Green’s function for cut points of chordal SLE attached with boundary arcs

Abstract

A technique of two-curve Green’s function is used to study the Green’s function of cut points of chordal SLEκ for κ∈(4,8). In order to apply the technique, we take the union of the SLE curve with two open boundary arcs, which share two boundary points other than the endpoints of the SLE curve. The Green’s function of interest is, for any z0 in the domain, the limit as r↓0 of the r−α times the probability that {|z−z0|<r} contains a cut point of the above union, where α=38κ−1. We prove the limit exists, obtain an exact formula up to a multiplicative constant, and derive a rate of convergence.