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

A $2$-SLE${}_{\kappa }$ ($\kappa \in (0,8)$) is a pair of random curves $({\eta }_1,{\eta }_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE${}_{\kappa }$ curve in a complement domain. In this paper we prove that for any $z_0\in D$, the limit $\lim_{r\to 0^+}r^{-\alpha_0} $\mathbb{P}$[\mbox{dist}(z_0,\eta_j)<r,j=1,2]$,where$\alpha_0=\frac{(12-\kappa)(\kappa+4)}{8\kappa}$, exists. Such limit is called a two-curve Green’s function. We find the convergence rate and the exact formula of the Green’s function in terms of a hypergeometric function up to a multiplicative constant. For $\kappa \in (4,8)$, we also prove the convergence of $\lim_{r\to 0^+}r^{-\alpha_0} $\mathbb{P}$[\mbox{dist}(z_0,\eta_1\cap \eta_2)<r]$, whose limit is a constant times the previous Green’s function. To derive these results, we work on two-time-parameter stochastic processes, and use orthogonal polynomials to derive the transition density of a two-dimensional diffusion process that satisfies some system of SDE.