Hausdorff Dimension of the SLE Curve Intersected with the Real Line

Tom Alberts (Courant Institute of Mathematical Sciences)
Scott Sheffield (Courant Institute of Mathematical Sciences)


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


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