Duke Mathematical Journal

Almost sure multifractal spectrum of Schramm–Loewner evolution

Ewain Gwynne, Jason Miller, and Xin Sun

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Suppose that η is a Schramm–Loewner evolution (SLEκ) in a smoothly bounded simply connected domain DC and that ϕ is a conformal map from D to a connected component of Dη([0,t]) for some t>0. The multifractal spectrum of η is the function (1,1)[0,) which, for each s(1,1), gives the Hausdorff dimension of the set of points xD such that |ϕ'((1ϵ)x)|=ϵs+o(1) as ϵ0. We rigorously compute the almost sure multifractal spectrum of 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 SLE, we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an SLE curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the SLE curve for κ4. Our results also hold for the SLEκ(ρ̲) processes with general vectors of weight ρ̲.

Article information

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

Dates
Received: 8 March 2016
Revised: 6 October 2017
First available in Project Euclid: 28 March 2018

Permanent link to this document
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

Keywords
Schramm–Loewner evolution multifractal spectrum Gaussian free field

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


Export citation

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.