Exit sets of the continuum Gaussian free field in two dimensions and related questions

Abstract

The topic of this thesis is the study of geometric properties of the two-dimensional continuum Gaussian free field (GFF), which is the analogue of Brownian motion when time is replaced by a two-dimensional domain. This is part of the wider field of two-dimensional and conformally invariant geometry, which is currently a very active area of probability theory. More specifically, the main theme of our work is the definition and properties of what we call {\em exit sets} of the GFF. These sets are the analogue of exit times of intervals by Brownian motion when one replace the one-dimensional time by a two-dimensional set. Because the GFF is not a continuous function, but only a generalised function, the definition and the study of these sets are somewhat challenging. We discuss the definition and characterizations of these sets, study their size, the connectivity properties of their complement, some of their approximations via discrete structures, their relations to conformal loop ensembles and Schramm-Loewner evolutions, and how they can be used to construct some Liouville quantum gravity measures.