Lecture notes on the gaussian free field (Relié)

The Gaussian Free Field (GFF) in the continuum appears to be the natural generalisation of Brownian motion, when one replaces time by a multidimensional continuous parameter. While Brownian motion can be viewed as the most natural random valued function defined on If, with B(0) = 0, the GPI' In a domain D for d >_ 2 is a natural random real-valued generalised function defined on D with zero boundary conditions on dD.
In particular, it is not a random continuous function. The goal of these lecture notes is to describe some aspects of the continuum GFF and of its discrete counterpart defined on lattices, with the aim of providing a gentle self-contained introduction to some recent developments on this topic, such as the relation between the continuum GFF, Brownian loop-soups, and the Conformal Loop Ensembles CLEc.
This is an updated and expanded version of the notes written by the first author (WW) for graduate courses at ETH Zürich (Swiss Federal Institute of Technology Zürich) in 2014 and 2018. It has benefited from the comments and corrections of students, as well as of a referee ; we thank them all very much. The exercises that are interspersed in the first half of these notes mostly originate from the exerccies sheets prepared by the second author (EP) for this course in 2018.