Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLE
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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.
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References
- 1.Astala, K., Duplantier, B., Zinsmeister, M.: Unpublished manuscript (2015)Google Scholar
- 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.Beliaev D., Smirnov S.: Harmonic measure and SLE. Commun. Math. Phys. 290, 577–595 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
- 5.Bettelheim E., Rushkin I., Gruzberg I.A., Wiegmann P.: Harmonic measure of critical curves. Phys. Rev. Lett. 95, 170602 (2005)ADSCrossRefMATHGoogle Scholar
- 6.Bieberbach L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. S-B. Preuss. Akad. Wiss. 1, 940–955 (1916)MATHGoogle Scholar
- 7.Binder, I., Duplantier, B.: Multifractal properties of harmonic measure and rotation for SLE (2017) (in preparation)Google Scholar
- 8.de Branges L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)MathSciNetCrossRefMATHGoogle Scholar
- 9.Duplantier B.: Conformally Invariant Fractals and Potential Theory. Phys. Rev. Lett. 84, 1363–1367 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
- 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.Duplantier B., Binder I.A.: Harmonic measure and winding of conformally invariant curves. Phys. Rev. Lett. 89, 264101 (2002)ADSCrossRefGoogle Scholar
- 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)ADSMathSciNetCrossRefMATHGoogle Scholar
- 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.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.Feng J., MacGregor T.H.: Estimates on the integral means of the derivatives of univalent functions. J. Anal. Math 29, 203–231 (1976)CrossRefMATHGoogle Scholar
- 16.Garnett, J.B., Marshall, D.E.: Harmonic measure. Cambridge University Press, Cambridge (2005)Google Scholar
- 17.Grunsky H.: Koeffizienten Bedingungen für schlicht abbidende meromorphe Funktionen. Math. Z. 45, 29–61 (1939)MathSciNetCrossRefMATHGoogle Scholar
- 18.Gwynne, E., Miller, J., Sun, X.: Almost sure multifractal spectrum of SLE (2014). Duke Math. J. (to appear). arXiv:1412.8764
- 19.Hastings M.B.: Exact multifractal spectra for arbitrary Laplacian random walks. Phys. Rev. Lett. 88, 055506 (2002)ADSCrossRefGoogle Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
- 21.Kemppainen A.: Stationarity of SLE. J. Stat. Phys. 139, 108–121 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
- 22.Kytölä K., Kemppainen A.: SLE local martingales, reversibility and duality. J. Phys. A Math. Gen. 39, L657–L666 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
- 23.Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187(2), 275–308 (2001)MathSciNetCrossRefMATHGoogle Scholar
- 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)MathSciNetCrossRefMATHGoogle Scholar
- 25.Le, T.B.: Around Milin’s conjecture and SLE maps, Mémoire de M2, Université d’Orléans (2010)Google Scholar
- 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.Loutsenko I.: SLE\({_{\kappa}}\): correlation functions in the coefficient problem. J. Phys. A Math. Theor. 45(26), 265001 (2012)MathSciNetCrossRefMATHGoogle Scholar
- 28.Loutsenko, I., Yermolayeva, O.: On exact multi-fractal spectrum of the whole-plane SLE (2012). arXiv:1203.2756
- 29.Loutsenko, I., Yermolayeva, O.: Average harmonic spectrum of the whole-plane SLE. J. Stat. Mech. p. 04007 (2013)Google Scholar
- 30.Loutsenko I., Yermolayeva O.: New exact results in spectra of stochastic Loewner evolution. J. Phys. A Math. Theor. 47(16), 165202 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
- 31.Löwner K.: Untersuchungen über schlichte konforme Abildungendes Einheitskreises. Math. Annalen 89, 103–121 (1923)MathSciNetCrossRefMATHGoogle Scholar
- 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.Milin I.M.: Estimation of coefficients of univalent functions. Dokl. Akad. Nauk SSSR 160, 196–198 (1965)MATHGoogle Scholar
- 34.Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. Grundlehren der mathematischen Wissenschaften, Vol. 299. Springer, Berlin (1992)Google Scholar
- 35.Robertson M.S.: On the theory of univalent functions. Ann. Math. 37, 374–408 (1936)MathSciNetCrossRefMATHGoogle Scholar
- 36.Rohde S., Schramm O.: Basic properties of SLE. Ann. Math. 161, 883–924 (2005)MathSciNetCrossRefMATHGoogle Scholar
- 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)ADSMathSciNetCrossRefMATHGoogle Scholar
- 38.Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math 118, 221–288 (2000)MathSciNetCrossRefMATHGoogle Scholar