# Random surfaces and Liouville quantum gravity

## TL;DR

This paper explores Liouville quantum gravity surfaces as models of random fractal two-dimensional manifolds, their relation to discrete random planar maps, and discusses convergence and open problems in the field.

## Contribution

It provides an accessible overview of the definitions, convergence, and motivations of LQG and random planar maps, highlighting recent developments and open questions.

## Key findings

- LQG surfaces serve as continuum limits of random planar maps
- Convergence of discrete maps to LQG is established in certain settings
- Open problems in the study of LQG and random surfaces are discussed

## Abstract

Liouville quantum gravity (LQG) surfaces are a family of random fractal surfaces which can be thought of as the canonical models of random two-dimensional Riemannian manifolds, in the same sense that Brownian motion is the canonical model of a random path. LQG surfaces are the continuum limits of discrete random surfaces called random planar maps. In this expository article, we discuss the definition of random planar maps and LQG, the sense in which random planar maps converge to LQG, and the motivations for studying these objects. We also mention several open problems. We do not assume any background knowledge beyond that of a second-year mathematics graduate student.