Boundary proximity of SLE

Original · EN

This paper examines how close the chordal κ 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 κ curve with the graph of the function y=x/(x)β, x>e, is a.s. bounded, while it is a.s. unbounded if β=βκ. The critical ₄ curve a.s. intersects the graph of y=x-(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 κ path with the graph of y to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the κ curve and real axis is 2-8/κ when 4<κ<8.