Duke Mathematical Journal

Almost sure multifractal spectrum of Schramm–Loewner evolution

Abstract

Suppose that $\eta$ is a Schramm–Loewner evolution ($\operatorname{SLE}_{\kappa}$) in a smoothly bounded simply connected domain $D\subset{\mathbf{C}}$ and that $\phi$ is a conformal map from $\mathbf{D}$ to a connected component of $D\setminus\eta([0,t])$ for some $t\gt 0$. The multifractal spectrum of $\eta$ is the function $(-1,1)\to[0,\infty)$ which, for each $s\in(-1,1)$, gives the Hausdorff dimension of the set of points $x\in\partial\mathbf{D}$ such that $|\phi'((1-\epsilon)x)|=\epsilon^{-s+o(1)}$ as $\epsilon\to0$. We rigorously compute the almost sure multifractal spectrum of $\operatorname{SLE}$, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure bulk integral means spectrum of $\operatorname{SLE}$, we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an $\operatorname{SLE}$ curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the $\operatorname{SLE}$ curve for $\kappa\leq4$. Our results also hold for the $\operatorname{SLE}_{\kappa}(\underline{\rho})$ processes with general vectors of weight $\underline{\rho}$.

Article information

Source
Duke Math. J., Volume 167, Number 6 (2018), 1099-1237.

Dates
Revised: 6 October 2017
First available in Project Euclid: 28 March 2018

https://projecteuclid.org/euclid.dmj/1522224103

Digital Object Identifier
doi:10.1215/00127094-2017-0049

Mathematical Reviews number (MathSciNet)
MR3786302

Zentralblatt MATH identifier
06870402

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE)
Secondary: 60G17: Sample path properties

Citation

Gwynne, Ewain; Miller, Jason; Sun, Xin. Almost sure multifractal spectrum of Schramm–Loewner evolution. Duke Math. J. 167 (2018), no. 6, 1099--1237. doi:10.1215/00127094-2017-0049. https://projecteuclid.org/euclid.dmj/1522224103

