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 $\lim_{r\to 0^+}r^{-\alpha} \mathbb{P}[\mbox{dist}(z_0,\eta_j)<r,j=1,2]$ 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.