Hausdorff Dimension of the SLE Curve Intersected with the Real Line
Scott Sheffield (Courant Institute of Mathematical Sciences)
Abstract
We establish an upper bound on the asymptotic probability of an $SLE(\kappa)$ curve hitting two small intervals on the real line as the interval width goes to zero, for the range $4 < \kappa < 8$. As a consequence we are able to prove that the random set of points in $R$ hit by the curve has Hausdorff dimension $2-8/\kappa$, almost surely.
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Pages: 1166-1188
Publication Date: July 29, 2008
DOI: 10.1214/EJP.v13-515
References
- Beffara, Vincent. Hausdorff dimensions for $rm SLEsb 6$. Ann. Probab. 32 (2004), no. 3B, 2606--2629. MR2078552 (2005k:60295)
- Beffara, Vincent. The dimension of the SLE curves. To appear in Ann. Prob. (2007)
- Dubédat, Julien. SLE and triangles. Electron. Comm. Probab. 8 (2003), 28--42 (electronic). MR1961287 (2004c:60266)
- Lawler, Gregory F. Geometric and fractal properties of Brownian motion and random walk paths in two and three dimensions. Random walks (Budapest, 1998), 219--258, Bolyai Soc. Math. Stud., 9, János Bolyai Math. Soc., Budapest, 1999. MR1752896 (2001d:60091)
- Lawler, Gregory F. Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI, 2005. xii+242 pp. ISBN: 0-8218-3677-3 MR2129588 (2006i:60003)
- Lawler, Gregory F. Dimension and natural parameterization for {SLE} curves. arXiv:0712.3263v1 [math.PR], 2007.
- Rohde, Steffen; Schramm, Oded. Basic properties of SLE. Ann. of Math. (2) 161 (2005), no. 2, 883--924. MR2153402 (2006f:60093)
- Schramm, Oded. Scaling limits of random processes and the outer boundary of planar Brownian motion. Current developments in mathematics, 2000, 233--253, Int. Press, Somerville, MA, 2001. MR1882537 (2002m:60160)

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