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

Research paper by **Dapeng Zhan**

Indexed on: **01 Jan '19**Published on: **01 Jan '19**Published in: **arXiv - Mathematics - Probability**

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#### Abstract

We prove that for a $2$-SLE$_\kappa$ pair $(\eta_1,\eta_2)$ in a simply
connected domain $D$, whose boundary is $C^1$ near $z_0\in \partial D$, there
is some $\alpha>0$ such that $\lim_{r\to 0^+}r^{-\alpha}
\mathbb{P}[\mbox{dist}(z_0,\eta_j)<r,j=1,2]$ converges to a positive number,
called the boundary two-curve Green's function. The exponent $\alpha$ equals
$2(\frac{12}{\kappa}-1)$ if $z_0$ is not one of the endpoints of $\eta_1$ and
$\eta_2$; and otherwise equals $\frac{12}{\kappa}-1$. We also prove the
existence of the boundary (one-curve) Green's function for a
single-boundary-force-point SLE$_\kappa(\rho)$ curve, for $\kappa$ and $\rho$
in some range. In addition, we find the convergence rate and the exact formula
of the above Green's functions up to multiplicative constants. To derive these
results, we construct a family of two-dimensional diffusion processes, and use
orthogonal polynomials to obtain their transition density.