The Escape Probability Of Sle_κ And The Dimension Intersected With A Ray

SLE (Stochastic Loewner Evolution) is a random process of growth of a set Kt,which is defined using the usual Loewner equation, where the driving function is a time-changed Brownian motion. In this paper our main work is as follows. First, the escapeprobability of SLE6is discussed. Let Ktbe an SLE6hull in the half-plane, and letτRbe the first time it reaches radius R. The estimate of probability that any point inthe unit circle doesn’t belong to KτRis obtained, which generalizes the known estimateof probability that1or i doesn’t belong to KτRto the more general case; Second, theHausdorf dimension of SLEκ(4<κ <8) carve intersected with a ray is investigated.We establish an upper bound on the asymptotic probability of an SLEkcurve hittingtwo small line segment on the ray as the length of line segment goes to zero. Then as aconsequence we conclude that the random set of points in the ray hit by the cure hasHausdorf dimension28/κ, almost surely.