Thick points of the planar Gaussian free field 

The Gaussian Free Field (GFF) in two dimensions is a random field which can be viewed as a multidimensional analogue of Brownian motion, and appears as a universal scaling limit of a class of discrete height functions. Thick points of the GFF are points where, roughly speaking, the field is atypically high. They provide key insights into the geometric properties of the field, and are the basis for construction of important associated objects in random planar geometry. The set of thick points with thickness level a is a fractal set with Hausdorff dimension 2-a^2/2. In this talk I will discuss another fundamental property, namely, that the set is almost surely disconnected for all non-zero a. This is based on joint work with Juhan Aru and Léonie Papon, and uses a remarkable relationship between the GFF and the "conformal loop ensemble" of parameter 4.