ABSTRACT:
The goal of this talk is to understand thin local sets of the continuous Gaussian free filed (GFF) in a domain of R^2, whose corresponding harmonic function takes only two values. We give a characterization of these sets and use it to show that in some sense they are maximal in a bigger class of local sets, where we only ask the function to be bounded. Important corollaries of this work are new constructions of the Conformal Loop Ensemble CLE_4 and a new perspective on the two known couplings between CLE_4 and the GFF. Joint work wiht JUHAN ARU and WENDELIN WERNER.