Dipolar Stochastic Loewner Evolution And Ising Model

Stochastic Loewner Evolution(SLE_?for short)is a family of random growth process based on Loewner differential equation with driving parameter a one-dimensional Brownian motion running with speed?.This process is intimately connected with scaling limits of percolation clusters and the outer boundary of Brownian motion.It is powerful tool to describe the continuous limit of lattice models in statistical physics at criticality.In this thesis,our main work is as follows.First,we investigate some properties of dipolar SLE and its phase transformations whose behaviors depend on the parameter?.Secondly,the crossing exponent of the dipolar SLE in the rectangle is discussed,and the equivalence of several common versions of SLE is proved.Finally,we discuss Ising model and its scaling limit.It is proved that crossing probability of the critical Ising model with free boundary conditions is conformally invariant in the scaling limit.