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

Abstract

We prove that for $\mathrm{\kappa }\in (0,8)$, if $({\mathrm{\eta }}_1,{\mathrm{\eta }}_2)$ is a 2-SLE${}_{\mathrm{\kappa }}$pair in a simply connected domain D with an analytic boundary point $z_0$, then as $r\to 0^{+}$, $\mathbb{P}[\mathrm{dist}(z_0,{\mathrm{\eta }}_j)<r,j=1,2]$ converges to a positive number for some $\mathrm{\alpha }>0$, which is called the two-curve Green’s function. The exponent α equals $\frac{12}{\mathrm{\kappa }}-1$ or $2(\frac{12}{\mathrm{\kappa }}-1)$ depending on whether $z_0$ is one of the endpoints of ${\mathrm{\eta }}_1$ or ${\mathrm{\eta }}_2$. We also find the convergence rate and the exact formula for 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.