BooksStandalone TitlesCollected Volumepp. 4212–4244

Introduction to the Liouville quantum gravity metric

  • Jian Ding

    The Wharton School, University of Pennsylvania, 3733 Spruce Street, Philadelphia, PA 19104, USA

  • Julien Dubédat

    Department of Mathematics, Columbia University, Room 509, MC 4406, 2990 Broadway, New York, NY 10027, USA

  • Ewain Gwynne

    Department of Mathematics, Eckhart Hall, 5734 S University Ave, Chicago, IL, 60637, USA
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Abstract

Liouville quantum gravity (LQG) is a one-parameter family of models of random fractal surfaces which first appeared in the physics literature in the 1980s. Recent works have constructed a metric (distance function) on an LQG surface. We give an overview of the construction of this metric and discuss some of its most important properties, such as the behavior of geodesics and the KPZ formula. We also discuss some of the main techniques for proving statements about the LQG metric, give examples of their use and discuss some open problems.