This survey paper is based on mini-courses given in July 2003 at the University of St-Andrews, and at the ICMS in Edinburgh. Its goal is to give a self-contained and heuristic overview of the recent (i.e. pre-2003) results concerning conformal restriction properties for random planar curves.
"Conformal restriction and related questions." Probab. Surveys 2 145 - 190, 2005. https://doi.org/10.1214/154957805100000113