Introduction to the Liouville quantum gravity metric

TL;DR

This paper provides an overview of the construction and properties of the Liouville quantum gravity metric, a fractal surface model, including geodesics, KPZ formula, and open problems.

Contribution

It summarizes recent developments in constructing and analyzing the LQG metric, highlighting key techniques and open questions.

Findings

Construction of the LQG metric overview

Properties of geodesics in LQG surfaces

Discussion of the KPZ formula and open problems

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.