| We present two results. The first result extends a theorem of Astala, Jones, Kupiainen and Saksman ([AJKS09]) to criticality. We construct a family of random Jordan curves in the plane by welding together two disks on their boundaries using a random homeomorphism. This homeomorphism arises from a random measure whose density, in a generalized sense, is the exponentiated Gaussian Free Field at criticality. We also introduce a representation of the Gaussian Free Field in terms of vaguelets, which may be of separate interest.;The second result is the classification of Holder graphs for continuous Sobolev removability. Holder alpha graphs for alpha > 2/3 are removable, while for alpha < 2/3 we construct examples of non-removable graphs. |
