Lecture notes on the Gaussian Free Field

Understand

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.

• 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 CLE(4).
• This is an updated and expanded version of the notes written by the first author for graduate courses at ETH Z\"urich.
• The exercises that are interspersed in the first half of these notes mostly originate from the exercise sheets prepared by the second author for this course in 2018.