The Probability Formula Associated With SLE_κand Conformal Invariance Of Self-avoid Walks

Stochastic Loewner evolution(SLE) is a random growth process of sets which isdefined using the usual Loewner equation, where the driving function is a tine-changedBrownian mation. This process is intimately connected with scaling limits of percolationclusters and with the outer boundary of Brownian motion. In this paper, our main workis as follows: first, the estimate of non-intersection probability for SLEκand Brownianmotion is discussed. For0<κ <4, using the excursion measure Poisson kernel andRiemann’s P-function, the formula of non-intersection probability for the trace of SLEκand planar Brownian motion is given by the solution of hypergeometric diferentialequation. This generalizes the known result for parameter κ=2to the general case of0<κ <4. Second, The conformal invariance of SAW in a finitely connected domain isinvestigated. Using the Riemann mapping theorem and the properties of SLE, we defineself-avoiding walk in finitely connected domains. It is proved that self-avoiding walk infinitely connected regions possess conformal invariance, which generalizes the conformalinvariance of SAW in simiply connected domains to the one in finitely connected regions.