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

Research paper by Dapeng Zhan

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



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.