Gaussian Free Field and Liouville Quantum Gravity

In this comprehensive volume, the authors introduce some of the most important recent developments at the intersection of probability theory and mathematical physics, including the Gaussian free field, Gaussian multiplicative chaos and Liouville quantum gravity. This is the first book to present these topics using a unified approach and language, drawing on a large array of multi-disciplinary techniques. These range from the combinatorial (discrete Gaussian free field, random planar maps) to the geometric (culminating in the path integral formulation of Liouville conformal field theory on the Riemann sphere) via the complex analytic (based on the couplings between Schramm-Loewner evolution and the Gaussian free field). The arguments (currently scattered over a vast literature) have been streamlined and the exposition very carefully thought out to present the theory as much as possible in a reader-friendly, pedagogical yet rigorous way, suitable for graduate students as well as researchers.

List of contents

1. Definition and properties of the GFF; 2. Liouville measure; 3. Gaussian multiplicative chaos; 4. Statistical physics on random planar maps; 5. Introduction to Liouville conformal field theory; 6. Gaussian free field with Neumann boundary conditions; 7. Quantum wedges and scale-invariant random surfaces; 8. SLE and the quantum zipper; 9. Liouville quantum gravity as a mating of trees; A. Chordal Loewner chains and chordal SLE; B. Reverse Loewner flow and reverse SLE; C. Radial Loewner chains and radial SLE; D. Convergence of random variables in the space of distributions; References; Notation and symbols; Subject index.

About the author

Nathanaël Berestycki has held the Chair of Stochastics at the University of Vienna since 2018. He has been an associate editor of numerous journals, including the 'Annals of Probability', and was an invited speaker at the International Congress of Mathematical Physics (ICMP) in 2024.Ellen Powell is Associate Professor in the probability group at Durham University, where she has been since 2019.

Product details

• Authors: Nathanaël Berestycki, Nathanael (Universitat Wien Berestycki, Ellen Powell
• Publisher: Cambridge University Press ELT
• Languages: English
• Product format: Hardback
• Released: 30.11.2025
• EAN: 9781009405508
• ISBN: 978-1-0-0940550-8
• No. of pages: 438
• Series: Cambridge Studies in Advanced Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; MATHEMATICS / Probability & Statistics / General, computer science, Quantum physics (quantum mechanics & quantum field theory), Quantum physics (quantum mechanics and quantum field theory), Mathematical physics, Probability & statistics, Stochastics, Probability and statistics, Quantum computing and information, Mathematical / Computational / Theoretical physics, Natural computing / Quantum computing