Scaling Limit Of Critical Percolation In Finitely Connected Regions And Left-passage Probabilities

The stochastic Loewner evolution?SLE?is a one-parameter family of random planar growth processes.It can be obtained by solving the Loewner differential equation whose driving term is a time-change of one-dimensional Brownian motion.The main work of this paper is as follows:Firstly,we use SLE₆to construct a family of non-simple random curves on a finitely connected region.Secondly,we prove that this process is consistent with the scaling limit of site critical percolation on a triangular grid of finitely connected region.Thirdly,the left passage probability of critical percolation in the finitely connected region is established using the relation between the scaling limit of critical percolation and SLE₆,and combined with the property of conformal mapping on the finitely connected region.