Annales Henri Poincaré

, Volume 16, Issue 6, pp 1311–1395 | Cite as

The Coefficient Problem and Multifractality of Whole-Plane SLE & LLE

  • Bertrand DuplantierEmail author
  • Chi Nguyen
  • Nga Nguyen
  • Michel Zinsmeister


Karl Löwner (later known as Charles Loewner) introduced his famous differential equation in 1923 to solve the Bieberbach conjecture for series expansion coefficients of univalent analytic functions at level n = 3. His method was revived in 1999 by Oded Schramm when he introduced the Stochastic Loewner Evolution (SLE), a conformally invariant process which made it possible to prove many predictions from conformal field theory for critical planar models in statistical mechanics. The aim of this paper is to revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, Lévy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the “oddified” or m-fold conformal maps of whole-plane SLEs or Lévy–Loewner Evolutions. We also study the (average) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a moment order p = p *(κ) > 0, at which one goes from the bulk SLE κ average integral means spectrum, as predicted by the first author (Duplantier Phys. Rev. Lett. 84:1363–1367, 2000) and established by Beliaev and Smirnov (Commun Math Phys 290:577–595, 2009) and valid for p ≤ p *(κ), to a new integral means spectrum for p ≥ p *(κ), as conjectured in part by Loutsenko (J Phys A Math Gen 45(26):265001, 2012). The latter spectrum is, furthermore, shown to be intimately related, via the associated packing spectrum, to the radial SLE derivative exponents obtained by Lawler, Schramm and Werner (Acta Math 187(2):237–273, 2001), and to the local SLE tip multifractal exponents obtained from quantum gravity by the first author (Duplantier Proc. Sympos. Pure Math. 72(2):365–482, 2004). This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map. A succinct, preliminary, version of this study first appeared in Duplantier et al. (Coefficient estimates for whole-plane SLE processes, Hal-00609774, 2011).


Harmonic Measure Multifractal Spectrum Boundary Equation Driving Function Loewner Chain 
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  1. 1.
    Adams D.A., Lin Y.T., Sander L.M., Ziff R.M.: Harmonic measure for critical Potts clusters. Phys. Rev. E 80, 031141 (2009)CrossRefADSGoogle Scholar
  2. 2.
    Adams D.A., Sander L.M., Ziff R.M.: Harmonic Measure for Percolation and Ising Clusters Including Rare Events. Phys. Rev. Lett. 101, 144102 (2008)CrossRefADSGoogle Scholar
  3. 3.
    Applebaum D.: Lévy Processes and Stochastic Calculus, Second edition. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  4. 4.
    Beliaev, D.: Harmonic measure on random fractals. Doctoral Thesis, Department of Mathematics, KTH, Stockholm (2005)Google Scholar
  5. 5.
    Beliaev, D., Duplantier, B., Zinsmeister, M.: Harmonic Measure and Whole-Plane SLE (2014). In preparationGoogle Scholar
  6. 6.
    Beliaev D., Smirnov S.: Harmonic Measure and SLE. Commun. Math. Phys. 290, 577–595 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  7. 7.
    Belikov A., Gruzberg I.A., Rushkin I.: Statistics of harmonic measure and winding of critical curves from conformal field theory. J. Phys. A: Math. Gen. 41, 285006 (2008)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Benjamini I., Schramm O.: KPZ in One Dimensional Random Geometry of Multiplicative Cascades. Commun. Math. Phys. 289, 46–56 (2009)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bettelheim E., Rushkin I., Gruzberg I.A., Wiegmann P.: Harmonic Measure of Critical Curves. Phys. Rev. Lett. 95, 170602 (2005)CrossRefADSGoogle Scholar
  10. 10.
    Bieberbach L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. S-B. Preuss. Akad. Wiss. 1, 940–955 (1916)Google Scholar
  11. 11.
    Carleson L., Makarov N.G.: Some results connected with Brennan’s conjecture. Ark. Mat. 32, 33–62 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189, 515–580 (2012)CrossRefADSzbMATHMathSciNetGoogle Scholar
  13. 13.
    Chen Z.-Q., Rohde S.: Schramm-Loewner Equations Driven by Symmetric Stable Processes. Commun. Math. Phys. 285, 799–824 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  14. 14.
