Green's function for chordal SLE curves

TL;DR

This paper proves the existence of Green's functions for multiple points on chordal SLE$_ppa$ curves, providing convergence rates, continuity properties, and bounds, advancing understanding of SLE path probabilities.

Contribution

It establishes the existence of multi-point Green's functions for chordal SLE$_ppa$, including convergence rates, continuity, and bounds, which were previously unknown.

Findings

Green's functions exist for any finite number of points.

Provided explicit convergence rates and modulus of continuity.

Derived up-to-constant bounds for the Green's functions.

Abstract

For a chordal SLE$_\kappa$ ($\kappa\in(0,8)$) curve in a domain $D$, the $n$-point Green's function valued at distinct points $z_1,\dots,z_n\in D$ is defined to be $$G(z_1,\dots,z_n)=\lim_{r_1,\dots,r_n\downarrow 0} \prod_{k=1}^n r_k^{d-2} \mathbb{P}[\mbox{dist}(\gamma,z_k)<r_k,1\le k\le n],$$ where $d=1+\frac{\kappa}{8}$ is the Hausdorff dimension of SLE$_\kappa$, provided that the limit converges. In this paper, we will show that such Green's functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-constant bounds for them.