Conformal Equivalence And Derivative Estimation Of Dipolar SLE

Stochastic Loewner evolution(SLEκ)is a family of random planar growth processes with one parameter κ,which can be constructed by solving the Loewner equation when the driving function is a one-dimensional Brownian motion.SLEK is a powerful tool to describe the scaling limits of statistical physics models at criticality.The main work of this thesis is as follows:First,conformal equivalence of dipolar SLEK with forced points is investigated.Consider dipolar Sπ-SLE(κ;ρ1,...,ρm)with forced points in the strip domain Sπ={x+iy:x∈R,y∈(0,π)}.Based on Loewner’s theorem it is first proved that,under the hypothesis ∑j=1m ρj=κ?2,if φ is a conformal map of Sπ onto the upper-plane H,then the image of Sπ-SLE(κ;ρ1,...,ρm)under φ becomes chordal H-SLE(κ;ρ1,...,ρm)with forced points,and vice versa.Next,it is showed that,under the hypothesis ∑j=1m ρj=κ-6,if ψ is a conformal map of Sπ,onto the unit disk D,then the image of Sπ-SLE(κ;ρ1,...,ρm)under ψ becomes radial D-SLE(κ;ρ1,...,ρm)with forced points,and vice versa.Secondly,the derivative of dipolar SLEK is estimated.The derivative estimation of the dipolor SLEK is first derived by conformal equivalence of dipolar and chordal SLEκ.From the conformal equivalence of dipolar and chordal SLEK we deduce that the derivative modulus of dipolar SLEK is less than or equal to the derivative modulus of chordal SLEK times a constant.This,combined with the derivative estimation of chordal SLEκ,implies the derivative estimation of dipolar SLEκ.Next,the derivative estimation of the dipolor SLEκ is obtained by conformal equivalence of dipolar and radial SLEκ.By the conformal equivalence of dipolar and radial SLEκ we conclude that the derivative modulus of dipolar SLEκ is less than or equal to the derivative modulus of radial SLEκ times a constant.This,together with the derivative estimation of the radial SLEκ,gives the derivative estimation of dipolar SLEκ.