This book, authored by Nathanaël Berestycki and Ellen Powell, is an introduction to the Gaussian free field (GFF) and Liouville quantum gravity (LQG), with a focus on providing a comprehensive and systematic treatment of the analytic subtleties involved. The book is divided into several chapters, each covering specific aspects of the GFF and LQG. **Chapter 1: Definition and Properties of the GFF** - **Discrete Case**: Introduces the discrete GFF on a finite, weighted graph, including the Green function and its properties. - **Continuous Green Function**: Discusses the continuous GFF, its definition, and conformal invariance in two dimensions. - **GFF as a Stochastic Process**: Explains the GFF as a random distribution and its stochastic process properties. - **Random Variables and Convergence**: Covers convergence in the space of distributions and integration by parts. - **Dirichlet Energy**: Discusses the Dirichlet energy and its role in the GFF. - **Function Spaces**: Reminds readers about function spaces relevant to the GFF. - **GFF as a Random Distribution**: Introduces the GFF as a random distribution and its properties. - **Itô's Isometry**: Discusses Itô's isometry for the GFF. - **Cameron–Martin Space**: Introduces the Cameron–Martin space of the Dirichlet GFF. - **Markov Property**: Explains the Markov property of the GFF. - **Conformal Invariance**: Discusses the conformal invariance of the GFF. - **Circle Averages**: Introduces circle averages and their properties. - **Thick Points**: Discusses thick points and their significance. - **Scaling Limit of the Discrete GFF**: Explains the scaling limit of the discrete GFF. **Chapter 2: Liouville Measure** - **Preliminaries**: Provides background on Liouville measure and its convergence properties. - **Weak Convergence to Liouville Measure**: Discusses weak convergence to the Liouville measure. - **GFF Viewed from a Liouville Typical Point**: Introduces the GFF from a Liouville perspective. - **Full \(L^1\) Phase**: Discusses the full \(L^1\) phase of the Liouville measure. - **Phase Transition for the Liouville Measure**: Explains the phase transition in the Liouville measure. - **Change of Coordinate Formula and Conformal Covariance**: Discusses the change of coordinate formula and conformal covariance. - **Random Surfaces**: Introduces random surfaces and their properties. - **Exercises**: Provides exercises for readers to practice the concepts. **Chapter 3: Gaussian Multiplicative Chaos** - **Motivation and Background**: Provides motivation and background on Gaussian multiplicative chaos. - **Setup for GaussianThis book, authored by Nathanaël Berestycki and Ellen Powell, is an introduction to the Gaussian free field (GFF) and Liouville quantum gravity (LQG), with a focus on providing a comprehensive and systematic treatment of the analytic subtleties involved. The book is divided into several chapters, each covering specific aspects of the GFF and LQG. **Chapter 1: Definition and Properties of the GFF** - **Discrete Case**: Introduces the discrete GFF on a finite, weighted graph, including the Green function and its properties. - **Continuous Green Function**: Discusses the continuous GFF, its definition, and conformal invariance in two dimensions. - **GFF as a Stochastic Process**: Explains the GFF as a random distribution and its stochastic process properties. - **Random Variables and Convergence**: Covers convergence in the space of distributions and integration by parts. - **Dirichlet Energy**: Discusses the Dirichlet energy and its role in the GFF. - **Function Spaces**: Reminds readers about function spaces relevant to the GFF. - **GFF as a Random Distribution**: Introduces the GFF as a random distribution and its properties. - **Itô's Isometry**: Discusses Itô's isometry for the GFF. - **Cameron–Martin Space**: Introduces the Cameron–Martin space of the Dirichlet GFF. - **Markov Property**: Explains the Markov property of the GFF. - **Conformal Invariance**: Discusses the conformal invariance of the GFF. - **Circle Averages**: Introduces circle averages and their properties. - **Thick Points**: Discusses thick points and their significance. - **Scaling Limit of the Discrete GFF**: Explains the scaling limit of the discrete GFF. **Chapter 2: Liouville Measure** - **Preliminaries**: Provides background on Liouville measure and its convergence properties. - **Weak Convergence to Liouville Measure**: Discusses weak convergence to the Liouville measure. - **GFF Viewed from a Liouville Typical Point**: Introduces the GFF from a Liouville perspective. - **Full \(L^1\) Phase**: Discusses the full \(L^1\) phase of the Liouville measure. - **Phase Transition for the Liouville Measure**: Explains the phase transition in the Liouville measure. - **Change of Coordinate Formula and Conformal Covariance**: Discusses the change of coordinate formula and conformal covariance. - **Random Surfaces**: Introduces random surfaces and their properties. - **Exercises**: Provides exercises for readers to practice the concepts. **Chapter 3: Gaussian Multiplicative Chaos** - **Motivation and Background**: Provides motivation and background on Gaussian multiplicative chaos. - **Setup for Gaussian