The Gaussian free field (GFF) is the analogue of Brownian motion when the time set is changed by a 2-dimensional domain. Exit sets are the 2-dimensional analogue of exit times of the Brownian motion. Together with collaborators, we show existence and uniqueness of these sets. Furthermore, we study many of their properties and we relate them to the Brownian loop soup and CLE$_4$. The main difficulties of these works lie in the fact that the GFF is not a function, but only a Schwartz distribution.

KT or topological phase transitions are a type of transition discovered by Kosterlitz and Thouless. Models that undergo this phenomenon are typically 2-dimensional and do not have a classical phase transition. We call them topological because at low temperatures the topological defects (vortices) of the model are local, however at large temperature these topological defects appear everywhere. The KT phase transition is in duality with the problem of localisation of integer-valued fields. This duality was famously used by Fröhlich and Spencer to formally prove the first case of topological phase transition in 1981. To do this, they show that the GFF conditioned to have integer values has fluctuations of order 1 at low temperatures, however it fluctuates as a constant times the GFF at large temperatures.

The theory of Gaussian multiplicative chaos (GMC) studies the exponential of the 2-dimensional continuum Gaussian free field (GFF). In fact, as the GFF is not a function the GMC cannot be defined pointwise, instead it is constructed as a measure by first discretising via mollification and then taking the limit. For a given level of mollification, the GMC measure has Radon-Nykodim derivative with respect to Lebesgue equal to the renormalised exponential of a GFF times a constant $\gamma$. GMC measures are fractal and they are non-trivial only when $\gamma\in (-2,2)$. Their importance comes from the fact that they are the conjectured scaling limit of the natural measure associated to planar maps weighted by statistical physics models.

Planar maps are a discrete approximation of a two-dimensional Riemmanian geometry. Mathematically, they are graph embedded on the sphere in such a way that edges do not cross. The metric structure of uniformly chosen planar maps have been deeply studied and their scaling limit correspond to a certain metric obtained from a GMC of parameter $\gamma=\sqrt{8/3}$. Recently, there has been a lot of interest in non-uniform laws of random planar maps. In particular, planar maps decorated by an FK-model. The simplest of this case correspond to the spanning-tree decorated planar map, which should converge, in an appropiate sense, to a GMC of parameter $\gamma=\sqrt 2$.