# Two-curve Green’s function for 2-SLE

Dapeng Zhan (Michigan State University), November 15, 2018

Abstract: 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. We prove that, for the exponent $\alpha=\frac{(12 - \kappa) (\kappa + 4)}{8 \kappa}$, for any $z_0 \in D$, the limit $\lim_{r \to 0^+} r^{-\alpha} \mathbb{P}[\mbox{dist}(\eta_j, z_0) < r,j=1,2]$ converges to a positive number, called the two-curve Green’s function. To prove the convergence, we transform the original problem into the study of a two-dimensional diffusion process, and use orthogonal polynomials to derive its transition density and invariant density.