Boundary proximity of SLE

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 −(log log x) , 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.