The Estimates Of Intersection Probabilities And Derivative Exponents For SLE

SLE (Stochastic Loewner Evolution) is a rigorous tool in mathematics and statis-tical physics for generating and studying scale invariant or fractal random curves in twodimensions (2D). In this paper, our main work is as follows. First, the probabilities ofSLEκintersected with circles and regular polygons, respectively, are discussed. The esti-mate of probability of a chordal SLEκ(0<κ <8) intersected with a circle in the upperhalf plane is presented. Moreover, its intersection probabilities with a regular polygonin the upper half plane is obtained. Second, the chordal SLEκin the upper half planeis extended to the case of whole plane. The SLEκin the whole plane is defined, and itsproperties are discussed. The derivative exponent for whole-plane SLEκis derived.