Left passage probability of Schramm-Loewner Evolution

SLE(kappa,(rho) over right arrow) is a variant of Schramm-Loewner Evolution (SLE) which describes the curves which are not conformal invariant, but are self-similar due to the presence of some other preferred points on the boundary. In this paper we study the left passage probability (LPP) of SLE(kappa,(rho) over right arrow) through field theoretical framework and find the differential equation governing this probability. This equation is numerically solved for the special case kappa = 2 and h(rho) = 0 in which h(rho) is the conformal weight of the boundary changing (bcc) operator. It may be referred to loop erased random walk (LERW) and Abelian sandpile model (ASM) with a sink on its boundary. For the curve which starts from xi(0) and conditioned by a change of boundary conditions at x(0), we find that this probability depends significantly on the factor x(0) -xi(0). We also present the perturbative general solution for large x(0). As a prototype, we apply this formalism to SLE(kappa,kappa - 6) which governs the curves that start from and end on the real axis.