References

• [1] T. Alberts, I. Binder, and F. Viklund, A dimension spectrum for SLE boundary collisions, Comm. Math. Phys. 343 (2016), 273–298.
• [2] V. Beffara, The dimension of the SLE curves, Ann. Probab. 36 (2008), 1421–1452.
• [3] D. Beliaev and S. Smirnov, “Harmonic measure on fractal sets,” in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, 41–59.
• [4] D. Beliaev and S. Smirnov, Harmonic measure and SLE, Comm. Math. Phys. 290 (2009), 577–595.
• [5] I. Binder and B. Duplantier, personal communication, December 2014.
• [6] J. Dubédat, Duality of Schramm-Loewner evolutions, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 697–724.
• [7] J. Dubédat, SLE and the free field: partition functions and couplings, J. Amer. Math. Soc. 22 (2009), 995–1054.
• [8] B. Duplantier, Harmonic measure exponents for two-dimensional percolation, Phys. Rev. Lett. 82, no. 20 (1999), 3940–3943.
• [9] B. Duplantier, Two-dimensional copolymers and exact conformal multifractality, Phys. Rev. Lett. 82, no. 5 (1999), 880–883.
• [10] B. Duplantier, Conformally invariant fractals and potential theory, Phys. Rev. Lett. 84, no. 7 (2000), 1363–1367.
• [11] B. Duplantier, Higher conformal multifractality, J. Statist. Phys. 110 (2003), 691–738.
• [12] B. Duplantier, “Conformal fractal geometry and boundary quantum gravity,” in Fractal Geometry and Applications: A Jubilee of Benôit Mandelbrot, Part 2, Proc. Sympos. Pure Math. 72, Amer. Math. Soc., Providence, 2004, 365–482.
• [13] B. Duplantier and I. Binder, Harmonic measure and winding of conformally invariant curves, Phys. Rev. Lett. 89, no. 26 (2002), art. ID 264101.
• [14] B. Duplantier and I. Binder, Harmonic measure and winding of random conformal paths: A Coulomb gas perspective, Nucl. Phys. B 802 (2008), 494–513.
• [15] B. Duplantier, X. Hieu Ho, T. Binh Le, and M. Zinsmeister, Logarithmic coefficients and generalized multifractality of whole-plane SLE, Comm. Math. Phys., published online 20 December 2017.
• [16] B. Duplantier, J. Miller, and S. Sheffield, Liouville quantum gravity as a mating of trees, preprint, arXiv:1409.7055v2 [math.PR].
• [17] B. Duplantier, C. Nguyen, N. Nguyen, and M. Zinsmeister, The coefficient problem and multifractality of whole-plane SLE & LLE, Ann. Henri Poincaré 16 (2015), 1311–1395.
• [18] B. Duplantier and S. Sheffield, Liouville quantum gravity and KPZ, Invent. Math. 185 (2011), 333–393.
• [19] H. Hedenmalm and A. Sola, Spectral notions for conformal maps: A survey, Comput. Methods Funct. Theory 8 (2008), 447–474.
• [20] X. Hu, J. Miller, and Y. Peres, Thick points of the Gaussian free field, Ann. Probab. 38 (2010), 896–926.
• [21] P. Kraetzer, Experimental bounds for the universal integral means spectrum of conformal maps, Complex Variables Theory Appl. 31 (1996), 305–309.
• [22] G. F. Lawler, The dimension of the frontier of planar Brownian motion, Electron. Comm. Probab. 1 (1996), 29–47.
• [23] G. F. Lawler, Conformally Invariant Processes in the Plane, Math. Surveys Monogr. 114, Amer. Math. Soc., Providence, 2005.
• [24] G. F. Lawler, “Multifractal analysis of the reverse flow for the Schramm-Loewner evolution,” in Fractal Geometry and Stochastics, IV, Progr. Probab. 61, Birkhäuser, Basel, 2009, 73–107.
• [25] G. F. Lawler, O. Schramm, and W. Werner, The dimension of the planar Brownian frontier is $4/3$, Math. Res. Lett. 8, no. 4 (2001), 401–411.
• [26] G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents, I: Half-plane exponents, Acta Math. 187 (2001), 237–273.
• [27] G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents, II: Plane exponents, Acta Math. 187 (2001), 275–308.
• [28] G. F. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents, III: Two-sided exponents, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), 109–123.
• [29] G. F. Lawler, O. Schramm, and W. Werner, Conformal restriction: The chordal case, J. Amer. Math. Soc. 16 (2003), 917–955.
• [30] G. F. Lawler, O. Schramm, and W. Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), 939–995.
• [31] J. R. Lind, Hölder regularity of the SLE trace, Trans. Amer. Math. Soc. 360, no. 7 (2008), 3557–3578.
• [32] I. Loutsenko and O. Yermolayeva, Average harmonic spectrum of the whole-plane SLE, J. Stat. Mech. Theory Exp. 2013, no. 4, art. ID P04007.
• [33] I. Loutsenko and O. Yermolayeva, New exact results in spectra of stochastic Loewner evolution, J. Phys. A 47, no. 16 (2014), art. ID 165202.
