Equivalence Of SLE_κand Left-passage Probability Of Critical Percolation

Stochastic Loewner evolution (SLE) is a random growth process of set Ktin time t,which is described by solving Loewner diferential equation with driving function equalto one-dimensional time-changed Brownain motion. SLEκhas two vision of chordalSLEκand radial one. In this paper our works are as follows. First, we prove that asκ=6, chordal SLE6in the upper half-plane H has the same law as radial SLE6in H,up to random time change; and that chordal SLE6and radial SLE6in H possesses aweaker form of equivalence for κ=6. This generalizes the problem of the equivalenceof SLEκin the unit disk to the upper half-plane H. Secondly, we consider critical per-colation on the triangular lattice with mesh δ in the polygonal domain D, where z0is afxed point in D. Based on the fact that the exploration process of critical percolationconverges to the trace of SLE6in the scaling limit and the left-passage probability for-mula of SLE6in the upper-half plane, using the Christofel-Schwarz formula we derivethe probability formula of the event that the exploration process in D passes to theleft of z0, as δâ†'0. This generalizes the problem of left-passage probability of criti-cal percolation on the triangular lattice in the unit disk to the case of polygonal domain.