Liouville quantum gravity as a metric space and a scaling limit

TL;DR

This paper reviews the connection between Liouville quantum gravity with parameter b3=6/3 and the Brownian map, establishing their equivalence and exploring implications for scaling limits of random planar maps and related stochastic processes.

Contribution

It demonstrates the equivalence of b3=6/3-LQG with the Brownian map and applies this to prove convergence of certain random walks and percolation models.

Findings

Liouville quantum gravity with b3=6/3 is equivalent to the Brownian map.

The b3=6/3-LQG metric is used to prove convergence of self-avoiding walks.

Percolation on random planar maps converges to SLE processes on a Brownian surface.

Abstract

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. This article surveys a series of works with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $\gamma=\sqrt{8/3}$ is equivalent to the Brownian map. We also briefly describe a series of works with Gwynne which use the $\sqrt{8/3}$-LQG metric to prove the convergence of self-avoiding walks and percolation on random planar maps towards SLE$_{8/3}$ and SLE$_6$, respectively, on a Brownian surface.