math.PR

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# Title: Boundary proximity of SLE

Abstract: This paper examines how close the chordal $\SLE_\kappa$ curve gets to the real line asymptotically far away from its starting point. In particular, when $\kappa\in(0,4)$, it is shown that if $\beta>\beta_\kappa:=1/(8/\kappa-2)$, then the intersection of the $\SLE_\kappa$ curve with the graph of the function $y=x/(\log x)^{\beta}$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $\beta=\beta_\kappa$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^\alpha}$, $x>e^e$, in an unbounded set if $\alpha\le 1$, but not if $\alpha>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient integrability condition for the intersection of the $\SLE_\kappa$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\SLE_{\kappa}$ curve and real axis is $2-8/\kappa$ when $4<\kappa<8$.
 Comments: 18 pages, new results are added, typos are corrected Subjects: Probability (math.PR); Complex Variables (math.CV) MSC classes: 60D05, 28A80 Cite as: arXiv:0711.3350 [math.PR] (or arXiv:0711.3350v3 [math.PR] for this version)

## Submission history

From: Wang Zhou [view email]
[v1] Wed, 21 Nov 2007 11:33:43 GMT (14kb)
[v2] Thu, 22 Nov 2007 03:50:54 GMT (14kb)
[v3] Thu, 6 Dec 2007 07:28:58 GMT (14kb)