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

Abstract

Let $\kappa \in (4,8)$. Let $\gamma$ be an SLE${}_{\kappa }$ curve in a Jordan domain $D$ connecting $a_1\ne a_2\in \partial D$. We attach $\gamma$ with two open boundary arcs $A_1,A_2$ of $D$, which share end points $b_1\ne b_2\in \partial D\setminus \{a_1,a_2\}$, and consider for each $z_0\in D$ the limit $$ \lim_{r \downarrow 0}r^{1-\frac 38κ} \mathbb{P}[γ\cup A_1\cup A_2 \mbox{ has a cut point in }\{|z-z_0|<r\}].$$ We prove that the limit converges, derive a rate of convergence, and obtain the exact formula of the limit up to a multiplicative constant depending only on $\kappa$.

45 pages

References in corpus (2)

• Duality of Chordal SLE
• Two-curve Green's function for $2$-SLE: the boundary case

Cited by in corpus (2)

• Two-curve Green's function for $2$-SLE: the boundary case
• Time-reversal of multiple-force-point SLE${}_{\kappa }(\underline{\rho })$ with all force points lying on the same side