Left Passage Probability of SLE($κ,ρ$)

SLE($κ,ρ$) 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($κ,ρ$) 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 $κ= 2$ and $h_ρ= 0$ 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($κ,κ-6$) which governs the curves that start from and end on the real axis.

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