The Crossing Probability Of Dipolar SLE And Its Related Estimation

Stochastic Loewner evolution(SLE for short)describes a collection of random curves which are related to scaling limits of two-dimensional statistical physics systems.In this thesis,our main work is as follows.First,The estimation of crossing probability for dipolar SLE_?is established.Consider a dipolar SLE_?process in a rectangle from the upper-left corner to the right edge.For?>4,let E denote the event that the SLE_?process has not hit the bottom edge before it reaches the right edge.The probability of the event E is estimated by terms of the length of bottom edge.Furthermore,the one-sided crossing exponent conditional on the event E is derived.This generalizes the corresponding crossing probability results of chordal and radial SLE_?to the case of dipolar SLE_?.Secondly,some estimates for the dipolar Loewner differential equation are investigated.An estimation of the solution of the dipolar Loewner differential equation for time-direction is derived by using Bieberbach theorem;based on the reverse-time Loewner equation the difference of two solutions to the dipolar Loewner equation with two different deriving functions is estimated in terms of the supremum norm of the difference of the two driving functions.This generalizes some related results for the chordal and radial Loewner equations to the dipolar setting.