Two-curve Green's function for $2$-SLE: the boundary case

Abstract

We prove that for $\kappa \in (0,8)$, if $({\eta }_1,{\eta }_2)$ is a $2$-SLE${}_{\kappa }$ pair in a simply connected domain $D$ with an analytic boundary point $z_0$, then $\lim_{r\to 0^+}r^{-α} \mathbb{P}[\mbox{dist}(z_0,η_j)<r,j=1,2]$ converges to a positive number for some $\alpha >0$, which is called the two-curve Green's function. The exponent $\alpha$ equals $\frac{12}{\kappa }-1$ or $2(\frac{12}{\kappa }-1)$ depending on whether $z_0$ is one of the endpoints of ${\eta }_1$ and ${\eta }_2$. We also find the convergence rate and the exact formula of the Green's function up to a multiplicative constant. To derive these results, we construct two-dimensional diffusion processes and use orthogonal polynomials to obtain their transition density.

62 pages

References in corpus (3)

• Duality of Chordal SLE
• Green's function for cut points of chordal SLE attached with boundary arcs
• Time-reversal of multiple-force-point SLE${}_{\kappa }(\underline{\rho })$ with all force points lying on the same side

Cited by in corpus (4)

• N-sided Radial Schramm-Loewner Evolution
• Green's function for cut points of chordal SLE attached with boundary arcs
• Time-reversal of multiple-force-point SLE${}_{\kappa }(\underline{\rho })$ with all force points lying on the same side
• Two-curve Green's function for $2$-SLE: the interior case