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

We prove that for $κ\in(0,8)$, if $(η_1,η_2)$ is a $2$-SLE$_κ$ 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 $α>0$, which is called the two-curve Green's function. The exponent $α$ equals $\frac{12}κ-1$ or $2(\frac{12}κ-1)$ depending on whether $z_0$ is one of the endpoints of $η_1$ and $η_2$. 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.

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