Left Passage Probability of SLE($\kappa,\rho$)

Abstract

SLE($\kappa,\rho$) is a variant of the Schramm-Loewner Evolution 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) for SLE($\kappa,\rho$) through field theoretical framework and find the differential equation which govern this probability. This equation is solved (up to two undetermined constants) for the special case $\kappa= 2$ and $h_\rho = 0$ for large x0 at which the boundary condition changes. This case may be referred to the Abelian sandpile model with a sink on the boundary. As an example, we apply this formalism to SLE($\kappa,\kappa-6$) which governs the curves that start from and end on the real axis.