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

Research paper by **Dapeng Zhan**

Indexed on: **25 Jun '18**Published on: **25 Jun '18**Published in: **arXiv - Mathematics - Probability**

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#### 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. 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.