2007/11/21 by Oded Schramm, Schramm, Oded, Wang Zhou +1
Mathematics · #28A80 #60D05 #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR) #math.CV #math.PR #msc:28A80 #msc:60D05
paper · pdf · doi:10.48550/arxiv.0711.3350
18 pages, new results are added, typos are corrected
arxiv created 2007/12/06 · arxiv updated 2009/12/01
This paper examines how close the chordal \SLEκ curve gets to the real line asymptotically far away from its starting point. In particular, when κ∈(0,4), it is shown that if β>βκ:=1/(8/κ-2), then the intersection of the \SLEκ curve with the graph of the function y=x/(log x)β, x>e, is a.s. bounded, while it is a.s. unbounded if β=βκ. The critical \SLE4 curve a.s. intersects the graph of y=x-(loglog x)α, x>ee, in an unbounded set if α≤ 1, but not if α>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 \SLEκ path with the graph of y to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the \SLEκ curve and real axis is 2-8/κ when 4<κ<8.