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.

‘Beautifully written and illustrated, this is a perfect introduction for anyone with a graduate-level probability background who wants to learn more about Gaussian free fields, random surfaces, conformal field theory, Liouville quantum gravity, SLE and the surrounding circle of ideas. The authors have spent years perfecting this exposition. I highly recommend this book to my own graduate students – and to interested researchers at any level.’

Scott Sheffield - Massachusetts Institute of Technology

‘For a technically difficult research area with a large body of literature it is essential to have a clean and pedagogical introductory text. Berestycki and Powell’s book fills exactly this need and it has been a tremendous help for all my students while learning this research area. Additionally, the book has quickly become a standard reference and it's an excellent text for looking up key facts of our field.’

Nina Holden - New York University

‘Berestycki and Powell succeed brilliantly in making the rapidly evolving field of Liouville Quantum Gravity accessible to the probability community. There is no better place to learn the basics of the theories of Gaussian Free Field and Multiplicative Chaos than this book and no better introduction to the recent research in Liouville CFT and its connections to SLE.’

Antti Kupiainen - University of Helsinki