On thin local sets of the Gaussian free field

Abstract

We study how small a local set of the Gaussian free field (GFF) in dimension d has to be in order to ensure that this set is thin, which loosely speaking means that it captures no GFF mass on itself. We provide a criteria on the size of the local set for this to happen, and on the other hand, we show that this criteria is sharp by constructing small local sets that are not thin.