Advertisement

Communications in Mathematical Physics

, Volume 359, Issue 3, pp 823–868 | Cite as

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE

  • Bertrand Duplantier
  • Xuan Hieu Ho
  • Thanh Binh Le
  • Michel Zinsmeister
Article
  • 80 Downloads

Abstract

It has been shown that for f an instance of the whole-plane SLE unbounded conformal map from the unit disk \({{\mathbb{D}}}\) to the slit plane, the derivative moments \({\mathbb{E}(\vert f'(z) \vert^p)}\) can be written in a closed form for certain values of p depending continuously on the SLE parameter \({\kappa\in (0,\infty)}\). We generalize this property to the mixed moments, \({\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)}\), along integrability curves in the moment plane \({(p,q) \in {\mathbb{R}}^2}\) depending continuously on \({\kappa}\), by extending the so-called Beliaev–Smirnov equation to this case. The generalization of this integrability property to the m-fold transform of f is also given. We define a novel generalized integral means spectrum, \({\beta(p,q;\kappa)}\), corresponding to the singular behavior of the above mixed moments. By inversion, it allows a unified description of the unbounded interior and bounded exterior versions of whole-plane SLE, and of their m-fold generalizations. The average generalized spectrum of whole-plane SLE is found to take four possible forms, separated by five phase transition lines in the moment plane \({{\mathbb{R}}^2}\). The average generalized spectrum of the m-fold whole-plane SLE is obtained directly from the m = 1 case by a linear map acting in the moment plane. We also give a conjecture for the precise form of the universal generalized integral means spectrum.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Astala, K., Duplantier, B., Zinsmeister, M.: Unpublished manuscript (2015)Google Scholar
  2. 2.
    Beliaev, D., Duplantier, B., Zinsmeister, M.: Integral means spectrum of whole-plane SLE. Commun. Math. Phys. 353(1), 119–133, (2017).  https://doi.org/10.1007/s00220-017-2868-z. arXiv:1605.03112
  3. 3.
    Beliaev D., Smirnov S.: Harmonic measure and SLE. Commun. Math. Phys. 290, 577–595 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Belikov A., Gruzberg I.A., Rushkin I.I.: Statistics of harmonic measure and winding of critical curves from conformal field theory. J. Phys. A Math. Theor. 41(28), 285006 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bettelheim E., Rushkin I., Gruzberg I.A., Wiegmann P.: Harmonic measure of critical curves. Phys. Rev. Lett. 95, 170602 (2005)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Bieberbach L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. S-B. Preuss. Akad. Wiss. 1, 940–955 (1916)zbMATHGoogle Scholar
  7. 7.
    Binder, I., Duplantier, B.: Multifractal properties of harmonic measure and rotation for SLE (2017) (in preparation)Google Scholar
  8. 8.
    de Branges L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duplantier B.: Conformally Invariant Fractals and Potential Theory. Phys. Rev. Lett. 84, 1363–1367 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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. In: Volume 72 of Proceedings of Symposia in Pure Mathematics, pp. 365–482. American Mathematical Society, Providence, RI (2004)Google Scholar
  11. 11.
    Duplantier B., Binder I.A.: Harmonic measure and winding of conformally invariant curves. Phys. Rev. Lett. 89, 264101 (2002)ADSCrossRefGoogle Scholar
  12. 12.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duplantier, B., Chi, N.T.P., Nga, N.T.T., Zinsmeister, M.: Coefficient estimates for whole-plane SLE processes (2011). http://hal.inria.fr/hal-00609774
  14. 14.
    Duplantier, B., Nguyen, C., Nguyen, N., Zinsmeister, M.: The coefficient problem and multifractality of whole-plane SLE and LLE. Ann. Henri Poincaré 16(6), 1311–1395 (2014). arXiv:1211.2451v2.pdf
  15. 15.
    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
  16. 16.
    Garnett, J.B., Marshall, D.E.: Harmonic measure. Cambridge University Press, Cambridge (2005)Google Scholar
  17. 17.
    Grunsky H.: Koeffizienten Bedingungen für schlicht abbidende meromorphe Funktionen. Math. Z. 45, 29–61 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gwynne, E., Miller, J., Sun, X.: Almost sure multifractal spectrum of SLE (2014). Duke Math. J. (to appear). arXiv:1412.8764
  19. 19.
    Hastings M.B.: Exact multifractal spectra for arbitrary Laplacian random walks. Phys. Rev. Lett. 88, 055506 (2002)ADSCrossRefGoogle Scholar
  20. 20.
    Johansson Viklund F., Lawler G.F.: Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math. 209(2), 265–322 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kemppainen A.: Stationarity of SLE. J. Stat. Phys. 139, 108–121 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kytölä K., Kemppainen A.: SLE local martingales, reversibility and duality. J. Phys. A Math. Gen. 39, L657–L666 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187(2), 275–308 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Le, T.B.: Around Milin’s conjecture and SLE maps, Mémoire de M2, Université d’Orléans (2010)Google Scholar
  26. 26.
    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
  27. 27.
    Loutsenko I.: SLE\({_{\kappa}}\): correlation functions in the coefficient problem. J. Phys. A Math. Theor. 45(26), 265001 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Loutsenko, I., Yermolayeva, O.: On exact multi-fractal spectrum of the whole-plane SLE (2012). arXiv:1203.2756
  29. 29.
    Loutsenko, I., Yermolayeva, O.: Average harmonic spectrum of the whole-plane SLE. J. Stat. Mech. p. 04007 (2013)Google Scholar
  30. 30.
    Loutsenko I., Yermolayeva O.: New exact results in spectra of stochastic Loewner evolution. J. Phys. A Math. Theor. 47(16), 165202 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Löwner K.: Untersuchungen über schlichte konforme Abildungendes Einheitskreises. Math. Annalen 89, 103–121 (1923)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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
  33. 33.
    Milin I.M.: Estimation of coefficients of univalent functions. Dokl. Akad. Nauk SSSR 160, 196–198 (1965)zbMATHGoogle Scholar
  34. 34.
    Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der mathematischen Wissenschaften, Vol. 299. Springer, Berlin (1992)Google Scholar
  35. 35.
    Robertson M.S.: On the theory of univalent functions. Ann. Math. 37, 374–408 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rohde S., Schramm O.: Basic properties of SLE. Ann. Math. 161, 883–924 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math 118, 221–288 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Bertrand Duplantier
    • 1
  • Xuan Hieu Ho
    • 2
  • Thanh Binh Le
    • 2
    • 3
  • Michel Zinsmeister
    • 2
  1. 1.Institut de Physique ThéoriqueUniversité Paris-Saclay, CEA, CNRSGif-sur- Yvette CedexFrance
  2. 2.MAPMOUniversité d’Orléans, Bâtiment de mathématiquesOrléans Cedex 2France
  3. 3.Department of MathematicsUniversity of Quy NhonQuy NhonVietnam

Personalised recommendations