A new approach to SLE phase transition

Abstract

It is well know that $SLE_\kappa$ curves exhibit a phase transition at $\kappa=4$. For $\kappa\le 4$ they are simple curves with probability one, for $\kappa>4$ they are not. The standard proof is based on the analysis of the Bessel SDE of dimension $d=1+4/\kappa$. We propose a different approach which is based on the analysis of the Bessel SDE with $d=1-4/\kappa$. This not only gives a new perspective, but also allows to describe the formation of the SLE `bubbles' for $\kappa>4$.