Lecture notes on the gaussian free field

Lecture notes on the gaussian free field

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 real-valued function defined on R₊ with B(0) = 0, the GFF in a domain D of R^d for d ≥ 2 is a natural random real-valued generalised function defined on D with zero boundary conditions on ∂D. 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 CLE_4.

This is an updated and expanded version of the notes written by the first author (WW) for graduate courses at ETH Zurich (Swiss Federal Institute of Technology in 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 exercise sheets prepared by the second author (EP) for this course in 2018.