    David F.: Conformal Field Theories Coupled to 2-D Gravity in the Conformal Gauge. Mod. Phys. Lett. A 3(17), 1651–1656 (1988)CrossRefADSGoogle Scholar
  15. 15.
    de Branges L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Dieudonné J.: Sur les fonctions univalentes. C. R. Acad. Sci. Paris 192, 1148–1150 (1931)Google Scholar
  17. 17.
    Distler J., Kawai H.: Conformal Field Theory and 2D Quantum Gravity. Nucl. Phys. B 321, 509–527 (1989)CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Duplantier B.: Harmonic Measure Exponents for Two-Dimensional Percolation. Phys. Rev. Lett. 82, 3940–3943 (1999)CrossRefADSzbMATHMathSciNetGoogle Scholar
  19. 19.
    Duplantier B.: Two-Dimensional Copolymers and Exact Conformal Multifractality. Phys. Rev. Lett. 82, 880–883 (1999)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Duplantier B.: Conformally Invariant Fractals and Potential Theory. Phys. Rev. Lett. 84, 1363–1367 (2000)CrossRefADSzbMATHMathSciNetGoogle Scholar
  21. 21.
    Duplantier, B.: Higher Conformal Multifractality. J. Stat. Phys. 110, 691–738 (2003). Special issue in honor of Michael E. Fisher’s 70th birthdayGoogle Scholar
  22. 22.
    Duplantier, B.: Conformal fractal geometry & boundary quantum gravity. In: Lapidus, M.L., van Frankenhuysen, M. (eds.) Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, volume 72 of Proc. Sympos. Pure Math., pp. 365–482. Amer. Math. Soc., Providence, RI (2004)Google Scholar
  23. 23.
    Duplantier, B.: Conformal Random Geometry. In: Bovier, A., Dunlop, F., den Hollander, F., van Enter, A., Dalibard, J. (eds.) Mathematical Statistical Physics (Les Houches Summer School, Session LXXXIII, 2005), pp. 101–217. Elsevier B.V., Amsterdam (2006)Google Scholar
  24. 24.
    Duplantier B., Binder I.A.: Harmonic Measure and Winding of Conformally Invariant Curves. Phys. Rev. Lett. 89, 264101 (2002)CrossRefADSGoogle Scholar
  25. 25.
    Duplantier B., Binder I.A.: Harmonic measure and winding of random conformal paths: A Coulomb gas perspective. Nucl. Phys. B [FS] 802, 494–513 (2008)CrossRefADSzbMATHMathSciNetGoogle Scholar
  26. 26.
    Duplantier, B., Nguyen, T.P.C., Nguyen T.T.N., Zinsmeister, M.: Coefficient estimates for whole-plane SLE processes. Hal-00609774, 20 (2011)Google Scholar
  27. 27.
    Duplantier B., Sheffield S.: Duality and KPZ in Liouville Quantum Gravity. Phys. Rev. Lett. 102, 150603 (2009)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Duplantier B., Sheffield S.: Liouville Quantum Gravity and KPZ. Invent. Math. 185, 333–393 (2011)CrossRefADSzbMATHMathSciNetGoogle Scholar
  29. 29.
    Duplantier B., Sheffield S.: Schramm-Loewner Evolution and Liouville Quantum Gravity. Phys. Rev. Lett. 107, 131305 (2011)CrossRefADSGoogle Scholar
  30. 30.
    Evans, L.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)Google Scholar
  31. 31.
    Fekete M., Szegö G.: Eine Bemerkung über ungerade schlichte Funktionen. J. London Math. Soc. 8, 85–89 (1933)CrossRefGoogle Scholar
  32. 32.
    Feng J., MacGregor T.H.: Estimates on the integral means of the derivatives of univalent functions. J. Anal. Math 29, 203–231 (1976)CrossRefzbMATHGoogle Scholar
  33. 33.
    Frisch, U., Parisi, G.: Turbulence and predictability in geophysical fluid dynamics and climate dynamics. In: Ghil, M., Benzi, R.R., Parisi, G. (eds.) Proceedings of the International School of Physics Enrico Fermi, course LXXXVIII, pp. 84–87. North Holland, New York (1985)Google Scholar
  34. 34.
