The convergence of circle packings to the Riemann mapping. J. Differential Geom. 26 349–360.
  • [103] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883–924.
  • [104] Sachs, H. (1994). Coin graphs, polyhedra, and conformal mapping. Discrete Math. 134 133–138.
  • [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.
  • [107] Schramm, O. (1993). How to cage an egg. Invent. Math. 107 543–560.
  • [108] Schramm, O. (1991). Existence and uniqueness of packings with specified combinatorics. Israel J. Math. 73 321–341.
  • [109] Schramm, O. (1993). Square tilings with prescribed combinatorics. Israel J. Math. 84 97–118.
  • [110] Schramm, O. (1995). Transboundary extremal length. J. Anal. Math. 66 307–329.
  • [111] Schramm, O. (1996). Conformal uniformization and packings. Israel J. Math. 93 399–428.
  • [112] Schramm, O. (1997). Circle patterns with the combinatorics of the square grid. Duke Math. J. 86 347–389.
  • [113] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  • [114] Schramm, O. (2001). A percolation formula. Electron. Comm. Probab. 6 115–120 (electronic).
  • [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.
  • [116] Schramm, O. and Sheffield, S. (2005). Harmonic explorer and its convergence to SLE4. Ann. Probab. 33 2127–2148.
  • [117] Schramm, O. and Sheffield, S. (2009). Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 21–137.
  • [118] Schramm, O. and Sheffield, S. (2010). A contour line of the continuum Gaussian free field. Available at arXiv:1008.2447.
  • [119] Schramm, O., Sheffield, S. and Wilson, D. B. (2009). Conformal radii for conformal loop ensembles. Comm. Math. Phys. 288 43–53.
  • [120] Schramm, O. and Wilson, D. B. (2005). SLE coordinate changes. New York J. Math. 11 659–669 (electronic).
  • [121] Schramm, O. and Zhou, W. (2010). Boundary proximity of SLE. Probab. Theory Related Fields 146 435–450.
  • [122] Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
  • [123] Sheffield, S. (2009). Exploration trees and conformal loop ensembles. Duke Math. J. 147 79–129.
  • [124] Sheffield, S. and Wilson, D. (2010). Schramm’s proof of Watts’ formula. Available at arXiv:1003.3271.
  • [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.
  • [126] Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians II 1421–1451. Eur. Math. Soc., Zürich.
  • [127] Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Ann. of Math. (2) 172 1435–1467.
  • [128] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744.
  • [129] Stephenson, K. (1996). A probabilistic proof of Thurston’s conjecture on circle packings. Rend. Sem. Mat. Fis. Milano 66 201–291 (1998).
  • [130] Stephenson, K. (2005). Introduction to Circle Packing: The Theory of Discrete Analytic Functions. Cambridge Univ. Press, Cambridge.
  • [131] Strebel, K. (1951). Über das Kreisnormierungsproblem der konformen Abbildung. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 101 22.
  • [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.
  • [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.
  • [135] Werner, W. (2005). Conformal restriction and related questions. Probab. Surv. 2 145–190 (electronic).
  • [136] Werner, W. (2008). The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc. 21 137–169.
  • [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.
  • [139] Zhan, D. (2008). Reversibility of chordal SLE. Ann. Probab. 36 1472–1494.
  • [140] Zhan, D. (2008). Duality of chordal SLE. Invent. Math. 174 309–353.