Lecture notes on the Gaussian Free Field

TL;DR

This paper provides a comprehensive, self-contained introduction to the Gaussian Free Field (GFF), exploring its continuum and discrete forms, and discusses recent developments including connections to Brownian loop-soups and Conformal Loop Ensembles.

Contribution

It offers an updated, expanded set of lecture notes that connect the GFF with recent advances in probability theory and conformal invariance, including exercises for deeper understanding.

Findings

Explains the relation between continuum GFF and Brownian loop-soups.

Describes the connection between GFF and Conformal Loop Ensembles CLE(4).

Provides a self-contained introduction suitable for graduate courses.

Abstract

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.