Cardy's Formula For SLE_κ And Brownian Intersection Exponents In Whole Plane

Stochastic Loewner evolution(SLE) is a random growth process of sets. It is a current hot topic of mathematical study, and the research involves complex analysis, partial differential equations, probability theory, statistical physics and stochastic analysis, and other important ideas. In this thesis, our main work is as follows:First, Cardy formula for SLEk is discussed. Using of the properties of SLEk and hypergeometric functions, we gave a probabilistic estima-tion for the SLEK trace crossing a rectangle without touching the upper and lower boundaries of the rectangle when the k> 4. Second, Brownian intersection exponents in the whole plane are discussed. By computing the index (?)(1,1,λ) (disintegrative exponent of intersection prob-ability about a beam of Brownian motion which starting at different points of half plane), using the brown cascade relations between Brownian intersection exponents in the whole plane and ones in the half plane and Half plane Brownian precise value of intersecting exponents, we get the precise value of whole plane brown precise value of intersecting exponents.