Lecture notes on the Gaussian Free Field

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 Zürich. 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.

arXiv commentFinal version, 145 pages, to appear in the "Cours Spécialisés" series published by the Société Mathématique de France

References in corpus (10)

• The dimension of the SLE curves
• Duality of Chordal SLE
• Conformal radii for conformal loop ensembles
• Random soups, carpets and fractal dimensions
• Convergence of the two-dimensional random walk loop soup clusters to CLE
• On bounded-type thin local sets of the two-dimensional Gaussian free field
• Decomposition of Brownian loop-soup clusters
• A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field
• The first passage sets of the 2D Gaussian free field: convergence and isomorphisms
• Lecture notes on Liouville theory and the DOZZ formula

Cited by in corpus (12)

• Critical Gaussian multiplicative chaos: a review
• Lozenge tilings and the Gaussian free field on a cylinder
• Multiplicative chaos of the Brownian loop soup
• Extremal distance and conformal radius of a CLE_4 loop
• The trunks of CLE(4) explorations
• Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees
• Level Set Percolation in Two-Dimensional Gaussian Free Field
• Relative Entropy and Mutual Information in Gaussian Statistical Field Theory
• A state space for 3D Euclidean Yang-Mills theories
• A characterisation of the continuum Gaussian free field in $d\ge 2$ dimensions
• Cluster explorations of the loop soup on a metric graph related to the Gaussian free field
• $(1+\epsilon )$-moments suffice to characterise the GFF