    Grunsky H.: Koeffizienten Bedingungen für schlicht abbidende meromorphe Funktionen. Math. Z. 45, 29–61 (1939)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Halsey T.C., Jensen M.H., Kadanoff L.P., Procaccia I., Shraiman B.I.: Fractal measures and their singularities - The characterization of strange sets. Phys. Rev. A 33, 1141–1151 (1986)CrossRefADSzbMATHMathSciNetGoogle Scholar
  36. 36.
    Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Shraiman, B.I.: Fractal measures and their singularities: The characterization of strange sets; Erratum: [Phys. Rev. A 33, 1141 (1986)]. Phys. Rev. A 34, 1601–1601 (1986)Google Scholar
  37. 37.
    Hastings M.B.: Exact Multifractal Spectra for Arbitrary Laplacian Random Walks. Phys. Rev. Lett. 88, 055506 (2002)CrossRefADSGoogle Scholar
  38. 38.
    Hentschel H.G.E., Procaccia I.: The infinite number of dimensions of probabilistic fractals and strange attractors. Physica D 8, 435–444 (1983)CrossRefADSzbMATHMathSciNetGoogle Scholar
  39. 39.
    Johansson F., Sola A.: Rescaled Lévy-Loewner hulls and random growth. B. Sci. Math. 133(3), 238–256 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Johansson Viklund.F., Lawler G.F.: Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math. 209(2), 265–322 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Jones P.W., Makarov N.G.: On coefficient problems for univalent functions and conformal dimensions. Duke Math. J. 66, 169–206 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Kemppainen A.: Stationarity of SLE. J. Stat. Phys. 139, 108–121 (2010)CrossRefADSzbMATHMathSciNetGoogle Scholar
  43. 43.
    Knizhnik V.G., Polyakov A.M., Zamolodchikov A.B.: Fractal Structure of 2D-quantum gravity. Mod. Phys. Lett. A 3, 819–826 (1988)CrossRefADSMathSciNetGoogle Scholar
  44. 44.
    Kytölä K., Kemppainen A.: SLE local martingales, reversibility and duality. J. Phys. A: Math. Gen. 39, L657–L666 (2006)CrossRefADSzbMATHGoogle Scholar
  45. 45.
    Lawler, G.F.: Multifractal nature of two dimensional simple random walk paths. In: Picardello, M.A., Woess, W. (eds.) Random walks and discrete potential theory. Proceedings of the conference, Cortona, Italy, June 1997. Cambridge: Cambridge University Press. Symp. Math. 39, 231–264 (1999)Google Scholar
  46. 46.
    Lawler, G.F.: Multifractal analysis of the reverse flow for the Schramm-Loewner evolution. In: Bandt, C., Mörters, P., Zähle, M. (eds.) Fractal geometry and stochastics IV. Proceedings of the 4th conference, Greifswald, Germany, September 8–12, 2008. Basel: Birkhäuser. Progress in Probability 61, 73–107 (2009)Google Scholar
  47. 47.
    Lawler, G.F.: Fractal and Multifractal Properties of SLE. In: Ellwood, D., Newman, C., Sidoravicius, V., Werner, W. (eds.) Probability and statistical physics in two and more dimensions. Proceedings of the Clay Mathematics Institute summer school and XIV Brazilian school of probability, Búzios, Brazil, July 11–August 7, 2010, vol. 15, pp. 277–318. American Mathematical Society (AMS); Clay Mathematics Institute (2012)Google Scholar
  48. 48.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187(2), 237–273 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187(2), 275–308 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38(1), 109–123 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Lawler G.F., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Lawler G.F., Werner W.: Intersection exponents for planar Brownian motion. Ann. Probab. 27(4), 1601–1642 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Lawler G.F., Werner W.: Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. 2(4), 291–328 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Lawler, G.F.: Conformally invariant processes in the plane. Volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2005)Google Scholar
  55. 55.
    Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Lapidus, M.L., van Frankenhuysen, M. (eds.) Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, vol. 72 of Proc. Sympos. Pure Math., pp. 339–364. Amer. Math. Soc., Providence, RI (2004)Google Scholar
  56. 56.
    Lebedev, N.A., Milin, I.M.: On the coefficients of certain classes of univalent functions. Mat. Sb. 28, 359–400 (1951). (In Russian)Google Scholar
  57. 57.
    Littlewood J.E.: On inequalities in the theory of functions. Proc. London Math. Soc. 23, 481–519 (1925)CrossRefGoogle Scholar
  58. 58.
