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A Dimension Spectrum for SLE Boundary Collisions

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Abstract

We consider chordal SLE\({_\kappa}\) curves for \({\kappa > 4}\), where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension \({{\rm min}\{2-8/\kappa,1\}}\). We study the random sets of points at which the curve collides with the real line at a specified “angle” and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov’s theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.

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References

  1. Alberts T., Kozdron Michael J.: Intersection probabilities for a chordal SLE path and a semicircle. Electron. Commun. Probab. 13, 448–460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alberts T., Sheffield S.: Hausdorff dimension of the SLE curve intersected with the real line. Electron. J. Probab. 13(40), 1166–1188 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Alberts T., Sheffield S.: The covariant measure of SLE on the boundary. Probab. Theory Rel. Fields. 149(3-4), 331–371 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beffara V.: The dimension of the SLE curves. Ann. Probab. 36(4), 1421–1452 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beliaev D., Smirnov S.: Harmonic measure and SLE. Commun. Math. Phys. 290(2), 577–596 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Gwynne, E., Miller, J., Sun, X.: Almost sure multifractal spectrum of SLE (2014). arXiv:1412.8764

  7. Johansson Viklund F., Lawler Gregory F.: Optimal Hölder exponent for the SLE path. Duke Math. J. 159(3), 351–383 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Johansson Viklund F., Lawler Gregory F.: Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math. 209(2), 265–322 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lawler, G.F.: Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions. In: Random Walks (Budapest, 1998). Bolyai Soc. Math. Stud., vol. 9, pp. 219–258. János Bolyai Math. Soc., Budapest (1999)

  10. Lawler, G.F.: Multifractal analysis of the reverse flow for the Schramm–Loewner evolution. In: Fractal Geometry and Stochastics IV. Progr. Probab., vol. 61, pp. 73–107. Birkhäuser Verlag, Basel (2009)

  11. Lawler, G.F.: Minkowski content of the intersection of a Schramm–Loewner evolution (SLE) curve with the real line (2014). http://www.math.uchicago.edu/~lawler/papers.html

  12. Lind J., Marshall D.E., Rohde S.: Collisions and spirals of Loewner traces. Duke Math. J. 154(3), 527–573 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Makarov N.: Fine structure of harmonic measure. J. St. Petersburg Math. 10(2), 217–268 (1999)

    MathSciNet  Google Scholar 

  14. Miller, J., Wu, H.: Intersections of SLE paths: the double and cut point dimension of SLE (2014). arXiv:1303.4725

  15. Revuz, D., Yor, M.: Continuous martingales and Brownian motion. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn, vol. 293. Springer, Berlin (1999)

  16. Schramm O., Zhou W.: Boundary proximity of SLE. Probab. Theory Rel. Fields 146(3–4), 435–450 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University of Utah, Salt Lake City, USA

    Tom Alberts

  2. University of Toronto, Toronto, Canada

    Ilia Binder

  3. KTH Royal Institute of Technology, Stockholm, Sweden

    Fredrik Viklund

Authors
  1. Tom Alberts
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  2. Ilia Binder
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  3. Fredrik Viklund
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Correspondence to Tom Alberts.

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Communicated by F. Toninelli

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Alberts, T., Binder, I. & Viklund, F. A Dimension Spectrum for SLE Boundary Collisions. Commun. Math. Phys. 343, 273–298 (2016). https://doi.org/10.1007/s00220-016-2587-x

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  • DOI: https://doi.org/10.1007/s00220-016-2587-x

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