Multi-point Green's functions for SLE and an estimate of Beffara

TL;DR

This paper introduces the multi-point Green's function for SLE, establishing its existence and providing a new proof of Beffara's bound, which is crucial for understanding the probability of SLE curves passing near multiple points.

Contribution

The paper defines and proves the existence of multi-point Green's functions for SLE and offers a new proof of Beffara's bound independent of previous methods.

Findings

Established the existence of multi-point Green's functions for SLE.

Provided a new, independent proof of Beffara's bound.

Connected the probability of SLE passing near points to two-sided radial SLE.

Abstract

In this paper we define and prove of the existence of the multi-point Green's function for SLE - a normalized limit of the probability that an $SLE_{\kappa}$ curve passes near to a pair of marked points in the interior of a domain. When $\kappa<8$ this probability is nontrivial, and an expression can be written in terms two-sided radial SLE. One of the main components to our proof is a refinement of a bound first provided by Beffara [Ann. Probab. 36 (2008) 1421-1452]. This work contains a proof of this bound independent from the original.