• [34] N. G. Makarov, Fine structure of harmonic measure (in Russian), Algebra i Analiz 10 (1998), 1–62; English translation in St. Petersburg Math. J. 10 (1999), 217–268.
• [35] J. Miller, Universality for SLE(4), preprint, arXiv:1010.1356v1 [math.PR].
• [36] J. Miller, Dimension of the SLE light cone, the SLE fan, and SLE$_{\kappa}(\rho)$ for $\kappa\in(0,4)$ and $\rho\in[\tfrac{\kappa}{2}-4,-2)$, Comm. Math. Phys., published online 20 February 2018.
• [37] J. Miller and S. Sheffield, Imaginary geometry, I: Interacting SLEs, Probab. Theory Related Fields 164 (2016), 553–705.
• [38] J. Miller and S. Sheffield, Imaginary geometry, II: Reversibility of $\operatorname{SLE}_{\kappa}(\rho_{1};\rho_{2})$ for $\kappa\in(0,4)$, Ann. Probab. 44 (2016), 1647–1722.
• [39] J. Miller and S. Sheffield, Imaginary geometry, III: Reversibility of $\operatorname{SLE}_{\kappa}$ for $\kappa\in(4,8)$, Ann. of Math. (2) 184 (2016), 455–486.
• [40] J. Miller and S. Sheffield, Quantum Loewner evolution, Duke Math. J. 165 (2016), 3241–3378.
• [41] J. Miller and S. Sheffield, Imaginary geometry, IV: Interior rays, whole-plane reversibility, and space-filling trees, Probab. Theory Related Fields 169 (2017), 729–869.
• [42] J. Miller and S. Sheffield, Gaussian free field light cones and $\operatorname{SLE}_{\kappa}(\rho)$, preprint, arXiv:1606.02260v2 [math.PR].
• [43] J. Miller, S. Sheffield, and W. Werner, CLE percolations, Forum Math. Pi 5 (2017), e4.
• [44] J. Miller, N. Sun, and D. B. Wilson, The Hausdorff dimension of the CLE gasket, Ann. Probab. 42 (2014), 1644–1665.
• [45] J. Miller, S. S. Watson, and D. B. Wilson, The conformal loop ensemble nesting field, Probab. Theory Related Fields 163 (2015), 769–801.
• [46] J. Miller and H. Wu, Intersections of SLE paths: The double and cut point dimension of SLE, Probab. Theory Related Fields 167 (2017), 45–105.
• [47] P. Mörters and Y. Peres, Brownian Motion, with an appendix by O. Schramm and W. Werner, Cambridge Ser. Stat. Probab. Math. 30, Cambridge Univ. Press, Cambridge, 2010.
• [48] C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss. 299, Springer, Berlin, 1992.
• [49] C. Pommerenke, The integral means spectrum of univalent functions (in Russian), Anal. Teor. Chisel i Teor. Funkts. 14 (1997), 119–128, 229; English translation in J. Math. Sci. (N.Y.) 95 (1999), 2249–2255.
• [50] S. Rohde and O. Schramm, Basic properties of SLE, Ann. of Math. (2) 161 (2005), 883–924.
• [51] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288.
• [52] O. Schramm and S. Sheffield, Harmonic explorer and its convergence to $\mathrm{SLE}_{4}$, Ann. Probab. 33 (2005), 2127–2148.
• [53] O. Schramm and S. Sheffield, Contour lines of the two-dimensional discrete Gaussian free field, Acta Math. 202 (2009), 21–137.
• [54] O. Schramm and S. Sheffield, A contour line of the continuum Gaussian free field, Probab. Theory Related Fields 157 (2013), 47–80.
• [55] O. Schramm and D. B. Wilson, SLE coordinate changes, New York J. Math. 11 (2005), 659–669.
• [56] S. Sheffield, Local sets of the Gaussian free field: slides and audio, 2005, www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield1, www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield2, www.fields.utoronto.ca/audio/05-06/percolation_SLE/sheffield3.
• [57] S. Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), 521–541.
• [58] S. Sheffield, Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab. 44 (2016), 3474–3545.
• [59] S. Smirnov, Conformal invariance in random cluster models, I: Holomorphic fermions in the Ising model, Ann. of Math. (2) 172 (2010), 1435–1467.
• [60] F. Viklund and G. F. Lawler, Optimal Hölder exponent for the SLE path, Duke Math. J. 159 (2011), 351–383.
• [61] F. Viklund and G. F. Lawler, Almost sure multifractal spectrum for the tip of an SLE curve, Acta Math. 209 (2012), 265–322.
• [62] M. Wang and H. Wu, Level lines of Gaussian free field, I: Zero-boundary GFF, Stochastic Process. Appl. 127 (2017), 1045–1124.
• [63] W. Werner, “Random planar curves and Schramm-Loewner evolutions,” in Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1840, Springer, Berlin, 2004, 107–195.
• [64] D. Zhan, Duality of chordal SLE, Invent. Math. 174 (2008), 309–353.
• [65] D. Zhan, Reversibility of chordal SLE, Ann. Probab. 36 (2008), 1472–1494.
• [66] D. Zhan, Duality of chordal SLE, II, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), 740–759.