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

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 $SL{E}_{\kappa }$ 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.

Published in at http://dx.doi.org/10.1214/11-AOP695 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)