    Littlewood J.E., Paley R.E.A.C.: A proof that an odd schlicht function has bounded coefficients. J. London Math. Soc. 7, 167–169 (1932)CrossRefGoogle Scholar
  59. 59.
    Loutsenko I.: SLE κ: correlation functions in the coefficient problem. J. Phys. A Math. Gen. 45(26), 265001 (2012)CrossRefADSMathSciNetGoogle Scholar
  60. 60.
    Loutsenko, I., Yermolayeva, O.: On exact multi-fractal spectrum of the whole-plane SLE. arXiv:1203.2756, (2012)
  61. 61.
    Loutsenko, I., Yermolayeva, O.: Average harmonic spectrum of the whole-plane SLE. J. Stat. Mech. page P04007 (2013)Google Scholar
  62. 62.
    Loutsenko, I., Yermolayeva, O.: On Harmonic Measure of the Whole Plane Lévy-Loewner Evolution. arXiv:1301.6508, (2013)
  63. 63.
    Löwner K.: Untersuchungen über schlichte konforme Abildungendes Einheitskreises. Math. Annalen 89, 103–121 (1923)CrossRefzbMATHGoogle Scholar
  64. 64.
    Makarov N.G.: Distorsion of boundary sets under conformal mapping. Proc. London Math. Soc. 51, 369–384 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    Makarov, N.G.: Fine structure of harmonic measure. Rossiĭskaya Akademiya Nauk. Algebra i Analiz 10, 1–62 (1998). English translation in St. Petersburg Math. J. 10, 217–268 (1999)Google Scholar
  66. 66.
    Mandelbrot B.B.: Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid. Mech. 62, 331–358 (1974)CrossRefADSzbMATHGoogle Scholar
  67. 67.
    Milin I.M.: Estimation of coefficients of univalent functions. Dokl. Akad. Nauk SSSR 160, 196–198 (1965)MathSciNetGoogle Scholar
  68. 68.
    Oikonomou, P., Rushkin, I., Gruzberg, I.A., Kadanoff, L.P.: Global properties of stochastic Loewner evolution driven by Lévy processes. J. Stat. Mech. page P01019 (2008)Google Scholar
  69. 69.
    Pommerenke, Ch.: Univalent functions. Van den Hoek and Ruprecht, Göttingen (1975)Google Scholar
  70. 70.
    Pommerenke Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der mathematischen Wissenschaften, vol. 299. Springer, Berlin (1992)CrossRefGoogle Scholar
  71. 71.
    Rhodes R., Vargas V.: KPZ formula for log-infinitely divisible multifractal random measures. ESAIM: Probability and Statistics 15, 358–371 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    Robertson M.S.: On the theory of univalent functions. Ann. of Math 37, 374–408 (1936)CrossRefMathSciNetGoogle Scholar
  73. 73.
    Rohde S., Schramm O.: Basic Properties of SLE. Ann. of Math. 161, 883–924 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  74. 74.
    Rushkin I., Bettelheim E., Gruzberg I.A., Wiegmann P.: Critical curves in conformally invariant statistical systems. J. Phys. A: Math. Gen. 40, 2165–2195 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  75. 75.
    Rushkin, I., Oikonomou, P., Kadanoff, L.P., Gruzberg, I.A.: Stochastic Loewner evolution driven by Lévy processes. J. Stat. Mech. page P01001 (2006)Google Scholar
  76. 76.
    Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math 118, 221–288 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  77. 77.
    Schramm O., Sheffield S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202(1), 21–137 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  78. 78.
    Smirnov S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)CrossRefADSzbMATHGoogle Scholar
  79. 79.
    Smirnov S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172(2), 1435–1467 (2010)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Bertrand Duplantier
    • 1
    • 2
    Email author
  • Chi Nguyen
    • 3
    • 4
  • Nga Nguyen
    • 4
  • Michel Zinsmeister
    • 4
    • 5
  1. 1.Institut de Physique Théorique, CEA/SaclayGif-sur-Yvette CedexFrance
  2. 2.The Mathematical Sciences Research InstituteBerkeleyUSA
  3. 3.College of SciencesHue UniversityHueVietnam
  4. 4.Bâtiment de mathématiquesMAPMO, Université d’OrléansOrléans Cedex 2France
  5. 5.Laboratoire de physique théorique de la matière condenséeUMR CNRS 7600, Tour 12-13/13-23, Boîte 121Paris Cedex 05France

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