Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity

TL;DR

This paper explains how Gaussian multiplicative chaos is used to construct Liouville Quantum Gravity, detailing the measures involved and their connection to large planar maps, suitable for newcomers.

Contribution

It provides a clear probabilistic construction of Liouville Quantum Gravity and discusses its relation to the scaling limits of planar maps.

Findings

Construction of Liouville measures using Gaussian multiplicative chaos

Description of the conjectured relation to large planar maps

Accessible introduction without prior knowledge required

Abstract

The purpose of these notes, based on a course given by the second author at Les Houches summer school, is to explain the probabilistic construction of Polyakov's Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, these notes contain a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. These notes are rather short and require no prior knowledge on the topic.