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

TL;DR

This paper investigates the left passage probability of SLE($, ho$) processes using field theory, deriving differential equations and solving specific cases, with applications to models like the Abelian sandpile.

Contribution

It introduces a field theoretical approach to analyze the left passage probability of SLE($, ho$) and derives governing differential equations, including solutions for special parameter cases.

Findings

Derived differential equations for LPP of SLE(, ho)

Solved the equations for =2, h_ ho=0 case

Applied formalism to SLE(,-6) processes

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.