Paths, fields and fractal geometry.

The 2D Brownian loop soup is a Poisson collection of Brownian loops in a 2D domain. It has been introduced by Lawler and Werner for its conformal invariance properties, and then used by Sheffield and Werner to give a construction of the Conformal Loop Ensembles (CLE).
For one particular intensity parameter (equal to 1/2), it satisfies remarkable properties, such as a spatial Markov property, and is related to the Gaussian free field (GFF). This has been first observed in the discrete setting by Le Jan, and is a particular instance of the random walk representations of the GFF. This kind of relations can be renormalized in the continuum limit in dimension 2. I will explain how in 2D continuum one gets the free field, its multiplicative chaos (Wick exponential), its Wick powers, its height gap, out of the Brownian loop soup with intensity parameter 1/2. Using Brownian loop soups with different intensity parameters, one gets other fields, a priori not related to the GFF, and I will mention that too.
This talk is based on different joint works with Juhan Aru, Avelio Sepulveda, Elie Aïdekon, Nathanaël Berestycki, Antoine Jégo and Wei Qian.