Acta Mathematica

Almost sure multifractal spectrum for the tip of an SLE curve

Fredrik Johansson Viklund and Gregory F. Lawler

Full-text: Open access


The tip multifractal spectrum of a 2-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm–Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability 1, and we give applications to the scaling of harmonic measure at the tip.

Article information

Acta Math., Volume 209, Number 2 (2012), 265-322.

Received: 12 November 2010
Revised: 23 April 2012
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2012 © Institut Mittag-Leffler


Viklund, Fredrik Johansson; Lawler, Gregory F. Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math. 209 (2012), no. 2, 265--322. doi:10.1007/s11511-012-0087-1.

Export citation


  • Beffara, V., The dimension of the SLE curves. Ann. Probab., 36 (2008), 1421–1452.
  • Beliaev, D. & Smirnov, S., Harmonic measure and SLE. Comm. Math. Phys., 290 (2009), 577–595.
  • Binder, I. & Duplantier, B., Harmonic measure and winding of conformally invariant curves. Phys. Rev. Lett., 89 (2002), 264101.
  • Duplantier, B., Conformal fractal geometry & boundary quantum gravity, in Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., 72, pp. 365–482. Amer. Math. Soc., Providence, RI, 2004.
  • Johansson Viklund, F. & Lawler, G. F., Optimal Hölder exponent for the SLE path. Duke Math. J., 159 (2011), 351–383.
  • Kang, N.-G., Boundary behavior of SLE. J. Amer. Math. Soc., 20 (2007), 185–210.
  • Karlin, S. & Taylor, H. M., A Second Course in Stochastic Processes. Academic Press, New York, 1981.
  • Lawler, G. F., Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, 114. Amer. Math. Soc., Providence, RI, 2005.
  • — Multifractal analysis of the reverse flow for the Schramm–Loewner evolution, in Fractal Geometry and Stochastics IV, Progr. Probab., 61, pp. 73–107. Birkhäuser, Basel, 2009.
  • — Schramm–Loewner evolution (SLE), in Statistical Mechanics, IAS/Park City Math. Ser., 16, pp. 231–295. Amer. Math. Soc., Providence, RI, 2009.
  • Lawler, G. F., Schramm, O. & Werner, W., Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math., 187 (2001), 237–273.
  • — Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187 (2001), 275–308.
  • — Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 109–123.
  • — Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32 (2004), 939–995.
  • Lawler, G. F. & Sheffield, S., A natural parametrization for the Schramm–Loewner evolution. Ann. Probab., 39 (2011), 1896–1937.
  • Lind, J. R., Hölder regularity of the SLE trace. Trans. Amer. Math. Soc., 360 (2008), 3557–3578.
  • Makarov, N. G., Fine structure of harmonic measure. Algebra i Analiz, 10 (1998), 1–62 (Russian); English translation in St. Petersburg Math. J., 10 (1999), 217–268.
  • Mattila, P., Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.
  • Pommerenke, C., Boundary Behaviour of Conformal Maps. Grundlehren der Mathematischen Wissenschaften, 299. Springer, Berlin–Heidelberg, 1992.
  • Rohde, S. & Schramm, O., Basic properties of SLE. Ann. of Math., 161 (2005), 883–924.
  • Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 (2000), 221–288.
  • Smirnov, S., Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 239–244.
  • — Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math., 172 (2010), 1435–1467.
  • Werner, W., Random planar curves and Schramm–Loewner evolutions, in Lectures on Probability Theory and Statistics, Lecture Notes in Math., 1840, pp. 107–195. Springer, Berlin–Heidelberg, 2004.